Engineering Design Applications V: Structures, Materials and Processes (Advanced Structured Materials, 171) 3031264657, 9783031264658

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Advanced Structured Materials

Andreas Öchsner Holm Altenbach   Editors

Engineering Design Applications V Structures, Materials and Processes

Advanced Structured Materials Volume 171

Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials are reaching their limits in many applications, and new developments are required to meet the increasing demands on engineering materials. The performance of materials can be improved by combining different materials to achieve better properties than with a single constituent, or by shaping the material or constituents into a specific structure. The interaction between material and structure can occur at different length scales, such as the micro, meso, or macro scale, and offers potential applications in very different fields. This book series addresses the fundamental relationships between materials and their structure on overall properties (e.g., mechanical, thermal, chemical, electrical, or magnetic properties, etc.). Experimental data and procedures are presented, as well as methods for modeling structures and materials using numerical and analytical approaches. In addition, the series shows how these materials engineering and design processes are implemented and how new technologies can be used to optimize materials and processes. Advanced Structured Materials is indexed in Google Scholar and Scopus.

Andreas Öchsner · Holm Altenbach Editors

Engineering Design Applications V Structures, Materials and Processes

Editors Andreas Öchsner Faculty of Mechanical and Systems Engineering Esslingen University Applied Sciences Esslingen, Baden-Württemberg, Germany

Holm Altenbach Institute of Mechanics Otto-von-Guericke University Magdeburg Magdeburg, Saxony-Anhalt, Germany

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-031-26465-8 ISBN 978-3-031-26466-5 (eBook) https://doi.org/10.1007/978-3-031-26466-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Different engineering disciplines such as mechanical, materials, computer and process engineering provide the foundation for the design and development of improved structures, materials and processes. The modern design cycle is characterized by an interaction of different disciplines and a strong shift to computer-based approaches where only a few experiments are performed for verification purposes. A major driver for this development is the increased demand for cost reduction, which is also connected to environmental demands. In the transportation industry (e.g. automotive), this is connected with the demand for higher fuel efficiency, which is related to the operational costs and the lower harm for the environment. One way to fulfil such requirements is lighter structures and/or improved processes for energy conversion. Another emerging area is the interaction of classical engineering with the health, medical and environmental sectors. This further volume in this series gives an update on recent developments in the mentioned areas of modern engineering design application. Esslingen, Germany Magdeburg, Germany

Andreas Öchsner Holm Altenbach

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Contents

1

2

Preparation of Thermoplastic Blends Filled with Polysaccharide and Study of Their Properties Before and After Ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Petra Skalková, Elena Nekorancová, Ivan Labaj, Andrej Dubec, Zuzana Miˇcicová, Slavomíra Božeková, Darina Ondrušová, and Mariana Pajtášová 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Characterization of LDPE/CMS Blends Before and After Ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Morphology of LDPE/CMS Blends Before and After Ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Vicat Softening Temperature of LDPE/CMS Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Visual Appearance of LDPE/CMS Blends . . . . . . . . . . . . 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheological and Mechanical Properties of Rubber Blends Filled with Modified Bentonite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zuzana Miˇcicová, Slavomíra Božeková, Mariana Pajtášová, and Petra Skalková 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Modification of Bentonite . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Preparation of Rubber Blends and Vulcanizates . . . . . . . 2.2.4 Characterization of the Modified Bentonite . . . . . . . . . . .

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1 3 3 3 4 4 8 13 17 18 19 21

21 22 22 23 23 24

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2.2.5

Characterization of the Rubber Blends and Vulcanizates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 FTIR Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Rheological and Curing Characteristics of Rubber Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Mechanical Properties of Vulcanizates . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Utilization of Microwave Radiation for Chemical Modification of Kaolin and Its Influence on the Curing, Mechanical and Surface Properties of Rubber Composites . . . . . . . . . . . . . . . . . . . Andrea Feriancová, Iveta Papuˇcová, Jana Pagáˇcová, Jana Šulcová, and Ivan Labaj 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Materials and Preparation of Modified Kaolin . . . . . . . . . 3.2.2 Characterization Techniques . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 NR/Kaolin Composites Preparation and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Characterization of Kaolin and Modified Kaolin Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Characterization of NR/Kaolin Composites . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulations of Tests of Polymeric Composites Based on Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan Krmela, Vladimíra Krmelová, Artem Artyukhov, Cornelia Lex, and Darina Ondrušová 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Materials and Methods and Computational Models . . . . . . . . . . . . 4.2.1 Materials Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Methods Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Computational Models for Shear Test Simulation . . . . . . 4.2.4 Computational Model for Tensile Test Simulation . . . . . 4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Shear Test Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Tensile Test Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24 24 25 25 27 28 29 31 32

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35 37 37 38 38 40 40 44 48 50 53

53 54 54 57 57 59 61 61 62 62 67

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5

Study of Influence of Calcium Carbonate Sedimentation on Electric Heater Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tatiana A. Kudryashova, Sergey V. Polyakov, and Nikita I. Tarasov 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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69 69 70 73 74 79 79

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A 3D Fiber-Based Strategy for Optimization of Tissue Materials Using a Combination of Liquid Absorbency/ Retention Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Flávia P. Morais, António O. Mendes, Ana M. M. S. Carta, Paulo T. Fiadeiro, Maria E. Amaral, and Joana M. R. Curto 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.1 Laboratory-Made Structures . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.2 Industrial Market Tissue Products . . . . . . . . . . . . . . . . . . . 97 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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Nanomaterials Based on Peptide Nanotubes with Graphene and Ferroelectric Polymers Layers: Modelling and Numerical Studies of Photoelectronic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir S. Bystrov, Ekaterina V. Paramonova, Pavel S. Zelenovskiy, Svitlana A. Kopyl, Xiangjian Meng, Hong Shen, Tie Lin, and Vladimir M. Fridkin 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Main Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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116 116 121 122

Mechanical Design of a Thermo-Mechanical-Cryogenic System to Evaluate Mechanical Properties of Samples of 3D Printing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Víctor Daniel Rodríguez-Gaspar, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa, Juan Atonal-Sánchez, Sebastián Arturo Medinilla-García, Luis Héctor Hernández-Gómez, and Teresa Berenice Uribe-Cortés 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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8.2 8.3 8.4 8.5

Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cooling System Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Test Results on 3D Printing Specimens in the Thermoelectric Cooling Chamber . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Investigation of Local Discontinuous Galerkin Method on the Solution of Convection–Diffusion Problems . . . . . . . . . . . . . . . . E. V. Shilnikov and I. R. Khaytaliev 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Linear Convection–Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Linear Convection–Diffusion Equation: Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Linear Convection–Diffusion Equation: Discontinuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Burgers’ Equation with Initial and Boundary Conditions . . . . . . . 9.4 Equation System for Gas Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Adjustment of Analytical Methods by Numerical Models for Determining Earth Pressures Behind Retaining Walls . . . . . . . . . Guillaume Puyhaubert, Ali Saeidi, Mahdiyeh Seifaddini, and Alain Rouleau 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Comparison Between Analytical and Numerical Results . . . . . . . 10.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Proposed Correction Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Regression Equation Method . . . . . . . . . . . . . . . . . . . . . . . 10.6 Validation of the Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126 127 127 130 130 140 140 141 141 142 147 150 153 156 159 159 161

161 164 165 166 167 168 168 169 170 171

11 Design of a System to Produce Rapid Biomedical Prototypes with Synthetic Materials: State of the Art . . . . . . . . . . . . . . . . . . . . . . . 173 Erik Omar Alvarado-Alcántara, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa, Juan Luis Cuevas Andrade, Luis Héctor Hernández-Gómez, Pablo Moreno-Garibaldi, Mauricio Rebattú y González, Alejandro Rebattú y González, Verónica Guzmán-Mercado, and Teresa Berenice Uribe-Cortés 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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11.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Biomodeling and Numerical Analysis of the Different Pathologies of the Upper Limb (Arm) that Limit Movement in Patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diego Ivan Islas-Jiménez, Guillermo Urriolagoitia-Sosa, Beatriz Romero-Ángeles, Dante Abel Islas-Jiménez, Misael Flores-Baez, Martha Eugenia Espinosa-Hernández, and Guillermo Manuel Urriolagoitia-Calderón 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Methodology/Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Basic Consideration for the Development of the Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Determination of External Agents and Frontier Conditions Applied into the Analyses . . . . . . . . . . . . . . . . 12.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Study for Validation and Implementation of Polymethyl Methacrylate in Neurocranium and Viscerocranium Prostheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carolina Alvarado Moreno, Juan Alfonso Beltrán-Fernández, Mauricio González Rebattú y González, Luis Héctor Hernández Gómez, Alejandro David González Peña, Edgar Alfonso Figueroa Rodríguez, José Enrique Rodríguez Miramar, Erik Omar Alvarado Alcántara, Fidel Romero Martínez, and Juan Luis Cuevas Andrade 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Parametrization and Assembly . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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174 174 175 178 178

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181 184 184 185 187 188 191

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195 196 196 197 198 199 199 201 201

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14 Numerical Biomechanical Analysis of the Fixation of Three Titanium Screws for Elbow Coronoid Fracture . . . . . . . . . . . . . . . . . . Dante Abel Islas-Jiménez, Beatriz Romero-Ángeles, Guillermo Urriolagoitia-Sosa, Diego Ivan Islas-Jiménez, Israel Flores-Baez, Juan Antonio Vargas-Bustos, and Guillermo Manuel Urriolagoitia-Calderón 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Methodology/Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Basic Considerations for the Development of the Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Determination of External Agents and Boundary Conditions Applied in the Analyses . . . . . . . . . . . . . . . . . 14.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Energetic Numerical Analysis of the Effect of Impact Loads into a Human Skull (Frontal and Lateral) . . . . . . . . . . . . . . . . . . . . . . . Fransciso Carrasco-Hernández, Guillermo Urriolagoitia-Sosa, Beatriz Romero-Ángeles, Diego Ivan Islas-Jiménez, José Luis Reyes-Reyes, Christian Díaz-León, Martha Eugenia Espinosa-Hernández, Iván González-Uribe, and Antonio Hernández-Cerón 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Methodology/Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Principle of Momentum and Movement Amount . . . . . . 15.2.2 Component Behavior Under Dynamic Loads . . . . . . . . . 15.2.3 Skull Numerical Model (Jones 1989) . . . . . . . . . . . . . . . . 15.2.4 General Conditions to Develop the Numerical Evaluation Under Impact Conditions . . . . . . . . . . . . . . . . 15.2.5 Study Particularities for the Development of Frontal Impact and Lateral Impact . . . . . . . . . . . . . . . . 15.3 Results for Frontal Impact in a Skull . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Results for Lateral Impact in a Skull . . . . . . . . . . . . . . . . . 15.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Design of Dynamic Systems and Electro-Assisted Immersion Connected with Fourth Generation Technology for the Use in Aquatic Therapy in Mexico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fidel Romero-Martinez, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa, Luis Héctor Hernández-Gómez, and Teresa Berenice Uribe-Cortés 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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203 206 206 208 210 212 213 215

216 218 218 219 221 223 225 227 230 231 237

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16.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 17 Dynamic and Experimental Testing of a Biomechanical System: Cadaveric Temporomandibular Specimen and a Multiaxial Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Juan Luis Cuevas-Andrade, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa, Mauricio González Rebattú y González, Luis Héctor Hernández-Gómez, Teresa Berenice Uribe-Cortés, Cesar Antonio Trujillo-Pérez, and Pablo Moreno-Garibaldi 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Digitization of Cadaveric Jaw . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.3 Analysis of TMJ Movement by Video Analysis . . . . . . . 17.2.4 Design of Adapters for Dynamic Testing . . . . . . . . . . . . . 17.2.5 Dynamic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.6 Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Numerical Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Video Results Using Kinovea® Software . . . . . . . . . . . . . 17.3.3 Digital Image Correlation and Dynamic Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Application of Generative Design and Reverse Engineering for the Improvement of a Dental Articulation System . . . . . . . . . . . . . Verónica Guzmán-Mercado, Juan Alfonso Beltrán-Fernández, Mauricio González-Rebattú y González, Erik Omar Alvarado Alcántara, Juan Carlos Hermida-Ochoa, Teresa Berenice Uribe-Cortés, and Alejandro García-Jarillo 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Redesign of the Dental Articulation System . . . . . . . . . . 18.2.3 Generative Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Static Analysis of the Prototype . . . . . . . . . . . . . . . . . . . . . 18.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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19 Buckling Analysis of an Origami-Inspired Structure with the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . José Javier Moctezuma-Reyes, Luis Héctor Hernández-Gómez, Alejan-dra Armenta-Molina, Lenin Ramos-Cantú, Tanya Nerina Arreola-Valles, Juan Alfonso Beltrán-Fernández, and Gilberto Soto-Mendoza 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Numerical Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1 Linear Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.2 Nonlinear Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . 19.4.3 Buckling Analysis with Arc Length Method (Riks) . . . . 19.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Accumulated Damage of the Main Steam Nozzle of a BWR-5 . . . . . . Tanya Nerina Arreola-Valles, Luis Héctor Hernández-Gómez, Alejandra Armenta-Molina, Salatiel Pérez-Montejo, Lenin Ramos-Cantú, José Javier Moctezuma-Reyes, and Irving Álvarez-Loya 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1 Analysis of the Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2 Analysis of the Sub Model of the Main Steam Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3 Evaluation of the Cumulative Usage Factor . . . . . . . . . . . 20.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Modeling and Design of Guillotine Cutting of a Cold Working Steel Sheet by Using FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmed Basem, Mahmoud A. Essam, and Ahmed Y. Shash 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Solid Works Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 ANSYS Analyses Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Results of Analysis of the Cutting Tool . . . . . . . . . . . . . . 21.3.2 Results of Analyses of Workpiece . . . . . . . . . . . . . . . . . . . 21.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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295 296 297 298 298 299 301 302 302 305

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22 Comparison of IR and Visual Stream in Maritime Zone Surveillance in Cases of Low and High Visibility Conditions . . . . . . . Igor Vujovi´c, Miro Petkovi´c, Ivica Kuzmani´c, and Joško Šoda 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Experimental Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Compressibility Effects on the Hydroelastic Vibration of a Plate at an Off-Center Position of a Rectangular Container Filled with Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuel Gascón-Pérez 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.1 Calculus of the Fluid Velocity Potential and Pressure Jump Over the Plate . . . . . . . . . . . . . . . . . . . 23.2.2 Calculus of the Plate Frequencies . . . . . . . . . . . . . . . . . . . 23.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Pedagogical Aspects of the Topic of Connection Properties in the Undergraduate Curriculum Using Catia v5 and Glulam Beam as the Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nader G. Zamani 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Catia v5 “Analysis Supports” and the “Connection Properties” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 Description of the Glulam Assembly and the Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 Results of Different Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Demand Forecasting, Production Planning, and Control: A Systematic Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . José Eduardo de Carvalho Lima, Paulo Renato Alves Firmino, and Luiz Alberto Oliveira Rocha 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 General Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Lexical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 Methodological Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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331 331 332 332 335 341 341

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361 361 363 366 368 373 375 377

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26 Assessment of Risks for Physical Calibration Labs . . . . . . . . . . . . . . . Mirna Osama and Ahmed Y. Shash 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.1 Points of Improvement the Calibration Lab Could Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.2 Key Factors for the Calibration Lab to Consider . . . . . . . 26.3.3 Minimum Requirements for Presence of Safe Accredited Physical Calibration Lab . . . . . . . . . . . . . . . . . 26.3.4 Minimum Requirements for Lab Proficiency . . . . . . . . . . 26.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.4 Future Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

401 401 402 428 428 429 430 430 431 432 432

Chapter 1

Preparation of Thermoplastic Blends Filled with Polysaccharide and Study of Their Properties Before and After Ageing Petra Skalková, Elena Nekorancová, Ivan Labaj, Andrej Dubec, Zuzana Miˇcicová, Slavomíra Božeková, Darina Ondrušová, and Mariana Pajtášová

1.1 Introduction During their lifetime, polymeric materials are subject to degradation processes due to the action of the environment. In general, these almost always lead to irreversible changes in properties, therefore it is very important to anticipate possible changes in advance. The emerging need to predict the long-term behavior and properties of polymers in certain environmental conditions gave the basis for the emergence of accelerated ageing tests. Testing of materials usually works on the principle of cyclic exposure of the tested material to different conditions (Hecksher et al. 2010; Robertson and Wilkes 2000; Simon et al. 2001; Slobodian et al. 2004).

P. Skalková (B) · E. Nekorancová · I. Labaj · A. Dubec · Z. Miˇcicová · S. Božeková · D. Ondrušová · M. Pajtášová Faculty of Industrial Technologies, A. Dubˇcek University of Trenˇcín, Ivana Krasku 491/30, 020 01 Púchov, Slovak Republic e-mail: [email protected] I. Labaj e-mail: [email protected] A. Dubec e-mail: [email protected] Z. Miˇcicová e-mail: [email protected] D. Ondrušová e-mail: [email protected] M. Pajtášová e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_1

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An example of accelerated ageing tests is exposure of the material to radiation and elevated temperature during light ageing at elevated temperature, or to cyclical changes of low and high temperature and phases with high air humidity during climatic tests. After ageing, the material can then be subjected to a wide range of tests and analyzes to determine the extent of degradation and changes in the studied properties. Based on the results, it is then possible to predict the service life of the investigated material, determine for which applications the material will be suitable (or unsuitable), or possibly obtain the information necessary to fine-tune the properties or formula of the material for its future use (Spoljaric et al. 2011; Calventus et al. 2001; Atkinson et al. 2002). A very useful tool for studying degradation is infrared spectroscopy or scanning electron microscopy. FTIR-ATR spectra provide a wide range of information about the structure of the investigated material. They can be used to calculate the carbonyl index, which has been proven to quantify the rate of degradation in the case of polyolefins (mainly polyethylene and polypropylene) (Caban 2022; Bredács et al. 2021). It also serves as evidence that photodegradation processes actually occurred due to heat and UV radiation during HLA (high-temperature light ageing). On the basis of SEM images, it is possible to observe the microstructure of the investigated blends (Bakshi et al. 2021; Lestari et al. 2022). It also helps to understand the effect of ageing on surface and fracture properties, or material appearance and interfacial adhesion. The appearance of the examined material can be subjected to colorimetric measurements. Tristimulus values translate color perception into a numerical world, on the basis of which it is relatively easy to compare and quantify color change before and after ageing. It is advantageous for practice to find out what changes have occurred in terms of thermal–mechanical properties. This is done by determining the Vicat softening temperature, which reflects the softening point that can be expected when the material is used in an environment with a raised temperature (Ding et al. 2022; YaoJerry et al. 2017). The main goal of the work was to analyze the ageing process of thermoplastic blends filled with polysaccharide by means of IR spectroscopy, scanning electron microscopy, colorimetry and Vicat softening temperature. At the same time, the influence of the presence and amount of filler on the properties of the blends before and after ageing were also studied. The investigated materials were low-density polyethylene and blends consisting of low-density polyethylene with carboxymethyl starch, which was added to the mixture in amounts of 5, 10, 15 and 20 wt%. The most likely area of use for this material could be found as a packaging material in the food industry or for the production of various types of films, where it would appear as a relatively fast-degrading, more ecological alternative to the materials used today.

1 Preparation of Thermoplastic Blends Filled with Polysaccharide …

3

1.2 Materials and Methods 1.2.1 Materials O-(carboxymethyl) starch with DS = 0.3 prepared by the reaction of potato starch in suspension of methanol with monochloroacetic acid after activation with 40% aqueous sodium hydroxide solution (Zhang et al. 2012) (Jena, Germany). LDPE BRALEN RB 2–62—non-additive, low-density, granulated, degradable polyethylene (Slovnaft Petrochemicals, Slovak Republic), according to EN 71, part 3, properties: melt flow index (MFI) 190 °C/ 2.16 kg: 2 g 10 min−1 (according to STN EN ISO 1133), density at 23 °C:0.918 g cm−3 (according to STN EN ISO 1183–2), tensile strength: 12 MPa (according to STN EN ISO 527), Vicat softening temperature: 95 °C (according to STN EN ISO 306), Shore D hardness: 46 (according to STN EN ISO 868).

1.2.2 Methods LDPE/CMS blends were prepared by mixing in the Brabender Plastograph EC Plus, heated to a temperature of 140 °C, at a speed of 30 min−1 . Different amounts of polysaccharide (0, 5, 10, 15 and 20 wt% CMS) were mixed into the polymer matrix for 7 min. The obtained blends were pressed in a hydraulic press at a temperature of 140 °C and a pressure of 40 kN for 2 min. The amounts of LDPE and CMS required for the preparation of blends were calculated according to: m=

xA ρA

V +

xB ρB

(1.1)

where m is the mass of the blend, V is the volume of the blend, x A is the mole fraction of the matrix (LDPE), x B is the mole fraction of the filler (CMS), ρ A is the density of the matrix (LDPE) and ρ B is the density of the filler (CMS). IR spectra were measured using a spectrometer Nicolet iS 50 FT-IR from Thermo Scientific. The measurement was performed using the ATR technique with a resolution of 4 cm−1 . A diamond crystal was used to measure the FTIR spectra, which was pressed onto the surface of the sample. Light ageing at elevated temperature was carried out in the test chamber of the Atlas Ci3000+ in the form of five 96 h exposure cycles according to the conditions: temperature of the black standard: 100 ± 3 °C, temperature of the test chamber: 65 ± 3 °C, relative humidity of the test chamber: 30 ± 5%, radiation intensity: 1.1–3.6 W m−2 . The source of radiation was an arc xenon lamp. Ageing in variations of climatic conditions was carried out in the climatic chamber CTS CW 70/1500 for 240 h (10 repetitions of a 24 h cycle). Each cycle consisted of

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a cold phase (4 h, −30 °C), a warm phase (4 h, 75 °C), and a high humidity phase (16 h, 38 °C, 95% RH). Vicat softening temperature was measured on a ZwickRoell HDT/Vicat Allround A1 device with an indenter force of 10 N. Color measurement was carried out using a spectrophotometer Datacolor Technology 500/8° with a xenon lamp. The morphology of the LDPE/CMS blends was examined using a TESCAN VEGA3 scanning electron microscope in the mode of secondary electrons (surface and fracture). Au–Pd mixture was used for plating the samples. The calculation of the carbonyl index was carried out using the MS Excel program according to relation (1.2). The content below the absorption bands was calculated using the INTEGRATE function, method A in the OPUS program. CI =

S AU B (1850 − 1650) S AU B (1500 − 1420)

(1.2)

where S AUB (1850–1650) indicates the content of the region under the absorption band in the wavenumber interval 1850 to 1650 cm−1 and S AUB (1500–1420) the content of the region under the absorption band in the wavenumber interval 1500 to 1420 cm−1 .

1.3 Results and Discussion 1.3.1 Characterization of LDPE/CMS Blends Before and After Ageing The effect of ageing on individual LDPE/CMS blends was studied by IR spectroscopy using the attenuated total reflectance (ATR) method. In Fig. 1.1 we can see the spectra of LDPE blends with different CMS contents before ageing. We observe four distinct absorption peaks in the spectra over the entire range of measured wavenumbers of 4000–400 cm−1 (Fig. 1.1a). A pair of very intense peaks at wavenumbers 2920– 2840 cm−1 are attributed to stretching vibrations of the C-H bond. A less intense peak, with a wavenumber around 1470–1450 cm−1 can be attributed to symmetric stretching vibrations of –CH2 –. The last peak is located in the region of skeletal vibrations (fingerprint region) with a wavenumber around 720–710 cm−1 and is attributed to the rocking bending of –CH2 –. All of the mentioned absorption peaks in the characteristic region can be attributed to the vibrations of the LDPE polymer matrix. In Fig. 1.1b, we observe a broad absorption band of the OH group, whose intensity increases with increasing CMS content in the blend, due to the increase in the amount of OH groups present in the starch (Aytunga 2014; Liu 2012; Gómez 2020; Amigo 2019; Corrales 2002; Wilpiszewska 2019).

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Fig. 1.1 IR spectra of LDPE blends with different CMS contents before ageing

In Fig. 1.1c, the part of the spectrum in the wavenumber range 1850–1490 cm−1 is shown. The low-intensity absorption bands in the interval from 1850 to 1650 cm−1 are mainly related to the presence of carboxyl groups, which are formed as a product of oxidation, for example, during conditioning and storage of the samples (Wypych 2018). In the region around 1600 cm−1 there are present C–O stretching deformation absorption bands related to the presence of OH groups. According to the author (Wilpiszewska 2019), the intensity of these bands directly correlates with the degree of substitution of OH groups of the starch (carboxymethylation). This can be observed as increasing intensity of the bands with increasing proportion of CMS in the blend with LDPE. In Fig. 1.1d we can see a part of the spectrum with the wavenumbers around 1420–790 cm−1 . The absorption bands with wavenumbers 1385–1375 cm−1 can be attributed to symmetric stretching vibrations of C-H. In the interval of wavenumbers 1200–800 cm−1 there are bands characteristic for starch molecules, i.e. C–O–C stretching vibrations of anhydro glucose unit at wavenumbers 1160–1150, 930 and 860 cm−1 and C–O stretching vibrations 1090–980 cm−1 (Aytunga 2014; Liu 2012; Amigo 2019; Wilpiszewska 2019). In Fig. 1.2, we can see the comparison of IR spectra of LDPE/CMS blends after heat light ageing (HLA). In the spectra of LDPE/CMS blends over the whole measured wavenumber range (Fig. 1.2a), we observe several pronounced absorption peaks. A pair of peaks with high intensity at wavenumber 2920–2840 cm−1 are attributed, as in the pre-ageing blends, to stretching vibrations of the C–H bond. The peaks with a wavenumber of 1750–1700 cm−1 are attributed to vibrations of the carbonyl group (esters, aldehydes, ketones, carboxylic acid), confirming that crosslinking occurred during HLA ageing due to degradation processes. In the wavenumber region around 1600 cm−1 there are again present the C–O stretching

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deformation absorption bands related to the presence of OH groups. Their intensity is higher compared to the LDPE/CMS blends before ageing due to the cleavage of the glycosidic bonds of the anhydro glucose unit of the starch. Another intense peak with a wavenumber of 1470–1450 cm−1 can be attributed to the symmetric stretching vibrations of the –CH2 – bond. The last peak with a wavenumber approximately 720– 710 cm−1 is located in the fingerprint region and is again attributed to the rocking bending vibration of –CH2 –. In Fig. 1.2b, we observe a broad absorption band of the OH group whose intensity increases with increasing CMS content in the blend with LDPE. In Fig. 1.2a, c part of the spectrum in the wavenumber range 1850–1490 cm−1 is observed. The bands in the range 1850–1650 cm−1 are related to the presence of carbonyl groups, which are formed as a product of oxidation, thermal degradation and photodegradation during ageing (Aytunga 2014; Liu 2012; Gómez 2020; Amigo 2019; Corrales 2002; Wilpiszewska 2019; Wypych 2018). During HLA ageing, photodegradation occurs due to oxygen, radiation and temperature. The first part of this mechanism is initiation by allyl hydroperoxides, which are formed by thermal oxidation of the vinylidene groups present in the LDPE matrix structure. Two free radicals can be formed during one initiation step. Secondary propagation processes give rise to compounds containing functional groups that are able to absorb radiation and undergo further degradation processes— chain scission and crosslinking. The main functional groups formed during degradation that are capable of absorbing radiation are ketones, therefore they are predominantly converted to vinyl groups and carboxylic acids. The resulting vinyl groups are no longer able to absorb radiation but can react photochemically with singlet oxygen formed in the LDPE matrix, making them more susceptible to free radical

Fig. 1.2 IR spectra of LDPE blends with different CMS contents after HLA

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Fig. 1.3 Carbonyl index of LDPE/CMS blends before and after HLA ageing

attack (Wypych 2018). Carboxylic acids are not susceptible to further reactions and therefore they accumulate in the LDPE structure during ageing. For this reason, we observe the occurrence of a pronounced absorption peak with a wavenumber around 1715 cm−1 in LDPE/CMS blends after HLA. It is used to calculate the carbonyl index (CI), that serves as an indicator of the degradation rate of LDPE/CMS blends (relation (1.2)). Comparison of the CI of different blends before and after HLA ageing is shown in Fig. 1.3. Due to the aforementioned formation and accumulation of carboxyl groups during the photodegradation process, we can observe a significant increase in CI after HLA ageing, suggesting that the LDPE/CMS blends truly underwent photodegradation during HLA (Amigo 2019; Wypych 2018; Almond 2020). In Fig. 1.4, we can see the comparison of the IR spectra of LDPE/CMS blends after ageing in the variation of climatic conditions (CL). In the spectra over the whole wavenumber range (Fig. 1.4a), we can observe the same pronounced absorption peaks as for the LDPE/CMS blends before ageing. The pair of most intense peaks at wavenumber 2920–2840 cm−1 are attributed to stretching vibrations of the C–H bond. The lower intensity peak with a wavenumber around 1470–1450 cm−1 can be attributed to the symmetric stretching vibrations of the –CH2 – bond. The last peak in the fingerprint region with a wavenumber around 720–710 cm−1 can be attributed to the rocking bending vibration of –CH2 –. In Fig. 1.4b, we observe a broad absorption band of the OH group whose intensity increases with increasing CMS content in the mixture with LDPE (Aytunga 2014; Liu 2012; Gómez 2020; Amigo 2019; Corrales 2002; Wilpiszewska 2019). In Fig. 1.4c, the spectral region in the wavenumber range 1850–1490 cm−1 is shown. The low-intensity absorption bands in the range 1850–1650 cm−1 are related to the presence of a small amount of carboxyl groups, which may have been formed as an oxidation product e.g., during conditioning and storage of the samples. During CL ageing, the samples are not exposed to UV radiation i.e., only thermo-oxidative degradation processes and the

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Fig. 1.4 IR spectra of LDPE/CMS blends after CL

formation of allyl hydroperoxides from the vinylidene groups present in the LDPE occurs. During ageing due to the action of the heat and oxygen, only the formation of ketone groups takes places and not their conversion to carboxylic acids. Also, due to the absence of UV irradiation, there was no photochemical reaction in the LDPE matrix, the product of which is singlet oxygen, indicating that the vinyl functional groups did not become susceptible to free radical attack. This results in a significantly lower degradation rate compared to LDPE/CMS blends after HLA, as confirmed by the reduction of the carbonyl index, Fig. 1.5 (Aytunga 2014; Liu 2012; Gómez 2020; Amigo 2019; Corrales 2002; Wilpiszewska 2019; Wypych 2018; Almond 2020).

1.3.2 Morphology of LDPE/CMS Blends Before and After Ageing The morphology of the prepared LDPE blends with different CMS contents was studied by scanning electron microscopy (SEM). SEM images of selected LDPE/CMS blends before ageing are shown in Fig. 1.6. The CMS particles are easily recognizable due to their oval and elliptical shape with smooth surface, which is characteristic for modified potato starch, as confirmed by studies (Akarsu and Dolaz 2019; Heinze et al. 1999; Cui 2013). The particle size of the CMS on average ranged from 10 to 38 µm. SEM images of the surface and fracture of pure LDPE after HLA and CL ageing are shown in Fig. 1.7. For the blends that were exposed to radiation and elevated

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Fig. 1.5 Carbonyl index of LDPE/CMS blends after HLA and CL ageing

Fig. 1.6 SEM images of LDPE blends with different CMS contents before ageing at 500 × magnification

temperature, surface photodegradation manifested by cracking can be observed. It probably also caused brittleness and reduced mechanical resistance of the blend. Increased temperature and radiation also caused the surface to melt, resulting in a smooth surface around the cracks. For the LDPE/CMS blends that were subjected to CL ageing, surface deterioration occurred due to the cyclic alternation of high and low temperature, which can be observed as a lighter appearance compared to the samples subjected to HLA (Quispe 2019). The difference in fracture appearance of the materials after HLA and CL ageing is mainly due to the difference in the length of the fibrils formed during tensile stressing of the material. The shortening of the fibrils after HLA was probably induced by UV irradiation. This shortening did not occur in LDPE/CMS blends subjected to CL ageing and thus only to cyclic alternation of high and low temperature and

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Fig. 1.7 SEM images of pure LDPE after ageing (surface and fracture)

humidity without radiation. Fibril shortening can also be interpreted as a reduced ability of the material to withstand stresses prior to failure, i.e., reduced strength and embrittlement of the material (during HLA) (Ferreira 2009). This fact is explained by the authors (Rodriguez 2020) by the term chemi-crystallization. When the material is heated, due to thermo-oxidation, the vinylidene bonds in the chain are cleaved and the resulting shorter chains migrate towards the surface. Decreasing the temperature, on the other hand, leads to a hardening of the material and therefore to a restriction of the movement of the shorter chains. Close to the surface, a structure is formed which resembles a network due to the increased density of ‘bonds’. The accumulated shorter chains cause a temporary thickening of the lamellae of the semi-crystalline LDPE matrix. As a result of these facts, pure LDPE appears lighter in SEM images after CL ageing than after HLA ageing. In the case of LDPE after HLA, which was in addition to the elevated temperature exposed to UV radiation for 480 h, it is likely that the radiation caused the elimination of the network formed. In fact, during the exposure, the adverse effect of chain scission started to prevail over chemicrystallisation, which contributed to the weakening of the material and the formation of the observed cracks, similar to the work of Rodriguez (2020). SEM images of the surface and fracture of the LDPE/5 wt% CMS blend after HLA and CL ageing are shown in Fig. 1.8. Also, for this blend exposed to radiation and elevated temperature, photodegradation of the surface manifested by the formation of cracks can be observed. The effect of elevated temperature and radiation also resulted in the formation of holes, which are probably associated with the presence of CMS particles in the blend. The LDPE/5 wt% CMS blend that was subjected to CL ageing

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Fig. 1.8 SEM images of LDPE/5 wt% CMS blend after ageing (surface and fracture)

developed voids around the CMS particles. This could be due to degradation of the polysaccharide associated with the cleavage of the glycosidic bonds, i.e., evaporation of the water present in the CMS. We believe that the cracking of the LDPE matrix combined with the formation of voids caused the CMS particles to fall out of the structure and the formation of the aforementioned holes. SEM images of the surface and fracture of the LDPE/10 wt% CMS blend after HLA and CL ageing are shown in Fig. 1.9. In the case that the LDPE/10 wt% CMS blend was exposed to HLA, it is again possible to observe photodegradation of the surface manifested by the formation of cracks and holes. During CL ageing, cavities were observed on the LDPE/10 wt% CFRP blend. SEM images of the surface and fracture of the LDPE/15 wt% CMS blend after HLA and CL ageing are shown in Fig. 1.10. For the LDPE/15 wt% CMS blend exposed to HLA, a hole can be seen which was caused by the degradation of the blend, especially the CMS filler. The effect of the elevated temperature at both HLA and CL was to split the OH bonds present in the CMS and hence to evaporate the water. Gas accumulation occurs initially only in the vicinity of the CMS particles but with increasing amounts of gas, cavities form and grow. Once a certain amount of gas is exceeded, the polymer matrix is disrupted, and the polysaccharide particle falls out of the structure. Figure 1.11 shows SEM images of the surface and fracture of the LDPE/20 wt% CMS blend after HLA and CL ageing. Due to the effect of UV irradiation and elevated temperature, more holes and cracks in the microstructure can be observed in LDPE blends with increasing polysaccharide content. The length of the fibrils

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Fig. 1.9 SEM images of LDPE/10 wt% CMS blend after ageing (surface and fracture)

Fig. 1.10 SEM images of LDPE/15 wt% CMS blend after ageing (surface and fracture)

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Fig. 1.11 SEM images of the LDPE/20 wt% CMS blend after ageing (surface and fracture)

formed when the material breaks also corresponds to Ferreira (2009). Thus, during HLA, significant photodegradation of the LDPE/20 wt% CMS blend occurred, which caused the embrittlement of the material and shortening of fibrils. Conversely, after CL ageing, the fibrils of the LDPE/20 wt% CMS blend were stiffened and elongated, probably due to the effect of chemi-crystallisation.

1.3.3 Vicat Softening Temperature of LDPE/CMS Blends The Vicat softening temperature (VST) represents the temperature at which softening of the material occurs when the material itself is in an environment of constant elevated temperature. VST was determined according to the standard (STN En ISO 1923). The results of the VST determination are presented in Fig. 1.12, that shows the graphical dependence of VST on the CMS content of the blend. In literature (Arndt and Lechner 2014), it is reported that as the filler content in the blend increases, there is a decrease in VST. In our case, pure, unfilled low- density polyethylene (LDPE) reached the highest VST value of 90.07 °C. The lowest VST value (86.60 °C) was obtained with the LDPE/20 wt% CMS blend. As the CMS content of the LDPE blend increases, the VST value decreases, which is in agreement with the literature (Arndt and Lechner 2014). In (Namhata et al. 1990), it is stated that the VST values determined by the A50 method of ISO 306 (2004) are equal to the temperature at which the shear

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Fig. 1.12 Dependence of the VST of LDPE blends on the CMS content before ageing

modulus of the material is equal to 1.7 × 108 Pa. The above fact is valid for all types of polymeric materials, regardless of whether they are filled or unfilled polymers or polymer blends. The above findings have led to a better understanding of the polymer structure and its thermo-mechanical properties under short-term loading. Thus, the VST represents on the shear modulus versus temperature curve the region where the polymer is in the glassy state, i.e., the VST is close to the temperature Tg . Thus, a decrease in the VST value can be equated to a decrease in the glass transition temperature Tg . The presence of a filler (CMS) in the thermoplastic matrix (LDPE) led to a decrease in the temperature at which the material changes from hard and brittle to soft and pliable. Consequently, less heat is required to transition the polymer from a plastic state to an elastic state. This fact could be particularly beneficial in the processing of polymeric materials in terms of energy consumption. However, if the starting polymeric material was used in a higher temperature environment the above statement could be considered as a negative (Namhata et al. 1990). The VST of LDPE blends with different CMS contents was determined after HLA ageing according to ISO 306 (2004). The VST dependence after HLA of LDPE blends with different CMS content is shown in Fig. 1.13. As with the LDPE/CMS blends before ageing, the VST after HLA decreases with increasing filler content in the LDPE matrix. Again, the unfilled LDPE had the highest VST after HLA, at 88.37 °C. A more pronounced difference can be observed for LDPE blends containing 15 and 20 wt% CMS. The LDPE/15 wt% CMS blend had a post-HLA VST value of 83.67 °C. The lowest VST after HLA (80.27 °C) was for the LDPE/20 wt% CMS blend. The VST of LDPE blends with different CMS contents after ageing in variations of climatic conditions was determined according to ISO 306 (2004). The dependence

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Fig. 1.13 Dependence of the VST of LDPE blends on the CMS content after HLA

of the VST of LDPE blends on the CMS content is shown in Fig. 1.14. Again, the highest VST value was achieved by pure, unfilled LDPE, namely 91.50 °C. Similar to the LDPE blends with different CMS contents before and after heat light ageing, we can observe a decreasing trend of VST with increasing CMS content in the LDPE blend. The LDPE blend with 15 wt% of CMS showed the lowest VST value, where there was a deviation from the expected dependence. This deviation was probably related to the presence of an air bubble in the LDPE blend, which caused the indenter to penetrate the sample faster at lower temperature. Carboxymethyl starch with a low degree of substitution (DS = 0.3) contains hydroxyl groups in its anhydro glucose unit. This means that physically bound water is also present in the CMS, which may have caused the formation of the aforementioned bubble during compression of the mixture. In Fig. 1.15 we can observe the effect of ageing on the Vicat softening temperature as a function of the filler content in the matrix. When comparing the LDPE/CMS blends before and after HLA ageing, we take into account the fact that during HLA, the blends were affected by UV radiation, oxygen present in the chamber and the ambient temperature in the chamber. Based on these factors, it can be argued that during HLA, initially thermo-oxidation and subsequently photodegradation processes occurred (Wypych 2018). As a consequence of the degradation processes, a decrease in VST was observed for all blends. Based on this, it can be concluded that the photodegradation process led directly to the deterioration of the mechanical properties—VST, and hence a decrease in the glass transition temperature is expected (Namhata et al. 1990; ISO 306 2004). Based on the determined VSTs, we can conclude that the amount of OH groups present in the polysaccharide structure is related to the degradation rate. For all LDPE blends containing CMS,

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Fig. 1.14 Dependence of the VST of LDPE blends on the CMS content after CL ageing

there was a decrease in the VST, i.e., less thermal energy is required for the phase transformation of these blends. For LDPE/CMS blends after CL ageing, we can conclude that only thermooxidative degradation processes occurred in the climate chamber when high and low temperature and humidity were alternated. The aforementioned breakdown of the vinylidene groups present in the LDPE structure led to the formation of allyl hydroperoxides and the subsequent initiation of degradation mechanisms. These

Fig. 1.15 Comparison of VST of LDPE blends with different CMS contents before and after ageing

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processes are mainly concentrated in the amorphous phase. The rate of thermal oxidation and oxygen permeation varies significantly above and below its melting point, which is due to the ratio of the amorphous and crystalline phases present (Wypych 2018). Oxygen diffusion into the crystalline domains is severely limited, and it is for this reason that it is likely that the rate of degradation was only minimal during the cold periods of ongoing CL ageing. With increasing temperature, oxygen more readily diffused into the amorphous phase. The maximum CL ageing temperature was only 75 °C and for this reason the investigated LDPE/CMS blends probably did not degrade to the same extent as they did during HLA. In contrast, CL ageing resulted in an increase in VST, which can be interpreted as an increase in strength due to chemi-crystallization (Wypych 2018; STN En ISO 1923).

1.3.4 Visual Appearance of LDPE/CMS Blends The effect of ageing of LDPE/CMS blends under different conditions on their visual appearance was studied by colorimetry. The color of the LDPE/CMS blends before ageing was determined to be light with a tinge of yellow and a red-yellow tint. An evaluation of the color change measured after the completion of each of the HLA cycles is shown in Fig. 1.16. The ΔE value determining the color change after each completed cycle increased for all blends. The highest ΔE value was reached by all LDPE/CMS blends after cycle 5, when the most significant color change occurred. The most pronounced color change occurred for the LDPE/15 wt% CMS and LDPE/20 wt% CMS blends (Fig. 1.17). Figure 1.18 shows the dependence of the color change of the LDPE/CMS blends after ageing in the variation of the climatic conditions depending on their CMS content. The overall color change of these blends after CL ageing can be considered negligible compared to the color change after HLA. Thus, degradation after CL

Fig. 1.16 Comparison of the color changes of the LDPE/CMS blends after each HLA cycle

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Fig. 1.17 Color changes of exposed and covered part of LDPE/15 wt% CMS (left) and LDPE/20 wt% CMS (right) blends

Fig. 1.18 Colour change of LDPE/CMS blends after CL

ageing occurred to a lesser extent than after HLA. This suggests that the main factor causing degradation from a visual point of view, i.e., deterioration in the appearance of the material, is UV radiation. It was also confirmed that all the filled blends with CMS degraded to a greater extent than pure low-density polyethylene (Wypych 2018).

1.4 Conclusion The blends of low-density polyethylene (LDPE) with carboxymethylated potato starch (CMS) were prepared, which was dosed into the blend in amounts of 5, 10, 15 and 20 wt%. Ageing of the blends took place in two ways, as light ageing at elevated temperature and ageing in variations of climatic conditions. With the help of analyses,

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we proved that degradation processes occurred as a result of ageing. During HLA, photodegradations occur, and during CL ageing, only thermo-oxidation processes occur. It was also found that the polysaccharide content in the blend has an effect on the properties, structure and, to a certain extent, the degradation of the materials. The knowledge of this work could find practical application in the use of blends in the food industry, for example as a packaging material. The results indicate that the material will be relatively reliable as long as there is no exposure to radiation. On the contrary, as a result of the action of radiation, relatively rapid and significant degradation occurs, which could be useful when the material becomes waste after the end of its useful life. Acknowledgements This research work has been supported by the Operational Program Integrated Infrastructure, co-financed by the European Regional Development Fund by the project: Advancement and support of R&D for "Centre for diagnostics and quality testing of materials" in the domains of the RIS3 SK specialization, Acronym: CEDITEK II., ITMS2014+ code 313011W442.

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Chapter 2

Rheological and Mechanical Properties of Rubber Blends Filled with Modified Bentonite Zuzana Miˇcicová, Slavomíra Božeková, Mariana Pajtášová, and Petra Skalková

2.1 Introduction The rubber is rarely used in commercial applications without fillers and chemical cross-linking agents and incorporating various inorganic fillers into the elastomer matrix is an effective strategy to improve mechanical properties (Valentin et al. 2010). Moreover, one of the most important characteristic parameters for fillers is their activity which depends on several factors: surface development, i.e., the general contact surface of the filler particles with the polymer, energy factor associated with the increase of energy, necessary to destroy the sample, geometric factor and the shape of grains, structure, porosity, agglomeration, and the ability to form adhesive bonds at the polymer-filler interface (Masłowski et al. 2019). The improvement of the properties of the resulting blend depends mainly on morphological aspects of these fillers, such as sizes and shapes, and sometimes on chemical modification, e.g., the use of a silane coupling agent to modify the inorganic filler (Basu et al. 2014). Chemical surface modification of nanofillers is commonly used to improve the interaction and bonding between filler and matrix while reducing the interaction between the filler and the matrix or increasing the interlayer distance (Valentin et al. 2010; Šupová et al. 2011). Such modifications also transform the Z. Miˇcicová (B) · S. Božeková · M. Pajtášová · P. Skalková Faculty of Industrial Technologies in Púchov, Alexander Dubˇcek University of Trenˇcín, I. Krasku 491/30, 02 001 Púchov, Slovakia e-mail: [email protected] S. Božeková e-mail: [email protected] M. Pajtášová e-mail: [email protected] P. Skalková e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_2

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surface of the silicate from being hydrophilic to being organophilic and reduce the swelling behavior of the modified clay (Andrunik and Bajda 2019). Bentonite is an aluminosilicate phyllosilicate, consisting mainly of montmorillonite, which is environmentally friendly and it occurs in nature as well as it is easily available in large amounts and it also has a potentially high aspect ratio as well as high surface area (Rezaei et al. 2016; Yalcin and Cakmak 2004). Montmorillonite belongs to the smectite group of minerals and has a 2:1 layered silicate structure that consists of inner octahedral sheets bound to outer tetrahedral sheets above and below, with a characteristic repeating distance (gallery spacing) between the tetrahedral–octahedral–tetrahedral layers (Barick and Tripathy 2011; Zhao et al. 2012). The basic composition of montmorillonite is Na1/3 (Al5/3 Mg1/3 )Si4 O10 (OH)2 . The layer thickness of each platelet is in the order of 1 nm, and the lateral dimension is approximately 200 nm (Rezaei et al. 2016). The lamellar structure of montmorillonite crystal cell contains some cations, such as Cu2+ , Mg2+ , Na+ , K+ , etc. However, the role of this cationic and montmorillonite crystal cell is very unstable and these cations are easier to be replaced by other cations (Jun et al. 2001; Cheng et al. 2006). A characteristic feature of the smectite group is that water and other polar molecules can cause the structure to expand in a direction that is normal to the ground plane by entering between the unit layers. Minerals belonging to this group have an overall negative surface charge, so cations can be easily adsorbed on their surface. This characteristic property is due to the presence of isomorphic substitutions in their structure. The surface charge can be changed by performing the modification process with organic surfactants. The organic surfactants interact with the hydroxylated surface groups and the organic moieties are covalently grafted to the mineral (Andrunik and Bajda 2019; Yalcin and Cakmak 2004; Barick and Tripathy 2011; Huang et al. 2017). The main objective of this study is to investigate experimentally the effect of modified bentonites on the rheological and curing characteristics and mechanical properties of rubber blends and vulcanizates. The aim of this modification is to increase the hydrophobicity of bentonite compared with the unmodified bentonite, which may lead to improvement of vulcanization properties as well as better dispersion and interaction of the modified bentonite in the rubber matrix.

2.2 Experimental 2.2.1 Materials In this study, we used bentonite as the initial material for the modified bentonite which was obtained from a site called Jelsový Potok. For all experiments, only the fraction containing a grain size of < 0.04 mm was used. Silane 3-(Trimethoxysilyl) propyl-methacrylate (TMSPM, ≥ 97%) and Bis[3(triethoxysilyl)propyl] tetrasulfide (TESPT, ≥ 90%) were purchased from SigmaAldrich. Styrene Butadiene Rubber (SBR 1723 and 1500), Stearic acid, Zinc oxide,

2 Rheological and Mechanical Properties of Rubber Blends Filled … Table 2.1 Chemical composition of the bentonite

Composition

Weight %

SiO2

68.0

Al2 O3

20.0

MgO

2.8

Fe2 O3

2.5

CaO

4.2

23

Carbon black (N339), N-(1,3-Dimethylbutyl)-N' -phenyl-p-phenylenediamine, N,N' (p-Phenylene)ditoluidine, N,N' -Di-(p-tolyl)-p-phenylenediamine, ozone protect wax, elemental sulfur, 1,3-Diphenylguanidine, N-cyclohexylbenzothiazole-2sulphenamide and residual aromatic extract oil were obtained from CMR Púchov Ltd., Slovak republic.

2.2.2 Modification of Bentonite The procedure of bentonite treatment was carried out as follows: firstly, 2 ml of TESPT silane was added to the 100 ml ethanol: water (4:1) solution. The prepared solution was stirred in a heater with a magnetic stirrer for 30 min at 60 °C. Then, 5 g bentonite was added to this solution while it was being vigorously stirred and after that, the subsequent stirring process was performed for 30 min. The solution was then filtered and washed with ethanol/water in order to remove excess of silanes. Prepared modified bentonites were being dried in a dryer at a temperature of 80 ± 5 °C for 3 h. The chemical composition of the used bentonite sample is given in Table 2.1

2.2.3 Preparation of Rubber Blends and Vulcanizates All ingredients were mixed in an internal mixer, Plasti–Corder Brabender®EC plus. The mixing process was performed at 50 rpm rotor speed, 0.75 fill factor, 20 min mixing time and 60 °C temperature. Synthetic rubber (SBR1723 and SBR1500) was first loaded to the mixer and mixed for period of 2 min, followed by the addition of curing activators. Stearic acid was added as the first and mixed for period of 0.5 min. Subsequently, oxide was added and this was mixed for period of 0.5 min. Then, the carbon black (N339) was added and mixed for period of 4 min. After that, the aminebased antioxidants and antiozonant (ozone protect wax) were added and mixed for period of 1 min. Subsequently, oil (residual aromatic extract) as added and mixed further for period of 3 min. Sulfur as a curing agent and N-cyclohexylbenzothiazole2-sulphenamide and 1.3-Diphenylguanidine as a cure accelerator were then loaded

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and mixed for period of 3 min. Bentonite and modified bentonites were added into the rubber blend, replacing the conventional filler in an amount of 5 phr. The prepared blends were molded by compression molding in a heated hydraulic press (LabEcon 600; Fontijne) to obtain optimum cure parameters (90% of the maximum cure—obtained by rheometrical curves) at 160 °C and at pressure of 20 MPa.

2.2.4 Characterization of the Modified Bentonite The bentonite and modified bentonite samples were characterized by spectral analysis (FTIR) and thermal analysis (TG/DSC). The FTIR analysis was recorded with a Thermo Scientfic Nicolet 50 FT-IR spectrometer. The samples were prepared in a KBr tablet and determined in transmittance mode over a 4000–400 cm−1 range at a resolution of 4 cm−1 . The thermal analysis was carried out with a TGA/DSC 2 STARe system METTLER TOLEDO thermal analyzer, in the 25–1000 °C temperature range, at a heating rate of 5 °C/min and under air flow of 100 ml/min.

2.2.5 Characterization of the Rubber Blends and Vulcanizates The PRPA 2000 rubber process analyzer with the lower die of the chamber oscillated sinusoidally at a fixed angle and frequency was used to characterize rheological and curing properties of rubber blends. The experiments were carried out at 160 °C, at a constant frequency of 100 CPM, according to ASTM D 5289. The weight of each used sample was approximately 6 g. The parameters: the optimum curing period (T c(90) ), pre-curing time (t s ), minimum torque (M L ), maximum torque (M H ) were calculated on the basis of the curing curves. Cure rate index (CRI) was calculated, using: C RI =

100 Tc(90) − ts

(2.1)

where T c(90) is the optimum curing period and t s is the pre-curing time.

2.2.6 Mechanical Properties The mechanical properties of vulcanized filled modified bentonite were studied and compared with standard sample which contains the conventional carbon black filler

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(N339). The tensile strength and elongation of the vulcanizates were measured by a universal testing machine—Shimadzu Autograph AG–X plus—5 kN operation with the crosshead speed of 500 mm/min according to ASTM D412 87 standard. The hardness of the vulcanizates was measured, using a Shore A Durometer hardness tester for 6 mm thick samples according to ASTM D 2240 86. Five different samples were tested and the average value for each formulation was reported.

2.3 Results and Discussion 2.3.1 FTIR Analysis The FTIR spectra of the bentonite and modified bentonites are shown in Figs. 2.1 and 2.2. The results obtained from FTIR analysis are listed in Table 2.2. In the case of the overall sample, the band visible in 3630–3631 cm−1 region is assigned to hydroxyl stretching vibrations of bentonite, whereas the bands visible in the 3448–3449 cm−1 and 1639 cm−1 regions are assigned to the with H–O–H vibrations of the water molecule (stretching vibrations and deformation vibrations, respectively) (Anirudhan and Ramachandran 2015). Major bands in the 1089–468 cm−1 regions correspond to bond vibrations in the structure of the examined montmorillonite and are associated with the stretching vibration of Si–O–Si groups in the 1037 cm−1 region. There are the absorption bands due to Al–O–Si group bending vibration (where Al is an octahedral cation) in 521 cm−1 and Al–Al–OH group bending vibration in 916 cm−1 . The bands in the 695 cm−1 and 625 cm−1 regions can be associated with the presence of stretching vibrations of Si–O–Mg and Si–O–Al bonds. A sharp absorption band in the 796 cm−1 region indicates quartz admixture in the sample (Andrunik and Bajda 2019; Parolo

Transmittance (%)

bentonite

bentonite/TESPT

100 80 60 40 20 0 3900

3400

2900

2400

1900

1400

Wavenumber (cm-1) Fig. 2.1 FTIR spectrum of bentonite and bentonite/TESPT samples

900

400

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bentonite

bentonite/TMSPM

Transmittance (%)

100 80 60 40 20 0 3900

3400

2900

2400

1900

1400

900

400

Wavenumber (cm-1) Fig. 2.2 FTIR spectrum of bentonite and bentonite/TMSPM samples

Table 2.2 Characteristic absorption peaks of bentonite and modified bentonites samples Wavenumber (cm−1 )

Bentonite

Bentonite/TESPT

Bentonite/MTSPM

Al–OH stretch

3631

3630

3631

H–O–H stretch

3448

3448

3449

CH3 asym. stretch

–

2976

2958

CH2 asym. stretch

–

2928

2931

CH3 sym. strech

–

–

2895

ν (C=O) stretch

–

–

1720

H–O–H bend

1639

1639

1639

– C–CO–O– skel

–

–

1324

– C–CO–O– skel

–

–

1300

Si–O stretch

1037

1038

1038

Al–Al–OH bend

916

917

916

Al–OH–Mg bend

796

796

796

Si–O–Mg bend

695

695

695

Si–O–Al bend

625

624

625

Al–O–Si bend

521

522

522

et al. 2014). In the case of modified bentonites, the new peaks were detected, indicating the presence of the silane. For bentonite/TMSPM sample, the peaks in 2958, 2931 and 2895 cm−1 can be assigned to the asymmetric and symmetric vibrations of CH3 and CH2 groups. The characteristic absorption peak in 1720 cm−1 corresponds to the presence of C=O group stretching vibration, whereas the peaks in 1324 cm−1 and 1300 cm−1

2 Rheological and Mechanical Properties of Rubber Blends Filled …

27

can be attributed to –C–CO–O– group skeletal vibrations. In the case of the bentonite/TESPT sample, the effectiveness of the modification of bentonite with TESPT silane is confirmed by the presence of the following weak absorption bands in 2976 and 2928 cm−1 regions which can be assigned to the asymmetric and symmetric vibrations of CH3 and CH2 groups (Shen et al. 2007; Volkov et al. 2021; Tabak et al. 2011).

2.3.2 Thermal Analysis Thermal analysis was used to determine the modified silane on bentonite. The thermal decomposition for bentonite sample and modified bentonites (bentonite/TESPT and bentonite/TMSPM) samples are shown in Fig. 2.3 and Table 2.3. According to Fig. 2.2, a weight loss of approximately 6.65% was observed at 102.81 °C in the temperature range for bentonite. This weight loss is attributed to the evaporation of physically adsorbed water on the bentonite surface (Abeywardena et al. 2017). In the case of modified bentonites, it is clearly possible to see that the weight loss of the modified particles was significantly lower in the temperature range bentonite

bentonite/TESPT

bentonite/TMSPM

Mass Loss (%)

100 95 90 85 80 75 70

0

200

400 600 Temperature (°C)

800

1000

Fig. 2.3 Thermal decomposition of bentonite and modified bentonites samples

Table 2.3 Thermal parameters of bentonite and modified bentonites samples Sample

Bentonite

First step

Second step

Third step

Mass loss (%)

T (°C)

Mass loss (%)

T (°C)

6.65

102.81

4.51

656.70

Mass loss (%)

T (°C)

Bentonite/TESPT

1.08

73.96

1.96

367.75

4.74

654.15

Bentonite/TMSPM

3.36

94.52

7.47

387.41

5.06

655.75

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of 70–100 °C. This observation shows that silane-modified bentonites adsorbed less water, compared with unmodified bentonite, confirming their hydrophobicity nature. The mass loss, observed at 300–500 °C, is probably due to vaporization of the products produced during condensation and caused by unreacted silanes. The loss of material, observed at 500–700 °C, is mainly attributed to oxidative thermal decomposition of grafted silane (Shen et al. 2007; Volkov et al. 2021). The mass loss corresponding to intercalated silane and grafted silane is higher for bentonite/TMSPM sample which could be due to a higher efficiency in the silanization reaction. The weight loss of approximately 4.51% was observed at 656.70 °C in the temperature range for bentonite. This weight loss can be attributed to the evaporation of dehydroxylation and breakdown of the mineral structure. In the case of samples for modified bentonites, the temperatures of dehydroxylation and breakdown of the mineral structure are lower than for natural bentonite sample and it can be attributed to bonding of the silane groups with the bentonite (Andrunik and Bajda 2019).

2.3.3 Rheological and Curing Characteristics of Rubber Blends Characterization of the curing characteristics of the rubber blends reveals the impact of the studied fillers on the properties of the final materials and the tendency of the filler particles to interact (to form a network in the elastomer). The vulcanization curves of the prepared rubber blends are show in Fig. 2.4. Table 2.4 shows the minimum torque (M L ), maximum torque (M H ), pre-curing time (t s ), optimum curing period (T c(90) ) and cure rate index (CRI) where the effect of modified bentonites on the rheological and curing properties of rubber blends was investigated. S

B

B/TESPT

B/TMSPM

Torque (dNm)

20 15 10 5 0 0

5

Fig. 2.4 Cure curves of rubber blends

10 Time (min)

15

20

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29

Table 2.4 Rheological and curing characteristics of rubber blends Blends

M L (dN.m)

M H (dN.m)

t s (min)

T c(90) (min)

CRI (min−1 )

S

2.91

17.06

2.83

6.91

24.51

B

2.80

15.43

3.04

9.03

16.69

B/TESPT

2.82

16.52

2.86

7.52

21.46

B/TMSPM

2.80

16.22

2.89

8.02

19.49

From Table 2.1, there is a decrease in the minimum and maximum torque values for all blends, compared with reference sample (S). The decrease of values indicates lower viscosity as well as lower stiffness of the tread blends at the beginning and at the end of the cure. In the case of the blend sample with bentonite (B), we can observe the lowest value of maximum torque, indicating weak interaction in the filler-matrix interphase. In the case of pre-curing time and curing period, blends exhibit the higher values in comparison with reference blend values. The value of the optimum cure period (T c(90) ) was higher for all rubber blends compared with that of the reference rubber blend. From a practical point of view, the increase in the t90 parameter can be an advantage but very short cure period for coarse rubber products can cause differences in the cure state between the center and surface of the rubber products (Lipsinska and Soszka 2019). Evidence that the cure reaction is enhanced by the application of modified bentonite filler was provided by the higher maximum torque and lower cure speed values for the blends containing modified bentonite compared with the blend containing unmodified bentonite. These results can lead to the assumption that the crosslinking density is increased and the curing reaction time is decreased due to the modification of bentonite, and the modified bentonite can more easily absorb the rubber molecules onto its surface, so it means, it can be considered as a good or suitable filler. In the case of the calculated curing rate index (CRI) (Fig. 2.5), all of the blends show the lower values, compared with the reference rubber blend, indicating a slower cure progress. The rate of cure can be recognized where the crosslinking and development of modulus of the blend occur after scorch point (Abidin et al. 2021).

2.3.4 Mechanical Properties of Vulcanizates The stress–strain curves for filler-filled rubber systems are affected by the crosslinking density of rubber matrix and the size of agglomerates formed by the filler and rubber-filler interactions (Mostafa et al. 2010). Figure 2.6 shows the tensile test curves for the vulcanizates. Table 2.5 presents the average values and the respective standard errors of the tensile strength, elongation, and Shore A.

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30

CRI (min-1)

25 20 15 10 5 0 S

B

B/TESPT Blends

B/TMSPM

Fig. 2.5 Cure rate index of rubber blends

S

B

B/TESPT

B/TMSPM

Stress (Mpa)

20 15 10 5 0

0

100

200

300 400 Strain (%)

500

600

Fig. 2.6 Tensile tests of vulcanizates

Table 2.5 Mechanical properties of vulcanizates Vulcanizates

Tensile strength (MPa)

Elongation (%)

Shore A hardness

S

17.57 ± 0.18

474 ± 7.51

58.14 ± 0.74

B

16.61 ± 0.58

542 ± 10.80

56.06 ± 0.27

B/TESPT

17.55 ± 0.48

521 ± 11.38

56.90 ± 0.41

B/TMSPM

15.94 ± 0.52

481 ± 12.61

57.02 ± 0.60

The tensile strength value of B/TESPT vulcanizate shows a comparable value to the reference vulcanizate, indicating a good interaction between the filler and the rubber. A reduction in TS values is observed for vulcanizates B and B/TSMPM. The reduction in strength is due to the agglomeration of bentonite, which limits the mobility of the rubber chain. The rubber-filler interaction is also weaker. As a result,

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31

Shore A Hardness

60 55 50 45 40 S

B

B/TESPT Vulcanizates

B/TMSPM

Fig. 2.7 Shore A hardness of vulcanizates

the stress under load cannot be distributed evenly throughout the matrix, reducing the strength of the vulcanizates. In addition, the tensile strength of polymers remains closely dependent on the adhesion between the filler and environment as well as the stress transfer mechanism in the filler–polymer system (Basu et al. 2014; Mat and Ismail 2016). All elongation values are higher compared with the reference vulcanizate (R). In the case of B/TESPT and B/TMSPM vulcanizates, the elongation values in the presence of modified bentonite are lower than those for unmodified bentonite due to the density of crosslinks in the rubber chain. The degree of hardness (Fig. 2.7) of the vulcanizate depends on the degree of crosslinking. When the crosslinking density is increased, the hardness is also increased (Ramesan 2005). Measurements of the hardness values for the vulcanizates show that all vulcanizates exhibit lower hardness. The obtained results confirmed that lower values of maximum torque reduce the hardness of the vulcanizate. Both results are due to the stiffness of the material, resulting from agglomeration of the filler in the elastomeric matrix and the crosslinking density, which directly influences the value of M H .

2.4 Conclusion In this study, the rheological and mechanical properties were investigated for rubber blends filled with modified bentonites. The results of FTIR and thermal analysis confirm the presence of silanes. The success of the modification can be confirmed by a group of absorption bands in the region of 3000–2800 cm−1 , which are assigned to the asymmetric and symmetric vibrations of the CH3 and CH2 groups for both modified bentonites. In the case of the benonite/TMSPM sample, additional new absorption peaks were observed, corresponding to the presence of an extensional vibration of the

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C=O group and a skeletal vibration of the –C–CO–O– group. From the vulcanization characteristics, we can observe the lowest value of maximum torque in the case of the sample blended with bentonite, indicating a weak filler-matrix interaction. There is the evidence that the curing reaction is enhanced by the application of modified bentonite filler and we have observed the higher maximum torque values and lower cure rates for the modified bentonite compared with the bentonite-containing blend. From the measured results of the mechanical properties, the tensile strength values of the B/TESPT vulcanizate were comparable to the reference vulcanizate, indicating a good or sufficient interaction between the filler and the rubber. From the overall comparison relating to all examined characteristics, we can conclude that the B/TESPT sample has the most comparable values to the reference rubber sample (S). Acknowledgements This research work has been supported by the Operational Programme Integrated Infrastructure, co-financed by the European Regional Development Fund by the project: Advancement and support of R&D for “Centre for diagnostics and quality testing of materials” in the domains of the RIS3 SK specialization, Acronym: CEDITEK II., ITMS2014+ code 313011W442.

References Abeywardena SBY, Perera S, Nalin de Silva KM et al (2017) A facile method to modify bentonite nanoclay with silane. Int Nano Lett 7:237–241 Abidin ZZ, Mamauod SNL, Khooi D et al (2021) Studies of carboxylated nitrile butadiene rubber/butyl reclaimed rubber (XNBR/BRR) blends for shoe soles application. J Mech Behav Mater 30:178–187 Andrunik M, Bajda T (2019) Modification of bentonite with cationic and nonionic surfactants: structural and textural features. Minerals. https://doi.org/10.3390/ma12223772 Anirudhan TS, Ramachandran M (2015) Adsorptive removal of basic dyes from aqueous solutions by surfactant modified bentonite clay (organoclay): Kinetic and competitive adsorption isotherm. Proc Saf Environ Prot 95:215–225 Barick AK, Tripathy DK (2011) Effect of organically modified layered silicate nanoclay on the dynamic viscoelastic properties of thermoplastic polyurethane nanocomposites. Appl Clay Sci 52:312–321 Basu D, Das A, Stöckelhuber KW et al (2014) Advances in layered double hydroxide (LDH)-based elastomer composites. Prog Polym Sci 39:594–626 Cheng Q, Li C, Pavlinek V et al (2006) Surface-modified antibacterial TiO2 /Ag+ nanoparticles: preparation and properties. Appl Surf Sci 252:4154–4160 Huang Z, Li Y, Chen W et al (2017) Modified bentonite adsorption of organic pollutants of dye wastewater. Mater Chem Phys 202:266–276 Jun L, Siddiqui JA, Ottenbrite RM (2001) Surface modification of inorganic oxide particles with silane coupling agent and organic dyes. Polym Adv Technol 12:285–292 Lipsinska M, Soszka K (2019) Viscoelastic behavior, curing and reinforcement mechanism of various silica and POSS filled methyl-vinyl polysiloxane MVQ rubber. SILICON 11:2293–2305 Masłowski M, Miedzianowska J, Strzelec K (2019) Natural rubber composites filled with crop residues as an alternative to vulcanizates with common fillers. Polymers. https://doi.org/10.3390/ polym11060972

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Mat NSC, Ismail H (2016) Curing characteristics and tear properties of bentonite filled ethylene propylene diene (EPDM) rubber composites. Procedia Chem 19:394–400 Mostafa A, Abouel-Kasem A, Bayoumi MR et al (2010) Rubber-filler interactions and its effect in rheological and mechanical properties of filled compounds. J Test Eval. https://doi.org/10.1520/ JTE101942 Parolo ME, Pettinari GR, Musso TB et al (2014) Characterization of organo-modified bentonite sorbents: the effect of modification conditions on adsorption performance. Appl Surf Sci 320:356– 363 Ramesan MT (2005) The effects of filler content on cure and mechanical properties of dichlorocarbene modified styrene butadiene rubber/carbon black composites. J Polym Res 11:333–340 Rezaei A, Abdi M, Mohebbi A et al (2016) Using surface modified clay nanoparticles to improve rheological behavior of hydrolized polyacrylamid (HPAM) solution for enhanced oil recovery with polymer flooding. J Mol Liq 222:1148–1156 Shen W, He H, Zhu J et al (2007) Grafting of montmorillonite with different functional silanes via two different reaction systems. J Colloid Interf Sci 313:268–273 Šupová M, Martynková GS, Barabaszová K (2011) Effect of nanofillers dispersion in polymer matrices: a review. Sci Adv Mater 3:1–25 Tabak A, Yilmaz N, Eren E et al (2011) Structural analysis of naproxen-intercalated bentonite (Unye). Chem Eng J 174:281–288 Valentin JL, Mora-Barrantes I, Gonzalez JC et al (2010) Novel experimental approach to evaluate filler-elastomer interactions. Macromolecules 43:334–346 Volkov DS, Rogova OB, Proskurnin MA (2021) Temperature dependences of IR spectra of humic substances of brown coal. Agronomy. https://doi.org/10.3390/agronomy11091822 Yalcin B, Cakmak M (2004) The role of plasticizer on the exfoliation and dispersion and fracture behavior of clay particles in PVC matrix: a comprehensive morphological study. Polymer 45:6623–6638 Zhao J, Milanova M, Warmoeskerken MMCG et al (2012) Surface modification of TiO2 nanoparticles with silane coupling agents. Colloid Surf A 413:273–279

Chapter 3

Utilization of Microwave Radiation for Chemical Modification of Kaolin and Its Influence on the Curing, Mechanical and Surface Properties of Rubber Composites Andrea Feriancová, Iveta Papuˇcová, Jana Pagáˇcová, Jana Šulcová, and Ivan Labaj

3.1 Introduction Kaolin (K) is one of the most economic white filler of natural origin in order to by employed in rubber composites and the polymers. Kaolin is an inorganic phyllosilicate which belongs to the group of clays with a natural hexagonal plate shape. The main component of kaolin is a mineral kaolinite which has a crystal structure composed 1:1 type of a two-layer structure stablished by an angle sharing tetrahedral layer [SiO4 ] and an edge sharing octahedral layer [AlO6 ] (Olaremu 2015; Zhang et al. 2012; Kumar et al. 2013). The water molecule separates each of these two layers by a monolayer and distance between two layers is about 0.72 nm. The surface properties and the laminar shape of kaolin particles determine also its industrial application (Panda et al. 2010). In practice, there is a growing interest in the use of modified kaolin for its capacity to adsorb the inorganic and organic molecules. Kaolin is important mineral fillers for rubber mixtures due to the low price and choice A. Feriancová (B) · I. Papuˇcová · J. Pagáˇcová · J. Šulcová · I. Labaj Department of Material Technologies and Environment, Faculty of Industrial Technologies, Alexander Dubˇcek University of Trenˇcín, Ivana Krasku 491/30, 020 01 Púchov, Slovakia e-mail: [email protected] I. Papuˇcová e-mail: [email protected] J. Pagáˇcová e-mail: [email protected] J. Šulcová e-mail: [email protected] I. Labaj e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_3

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of reinforcement (hard kaolin) (Trabelsi and Tlili 2017). It can be used as an environmentally friendly filler, as a substitute for activated carbon black for the preparation of rubber blends with the desired properties (Domka et al. 2003). Carbon black (CB) and precipitated silica are used for their reinforcing properties of the rubber composites. The reinforcing effect is manifested by the improvement of rubber properties, such as tensile strength, abrasion resistance, thermal stability, processing properties and others (Andrews 1963). Several studies suggest that clay minerals could improve the mechanical and dynamic properties of rubber particularly at low filler loading (up to 10 phr) (Sheikh et al. 2017; Roshin et al. 2018; Ogbebor et al. 2015; Raji et al. 2016). The intercalation of kaolin are more desirable due to its high surface area, which is important for kaolin dispersion in the clay/polymer composites and could affect required rubber reinforcement (Valášková et al. 2007a, b; Paul and Robeson 2008). The kaolin surface possesses OH groups, which makes it polar and moisture adsorbing. To improve dispersion of the clay particles in rubber, the filler surface was treated with different reagents, such as silanes (Zhang et al. 2016; Yang et al. 2020; Dai and Huang 1999), quarternary ammonium salts (Roshin et al. 2018; Ogbebor et al. 2015; Shah et al. 2013; Kim et al. 2007) or sodium salts of rubber seed oil (Raji et al. 2016; Yahaya et al. 2009; Yang et al. 2017). The intercalation reactions with kaolin often take place by applying the guest molecules of reagents in the form of liquids, melts or concentrated solutions, at temperatures from 20 to 100 °C. Some authors reported an intercalation time of tens of hours or several days using conventional heating of the reaction mixture (Yang et al. 2020; Avila et al. 2010; Sukumar and Menon 2008; Feriancová et al. 2019). In general, if the intercalation or chemical treatment of kaolin is conducted near room temperature or by traditional convection heating, the methods are usually complex and time-consuming. Some authors (Mo et al. 2019; Li et al. 2007; Pi et al. 2007a) used the microwave heating to shorten intercalation time of kaolin. According to Zhang and Xu (2007) and Feriancová et al. (2021) reports the microwave heating shorted the reaction time from a few days to hours or even minutes, and a modified kaolin with a higher rate of intercalation of dimethyl sulfoxide (DMSO) was also obtained. The microwave heating at 2.45 GHz has gained popularity in many fields of industry and science recently. The application of microwave irradiation (wavelength range of 0.001–1.0 m) has been reported in surface activation and in desorption/reactivation/recycling studies (Bakain et al. 2014). The activity of kaolin fillers depends upon the specific surface area, the extent of dispersion in the elastomer matrix, the filler–polymer interactions and the fillerfiller interactions. For the increasing of interactions in the rubber composites, a kaolin surface has to be modified or surface treated with reagents such as ammonium acetate (AAc), urea (U) and hydrolyzed silane (3-aminopropyl)triethoxysilane (ATS) in various combinations with distilled water, dimethyl sulphoxide, ethanol and toluene. The experiments were carried out in the microwave reactor FlexiWAVE (designed for chemical synthesis in laboratory) at a constant set temperature (80 °C). Modified kaolin has been reported as the most common inorganic additive as dispersed phase in the rubber composites, improving required processability and physicomechanical properties which may enable their applications for automobile tires

3 Utilization of Microwave Radiation for Chemical Modification …

37

(Feriancová et al. 2019; Zhang et al. 2010; Sreelekshmi et al. 2017). Some of the demerits of carbon blacks, such as environmental pollution or high heat build-up at higher dosages, prompt to the use of nonblack fillers, such as silica, calcium carbonate, kaolin, mica or combinations of different light fillers for various applications (Gopi et al. 2011). Based on mentioned facts, the prepared modified kaolinite samples were applied as fillers in natural rubber compounds in amount of 10 phr and/or 30 phr in combination with carbon black. By microwave assisted irradiation, the effect of modified kaolinite on cure and mechanical properties of rubber compounds was evaluated. Subsequently, the influence of modified kaolin samples on the surface properties of vulcanizates was investigated by measuring the contact angle and free surface energy on rubber composites.

3.2 Material and Methods 3.2.1 Materials and Preparation of Modified Kaolin Commercial sample of kaolin, supplied by LB Minerals, Ltd. (Kaznˇejov) with a composition of 61% of SiO2 , 35% of Al2 O3 , 2% of K2 O, 0.6% of Fe2 O3 was used. Its loss on ignition is 8.87%wt. The 25 μm sieving undersize was used for treatment as well as for all experiments. The silane coupling agent (3-aminopropyl)triethoxysilane (ATS), ammonium acetate (AAc) and urea (U) were purchased from Sigma-Aldrich, Ltd. Ethanol and toluene were purchased from Merck, Ltd. All chemicals were of analytical grade and used as received. In a typical process, 20.0 g of kaolin was dispersed in 100 mL of selected chemical solution: ethanol + water + ATS + DMSO in ratio of 40:5:5:50 mL (K1 sample); DMSO + water + ATS in ratio of 87:8:5 mL (K2 sample); toluene + ATS in ratio of 95:5 mL (K3 sample); 20%wt solution of urea (K4 sample); 2 M solution of AAc (K5 sample). Dispersions of kaolin with chemical reagents were all reacted under the vigorous stirring in FlexiWAVE (MILESTONE) microwave reactor for 120 min, wherein the microwave power of the reactor was controlled and automatically changed by the set temperature of 80 °C. The labeling of the samples and the modification process are shown in Table 3.1. During the reactions, the average power of the reactor was around 100–160 W. All modified-kaolin samples were washed three times with ethanol and dispersion was centrifuged for 20 min at 3000 rpm. All modified kaolin samples were dried in a vacuum oven at 60 °C for 12 h.

38 Table 3.1 Samples and modification procedure

A. Feriancová et al. Sample Procedure under the microwave radiation K1

Kaolin + Ethanol + H2 O + ATS + DMSO → 80 °C, 120 min

K2

Kaolin + DMSO + H2 O + ATS → 80 °C, 120 min

K3

Kaolin + Toluene + ATS → 80 °C, 120 min

K4

Kaolin + Urea + H2 O → 80 °C, 120 min

K5

Kaolin + AAc + H2 O → 80 °C, 120 min

3.2.2 Characterization Techniques Kaolin samples were characterized by elemental analysis (EDX), Fourier transform infrared spectroscopy (FT-IR), simultaneous thermogravimetry and differential scanning calorimetry (TG/DSC). The EDX analysis was carried out by the energy dispersive fluorescence X-ray spectrometer (Shimadzu EDX-7000). Using FT-IR Nicolet iS50 Thermo Scientific spectrometer, the infrared spectra were recorded in the 4000– 400 cm−1 range, with spectral resolution of 4 cm−1 , and the measurements were made by ATR technique using diamond crystal. The thermal stability of kaolinite samples was studied, using TGA/DSC 2 STARe System METTLER TOLEDO. The TG/DSC measurements were carried out in nitrogen atmosphere with heating rate of 10 °C min−1 from room temperature to 900 °C, and sample mass was about 20–24 mg.

3.2.3 NR/Kaolin Composites Preparation and Characterization Natural rubber used as an elastomer matrix was Standard Malaysian Rubber (SMR10). Carbon black of N339 type was used as a commercial filler. Table 3.2 lists the recipe used for NR/kaolin composites: 100 phr (parts per hundred rubber) of SMR10, 2.5 phr of zinc oxide (ZnO, activator), 1.5 phr of stearic acid (SA, activator), 60.0 phr of filler (50.0/30.0 phr of N339 in combination with modified kaolin with designation of K1, K2, K3, K4, K5 in amount of 10.0/30.0 phr), 3.0 phr of antidegradants, 1.0 phr of N-cyclohexyl-2-benzothiazolesulfenamide (CBS, accelerator), and 1.7 phr of sulphur (curing agent). Kaolin and modified kaolin samples were used as a filler replacement for carbon black in the amount of 10.0 and or 30 phr. The rubber composites were prepared by two-step mixing in Plasti–Corder Brabender® EC plus (chamber volume of 80 cm3 ), with constant speed of 50 rpm, at temperature of 110 °C in I step (mixing time—7.5 min) and at temperature of 90 °C in II step (mixing time—5 min). The curing characteristics (t 90 , t s2 , CRI, M L , M H ) of the rubber composites were measured by an RPA-2000 rheometer at a test temperature of 160 °C for 25 min. Cure

3 Utilization of Microwave Radiation for Chemical Modification … Table 3.2 Prescription for NR composites filled with kaolin samples

39

Composites/ingredients

ST (phr* )

(K1–K5)–10 (phr* )

(K1–K5)–30 (phr* )

SMR 10

100

100

100

Activators

4.0

4.0

4.0

Antidegradants

3.0

3.0

3.0

Carbon black

60.0

50.0

30.0

Kaolin (K1–K5)

–

10.0

30.0

Sulfur

1.7

1.7

1.7

Accelerant

1.0

1.0

1.0

phr *

parts per hundred rubber

rate index was calculated as CRI = [100/(t 90 – t s2 )], where t 90 is the optimum cure time and t s2 is the scorch time. Prespecified vulcanization conditions were 160 °C and 20 MPa for the respective optimum cure time (t 90 ). The mechanical performance of the vulcanized composites (TS b and E b ) was examined through Autograph AG–X plus 5 kN, Shimadzu at a crosshead speed of 500 mm min−1 . Shore A Durometer hardness was tested based on the standard of ASTM D2240. The surface free energy (SFE) values of solid materials were determined using the Fowkes method, which assumes that the surface free energy of a solid (γs ) is the p ˙ 2007): sum of polar (γs ) and dispersive (γsd ) components of SFE (Zenkiewicz γs = γsd + γsp

(3.1)

The polar component of surface free energy is the sum of the components of interparticle interactions (polar, hydrogen bond type, inductive and acid–base, excluding dispersive interactions). Dispersion interactions define the value of the surface free energy dispersive component. Surface free energy values are determined by measuring the contact angle θ (CA) of samples of polymeric composite materials using polar and non-polar liquids. From the calculated contact angles, the values of the surface energy of the studied layers and its polar and dispersive components were calculated, using the Fowkes ˙ 2007). First, the contact angle θ for the solid surface was method (Zenkiewicz determined using a non-polar liquid. Then the value of γsd was calculated from the following equation: γsd = 0.25γl (1 + cos θ )2

(3.2)

where γl is the surface energy for a non-polar liquid for which γl = γld . The value p of the contact angle γl = γld determined for a polar liquid for which γl = γld + γl , p p and the calculated value of γs were used to calculate the value of γs according to equation:

40

A. Feriancová et al.

    0.5 2 p γsp = 0.5γl 1 + cos θ p − γsd .γld /γl

(3.3)

Diiodomethane (DIM) was chosen as a non-polar liquid (γl = γld = 50.8 mJ m−2 ) p and distilled water (DW) as a polar liquid (γsd = 21.8 mJ m−2 , γl = 51.0 mJ m−2 ). Contact angle measurements were made under laboratory conditions (temperature of 23 ± 2 °C and humidity of 32 ± 2%). The volume of test fluid drops was 5 μl. The contact angle was measured immediately after applying a drop of test fluid to the sample surface (after a few seconds). Measurements were performed using a contact angle measuring device (light source, zoom camera and fixed horizontal pad) and a computer image analysis program. 6–10 drops of each test liquid were applied to the surface of each test sample. The surface free energy was determined from the individual values of the contact angles of DW and DIM. Statistical analysis of DW and DIM contact angles was performed according to the literature (Rudawska et al. 2017). The Grubss test was used to test outliers, which out of a set of contact angle DW and DIM values marked 5 outliers (significance level is α = 0.5).

3.3 Results and Discussion 3.3.1 Characterization of Kaolin and Modified Kaolin Samples 3.3.1.1

Elemental Analysis of Kaolin

The elemental analysis was carried out to analyze the chemical composition of the kaolin and the subsequent chemical changes that occurred due to chemical treatment under MW radiation. The chemical composition of the kaolin is listed in Table 3.3. It is evident that SiO2 is the major component with 61.01%wt followed by alumina (Al2 O3 ) with 34.54%wt. Kaolin has a sufficient number of surface –OH groups which can continue to react with binding agents such as silanes, quarter ammonium salts and others (Reinosa et al. 2019; Kloprogge 2019). Table 3.3 Chemical composition of raw kaolin and modified kaolin (K1–K5) sample Oxides

Kaolin %

K1%

K2%

K3%

K4%

K5%

SiO2

61.01

62.06

54.03

60.96

60.92

60.87

Al2 O3

34.54

33.22

30.97

34.62

34.35

34.58

K2 O

1.84

1.82

1.57

1.79

1.83

1.78

SO3

0.86

1.17

11.96

0.89

1.17

1.10

Fe2 O3

0.57

0.56

0.49

0.57

0.56

0.56

Si/Al ratio

1.43

1.5

1.3

1.42

1.45

1.42

3 Utilization of Microwave Radiation for Chemical Modification …

41

After silane treatment, it was observed the highest content of silica (62.06%wt. of SiO2 ) in the K1 sample, when the kaolin was exposed to ATS in combination with ethanol and DMSO in ratio of 40:50. At the same time, the Si/Al ratio increased slightly. During ATS treatment in DMSO and water in ratio of 87:8 (K2 sample), a slight decrease in silica and alumina content was observed. In contrast, an increasing sulphur content (expressed as SO3 ) was observed. With increasing content of SO3 at the same time, it reduces the content of silica and alumina. However, we can state that the Si/Al ratio of K2 sample, treated with combination of reagents, after the assistance of microwave irradiation changed slightly. The small changed Si/Al ratio means that there was no leaching of Al3+ ions from octahedral layer due to chemical treatment (Feriancová et al. 2021).

3.3.1.2

Infrared Spectroscopy

Selected bands of the FT-IR spectra in the kaolin and modified kaolin K1–K5 samples are shown in Table 3.4. In the raw kaolin, the characteristic stretching vibration bands of external surface –OH (ν(ouOH)) groups and inner –OH ν(inOH) groups of kaolin were observed at 3687, 3669, 3650 cm–1 , and 3618 cm–1 respectively (Yang et al. 2012; Makó et al. 2014). Bending vibration of –OH group is localized at 910 cm–1 . The strong bands at 1025 and 1001 cm–1 correspond to the external and in plane stretching vibration of Si–O group (Madejová and Komadel 2001; Djomgoue and Njopwouo 2013). FT-IR spectra of the modified kaolin samples (K1–K5) after microwave irradiation are shown in Fig. 3.1 for the 3800–3400 cm–1 region. In Fig. 3.1, the IR spectra of modified K1, K3, K4, and K5 samples show the decrease in relative intensity of ν(ouOH) vibration and the band of external hydroxyl group ν(ouOH) is slightly shifted to higher wavenumber. FT-IR analysis of K1 and K3 samples showed that the spectra were not significantly altered after the modification reactions. We observed the decrease of intensities of the stretching vibrations of silica groups. In comparison with raw kaolin, K1 and K3 samples exhibit several rather small vibration bands assigned to C–H stretching of methylene groups around ~ 2920–2850 cm–1 , and deformation vibration band of CH3 group at ~ 1440–1420 cm–1 (Mbey et al. 2013). A weak vibration band at ~ 1570 cm–1 could be assigned to the vibration of the N–H group (Yang et al. 2020). These findings evidence the presence of the silane moieties in silane modified samples. Moreover, based on the unaltered frequency of stretching bands of the hydroxyl of the inner-surface Al–OH groups, we can conclude that grafting has taken place only on the external surface of kaolin (Fig. 3.1). In FT-IR spectrum of K2 sample, the modification with DMSO and ATS is confirmed by the weaker intensities of the –OH bands at 3693 and 3619 cm–1 , the disappearance of the bands at 3668 and 3650 cm–1 , and the appearance of the new OH stretching band at 3659 cm–1 . The intensity of external inner hydroxyl groups decreases simultaneously with the occurrence of new bands. Sulphonyl group in DMSO creates new bonds (S=O–OH) at 3535 and 3502 cm–1 and changed of

42

A. Feriancová et al.

Table 3.4 Selected band of the FT–IR spectra for raw kaolin and K1–K5 samples (4000–400 cm–1 ) Typ of vibration

K1 (cm–1 )

K2 (cm–1 )

K3 (cm–1 )

K4 (cm–1 )

K5 (cm–1 )

Kaolin (cm–1 )

ν1-3 (O–H)

3686

3693

3686

3685

3685

3687

3669

3659

3669

3669

3669

3669

3650

3619

3650

3650

3650

3650

3535

–

–

–

–

–

–

–

ν (S=O–OH)

–

3502 ν (CH3 )

2926

3022

2940

2854

2936

2929

ν (C=O)

–

–

–

1663 1624

–

–

δ (CH3 )

1440

1428

1420

–

1430

–

1570

–

1448

–

1318 δ (N–H) or (COO– )

1570

1550

1560 ν (–CN)

–

–

–

1460 1430

–

–

ν (Si–O)

1025

–

1025

1024

1023

1025

1001

1001

1001

997

998

1001

529

530

527

526

526

529

δ (Si–O–Al)

Fig. 3.1 FT-IR spectra of modified kaolin K1-K5 samples in the 3750–3350 cm–1 region

3 Utilization of Microwave Radiation for Chemical Modification …

43

the intensity and location of the characteristic band of external inner surface – OH. The Si–O–Si stretching vibrations were transformed into one broad vibration band at 1001 cm–1 , due to interaction of inner surface oxygen atom with DMSO (Mbey et al. 2013). In FT-IR spectrum of K4 sample, which was modified with urea solution, the formation of weaker vibration bands in the region of 1663–1624 cm–1 was observed, which belong to the stretching vibration of the C=O group of urea, as well as a bands in the region of 1460–1430 cm–1 , which belong to the stretching vibration of –CN (Pi et al. 2007b). The FT-IR spectrum of K5 sample show the presence of ammonium ions within the kaolin through the bending vibration of NH4 + observed at 1448 cm–1 and carbonyl stretching vibration at 1560 cm–1 . Stretching mode of Si–O is shifted to lower values (Yang et al. 2020; Mbey et al. 2013).

3.3.1.3

Thermal Analysis of Kaolin

Thermal analysis is an appropriate technique to evaluate the loading of organic molecules grafted onto kaolin particles surfaces as well as the thermal stability of kaolin samples. The major endothermic mass loss of the kaolin sample was observed at 430–720 °C. The maximum mass loss occurs at 521 °C and it is attributed to the dehydroxylation of kaolin. The dehydroxylation of structural water might result in the disturbance of the Al–OH octahedral sheets by the outer –OH groups, but it does not have much effect on the SiO4 tetrahedral sheets due to the more stable inner –OH groups (Gasparini et al. 2013; Horváth et al. 2003). The total mass loss of the kaolin sample, recorded at 900 °C, was 8.87%wt. DSC/TG measurements of K1, K3, K4, and K5 samples did not exhibit any significant differences between the studied samples and the raw kaolin sample. Higher mass loss was noted for mentioned samples (Table 3.5). For K4 sample, the peak attributed to the dehydroxylation of kaolin occur at the lower temperature (510 °C) than for raw kaolin. A higher mass loss in the modified samples indicates the degradation of kaolin crystal structure in some extent resulting from the attachment of functional groups to the surface of kaolin. K2 sample exibited a large mass loss (8.87%wt.) at about 164 °C. It is attributed to the volatilization of the intercalated organic compound. They mainly consisted of residual DMSO molecules, as well as small amount of ATS bounded to kaolin. The dehydroxylation temperature of K2 sample takes place at 506 °C and it is slightly lower than the dehydroxylation temperature of raw kaolin, and this fact is connected with the reduced cohesion of the kaolin sheets (Mbey et al. 2013). The dehydroxylation for K2 sample led to mass loss of 8.7%wt. (Fig. 3.2).

44

A. Feriancová et al.

Table 3.5 Thermal analysis of raw kaolin and modified kaolin (K1–K5) samples

Sample

Dehydroxylation (°C)

Total mass loss (%)

Kaolin

521

8.87

K1

518

9.54

K2

506

17.57

K3

517

9.40

K4

510

9.50

K5

522

9.51

Fig. 3.2 DSC and TG analysis of K2 sample

3.3.2 Characterization of NR/Kaolin Composites 3.3.2.1

Vulcanization Characteristics

Two of the most critical properties of rubber composites are processibility and vulcanization. Table 3.6 shows the curing features of NR/kaolin composites containing 10 phr and/or 30 phr of modified kaolin samples (K1–K5) and their comparison to standard blend with carbon black (ST). It is evident that for NR/kaolin composites containing of 10 phr of K1–K5 filler, the occurrence of vulcanization is accelerated. The composites showed an improvement in vulcanization characteristics—reduction of cure time (t 90 ) (Fig. 3.3a) and accelerated the vulcanization rate CRI (Fig. 3.3b). The vulcanization time was the shortest when kaolin with DMSO (K2–30) and urea (K4–30), at a loading of 30 phr was used. This result indicates that modified K1, K2 and/or K4 kaolins can act as an efficient ingredient for NR compounds. The formed

3 Utilization of Microwave Radiation for Chemical Modification …

45

bond (S=O–OH) from DMSO (K2 sample) or formed silica bond from silane could participate in the process of forming an active sulphur vulcanizing agent along with Zn2+ cations from the ZnO and contributes to the faster crosslinking of network. Cure rate index reveals that K2-10, K2-30, K4-30 compounds possess the highest CRI. Therefore, we can suppose, the higher the content of the active groups, such as S=O–OH, COO– or N–H the greater the possibility of the formation of active ZnSulphur complex in the rubber compounds (Liu et al. 2008). The increase in scorch time is the benefit also regarding mixing, extrusion, molding, and filling into molds. Table 3.6 shows the decreasing of the minimum torque of all composites with modified kaolin in comparison with standard blend. Lower value of M L corresponds to the high viscosity of the composites, and it is indicative of a higher extent of polymer-filler interaction for modified kaolin (Leblanc 2000). Viscosity increases as a function of bound rubber, and bound rubber improves as a function of filler surface area and filler loading. This property of kaolin filler has great advantages in the case of high viscosity of rubber matrix because it is easy to mix and process it (Liu et al. Table 3.6 Cure characteristics of NR/kaolin composites Composite/phr

ML (dNm)

MH (dNm)

t s2 (min)

t 90 (min)

CRI (min−1 )

K1–10

2.48

28.04

0.94

2.31

72.99

K2–10

2.47

27.00

0.88

2.04

86.21

K3–10

2.33

27.65

1.02

2.38

73.53

K4–10

2.25

28.19

1.02

2.46

69.44

K5–10

2.52

29.20

0.90

2.42

65.79

ST

3.11

30.67

0.79

2.92

46.95

K1–30

2.11

25.31

1.15

2.66

66.23

K2–30

2.59

23.88

0.68

1.60

108.70

K3–30

1.92

24.37

1.24

2.52

78.13

K4–30

2.17

34.12

0.57

1.69

89.29

K5–30

1.89

25.66

1.39

2.87

67.57

(a)

(b)

Fig. 3.3 Optimal cure time t 90 a and vulcanization rate CRI b of NR/kaolin composites

46

A. Feriancová et al.

Fig. 3.4 Maximum torque M H of NR/kaolin composites

2008; Raji et al. 2015). However, the maximum torque (M H ) which is related to stiffness, presented a small reduction in all samples, while decreasing with higher filler loading, apart from K4–30 sample. The composite with 10 phr of kaolin filler (Fig. 3.4) showed comprehensive performance, comparable to standard and high value of M H indicates the formation of good filler network (Zhang et al. 2016). The presence of the kaolin-urea in K4–30 sample resulted in increase of the M H value caused by the formation of a higher number of crosslinks, which was attributed to the closure of the rubber chains within the urea moieties and hence to high interaction between the filler and the rubber.

3.3.2.2

Mechanical Properties

The mechanical properties strongly depend on the morphology and nature of the using filler. The interaction of rubber-kaolin composites can be improved by applying an appropriate agent for clay modification because it could form a film on the mineral surface and reduce the surface energy. The results from mechanical testing of NR/kaolin composites are given in Table 3.7 and Figs. 3.5 and 3.6. The reinforcement effect of modified kaolin on rubber composites even at low filler content (10 phr), as evidenced by the comparable values to the standard in the tensile strength and elongation at break, may be related to the occurrence of the good interaction between modified kaolin and natural rubber. The rubber trapped between the kaolin particles forming the bonded rubber, possessed the rigidity similar to the standard. Therefore, increased filler content (30 phr) in the composites could

3 Utilization of Microwave Radiation for Chemical Modification … Table 3.7 Mechanical properties of NR/kaolin composites

47

Composite

TS b (MPa)

E b (%)

Shore A hardness (IRHD)

K1–10

24.36 ± 0.3

409.16 ± 4.4

62.5 ± 0.2

K2–10

24.43 ± 1.8

381.09 ± 30.3

61.5 ± 0.4

K3–10

26.00 ± 0.7

420.84 ± 10.8

62.2 ± 0.1

K4–10

24.53 ± 0.7

423.16 ± 8.2

61.6 ± 0.3

K5–10

24.07 ± 1.2

412.42 ± 15.2

63.2 ± 0.3

ST

26.21 ± 0.6

398.98 ± 8.2

64.9 ± 0.4

K1–30

23.21 ± 1.5

462.09 ± 38.5

57.4 ± 0.3

K2–30

25.40 ± 0.4

441.83 ± 3.3

56.7 ± 0.1

K3–30

22.29 ± 0.9

441.73 ± 11.5

57.2 ± 0.1

K4–30

10.61 ± 2.6

220.53 ± 46.3

63.7 ± 0.3

K5–30

20.54 ± 0.6

462.86 ± 6.7

57.8 ± 0.2

(a)

(b)

Fig. 3.5 Tensile strength TS b a and elongation at break E b b of NR/kaolin composites

Fig. 3.6 Hardness of NR/kaolin composites

48

A. Feriancová et al.

carry stress and good TS b value. It may be considered as a direct indication of the reinforcing effect of modified kaolin filler. Relating to kind of modification agent, addition of the urea (K4–30 sample) led to the significantly lower value of TS b and hardness at 30 phr loading (Fig. 3.6) reached comparable value in comparison with standard. It means, that this composite is strong but fragile at the same time. The ability of the rubber composites to recover after being mechanically loaded was evaluated through elongation at break (E b ). The results in Fig. 3.5b confirmed that the elasticity of the NR/kaolin composites was improved with the addition of 30 phr of kaolin samples (K1, K2, K3, K5) and it provides the direct evidence of the enhanced polymer-filler interactions (Sheikh et al. 2017) with the exception of K4, when the elasticity of the K4–30 composite significantly decreased. The hardness is the most obvious property to be influenced and could be increased with increased cross-link density (Leblanc 2000). Figure 3.6 shows the hardness of the NR composites by varying modification of kaolin. Results show that the hardness of NR/kaolin composites reached the standard value at loading 10 phr of kaolin. With a higher loading 30 dsk, the values of hardness are reduced by 5%. The organically modified kaolin particles could interact with rubber molecules by forming chemical bonds or physical adsorption, these interactions could restrict the motion of rubber chains (Zhang et al. 2016). It is possible to assume that there could be interactions between the organo-kaolin complex and rubber due to increased hydrophobicity of modified kaolin (Sreelekshmi et al. 2017). Such findings could be attributed to the fine dispersion of kaolin particles in the rubber matrix and good interactions between kaolin particles and rubber chains, which constrained the motion of the rubber chains.

3.3.2.3

Surface Properties of NR/Kaolin Composites

The analysis of surface free energy was performed for samples of rubber composites with modified kaolin K1–K5 samples as a filler in amount of 10.0/30.0 phr and for 1 standard. The surface free energy was determined from the individual values of the contact angles of DW and DIM. The lower values of the surface energy correspond to the more hydrophobic surface of the composite. Results are sumarised in Table 3.8 and Fig. 3.7. By comparing the composites filled with the modified kaolins (K1–K5), with two different fillings, with the standard (ST), we can state that results for sample K1–10 are closest to the required values of the standard.

3.4 Conclusions The influence of ammonium acetate, urea and silane modification in various combinations with different solvents on kaolin under microwave radiation was studied. Microwave radiation and the time of its action on kaolin has significant influence

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49

Table 3.8 Surface properties of NR/kaolin composites Composite

DIM (°)

DV (°)

SFE (mJ m–2 )

Polar SFE (mJ m–2 )

Dispersive SFE (mJ m–2 )

K1–10

54.46 ± 2.6

108.8 ± 1.9

31.8 ± 1.5

0.08 ± 0.09

31.8 ± 1.4

K2–10

56.45 ± 1.5

106.52 ± 1.6

30.6 ± 0.8

0.02 ± 0.02

30.6 ± 0.8

K3–10

55.22 ± 0.7

108.95 ± 1.6

31.4 ± 0.4

0.07 ± 0.08

31.3 ± 0.4

K4–10

55.51 ± 3.3

107.69 ± 1.8

31.2 ± 1.8

0.04 ± 0.04

31.1 ± 1.8

K5–10

56.00 ± 2.4

104.83 ± 3.2

31.0 ± 1.3

0.10 ± 0.12

30.9 ± 1.3

ST

54.33 ± 1.1

103.73 ± 1.9

31.9 ± 0.6

0.06 ± 0.06

31.8 ± 0.6

K1–30

56.45 ± 1.5

106.52 ± 1.6

30.6 ± 0.8

0.02 ± 0.02

30.6 ± 0.8

K2–30

56.45 ± 1.5

106.52 ± 1.6

30.6 ± 0.8

0.02 ± 0.02

30.6 ± 0.8

K3–30

56.45 ± 1.5

106.52 ± 1.6

30.6 ± 0.8

0.02 ± 0.02

30.6 ± 0.8

K4–30

56.45 ± 1.5

106.52 ± 1.6

30.6 ± 0.8

0.02 ± 0.02

30.6 ± 0.8

K5–30

56.45 ± 1.5

106.52 ± 1.6

30.6 ± 0.8

0.02 ± 0.02

30.6 ± 0.8

Fig. 3.7 Surface properties of NR/kaolin composites

on shortening of the time needed for modification of kaolin from days to minutes. The results of FT-IR, EDX and thermal analyses showed that the observed properties of kaolin have been already affected even after 120 min of microwave assisted modification. Modified kaolin samples were used as a filler replacement for carbon black in the amount of 10.0 and 30 phr. Processability characteristics as well as physical and mechanical properties of modified NR/kaolin composites were studied. Partial substitution of carbon black with modified kaolin led to the lower vulcanization rate and to the acceleration of the curing process, improving the processing characteristics of the NR/kaolin composites. Moreover, there is the enhancement in the productive efficiency. The consumption energy is also decreased significantly. In

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addition, the NR/kaolin composites containing modified kaolin showed comparable values to standard of tensile strength and hardness along with high maximum torque, which are indicators of the reinforcing effect of the modified kaolin. The composites containing modified kaolin could be advantageous for its application in tire tread compounds. In practice, the kaolin is commonly used in rubber compounds for the bead area of car tires. Acknowledgements This research work has been supported by the Operational Program Integrated Infrastructure, co-financed by the European Regional Development Fund by the project: Advancement and support of R&D for “Centre for diagnostics and quality testing of materials” in the domains of the RIS3 SK specialization, Acronym: CEDITEK II., ITMS2014+ code 313011W442.

References Andrews EH (1963) Reinforcing of rubber by fillers. Rubber Chem Technol 36:325–336 Avila RA, Faria EH, Ciuffi KJ, Nassar EJ (2010) New synthesis strategies for effective functionalization of kaolin and saponite with silylating agents. J Colloid Interf Sci 341:186–193 Bakain RZ, Al-Degs YS, Issa AA, Jawad SA, Abu Safieh KA, Al-Ghouti MA (2014) Activation of kaolin with minimum solvent consumption by microwave heating. Clay Miner 49:667–681 Dai JC, Huang JT (1999) Surface modification of clays and clay–rubber composite. Appl Clay Sci 15:51–65 Djomgoue P, Njopwouo D (2013) FT-IR spectroscopy applied for surface clays characterization. J Surf Eng Mater Adv Technol 3:275–282 Domka L, Foltynowicz Z, Jurga S, Kozak M (2003) Influence of silane modification of kaolins on physicomechanical and structural properties of filled PVC composites. Polym Polym Compos 11(5):397–406 Feriancová A, Pajtášová M, Pecušová B, Ondrušová D (2019) The effect of modified Cu(II) kaolinite on interactions with rubberizing components. Appl Clay Sci. https://doi.org/10.1016/j.clay.2019. 105313 Feriancová A, Dubec A, Pagáˇcová J, Papuˇcová I, Moricová K, Žitˇnan M (2021) Preparation and application of modified organo-kaolinite by microwave-sssisted irradiation. Appl Clay Sci. https:// doi.org/10.1016/j.clay.2021.106259 Gasparini E, Tarantino SC, Ghigna P, Riccardi MP, Cedillo-González EI, Siligardi C, Zema M (2013) Thermal dehydroxylation of kaolinite under isothermal conditions. Appl Clay Sci 80–81:417–425 Gopi JA, Patel SK, Chandra AK, Tripathy DK (2011) SBR-clay-carbon black hybrid nanocomposites for tire tread application. J Polym 18:1625–1634 Horváth E, Frost RL, Makó É, Kristóf J, Cseh T (2003) Thermal treatment of mechanochemically activated kaolinite. Thermochim Acta 404(1–2):227–234 Kim MS, Kim GH, Chowdhury SR (2007) Polybutadiene rubber/ organoclaynanocomposites: effect of organoclay with various modifier concentrations on the vulcanization behavior and mechanical properties. Polym Eng Sci 47:308–313 Kloprogge JT (2019) Spectroscopic methods in the study of kaolin minerals and their modifications. Springer Nature Switzerland, Cham Kumar S, Panda AK, Singh RK (2013) Preparation and characterization of acids and alkali treated kaolin clay. Bull Chem React Eng Catal 8(1):61–69 Leblanc JL (2000) Elastomer–filler interaction and the rheology of filled rubber compounds. J Appl Sci 78:1541–1550 Li ZJ, Zhang XR, Xu Z (2007) Novel method for preparation of kaolinite intercalation composite. Matl Tech Adv Perf Matl 22(4):205–208

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Liu Q, Zhang Y, Xu H (2008) Properties of vulcanized rubber nanocomposites filled with nanokaolin and precipitated silica. Appl Clay Sci 42:232–237 Madejová J, Komadel P (2001) Baseline studies of the clay minerals society source clays: infrared methods. Clays Clay Miner 49:410–432 Makó É, Kovács A, Horváth E, Kristóf J (2014) Kaolinite-potassium acetate and halloysitepotassium acetate complexes prepared by mechanochemical, solution and homogenization techniques: a comparative study. Clay Miner 49:457–471 Mbey JA, Thomas F, Ngally Sabouang CJ, Liboum, Njopwouo D (2013) An insight on the weakening of the interlayer bonds in a Cameroonian kaolinite through DMSO intercalation. Appl Clay Sci 83–84:327–335 Mo S, Pan T, Wu F, Zeng M, Huang D, Zhang L, Jia L, Chen Y, Cheng Z (2019) Facile one-step microwave-assisted modification of kaolinite and performance evaluation of pickering emulsion stabilization for oil recovery application. J Environ Manage 238:257–262 Ogbebor OJ, Okieimen FE, Ogbeifun DE, Okwu UN (2015) Organomodified kaolin as filler for natural rubber. Chem Ind Chem Eng Q 21(4):477–484 Olaremu AG (2015) Physico-chemical characterization of Akoko mined kaolin clay. J Miner Mater Char Eng 3(5):353–361 Panda K, Mishra BG, Mishra DK, Singh RK (2010) Effect of sulfuric acid treatment on the physicochemical characteristics of kaolin clay. Colloids Surf A Physicochem Eng 363(1–3):98–104 Paul DR, Robeson LM (2008) Polymer nanotechnology: nanocomposites. Polymer 49:3187–3204 Pi Z, Liu Z, Yang C, Tian X, Fei J, Zheng J (2007a) Exfoliation of kaolinite by urea-intercalation precursor and microwave irradiation assistance process. Front Earth Sci China 1(1):26–29 Pi Z et al (2007b) Exfoliation of kaolinite by urea-intercalation precursor and microwave irradiation assistance process. Front Earth Sci China 1:26–29 Raji VV, Surya R, Rugmini S, Brahmakumar M, Menon ARR (2015) Kaolin modified with sodium salt of rubber seed oil as a reinforcing filler for blends of natural rubber polybutadiene rubber and acrylonitrile butadiene rubber. Polym Int 64(11):1585–1593 Raji VV, Anitha AM, Menon ARR (2016) Studies on blends of natural rubber and butadiene rubber containing silica—organomodified kaolin hybrid filler systems. Polymer 89:135–142 Reinosa JJ, Baños BG, Catalá-Civera JM, Fernández JF (2019) A step ahead on efficient microwave heating for kaolinite. Appl Clay Sci 168:237–243 Roshin P, Sreelekshmi RV, Menon ARR (2018) Cetyltrimethyl ammonium bromide modified kaolin as a reinforcing filler for natural rubber. J Polym Environ 26:39–47 Rudawska A, Jakubowska P, Klozi´nski A (2017) Surface free energy of composite materials with high calcium carbonate filler content. Polimery 62(6):434–440 Shah KJ, Mishra MK, Shukla AD, Imae T, Shah DO (2013) Controlling wettability and hydrophobicity of organoclays modified with quaternary ammonium surfactants. J Colloid Interf Sci 407:493–499 Sheikh SH, Yin X, Ansarifar A, Yendall K (2017) The potential of kaolin as a reinforcing filler for rubber composites with new sulfur cure systems. J Reinf Plast Comp 36:1132–1145 Sreelekshmi RV, Sudha JD, Menon ARR (2017) Novel organomodified kaolin/silica hybrid fillers in natural rubber and its blend with polybutadiene rubber. Polym 74:783–801 Sukumar R, Menon ARR (2008) Organomodified kaolin as a reinforcing filler for natural rubber. J Appl Polym Sci 107:3476–3483 Trabelsi W, Tlili A (2017) Phosphoric acid purification through different raw and activated clay materials (Southern Tunisia). J Afr Earth Sci 129:647–658 ˇ Valášková M, Rieder M, Matˇejka V, Capková P, Slíva A (2007a) Exfoliation/delamination of kaolinite by low-temperature washing of kaolinite-urea intercalates. Appl Clay Sci 35(1–2):108–118 Valášková M, Martynková GM, Matìjka M (2007b) Chemically activated kaolinites after deintercalation of formamide. Ceramics 51(1):24–29 Yahaya LE, Adebowale KO, Menon ARR (2009) Mechanical properties of organomodified kaolin/natural rubber vulcanizates. Appl Clay Sci 46:283–288

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Yang SQ, Yuan P, He HP, Qin ZH, Zhou Q, Zhu JX, Liu D (2012) Effect of reaction temperature on grafting of γ-aminopropyl triethoxysilane (APTES) onto kaolinite. Appl Clay Sci 62–63:8–14 Yang N, Zhang ZC, Ma N, Liu HL, Zhan XQ, Li B, Shi D (2017) Effect of surface modified kaolin on properties of polypropylene grafted maleic anhydride. Results Phys 7:969–974 Yang Y, Zhang H, Zhang K, Liu L, Jib L, Liua Q (2020) Vulcanization, interfacial interaction, and dynamic mechanical properties of in-situ organic amino modified kaolinite/SBR nanocomposites based on latex compounding method. Appl Clay Sci. https://doi.org/10.1016/j.clay.2019.105366 ˙ Zenkiewicz M (2007) Methods for the calculation of surface free energy of solids. J Achiev Mater Manuf Eng 24(1):137–145 Zhang XR, Xu Z (2007) The effect of microwave on preparation of kaolinite/dimethylsulfoxide composite during intercalation process. Mater Lett 61:1478–1482 Zhang Y, Liu Q, Zhang Q, Lu Y (2010) Gas barrier properties of natural rubber/kaolin composites prepared by melt blending. Appl Clay Sci 50:255–259 Zhang B, Pan L, Zhang HY, Liu ST, Ye Y, Xia MS, Chen XG (2012) Effects of acid treatment on the physico-chemical and pore characteristics of halloysite. Colloids Surf A Physicochem Eng 396:182–188 Zhang Y, Liu Q, Zhang S, Zhang Y, Zhang Y, Liang P (2016) Characterization of kaolinite/styrene butadiene rubber composite: Mechanical properties and thermal stability. Appl Clay Sci 124– 125:167–174

Chapter 4

Simulations of Tests of Polymeric Composites Based on Experimental Data Jan Krmela, Vladimíra Krmelová, Artem Artyukhov, Cornelia Lex, and Darina Ondrušová

4.1 Introduction It is necessary to have knowledge about geometry, material parameters, cross-section, and structure of a tire casing (number of layers of a belt and carcass, information about a bead and cap ply) and other parameters for creations of computational models of a tire casings as polymeric composites for the strain–stress analysis of a tire under the vertical load, modal analysis etc. Tire casings have different constructions depending on the type of transport means. The construction of tires is different for passenger cars, trucks, off-highway cars and sport cars. A standard automobile radial tire casing consists of elastomer parts and parts with textile-cords and steel-cords in a tire tread as reinforcements (Krmela 2017). The structure parts applied into passenger car radial tire casings are textile carcass plies, a textile cap ply (called an overlap belt) and steel-cord belts. These structures of a tire have a different cord material, cord J. Krmela (B) · V. Krmelová · D. Ondrušová Faculty of Industrial Technologies in Púchov, Alexander Dubˇcek University of Trenˇcín, Ivana Krasku 491/30, 020 01 Púchov, Slovak Republic e-mail: [email protected] V. Krmelová e-mail: [email protected] D. Ondrušová e-mail: [email protected] A. Artyukhov Sumy State University, Academic and Research Institute of Business, Economics and Management, Sumy State University, 2, Rymskogo-Korsakova St, Sumy 40007, Ukraine e-mail: [email protected] C. Lex Mechanical Engineering and Economic Sciences, Technical University of Graz, Inffeldgasse 11/II, 8010 Graz, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_4

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angle (e.g., for a steel belt there is applied an angle of 21–33° in a radial tire casing for a passenger car), the construction of cord and number of layers (single-layer or multi-layer). Therefore, tires have specific deformation properties. Data about crosssections, construction-reinforcing plies, etc. are a necessary input for the creation of computational models of tires. The image analysis is applied for obtaining the information about geometric parameters of cords such as distances between cords and EPM (EPM is density—a number of ends (filaments) per one meter of width of layer), ply thickness, cord diameters, etc. The steel-cord of typical tire casings has the construction of 2 + 2 × 0.28 mm (the cord consists of four filaments) or 2 × 0.30 mm (two filaments). The steel-cord belt consists of two symmetrical layers. The geometric parameters, which are obtained by an image analysis of the cross-section, will be used as necessary geometric input data in order to create a computational model with a real configuration of cords. For example, the simulation of tensile testing of composite is described in (Nirbhay et al. 2014), in which the authors used the FEM software Abaqus. In the case of cyclic loading of composites and computational simulations of these tests (Muc 2000; Krmela 2021), which is a more specific area, many simplifying assumptions are used with can lead to inaccurate results. In this article, the authors used FEA (Finite Element Method) software ANSYS Mechanical for the computational simulation of composites. The APDL (Ansys Parametric Design Language) in ANSYS is used for created of 3D computational models. For the description of rubber matrix of composites, several material constitutive models have been considered in existing works to capture the hyperelasticity of elastomers for determination of material parameters, based on the tensile test or Shore A hardness (Krmela et al. 2021) for elastomer specimens for example. Among these the Mooney-Rivlin model (Krmela et al. 2021) is the most used for computational modelling of tires. It shows satisfactory results for the range of elongation (strain) up to 150%. There exists a large family of Mooney-Rivlin material models developed for a pool of hyperelastic materials. In the scope of this article and for the description of the behaviour of elastomers in tire casings, the Mooney-Rivlin material model used is the one with two parameters, which is sufficient to cover the range of strain abovementioned.

4.2 Materials and Methods and Computational Models 4.2.1 Materials Used The PA 66 and PES textile fibres are used for passenger tires especially for common purposes. The sidewall also shows the material of cords and number of plies in the sidewall and under the tread of tire casing. The tire with symbol Extra load may have two polyester or two polyamide plies. The textile carcass EPM is from 700 to 1150 m−1 . The EPM of textile cap ply is 1100–1200 m−1 . For computational

4 Simulations of Tests of Polymeric Composites Based on Experimental Data

55

modelling the elastic modulus of elasticity and Poisson ratio are used as material input parameters of textile reinforcements. From producer of textile cord the LASE modulus is used. LASE is modulus of elasticity for elongation 5% (LASE is acronym from Load At Specific Elongation 5%). Other way is determination of modulus as stress necessary on elongation 100% obtained by extrapolation for elongation 2%. In this article, the PA 66 is used as textile cords. Statical tensile modulus of elasticity of PA 66 (Shiguo 2004) is from 900 to 3450 MPa or from 9 to 50 cN/dtex. Typically values of moduli of elasticity are in Table 4.1. Usually, the experimental conditions are temperature 20 ± 2 °C, humidity 65 ± 5% and initial length between clamps of test machine (gauge length) 500 mm based on standard (ISO 2062:2009). For the simulations, the modulus values of 3400 MPa and the Poisson ratio of 0.4 are entered. The two cord diameters are used for simulation, 0.4 mm for shear test simulation and 0.5 mm for tensile test simulation. The first variant of computational model with 0.4 mm cord diameter is intended for verification of the reinforcement modelling method based on shear test simulation. The second variant of computational model with 0.5 mm cord diameter is given by real data from tensile experiments. The composite consists of a rubber matrix (elastomer drift for a textile cap is taken to produce the composite samples as matrix) with modulus of elasticity of 3.96 MPa. For the description of matrix is used the hyperelastic Mooney-Rivlin model, which is described by the two Mooney-Rivlin parameters. To determine the basic MooneyRivlin parameters, it is necessary to carry out the tensile test for elastomer specimens. The values of the Mooney-Rivlin parameters of elastomer parts and elastomer matrix (drift) are depicted in Table 4.2 for the Matador 165/65 R13 tire casing as a sample of the results. The Mooney-Rivlin (MR) parameters obtained by the tests are 0.548 MPa (as C10 in ANSYS) and 0.112 MPa (as C01 in ANSYS) and incompressibility parameter d is 0.056 MPa−1 . The geometrical parameters of specimens for tests are the specimen’s length of 140 mm, width of 35 mm, initial length between the clamps of test machine is 100 mm, specimen’s thickness is 1.1 mm and cord angle is 45°. The EPM is 870 m−1 . The Table 4.1 Moduli of elasticity of textile cords Mooney-Rivlin parameters

C10 [MPa]

C01 [MPa]

d [MPa−1 ]

Tread

0.417

0.519

0.103

Inner liner

0.109

0.259

0.206

Bead elastomer

0.692

0.371

0.267

Sidewall with a tread side edge

0.532

0.065

0.138

−0.111

1.945

0.088

Elastomer drift for a steel-cord belt

0.638

0.284

0.151

Elastomer drift for a textile cap

0.548

0.112

0.056

Elastomer drift for a textile carcass

0.328

0.119

0.101

Bead bundle

56 Table 4.2 Mooney-Rivlin parameters for elastomer parts of the Matador 165/65 R13

J. Krmela et al. Material

Labeling of construction [tex]

Modulus of elasticity [GPa]

Polyester (PES)

PES 110 × 1 × 2 4

Polyamide 66 (PA 66)

PA 66 94 × 1 × 2 3.4

Rayon (viscose)

VS 184 × 1 × 3

11

Aramid

Aramid 167 × 1 ×2

25

initial length between the two points for video-extensometer is 50 mm (Fig. 4.1). Other cord angles such as 0° and 60° were also used in tests. This article focuses on the 45° cord angle.

Fig. 4.1 Specimens with geometric parameters (in mm)

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57

4.2.2 Methods Used The test machine Autograph AG–X plus 5 kN—Shimadzu with video-extensometer for large strain as tensile tests of composite materials with elastomer and viscoelastic materials with test mode Control of the software Trapezium X is used for tests. Before the tests, calibration of the video-extensometer is required. For computational modelling, the program ANSYS Mechanical is used. It is possible to rely on publications which deal with computational modelling of composites in tires Krmela and Krmelová, shear computational simulation (Kießling et al. 2016; Basri et al. 2021), and standard ASTM D5379 ASTM (2019) focused on shear composites testing for the application of test specimen geometry for polymers and Ansys tutorials for composites (Kanani 2021).

4.2.3 Computational Models for Shear Test Simulation The purpose of the models is to verify which method of modelling the reinforcement is ideal and to compare the results with each other. The models are also used to verify the calculation settings for large deformations in terms of the convergence of the calculation. The volume SOLID186 element type with the Mixed U/P (meant for a mixed variational formulation with two fields: the displacement U and hydraulic pressure P. Recalling that the second field is introduced in the formulation to enforce the incompressibility condition to the potential energy of the variational problem) setting is used for computational models with MR parameters. One layer had a length and a width of 20 mm. EPM is 420 m−1 based on real geometric parameters for Matador tire casing, Table 4.3. Therefore, the distance between each cord is 2.38 mm. The APDL procedure includes parameterization with the following parameters: *cset,1,3,Distance,’Distance between cord [mm] ’,2.38 (based on EPM value) *cset,4,6,Diameter,’Cord diameter [mm]’,0.5 *cset,7,9,Thickness,’Thickness of layer [mm]’,1.1 *cset,10,12,Width,’Width of layer [mm]’,20 *cset,13,15,Lengh,’Lengh of layer [mm]’,20 *cset,16,18,Angle,’Cord angle [degree]’,0 *cset,19,21,E,’Modulus of elasticity of cord [GPa] ’,3.4 *cset,22,24,PR,’Poisson ratio [-]’,0.4.

The APDL procedure includes the computation of rubber modulus based on MR parameters which can be entered directly or are determined based on data from a tensile test: D = (2*(1-2*PR_E))/(CONST1(1)*(5*PR_E-2)+CONST1(2)*(11*PR_E-5)) !parameter of incompressibility TB,HYPE,2,1,2,MOON TBDATA„CONST1(1),CONST1(2),D„, !parameters are in MPa;

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Table 4.3 Geometric and material parameters of composite parts of Matador 165/65 R13

Composite part

Steel-cord belt

Textile cap

Textile carcass

Material of cord

Steel HT

PA66

PES

Number of layers

2

1

1

0.8

0.95

0.4

0.48

Thickness of 0.95 one layer [mm] Diameter of cord [mm]

0.6

EPM [m−1 ]

961

420

1 160

Spacing between cords [mm]

1.04

2.38

0.86

Modulus of elasticity [GPa]

190

3.4

4

Poisson’s ratio 0.3 [–]

0.4

0.4

E_E = 6*(CONST1(1)+CONST1(2)) !modulus of elasticity G_E = 2*(CONST1(1)+CONST1(2)) !shear modulus K_E = 2/D !volume modulus.

The models consist of two plus two composite layers between three steel sheets. The models are reverse loaded, the displacement in the z-axis is defined and the summary reaction forces at the area of steel edges (using these edges, the specimen will be clamped in the jaws of the testing machine) are searched. The initial condition is 5 mm displacement of the middle steel sheet (Fig. 4.2). Solution control settings: calculated prestress is switched on, nonlinear geometric effects are in the on state, time at the end of load step is 5 as defined displacement, number of substeps is 20 (it means that the increment of every substep is 0.25 mm in the z-axis). Material parameters are: The first model included hyperelastic MR model for matrix with two MR parameters for rubber matrix and linear isotropic material for PA 66 cords with modulus values of 3400 MPa and the Poisson ratio of 0.4. The second model is based on the homogenization of the entire cord and rubber composite system expressed by the parameters of the linear orthotropic material: moduli of elasticity Ex = 228 MPa, Ey = Ez = 4.24 MPa and Poisson main ratios PRxy = 0.486, PRyz = 0.581, PRxz = 0.49. Shear moduli Gxy = Gyz = Gxz = 1.506 MPa. These parameters are obtained based on freeware CADEC (Computer Aided Design Environment for Composites) (Barbero 2022) based on material parameters of cord and matrix. The third model is a model with the concrete element SOLID65 with real constant 0.07 volume ratio of PA 66 cords. The linear isotropic material is used for PA 66

4 Simulations of Tests of Polymeric Composites Based on Experimental Data

59

Fig. 4.2 Computational models for shear test simulation

cords with modulus values of 3400 MPa and the Poisson’s ratio of 0.4 and for rubber matrix with modulus values of 3.96 MPa and the Poisson’s ratio of 0.495. The fourth model is a model with the element BEAM189 for PA 66 cord with beam section 0.2 mm of radius. The linear isotropic material is used for PA 66 cords and matrix such as 3rd model. The computational models are presented in Fig. 4.2.

4.2.4 Computational Model for Tensile Test Simulation The diameter of cords is 0.5 mm, EPM is 870 m−1 and thickness is 1.1 mm. The model consists of half of one layer with symmetry boundary conditions. The SOLID186 element type is used. The computational model consists of 22,836 elements and 106,053 nodes. The model has length 140 mm and the initial length between the

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jaws of test machine 100 mm is modelling as remove of degrees of freedom in all axes (application of displacements on the selected nodes in areas of jaws). The model is reverse loaded, the displacement in x-axis is defined and reaction force at area of sliding clamp of test machine is searched. It is better for quickly convergence and speed solution. The initial condition is 30 mm displacement (this corresponds to an elongation (strain) of 30%). The solver PCG is used. The increment of every substep is 0.2 mm. A preload with a force of 2 N was considered, as in the experiment. A force of 2 N caused a deformation of 0.8 mm, similar to the case of the experiment. The model included the hyperelastic MR model for the matrix with two MR parameters for rubber matrix (C10 = 0.548 MPa, C01 = 0.112 MPa and d is 0.056 MPa−1 as was the case the first model for shear test simulation) and linear isotropic material for PA 66 cords with modulus values of 3400 MPa and the Poisson’s ratio of 0.4. The elongation measurement points that represent the points for the video-extensometer are shown in the Fig. 4.3 with detail of the cross-section.

Fig. 4.3 Computational models for tensile test simulation with measurement points and detail of cross-section

4 Simulations of Tests of Polymeric Composites Based on Experimental Data

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4.3 Results and Discussion 4.3.1 Shear Test Simulations The results from the computational modelling of strain–stress state of all variants are represented as the summary of displacement in Fig. 4.4. Reaction force–displacement dependences are shown in Fig. 4.5. The value of force in z-axis for deformation of 5 mm is 3243 N for the first model. The second model gives reaction forces in z-axis of 3452 N. The value of the third model is 2887 N. The resulting reaction force in z-axis for the fourth model is 3017 N. The rebar element (the fourth model) makes the model more flexible about 7% compared to the first model. If we were to compare the results for a half deformation of 2.5 mm, we would reach similar results in percentage.

Fig. 4.4 Displacement from computational simulation of shear test

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Fig. 4.5 Reaction force–displacement dependences from computational simulation of shear test

Legend 1st model 2st model 3st model 4st model

3200 2800

Force [N]

2400 2000 1600 1200 800 400 0

0

1

2

3

Displacement [mm]

4

5

4.3.2 Tensile Test Simulation The results from computational modelling as length displacement and width displacement for elongation 30% are in Figs. 4.6 and 4.7. The result as stress sigma1 (1st principal stress) in the PA 66 cords for elongation 30% is in Fig. 4.8. The value of reaction forces is 63.3 N. From experiment data, the tensile force 64.7 N causes an elongation of 30%. The computational model has good tensile stiffness because the forces are very similar. The force value differences are to 2.2%. A comparison of calculation results with experimental data in terms of dependences of force on elongation, stress on elongation (between jaws) and real stress on elongation between measuring points for the video-extensometer are shown in Figs. 4.9 and 4.10.

4.4 Conclusions The different computational models with polymer cords were created. The simulations were created for different load states. The APDL procedure with parameterization of geometrical and material parameters such as cord distance and angle, thickness of the layer and number of layers for the creation of computational models was designed and programmed. The proposed APDL procedure is ready to create multi-layer models. The APDL procedure allows the user the choice of whether to use volume or beam elements to model cords and purpose of model—tensile or bend test simulations.

4 Simulations of Tests of Polymeric Composites Based on Experimental Data

Fig. 4.6 Deformations and displacements from computational simulation of tensile test

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Fig. 4.7 Deformations and displacements of cords from computational simulation of tensile test

Fig. 4.8 Stress in the cords from computational simulation of tensile test

4 Simulations of Tests of Polymeric Composites Based on Experimental Data 70.0

Fig. 4.9 Dependences of force on elongation

65

Force - Elongation cord angle 45° test data

60.0

simulation

Force [N]

50.0

40.0

30.0

20.0

10.0

0

10

20

30

Elongation [%]

The results from tests and computational simulations of composites which represented parts of tire casings provide a better understanding of the mechanical properties of composites with textile reinforcements under static and specific loading. Based on results, the best way is of reinforcement modelling is the first variant in which MR parameters for rubber matrix. The calculation time was a little longer than other models. The second way requires a good determination of material parameters. A slight change in the parameters of the moduli in tension and in shear can cause inaccurate results—there is a high sensitivity to the input material parameters. The model for tensile test simulation gave quality results—force–elongation dependences almost overlap with an error of 5%. The model was prepared quickly by the APDL procedure, and the calculation setup guaranteed fast convergence. Next, the simulations of multiaxial loading and the inclusion of the effect of temperature will be realized because these simulations are important for practice, as these calculations simulate to some extent the real states of loading and composites for engineering applications such as their use in vehicles not only in automobiles.

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Fig. 4.10 Dependencies of stress on elongation (between jaws) and real stress on elongation between measuring points for the video-extensometer elongation

Stress - Elongation cord angle 45° test data

simulation

Stress [MPa]

2.0

1.5

1.0

0.5

0 2.5

10

Elongation [%]

20

30

True stress - Elongation on video-extensometer cord angle 45° test data video-extensometer simulation

True stress [MPa]

2.0

1.5

1.0

0.5

0

5

10

15

20

25

30

Elongation ex. [%]

Acknowledgements This research work has been supported by the Operational Programme Integrated Infrastructure, co-financed by the European Regional Development Fund by the project: Advancement and support of R&D for "Centre for diagnostics and quality testing of materials" in the domains of the RIS3 SK specialization, Acronym: CEDITEK II., ITMS2014+ code 313011W442 and the Aktion Austria–Slovakia, project No. 2019-05-15-001 and the Cultural and Educational Grant Agency of the Slovak Republic (KEGA), project No. 003TnUAD-4/2022 “Simulations of basic and specific experiments of polymers and composites based on experimental data in order to create a virtual computational-experimental laboratory for mechanical testing”.

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References ASTM D5379/D5379M–19: Standard test method for shear properties of composite materials by the V-notched beam method Barbero EJ (2022) CADEC. https://barbero.cadec-online.com/. Accessed 1 July 2022 Basri EI, Sultan MTH, Basri AA, Mustapha F, Ahmad KA (2021) Consideration of lamination structural analysis in a multi-layered composite and failure analysis on wing design application. Materials 14:3705 ISO 2062:2009 Textiles—Yarns from packages—determination of single-end breaking force and elongation at break using constant rate of extension (CRE) tester Kanani AY (2021) Ansys tutorial for ACP (Full composite tutorial in ANSYS). https://www.res earchgate.net/publication/332766462_Ansys_Tutorial_for_ACP_Full_composite_tutorial_in_ ANSYS. Accessed 1 July 2022 Kießling R et al (2016) On the development of an intrinsic hybrid composite. In: IOP Conference series: materials science and engineering, vol 118. pp 012017 Krmela J (2017) Tire casings and their material characteristics for computational modeling. Oficyna wydawnicza slowarzyszenia menadzerów jakošci i produkcji, Czestochowa Krmela J (2021) The influence of temperature and other parameters on the tensile properties of polymer composites and polymers under cyclic loading. Oficyna wydawnicza slowarzyszenia menadzerów jakošci i produkcji, Czestochowa Krmela J, Artyukhov A, Krmelová V, Pozovnyi O (2021) Determination of material parameters of rubber and composites for computational modeling based on experiment data. J Phys: Conf Ser 1741:012047 Krmela J, Krmelová V, Artyukhov A, Sadjiep S, Bakošová A (2021) Computational simulation of the shear test of a multi-layered long-fibre composite with a polymer matrix. In: IOP conference series: materials science engineering, vol 1199. pp 012075 Krmela J, Krmelová V (2016) Replacement of belt structure for FEA of tire. Proc Eng 136:132–136 Muc A (2000) Design of composite structures under cyclic loads. Comput Struct 76:211–218 Nirbhay M, Dixit A, Misra RK, Singh Mali H (2014) Tensile test simulation of CFRP test specimen using finite elements. Proc Mater Sci 5:267–273 Shiguo R et al (2004) Mechanical properties and failure behaviour of cord/rubber composites. Appl Compos Mater 11:353–357

Chapter 5

Study of Influence of Calcium Carbonate Sedimentation on Electric Heater Efficiency Tatiana A. Kudryashova, Sergey V. Polyakov, and Nikita I. Tarasov

5.1 Introduction The issue of energy saving in the modern world is very acute. The use of energy saving technologies is a necessity. Energy saving technologies help not only reduce energy costs, but also reduce the negative impact that humans have on the environment. Energy consumption has increased greatly since the nineteenth century. Firstly, this is due to the growth of the world’s population from 2 billion at the beginning of the nineteenth century to almost 8 billion now (see Fig. 5.1). The analysis shows that the most significant factors influencing electricity consumption are: mode of operation of enterprises; household way of life of the population; duration of the working week and days off; climatic conditions, etc. Energy saving is one of the priority tasks due to the deficit of basic energy resources, the increasing cost of their production, as well as global environmental problems. It is predicted that by 2040 energy consumption in Russia will grow by 20%, energy consumption growth in India is estimated at 165%, in Brazil—60%, in China—40% (Bezrukih 2004; Yurina et al. 2021; Barzykina 2014). Experts from all countries work on changing the structure of energy consumption, introducing solutions to reduce the amount of consumed fuel, improve the energy efficiency of cars and household appliances. Energy saving in any area is the reduction useless energy losses. The proposed study is limited to influence of calcium carbonate precipitation on electric heater efficiency (Elistratova et al. 2020; Davidzon 2007; Prisyazhnyuk 2003; Telin 2015). The experiments show that at 2 mm thick deposit, depending on the chemical composition of water scale, the thermal flow is reduced from 10 to 40%. T. A. Kudryashova (B) · S. V. Polyakov · N. I. Tarasov Keldysh Institute of Applied Mathematics, RAS, Miusskaya sq. 4, Moscow, Russia 125047 e-mail: [email protected] S. V. Polyakov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_5

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Fig. 5.1 Earth’s population and power consumption

As a result, the consumption of electricity by such element increases (Dobersek and Goricanec 2007). Hard water with a high content of salts of calcium, magnesium, sulfates and chlorides causes pollution of the heating elements and their damage. Dissolved calcium and magnesium compounds are converted into solid particles when the temperature rises. These impurities are fixed on the surface of the heating elements and create porous layer that is characterized by poor thermal conductivity, and therefore prevents normal heat transfer. Thus, intensity of the formation of scale depends on water hardness and temperature (Sergˇej et al. 2021; Puchina et al. 2020; Linnikov 2014). In this paper, impact of calcium carbonate precipitation on electric heater efficiency is discussed. To study this problem, the formation of scale during the operation of the heating element and the removal of scale with hydrochloric acid during its regeneration is examined. For simulation of these processes, an original mathematical model was proposed. A numerical method was developed based on the use of unstructured grids and finite-volume approximations of differential equations on these grids. A parallel algorithm and a code for calculating selected physical processes under conditions close to real were designed. With the help of the developed computing technology, a number of computational experiments was carried out to verify work of a heating system with a heating element of complex threedimensional geometry. During the experiments, the dependence of the processes of formation and removal of scale from the heating element on the flow parameters was analyzed.

5.2 Mathematical Model Mathematical model includes the processes of formation and removal of calcium carbonate sedimentation on the electric heaters. The computational domain with the heating element is shown in Fig. 5.2. 3D geometry is applied for calculations. Such

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Fig. 5.2 Computational geometry, 3D (x, y, z)

domain is model of a tube and it can be part of heat exchangers, piping and plumbing systems. Mathematical modeling of this problem is divided into three stages: 1. First stage is setting the flow and the required temperature mode in the tank. The temperature mode at all three stages is regulated by an electromagnetic method. This approach was presented in our papers (Mosin and Ignatov 2015; Saksono et al. 2008; Kudryashova et al. 2021; Kamaletdinov 2009). 2. The second stage is the passage of hard water through the tank and the formation of sediment on the heating element; due to the decrease in solubility calcium bicarbonate with increasing temperature, calcium salts are actively formed from an aqueous solution as a result of heating hard water in accordance with the following chemical reaction (Zarga et al. 2013): Ca(HCO3 )2 = CaCO3 + CO2 + H2 O.

(5.1)

At the same time, the heat flow released into the medium by the heating element is reduced. 3. The last stage is cleaning the heating element by passing a solution of hydrochloric acid. As a result of the interaction of calcium carbonate with hydrochloric acid, the aqueous solutions carbon dioxide and calcium chloride are formed. They freely leave the contaminated reservoir of the system. The formula for a chemical reaction has the following view: CaCO3 + 2HCl = CaCl2 + CO2 + H2 O.

(5.2)

To simulate the flow of a viscous incompressible fluid in a reservoir, taking into account the processes of heat conduction at all stages of modeling, quasihydrodynamic (QHD) system of equations is used (Elizarova 2009; Sheretov 2004; Chetverushkin 2008; Tarasov et al. 2020). In dimensionless form, it has the following view with common notation: ∂ (ρu) + (ρu · ∇)u = (∇ · P), ∂t

(5.3)

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∂ (ρε) + div(u(ρε + p)) + div(q) = (P · ∇) · u + Q, ∂t

(5.4)

here ρ is medium density, u is velocity vector, P is of the Navier–Stokes pressure tensor with additive QHD corrections, p is pressure, ε is the internal energy, q is the heat flux vector with QHD corrections, Q is volumetric density of heat sources, t is time. These equations are supplemented by the conditions of the medium continuity: div(u) = div(w), w = τ [(u, ∇)u + ∇ p],

(5.5)

here u is velocity vector, and w is regularizing correction, τ—regularization parameter, (•,•) is scalar product. In the case of an incompressible medium, the momentum equation is transformed into the equation: ∂u 1 = div[∇ ⊗ u + (∇ ⊗ u)T ] + div[w ⊗ u + u ⊗ w] − div(u ⊗ u) − ∇ p, ∂t Re (5.6) here Re = U0 D0 ρ/μ0 is the Reynolds number, (· ⊗ ·) is direct product of vectors, U 0 is maximum speed at input, D0 is hydraulic diameter, ρ is density, μ0 is dynamic viscosity of the medium. The expression to calculate the pressures is used in the form: Δp =

1 div(u) − div[(u, ∇)u], τ

(5.7)

here Δ is the Laplace operator. The energy equation is converted into the heat conduction equation: ρcV

∂T + div(u(ρcV T + p)) + div(q) = (P · ∇) · u + Q. ∂t

(5.8)

The boundary conditions for hydrodynamic equations are formulated as follows. The Poiseuille flow is set at the inlet. ) ( ∂p (y − 0.5)2 + z 2 = Const; T = 0 (5.9) , 0, 0 ; u→ = 1 − r2 ∂n CCa(HCO3 )2 = 1; CCO2 = CCaCO3 = 0

(5.10)

soft boundary conditions are applied at the outlet. ∂T ∂Ck ∂ u→ = 0; p = 0.5; = 0; = 0. ∂n ∂n ∂n

(5.11)

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At the walls: → ∂ p = 0; ∂ T = 0; ∂Ck = 0. u→ = 0; ∂n ∂n ∂n

(5.12)

Sticking conditions are used on the walls of the tank and on the surface of the heating element. The conditions for heat exchange with the environment are set on the walls of the tank, the specified fixed temperature is set on heat-generating element. At the heating element: ( 0 ) → ∂ p = 0; T = T0 + αT CCaCO u→ = 0; − CCaCO3 3 ∂n −

(5.13)

∂CCa(HCO3 )2 ∂CCaCO3 ∂CCO2 = 0.5 = 0.5 ∂t ∂n (∂n 0 ) = α1 CCaCO C − C − T0 ). (T CaCO Ca(HCO ) 3 3 2 3

(5.14)

Initial conditions were taken from the article (Martos et al. 2012): P = 1 bar, pH = 8.31, T = 25 °C, CCa2+ = 5.26 × 10−4 mol/l, CHCO3− = 2 ∗ CCa2+ = 1.052 × 10−3 mol/l. Dimensionless values of parameters are used for the code.

5.3 Numerical Method The well-known finite-volume method is applied to discretize spatial derivatives on tetrahedral meshes and it is combined with explicit( time discretization. The scheme ) has the order of approximation and accuracy O h2 + Δt . For the calculations, two tetrahedral meshes were used (see Table 5.1). The meshes were constructed by GMSH: a three-dimensional finite element mesh generator. The example of computational meshes is shown in Fig. 5.3. The parallel code was developed for calculations (Basic meshing algorithms; Karamzin et al. 2016). For parallelization domain decomposition technique was realized. The mesh decomposition by computational domains was carried out using the Metis package. An example of the resulting partitions is shown in Fig. 5.4. For selected computational meshes, the acceleration and efficiency of the parallel algorithm were measured and shown in Fig. 5.5. The calculations were fulfilled by work cluster with the Intel(R) Xeon Phi(TM) CPU 7250 processors. Table 5.1 Meshes

S. No.

Type of element

Number of element

Max. length of the rib

1

Tetrahedron

116,762

0.1

2

Tetrahedron

222,626

0.08

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Fig. 5.3 The example of computational grids

Fig. 5.4 Splitting the computational grid into 4 subdomains (left) and 64 (on the right)

Fig. 5.5 Speedup (left) and efficiency (on the right, logarithmic scale) of the parallel implementation

5.4 Results In this paper, we consider the drop in heat release on the heating element due to scale formation, as well as its regeneration under several flow regimes: for Reynolds

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numbers of 100, 200, 300, 500, and 700. The Prandtl number is 10 in each of the formulations. The general character of the flow for Re = 100 and Re = 700 is shown in Fig. 5.6 by arrows. When modeling the flow of a polluted flow with the chemical reaction: Ca(HCO3 )2 = CaCO3 + CO2 + H2 O.

(5.15)

at time t = 50, the distribution of CaCO3 scale on the heating element is obtained, shown in Fig. 5.7. The corresponding drop in heat release is shown in Fig. 5.8. The temperature distribution in the central section is shown in Fig. 5.9. Figures 5.10 and 5.11 show the evolution of CaCO3 concentration and temperature for calculated flows with Reynolds numbers: 100, 200, 300, 500, and 700.

Fig. 5.6 Velocity modulus for Re = 100 (left) and Re = 700 (on the right)

Fig. 5.7 Distribution of CaCO3 on the heating element (left Re = 100, on the right Re = 700)

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Fig. 5.8 Distribution of temperature on the heating element (left Re = 100, on the right Re = 700)

Fig. 5.9 Distribution of temperature in section Z = 0 (left Re = 100, on the right Re = 700)

Table 5.2 presents the integral characteristics of the heating element during the formation of scale from the beginning of the process t = 0: the heating element is not dirty and the data at the end of the process t = 50: how polluted it is. The regeneration process was carried out for the same flows in the chemical reaction: 2HCl + CaCO3 = CaCl2 + CO2 + H2 O

(5.16)

The evolution of scale concentration and heat release are shown in Figs. 5.12 and 5.13. Table 5.3 shows the integral characteristics of the heating element during regeneration. Data are given at the beginning of the process t = 0: the heating element is contaminated and data at the end of the process t = 50: how clean it is.

5 Study of Influence of Calcium Carbonate Sedimentation on Electric …

Fig. 5.10 Evolution of the integral concentration of CaCO3 on the heating element

Fig. 5.11 Evolution of drop of heat release on the heating element Table 5.2 Integral characteristics of the heating element in case of contamination Re 100 200

Concentration of CaCO3

Temperature

t=0

t=0

0.0

t = 50

Drop in temperature (%) t = 50

4.3693

4.0302

1.9213

47.7

3.4181

4.0631

2.3917

58.9

300

3.1915

4.0598

2.5

61.6

500

3.4404

4.0688

2.372

58.3

700

3.53

4.07

2.33

57.2

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Fig. 5.12 Evolution of the integral concentration of CaCO3 on the heating element during regeneration

Fig. 5.13 Evolution of the recovery of heat release on the heating element during regeneration Table 5.3 Integral characteristics of the heating element during regeneration Re

Concentration of CaCO3

Temperature

t=0

t = 50

t=0

t = 50

Purification (%)

100

4.3693

0.07804

1.9213

4.0396

98.2

200

3.4181

0.15325

2.3917

3.9995

95.6

300

3.1915

0.15696

2.5

3.9965

95.1

500

3.4404

0.13174

2.372

4.0053

96.2

700

3.53

0.12493

2.33

4.0085

96.5

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Let’s consider the obtained results in more detail. When modeling the formation of scale from the Table. 5.2 and Figs. 5.10 and 5.11, it can be seen that with an increase of the Reynolds number (100–300), the amount of captured CaCO3 decreases. The main reason for this effect is the localization of the pollutant flow near the upper part of the heating element at Re = 200, 300. At Re = 100, the impurity concentration spreads more actively over the studied volume due to the presence of a diffusion additive. A further increase in the flow intensity (Re = 500, 700) leads to the formation of a vortex flow at the outlet (Fig. 5.6, right), as a result of which the contact surface of the heating element with the pollutant increases. A similar situation arises when modeling the regeneration process (Table 5.3, Fig. 5.12 and Fig. 5.13). Thus, the maximum percentage of scale removal at Re = 100 decreases with an increase in the Reynolds number, and then increases as a result of a change in the nature of the flow. Such model calculations already allow us to draw certain conclusions about the optimization of the regeneration process. For example, to improve the efficiency of the regeneration process, the laminar regime of flow will be optimal.

5.5 Conclusion The paper studies influence of calcium carbonate precipitation on a water heating element on electric heater efficiency and the regeneration of such an element by a chemical method. To analyze the problem, the approach based on mathematical modeling is offered. As a model for the physical processes of the problem, it is proposed to use the system of quasi-hydrodynamics equations. This system is completed by the equations of convection–diffusion-reaction. The numerical implementation of the model for three-dimensional geometry is based on the grid finte-volume method and parallel computations. The performed numerical experiments show a significant change in the scale localization on the heating element under different flow regimes. The quantitative estimates and localization of calcium carbonate sedimentation influence differences in heat release in the volume under study. Similar changes occur during heater regeneration. The calculations were performed on the hybrid supercomputer K60 installed in the Supercomputer Ceptre of Collective Usage of KIAM RAS.

References Barzykina G (2014) Energy security as a part of economic security in the national security system of the country. J Electricity 3:20–22 (in Russian) Basic meshing algorithms, Mesh 9.6.0 documentation, Information on https://docs.salome-pla tform.org/latest/gui/SMESH/basic_meshing_algos.html Bezrukih P (2004) The role of renewable energy in energy saving in the world and in Russia. J Electricity 4:3–5 (in Russian)

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Chetverushkin BN (2008) Kinetic schemes and quasi-gasdynamic system of equations. CIMNE, Barcelona Davidzon M (2007) Formation of scale inside the tubes of heat exchangers at a constant wall temperature. J Therm Eng 54(9):739–742 Dobersek D, Goricanec D (2007) Influence of water scale on thermal flow losses of domestic appliances. Int J Math Models Methods Appl Sci 2(1):55–61 Elistratova Y, Seminenko A, Minko V (2020) Relevance of contamination models for diagnostics of plate heat exchangers. J Bulletin of Belgorod State Technological University named after V. G. Shukhov. https://doi.org/10.34031/2071-7318-2020-5-10-33-40 Elizarova TG (2009) Quasi-gas dynamic equations. Springer, Berlin Kamaletdinov RS (2009) Review of existing methods of prevention and control of scale in submersible equipment. J Eng Pract 12:12–15 (in Russian) Yu K, Kudryashova T, Podryga V et al (2016) Two-scale computation of N2 –H2 Jet flow based on QGD and MMD on heterogeneous multi-core hardware. J Adv Eng Software. ISSN 0965-9978. https://doi.org/10.1016/j.advengsoft.2016.02.005 Kudryashova TA, Polyakov SV, Tarasov NI (2021) Mathematical modelling of electrophysical water treatment. Defect Diffus Forum Trans Tech Publ Ltd. https://doi.org/10.4028/www.scientific.net/ ddf.412.149 Linnikov OD (2014) Mechanism of precipitate formation during spontaneous crystallization from supersaturated aqueous solutions. J Russ Chem Rev 83(4):343–364 C. Martos JL, Peña R, Rodríguez G et al (2012) Effects in the solubility of CaCO3 : experimental study and model description. J Fluid Phase Equilib. https://doi.org/10.1016/j.fluid.2012.03.020 Mosin O, Ignatov I (2015) Magnetic water treatment for elimination scaling salts. J Med Physiol Biophy 11:86–100 Prisyazhnyuk V (2003) Physical and chemical bases for preventing salt crystallization on heat exchange surfaces. J Plumbing Heating Air Conditioning 10:26–30 Puchina GR, Ragulin VV, Telin AG et al (2020) Modern practice of salt deposition preventing and removing. J Pet Eng. https://doi.org/10.17122/ngdelo-2020-2-72-80 Saksono N, Gozan M, Bismo S et al (2008) Effect of magnetic field on calcium carbonate precipitation: Ionic and particle mechanisms. Korean J Chem Eng 25(5):1145–1150 Sergˇej YM, Seepma H, Sergio E et al (2021) Controlling CaCO3 particle size with Ca2+ }:{CO3 2– ratios in aqueous environments. J Crystal Growth Des. https://doi.org/10.1021/acs.cgd.0c01403 Sheretov YV (2004) Mathematical models of hydrodynamics. TvSU, Tver (in Rusian) Tarasov N, Polyakov S, Yu K et al (2020) Incompressible viscous flow simulation using the Quasi— hydrodynamic equations’ system. J Math Models and Comput Simul. ISSN 2070-0482. https:// doi.org/10.1134/S2070048220040183 Telin N (2015) Kinetics of scale formation on the heat exchange surface. J Bull Cherepovets State Univ 8(69):35–37 Yurina E, Ya K, Pustovalov D (2021) Main problems related to energy saving and possible solutions. J Bulletin of the Volga University named after V. N. Tatishchev. https://doi.org/10.51965/20767919_2021_2_1_144 (in Russian) Zarga Y, Ben Boubaker H, Ghaffour N et al (2013) Study of calcium carbonate and sulfate coprecipitation. J Chem Eng Sci 96:33–41

Chapter 6

A 3D Fiber-Based Strategy for Optimization of Tissue Materials Using a Combination of Liquid Absorbency/Retention Methods Flávia P. Morais, António O. Mendes, Ana M. M. S. Carta, Paulo T. Fiadeiro, Maria E. Amaral, and Joana M. R. Curto

6.1 Introduction The design and development of advanced fiber-based biomaterials, the enhancement of natural resources, and the development of sustainable processes and products are the core of our research. The development of biopolymer-based materials follows the guidelines of EU sustainability societal challenges. The optimization of the 3D network structure, porosity and pore dimensions, and distribution is decisive in the development of porous polymeric biomaterials for liquid retention applications such as tissue products, delivery systems, scaffolds, etc. Cellulose-based fibrous materials F. P. Morais (B) · A. O. Mendes · P. T. Fiadeiro · M. E. Amaral · J. M. R. Curto (B) Fiber Materials and Environmental Technologies (FibEnTech-UBI), Universidade da Beira Interior, R. Marquês de Ávila E Bolama, 6201-001 Covilhã, Portugal e-mail: [email protected] J. M. R. Curto e-mail: [email protected] A. O. Mendes e-mail: [email protected] P. T. Fiadeiro e-mail: [email protected] M. E. Amaral e-mail: [email protected] A. M. M. S. Carta Forest and Paper Research Institute (RAIZ), R. José Estevão, 3800-783 Eixo, Aveiro, Portugal e-mail: [email protected] J. M. R. Curto Chemical Process Engineering and Forest Products Research Centre (CIEPQPF), Universidade de Coimbra, R. Sílvio Lima, Polo II, 3004-531 Coimbra, Portugal © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_6

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have grown over the years in several applications since they present advantages as a renewable, sustainable, and economical resource, and also have shown interesting properties for each type of application (Kurimoto et al. 2016; Hu et al. 2020). These materials present unique characteristics of biocompatibility, biodegradability, safety, controllable porosity, the capacity of being functionalized, ability to incorporate, retain and release molecules (Kurimoto et al. 2016; Hu et al. 2020). The development of engineering advanced structural materials, such as tissue paper materials, allows the combination of multi-structured materials and their interaction with liquids, achieving better performance and functional properties in quite diverse fields, at the micro and nanoscale. The use of computational tools constitutes an innovative strategy to enhance biopolymeric materials, through the modeling of fibers and structures simultaneously and in three dimensions. The porosity optimization presents a relevant impact on release kinetics, through experimental and computational planning. The design of new materials is done to achieve optimized properties considering the selection of the fibrous material (morphology vs properties), the fiber modification treatments, and/or the incorporation of the additive in suspensions (Boucher 1976). The desired porosity for a specific application of these materials is one of the fundamental properties to be optimized. The porosity and pore size distribution in different fibrous materials presents an essential role in liquid absorption. Besides, the inter-fiber pores and the blind pores of the fiber surface structure play a crucial role in the liquids flow through the fibrous network (Hassan et al. 1998; Lang et al. 2013; Gigac et al. 2017; Stankovská et al. 2019). The complexity of fibrous materials lies in the fact that they do not have a regular pore size distribution, and in the presence of tortuous paths (tortuosity) that the pores provide for the flow-through of liquids or active substances, such as essential oils (Neumann et al. 2021). These molecules need to travel a path higher than a straight line between the original source and its active local (Ghanbarian et al. 2013). The fibrous structures and materials’ porosity structure knowledge is important to predict and develop liquid absorption properties (Axelsson and Svensson 2010). Tissue paper materials are examples of important applications for optimizing the properties of liquid or molecules’ absorption, deposition time, spreading area, and kinetics. The liquid interaction properties, through the understanding of liquid transport processes in porous media, are important variables to optimize the performance of tissue paper materials. Tissue paper materials are recognized for their low basis weight, high bulk, softness, strength, and water absorption capacity, although the priority given to each of these properties varies according to the desired final product, such as napkins, towel papers, toilet papers, among others (Beuther et al. 2010; Assis et al. 2018a). Overall, what distinguishes tissue papers from other paper materials is their production, namely, the creping process, used essentially to change tissue paper properties, and the converting process in which paper is transformed into different types of tissue products. Due to the low basis weight and density of these fibrous materials, the sheets cannot be subjected to drying with multiple cylinders, as it happens for other conventional types of paper. Instead, the tissue paper sheet just goes through a large steel dryer cylinder, called the Yankee dryer. This dryer removes the remaining water,

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and the sheet adheres to the Yankee’s surface through specific additives, which are applied directly to the dryer. The tissue sheet is then removed from the surface of this cylinder with the aid of a creping blade, to be wound in reels (initial industrial base tissue paper) (Raunio and Ritala 2012; Rezaei-Arhomand et al. 2013; Boudreau and Germgård 2014; Raunio and Ritala 2018). These tissue paper reels feed afterward the converting process that includes the following steps: unwinding, embossing, printing, perforating, rewinding, cutting, packaging, and palletizing (Spina and Cavalcante 2018; Mendes et al. 2020; Vieira et al. 2020). In the laboratory these steps are not easy to replicate, hence the quantification of differences between the finished tissue paper product properties and the laboratory-made structures that were produced from its disintegration becomes essential. Similarly, isotropic structures (laboratory handsheets) using hardwood and softwood pulps were made. In tissue paper materials, liquid absorption properties, such as absorption capacity, absorption rate, and absorption time are the most significant variables to predict the shelf price of napkins and towel papers (Assis et al. 2018b). These properties are related to fiber type, fine contents, mechanical treatments, additives, creping process, embossing operation, and the stacking sequence of tissue paper sheets, due to their bulk, all have an impact on the liquid absorption capacity (Vieira et al. 2020; Assis et al. 2018b; Vieira et al. 2019). Therefore, the know-how and optimization of the tissue paper absorption properties can be carried out using a combination of different types of fibers, with different treatments (mechanical or enzymatic), and additives incorporation (Assis et al. 2018a; Gigac et al. 2019; Morais et al. 2019, 2021a, b; Morais and Curto 2022a), which consequently affect the porosity and relative bonding area, used to predict the initial liquid absorption (Gigac et al. 2019). The need to reduce costs and increase industrial tissue paper productivity makes furnish optimization and management an important milestone that correlates computational and experimental approaches for the developing and designing of high-quality products with the improved functional properties demanded by the consumers. Examples of these premium tissue products are antimicrobial tissue papers which can prevent infectious diseases and provide a low-cost and sustainable personal hygiene product; individual protection tissue masks for COVID-19 disease that allow absorbing the water vapor exhaled from the mouth and nose and reduce the compression of the skin on the nose; and facial tissue masks, which are capable of retaining natural products such as essential oils, transporting them to the skin, promoting their release, penetration, and permeation. Our approach includes the combination of experimental and computational design aiming the optimization of tissue paper materials through advanced computational tools (Raunio and Ritala 2012; Morais et al. 2020a). An innovative approach of combining different 3D fiber models that define raw materials by a fiber population was used to simulate tissue paper structures, predicting their properties (Morais et al. 2020a). An experimental planning of fiber characterization and fiber and structure modification process steps, with the development of our own computational simulator, the SimTissue, allow us to establish relationships between the process inputs and the tissue paper end-use properties, to support tissue paper furnish management (Morais et al. 2019, 2020a, 2021a, b; Morais and Curto 2022a).

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Additionally, a fibrous materials simulator, the voxelfiber, allows us to model 3D structures according to the fiber dimensions and properties, such as length and width ratio, flexibility, wall thickness, lumen, among others (Conceição et al. 2010; Curto et al. 2011), to simulate and optimize different fibrous materials at different scales (Curto 2012; Morais and Curto 2022b; Curto et al. 2015; Morais et al. 2020b; Martins et al. 2018). The laboratory-made structures can be modeled with accuracy using our 3D computational simulations for tissue paper materials, obtaining data of their structural properties such as thickness, intra-, and inter-fiber porosity, relative bonding area, among others. On the other hand, an optical prototype, developed by our research team (Mendes et al. 2013), has been used to collect data of the liquid deposition in the fibrous structures over time, to measure the spreading area and kinetics, through image analysis (Curto et al. 2015; Mendes et al. 2013; Fiadeiro et al. 2013; Sousa et al. 2014). Hydrophobic and hydrophilic interactions between fibrous structures and liquid droplets are also essential aspects to be considered when studying the absorbency properties (Wågberg and Westerlind 2000; Modaressi and Garnier 2002; Kannangara et al. 2006; Rosenholm 2015). Additionally, different studies of deposition and spreading of liquid droplets have indicated that porous structure knowledge is one of the key parameters for optimizing properties, since the pore size distribution and dimensions influence the spreading and equilibrium state of the droplets, contributing to the design of these materials for various industrial advanced applications (Curto et al. 2015; Senden et al. 2000; Clarke et al. 2002; Starov et al. 2003; Hilpert and Ben-David 2009; Chen et al. 2020). For these reasons, in future works, the combination of simulations of liquid droplets and 3D fibrous materials is essential for optimization and modeling the spreading area and kinetics, the time of deposition, and other absorption properties, in order to develop products with innovative and advanced features (Curto et al. 2015; Zhang et al. 2020). To the best of our knowledge, the approach of combining absorbency characterization methods and liquid retention optimization methodologies, that captures the liquid interaction with the structures over time, applied to low basis weight and high porosity materials, such as tissue papers, has not been studied in the literature. The main goal of this work was to develop a methodology aiming the optimization of liquid retention properties, combining water absorption capacity, Klemm capillary rise, through the experimental characterization of different tissue paper fibrous structures, and liquid spreading area and kinetics measurements, which uses liquid droplets deposition on these structures. Computational simulations of these fibrous materials were also used to present more information about its structural properties and predict their functional properties. The experimental design plan consisted of the characterization of different industrial tissue papers and different structures, with different treatments that mimic the production of tissue papers, such as the fiber furnish mixture, enzymatic treatments, and additive incorporation, in this case at micro- and nanoscales.

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6.2 Materials and Methods 6.2.1 Materials The hardwood samples include a never dried, or slush, total chlorine free (TCF) bleached eucalyptus kraft pulp, and two dried bleached eucalyptus pulps, one produced using the elemental chlorine free (ECF) bleaching and kraft cooking processes and the other the TCF bleaching and sulfite cooking processes. The softwood samples include three dried bleached pinus kraft pulps. The nano/ microfibers were obtained by mechanical and enzymatic processes applied to a dried bleached eucalyptus pulp. The selected industrial tissue products were napkins, towel papers, and toilet papers, supplied by a Portuguese tissue paper manufacturer.

6.2.2 Experimental Methods Different experimental design plans were performed in order to evaluate the influence of liquid retention properties in all the paper samples, as shown in Table 6.1. The first four experimental sets included paper samples produced in the laboratory using cellulosic fibers from eucalyptus, fiber mixtures with softwood reinforcement fibers, that differ the type of bleaching from each other, enzymatic treatments, and also the addition of micro/ nanofibrilated cellulose (MFC/NFC). To evaluate the impact of the different never dried and dried bleached eucalyptus kraft pulps, isotropic laboratory structures with basis weights of 20, 40, and 60 g/m2 , without pressing, were produced according to the adaptation of ISO 5269/1. Different mixtures of eucalyptus and softwood fibers pulps, with different percentages, were also prepared and studied, as well as structures with a basis weight of 20 and 60 g/m2 . To evaluate the enzymatic effect, two mixtures of 80% of kraft eucalyptus enzymatically treated and 20% of sulfite eucalyptus pulp without enzymatic treatment were selected (Morais et al. 2021a). The kraft eucalyptus pulp was treated with 10 g of enzyme per ton of pulp for 30 min (E1) and 60 min (E2). Finally, to evaluate the influence of micro/ nanofibers, a furnish mixture of eucalyptus fiber, reinforcement fiber, MFC/NFC was carried out (Morais et al. 2021b). For these last two assays, isotropic laboratory structures of 20 g/m2 were also produced as reported above. The second experimental set, corresponding to the last three rows in Table 6.1, refers to industrial tissue products. These samples were disintegrated according to ISO 5263/1 (named as repulping in Table 6.1) to produce isotropic laboratory structures with a basis weight of 20 g/m2 , without pressing, similarly to the previous ones, where the effect of creping and embossing was attenuated or removed. The structural characterization of fibrous structures was carried out using ISO 12625-6 for the basis weight, or grammage, ISO 12625-3 for the thickness, using the universal micrometer (FRANK-PTI—Birkenau, Germany), and the bulk was determined by using these two parameters in accordance with ISO 12625-3. Apparent

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Table 6.1 Identification and description of all the selected and prepared samples Samples ID

Experimental and Design Plan Description

REF_LAB_SS_20

Reference of laboratory-made structures of Eucalyptus slush pulp with 20 g/m2

REF_LAB_SS_40

Reference of laboratory-made structures of Eucalyptus slush pulp with 40 g/m2

REF_LAB_SS_60

Reference of laboratory-made structures of Eucalyptus slush pulp with 60 g/m2

REF_LAB_T_20

Reference of laboratory-made structures of Eucalyptus dried pulp with 20 g/m2

REF_LAB_T_40

Reference of laboratory-made structures of Eucalyptus dried pulp with 40 g/m2

REF_LAB_T_60

Reference of laboratory-made structures of Eucalyptus dried pulp with 60 g/m2

LAB_T_30SW1_20

Laboratory-made structures of a blend of Eucalyptus dried pulp (70%) and softwood pulp_1 (30%) with 20 g/m2

LAB_T_30SW1_60

Laboratory-made structures of a blend of Eucalyptus dried pulp (70%) and softwood pulp_1 (30%) with 60 g/m2

LAB_T_10SW1_20

Laboratory-made structures of a blend of Eucalyptus dried pulp (90%) and softwood pulp_1(10%) with 20 g/m2

LAB_T_10SW1_60

Laboratory-made structures of a blend of Eucalyptus dried pulp (90%) and softwood pulp_1 (10%) with 60 g/m2

LAB_T_30SW2_20

Laboratory-made structures of a blend of Eucalyptus dried pulp (70%) and softwood pulp_2 (30%) with 20 g/m2

LAB_80TE1_20S

Laboratory-made structures of 80% enzyme-treated (1) Eucalyptus kraft dried pulp and 20% Eucalyptus sulfite pulp, with 20 g/m2

LAB_80TE2_20S

Laboratory-made structures of 80% enzyme-treated (2) Eucalyptus kraft dried pulp and 20% Eucalyptus sulfite pulp, with 20 g/m2

LAB_SS_5SW3_5MFC/NFC Laboratory-made structures of 90% Eucalyptus slush pulp, 5% softwood pulp_3, and 5% micro/ nanofibrillated cellulose, with 20 g/m2 NAP_IND_A

Industrial-made napkin A, with 34 g/m2

NAP_IND_B

Industrial-made napkin B, with 40 g/m2

NAP_LAB_RP_A

Laboratory-made structures of the repulping of the napkin A, with 20 g/m2

NAP_LAB_RP_B

Laboratory-made structures of the repulping of the napkin B, with 20 g/m2

TW_IND_A

Industrial-made towel paper A, with 64 g/m2

TW_IND_B

Industrial-made towel paper B, with 41 g/m2

TW_IND_C

Industrial-made towel paper C, with 66 g/m2 (continued)

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Table 6.1 (continued) Samples ID

Experimental and Design Plan Description

T_IND_I

Industrial base tissue paper (reel) for toilet papers, with 16 g/m2

T_IND_F

Industrial-made toilet paper, with 33 g/m2

T_LAB_RP

Laboratory-made structures of the repulping of the toilet paper, with 20 g/m2

porosity (theoretical) was calculated according to the following equation:   ρsample × 100 Por osit y(%) = 1 − ρcellulose

(6.1)

where ρsample is the sample density (g/cm3 ) and ρcellulose is the cellulose density (which is assumed to be 1.5 g/cm3 ) (Morais et al. 2021a, b). The fibers morphological properties were determined automatically by image analysis using a diluted suspension of 20 mg/L, in a flow chamber, using the fiber and shive analyzer system LB01 166 (MorFi® TECHPAP—Grenoble, France). The drainability of a pulp suspension in water was measured in accordance with ISO 5267/1 using by Schopper-Riegler (ºSR) method. The surface morphology of the samples was analyzed using scanning electron microscopies S-2700 and S-3400N (SEM Hitachi–Tokyo, Japan), operating at 20 kV and at different magnifications. All paper samples were coated with gold using a sputter coater Q 150 R ES (Quorum–Quorum Technologies Ltd., Ashford, United Kingdom). Finally, the tensile index was evaluated using ISO 12625-4, the water absorption capacity was measured by the immersion method according to ISO 12625-8, and an adaptation of ISO 8787 was used for the measurement of Klemm capillary rise. Using tissue softness analyzer (TSA Emtec–Leipzig, Germany), the TSA-softness properties were determined, with the QAI algorithm to measure the handfeel (HF) parameter. The liquid spreading area was studied using an optical system (Curto et al. 2015; Mendes et al. 2013; Fiadeiro et al. 2013; Sousa et al. 2014) specifically developed to investigate the interaction over time of liquid droplets on different furnish fibrous structures (Mendes et al. 2013) and tissue paper materials. This experimental system works by ejecting microliter droplets, through a syringe filled with dyed water, toward the fibrous structure surface. The droplets spreading images were captured for a given time. We considered the spreading kinetics from 35.7 ms to 3 s. For the different assays, only the spreading images of the initial time (35.7 ms), intermediate time (1 s), and maximum stabilization time (3 s) are shown.

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6.2.3 Computational Methods Different fibrous structures were modeled in a 3D fiber-based computational simulator, named voxelfiber, in which the code is available on GitHub (https://github. com/eduardotrincaoconceicao/voxelfiber). This computational tool allows modeling tissue paper materials as planar random networks. The fibers are represented by a voxel chain and modeled according to their dimensions and properties. A more detailed description of the 3D simulator can be found in (Morais and Curto 2022a; Conceição 2010; Curto et al. 2011). The fibers can also be modeled according to different models proposed including the fiber wall thickness and lumen dimensions (Morais et al. 2020a; Curto et al. 2011). The 3D computational structures are processed to evaluate different structural properties. Computational studies were carried out using the programming platform MATLAB® (R 2020a—MathWorks, Natick, United State of America). Additionally, a computational tissue simulator, the SimTissue, was developed and used to predict the influence of furnish on TSAsoftness, tensile strength, and absorption properties. The prediction and establishment of relationships between these parameters were achieved through the programming of calculation engine algorithms and database integration, and mathematical and statistical models including clusters, multiple linear regressions, and artificial neural networks analysis. More detailed information about this computational methodology can be found in our previous publications (Morais and Curto 2022a, b, Morais et al. 2020a, 2021a, b).

6.3 Results and Discussion The complete set of results that correspond to each experiment are presented and can be found in the appendix section on Tables 6.9, 6.10, 6.11 and 6.12.

6.3.1 Laboratory-Made Structures 6.3.1.1

Influence of Using Different Types of Eucalyptus Pulps, with Different Basis Weights

Eucalyptus fibers are considered ideal to produce tissue papers due to their morphological and biometric properties that allow improving the properties of bulk, porosity, softness, and absorption (Assis et al. 2018a). Two eucalyptus pulps were selected for the study since the dried pulp present good strength properties (10 Nm/g of tensile index) and the never dried pulp presented good softness properties, with a HF of 82 units, according to our previous studies (Morais et al. 2019; Morais and Curto 2022a). Tissue products can present different basis weights, depending

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on the type of tissue paper to produce. For this reason, our goal was to produce isotropic laboratory structures of these eucalyptus pulps with different basis weights, in order to evaluate the liquid retention properties. The morphological analysis for the selected eucalyptus pulp fibers presented a fiber length weighted by length range between 0.73 and 0.80 mm, fiber width of 19.1 µm, fiber coarseness between 6.31 and 6.83 mg/100 m, fines content between 37.1 and 38.5%, and drainability property accessed by Schopper-Riegler between 20 and 25ºSR (Table 6.9). Figure 6.1 presents the structural properties behavior for the dried eucalyptus pulp (REF_LAB_T) and the eucalyptus slush pulp (REF_LAB_SS) structures with 20, 40, and 60 g/m2 . The structure thickness increases with their basis weight (Bloch et al. 2019), in contrast to the bulk and apparent porosity properties. Compared to the REF_LAB_T, the REF_LAB_SS showed, on average, an increase of 15% in thickness, 28% in bulk, and 4% in apparent porosity (Table 6.10). REF_LAB_SS structures between 20 and 60 g/m2 showed bulk variation between 6.5 and 4.1 cm3 /g, and porosities between 89.4 and 83.5%, while REF_LAB_T structures between 20 and 60 g/m2 showed bulk variation between 4.5 and 3.7 cm3 /g, and porosities between 85.2 and 81.7%. Apart from the fact that these two samples are either dry or never-dried pulp, these differences are also due to the different bleaching procedures applied to produce these eucalyptus pulps. The REF_LAB_T samples was treated with an ECF bleaching sequence, while the REF_LAB_SS sample with a TCF bleaching sequence. For this reason, the REF_LAB_T and REF_LAB_SS samples presented viscosity values in the order of 915 and 453 mL/g, respectively. The laboratory-made structures were simulated computationally in 3D, and the same structural properties were obtained (Table 6.11). In 3D fiber modeling, the input variables of fiber properties and structures are considered, including the length/width ratio, fiber population, fiber wall thickness, fiber lumen, grammage, among others. The 3D computational simulation of these tissue structures allows, from experimental and simulated results, the modeling of any type of material by adjusting the parameters of the fiber dimensions and the structures formed by them. This approach is considered a good alternative to model and optimize new innovative materials with a wide field of applications (Morais and Curto 2022a, b).

Fig. 6.1 Basis weight as a function of thickness a and bulk and apparent porosity b for the REF_LAB_SS and REF_LAB_T structures

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For the absorbency properties, the REF_LAB_SS structures presented a slightly higher water absorption capacity (8.13 vs. 8.05 g/g), and Klemm capillary rise at 10 min (132 vs. 113 mm), compared to the REF_LAB_T. The kinetics of the liquid droplet spreading area over time for the REF_LAB_SS and REF_LAB_T paper samples with different basis weights is presented in Fig. 6.2 and Table 6.2. The results indicated that the liquid droplet deposition into these fibrous structures showed a smaller spreading area with the basis weight increase, corresponding to a less porous structure, and consequently, a structure bulk decrease with the basis weight, as shown in Fig. 6.1b. The spreading rate of droplet absorption decreased with increasing basis weight, using an identical volume of liquid released. The liquid scattering in structures with 20 g/m2 occurs essentially in the XY direction, while in structures with 60 g/m2 , this phenomenon also appears along the Z direction. Therefore, a smaller volume is available to be absorbed in the XY direction since these structures present higher fiber layers in the Z direction, compared to low basis weight samples. It was also possible to verify that the REF_LAB_SS structures presented a higher liquid droplet spreading area compared with the REF_LAB_T structures. This result is in accordance with the water absorption capacity and Klemm capillary rise properties, as previously mentioned. This result is also due to the fiber hornification in higher basis weights, and structure bulk and porosity differences (Fig. 6.1b). Furthermore, for these samples (Table 6.9), the viscosity, the ºSR, and the biometric properties, such as curl and kink deformations and coarseness may also have contributed to these differences. The REF_LAB_SS sample was produced with a more degraded pulp presenting more deformations and higher ºSR when compared with the REF_LAB_T sample. REF_LAB_SS fibers present more fibrillation as a consequence of their production process (bleaching sequence), unlike REF_LAB_T pulp.

Fig. 6.2 The normalized spreading area as a function of time for the samples REF_LAB_SS (a) and REF_LAB_T (b). The structures with basis weights of 20, 40, and 60 g/m2 are represented in blue, red, and green, respectively

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Table 6.2 Evolution of the liquid droplet spreading area over time in laboratory-made structures with two Eucalyptus pulps and different basis weights (20, 40, and 60 g/m2 ). One set suitable for softness properties (REF_LAB_SS) and the other set for strength (REF_LAB_T)

t = 35.7 ms

t = 1.0 s

t = 3.0 s

REF_LAB_SS_20

REF_LAB_SS_40

REF_LAB_SS_60

REF_LAB_T_20

REF_LAB_T_40

REF_LAB_T_60

6.3.1.2

Influence of Fiber Mixtures and Two Different Basis Weights

Tissue products are produced with a high percentage of bleached hardwood pulps, which confer structural, softness, and absorption properties, and with a low percentage of bleached softwood pulps, which provide strength properties and ensure the tissue machine runnability (Assis et al. 2018a). Isotropic laboratory-made structures were produced with different softwood reinforcement fibers incorporation in percentages of 10 and 30%, with basis weights between 20 and 60 g/m2 . Previous studies have proven that mixtures of eucalyptus fibers and reinforcement fibers between these percentages can be suitable to produce tissue papers with high bulk

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Fig. 6.3 SEM images of fibrous structures. Top view of sample LAB_T_30SW1_20 (a) and thickness of sample LAB_T_30SW1_60 (b)

softness, tensile index, absorption, and brightness (Stankovská et al. 2020). The isotropic structures with 20 and 60 g/m2 showed, respectively, a thickness of 99 and 269 µm, bulk of 5.30 and 3.66 cm3 /g, and apparent porosity of 87.4 and 81.7% (Table 6.10). SEM images were obtained for the different fibrous structures (Fig. 6.3) to represent, as an example the top view and thickness cross-section of the laboratorymade structures LAB_T_30SW1_20 and LAB_T_30SW1_60. These laboratorymade structures considering the corresponding fiber mixtures were also modeled computationally with the same fibrous elements, and the same structural properties were obtained. The response of the liquid droplet deposition into the fibrous structure made from fiber mixtures having 30% of softwood fibers (LAB_T_30SW1) indicates that the spreading area is smaller compared to 10% of softwood fibers (LAB_T_10SW1), as shown in Fig. 6.4a, b and Table 6.3. The spreading area decreases when the basis weight increases, as previously verified. Additionally, less porous and bulky structures are produced with a higher percentage of softwood pulp incorporation, also enhancing a decrease spreading area. These structures were produced with a softwood fiber with an aspect ratio of 53, coarseness of 17.4 mg/100 m, fine content of 31.6% and water absorption capacity of 8.4 g/g (Morais et al. 2019; Morais and Curto 2022a). The change of softwood fibers with higher aspect ratio (55), coarseness (18.8 mg/100 m) and absorption properties (8.6 g/g), and lower fine content (20.3%) also modifies the liquid droplet spreading area, as shown in Fig. 6.4c and Table 6.3. The sample LAB_T_30SW2_20 showed a higher spreading area when compared with the samples LAB_T_30SW1_20 and LAB_T_10SW1_20. The results presented that the SW2 incorporation of 30% increased the spreading kinetics compared to the SW1 incorporation of 30%, and even of 10%. This evidence can be justified by the differences between morphological and absorption properties of these two reinforcing fibers. With this study, it was possible to conclude that not only the percentage of softwood fibers incorporated in tissue materials influences the liquid retention properties but also the type of softwood fibers used.

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Fig. 6.4 The normalized spreading area as a function of time for the samples LAB_T_30SW1 (a), LAB_T_10SW1 (b), and LAB_T_30SW2 (c). The structures with basis weights of 20 and 60 g/m2 are represented in blue and red, respectively Table 6.3 Evolution of the liquid droplet spreading area over time in laboratory-made structures of a blend of eucalyptus pulp and softwood pulp_1 at 30 and 10%, with a basis weight of 20 and 60 g/m2 , and another softwood pulp_2 at 30%, with 20 g/m2 t = 35.7 ms

LAB_T_30SW1_20

LAB_T_30SW1_60

LAB_T_10SW1_20

LAB_T_10SW1_60

LAB_T_30SW2_20

t = 1.0 s

t = 3.0 s

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Influence of Enzymatic Reaction Time Treatments

To enable the great potential of eucalyptus fibers in tissue papers, it is necessary not only to identify the fiber properties, their relationship with the process, and the tissue final properties but also to modify the fibers, minimizing the impact on the functional properties. Biorefining allows modifying the fibers with enzymes, in order to improve the softness by fiber cellulose hydrolysis, producing weak points that make them flexible. The application of enzymes under ideal conditions presents a higher potential to produce materials with high added value, at low costs, being considered environmentally and sustainability friendly (Morais et al. 2021a; Gil et al. 2009). We consider that an innovative feature of biorefining is the mixture of two eucalyptus pulps with different enzymatic treatments, one pulp ensures strength properties (80%) and the other softness properties (20%). Through this methodology, the fibrous structures, that take advantage of all the eucalyptus pulps potential, can be obtained laboratory and computationally. In these two blends, only eucalyptus fibers suitable for strength properties were subjected to two enzymatic treatments with the same enzyme dosage and different reaction times of 30 and 60 min, corresponding to the LAB_80TE1_20S and LAB_80TE2_20S, respectively. The LAB_80TE1_20S and LAB_80TE2_20S presented, respectively, a fiber length weighted by length range of 0.78 mm, fiber width of 19 µm, fiber coarseness of 6.82 and 6.72 mg/100 m, fines content of 36 and 38%, and 21 and 22º ºSR (Table 6.9). These isotropic laboratory-made structures presented, respectively, a thickness of 100 and 96 µm, bulk between 4.76 and 4.72 cm3 /g, and apparent porosity of 86% (Table 6.10). Regarding tissue properties, these structures presented, respectively, a softness HF of 58 and 63 units, tensile strength index of 18 and 16 Nm/g, water absorption capacity of 7.0 and 6.4 g/g, and Klemm capillary rise at 10 min of 120 and 119 mm (Table 6.12) (Morais et al. 2021a). Additionally, the evolution of the liquid droplet spreading area over time in LAB_80TE1_20S, and LAB_80TE2_20S samples is presented in Fig. 6.5 and Table 6.4. The results indicated that the spreading area decreased as a function of the enzyme reaction time. The sample LAB_80TE2_20S presented more flexible fibers, with less porosity and bulk, corresponding to the liquid retention properties decreases, which were also predicted for different furnish scenarios obtained by our tissue simulator, SimTissue.

6.3.1.4

Influence of Micro/Nanofibers as an Additive

In addition to the fiber modification processes, it is important to find several solutions that allow obtaining improved tissue properties to introduce innovative formulations on the market and produce premium tissue materials. An example of a raw material that can be used in tissue papers as an additive in order to improve the tissue final properties is the MFC/NFC (Morais et al. 2021b; Zambrano et al. 2021; Guan et al. 2019). We developed an experimental design to produce multi-structured fibrous structures with eucalyptus fibers (90%), softwood fibers (5%), and MFC/NFC (5%). The morphological characterization shows for this sample a fiber length weighted by

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Fig. 6.5 The normalized spreading area as a function of time for the structures LAB_80TE1_20S (red) and LAB_80TE2_20S (blue)

Table 6.4 Evolution of the liquid droplet spreading area over time in laboratory-made structures of a blend of two Eucalyptus pulp (80:20) with different enzymatic reaction time treatments

t = 35.7 ms

t = 1.0 s

t = 3.0 s

LAB_80TE1_20S

LAB_80TE2_20S

length of 0.74 mm, fiber width of 19.4 µm, fiber coarseness of 7.32 mg/100 m, fines content of 41.7%, and 33 ºSR (Table 6.9). The isotropic laboratory-made structures present a thickness of 110 µm, bulk of 5.34 cm3 /g, and apparent porosity of 87.5% (Table 6.10). Regarding tissue properties, these structures presented a softness HF of 70.5 units, tensile strength index of 14.31 Nm/g, water absorption capacity of 7.4 g/g, and Klemm capillary rise at 10 min of 117 mm (Table 6.12). From SEM images of the multi-structured fibrous structures, it is possible to identify the different fiber fibrillation degrees, with inter-fiber bonding resulting from MFC/NFC as an additive, as exemplified in Fig. 6.6.

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Fig. 6.6 SEM image of LAB_SS_5SW3_5MFC/NFC

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multi-structured

fibrous

structures

of

the

sample

Furthermore, the incorporation of MFC/NFC, as an additive, enhanced the liquid droplet spreading area, in the formulation of 90% eucalyptus slush pulp, 5% softwood pulp_3, and 5% MFC/NFC, as shown in Fig. 6.7 and Table 6.5. This result was also improved compared to fiber mixtures (Sect. 3.1.2) and enzymatic treatments (Sect. 3.1.3). An approach of combining these three cellulose sources allowed to obtain a multi-structured fibrous material, at different scales, with improved liquid retention properties obtained by SimTissue, compared to the processes currently used in the tissue industry. The potential to combine the different properties of commonly used raw materials with the micro/ nanofiber properties could enhance the production of tissue materials with added value and innovative functionalities. Fig. 6.7 The normalized spreading area as a function of time for the sample LAB_SS_5SW3_5MFC/NFC

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Table 6.5 Evolution of the liquid droplet spreading area over time in laboratory-made structures of a blend of Eucalyptus pulp (90%), softwood fiber pulps (5%), and MFC/NFC (5%)

t = 35.7 ms

t = 1.0 s

t = 3.0 s

LAB_SS_5SW3_5MFC/NFC

6.3.2 Industrial Market Tissue Products 6.3.2.1

Influence of Industrial Napkins and Repulping

Industrial napkins are usually produced with a content of eucalyptus pulp between 50 and 60% (Assis et al. 2018a). Table 6.9 in supplementary materials presents slight differences between the morphological and drainability properties of the two napkins used in this work, namely the samples NAP_IND_A, and NAP_IND_B. The fiber curl and kink deformations appear throughout the pulp production process, being directly related to the strength properties (Morais and Curto 2022a). Figure 6.8 shows that by the fiber length and width distributions, the presence of reinforcing fibers (softwood fibers) is evident, since fiber lengths and widths higher than 1.2 mm and 20 µm, respectively, were observed, contrary to what happens in eucalyptus fiber pulps (Morais and Curto 2022a). In these industrial napkin samples, the percentage of reinforcement fiber can vary from 30 to 40%. A lower value of ºSR implies the presence of reinforcement fibers, being more pronounced in the sample NAP_IND_B. The production of isotropic laboratory-made structures of 20 g/m2 , samples NAP_LAB_RP_A and NAP_LAB_RP_B, from the industrial napkin samples allowed quantifying the limits of the creping process conditions. Both napkin samples are made up of 2-ply tissue paper sheets. Each ply of the NAP_IND_A sample

Fig. 6.8 Fiber length weighted by length and width distributions for samples NAP_IND_A (a) and NAP_IND_B (b)

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(34 g/m2 ) presented a basis weight of 16 g/m2 , while each ply of the NAP_IND_B sample (40 g/m2 ) presented a basis weight of 20 g/m2 (Table 6.10). Then, it was produced laboratory-made structures with 20 g/m2 of the disintegrated commercial products, since this basis weight is a standard value used in all previously reported experimental studies to mimic tissue papers. Regarding tissue properties, a combination of absorption and strength properties is required for napkins (Assis et al. 2019). Overall, under these conditions, the sample NAP_IND_A stood out, presenting additionally less softness properties. The opposite happened for the sample NAP_IND_B. The same trend was observed for isotropic laboratory-made structures. Creping conditions positively affected the softness HF properties by 11% and 12%, and water absorption capacity by 30 and 9%, and negatively affected the tensile strength index properties by 31 and 40% for the samples NAP_IND_A and NAP_IND_B, respectively. Contrary to the water absorption capacity properties, the sample NAP_IND_B showed an improved performance in the Klemm capillary rise properties. Capillarity is related not only to the global porosity of the fibrous structure but also to the pore dimensions and distribution (Mullins and Braddock 2012; Kolesnikov and Gavrilov 2020; Baek et al. 2021), together with the presence of fibers with good water affinity. Compared with the isotropic laboratory-made structures obtained by repulping of these napkins, the same behavior was verified for both samples. At 10 min, creping conditions negatively affect Klemm capillary rise properties by 6% for the samples NAP_IND_A (Table 6.12). However, for sample NAP_IND_B, a variation of only 1% was observed, and consequently, this result does not ensure that it was the creping conditions that affected Klemm capillary rise. The liquid droplet spreading area of the sample NAP_IND_A was higher than the laboratory-made sample NAP_LAB_RP_A, which was structures produced by repulping the original sample, as shown in Fig. 6.9a, being in accordance with the water absorption capacity properties. However, the opposite was verified for the samples NAP_IND_B and NAP_LAB_RP_B, presenting similar spreading area values, as shown in Fig. 6.9b). This result is following the capillary rise properties. Additionally, the phenomenon of anisotropy was observed in samples NAP_IND_A and NAP_IND_B, and isotropy in samples NAP_LAB_RP_A and NAP_LAB_RP_B, according to Table 6.6. The liquid droplet spreading area was direction-dependent of the creping lines in the napkin samples, presenting an elliptical shape. In the case of NAP_IND_A, the anisotropy of the creping lines enhances a liquid spreading area increase, while the isotropy in the NAP_LAB_RP_A sample can cause higher capillary pressures and lower permeability constants. The creping lines enhance the tortuosity phenomenon, since the liquid droplet presents higher difficulty to intrude into or pass through the cellulosic fibers (Ashari et al. 2010; Coutelieris and Delgado 2012). This result is also following the structure porosity (Table 6.10), as the amount and structure of the void spaces (pores) influence the liquid flow through these porous structures. Regarding the samples NAP_IND_B and NAP_LAB_RP_B, the spreading area variation was not as pronounced. This result is in accordance with the Klemm capillary rise properties, as shown in Table 6.12.

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Fig. 6.9 The normalized spreading area as a function of time for the samples NAP_IND A (blue) and NAP_LAB_RP_A (red) (a), and samples NAP_IND B (blue) and NAP_LAB_RP_B (red) (b)

Table 6.6 Evolution of the liquid droplet spreading area overtime in industrial napkins and the corresponding laboratory-made structures

t = 35.7 ms

t = 1.0 s

t = 3.0 s

NAP_IND_A

NAP_LAB_RP_A

NAP_IND_B

NAP_LAB_RP_B

6.3.2.2

Influence of Industrial Towel Papers

Industrial towel papers, just as napkins, are also produced with a content of eucalyptus pulp between 50 and 60%. Towel papers present a higher percentage of reinforcing fiber compared to other tissue products to confer strength properties. Besides, a peculiarity of these types of papers is the presence of additives, such as wet-strength

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additives, with a higher capacity of inter-fiber bonding, in order to ensure good absorption and strength properties (Assis et al. 2018a). For this reason, the repulping of these industrial papers is a challenge, therefore the preparation of its isotropic laboratory-made structures was not possible. However, it was possible to obtain a sufficient amount of fiber suspension from these samples for the morphological properties analysis. The industrial towel papers selected for this study presented a fiber length weighted by length range between 0.79 and 0.85 mm, fiber width between 24.8 and 25.9 µm, fiber coarseness between 42.79 and 55.03 mg/100 m, and fines content between 74.1 and 78.8% (Table 6.9). As mentioned before, the fiber curl and kink deformations were directly related to the structure strength properties. The selected industrial towel papers can present 30 to 40% of reinforcement fibers, according to the fiber length and width distributions, as shown in Fig. 6.10. The sample TW_IND_B had a high content of long fibers, presenting fiber lengths that were longer than 1.74 mm, and the samples TW_IND_A and TW_IND_C presented an intermediate content of long fibers, with lengths measured between 1.23 and 1.47 mm. Additionally, the sample TW_IND_B showed a lower value of ºSR, followed by the sample TW_IND_A and the sample TW_IND_C. Figure 6.11 also presented an example of a SEM image of the industrial towel paper sample TW_IND_A, in which an a detail of the embossing pattern and the fiber furnish mixture are visible. This furnish presents different flexibility degrees due to the beating or enzymatic processes applied to the fibers for the tissue paper production. These samples are made up of 2-ply and 3-ply tissue paper sheets, consequently, their structural properties differ in terms of basis weight, thickness, bulk, and apparent porosity (Table 6.10). Regarding tissue properties, the selected industrial towel papers showed softness HF in the range between 60.8 and 75.5 units, tensile index between 3.28 and 7.96 Nm/g and water absorption capacity between 10.14 and 12.46 g/g. The properties of Klemm capillary rise agree with the water absorption capacity properties. The sample TW_IND_B presented higher water absorption capacity and Klemm capillary rise properties, followed by samples TW_IND_A and TW_IND_C. At 10 min, the samples TW_IND_A, TW_IND_B, and TW_IND_C showed 103, 132, and 90 mm of water rise, respectively (Table 6.12). The sample TW_IND_B presented the highest liquid spreading area, followed by the samples TW_IND_A and TW_IND_C, as shown in Fig. 6.12. This result agrees with the water absorption capacity and capillary rise properties. Additionally, for the sample TW_IND_B, the high paper bulk and porosity increased the absorbency properties. The bulk increment resulted from the thickness increase induced by the 2-ply tissue paper sheets and the creping and embossing process. However, the relationship between structural and absorption properties was not observed for the other two samples, because TW_IND_C consists of 3-ply tissue paper sheets and TW_IND_A by 2-ply tissue paper sheets. For this reason, the results indicate that a comparison of tissue absorption properties between paper towel samples can only be performed on papers with the same number of plies. Besides this factor, the major influence on these absorbency results is also the embossing pattern applied to towel papers, as shown in Table 6.7. The deeper the embossing pattern is engraved on the towel

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Fig. 6.10 Fiber length weighted by length and width distributions of samples TW_IND_A (a), TW_IND_B (b) and TW_IND_C (c)

Fig. 6.11 SEM image of the industrial towel paper sample TW_IND_A, highlighting an embossing pattern (a) and the fiber mixture with different fibrillation degrees (b)

papers, the higher the water displacement towards these spaces (Morais et al. 2021c; Vieira et al. 2022), being more noticeable in the sample TW_IND_B, followed by samples TW_IND_A and TW_IND_C.

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Fig. 6.12 The normalized spreading area as a function of time for the samples TW_IND_A (blue), TW_IND_B (red) and TW_IND_C (green)

Table 6.7 Evolution of the liquid droplet spreading area over time in industrial-made towel papers

t = 35.7 ms

t = 1.0 s

t = 3.0 s

TW_IND_A

TW_IND_B

TW_IND_C

6.3.2.3

Influence of Initial and Final Industrial Toilet Paper and Repulping

Finally, in this section, our approach was to characterize the initial creped industrial base tissue toilet paper (reel), its final creped and embossed industrial toilet paper, and laboratory-made structures prepared with the repulping from the last one. The creped industrial base toilet paper sample T_IND_I and the final creped and embossed industrial toilet paper sample T_IND_F were characterized. The sample

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T_IND_F has been repulped to produce isotropic laboratory-made structures, identified by T_LAB_RP, without the influence of the creping and embossing processes. Figure 6.13 shows a scheme of the methodology used. In SEM images, the creping and embossing operations are visible in the sample T_IND_F, while in the sample T_IND_I, only the creping process is observed. The anisotropy effect of the fibers is also seen in these samples. The perpendicularity between the creping lines and the fiber orientation is remarkable in these tissue samples. The fiber orientation, creping, and embossing processes were suppressed with the repulping of the final industrial toilet paper. Fiber fibrillation is also present in these products due to the fiber modification processes applied to the tissue paper production. Industrial toilet papers are produced with a content of eucalyptus pulp between 60 and 100% (Assis et al. 2018a). The industrial toilet paper samples presented a fiber length weighted in length of 0.81 mm, fiber width of 20.8 µm, fiber coarseness of 9.02 mg/100 m, and fines content of 47.8% (Table 6.9). The selected industrial toilet paper can present 25–30% of reinforcement fibers, according to the fiber length and width distributions, as shown in Fig. 6.14. From all the studied industrial tissue paper samples, the industrial toilet paper was the one with the highest ºSR, indicative of the high percentage of hardwood fibers in its composition. Creping and converting conditions increased the structural properties of industrial toilet papers, by comparison with isotropic laboratory structures (Table 6.10). The structure thickness, bulk, and apparent porosity were increased by 368, 191, and 9%, respectively. However, the final industrial toilet paper consists of 2-ply tissue paper sheets, each with approximately 16 g/m2 . Converting conditions increase these properties by 281, 80, and 4%, respectively, by comparing the initial and final industrial tissue papers. Regarding tissue properties, creping and converting conditions positively affected the softness HF properties by 10%, and water absorption capacity by 77%, but negatively affected the tensile strength index properties by 35%, and the Klemm capillary rise by 16% at 10 min. Besides, the embossing process increased the properties of softness HF by 2%, tensile index by 29%, and water absorption capacity by 31%, when comparing the samples T_IND_I and T_IND_F. In contrast, the Klemm capillary rise in sample T_IND_I was lower than in the other samples. The creping suppression and the embossing process increase the Klemm capillary rise by 107 and 74%, respectively (Table 6.12). The response of the liquid droplet deposition into the samples is higher for sample T_IND_I, followed by samples T_LAB_RP and T_IND_F, as shown in Fig. 6.15. In this case, the enhanced anisotropy effect of the liquid droplet spreading area result from the creping operation, as it can be seen in Table 6.8. The largest axis of the elliptical shape of the spreading area in the sample T_IND_I is directiondependent of the creping lines, indicating that this operation influences the tissue paper structure and water spreading. Additionally, when this sample was subjected to the converting process, the spreading area was reduced. The results indicated that in the sample with 2-ply tissue paper sheets (T_IND_F), the droplet spreading area in the XY direction was reduced since an additional absorption occurs in the Z direction compared to the single-ply sample (T_IND_I). The number of sheets and structural

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Fig. 6.13 Scheme of the methodology used for the industrial toilet paper characterization. The SEM images are representative of each paper

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properties (higher bulk and porosity) also presented the most impact on the global water absorption properties. In Table 6.8, it is also possible to observe a deformation of the embossing pattern in the sample T_IND_F that promoted the water flow in its direction, conditioning the spreading area by the creping lines, compared with the sample T_IND_I. Regarding the sample T_LAB_RP, the isotropy effect of the fibers promoted a circular shape of the liquid droplet spreading area, as depicted in Table 6.8. The process operations suppression gave rise to a less porous structure, conditioning the spreading area, comparing the samples T_IND_I and T_LAB_RP.

Fig. 6.14 Fiber length weighted by length and width distributions of industrial toilet paper sample

Fig. 6.15 The normalized spreading area as a function of time for the samples T_IND_I (blue), T_IND_F (red), and T_LAB_RP (green)

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Table 6.8 Evolution of the liquid droplet spreading area overtime in industrial toilet papers and the corresponding laboratory-made structure

t = 35.7 ms

t = 1.0 s

t = 3.0 s

T_IND_I

T_IND_F

T_LAB_RP

6.4 Conclusions The purpose of this work was to investigate the liquid retention properties on different fibrous furnish structures. The materials were selected to be representative of the different tissue process stages, including eucalyptus fibers, blend of eucalyptus and softwood fibers, enzymatic treatments, additives incorporation, and even different tissue papers, such as napkins, towel papers, and toilet papers. A combination of experimental and computational approaches aiming the optimization of the absorbency properties was presented. The structures were also modeled according to the 3D simulation model, resulting in a good estimative for the laboratory-made fibrous structures, to be optimized for the desired softness, strength, and absorption properties. The model included the fiber morphology and the behavior in the Zdirection, building a 3D structure with the sequential deposition of individual fibers, and consequently, having representative simulations of the laboratory-made fibrous porous tissue structures. The integration of liquid droplet deposition data into tissue structures with the data of our tissue simulator (SimTissue) proved to be a good tool for studying the influence of fiber properties and process steps, on the spreading area of structured fibrous materials. The spreading area kinetics and its spreading shape is influenced by the type of structure in which the liquid droplet is deposited, being an isotropic or anisotropic structure, and by the process operations, including furnish and fiber modifications, creping, and embossing. This work also presented several experimental and computational design plans for the water interaction influence in industrial, and laboratory porous tissue structures produced with different industrial strategies. A combination of absorption methods and spreading area kinetics were applied to measure these

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functional properties, aiming the optimization of industrial furnish processes for tissue products. This work is a proof of concept that combines a droplet optical system, a 3D fiberbased simulator, and the SimTissue which has a predictive capacity not only of tissue properties but also of trends for changes in the softness, strength, and absorption measurements. This approach contributed to the design of multi-structured, innovative biomaterials with advanced features, at the micro and nanoscale. This strategy was decisive for optimizing the tissue functional properties, with more economical and sustainable resources. Due to the good absorption and retention profiles, tissue products can be used to provide rapid and deep skin hydration, as is the case of facial masks. Sustainable cosmetic substances, such as essential oils, can be incorporated into these single-use materials, promoting their penetration and permeation. The design and development of tissue products with optimized structural and absorption properties can also allow the replacement of plastic-based synthetic filters used, for example, in individual protection masks, being a sustainable and affordable alternative. Acknowledgements This research was supported by Project InPaCTus – Innovative Products and Technologies from euca-lyptus, Project Nº 21 874 funded by Portugal 2020 through European Regional Development Fund (ERDF) in the frame of COMPETE 2020 nº 246/AXIS II/2017. The authors are also very grateful for the support given by research unit Fiber Materials and Environmental Technologies (FibEnTech-UBI), on the extent of the project reference UIDB/00195/2020, funded by the Fundação para a Ciência e a Tecnologia, IP/MCTES through national funds (PIDDAC).

Appendix 1 See Tables 6.9, 6.10, 6.11 and 6.12.

19.7 ± 0.1

0.767 ± 0.010

NAP_IND_A

24.8 ± 0.5 25.7 ± 0.3 20.8 ± 0.1

0.850 ± 0.018

0.796 ± 0.001

0.813 ± 0.000

TW_IND_B

TW_IND_C

T_IND_I

9.02 ± 0.13

55.00 ± 2.90

47.79 ± 4.23

55.03 ± 2.3

9.50 ± 0.37

9.61 ± 1.00

7.32 ± 0.10

6.72 ± 0.19

6.82 ± 0.02

6.83 ± 0.10

6.31 ± 0.00

C (mg/100 m)

47.8 ± 0.1

78.6 ± 0.8

74.1 ± 2.1

78.8 ± 0.99

43.7 ± 1.1

45.6 ± 2.4

41.7 ± 0.1

38.0 ± 0.3

36.4 ± 0.6

37.1 ± 0.3

38.5 ± 0.4

FC (%)

46.0 ± 0.4

49.8 ± 0.1

49.1 ± 1.2

51.7 ± 0.8

49.2 ± 0.1

53.0 ± 0.3

43.1 ± 0.4

39.2 ± 0.3

40.0 ± 0.1

35.5 ± 0.2

44.2 ± 0.3

Kinks (%)

LL length weighted by length, W width, C coarseness, FC fines content (% in length), ºSR Schöpper-Riegler degree

T_LAB_RP

T_IND_F

25.9 ± 0.2

0.800 ± 0.010

TW_IND_A

NAP_LAB_RP_B

NAP_IND_B

20.1 ± 0.1

19.4 ± 0.0

0.737 ± 0.001

LAB_SS_5SW3_5MFC/NFC

0.757 ± 0.002

19.0 ± 0.1

0.778 ± 0.002

NAP_LAB_RP_A

19.0 ± 0.1

0.780 ± 0.000

LAB_80TE2_20S

19.1 ± 0.1

LAB_80TE1_20S

REF_LAB_T_60

REF_LAB_T_40

REF_LAB_T_20

REF_LAB_SS_60

0.798 ± 0.004

19.1 ± 0.0

0.729 ± 0.001

REF_LAB_SS_20

REF_LAB_SS_40

W (µm)

LL (mm)

Samples

10.3 ± 0.1

11.9 ± 0.0

11.4 ± 0.3

12.1 ± 0.1

11.2 ± 0.0

11.6 ± 0.1

9.4 ± 0.1

9.7 ± 0.0

9.8 ± 0.1

8.4 ± 0.1

9.7 ± 0.0

Curl (%)

21 ± 1

16 ± 1

14 ± 1

15 ± 1

15 ± 0

18 ± 0

33 ± 1

22 ± 0

21 ± 0

20 ± 0

25 ± 0

ºSR

Table 6.9 Morphological and suspension drainability of laboratory-made structures produced and industrial market tissue products. Values expressed as mean ± standard deviation

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Table 6.10 Structural properties of laboratory-made structures produced and industrial market tissue products. Values expressed as mean ± standard deviation Samples

Basis weight (g/m2 )

Thickness (µm)

Bulk (cm3 /g)

Apparent Porosity (%)

REF_LAB_SS_20

20.1 ± 0.4

114 ± 2

6.5 ± 0.0

89.4 ± 0.1

REF_LAB_SS_40

43.3 ± 0.4

175 ± 2

4.1 ± 0.1

83.6 ± 0.3

REF_LAB_SS_60

60.9 ± 0.4

249 ± 6

4.0 ± 0.1

83.4 ± 0.5

REF_LAB_T_20

20.1 ± 0.1

90 ± 2

4.5 ± 0.1

85.2 ± 0.3

REF_LAB_T_40

40.2 ± 0.1

160 ± 3

3.9 ± 0.1

83.3 ± 0.3

REF_LAB_T_60

63.1 ± 0.6

230 ± 3

3.6 ± 0.0

81.7 ± 0.1

LAB_T_30SW1_20

23.6 ± 0.5

116 ± 5

5.6 ± 0.2

87.0 ± 0.6

LAB_T_30SW1_60

68.6 ± 1.0

252 ± 6

3.7 ± 0.2

81.7 ± 1.2

LAB_T_10SW1_20

20.4 ± 0.4

99 ± 6

4.9 ± 0.3

86.3 ± 0.7

LAB_T_10SW1_60

67.5 ± 1.2

269 ± 4

3.9 ± 0.3

83.2 ± 1.2

LAB_T_30SW2_20

21.5 ± 0.8

114 ± 3

5.3 ± 0.2

87.4 ± 0.6

LAB_80TE1_20S

21.0 ± 0.4

100 ± 5

4.8 ± 0.2

86.0 ± 0.7

LAB_80TE2_20S

20.3 ± 0.4

96 ± 5

4.7 ± 0.3

85.9 ± 0.8

LAB_SS_5SW3_5MFC/NFC

20.1 ± 0.3

110 ± 4

5.3 ± 0.2

88.4 ± 0.0

NAP_IND_A

33.9 ± 0.1

225 ± 3

6.6 ± 0.1

89.9 ± 0.2

NAP_LAB_RP_A

20.5 ± 0.3

122 ± 4

5.9 ± 0.2

88.7 ± 0.5

NAP_IND_B

40.2 ± 0.1

213 ± 3

5.3 ± 0.1

87.4 ± 0.1

NAP_LAB_RP_B

20.8 ± 0.6

116 ± 4

5.6 ± 0.2

88.0 ± 0.5

TW_IND_A

64.1 ± 0.1

755 ± 3

11.8 ± 0.2

94.3 ± 0.1

TW_IND_B

41.4 ± 0.2

710 ± 4

17.1 ± 1.1

96.1 ± 0.3

TW_IND_C

65.6 ± 0.9

892 ± 3

14.3 ± 0.5

95.3 ± 0.2

T_IND_I

15.7 ± 0.1

139 ± 3

8.9 ± 0.9

92.4 ± 0.7

T_IND_F

32.9 ± 0.2

529 ± 9

16.0 ± 0.3

95.8 ± 0.1

T_LAB_RP

20.6 ± 0.3

113 ± 5

5.5 ± 0.2

87.8 ± 0.4

Table 6.11 Experimental and computational characterization of 3D fibrous structures. Values expressed as mean ± standard deviation Experimental data Samples

Thickness Bulk (µm) (cm3 /g)

Computational data Apparent Porosity (%)

Thickness Bulk (µm) (cm3 /g)

Apparent Porosity (%)

REF_LAB_SS_20 114 ± 2

6.5 ± 0.0 89.4 ± 0.1 100 ± 2

5.9 ± 0.3

89.0 ± 0.3

REF_LAB_SS_40 175 ± 2

4.1 ± 0.1 83.6 ± 0.5 175 ± 3

4.0 ± 0.6

78.4 ± 0.3

REF_LAB_SS_60 249 ± 6

4.0 ± 0.1 83.4 ± 0.3 248 ± 4

4.1 ± 0.8

81.4 ± 0.6

4.5 ± 0.1 85.2 ± 0.3

4.8 ± 0.1

81.5 ± 0.1

REF_LAB_T_20

90 ± 2

99 ± 1

REF_LAB_T_40

160 ± 3

3.9 ± 0.1 83.3 ± 0.3 161 ± 3

4.0 ± 0.0.4 77.4 ± 0.2

REF_LAB_T_60

230 ± 3

3.6 ± 0.0 81.7 ± 0.1 227 ± 4

3.6 ± 0.3

80.8 ± 0.1

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Table 6.12 Tissue properties of laboratory-made structures produced and industrial market tissue products. Values expressed as mean ± standard deviation Samples

TSA-softness HF

Tensile index (Nm/g)

Water absorption capacity (g/g)

Klemm capillary rise (mm at 10 min)

REF_LAB_SS_20

82.7 ± 4.0

4.60 ± 0.61

8.13 ± 0.29

132 ± 3

REF_LAB_T_20

63.2 ± 3.0

7.86 ± 1.58

8.10 ± 0.36

113 ± 2

LAB_80TE1_20S

57.8 ± 3.3

17.68 ± 3.34

6.63 ± 0.32

120 ± 2

LAB_80TE2_20S

63.0 ± 7.1

15.93 ± 2.64

6.49 ± 0.28

119 ± 1

LAB_SS_5SW3_5MFC/NFC

70.5 ± 1.9

14.31 ± 1.05

7.41 ± 0.22

1174

NAP_IND_A

63.7 ± 2.2

6.09 ± 0.62

9.23 ± 0.68

78 ± 3

NAP_LAB_RP_A

57.3 ± 3.3

8.77 ± 1.16

7.12 ± 0.12

83 ± 2

NAP_IND_B

72.9 ± 1.5

4.24 ± 0.40

7.62 ± 0.17

90 ± 1

NAP_LAB_RP_B

65.2 ± 1.8

7.07 ± 1.48

6.97 ± 0.41

91 ± 3

TW_IND_A

60.8 ± 2.0

7.96 ± 3.79

10.87 ± 0.11

103 ± 5

TW_IND_B

75.5 ± 1.8

3.28 ± 0.25

12.46 ± 0.22

132 ± 3

TW_IND_C

64.3 ± 1.6

4.35 ± 0.61

10.14 ± 0.24

90 ± 1

T_IND_I

69.6 ± 3.7

11.04 ± 0.45

9.42 ± 0.33

43 ± 1

T_IND_F

71.3 ± 2.3

5.51 ± 0.25

12.74 ± 0.40

75 ± 5

T_LAB_RP

64.9 ± 3.0

8.51 ± 0.99

7.21 ± 0.35

89 ± 4

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Chapter 7

Nanomaterials Based on Peptide Nanotubes with Graphene and Ferroelectric Polymers Layers: Modelling and Numerical Studies of Photoelectronic Properties Vladimir S. Bystrov, Ekaterina V. Paramonova, Pavel S. Zelenovskiy, Svitlana A. Kopyl, Xiangjian Meng, Hong Shen, Tie Lin, and Vladimir M. Fridkin

V. S. Bystrov (B) · E. V. Paramonova Institute of Mathematical Problems of Biology—branch of Keldysh Institute of Applied Mathematics, RAS, 142290 Pushchino, Russia e-mail: [email protected] P. S. Zelenovskiy School of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg 620000, Russia e-mail: [email protected]; [email protected] P. S. Zelenovskiy · S. A. Kopyl Department of Physics and CICECO-Aveiro Institute of Materials, University of Aveiro, 3810-193 Aveiro, Portugal e-mail: [email protected] X. Meng · H. Shen · T. Lin Shanghai Institute of Technical Physics, CAS, Shanghai 200083, China e-mail: [email protected] H. Shen e-mail: [email protected] T. Lin e-mail: [email protected] V. M. Fridkin Federal Center of Photonics and Crystallography RAS, Shubnikov Institute of Crystallography RAS, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_7

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7.1 Introduction Peptide nanotubes (PNT) based on diphenylalanine (FF) were previously studied by various authors, including us, using various modeling methods and numerical methods of calculations, as well as experimentally. In our recent work on the modeling and numerical studies of the FF of PNT structures, we based ourselves on experimental crystallographic data obtained by X-ray methods (Bystrov et al. 2020; Zelenovskiy et al. 2019; Bystrov and Filippov 2022). The models constructed and studied by us on this basis have shown their high efficiency in describing and calculating the polar and piezoelectric properties of materials based on FF PNT (Bystrov et al. 2020, 2012, 2021; Zelenovskiy et al. 2019; Bystrov and Filippov 2022). However, the photoelectronic properties of the FF PNT have not yet been studied sufficiently and systematically. Although there are some interesting experimental works here (Amdursky et al. 2009; Gan et al. 2013a, b; Nikitin et al. 2016; Akdim et al. 2015; Bystrov 2018), as well as a number of theoretical calculations (Nikitin et al. 2016; Akdim et al. 2015). For example, in the work (Bystrov 2018) even photo-ferroelectric effects were considered, but the model used there was based on the β-sheet conformation of phenylalanine amino acid (F), while the latest experimental X-ray data convincingly showed that the initial F conformation in FF PNT is an α-helix. Therefore, below in this paper, we carry out all calculations of the photoelectronic properties of FF PNTs based on these recent new models with α-helix conformation, which are built on the basis of experimental X-ray crystallographic data, with subsequent transformation of their structures into computer work spaces for modelling and calculations of their properties (using such software as HyperChem (HyperChem 2011), MOPAC (Stewart 2016), etc.).

7.2 Main Results and Discussion The results obtained for the photoelectronic properties of FF PNTs with different initial chirality L-FF (left) and D-FF (right) of the phenylalanine amino acid and based on the calculations using various quantum semi-empirical methods (AM1, PM3, PM7, P6-D3H4) (HyperChem 2011; Stewart 2016) are presented in the Tables 7.1, 7.2 and 7.3 and in the graphs on Figs. 7.1 and 7.2. Figure 7.1 describe the molecular models of the structures of FF PNTs with different initial chirality L-FF and D-FF and space localized positions of the electronic molecular orbitals for ground state (highest occupied molecular orbital— HOMO) and excited state (lowest unoccupied molecular orbital—LUMO), computed by AM1/PM3 methods in restricted Hartree–Fock (RHF) approximation. These molecular models present two coils of helical FF PNT structures, corresponding to its one crystallographic unit cell obtained from X-ray data (Bystrov et al. 2020; Zelenovskiy et al. 2019; Gorbitz 2001). These molecular models present two coils of helical FF PNT structures, corresponding to its one crystallographic unit cell

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Table 7.1 Electron energy levels and Eg for PNT FF (D & L) in external electric field Ez = 0, calculated by HyperChem methods Type/ method

E HOMO, eV

E LUMO, eV

Eg, eV

λ, nm

Dt, Debye

Pt, C/m2

EP = = P/εε0, GV/m (ε = 4)

D-FF/ PM3/AM1

−5.924

−2.349

3.575

347

137.7

0.137

3.875

L-FF/ PM3/AM1

−5.941

−2.499

3.441

360

139.7/140.7

0.138/ 0.139

3.909/ 3.938

Volume: V(L-FF) = 3365.6 Å3 , V(D-FF) = 3346.49 Å3 Table 7.2 Electron energy levels and Eg for PNT FF (D & L) in external electric field Ez = 0, calculated by MOPAC methods Type/method

E HOMO, eV

E LUMO, eV

Eg, eV

λ, nm

Dt, Debye

Pt, C/m2

E P = P/εε0, GV/m (ε = 4)

D-FF/ PM7

−6.085

−2.104

3.981

312

146.60

0.146

4.126

L-FF/ PM7

−6.167

−2.201

3.966

313

146.24

0.145

4.092

D-FF/ PM6-D3H4

−6.351

−2.731

3.620

343

147.54

0.147

4.152

L-FF/ PM6-D3H4

−6.407

−2.835

3.572

348

147.85

0.149

4.194

Volume: V(L-FF) = 3365.6 Å3 , V(D-FF) = 3346.49 Å3 Table 7.3 Electron energy levels and Eg for PNT FF (D&L) depending on the electric field Ez, calculated by AM1 method from HyperChem Ez, GV/m

Ez, V/Å

L-FF AM1 RHF

D-FF AM1 RHF

E HOMO, E LUMO, Eg, eV eV eV

λ, nm E HOMO, E LUMO, Eg, eV eV eV

λ, nm

1.542

0.15

−6.465

−1.967

4.50 276

−6.494

−1.844

4.65 267

1.028

0.10

−6.311

−2.134

4.18 297

−6.327

−1.996

4.33 287

0.514

0.05

−6.136

−2.315

3.82 325

−6.140

−2.153

3.99 312

0.257

0.025

−6.038

−2.407

3.63 342

−6.032

−2.250

3.78 328

0

0

−5.940

−2.499

3.44 361

−5.924

−2.349

3.58 347

−0.257 −0.025 −5.843

−2.592

3.25 382

−5.816

−2.448

3.37 369

−0.386 −0.039 −5.794

−2.638

3.16 394

−5.762

−2.497

3.26 380

−0.401 −0.040 −5.703

−2.761

2.94 422

−5.751

−2.507

3.24 382

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Fig. 7.1 Molecular models of FF PNT with different initial chirality L-FF and D-FF presented in the HyperChem workspace with electron molecular orbitals HOMO and LUMO

obtained from X-ray data (Zelenovskiy et al. 2019; Gorbitz 2001). The distances between each coils (helix pitch) are equal to unit cell parameter along main c-axis of FF PNT: for L-FF PNT it is c = 5.456 Å; for D-FF PNT it is c = 5.441 Å (Bystrov et al. 2020; Zelenovskiy et al. 2019; Bystrov and Filippov 2022; Gorbitz 2001). As results, the photoexcitation of electron leads to shift of the electronic wave cloud (during electron transition from HOMO to LUMO) on the space distance equal these parameters c (per unit cell) along main PNT axis for each chirality type of the FF PNT structures. Analysing the obtained data of our calculations, we see that there is a strong dependence of all electronic energy levels and the band gap on the magnitude of the electric field (Figs. 7.1 and 7.2; Tables 7.1, 7.2, 7.3 and 7.4). These dependences are somewhat different when calculated by different methods, while the PM7 method is more developed and gives data that may be more adequate to experimental observations. Specifically, the values of the coefficient α of change in the band gap Eg with increasing values of the electric field E, from the calculated data, can be determined as α = Δ(Eg)/Δ(E). In this case, we get the following values: (1) for AM1 method α1 = 7.046 eV/(V/Å) for L-FF case and α2 = 7.166 eV/(V/Å) for D-FF case; an average value for both chirality cases here is α12 = 7.05 eV/(V/Å); 2) for PM7 method α34 = 7.733 eV/(V/Å) and have very small deviation between L-FF and D-FF chirality cases.

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Fig. 7.2 Dependence of the electron energy levels HOMO and LUMO as well as the energy band gap Eg = E_LUMO—E_HOMO for FF PNTs structures (with chirality L-FF and D-FF) calculated by semi-empirical quantum method PM7 from MOPAC package

Table 7.4 Electron energy levels and Eg for PNT FF (D&L) depending on the electric field Ez, calculated by PM7 method from MOPAC Ez, GV/m

Ez, V/Å

L-FF PM7 RHF

D-FF PM7 RHF

E HOMO, E LUMO, Eg, eV eV eV

λ, nm E HOMO, E LUMO, Eg, eV eV eV

λ, nm

1.542

0.150

−6.769

−1.663

5.106 243

−6.684

−1.623

5.061 245

1.028

0.10

−6.593

−1.842

4.751 261

−6.512

−1.759

4.753 261

0.514

0.05

−6.380

−2.021

4.359 285

−6.312

−1.919

4.393 283

0.257

0.025

−6.274

−2.111

4.163 298

−6.199

−2.012

4.187 297

0

0

−6.167

−2.201

3.966 313

−6.085

−2.104

3.981 312

−0.257 −0.025 −6.061

−2.291

3.770 329

−5.972

−2.198

3.774 329

−0.514 −0.05

−5.955

−2.381

3.574 348

−5.858

−2.291

3.567 348

−1.028 −0.10

−5.742

−2.561

3.181 390

−5.631

−2.477

3.154 394

−1.542 −0.15

−5.528

−2.741

2.787 446

−5.404

−2.663

2.741 453

The values of the band gap Eg without influence of the electric field E (it is for Ez = 0, where z is axis along the main nanotube axis with unit cell parameter c): (1) for AM1/PM3 methods Eg01 = 3.441 eV for L-FF case, and Eg02 = 3.575 eV for D-FF

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case; (2) for PM7 method Eg03 = 3.966 eV for L-FF case and Eg04 = 3.981 eV for D-FF case. These data show that the Eg values calculated by the PM7 method turn out to be larger and their values are closer to the data observed in the experiment, that usually is about Eg ~ 4 eV and higher (Amdursky et al. 2009; Gan et al. 2013a; Nikitin et al. 2016; Akdim et al. 2015) (without influence of an external electric field). In this case, the difference in Eg values between L-FF and D-FF cases, which for the AM1 method is about 0.134 eV, for PM7 decreases to 0.015 eV, i.e. almost indistinguishable. Let us now consider the influence of the polar properties of the FF PNT on the process of electron photo-excitation and its transition from the ground HOMO state to the LUMO electron state. It is known that in the internal environment of the FF PNT structures there is a rather strong internal total dipole momentum Dt and spontaneous polarization P, directed along the main axis of the FF PNT nanotube OZ (Bystrov et al. 2020, 2012, 2021; Zelenovskiy et al. 2019; Bystrov and Filippov 2022). This spontaneous polarization, which is similar in nature to ferroelectrics, also creates a significant electric field E P = P/εε0 (Bystrov 2018; Fridkin 1979) (where ε is dielectric permittivity of the FF PNT, which is ε = 4 for usual known protein structures, and ε0 is dielectric vacuum constant). As a result, a number of specific photoferroelectric and photovoltaic effects appear here, which are characteristic of ferroelectrics (Bystrov et al. 2012; Bystrov 2018; Fridkin 1979). Accordingly, this electric field will also affect the energy levels during photoexcitation of electrons, and as a result, the real (or effective) band gap changes and increases. Considering that one direction of polarization is important here (which can be considered as directed along the axis of the nanotube from one of its sides as from the wall perpendicular to this axis), we need to take half of the value of the total internal electric field created by polarization: Eg* = Eg0 + α*E P /2 = Eg0 + α*P/(2εε0 ); or ΔEg = α*E P /2. Therefore, the real data for Eg* observed after photoexcitation could be shifted for upper values. In our calculated cases (see data in Tables 7.1 and 7.2, and take E P in V/Å unit), this will be the following values: (1) for AM1/PM3 methods Eg01 * = 3.441 + 7.046*0.394/2 = ~ 4.829 eV, shift ΔEg01 ~ 1.388 eV for L-FF case; Eg02 * = 3.575 + 7.166*0.3875/2 = ~ 4.965 eV, shift ΔEg02 ~ 1.390 eV for D-FF case; (2) for PM7 method Eg03 * = 3.966 + 7.733*0.4092/2 ~ 5.548 eV, shift ΔEg03 ~ 1.582 eV for L-FF, Eg04 * = 3.981 + 7.733*0.4126/2 = ~ 5.576 eV, shit ΔEg04 ~ 1.595 eV for D-FF. Interestingly, and it should be emphasized here, these data correspond even more closely to the observed values in various experiments (Amdursky et al. 2009; Gan et al. 2013a, b; Nikitin et al. 2016; Akdim et al. 2015; Bystrov 2018). Next step connected with following photoluminescence processes. It should be noted here that as soon as the electron is in an excited state (LUMO or at the bottom of the conduction band), and a “hole” appears in the HOMO state, the inevitable separation of charges in the intrinsic electric field E e occurs at a distance r = z = c ~ 5.45 Å (and an exciton—pair electron–hole). At the same time, an electric

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field E Q = q/(4πεε0 r2 ) ∼ 1.2 GV/m = 0.12 V/Å also arises between these charges q (electron and hole), directed against the field E P /2 and this again affects the band gap—it decreases. Accordingly, photoluminescence occurs in a smaller emission of photon energy, which now turns out to be equal to less Eg** data. As a result, the total intrinsic electric field E e decreases up to value E e = E P /2−E Q : (1) for AM1/PM3 the average E e = E P /2−E Q ~ 0.195 V/Å−0.12 V/Å = ~ 0.075 V/Å and corresponding back “red” shift of the band gap Eg for this case is ΔEg12 ** = α1 *E e = 7.1 eV/(V/Å)*0.075 V/Å = ~ 0.53 eV; as results, the values of band gap Eg could be Eg01 ** = Eg01 * −ΔEg12 ** = 4.829 eV−0.53 eV = ~ 4.3 eV for L-FF case and similarly Eg02 ** = 4.965 eV−0.53 eV = ~ 4.43 eV for D-FF case; (2) for PM7 the average E e = E P /2−E Q ~ 0.205 V/Å−0.12 V/Å = ~ 0.085 V/Å and corresponding back “red” shift of the band gap Eg for this case is obtained as ΔEg34 ** = 0.085 V/Å*7.733 eV/(V/Å) = ~ 0.6573 eV; as results, the values of band gap Eg could be Eg03 ** = 5.548 eV–0.657 eV = ~ 4.891 eV for L-FF case and Eg04 ** = 5.576 eV–0.657 eV = ~ 4.919 eV for D-FF case. The values obtained of ΔEg12 ** ~ 0.53 eV and ΔEg34 ** ~ 0.6573 eV correspond in our case to the exciton binding energy (that is determined at the level of ~ 0.98 eV in work (Amdursky et al. 2009) and at the value of ~ 0.34 eV in Nikitin et al. (2016)). Corresponding change of the photoluminescense wave length can be estimated for AM1/PM3 data as “red” shift from of the average value (for both L-FF and DFF) from Eg12 * ~ 4.9 eV (that is equal to λ~254 nm) up to Eg12 ** ~ 4.3 eV (λ ~ 289 nm) with total wave length change on Δλ ~ 35 nm. The same estimation for PM7 data show value of the average “red” shift from E34 * ~ 5.55 eV (λ ~ 224 nm) up to E34 ** ~ 4.9 eV (λ ~ 254 nm) and Δλ ~ 30 nm total. All these data for photoluminescence phenomena are in line with various observed and theoretically estimated data (Amdursky et al. 2009; Gan et al. 2013a, b; Nikitin et al. 2016; Akdim et al. 2015; Bystrov 2018).

7.3 Conclusion The main new conclusion obtained from this our study is that these phenomena could be explained due the ferroelectric nature of the FF PNT structures, existing the intrinsic electric field arising from internal ferroelectric-like polarization inside FF PNT structures, and that the photoelectronic and photovoltaic processes occurring here can be described on the photoferroelectric phenomena base (Bystrov 2018; Fridkin 1979) in these ferroelectric nanomaterials. The established range of Eg = 3.1–4.9 eV for these FF PNT corresponds to the ultraviolet range ~ 400–253 nm

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and turns out to be in the solar-blind ultraviolet absorbed by ozone layer. Therefore, it is possible to use FF PNTs for solar-blind ultraviolet detection (“solarblind ultraviolet”—SBUV) is a new direction in the creation of such photodetectors. The resulting Eg changes in FF PNT can be adjusted also by the electric field E created by the polarization in layers of ferroelectric polymer PVDF and P(VDF-TrFE), similarly to photodetectors based on dichalcogenides (MoS2 type), with PVDF/P(VDF-TrFE) control layers (Wang et al. 2015). Thanks to the large photovoltaic effect in nanotubes FF PNTs new composite heterostructures based on FF PNT/PVDF/P(VDFTrFE)/Graphene can be used to create cost-effective self-powered and flexible ultraviolet photodetectors area solar-blind range (SBUV) (Bystrov et al. 2022). Similarly, various nanoscale devices based on such hybrid nanostructures FF PNT, PVDF/P(VDF-TrFE) and also graphene layers, can become prototypes of the photodetectors and other photosensitive elements for many new practical applications.

References Akdim B, Pachter R, Naik RR (2015) Self-assembled peptide nanotubes as electronic materials: an evaluation from first-principles calculations. Appl Pnys Lett 106:183707 Amdursky N, Molotskii M, Aronov DL et al (2009) Blue luminescence based on quantum confinement at peptide nanotubes. Nano Lett 9(9):3111–3115 Bystrov VS (2018) Photo-ferroelectricity in di-phenylalanine peptide nanotube. Comp Cond Matter 14:94–100 Bystrov VS, Filippov SV (2022) Molecular modelling and computational studies of peptide diphenylalanine nanotubes, containing waters: structural and interactions analysis. J Mol Mod 28:81 Bystrov VS, Paramonova E, Bdikin I et al (2012) BioFerroelectricity: diphenylalanine peptide nanotubes computational modeling and ferroelectric properties at the nanoscale (Review paper). Ferroelectrics 440(01):3–24 Bystrov VS, Coutinho J, Zhulyabina OA et al (2021) Modeling and physical properties of diphenylalanine peptide nanotubes containing water molecules. Ferroelectrics 574(1):78–91 Bystrov V, Coutinho J, Zelenovskiy P et al (2020) Structures and properties of the self-assembling diphenylalanine peptide nanotubes containing water molecules: modeling and data analysis. Nanomaterials 10(10):1999 Bystrov VS, Paramonova EV, Zelenovsky PS, Fridkin VM, Lin T, Shen H, Meng X (2022) Photoelectronic properties of diphenylalanine peptide nanotubes. In: Lakhno V (ed) Proceedings of the international conference “Mathematical Biology and Bioinformatics”. vol 9. IMPB RAS, Pushchino, Moscow Region. Paper No. e18. https://doi.org/10.17537/icmbb22.24 Fridkin VM (1979) Photoferroelectrics. Springer, New York Gan Z, Xinglong W, Zhu X, Shen J (2013a) Light-induced ferroelectricity in bioinspired selfassembled diphenylalanine nanotubes/microtubes. Angew Chem Int Ed Engl 52(7):2055–2059 Gan Z, Xinglong W, Zhang J et al (2013b) In Situ thermal imaging and absolute temperature monitoring by luminescent diphenylalanine nanotubes. Biomacromol 14:2112–2116 Gorbitz CH (2001) Nanotube formation by hydrophobic dipeptides. Chem Eur J 7:5153 HyperChem (2011) Tools for molecular modeling (Release 8.0/01 USB). Hypercube, Inc., Gainnesville, FL, USA Nikitin T, Kopyl S, Shur VYa et al (2016) Low-temperature photoluminescence in self-assembled diphenylalanine microtubes. Phys Lett A 380(18):1658–1662

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Stewart JJP (2016) Stewart computational chemistry. MOPAC2016 [Electronic resource], Colorado Springs, CO, USA. http://openmopac.net/MOPAC2016.html. Accessed 30 May 2022 Wang X, Wang P, Wang J et al (2015) Ultrasensitive and broadband MoS2 photodetector driven by ferroelectrics. Adv Mater 27:6575–6581 Zelenovskiy PS, Nuraeva AS, Kopyl S et al (2019) Chirality-dependent growth of self-assembled diphenylalanine microtubes. Cryst Growth Des 19(11):6414–6421

Chapter 8

Mechanical Design of a Thermo-Mechanical-Cryogenic System to Evaluate Mechanical Properties of Samples of 3D Printing Systems Víctor Daniel Rodríguez-Gaspar, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa, Juan Atonal-Sánchez, Sebastián Arturo Medinilla-García, Luis Héctor Hernández-Gómez, and Teresa Berenice Uribe-Cortés

8.1 Introduction The development of societies could not be understood without their historical relationship with the evolution of materials, which have served mainly to survive and improve their quality of life. In the evolutionary line, man has used, produced and shaped various types of materials to meet their needs and improve their conditions, starting from a natural origin to the implementation of new technological materials (stainless steel, polymers and nanocomposites). The implementation of these materials as final structural elements allows technological advance, generating better performance when used in specific applications, replacing materials such as concrete metals and glass, the field of research is very limited in relation to these implementations, specifically in composite 3D printing materials, this represents a challenge for obtaining data in V. D. Rodríguez-Gaspar · J. A. Beltrán-Fernández (B) · J. Atonal-Sánchez · S. A. Medinilla-García · L. H. Hernández-Gómez · T. B. Uribe-Cortés Instituto Politécnico Nacional—Escuela Superior de Ingeniería Mecánica y Eléctrica—Sección de Estudios de Posgrado e Investigación Edificio 5, Unidad Profesional Adolfo López Mateos “Zacatenco” Col. Lindavista, 2do Piso, C.P. 07738 Ciudad de México, México e-mail: [email protected] V. D. Rodríguez-Gaspar e-mail: [email protected] J. C. Hermida-Ochoa Centro de Investigación Y Laboratorio Biomecánico (CILAB), Del Carmen 18, Chimalistac, 01070 Ciudad de México, México © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_8

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relation to their mechanical properties under operating conditions where temperature plays a fundamental role as it has a direct influence on these properties.

8.2 Materials and Methods Thermoelectricity is a branch of thermodynamics superimposed on electricity, the best known phenomenon is that of electricity generated by the application of heat in the union of two different materials, if two wires of different material (thermocouple circuit) are joined at both ends and one of the joints is maintained with a temperature higher than the other, a voltage difference arises that causes an electric current to flow between the hot and cold junctions (Failed 2012). The main component of the cooling system consists of a Peltier module, whose operation is based on thermoelectric cooling which uses the Peltier effect to create a thermal flow through the union of two different materials, such as metals or semiconductors. Simply connecting it with a continuous voltage source causes cooling of one of the parts, while the other heats up (Julian Goldsmid 2016) (Fig. 8.1). The main objective of this work is to design and build a chamber for cooling specimens printed in 3D of polymeric material with the use of Peltier cells. This system has an automatic control that allows to maintain the desired temperature, for the realization of thermo-mechanical tests at sub-ambient temperatures.

Fig. 8.1 Peltier effect (Venkatesan and Venkataramanan 2020). Reprinted with permission from Springer Nature publishers

8 Mechanical Design of a Thermo-Mechanical-Cryogenic System … Fig. 8.2 Thermo-electric cooling device for thermo-mechanical testing on 3D printed specimens (Chen et al. 2020). Reprinted with permission from Springer Nature publishers

127

Circulation of

Dissipators

Cold Air

Hot side

Figure 8.2 shows the basic operating scheme of the thermoelectric system for controlling the internal temperature of the inside the chamber.

8.3 Cooling System Components The following lists contains the components of the cooling system: • Cabin: it is the place or space destined for the preservation of the desired temperature in our particular case. • Thermal insulator: Material used in the construction of the chamber, characterized by its high thermal resistance. It establishes a barrier to the passage of heat between two media that would naturally tend to equalize in temperature. • Heatsinks: Composed mainly of aluminum fins, a heatsink extracts heat from the component it cools and evacuates it to the outside, usually to the air. • Fans: this element is intended to remove excess heat from any component, transferring heat from the hot part to be dissipated into the air. • Peltier cell: Thermoelectric cooling device based on the Peltier effect. Figure 8.3 shows the module or Peltier assembly consisting of two aluminum heatsinks and two fans. These elements circulate the flow of cold air inside the device and extract the heat caused on the other side of the cell. This system has an automatic control that allows maintaining the desired temperature during the voltage test.

8.4 Materials and Methods The characteristics proposed for the system are the following: • Camera dimensions 20 × 15 cm base and 30 cm height. This is sufficient for implementation in the testing machine. • Temperature range between 20 and − 10 °C. • Internal temperature display and setpoint temperature control.

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Heat

Aluminium dissipators

Cold

Peltier Fans

Fig. 8.3 Module or Peltier assembly

• Control of the Peltier cell by means of a thermostat, start and stop at a certain temperature. • Possibility of maneuvering for stress tests. The materials that were used in the realization of the housing of the mini cold room will be listed below, which will serve as an insulating auxiliary, since the material used has these important characteristics (fiberglass). The main technical properties of fiberglass that were used are: • Insulation: fiberglass is a well-known insulating material. In this way, it avoids electrical conduction and “sparkling”. These two characteristics make its use very safe in electrical installations or areas with combustible materials nearby. • Strength: Fiberglass has a higher specific strength (tensile strength/volumetric mass) than steel. This characteristic is the main reason for the use of fiberglass in the production of high-performance composites. • Corrosion resistance: Fiberglass reinforced with plastic and makes it resistant to corrosion caused by chemical agents (such as oils and solvents) or salt water. That is why it is recommended for use in most industrial areas (Zu et al. 2021). To carry out the tests it is necessary to implement a fastening system for 3D printed specimens (auxiliary jaws) that in relation to the size of the original jaws allows us a better use of the cryogenic environment. The adaptation of the auxiliary jaws and the complete assembly of the system can be observed in the universal testing machine (Figs. 8.4, 8.5 and 8.6).

8 Mechanical Design of a Thermo-Mechanical-Cryogenic System …

a)

129

b)

Fig. 8.4 Auxiliary jaws with specimen

Fig. 8.5 Complete and interior assembly of the cold room

For the installation of the specimen in the auxiliary jaws it is recommended to adapt separately from the machine and tighten the screws in order to avoid slippage between jaws and specimen (Fig. 8.4b). Figure 8.5 represents the installation of the adjustable base where one will place the cooling chamber, trying to adjust all the components, as well as the installation of the set of auxiliary jaws previously assembled with the specimen to be tested. It is recommended at this point to take the appropriate precautions, and any accidental bending of the specimen could affect the properties of it. The lower jaw is installed by inserting it into the lower hole of the cold room and inserting it into the original

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Fig. 8.6 Detailed view: device elements

jaw of the testing machine, focusing very carefully on this step the set of auxiliary and main jaws. Once all the fastening elements of the thermoelectric cooling device have been adjusted, it is closed and turned on, the whole assembly works with a 12 V DC supply.

8.5 Results Once the cooling system is perfectly arranged, it proceeds to perform stress tests at controlled temperatures starting from 20 °C decreasing until reaching the target of −10 °C, with an operating range of 5 in 5° in descending order for each test with a total of 5 specimens per sample according to ASTM DS638 resulting in stress–strain graphs, and load-elongation.

8.5.1 Test Results on 3D Printing Specimens in the Thermoelectric Cooling Chamber After performing the stress tests, results were obtained and recorded in the computer of the universal test machine, which were then graphed to observe the thermomechanical behavior of PLA and ABS at different temperatures starting from 20 °C to a temperature of − 10 °C. The tests shown in Fig. 8.7, performed at 15 °C with constant velocity compared to the tests performed at 20 °C, show as a main characteristic the beginning of increased yield stress, by an average of 5 MPa compared to tests at 20 °C.

8 Mechanical Design of a Thermo-Mechanical-Cryogenic System …

Fig. 8.7 Stress-deformation graph of specimens at 20 and 15 °C (PLA)

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In Fig. 8.8 the decrease in the plastic zone is remarkable, as well as the constant increase in the yield stress.

Fig. 8.8 Stress-deformation graph of specimens at 10 and 5 °C (PLA)

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The graphs of Fig. 8.9 show a percentage of unit deformation higher than the graphs of the previous tests, however the considerable decrease experienced by the plastic zone is also observed, these results were obtained by tests carried out at 0 and − 5 °C.

Fig. 8.9 Stress-deformation graph of specimens at 0 and −5 °C (PLA)

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In Fig. 8.10 it is noticeable the constant behavior of the material subjected to a temperature of −10 °C where it is remarkable how in some cases the plastic zone decreases in such a way that the yield stress and the ultimate stress converge in the same area of the graph. Results and graphs obtained from the tests carried out on 3D compression specimens in ABS (Fig. 8.11). Tests performed on ABS specimens in the temperature ranges of 20 and 15 °C compared to PLA tests show evidence of the higher ductility of the material analyzed. At this operating temperature (5 °C) for the performance of the ABS specimen tests, a decrease in the plastic zone and an increase in yield stress can be observed in Fig. 8.13. It is at this point where it can be seen the constant behavior in the elastic zone of the material up to approximately 30 MPa, except for a particular case of test, it is observed how the plastic zone decreases to such an extent that they tend to converge the yield stress and the ultimate or breaking stress.

Fig. 8.10 Stress-deformation graph of specimens at −10 °C (PLA)

8 Mechanical Design of a Thermo-Mechanical-Cryogenic System … Fig. 8.11 Stress-deformation graph of specimens at 20 and 15 °C (ABS)

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Fig. 8.12 Stress-deformation graph of specimens at 10 and 5 °C (ABS)

8 Mechanical Design of a Thermo-Mechanical-Cryogenic System …

Fig. 8.13 Stress-deformation graph of specimens at 0 and −5 °C (ABS)

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In Fig. 8.14 a significant increase in yield stress, being a material with greater ductility than the PLA material at the lowest application temperature and no convergences are observed between the yield stress and the ultimate stress. Tables 8.2 and 8.3 show the comparative load–displacement graphs. Tables 8.2 and 8.3 show the results of the averages of the tests of PLA and ABS materials, which were subjected to a decrease in temperature using the cold room. In the past graphs it is possible to see how the specimens of both materials begin to transit or change to a state of glass transition, that is, they present a change from a ductile to brittle behavior.

Fig. 8.14 Stress-deformation graph of specimens at −10 °C (ABS)

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139

Table 8.1 Main materials for the elaboration of the refrigeration system (a) Casing

(b) Peltier module

Polyester resin

Peltier TEC 12,715 Cell

Fiberglass

Aluminium heatsinks

Catalyst

Fans

Industrial talc

Thermal plaster paste

Calcite

Installation cable

Cardboard

Power supply with 12 V DC output

Brush

W1209 digital thermostat (temperature control.)

Containers

–

Aluminium insulation material

–

Glue

–

Scissors

–

Table 8.2 Results of PLA tests Temperature °C

Yield stress (Sand ) MPa

Ultimate stress (Su ) MPa

Modulus of elasticity (E) MPa

Elongation (L) mm

Strain (ε) %

20

43.99

40.54

922.222

1.133

4.51

15

48.61

45.19

612.216

1.137

7.94

10

50.31

48.94

582.291

1.355

9.39

5

56.35

53.30

552.450

1.458

10.20

0

63.65

61.29

461.566

1.972

13.79

−5

67.61

65.95

557.742

1.734

12.14

−10

63.06

61.35

519.012

1.763

12.15

Table 8.3 Results of ABS tests Temperature °C

Yield stress (Sand ) MPa

Ultimate stress (Su ) MPa

Modulus of elasticity (E) MPa

Elongation (L) mm

Strain (ε)%

20

35.72

29.13

502.309

1.019

7.11

15

30.25

23.43

490.275

0.873

6.17

10

24.68

18.37

496.579

0.713

4.97

5

39.05

32.54

601.694

0.9353

6.49

0

36.16

31.15

554.601

0.9356

6.52

−5

45.81

33.32

562.776

1.166

8.14

−10

41.15

30.10

519.570

1.152

7.90

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8.6 Conclusions The thermo-electric cooling device operates adequately in the contemplated temperature ranges 20–10 °C allowing thermo-mechanical testing in a controlled environment and obtaining its main properties from 3D printing specimens (PLA and ABS). A problem was identified with the clamping system (jaws) in the universal testing machine for 3D printing specimens. The adaptation of auxiliary jaws allowed a better use of the decrease in temperature (Fig. 8.3). The data presented in this work of ABS and PLA printed materials are a fundamental basis for the characterization of the behavior of composite materials at sub-ambient temperatures, allowing a better understanding for designers. To follow up on the project, the objective is the implementation of an extra cooling module for a greater decrease in temperature inside the device and the performance of thermo-mechanical tests on composite materials, as well as the application of a 3D printing pattern (Voronoi) to the specimens to optimize the printing geometry, retaining a degree of utility in relation to its mechanical capacity. Finally, the use of thermomechanical chambers allows to analyze the main geometric changes and influencing on the mechanical behavior and this project will continue performing the environment and thermal parameters [6].

References Bhushan B et al (2012) Peltier effect. In: Encyclopedia of nanotechnology. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9751-4_100632 Chen L, Meng F, Xie Z et al (2020) Thermodynamic modeling and analysis of an air-cooled small space thermoelectric cooler. Eur Phys J plus 135:80. https://doi.org/10.1140/epjp/s13360-01900020-3 Julian Goldsmid H (2016) In: Introduction to thermoelectricity. Springer, Berlin, Heidelberg.https:// doi.org/10.1007/978-3-662-49256-7 Venkatesan K, Venkataramanan M (2020) Experimental and simulation studies on thermoelectric cooler: a performance study approach. Int J Thermophys 41:38. https://doi.org/10.1007/s10765020-2613-2 Zu Q, Solvang M, Li H (2021) Commercial glass fibers. In: Li H (eds) Fiberglass science and technology. Springer, Cham. https://doi.org/10.1007/978-3-030-72200-5_1

Chapter 9

Investigation of Local Discontinuous Galerkin Method on the Solution of Convection–Diffusion Problems E. V. Shilnikov and I. R. Khaytaliev

9.1 Introduction Nowadays, the Galerkin method with discontinuous basis functions (DGM) is widely used to solve problems of computational gas dynamics (Cockburn 1998; Cockburn and Shu 2001, 2005). This method is characterized by a high order of accuracy on smooth solutions. As is known, in order to ensure the monotony of the solution obtained by this method, that it is necessary to introduce so-called slope limiters, especially if the solution contains strong discontinuities, such as, for example, shock waves. However, the use of limiters can negatively affect the accuracy of the resulting solution. Therefore, the study of preserving the order of accuracy of the solution and ensuring the monotony of the solution remains relevant now. The issues of the influence of limiters on maintaining or decreasing the order of accuracy and monotony of the solution are investigated in (Ladonkina et al. 2012, 2013, 2014). DGM faces problems that are even more serious when modeling viscous flows based on the Navier– Stokes system of equations, which, unlike Euler’s equations, contains second order spatial derivatives. In this case, the fluxes at the cell boundaries contain derivatives of the desired functions. This causes serious difficulties in their approximation with the help of basic functions having discontinuities at the boundaries of the computational grid cells. In this regard, a class of DGM was proposed for solving partial differential equations (PDEs) with higher derivatives, which was called the local discontinuous Galerkin (LDG) method. The idea of the LDG methods is to rewrite high-order PDEs appropriately into a system of first-order equations, and then apply the classical DG I. R. Khaytaliev State Technical University MADI (STU-MADI), Moscow, Russia e-mail: [email protected] E. V. Shilnikov (B) Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_9

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method to this system. As a result, there are no derivatives of the desired variables in the expressions for fluxes at the cell boundaries. There is an opinion that with this approach, limiters are not necessary. The first LDG method was designed to solve a convection diffusion equation (containing second derivatives) in (Cockburn and Shu 1998). Later, this approach was generalized to the cases of equations of higher orders (Yan and Shu 2002a, b). Usually, the functions of the Taylor basis {(x − xc )n , n = 0, 1, 2 . . .} are chosen as the basis functions in each grid cell. Here x c is the center of grid cell. However, other basis functions are also used, such as exponential or trigonometric functions (Yuan and Shu 2006). Note that as a result of using the LDG method, a system of time-dependent ODEs is obtained, which is solved by the Runge–Kutta method. This system has the simplest form in the case of an orthogonal system of basis functions. The aim of this work is to study the accuracy and stability of the LDG method by the example of solving linear and nonlinear convection–diffusion problems. An orthogonal system of Legendre polynomials is chosen as a system of basis functions. The cases of both continuous and discontinuous solutions of the problem are considered. In the future, it is proposed to apply the LDG method to solving problems of modeling gas mixtures flows based on a quasi-gas dynamic (QGD) system of equations.

9.2 Linear Convection–Diffusion Equation The development of the local discontinuous Galerkin method at the first stage is carried out by the example of solving the initial boundary value problem for the linear convection–diffusion equation: ∂u ∂ 2u ∂u +c∗ − a ∗ 2 = 0, ∂t ∂x ∂x

(9.1)

u(x, 0) = F(x),

(9.2)

u(x L , t) = v L (t),

(9.3)

u(x R , t) = v R (t),

(9.4)

t ∈ (0, T ), x ∈ (x L , x R ).

(9.5)

Here a ≥ 0, c ≥ 0 are constants; v L (t) and v R (t) are boundary conditions; x L = 0, x R = 2π ; t is time; x is spatial coordinate; u(x, t) is the desired solution.

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

143

To begin with, we transform a second-order equation into a system of first-order equations. To do this, we introduce a new variable: q(x, t) = −cu + a

∂u . ∂x

(9.6)

Then (9.1) takes the form: ∂u ∂q = . ∂t ∂x

(9.7)

Equations (9.6) and (9.7) form a system that we will solve. Note that in this case q(x, t = 0) = −c ∗ F(x) + a ∗ F ' (x).

(9.8)

In the computational domain x ∈ (0, 2π ), we introduce a] difference grid 0 = [ x0 < x1 < . . . < x M = 2π. On the segment Ik : xk , xk+1 , k = 0, M − 1 we are making a replacement x = xk + h2k (ξ + 1), where h k = xk+1 − xk . Then Ik is displayed in a segment −1 ≤ ξ ≤ 1, and the system (9.6–9.7) takes the following form on this segment: 2 ∂q ∂u = , ∂t h k ∂ξ q = −cu + a

2 ∂u . h k ∂ξ

(9.9) (9.10)

Let ϕ j (ξ ) is a basis in the space of functions continuous on the segment [−1, 1]. We are looking for an approximate solution of the system (9.9–9.10) in the form of a projection of the exact solution onto a linear shell of a finite number of the first basis functions: uk =

N ∑

α kj (t) ∗ ϕ j (ξ ),

(9.11)

β kj (t) ∗ ϕ j (ξ ).

(9.12)

j=0

q = k

N ∑ j=0

Multiplying each equation of this system by ϕi (ξ ) and integrating it by ξ from −1 to 1, we obtain a system of equations connecting the coefficients α kj (t) and β kj (t).

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∂ ∂t

∫1 −1

2 uϕi dξ = hk ∫1 (

∫1 qϕi dξ = −1

−1

∫1 −1

∂q ϕi dξ , ∂ξ

) 2 ∂u −cu + a ϕi dξ . h k ∂ξ

(9.13)

(9.14)

Consider Eq. (9.13). Substituting representations (9.11) and (9.12) into it, for the left part we get: ∂ ∂t

∫1 −1

⎛ ⎝

N ∑

⎞ α kj (t) ∗ ϕ j (ξ )⎠ ∗ ϕi dξ

j=0

∫ N ∂ ∑ k α j (t) ϕ j (ξ )ϕi (ξ )dξ = ∂t j=0 1

−1

=

∂ ∂t

N ∑

α kj (t) ∗ Di j .

(9.15)

j=0

In the right part there is a derivative of the function, represented as a series. It is known that differentiation worsens the convergence of series. Therefore, in order to avoid differentiation of the series (9.12), we apply integration in parts: ∫1 −1

)|1 ( ∂q ϕi dξ = q k ϕi |−1 − ∂ξ

= q+k ϕi (1) − q−k ϕi (−1) −

∫1 ∑ N

β kj (t)ϕ j ϕi' dξ

−1 j=0

N ∑

β kj (t) ∗ Ri j .

(9.16)

j=0

∫1 ∫1 Here Di j = −1 ϕi (ξ )ϕ j (ξ )dξ , Ri j = −1 ϕ j (ξ )ϕi' (ξ )dξ . As a result, instead of Eq. (9.13), we obtain the following system (i = 0, N ): N N ∑ hk ∂ ∑ k α j (t) ∗ Di j = q+k ϕi (1) − q−k ϕi (−1) − β kj (t) ∗ Ri j . 2 ∂t j=0 j=0

(9.17)

Having carried out similar transformations with Eqs. (9.14), we get instead the equations:

9 Investigation of Local Discontinuous Galerkin Method on the Solution … N ∑ j=0

β kj (t) ∗ Di j = −c

N ∑

145

α kj (t) ∗ Di j

j=0

⎛

⎞ N ∑ 2⎝ k +a α kj (t) ∗ Ri j ⎠. u + ϕi (1) − u k− ϕi (−1) − hk j=0

(9.18)

The system (9.17–9.18) has the simplest form in the case of an orthogonal basis. Therefore, as the basis functions ϕ j (ξ ) we choose an orthogonal system of Legendre polynomials P j (ξ ) on the segment [−1, 1]. P0 = 1, P1 = ξ, Pn+1 (ξ ) =

n 2n + 1 ξ ∗ Pn (ξ ) − ∗ Pn−1 (ξ ). n+1 n+1

∫1 Since Di j = −1 Pi (ξ )P j (ξ )dξ = obtain the basic equations:

2δi j , 2i+1

(9.19)

P j (1) = 1, P j (−1) = (−1) j , we

∑ hk ∂ k αi (t) = q+k − q−k (−1)i − β kj (t) ∗ Ri j , 2i + 1 ∂t j=0 N

⎛ ⎞ N ∑ 2i + 1 ⎝u k+ − u k− (−1)i − βik = −c ∗ αik (t) + a α kj (t) ∗ Ri j ⎠. hk j=0

(9.20)

(9.21)

For derivatives of Legendre polynomials, we can write the recurrence relation (Koshlyakov et al. 1964): ' ' Pn+1 (ξ ) − Pn−1 (ξ ) = (2n + 1)Pn (ξ ).

(9.22)

' If we move Pn−1 (ξ ) to the right side of the equation, multiply both parts by Pl (ξ ) and integrate from –1 to 1, we get:

∫1 Rn+1,l = (2n + 1)

Pn (ξ )Pl (ξ )dξ

−1

∫1 +

' Pn−1 (ξ )Pl (ξ )dξ

−1

= 2δn,l + Rn−1,l , l ≤ n.

(9.23)

For l > n Rn+1,l = 0. Given that R0,l = 0 and assuming formally R−1,l = 0 for all l, we have a recurrent formula for calculating Rn,l .

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The values of the functions at the cell boundaries are calculated as follows, using the representations (9.11) and (9.12): u k (xk ) + u k−1 (xk ) u k− = 2 ∑ ∑N k α (t) ∗ (−1) j + Nj=0 α k−1 j=0 j j (t) , = 2 q k (xk ) + q k−1 (xk ) q−k = 2 ∑N ∑N k−1 j k (t) j=0 β j (t) ∗ (−1) + j=0 β j . = 2

(9.24)

The integration of the system of Eqs. (9.20–9.21) is carried out by the second order Runge–Kutta method. Thus, the general algorithm for solving the problem looks like this: (1) For t = 0: calculate all αik (0) and βik (0) for k = 0, M − 1 from the initial conditions (9.2) and (9.6) by the formulas: ∫1 F(xk + −1

hk 2 (ξ + 1))ϕi dξ = αk , 2 2i + 1 i

∫1 (−c ∗ F(xk + −1

+ a ∗ F ' (xk +

(9.25)

hk (ξ + 1)) 2 hk 2 (ξ + 1)))ϕi dξ = βk . 2 2i + 1 i

(9.26)

(2) Calculate the values of q at time t at the inner boundaries of the cells according to the formulas (9.24). We take the values at the boundaries of the computational domain from the boundary conditions: q−0 =

N ∑

β 0j (t) ∗ ϕ j (−1),

j=0

q+M−1 =

N ∑

β jM−1 (t) ∗ ϕ j (1).

(9.27)

j=0

(3) The first stage of the Runge–Kutta method: we find the values of all coefficients α˜ ik at a time t + τ/2, where τ is the time step:

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

⎛ ⎞ N ∑ 2i + 1 τ ⎝q+k − q−k (−1)i − α˜ ik = αik + β kj (t) ∗ Ri j ⎠. 2 hk j=0

147

(9.28)

Recalculate u k− by formulas (9.24) with new values α˜ ik , and at the boundaries by formulas u 0− = v L (t), u +M−1 = v R (t) for t + τ/2. Now, using the formula (9.21), you can calculate all β˜ik . ⎛ ⎞ N ∑ 2i + 1 ⎝u k+ − u k− (−1)i − α˜ kj (t) ∗ Ri j ⎠. β˜ik = −c ∗ α˜ ik + a hk j=0

(9.29)

And recalculate q−k by formulas from (9.24) with new values β˜ik , and at the boundaries by formulas (9.27) for t + τ/2. Now we have all the values u and q on the borders of the cells at the time t + τ/2 (values with a tilde). (4) The second stage is the Runge–Kutta method. Similarly, using formulas (9.30) and (9.31), we calculate all the values at the time t + τ (values with a lid). ⎛ ⎞ N ∑ 2i + 1 ⎝q+k − q−k (−1)i − β˜ kj (t) ∗ Ri j ⎠, αˆ ik = αik + τ hk j=0 ⎛ ⎞ N ∑ 2i + 1 ⎝u k+ − u k− (−1)i − βˆik = −c ∗ αˆ ik + a αˆ kj (t) ∗ Ri j ⎠. hk j=0

(9.30)

(9.31)

(5) After that, it is necessary to update all the values αik and βik , this is the end of the time cycle. The values u and q have already been updated during calculations 4).

9.2.1 Linear Convection–Diffusion Equation: Continuous Case Consider the case of a continuous initial condition: F(x) = sin(x).

(9.32)

We establish the following boundary conditions: v L (t) = v R (t) = −e−at sin(ct).

(9.33)

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Then our problem has the following analytical solution: u(x, t) = e−at sin(x − ct).

(9.34)

q(x, t = 0) = −c ∗ sin(x) + a ∗ cos(x).

(9.35)

Note that in this case

From (9.25) and (9.26) we obtain for the first coefficients: 2α0k =

( ) 2xk + h k hk 4 sin sin , hk 2 2

2 k 4 2xk + h k 2xk + h k hk 4 hk cos α = 2 cos sin − cos , 3 1 2 2 hk 2 2 hk hk a hk 2xk + h k 2xk + h k c sin + 4 cos sin , sin hk 2 2 hk 2 2 ) ( ) ( xk 2 k xk β1 = c −4 cos + 2 sin xk − a sin − 2 cos xk . 3 hk hk

2β0k = −4

(9.36) (9.37) (9.38) (9.39)

This problem has an analytical solution (9.34), therefore, substituting it into (9.6), it is possible to obtain exact values at the boundaries: q−0 = q−M−1 = e−a∗t (c ∗ sin(c ∗ t) + a ∗ cos(c ∗ t)).

(9.40)

The calculation result for M = 100, τ = 0.000494 at parameters a = 1 and c = 1 is shown in Fig. 9.1. Comparison with the exact solution (9.34) was carried out at the points xk + h k /2, k = 0, M − 1 at which the obtained numerical solution u numerical = α0k . The error in the norm C for this case was δ = 6.2 ∗ 10−5 . With increasing spatial step, the error increases almost quadratically, in particular: with M = 50 it turned out δ = 2.48 ∗ 10−4 , with M = 25 it turned out δ = 9.85 ∗ 10−4 . Thus, it can be concluded that the local discontinuous Galerkin method provides good accuracy for a problem with a continuous initial condition. Figure 9.2 shows the results of the algorithm stability study. For different values of the dissipation parameter a and the speed of sound c, the dependences in the double logarithmic scale of the maximum allowable time step τ on the number of spatial grid cells M are given. At a = 0, c = 1, this dependence, shown in Fig. 9.2 (1), turned out to be almost linear. This seems natural, since Eq. (9.1) in this case becomes the transfer equation. With a small dissipation parameter, the linear nature of the dependence is violated, but in general, at a = 0.01, it does not change very significantly. Figure 9.2 (2) corresponds to this case. With a further increase in the parameter a (the parameter c does not change at the same time), the maximum allowable values τ decrease

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

149

Fig. 9.1 Continuous initial condition. Comparison of numerical and exact solutions for T = 2; M = 100; τ = 0.000494

Fig. 9.2 Continuous initial condition. The dependence of the maximum τ, at which stability is maintained on the number of cells M

faster with growth M. At a = 1, the dependence becomes almost quadratic. This case is shown in Fig. 9.2 (3). For comparison, Fig. 9.2 (4) presents the results of calculations for c = 0, the case in which Eq. (9.1) becomes the equation of thermal conductivity. The graphs in Fig. 9.2 of (3) and 2 (4) practically coincide. Note that the characteristic condition for the stability of explicit schemes for a parabolic equation is just the condition τ ∼ h 2 . Thus, we can conclude that the stability condition of the algorithm used adapts to the type of equation. In the case of prevailing convection, it turns out to be close to the characteristic condition for hyperbolic equations, and in the case of strong diffusion—for parabolic ones.

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9.2.2 Linear Convection–Diffusion Equation: Discontinuous Case Now consider the case of a discontinuous initial condition: ⎧ ⎪ ⎨ 0.2, x < 1; F(x) = 1, 1 ≤ x ≤ 4; ⎪ ⎩ 0.2, x > 4.

(9.41)

Replacing the variables y = x − c ∗ t, τ = t in Eq. (9.1) turns it into the equation of thermal conductivity. The analytical solution of the Cauchy problem for this equation with the initial condition (9.41) is given by the Poisson integral. Thus, the analytical solution for the problem (9.1–9.2) has the following form: ) ( 4−x +c∗t 1 − 0.2 1−x +c∗t . er f u ex (x, t) = 0.2 + − er f √ √ 2 2 at 2 at

(9.42)

We are looking for a solution in the domain 0 ≤ x ≤ 8, 0 < t ≤ T . Then expression (9.42) gives an analytical solution to the initial boundary value problem (9.1–9.4) under the following boundary conditions: ν L (t) = u ex (0, t), ν R (t) = u ex (8, t). However, if you choose T = 2, then the perturbation practically does not reach the boundary and the boundary conditions can be considered constant and equal: u(0, t) = u(8, t) = 0.2 and q(0, t) = q(8, t) = −c ∗ 0.2. From (9.30) and (9.31) we obtain for the first coefficients: α0k = u(x, 0),

(9.43)

α1k = 0,

(9.44)

β0k = q(x, 0),

(9.45)

β1k = 0.

(9.46)

As can be seen from the calculation results presented in Figs. 9.3, 9.4 and 9.5, the solution of the convection–diffusion Eq. (9.1) with a discontinuous initial condition strongly depends on the diffusion parameter a, in contrast to the case of a continuous initial condition, in which the differences are not so great. With an increase in the a parameter, the initial vertical boundaries of the perturbation “blur” more and more, and the solution differs more and more from the solution of the transfer equation—Eq. (9.1) at a = 0. At the same time, the shape of the perturbation remains symmetrical relative to its middle, and the velocity of the perturbation turns out to be correct.

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

151

Fig. 9.3 Discontinuous initial condition. Comparison of numerical and exact solutions at T = 2; M = 640; τ = 0.00001; a = 0.1; c = 1

Fig. 9.4 Discontinuous initial condition. Comparison of numerical and exact solutions at T = 2, M = 640, τ = 0.0001201, a = 0.01, c = 1

Fig. 9.5 Discontinuous initial condition. Comparison of numerical and exact solutions at T = 2, M = 640, τ = 0.000407, a = 0.001, c = 1

In the same way as it was done in the previous section, a study of the stability of the scheme was carried out. Its results are shown in Fig. 9.6. In general, the stability condition turned out to be approximately the same as in the case of continuous initial data. Thus, for sufficiently large values of a, the dependence of the maximum allowable τ on the spatial step is almost quadratic (see Fig. 9.6a). With a decrease in a, the dependence becomes more complex, but on detailed spatial grids it also becomes almost linear (the right parts of the graphs in Fig. 9.6b and c). The absolute values of τ at large M in these figures are noticeably larger than in Fig. 9.6a. The non-monotonic behavior of graphs at small M is explained, apparently, by the method of determining the maximum allowable value of τ. We

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Fig. 9.6 The dependence of the maximum τ, at which stability is maintained on the number of cells M. The parameter c = 1

demanded the absence of oscillations. On a coarse grid, the accuracy of the solution is low, and oscillations can occur not because of instability, but because of poor quality of the solution. It is not easy to distinguish these two cases, so the allowable τ for small M numbers are underestimated. Figures 9.7 and 9.8 shows the dependence of the absolute error (δ) on the number of grid cells (M) for a different number of basis functions. Error in the norm C is presented in Fig. 9.7 and in the norm L 1 in Fig. 9.8. It is seen that the error decreases both with an increase in the number of grid cells and with an increase in the number of basis functions. Thus, the LDG method also shows good accuracy for the case of a discontinuous initial condition. Fig. 9.7 The dependence of the error δ in the norm C on the number of cells M for different number of basis functions N . Parameters T = 2.0, a = 0.001, c = 1

Fig. 9.8 The dependence of the error δ in the norm L 1 on the number of cells M for different number of basis functions N . Parameters T = 2.0, a = 0.001, c = 1

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

153

9.3 Burgers’ Equation with Initial and Boundary Conditions The development of the local discontinuous Galerkin method at the second stage is carried out by the example of solving the initial boundary value problem for the quasi-linear Burgers’ equation: ∂u ∂ 2u ∂u +u∗ − a ∗ 2 = 0, ∂t ∂x ∂x

(9.47)

u(x, 0) = F(x),

(9.48)

u(−10, t) = v L (t) = 0,

(9.49)

u(10, t) = v R (t) = 0,

(9.50)

t ∈ (0, T ), x ∈ (−10, 10).

(9.51)

Here a ≥ 0 is the viscosity parameter; v L (t) and v R (t) are boundary conditions; t is time; x is the spatial coordinate; u(x, t) is the desired solution. In formula (9.48), an initial condition is given, which has the form: ⎧ 0, x ∈ [−10, −1); ⎪ ⎪ ⎪ ⎨ x + 1, x ∈ [−1, 0]; F(x) = ⎪ −x + 1, x ∈ (0, 1]; ⎪ ⎪ ⎩ 0, x ∈ (1, 10].

(9.52)

The Burgers’ equation can be linearized by the Hopf-Cole transform (Liu 2017). To do this, you need to replace the function: u=a

wx ∂ ln w =a . ∂x w

(9.53)

As a result of this substitution, Eq. (9.47) turns into a thermal conductivity equation, whose solution is written as a Poisson integral: 1 w(x, t) = √ 2 aπ Here

∫+∞ (x−y)2 e− 4at w0 (y)dy. −∞

(9.54)

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w0 (x) = w(x, 0) = e a

∫x −∞

u 0 (y)dy

.

(9.55)

Substituting Eq. (9.54) into Eq. (9.53), we obtain an analytical solution of the Cauchy problem for Eq. (9.47). We are looking for a solution in the domain −10 ≤ x ≤ 10, 0 < t ≤ T . Provided that the perturbation does not reach the boundary of the domain, the solution obtained as a result of substituting (9.54) into (9.53) practically coincides with the analytical solution of the initial boundary value problem (9.47–9.50) under the following boundary conditions: ν L (t) = 0, ν R (t) = 0 for T < 1 (before the onset of the gradient catastrophe). We transform the second-order Eq. (9.47) to a system of first-order equations. To do this, we introduce a new variable: q(x, t) = −

∂u u2 +a∗ . 2 ∂x

(9.56)

Then we proceed in the same way as in Chap. 2. Equations (9.7) and (9.56) form a system that we solve. Note that in this case q(x, t = 0) = −

F 2 (x) + a ∗ F ' (x), 2

(9.57)

and Eq. (9.14) is replaced by ∫1 −1

) ∫1 ( 2 u 2 ∂u − +a∗ ϕi dξ . qϕi dξ = 2 h k ∂ξ

(9.58)

−1

Then Eq. (9.18) takes the form N ∑

β kj (t) ∗ Di j = Q ik

j=0

⎛ ⎞ N ∑ 2⎝ k +a α kj (t) ∗ Ri j ⎠. u + ϕi (1) − u k− ϕi (−1) − hk j=0

(9.59)

Here ⎛ ⎞2 ⎞ ⎛ ∫1 ( 2 ) ∫1 N ∑ u 1 − ϕi dξ = ⎝− ⎝ α k (t) ∗ ϕ j (ξ )⎠ ⎠ϕi dξ . Q ik = 2 2 j=0 j −1

Instead of Eq. (9.21) we get

−1

(9.60)

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

⎛ ⎞ N ∑ 2i + 1 ⎝u k+ − u k− (−1)i − βik = Q ik + a α kj (t) ∗ Ri j ⎠. hk j=0

155

(9.61)

The integration of the system of Eqs. (9.20) and (9.61) in time is carried out in the same way as in the case of a linear equation. The numerical solutions for two different time moments are shown in Figs. 9.9 and 9.10. Calculations were held with N = 1. In this figures, the legend contains information about the area under the curve. The conservation of this domain corresponds to the conservation law, which is valid for the Burgers’ equation in differential form. Up to the moment of time t = 1 corresponding to the moment of the gradient catastrophe, we have an analytical solution of the problem. For the moment of time shown on the Fig. 9.9, the error of the solution is about 6.5 ∗ 10−4 in the norm C. In the Fig. 9.10 the solution is presented for time moment t = 10. The vertical line corresponds to the theoretical position and amplitude of the shock wave, if the viscosity parameter a is equal to zero. The stability investigation results for the quasi-linear case is presented in Fig. 9.11. One can see from the results that τ is proportional to the spatial step to an almost quadratic degree for the viscosity parameter equal to one. For the case of prevailing convection (the viscosity parameter equal to 0.001) this dependence is almost linear. Figure 9.12 presents solutions for different values of the viscosity parameter a. In Fig. 9.12a, the parameter a = 0.1. In Fig. 9.12b, the parameter a = 0.01. In Fig. 9.9 The solution profile before gradient catastrophe. Kinematic viscosity a = 0.01; number of cells M = 500; time step τ = 0.01

Fig. 9.10 The solution profile after gradient catastrophe. Kinematic viscosity a = 0.01; number of cells M = 500; time step τ = 0.01

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Fig. 9.11 The dependence of the maximum τ, at which stability is maintained, on the number of cells M

Fig. 9.12c, the parameter a = 0.001. It can be seen that the larger the viscosity parameter, the more blurred the solution profile turns out to be. Figures 9.13 and 9.14 shows graphs with different numbers of basic functions. Figure 9.13 with parameters M = 640, a = 0.01, τ = 0.001, T = 0.7. Figure 9.14 with parameters M = 500, a = 0.01, τ = 0.0001, T = 10. It is shown that solutions for different number of basis functions coincide well for both time points T.

9.4 Equation System for Gas Mixture Based on a model from the paper of Elizarova and Shilnikov (2021), we consider a mixture consisting of two gases. If there are more gases, then more continuity equations will be added. External force F and external heat sources Q are considered equal to 0. ∂ρa (u − w) ∂ρa + = 0, ∂t ∂x ∂ρb ∂ρb (u − w) + = 0, ∂t ∂x ∂ρu ∂ ∂∏ ∂p + = , (ρ(u − w)u) + ∂t ∂x ∂x ∂x ∂E ∂ ∂∏u ∂q + + . ((E + p)(u − w)) = − ∂t ∂x ∂x ∂x

(9.62)

These are the terms of the equation that differ from the Navier Stokes equation. ( ) ( ) ∂p τ ∂u ∂p τ ∂ρu 2 + , wˆ = ρu + , q = qN S + q τ , w= ρ ∂x ∂x ρ ∂x ∂x ( ) ∂p ∂u ∂w +τ u +γp , ∏ = ∏ N S + ρu ∂x ∂x ∂x

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157

Fig. 9.12 The numerical solutions for N = 1 with different a at M = 500, τ = 0.001,T = 0.7

) ∂u 1 ∂ε +p . − q = τρu u ∂x ∂x p τ

(

(9.63)

A single-fluid model of a mixture of gases is considered, that is, it has a uniform velocity and temperature but different densities. The gas components and the mixture satisfy the equation of state of an ideal gas. ρ = ρa + ρb , p = pa + pb , E = ρε + ρ

u2 . 2

(9.64)

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Fig. 9.13 Graphs of the numerical solution for different N = 1 . . . 3. Parameters: M = 500, a = 0.01, τ = 0.001, T = 0.7

Fig. 9.14 Graphs of the numerical solution for different N = 1 . . . 3. Parameters: M = 640, a = 0.01, τ = 0.0001, T = 10

The adiabatic index of the mixture, the gas constant and the specific internal energy of the mixture are calculated with weights equal to the densities of the components. p = ρ RT = ρε(γ − 1).

(9.65)

9 Investigation of Local Discontinuous Galerkin Method on the Solution …

Ra ρa + Rb ρb = c p − cV , ρ cV a ρa + cV b ρb εa ρa + εb ρb ε= = cV T , cV = . ρ ρ

159

R=

(9.66)

It is described in more detail in the article. Equations in the system of equations have the same type as the Burgers’ equation considered earlier. Therefore, the performed research will be useful in the future, when applying the local discontinuous Galerkin method to the system of quasi-gas dynamic equations.

9.5 Conclusion A study of the accuracy and stability of the LDG method is carried out on solving initial boundary value problems for the linear convection–diffusion equation and for the quasi-linear Burgers’ equation. Errors are estimated when changing the number of basis functions. It is shown that as for continuous, as for discontinuous solutions, increasing the number of basis functions improves the quality of the numerical solution even on sufficiently coarse grids. In the future, it is necessary to comprehensively study and develop an algorithm for modeling gas mixture flows based on quasi-gas dynamic equation system using the local discontinuous Galerkin method in the multidimensional case. A separate problem that arises in this case is the construction of an orthogonal system of polynomials in the multidimensional case, especially on non-orthogonal difference grids. It is assumed that due to the complexity of calculations, parallel computing technologies will be used in the work. Note that the LDG method is very convenient for parallel implementation due to data locality. For any number of basic functions, the values of the required quantities are related to each other only in neighboring cells of the difference grid. This distinguishes the LDG method from other methods of constructing schemes of high order of accuracy, in which the pattern increases with increasing order of the difference scheme.

References Cockburn B (1998) An introduction to the discontinuous Galerkin method for convection—dominated problems, advanced numerical approximation of nonlinear hyperbolic equations. Lecture Notes in Math 1697:151–268 Cockburn B, Shu C-W (2001) Runge-Kutta discontinuous Galerkin methods for convection dominated problems. J Sci Comput 16:173–261 Cockburn B, Shu C-W (2005) Foreword for the special issue on discontinuous Galerkin method. J Sci Comput 22:1–3

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Cockburn B, Shu C-W (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal 35:2440–2463 Elizarova TG, Shil’nikov EV (2021) Numerical simulation of gas mixtures based on the quasigasdynamic approach as applied to the interaction of a shock wave with a gas bubble. Comput Math Math Phys 61:124–135 Koshlyakov NS, Smirnov MM, Gliner EB (1964) Differential equations of mathematical physics. Elsevier, Amsterdam Ladonkina ME, Neklyudova OA, Tishkin VF (2012) Studying the effect of limiter on the order of accuracy of a solution obtained by the discontinuous Galerkin method. Keldysh Institute of Applied Mathematics, Preprint No. 34 [in Russian] Ladonkina ME, Neklyudova OA, Tishkin VF (2013) Impact of different limiting functions on the order of solution obtained by RKDG. Math Models Comput Simul 5:346–349 Ladonkina ME, Neklyudova OA, Tishkin VF (2014) Application of the RKDG method for gas dynamics problems. Math Models Comput Simul 6:397–407 Liu T-P (2017) Hopf-Cole transformation. Bulletin of the Institute of Mathematics Academia Sinica (New Series). https://doi.org/10.21915/BIMAS.2017103 Yan J, Shu C-W (2002a) A local discontinuous Galerkin method for KdV type equations. SIAM J Numer Anal 40:769–791 Yan J, Shu C-W (2002b) Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J Sci Comput 17:27–47 Yuan L, Shu C-W (2006) Discontinuous Galerkin method based on non-polynomial approximation spaces. J Comput Phys 218:295–323

Chapter 10

Adjustment of Analytical Methods by Numerical Models for Determining Earth Pressures Behind Retaining Walls Guillaume Puyhaubert, Ali Saeidi, Mahdiyeh Seifaddini, and Alain Rouleau

Abbreviations A c' E H K Ka z φ' α β β' γ δ' ν σh σ h corr σv

Modifying coefficient for the Coulomb’s equation Soil cohesion [kPa] Young’s modulus [kPa] Height of wall [m] Earth pressure coefficient Active earth pressure coefficient Selected depth [m] Soil friction angle [°] Slope angle of the soil surface [°] Right corner angle of the bottom of the wall [°] Left corner angle of the bottom of the wall [°] Soil volumetric weight [kN/m3 ] Joint friction angle [°] Poisson’s ratio Horizontal earth pressure [kPa] Corrected horizontal earth pressure [kPa] Vertical earth pressure [kPa]

10.1 Introduction Retaining walls are one of the most common support systems for several purposes like road protection or support, landslide prevention, quay construction, and many G. Puyhaubert · A. Saeidi (B) · M. Seifaddini · A. Rouleau Department of Applied Sciences, Université du Québec À Chicoutimi, Saguenay, QC, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_10

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others. A variety of building materials may be used to construct retaining walls, such as stones, rubbles, bricks, gabion, reinforced concrete, or steel. The key point for designing these structures in geotechnical engineering is the estimation of the horizontal (lateral) earth pressure behind a wall, and many studies to estimate this pressure have been reported in the literature. The capital letter “K” is used to indicate the horizontal earth pressure coefficient, the most famous formulations of the “K” coefficient are Rankine’s (1857) and Coulomb’s (1776) methods. In each one of the formulations, the horizontal stress, σ h , is expressed as the product of the vertical stress, σ v , and the pressure coefficient, “K”. This later coefficient may be expressed by the Rankine’s active earth pressure coefficient: ) ( ϕ' K a,Rankine = tan2 45 − 2

(10.1)

For an inclined soil surface, this equation becomes: K a,Rankine

√ cos2 (α) − cos2 (ϕ ' ) √ = cos α × cos(α) + cos2 (α) − cos2 (ϕ ' ) cos(α) −

(10.2)

where α is slope angle of the soil surface [°] and φ ' is friction angle of the soil [°]. With the Coulomb’s method, the “K” coefficient is expressed by: K a,Coulomb

( ) sin2 β + ϕ ' = √ )2 ( ' +δ ' )×sin(ϕ ' −α) sin2 (β) × sin2 (β − δ ' ) × 1 + sin(ϕ ' sin(β−δ )×sin(α+β)

(10.3)

In the equation above β is angle of the external face of the wall with the horizontal [°] (Fig. 10.1) and δ ' is friction angle of the interface between the wall and the soil [°]. As mentioned above, the horizontal stress is: σh = K × σv

(10.4)

σv = γ × z

(10.5)

with:

That γ is unit weight of the soil [kN/m3 ] and z is depth below the soil surface along the wall [m]. Each of these methods is developed analytically considering several simplifying assumptions: For instance, Rankine’s method assumes a planar soil-wall contact surface and does not consider the friction along that plane (Retaining wall is frictionless). In the Coulomb’s method, the force acting on the wall is calculated by considering static equilibrium of the ground corner delimited by the wall and the

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Fig. 10.1 Finite element model of the retaining wall domains

plane failure surface that passed through the foot of the wall. This force has a known direction according to δ ' , the friction angle between the soil and the wall. Finally, the soil fails according to a planar failure surface passing through the bottom of the wall and the failure wedge is a rigid body undergoing translation. Besides the simplifying assumptions required for their development, these methods do not consider the arching effect which occurs behind the wall specially in deeper depth also the real failure type behind of the wall isn’t planar. Also, the arching effect phenomenon results from internal stresses which distribute the weight of a granular layer on the sides rather than under the layer, one of the sides being the retaining wall itself. A number of researchers, such as Aubertin et al. (2003), have studied this phenomenon as a factor in the earth pressure, including the horizontal pressure in the backfilling material of underground mine stopes. Few studies however have analysed the arching effect on a retaining wall (Salgado and Paik 2003). Numerical models can be used to reduce the number of simplifying assumptions that are required to develop analytical equations. These models can also be used to consider the arching effects, as well as a more realistic failure type, resulting in a more accurate estimation of earth pressure. A number of studies have used numerical methods to estimate horizontal pressure in soil material, but most of them have analysed the cases of infilling material in underground mining stopes (e.g., Ting et al. 2011; Fahey et al. 2009), and the arching effects appear as one of the important phenomena for determining this pressure. Few studies have analyzed similar effects in the case of a retaining wall; the works by Yap et al. (2012), and by Nadukuru Michalowski (2012) are exceptions. Yap et al. (2012) have conducted a sensitivity analysis on the effects of several parameters such as the friction angle and soil

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cohesion on the horizontal stress behind a retaining wall. But they did not develop a solution for estimating these pressures in different situations and they did not consider the arching effect. Also, the number of considered parameters was limited, and some parameters such as the Young’s modulus, the Poisson’s ratio and the slope angle of earth surface were not considered in the sensitivity analysis because it is a limit equilibrium-type analysis. Nadukuru Michalowski (2012) have demonstrated that the pressure coefficient “K” can take different values according to three modes of wall movements, and for a smooth wall and a rough one. Moreover, they determined the stress distribution centroid location of these three modes, but no sensitivity analysis on parameters is mentioned. This study presents first a reference case to carry a sensitivity analysis of the effects of geotechnical parameters on the stress’s distribution behind a retaining wall. The most sensitive parameters are then selected to develop the adjustment coefficients for analytical methods of horizontal earth pressure.

10.2 The Numerical Model The model is developed using the 2D finite element program RS2 (Rocscience 2013); it considers three distinct domains (Fig. 10.1), the foundation, the retaining wall and the soil material. The material properties and the geometrical parameters for the reference case are described in Tables 10.1 and 10.2 respectively. The Z-axis begins at the top of the wall (z = 0) and increases toward the bottom of the wall. Roller boundary conditions (i.e., fixed horizontal displacement) are applied to the sides of the model, and the base is fixed (i.e., zero displacement). A 6 m wall is considered with a horizontal ground surface (α = 0; Table 10.2). The loading soil and the foundation material are elasto-plastic, but the concrete wall is elastic and isotropic. The Mohr–Coulomb failure criterion is considered for the Table 10.1 Material properties for the reference case

Table 10.2 Geometrical parameters for the reference case

Main soil (1)

Foundation (2)

Wall (3)

Unit weight (γ ) [kN/m3 ]

20

20

24

Young’s modulus (E) [MPa]

70

800

25 000

Poisson’s ratio (ν) 0.2

0.2

0.2

H: Height of the wall [m]

6

β: Angle on the right corner at the bottom of the wall [°] 90 β' : Angle on the left corner at the bottom of the wall [°]

71.6

α: Slope angle of the soil surface [°]

0

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materials, as well as for the interfaces between the wall and the foundation, and between the wall and the loading soil. As the wall is in concrete, the friction angle between the concrete wall and the soil is 2/3 of the friction angle of the soil, i.e., 23°, as commonly assumed in the geotechnics literature.

10.3 Comparison Between Analytical and Numerical Results The distribution of horizontal stresses (σ h ) behind a retaining wall obtained using three analytical methods (Rankine 1857 and Coulomb 1776), and by numerical modeling, are presented in Fig. 10.2 for the reference case. The X-axis is defined as the relative depth (z/H ratio) to facilitate comparison with results using other input values (Fig. 10.2). The numerical model results are very different from the results of the two (Rankin’s and Coulomb) analytical models for z/H values above 0.6. This is a result of the arching effect and consideration the other neglected parameters, which reaches its maximum at a z/H value of about 0.8; and goes down to a minimum value at the soil base (z/H = 1). For z/H between 0 and 0.6, the curve from the numerical simulations is almost linear and follows approximately the same values as the two analytical models. Moreover, the increasing rate of the horizontal stress shows a slight decline in the z/H interval between 0.4 and 0.6, which results in lower values of σ ah than predicted by the two analytical models in this depth range. This diminution rate is

Fig. 10.2 Comparison of analytical and numerical methods

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more apparent in several graphs in the next figures; and its effects will be discussed thereafter.

10.4 Sensitivity Analysis A sensitivity analysis is conducted to determine the most important parameters that have an effect on horizontal stress. This analysis is carried out for geotechnical soil parameters of the slope i.e., the friction angle, the friction coefficient between the wall and soil, as well as Young’s modulus and Poisson’s ratio of the soil. A single parameter is varied at a time, every other parameter being kept constant at the value used in the reference case. The effect of the effective friction angle of the soil (φ ' ) is considered between 25 and 40°. The values of all other parameters are kept the same as in Tables 10.1 and 10.2, even for the friction angle of the joint, i.e., the soil-concrete interface, which is kept at 23° (Fig. 10.3a). As we can see at the Fig. 10.3a, the lateral earth pressure decreases with increasing frictions angle (Fig. 10.3a) as expected, and we have the pick values at relative depth around z/H = 0.8. Also, the effect of the friction angle of the joint δ ' (soil-concrete interface) is analyzed by having its value to vary between 20° and 35° (Fig. 10.3b). The effect of δ ' on the lateral earth pressure appears to be lesser than for φ ' analyzed above, suggesting that the earth pressures is less dependent on this parameter. Variations of the soil Young’s modulus (E) has little effects on the horizontal stress (Fig. 10.4a), much like the variations of δ ', as shown above (Fig. 10.4a). Consequently, the Young’s modulus (E) will not be considered in the method development presented below. The effect of the Poisson’s ratio of the soil (ν) is analyzed by assigning values of 0.2, 0.25 and 0.3 to this parameter (Fig. 10.4b). The Poisson’s ratio of the other material is kept constant at 0.2, like the other constant parameters which keep values

Fig. 10.3 Effects of a Soil friction angle(φ ' ) and b soil-wall interface (δ ' ) on the earth pressure

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Fig. 10.4 The effect of a Young’s modulus and b poisson’s ratio of the earth pressure

of the reference case (Tables 10.1 and 10.2). Variations of “ν” does not significantly affect the horizontal earth pressure. The sensitivity analysis presented above indicates that the most important parameters affecting the horizontal pressure behind a retaining wall is the soil friction angle. Hereafter a methodology is developed for the estimating the horizontal stress distribution behind a retaining wall. A retaining wall with a flat soil surface is used to explain the methodology, which is applicable for different values for the slope of the soil surface. The methodology could be applied to numerous other cases.

10.5 Proposed Correction Approaches The proposed method consists of adjusting the analytical methods for determining the horizontal pressures using a correction coefficient “a” based on the numerical methods. We chose the Coulomb’s method because its results seem to be slightly closer to the numerical ones than the other analytical curves (Fig. 10.2). Consequently, it is assumed that the Coulomb’s active earth pressure coefficient will provide a better starting value to obtain a corrected estimate when multiplied by the coefficient “a”, as in Eq. 10.6. σhcorr = a × K a,Coulomb × γ × z

(10.6)

Considering that the numerical curve in Fig. 10.2 is relatively close to the Coulomb’s curve for z/H values between 0 and 0.75, the value of 1 is assigned to the coefficient “a” in this range of z/H values. The coefficient “a” is different than 1 for z/H values between 0.75 and 1; “a” is then expressed as the Numerical/Coulomb ratio (Eqs. 10.7). Two correction methods are presented: one by graphical reading and the other one by using regression equations.

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Fig. 10.5 Graphical method for determination adjustment coefficient for Coulomb method (a)

⎧

a = 1; f or 0 ≤ Hz < 0.75 umerical ; f or 0.75 ≤ Hz ≤ 1 a = NCoulomb

(10.7)

10.5.1 Graphical Method Horizontal earth pressures have been estimated by numerical simulation using different values of friction angle φ ' (25°, 30°, 35°, and 40°; Fig. 10.3), with other input parameters kept constant to the reference case values (Tables 10.1 and 10.2). The horizontal earth pressures have also been calculated analytically using the Coulomb’s method (Eq. 10.6) for the same four friction angle values. The values of horizontal earth pressure obtained by numerical simulation has been divided by the Coulomb’s analytical values for the corresponding friction angles, and for several z/H values; this yields a series of values for the coefficient “a”. The resulting graph (Fig. 10.5) can be read to estimate values of the coefficient “a” for 0.75 < z/H < 1, and for different values of the friction angle. These values of “a” could be used to estimate the horizontal ground stresses without using numerical methods. Interpolation between the curves shown on Fig. 10.5 would provide estimates of “a” for intermediate friction angle values.

10.5.2 Regression Equation Method With the regression equation method, two values of the coefficient “a” is estimated for each case, one value for z/H values between 0.75 and 0.9 and the other for z/H

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Fig. 10.6 Curves of coefficient “a” for a values of z/H between 0.75 and 0.9 b values of z/H between 0.9 and 0.1

between 0.9 and 1. A polynomial trend curve can be fit to the data points for each value of friction angle, for z/H values between 0.75 and 0.9, as shown in Fig. 10.6 with the corresponding quadratic equations. An average curve is also drawn. Moreover, the distance between the regression and the numerical curves is pretty similar from one value of φ ' to the other, supporting the validity of this approach. Linear trend curves are calculated for z/H between 0.9 and 1 (Fig. 10.6b). An average curve is also drawn. The four equations in Fig. 10.6 can be used separately to obtain values of the coefficient “a”. However, the following equations of the average curves shown in Fig. 10.6 can also be used to obtain first approximation values of “a” in the corresponding ranges of z/H values. Three different values may be used for the coefficient “a”, depending on the z/H values, as expressed in the series of equations: ⎧ ( z )2 (z) z ⎪ ⎪ a = 1 f or 0 ≤ < 0.75 a − 8.4 = −13.5 × + 22.7 × ⎪ av1 ⎪ H H H ⎪ ⎨ z ≤ 0.9 a = aav1 f or 0.75 ≤ ⎪ H ⎪ (z) ⎪ ⎪ z ⎪ ⎩ a = aav2 f or 0.9 < ≤1 aav2 = −5.8 × + 6.3 H H (10.8)

10.6 Validation of the Graphical Method To validate this methodology, a specific case is considered, with φ ' = 33° and H = 6 m. The coefficient “a” can be determined by reading Fig. 10.7. Then the earth pressure can be estimated by multiplying the value of “K a , Coulomb” by this “a”

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Fig. 10.7 Method validation with H = 6 m and φ ' = 33°

value. Figure 10.7 shows a graph validation comparing the modified Coulomb method and the numerical curve simulated with H = 6 m and φ ' = 33°. This figure shows that the modified Coulomb method is pretty close to the numerical curve. Therefore, this graphical method appears as a convenient way to estimate realistic earth pressures values that consider the arching effect and other neglected parameters.

10.7 Conclusions This study has shown that the commonly used analytical equations proposed by Rankine and by Coulomb, are not accurate enough in many actual field conditions. Numerical modeling results have demonstrated that the differences between the results are significant and occurs at a characteristic normalized depth (z/H) value around 0.75. Moreover, a sensitivity analysis has indicated that the friction angle of the soil has the most important effects on the horizontal earth pressure. Comparison of numerical modeling results to the two commonly used analytical curves for determination the active pressure has indicated that the Coulomb’s curve performs better than the others in most cases. Therefore, the Coulomb’s equation has been kept unmodified for 0 ≤ z/H < 0.75. For higher z/H values, a modifying coefficient “a”, is applied to the Coulomb’s active earth pressure coefficient. The value of “a” is equal to the Numerical/Coulomb ratio. Two equations are proposed to determine values of the adjusting coefficient “a”. A quadratic regression is proposed for z/H values between 0.75 and 0.9, and a linear regression equation is used to obtain values of coefficient “a” for the depth range

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between 0.9 and 1. Finally this study does not consider the presence of groundwater in the soil, which could cause more important horizontal stresses on the wall. The other limitation of the results is neglection of the effect of slop of ground surface, therefore a more comprehensive methods should be applied for adjustment the Coulomb’s active earth pressure distribution behind of a retaining wall.

References Aubertin M, Li L, Arnoldi S, Belem T, Bussiere B, Benzaazoua M, Simon R (2003) Interaction between backfill and rock mass in narrow stopes. Soil Rock Am 1:1157–1164 Coulomb CA (1776) Essai sur une application des règles de maximis & minimis à quelques problèmes de statique, relatifs à l’architecture. Mémoires de mathématique & de physique, présentés à l’Académie Royale des Sciences par divers savants, 7 :343–382 Fahey M, Helinski M, Fourie A (2009) Some aspects of the mechanics of arching in backfilled stopes. Can Geotech J 46:1322–1336 Inc R (2013) Software tools for rock and soil. Ontario, Canada, Toronto Nadukuru SS, Michalowski RL (2012) Arching in distribution of active load on retaining walls. J Geotech Geoenviron Eng. 138:575–584 Rankine WJM (1857) On the stability of loose earth. Phil Trans 1857:9–27 Salgado R, Paik KH (2003) Estimation of active earth pressure against rigid retaining walls considering arching effects. Géotechnique 53:643–653 Ting CH, Shukla SK, Sivakugan N (2011) Arching in Soils Applied to Inclined Mine Stopes. Int J Geomech 11:29–35 Yap SP, Salman FA, Shirazi SM (2012) Comparative study of different theories on active earth pressure. J Cent South Uni 19:2933–2939

Chapter 11

Design of a System to Produce Rapid Biomedical Prototypes with Synthetic Materials: State of the Art Erik Omar Alvarado-Alcántara, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa, Juan Luis Cuevas Andrade, Luis Héctor Hernández-Gómez, Pablo Moreno-Garibaldi, Mauricio Rebattú y González, Alejandro Rebattú y González, Verónica Guzmán-Mercado, and Teresa Berenice Uribe-Cortés

11.1 Introduction Additive manufacturing is a relatively new reality that has been well received in countries such as The United States and the European Union. It has been developed to a degree that has forced a rethinking in the manufacturing process. It has grown from an expediting agent in the traditional manufacturing process, in creating molds, to the production of functional objects in small and medium batches. This change in paradigm has been better understood under the current health emergency of SARS-COV-2 when personal protective equipment (PPE) became scarce. This setting acknowledges the effectiveness of additive manufacturing in the supply chain. It has also proven its significance in industry 4.0 during a similar pandemic setting.

E. O. Alvarado-Alcántara · J. A. Beltrán-Fernández (B) · J. L. C. Andrade · L. H. Hernández-Gómez · P. Moreno-Garibaldi · V. Guzmán-Mercado Instituto Politécnico Nacional, Sección de Estudios de Posgrado E Investigación, Escuela Superior de Ingeniería Mecánica Y Eléctrica, Unidad Zacatenco, “Unidad Profesional Adolfo López Mateos”Edificio 5, Segundo Piso, Colonia Lindavista, Alcaldía Gustavo A. Madero, 07738 Ciudad de México, Mexico e-mail: [email protected] J. L. C. Andrade e-mail: [email protected] J. C. Hermida-Ochoa Centro de Investigación Y Laboratorio Biomecánico (CILAB), Del Carmen 18, Chimalistac, 01070 Ciudad de México, Mexico M. R. González · A. R. González · T. B. Uribe-Cortés Hospital Regional “1 de Octubre” del ISSSTE, Av. Instituto Politécnico Nacional 1669, Revolución IMSS, Gustavo A. Madero, 07300 Ciudad de México, CDMX, Mexico © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_11

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11.2 Materials and Methods The search for information corresponding to the subject consisted of using different official platforms that provided reliable information, such as: Scopus, Mendeley, Springer Nature.

11.2.1 State of the Art Manufacture of free-form solids, (Chua et al. 2010) rapid prototyping, 3D printing or additive manufacturing, which is the standardized term, is the process of creating a digital three-dimensional object through computer-aided design (CAD) tools. It consists in adding thin layers of material to solidify and, in consequence, create such models (Failed 2013; Beaman et al. 1997; Khorram Niaki and Nonino 2018; Jyothish Kumar et al. 2018). The methodology of additive manufacturing consists of starting from a CAD model created in a specialized computer software Fig. 11.1. The file is prepared using an STL extension or a 3MF. A code is then generated to be interpreted by a 3D printer (CAM). Next, it is manufactured, and the piece goes through a “cleaning process” or fine-tuning. The concept of layered manufacturing has its roots at the beginning of the XIX century with the advancement of topography to reproduce models of elevation contour lines and photo sculpture. In the early part of the 1990s the use of additive manufacturing focused on expediting traditional production processes, mainly, mold production (Arcos-Novillo and Güemes-Castorena 2017). This technique was not expected to substitute traditional production processes, nor to produce products that are used on day-to-day functions, however, it was subtly foreseen, this current trend faces the consumer who acquires the final product, even if it is with limited mechanics from the photo polymeric materials. Additive manufacturing categories evolve (Fig. 11.2), new approaches are gradually incorporated; new materials, processes and applications are discovered, it has been exponentially growing over the last years. For example, from 2011 to 2012, the number of articles published increased 10 times, from 1600 to 16,000; according to Rodriguez (Mota et al. 2015) a similar analysis for additive manufacturing with multi-materials shows similar growth according to which is shown in the following chart. From June 2012 to May 2013, it can be noted the association of the increase of scientific published articles and the dates the patents expired. It is observed in Fig. 11.3 that from 2013 to 2022 the interest held on, registering more attention in March 2020 during the SARS-CoV-2 pandemic, time frame during which it was highlighted how 3D printing can aid during times of crisis such as the ones due to lack of PPE (Fig. 11.4).

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Fig. 11.1 Additive manufacturing methodology. Reprinted with permission from Springer Nature publishers (Medellin-Castillo 2019)

The tendency of this new technology has increased because the legal protection of the earlier inventions had gradually expired since 2014, therefore, the use of these technologies was propelled by enthusiasts, hobbyists, independent creators, researchers, and industries that were attempting to make their own 3D printers (Jyothish Kumar et al. 2018; Arcos-Novillo and Güemes-Castorena 2017).

11.3 Discussion The fused deposition modeling patent (FDM) registered to Stratasys birthed the surge of new machines with this technology due to its low cost. Because of this, the RepRap project can quickly make prototypes of itself. The success of the project is part of a community that shares knowledge needed to fabricate this equipment. Moreover, there are also other projects under the open-source scheme with other technologies like SLS (Yusuf et al. 2019; Wohlers and Gornet 2016; RepRap contributor. 2019). This project kick started the use of FDM technology (Stratasys Brand), also known as Plastic Jet Printing (PJP) (Alomar 2020), Layer Plastic Deposition BY Zortrax (LPD) (Ciobota et al. 2018) and renamed by RepRap project as Fused Filament Fabrication (FFF) encouraged the launch of many diverse 3D printers to the market. The Wohlers report (Fig. 11.5) shows that in 2020 only Brazil appears in the list on the top manufacturers of additive manufacturing systems. Therefore, Mexico can potentially debut in this field as it has the technology, geopolitics and intellectual

Fig. 11.2 Hierarchy of manufacturing process and their categories (Yusuf et al. 2019)

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Fig. 11.3 Number of published articles about multi-material additive manufacturing and their references from 2015 to 2020 (García-Collado et al. 2021)

Fig. 11.4 World trend searches for the term “3D printing” (Trends 2015)

conditions to develop this technology that is promised to be “the great next thing” (Arcos-Novillo and Güemes-Castorena 2017).

11.4 Conclusions The conditions for Latin America and Mexico are beneficial if, the focus is on technology development for additive manufacturing, to open opportunities for research, education and establishing new employment opportunities. As a result, there would be a growth in economic, educative, social, and technological development. There is no doubt that additive manufacturing is a great opportunity of entrepreneurship, especially for emerging economics like the ones in Latin America as only the services of 3D printing, reverse engineering and product development are offered. By consequence, equipment development could strengthen research and development areas for the creation of new products.

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Fig. 11.5 Lead countries on 3D printer fabrication (Home (n.d.))

References Alomar A (2020) Tecnologías de Impresión 3D—Parte 2: FDM (Modelado por Deposición Fundida). https://www.youtube.com/watch?v=aYFsLCtPMg&ab_channel=TecnologiasDisrupt ivas Arcos-Novillo DA, Güemes-Castorena D (2017) Development of an additive manufacturing technology scenario for opportunity identification-The case of Mexico. Futures 90:1–15. https://doi. org/10.1016/j.futures.2017.05.001 ASTM INTERNATIONAL (2013) ASTM F2792-12a Rapid Manufacturing Association, pp 1–3. https://doi.org/10.1520/F2792-12A.2 Beaman JJ, Barlow JW, Bourell DL, Crawford RH, Marcus HL, McAlea KP (1997) Solid freeform fabrication: a new direction in manufacturing. In: Solid freeform fabrication: a new direction in manufacturing. Springer US. https://doi.org/10.1007/978-1-4615-6327-3 Chua CK, Leong KF, Lim CS (2010) Rapid prototyping: principles and applications, 3 edn. World Scientific Publishing Co. https://doi.org/10.1142/6665 Ciobota ND, Stanciu P, Gheorghe GI (2018) 3D Complex structures through layer plastic deposition designed for carbon material impregnation. INCAS Bulletin 10(3):65–74. https://doi.org/10. 13111/2066-8201.2018.10.3.6 García-Collado A, Blanco JM, Gupta MK, Dorado-Vicente R (2021) Advances in polymers based multi-material additive-manufacturing techniques: state-of-art review on properties and applications. Additive Manuf 102577. https://doi.org/10.1016/j.addma.2021.102577 Home—Wohlers Associates (n.d) Wohlers Associates. https://wohlersassociates.com/ Jyothish Kumar L, Pandey PM, Wimpenny DI (2018) 3D printing and additive manufacturing technologies. In: 3D Printing and additive manufacturing technologies. https://doi.org/10.1007/ 978-981-13-0305-0 Khorram Niaki M, Nonino F (2018) Strategic alignment of additive manufacturing. in: the management of additive manufacturing. Springer Series in Advanced Manufacturing. Springer, Cham. https://doi.org/10.1007/978-3-319-56309-1_6 Medellin-Castillo, Chin ZS (2019) Design and manufacturing strategies for fused deposition modelling in additive manufacturing: a review. Chin J Mech Eng 32:53. https://doi.org/10.1186/ s10033-019-0368-0 Mota C, Puppi D, Chiellini F, Chiellini E (2015) Additive manufacturing techniques for the production of tissue engineering constructs. J Tissue Eng Regen Med 9(3):174–190

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RepRap contributor (2019) DIY Selective Laser Sintering FAQ. RepRap. https://reprap.org/med iawiki/index.php?title=DIYSelective_Laser_Sintering_FAQ&oldid=185544 Google Trends (2015) Google Trends. https://trends.google.com/trends/?geo=MX Wohlers T, Gornet T (2016) History of additive manufacturing Yusuf SM, Cutler S, Gao N (2019) Review: the impact of metal additive manufacturing on the aerospace industry. Metals 9(12):1286. https://doi.org/10.3390/met9121286

Chapter 12

Biomodeling and Numerical Analysis of the Different Pathologies of the Upper Limb (Arm) that Limit Movement in Patients Diego Ivan Islas-Jiménez, Guillermo Urriolagoitia-Sosa, Beatriz Romero-Ángeles, Dante Abel Islas-Jiménez, Misael Flores-Baez, Martha Eugenia Espinosa-Hernández, and Guillermo Manuel Urriolagoitia-Calderón

12.1 Introduction Since prehistoric times for mankind, bone fractures have proven to be a problem, which has been increasing over the years. Subsequently, pieces of evidence were found where it was described how ancient’s humans treated bone fractures and skeleton malformations (Sözen et al. 2017). For individuals, fractures in the distal radius are one of the most common, over the years and with developing technological D. I. Islas-Jiménez (B) · G. Urriolagoitia-Sosa · B. Romero-Ángeles · D. A. Islas-Jiménez · M. Flores-Baez · M. E. Espinosa-Hernández · G. M. Urriolagoitia-Calderón Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica Y Eléctrica, Sección de Estudios de Posgrados E Investigación, Unidad Profesional Adolfo López Mateos “Zacatenco” Edificio 5, 2° Piso, Grupo de Biomecánica, Col. Lindavista, 07300 Ciudad de México, México e-mail: [email protected] G. Urriolagoitia-Sosa e-mail: [email protected] B. Romero-Ángeles e-mail: [email protected] D. A. Islas-Jiménez e-mail: [email protected] M. E. Espinosa-Hernández e-mail: [email protected] G. M. Urriolagoitia-Calderón e-mail: [email protected] M. Flores-Baez Universidad Politécnica de Tecámac, Centro de Ingeniería Avanzada Y Manufactura, Prolongación 5 de Mayo, No. 10, Tecámac de Felipe Villanueva, Km 24.7, 55740 Estado de México, México © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_12

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advances, these fractures can be cured more easily since could be treated by several different means. For example, surgical interventions can apply orthopedic material (from the simplest splints to the most complicated and expensive titanium plate) (Redfern et al. 2004). This kind of bone fracture, in most cases, are the consequence of daily activity and has sought to heal alternatives within Medicine and Surgery. The first written findings to explain orthopedic and traumatic injuries can be seen in the Smith Papyrus dating from around 2000 b. C. Where it is described that the fractures are treated with palm fiber splints, which were gripped by pieces of cane (Failed 1933). Hippocrates was the first to present the five concepts for the treatment of fractures; antisepsis, bandage, reduction, splinting, and traction, which reinforced the healing techniques of nature (Martinez-Dubois 2013). Hippocrates focused more especially on bone fractures and dislocations, he managed to have a wide experience and practice in the treatment of broken arms (Stathopoulos 2017). Later, Rhazes was the first who managed to make a mixture (the closest thing to plaster, his mixture consisted of stirring lime with egg until obtaining a homogeneity between both elements) to immobilize fracture and treat the injury (Monro 1935). Additionally, the study of treatments aimed at the rehabilitation of radio-distal fractures (FRD) has been in existence for more than 300 years (Rogelio et al. 2010). By the XVIII century, Poteau referred to these fractures as dislocations, which were treated with conventional methods of fixation; plasters, bandages, and splints (Cerro et al. 2007). Colles, in the early part of the 1800s, established the diagnosis of the first fracture for these conditions (Lester et al. 1990). It was recognized the need for a deeper study of these types of injuries and later began the implementation of the fixation with an external bar held by proximal and distal needles proposed by Anderson and O’Neil (Anderson 1944; Cooney et al. 1979). Meanwhile, Cole and Obletz proposed the use of needles with plaster (Cole and Obletz 1966; Green 1975). Nevertheless, it was until 1965 that Ellis began applying the flying support plate (Ellis 1965). The use of the blocked fly plate, for the treatment of FRD with osteosynthesis, has become the trend in the last century, because it was considered the most adaptable procedure for the anatomy of the area to be treated, even in cases of spongy bone and osteoporotic, since it provides sufficient support to the fracture zone, reduces the shortening and displacement of joint fragments (Bradway et al. 1989). Plate implants are an optimal treatment for all types of fractures as described by Fernández (Kural et al. 2010). The fine geometry of the implant and the fixed angle it presents serve to decrease irritative tendinitis and tendon ruptures. Since it allows its insertion preserving a correct separation between the implant and tendons, through the Pronator Square (PC) muscle while promoting internal fixation with few complications, even in fractures with dorsal displacements (Rodriguez et al. 2016; Pereira et al. 2007). Also, the need for a bone graft is diminished, as it contributes to the restoration of the anatomical continuity of the dorsal. All of this leads to early mobilization (second week) and an early return of normal wrist strength function (from the sixth week). Open reduction with internal fixation (RAFI) with a blocked flying plate offers the best results both in the short and long term, with more than 90% satisfactory results, with a low DASH score (Auñón et al. 2011). Distal radius fractures Fig. 12.1a, have now been increasing in society, the patient main problem during recovery from a surgical

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23

23-A

23-B

23-C

Fig. 12.1 Anatomic representation of the study case. a Hand and wrist radiography b Müller and AO/ASIF distal radius fracture classification

intervention occurs when not having the proper rest. On many occasions, patients unconsciously commit certain excessive or exaggerated activities, which promote injury or delay the recovery of the biological system. The main idea behind the development of this research is to ensure that the design of a support plate for bone fracture healing will be adequate during the patient’s recovery process. The solution will be implemented by the development of a CAD model of a 6AL-4V titanium plate, which will be established to treat 2B fractures type (classification developed by Müller and AO/ASIF (Illarramendi et al. 1998; Jupiter and Fernandez 1997)) for the radius in the distal area. This model will have the following specifications Fig. 12.1a, b 12° inclination, a locking hole, and 11 holes perfectly distributed over the entire area of the plate, which makes it different from those already available for the communities Fig. 12.2. Subsequently, an analytical and numerical study of the plate will be carried out, for its numerical development the commercial computer program with algorithms of the finite element method will be applied. By reconstructing the bone, a three-dimensional model will be obtained, which will serve to generate a bone-plate system. With the help of the system generated above, a compressive, bending, and tensile study will be carried out, which will be obtained from specific cases that patients may present during their recovery process. Through an analytical study, the external agents for each case will be obtained and thus be able to carry out a numerical simulation for each one. Finally, with the results obtained, it will be possible to corroborate that the design of the plate model is acceptable to treat these malformations in the human body and guarantee a solution to the main problem so that people can enjoy a recovery without so many limitations.

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17.99 A

Material; 6AL-4V Titanium

Acot; mm

5

B

65.10

R 2.50 Detail A

50.05

168

R

R 1.75 R3

Detail B 2.30

Fig. 12.2 CAD design of the locking plate

12.2 Methodology/Development 12.2.1 Basic Consideration for the Development of the Numerical Analysis The use of orthopedic forearm implants has shown satisfactory characteristics. However, some of the problems that are currently not considered are the way to adapt to the radio bone, as these are limited by a tilt angle. The osteosynthesis implant evaluated in this research consists of a radio-district fracture fixation plate (FRD), which allows for stabilizing the damaged area of the bone, increasing the bone fixation of the fracture, and reducing the healing time and recovery of the patient. As the geometry of the implant has an angle of inclination that allows adjusting to the morphology of the average Mexican, based on the Mexican Anthropometry, this orthopedic implant offers the advantage of being able to be placed in any type of fracture distal radius, as it has a locking orifice that guarantees a better fit for this type of interventions Fig. 12.2 (Islas-Jiménez 2019). Reconstruction of the human radius bone (considering cortical and trabecular tissues) was developed by obtaining a computerized axial tomography (Hernández-Vázquez et al. 2020; Marquet-Rivera et al. 2021) with the biggest number of cuts from a DICOM file because from the cut it is possible to smooth the roughness of the model Fig. 12.3 (Islas-Jiménez 2019). Then, the model is exported into a Scan IP program to solidify the model by uniting the scan cuts. The model is saved in an STL format, which will be an excellent decision and will help to develop the analysis into the finite element method program. The objective of the numerical simulation of a distal radius fracture is to quantify the effect that a crack can produce on the structural integration of the radius bone

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Fracture

12°

185

Trabecular bone

Cortical bone Screws Plate

Fig. 12.3 Intra articular fracture with displacement in the distal area of the radius which is reduced and fixed with screws to a locking plate (Frykman 1969)

and observe the outcome that the orthopedic implant produces when applying an external agent, to obtain the expected results by the numerical analysis as closer to reality. Subsequently, the bone-plate system was designed Fig. 12.4 (Islas-Jiménez 2019). To perform the numerical simulation, a commercial computational program that applies finite element method algorithms was used, when applying this methodology, a three-dimensional model is imported and a file is loaded. The radius bone numerical model is prepared for numerical simulation. The study methodology was based on the basic four dogmas of Mechanics, which consider the materials to have a homogeneous, linear, continuous, and isotropic behavior. The mechanical properties of the materials for this simulation are the Young’s Modulus and Poisson’s Relationship (for all the materials titanium 6AL-4V alloy, cortical and trabecular bone) Table 12.1. The fracture in the model was introduced by creating a discontinuity in the solid by removing elements, as how is presented in Fig. 12.4. The next step was to discretize the continuous (solid bone) by selecting a high-order element (Solid Brick 186, 20 nodes and 6 degrees of freedom per element), and because of the irregularity of the shape in the model, was performed a freely discretized (generating 182,228 nodes and 105,249 elements) Fig. 12.4. The titanium plate was attached to the radius bone by 5 bolts, which will be represented in the system by incorporating the plate into the radius bone in these five places.

12.2.2 Determination of External Agents and Frontier Conditions Applied into the Analyses Figure 12.5, presents the free body diagrams for each one of the cases of study (tensile, compressive, and bending loading), where can be observed the manner the

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Fig. 12.4 DICOM file exportation to Scan IP® program to develop the model

Table 12.1 Mechanical properties applied into the analyses

Radius bone

Titanium 6AL-4V alloy

Cortical bone

Trabecular bone

Young’s modulus

15 GPa

2 GPa

112 GPa

Poisson’s relation

0.31

0.3

0.33

12.7 MPa

1 140 MPa

Yield stress 49 MPa

load and movement restriction are applied. Additionally, the magnitude of each kind of loading is to be determined. To define the external agent acting in the biological system, initially, it is taking into consideration the system in a compressive loading condition. From the compressive loading condition, it is possible to consider biomechanical principles, such as the radius bone transfers of 80% of the load. While the ulna takes the other 20%. For the development of these analyses, it is considered a male patient with a 68 kg weight, with a load coefficient (0.8) f1 . Also, a dynamic coefficient should be considered (1.5) f2 , and finally the gravity acceleration g = 9.817 m/s2 (Frydrýšek et al. 2022). The total compressive force applied to the biological systems is as follows: Fcom = m × g × f1 × f2   Fcom = (27.2 kg ) 9.81 m/s2 (0.8)(1.5) = 320.18 N

(12.1)

With the corresponding force it is possible to determine the acting pressure into the study, which it is through the following equation:

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(a)

(c)

(b)

Ften

187

Y

Y

X Z X

Z

F

Ux = Uy = Uz = 0 Rotx = Roty = Rotz = 0 Z

Ux = Uy = Uz = 0 Rotx = Roty = Rotz = 0

g

X Y

Fcom

Fig. 12.5 Application of external agent and movement restrictions. a Tensile loading. b Compression loading. c Bending loading

σ = Fcom /A; A = 15,134.57 mm2 σ = Fcom /A; A = 15,134.57 mm2

(12.2)

σ = 0.0211 MPa The numerical simulation was carried out using the finite element method (FEM) through a commercial computer program where the biomodel shown in Fig. 12.3 was considered, which represents a three-dimensional bone-plate system. Subsequently, a higher order solid element is selected (20 nodes) structural type (Solid 186), with 6 degrees of freedom per node. An isotropic-linear-elastic analysis was performed, applying a free discretized technique and submitting an external agent for the compression case of 0.0211 N/mm2 . Subsequently, the mechanical system is solved and the solutions are visualized. Finally, this methodology must be repeated to solve each case of an external agent, depending on the case study (compression, tensile, and bending) the direction and position of the load in the biological system will be introduced. Movement restrictions in the biological system are very important when simulating loading effects on the radius bone since the mechanical effects will have a direct relationship when the titanium plate is introduced into the biological system. For all case studies, movement restrictions are applied to the elbow and all types of movements are blocked, Fig. 12.5.

12.3 Results The numerical results obtained during the first numerical evaluation (compressive load application) show that the plate considered for the rehabilitation of the patient offers adequate mechanical support for the effectiveness of the bone fixation.

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(a)

mm 3.17 x 10-1 3.05 x 10-1 2.67 x 10-1 2.29 x 10-1 1.91 x10-1 1.52 x 10-1 1.14 x 10-1 7.64 x 10-2 3.82 x 10-2 7.40 x 10-2 4.35 x 10-3

MPa 1.14 x 102 3.57 x 100 3.21 x 100 2.85 x 100 2.50 x 100 2.14 x 100 1.78 x 100 1.42 x 100 1.01 x 100 5.40 x 10-3 6.35 x 10-3

(b)

Fig. 12.6 First case of study (compressive loading). a Maximum elongation of the bone-plate system. b Total von Mises stress (elastic failure criterion)

The proposed endoprosthesis in this research behaved, mechanically speaking, in a linear-elastic manner, since a maximum elongation value is 0.317 mm, which is the direct effect that the endoprosthesis suffers during the numerical simulation under a compressive condition and is not considered to cause affectation in the patient. Additionally, it can be observed that there is no loosening between the plate and the bone, which is a direct consequence of its correct fixation (by the screws into the bone). Also, it can be realized there is no possibility of a new splintering of the bone, which can be related to the von Mises stress value of 1.14 × 102 N/mm2 , which is well below the bone yielding stress (cortical bone yield stress 49 MPa and trabecular bone yield stress 12.7 MPa). Furthermore, it is guaranteed that the prosthesis works within the linear-elastic zone, the maximum stress generated in the prosthesis is 1.17% concerning the yield strength of the material. This ensures that there will be no permanent deformations or stresses that could cause any kind of failure in the bone-plate system Fig. 12.6. In Figs. 12.7 and 12.8 can be observed the results from the second case (tensile loading) and third case (bending loading) of studies. The numerical evaluations were carried out under the same conditions as mentioned above (changing only the direction of the load). The results were as expected for this investigation. These results ensure the continuity and correct alignment of the fractured parts, as well as the correct transmission of external loads, applied during rehabilitation and/or daily activities, allowing a primary bone remodeling of the bone, i.e., without bone callus formation, allowing the reduction in recovery time of patients.

12.4 Discussion and Conclusion Virtual biomodelling is reliable proof of the impact of technology in Medicine (Marquet-Rivera et al. 2021). Thanks to biomodelling, it is a reality to perform rapid

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(a)

mm 2.18 x 10-1 1.95 x 10-1 1.67 x 10-1 1.39 x 10-1 1.09 x 10-1 2.52 x 10-2 3.14 x 10-2 5.64 x 10-2 7.82 x 10-2 8.40 x 10-2 4.35 x 10-3

MPa 4.51 x 102 3.96 x 102 3.36 x 102 3.05 x 102 2.75 x 100 2.46 x 100 2.21 x 100 1.83 x 100 1.52 x 100 1.22 x 10-3 2.45 x 10-3

189

(b)

Fig. 12.7 Second case of study (tensile loading). a Maximum elongation of the bone-plate system. b Total von Mises stress (elastic failure criterion)

(a)

mm 3.09 x 10-2 2.53 x 10-2 1.85 x 10-2 7.53 x 10-3 6.53 x 10-3 4.05 x 10-3 2.86 x 10-3 1.52 x 10-3 0.62 x 10-4 0.59 x 10-4

MPa 8.77 x 101 7.31 x 101 5.85 x 101 4.38 x 101 3.65 x 101 2.92 x 101 2.19 x 101 1.46 x 101 7.31 x 100 8.04 x 100

(b)

Fig. 12.8 Third case of study (bending loading). a Maximum elongation of the bone-plate system. b Total von Mises stress (elastic failure criterion)

prototyping in a three-dimensional printing manner that enhances the performing a numerical simulation of a surgical implant, and the customization of prostheses is a reality (Marquet et al. 2018; Miranda et al. 2019; Rodriguez-Martínez et al. 2012). The implementation of this kind of technology, not only has an impact on reducing costs but also achieves more efficient treatments and improves the patient quality of life since the possibility of any kind of failure in the implanted prostheses is significantly reduced and recovery time is noticeably reduced. The virtual biomodels are obtained from tomographic studies of healthy individuals or by patients with conditions that need to be studied or/and treated. The computer programs, which are currently applied, can achieve biological systems very close to reality (in terms of geometry and mechanical characterization). In addition to this, the implementation of the finite element method offers the possibility of carrying out numerical simulations to understand the mechanical behavior of the integrated elements under the real working conditions in which they must perform, allowing for error

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corrections in; geometry, materials, and mechanical resistance. All of this is to avoid catastrophic failures during operation and to implement improvements in processes and designs that allow reducing costs (Islas-Jiménez 2019; Beristain-Lima 2015; Mastache-Miranda 2016; Marquet-Rivera 2017). Nowadays, 3D printing has had a considerable impact on the development of different Medical and Biological areas, since with this technology it is possible to personalize or customize at a high degree of precision. The latter is a feature of vital importance in the manufacture of prostheses and medical transplants (Rodríguez et al. 2015). In other sense of ideas, powder metallurgy has played a very important role in obtaining new materials for 3D printing, since printed models can be obtained in almost any material available in powder form. Additionally, the fact that with this kind of technology allows materials to have the characteristics desired in medical supplies at a lower cost (Romero-Ángeles et al. 2019). The biomodel analyzed in this document corresponds to a part of the forearm, specifically the trabecular and cortical bone that makes up the radius bone. In this biomodel, an intra-articular fracture is simulated with displacement in the distal area of the radius bone and treated through a Ti6Al-4V locking plate (modified in the angle of inclination and locking holes for a better grip). The modifications implemented to the plate are due to the need to adapt the implants used for the treatment of this type of injury to the physical characteristics of the average Mexican male. The applied load in each case study was based on the morphology of the body of a male person between 20 and 27 years of age, weighing 68 kg and with an average height of 1.65– 1.72 m, which obeys the Mexican anthropometry. The numerical simulations carried out in the bone-plate system were based on three different forms of load; tensile, compressive and bending into the damage system, since these encompass most of the stresses to which the forearm is subjected in different positions. The tensile stress analysis simulates the behavior of the bone-plate system when the patient’s weight is supported only by the arms since it is one of the procedures in the rehabilitation of the patient, where there is a greater possibility of implant failure. In the case of compressive stress analysis, the patient’s weight is supported on the palm of the fractured forearm, which is exemplified, this being a critical case of analysis. Finally, the bending case represents the lifting of an object, whose weight is concentrated in the palm, since it is a case that occurs regularly in the use of a person’s forearm. In the results obtained in each of the different case studies, it can be observed that the deformation (strain) that undergoes the biological system is minimal, so it can be concluded that the interaction between the metallic plate and the forearm bone will not be affected by loosening. By applying the von Mises theory of failure, it is possible to observe that the maximum stress obtained is 114 MPa, which is below the limit of material yielding (which does not enhance plastic deformation and propagation of the active crack in the bone) and the system is working in the linear-elastic zone (which mean that any kind of effect will be erased when the external agent (load) is removed from the system). This implies, that the modifications performed to the design of the plate are adequate. The modification of the angle in the plate allows a better fit of the metallic component in the distal area, while the modification of the plate locking hole permits the necessary clearance to be adjusted in the correct

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position at the bone (which has been done, taking into consideration the physical characteristics of the Mexican patient). Plates with locked screws or internal fixing systems, where the thread of the screw head and the plate hole is at a fixed angle inside the plate, preventing its slippage. They are implanted by surgical surgeries and make it less complex by reducing the area of implant placement in less time getting better recovery and healing. One of the main advantages of the implant is that it offers better malleability, due to its angular inclination and makes its surgical implementation not so prolonged. Although it is true, that there are still many particular cases in which to analyze the mechanical behavior of the proposed endoprosthesis, the results obtained in this investigation give the green light to the manufacture and application of the plate (Islas-Jiménez 2019). Acknowledgements The authors gratefully acknowledge the financial support from the Mexican government by the Instituto Politécnico Nacional and the Consejo Nacional de Ciencia y Tecnología provided to this research.

References Anderson R (1944) Comminuted fractures of the distal end of the radius. Surg Gynecol Obstet 78:434–440 Auñón M, Cecilia L, Rodríguez V, Resines E (2011) Evolution of the treatment of distal radius fractures in Spain. Is this the right way to go? Acta Ortopédica Mexicana 25(5):289–293 Beristain-Lima S (2015) Evaluation of defects in the vertebrae and their contribution to spinal instability. M. Sc. Thesis Instituto Politécnico Nacional SEPI ESIME Zacatenco, CDMX Bradway JK, Amadio PC, Cooney WP (1989) Open reduction and internal fixation of displaced, comminuted intra-articular fractures of the distal end of the radius. J Bone Joint Surg Am 71(6):839–847 Cole JM, Obletz BE (1966) Comminuted fractures of the distal end of the radius treated by skeletal transfixion in plaster cast: an end-result study of thirty-three cases. J Bone Joint Surg 48(5):931– 945 Cooney W3, Linscheid RL, Dobyns JH (1979) External pin fixation for unstable Colles’ fractures. J Bone Joint Surg Am 61(6A):840–845 Del Cerro M, De las Heras J, García D, Martín A, Vaquero J (2007) Utilidad de la artroscopia en el tratamiento de las fracturas de la extremidad distal del radio. Patología del Aparato Locomotor 5:64–71 Ellis J (1965) Smith’s and Barton’s fractures: a method of treatment. J Bone Joint Surg Br 47(4):724– 727 ˇ Frydrýšek K, Halo T, Cepica D, Machalla V, Šimeˇcková K, Skoupý O, Madeja R, Havlíˇcek M, Dostálová K, Trefil A et al (2022) Biomechanical assessment of cannulated nails for the treatment of proximal femur fractures Frykman G (1969) Fracture of the distal radius including sequelae. Acta Orthop Scand 1(153):108 Green DP (1975) Pins and plaster treatment of comminuted fractures of the distal end of the radius. J Bone Joint Surg Am 57(3):304–310 Hernández-Vázquez RA, Urriolagoitia-Sosa G, Marquet-Rivera RA, Romero-Ángeles B, MastacheMiranda OA, Vázquez-Feijoo JA and Urriolagoitia-Calderón G (2020) High-biofidelity biomodel generated from three-dimensional imaging (cone-beam computed tomography): a methodological proposal. Comput Math Method M

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Illarramendi A, González Della Valle A, Segal E, De Carli P, Maignon G, Gallucci G (1998) Evaluation of simplified Frykman and AO classifications of fractures of the distal radius. Int Orthop 22(2):111–115 Islas-Jiménez DI (2019) Mechanical design and numerical analysis of the application of a titanium plate for distal radio fracture. M. Sc. Thesis Instituto Politécnico Nacional SEPI ESIME Zacatenco, CDMX Jupiter J, Fernandez DL (1997) Comparative classification for fractures of the distal end of the radius. J Hand Surg Am 22(4):563–571 Kural C, Sungar I, Ugras A, Ertürk A, Cetinus E (2010) Controversies in the management of distal radius fractures. J Am Acad Orthop Sur 22(9):566–575 Lester GE, Anderson JJ, Tylavsky FA, Sutton WR, Stinnett SS, DeMasi RA, Talmage RV (1990) Update on the use of distal radial bone density measurements in prediction of hip and Colles’ fracture risk. J Orthop Res 8(2):220–226 Marquet-Rivera RA (2017) Numerical analysis of anterior cruciate ligament injury at three different degrees of damage. M. Sc. Thesis Instituto Politécnico Nacional SEPI ESIME Zacatenco, CDMX Marquet RRA, Urriolagoitia SG, Hernández VRA et al (2018) The importance of bio-fidelity in the biomodelling for a biomechanical analysis. MOJ App Bio Biomech. 2(3):174–175 Marquet-Rivera RA, Urriolagoitia-Sosa G, Hernández-Vázquez RA, Romero-Ángeles B, MastacheMiranda OA and Urriolagoitia-Calderón G. (2021) High Biofidelity 3D biomodel reconstruction from soft and hard tissues (knee), FEM, and 3D printing: a three-dimensional methodological proposal. Biomed Res Int Martinez-Dubois S (2013) Cirugía bases del conocimiento quirúrgico y apoyo en trauma. McGrawHill Education, New York Mastache-Miranda OA (2016) Volume tomography modeling of bone structures for numerical simulation under the action of loads and/or external agents. M. Sc. Thesis Instituto Politécnico Nacional SEPI ESIME Zacatenco, CDMX Miranda MOA, Sosa UG, Vázquez HRA et al (2019) Biomodels for the medicine teaching. MOJ App Bio Biomech. 3(5):122–123 Monro JK (1935) The history of plaster-of-Paris in the treatment of fractures. Brit J Surg 23(90):257– 266 Myres J (1933) The Edwin Smith Surgical Papyrus: published in facsimile and hieroglyphic transliteration with translation and commentary, in two volumes. In: Breasted JH (ed), vol III–IV. University of Chicago Oriental Institute Publications, Chicago 7(26):244–246 Pereira E, Seré I, Miranda D, Arce G, Rodríguez Castells F (2007) Osteosíntesis con placa bloqueada palmar de ángulo fijo en fracturas del radio distal. Rev Asoc Argent Ortop Traumatol 72(1):24–31 Redfern DJ, Syed SU, Davies SJM (2004) Fractures of the distal tibia: minimally invasive plate osteosynthesis. Injury 35(6):615–620 Rodríguez R, Aguilar LA, Torres CR, Lugo E, Urriolagoitia-Sosa G, Hernández LH, UrriolagoitiaCalderón G (2015) Design and development of an experimental claw-grip, forefinger simulator. Part I: kinematics. In: Applications of computational tools in biosciences and medical engineering. Springer, Cham, pp 19–36 Rodriguez GG, Sarmiento H, Clembosky G (2016) Minimally invasive approach with pronator quadratus preservation for distal radius fracture a prospective study. HAND 43S–43S Rodriguez-Martínez R et al (2012) Development of an experimental apparatus for testing a total knee prosthesis focused on Mexican phenotype. Int J Phys Sci 7(43):5779–5786 Rogelio RR, Martínez ND, Jiménez JM (2010) Clinico-radiologic evaluation of distal radius fractures treated with a percutaneous technique. Acta Ortop Mex 24(3):169–176 Romero-Ángeles B, Hernández-Campos D, Urriolagoitia-Sosa G, Miguel T, S René C, RodríguezMartínez R and Urriolagoitia-Calderón G (2019) Design and manufacture of a forearm prosthesis by plastic 3D impression for a patient with transradial amputation applied for strum of a guitar. In: Engineering design applications. Springer, Cham, pp 97–121

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Sözen T, Özı¸sık L, Ba¸saran NÇ (2017) An overview and management of osteoporosis. Eur J Rheumatol Infl 4(1):46–49 Stathopoulos P (2017) Corpus Hippocraticum: historical source of treatment of craniomaxillofacial trauma. Brit J Oral Max Surg 55(3):296–297

Chapter 13

Study for Validation and Implementation of Polymethyl Methacrylate in Neurocranium and Viscerocranium Prostheses Carolina Alvarado Moreno, Juan Alfonso Beltrán-Fernández, Mauricio González Rebattú y González, Luis Héctor Hernández Gómez, Alejandro David González Peña, Edgar Alfonso Figueroa Rodríguez, José Enrique Rodríguez Miramar, Erik Omar Alvarado Alcántara, Fidel Romero Martínez, and Juan Luis Cuevas Andrade

13.1 Introduction Craniofacial bone loss due to trauma, congenital malformations or aggressive pathologies, leads to a malfunction of the stomatognathic system and the central nervous system, altering the human masticatory apparatus and unbalancing the intracranial pressure (ICP). With proper prosthetic management, the discomfort of the trepanation syndrome and temporomandibular disorders is reduced. Nowadays, the use of implants made of polymethylmethacrylate (PMMA) is a viable alternative, since it is a material with physicochemical characteristics that promote the biocompatibility between the implant and living tissue (Cruz Ramos and García Becerra 2009).

C. A. Moreno · J. A. Beltrán-Fernández (B) · L. H. H. Gómez · A. D. G. Peña · E. A. F. Rodríguez · J. E. R. Miramar · E. O. A. Alcántara · F. R. Martínez · J. L. C. Andrade Instituto Politécnico Nacional - Escuela Superior de Ingeniería Mecánica Y Eléctrica - Sección de Estudios de Posgrado E Investigación Edificio 5, 2do Piso, Unidad Profesional Adolfo López Mateos “Zacatenco” Col. Lindavista, 07738 Ciudad de México, México e-mail: [email protected] C. A. Moreno e-mail: [email protected] J. L. C. Andrade e-mail: [email protected] M. G. R. González Hospital Regional “1 de Octubre” del ISSSTE, Av Instituto Politécnico Nacional 1669, Revolución IMSS, Gustavo A. Madero, 07300 Ciudad de México, Mexico © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_13

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Segmentation

•Generate a .stl file of the computed axial tomography of the skull.

Parametrization

.Conversion of volumetric objects to solid of the bone segments and development of the corresponding prostheses.

Assembly

.Perform the virtual placement of the prosthesis, of the case study.

Numerical Analysis

.Generate numerical studies through the Finite Element Method.

Fig. 13.1 Methodology to generate a FEM study

The beginning of the application of PMMA in medicine dates back to 1940 as a bone replacement. Since then, PMMA implants have remained as an alternative for craniofacial reconstruction. PMMA implants, due to their physical–chemical reaction, initially have a plastic behavior that allows the material to mold. Adopting the shape of the bone segment to be reconstructed preoperatively, it will subsequently become a rigid and solid material, with high levels of fatigue resistance, avoiding plastic deformations (Russo 2005; Parra Castañeda and Ribera Gonzaga 2018). The finite element method can display biomechanical characteristics of the bone in the generated models, applying mechanical properties such as Poisson’s ratio and Young’s modulus, and multiple loading conditions. This allows visualization of stresses and deformations in the bone segment and in PMMA prostheses. The methodology is shown in Fig. 13.1.

13.2 Methodology 13.2.1 Segmentation The function of the computer program used to generate the three-dimensional model detects each set of elements as a type of mass. That is, the program detects and differentiates the bone mass from the adipose and the muscular mass. For this case of study, it was worked on the particularity of the bone system, in order that each pixel highlighted in all of the slices, in any anatomical view, obtained from the medical tomographic equipment, is converted into a stereolithography (stl) file (Fig. 13.2). These files do not include physical properties and are defined by a mesh of triangles.

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Fig. 13.2 Segmentation and preview of the complete skull

13.2.2 Parametrization and Assembly For the cases with a problem found in the triangle mesh. The Autodesk Netfabb® and Autodesk Meshmixer® programs were used, with the tools “automatic repair”, “smooth texture” and “hole detection”. Subsequently within ANSYS® SpaceClaim® , the generated files were converted into solid objects. With the “automatic skin” function, similarly, within the ANSYS® SpaceClaim design module, the plates corresponding to each case are placed by means of assemblies (Fig. 13.3). Finally, the mechanical properties of the materials are set in ANSYS® Workbench (Autodesk 1982; Autodesk 1982; Swanson Analysis Systems 1970a, b).

Fig. 13.3 Conversion to solid objects

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13.2.3 Numerical Analysis In the case of the cranial vault, a simulation was performed on the area with the highest incidence of fractures, considering that it will be a linear fracture, derived from a translational acceleration with an impact force (Fig. 13.4. and Table 13.1). In the case of the lower jaw, the insertion of the masticatory muscles and the actions they perform (e.g., elevation, descent, protrusion, retrusion and laterality) were taken as a reference (Fig. 13.5). These contacts will be placed as forces in ANSYS® Mechanical (Swanson Analysis Systems 1970c). The forces correspond to the values found in Table 13.2. For the subsequent analysis, the Young’s modulus of PMMA and of cortical bone at 2400 MPa and 20,700 MPa will be taken into account (Beltrán Fernández 2019).

Fig. 13.4 Placement of force in the cranial vault

Table 13.1 Fracture impact conditions (Ruiz et al. 2018)

Speed

Impact force (N)

5.46 m/s

9135

Fig. 13.5 Placement of forces in the jaw

13 Study for Validation and Implementation of Polymethyl Methacrylate … Table 13.2 Maximum muscle strength (Cansel Dogru and Cansiz 2018; Kang and Updike 1990)

199

Muscle

Maximum strength (N)

Masseter

408

Temporal

468

Internal Pterygoid

228

External Pterygoid

252

Digastric (accessory)

120

Mylohyoid (accessory)

20

Maximum occlusion force on teeth

300

Occlusal force in the plate

150

Bulging of the skull

Chewing

Fig. 13.6 Healthy bone cases

13.3 Results The cases in which a FEM study was carried out with the values of healthy cortical bone are shown in Fig. 13.6. In the case of the neurocranium, the bulging of the skull under external force was reproduced, and in the case of the viscerocranium, the biomechanics of chewing (Figs. 13.7 and 13.8). The Maximum values obtained from the FEM forces can be found in Table 13.3.

13.4 Discussion of Results In the case where the simulation of the neurocranium was performed as a healthy case, it can be observed that a bulging of the skull was obtained. When an external force is applied, the exposed area is depressed and the distant areas are elevated, this event occurs in the diploe bone tissue. If the limit of elasticity of the bone is exceeded, a fracture occurs. On the other hand, for the viscerocranium area, in the jaw, the natural biomechanical behavior was obtained, showing the areas with the greatest absorption of stress during chewing. In other words, the areas highlighted in the FEM

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Fig. 13.7 PMMA plate case in parietal and temporal bone

Fig. 13.8 PMMA plates cases in the jaw angle and body

Table 13.3 Maximum values obtained from the FEM (MPa)

Simulation

Healthy case

Skull

20.777

Jaw

0.0005825

Model 1—plate

Model 2

51.854

N.A

11.297

15.054

studies were: the mandibular body, angle and condyle. This is also consistent with the literature on the incidences of fracture in the jaw mentioning the same areas, which were obtained in the present study. Therefore, in both studies, it is suggested that the natural mechanisms were obtained and subsequently the prosthesis simulations can be started, using the same parameters. As a result, in the areas where a PMMA plate was placed in none of the areas of the skull, the elasticity limit of polymethylmethacrylate was exceeded. In addition, in the proposal where half of the bone segment of the jaw is entirely made of PMMA,

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it remains below the elastic limit. Highlighting that, for these cases within the simulation module environment, the maximum values are located in the cortical bone. In no case, are the areas of highest stress concentration on the proposed implants of PMMA material.

13.5 Conclusions Based on the simulations of the healthy cases, the natural biomechanical behavior of the areas of interest was corroborated. This work ideally reproduced the bulging of the skull under an external force and the location of the critical zones in the jaw, with the purpose of being contrasted with the different proposals of PMMA prosthesis designs. Showing that in all cases, the maximum value obtained is below the elastic limit of polymethylmethacrylate. Therefore, they maintain their structural integrity and are not plastically deformed. Acknowledgements The grants of National Council of Science and Technology of Mexico, which support this research.

References Autodesk (1982) Software autodesk netfabb premium 2020. San Rafael, California. Autodesk, Inc. Autodesk (1982) Software autodesk meshmixer 2018. San Rafael, California. Autodesk, Inc. Beltrán Fernández JA (2019) Design and characterization of a mandibular prosthesis prototype by Hemimandibulectomy. Adv Eng Mater 92:313–343. https://doi.org/10.1007/978-3-319-790053_22 Cansel Dogru S, Cansiz E et al (2018) A review of finite element applications in oral and maxillofacial biomechanic. J Mech Med Biol. https://doi.org/10.1142/S0219519418300028 Cruz Ramos ME, García Becerra RM (2009) Rehabilitación mandibular. GAMO 8:75–79 Kang QS, Updike DP et al (1990) Theorical prediction of muscle forces on the mandible during bite. J Biomech Eng. ASME Parra Castañeda R, Ribera Gonzaga A et al (2018) Polimetilmetacrilato, una alternativa viable para la fabricación de prótesis craneal. ICSA 13:49–53 Ruiz EA, Ramírez EJ, Ruiz O et al (2018) Modelado de fractura del cráneo bajo condiciones de impacto. SOMIM 42–48 Russo C (2005) Rehabilitación mandibular. Diseño y realización. RSM 27:18–38 Swanson Analysis Systems (1970a). Software space claim. Canonsburg, Pennsylvania. Ansys, Inc. Swanson Analysis Systems (1970b) Software workbench. Canonsburg, Pennsylvania. Ansys, Inc. Swanson Analysis Systems (1970c) Software mechanical. Canonsburg, Pennsylvania. Ansys, Inc.

Chapter 14

Numerical Biomechanical Analysis of the Fixation of Three Titanium Screws for Elbow Coronoid Fracture Dante Abel Islas-Jiménez, Beatriz Romero-Ángeles, Guillermo Urriolagoitia-Sosa, Diego Ivan Islas-Jiménez, Israel Flores-Baez, Juan Antonio Vargas-Bustos, and Guillermo Manuel Urriolagoitia-Calderón

14.1 Introduction Throughout its history, mankind has faced a recurrent and determining problem, which has been the different types of malformations and fractures in its bone system. This has been gradually increasing over the years, historical antecedents have been D. A. Islas-Jiménez · B. Romero-Ángeles (B) · G. Urriolagoitia-Sosa · D. I. Islas-Jiménez · I. Flores-Baez · G. M. Urriolagoitia-Calderón Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica Y Eléctrica, Sección de Estudios de Posgrados E Investigación, Unidad Profesional Adolfo López Mateos “Zacatenco” Edificio 5, 2° Piso, Grupo de Biomecánica, Col. Lindavista, 07300 Ciudad de México, México e-mail: [email protected] D. A. Islas-Jiménez e-mail: [email protected] G. Urriolagoitia-Sosa e-mail: [email protected] D. I. Islas-Jiménez e-mail: [email protected] I. Flores-Baez e-mail: [email protected] G. M. Urriolagoitia-Calderón e-mail: [email protected] I. Flores-Baez Universidad Politécnica de Tecámac, Centro de Ingeniería Avanzada Y Manufactura, Prolongación 5 de Mayo, No. 10Tecámac de Felipe Villanueva, Km 24.7, 55740 Estado de México, México J. A. Vargas-Bustos Universidad Nacional Autónoma de México, Facultad de Estudios Superiores, Iztacala, Departamento de Psicología Clínica, Av. de los Barrios S/N, Los Reyes Iztacala, 54090 Estado de México, México © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_14

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found describing how to treat the different malformations that the human being has had to face. These findings are of utmost importance in the area of Traumatology and Orthopedics. The elbow fracture is one of the most frequent fractures that usually occur in the human skeleton, since it represents a great surgical challenge for the orthopedist. The orthopedist must understand and know extensively the complex anatomy of the region of study. But over the years and with technological advances that have been increasing, elbow fractures can be treated in an easier and more efficient way, due to new trends in specialized fixation, for surgical interventions to treat fractures osteosynthesis material is used, such as plates, screws, wire, needles, among others. In order to carry out a surgical intervention, a patient must have a malformation in the mentioned area, which requires orthopedic treatment based on titanium screws, where it has been observed that most of them have a favorable recovery. This is due to the surgical material, which has a degree of compatibility with the human body (Islas-Jiménez 2019). In the year 1850, the first internal fixation procedure was performed, which consisted of the reduction of an olecranon fracture with two transcutaneous screws attached with a string. This was performed by Surgeons Cucel and Rigaud. By 1886, German surgeon Carl Hansmann performed the first internal plate fixation, which used a removable steel plate and nickel-plated screws. These screws were previously welded to his screwdrivers to achieve fixation (Sauerbier et al. 2008). However, in 1912, Surgeon William O. Sherman published several recommendations on the most effective properties of orthopedic screws, ranging from their alloy composition to the width of the driving heads. But until the 1940s, several surgeons advocated the development of screws specially adapted to human bone (Roberts et al. 2013). Screws are devices that are used in internal fixations for the correction or repair of bone fractures. There are different models, which are made of stainless steel, titanium or biodegradable. Currently there is a wide range of bone screws (Yuehuei-H 2002). The type of screw to be chosen depends on the density of the bone (cortical or trabecular bone), the location of the fracture (epiphysis, metaphysis or diaphysis) and the type of fracture classification (Yañez-Santana 2009). An important parameter is the pull-out strength of a screw in the bone, it depends on its outer diameter, its grip length, the density of the bone and the shape of the thread (Yañez-Santana 2009; Cuadrado 2013). This type of screw belongs to the AO group or more accredited in English-speaking countries as ASIF (Association for the Study of Internal Fixation) (Yañez-Santana 2009; Osorio et al. 2010). Cortical screws have thread or thread along their entire length, their main function is the fixation of plates to the bone Fig. 14.1. They are possibly the most widely used of the internal fixation devices (Anderson 1944; Yañez-Santana 2009; Osorio et al. 2010). These can also function as compression screws. In addition, these are less prone to breakage when used in fractures with fatigue. Different sizes of this screw are available in the catalog, both in external diameter and length (Yañez-Santana 2009; Ortiz EF and Blasnik 2004). Cancellous bone screws are designed to transfer long fragments of trabecular (cancellous) bone. For this purpose, they have a wider and thicker thread, which produces a drilling effect Fig. 14.2 (Wang et al. 2016). Cancellous AO screws can have a partial thread (for use as a compression screw) or a full thread (used to fix

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plates in the metaphyseal regions of long bones) (Yañez-Santana 2009; Taljanovic et al. 2019). Coronoid fractures of the elbow may result from a fall, direct blow to the elbow, or twisting of the arm. In addition to the fracture, there are often sprains, strains, or dislocations (Domínguez 2018). To confirm if there is a fracture, x-rays are used to see if the bones are out of place. Sometimes, a computed tomography (CT) scan may be needed to get more information and be more accurate to the diagnosis. A case is presented with a diagnosis of right elbow fracture Fig. 14.1, secondary to a fall from its support plane with mechanism of injury, axial load in the right thoracic extremity which causes deformity and intense pain at the elbow. The treatment of

(a) (b)

Type II radial head fracture

Fixation of three osteosynthesis screws

Fig. 14.1 Radiographic representation of case study. a Type of fracture. b Treatment with osteosynthesis trunnion fixation

(b)

(a) Head

Head Core diameter

Core diameter Pitch

Pitch Outside

Outside

Fig. 14.2 Properties and CAD model. a Cancellous screw. b Cortical screw

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Table 14.1 Osteosynthesis screw specifications Material

Outside diameter (mm)

Pitch (mm)

Length (mm)

Head diameter (mm)

Internal hexagon (mm)

Core diameter (mm)

Sponge screw

Titanium 6AL-4V

4.0

1.75

18

6.0

2.5

2.3

Cortical screw

Titanium 6AL-4V

2.7

1

18

5.0

2.5

1.9

the injury is based on stabilization of the radial head, by means of open reduction and osteosynthesis material. Figure 14.1 shows the fixation of three osteosynthesis screws for fracture of the coronoid of the radial head. Two types of fixation screws were used to treat this fracture. Cortical screws are commonly used as a fixation system for osteosynthesis in general; they have a thread profile with special characteristics for fixation on the cortical bone (Cognet et al. 2009). The development of the screw design was carried out according to the established standards of the AO of Osteosynthesis and Traumatology. Cancellous bone screws are typically used for internal fixation, either for bone fractures or for implants. This type of screw appears to be simple, but has a great deal of engineering technology, which contributes to its design and its optimal performance is seen in Fig. 14.2, (Lucas et al. 1999). These screws are usually fully or partially threaded. In Table 14.1, some specifications of the dimensions of the cancellous and cortical screws can be seen.

14.2 Methodology/Development 14.2.1 Basic Considerations for the Development of the Numerical Analysis To carry out the construction of the real elbow model, a computer program called Simpleware ScanIP® was used. This program offers the implementation of 3D models through DICOM (Digital Imaging and Communication On Medicine) images. These files offer a wide selection of computational tools for viewing images or 3D files. The reconstruction of the elbow bones was carried out through the Scan IP® program, which provides a complete segmentation computer program environment to process 3D image data, different masks are created which represent the parts that make up the human elbow, once each of the layers is filled with the different masks, the 3D model is visualized as shown in Fig. 14.3. The model was saved in a standard STL format for computer-aided design that defines 3D object geometry. Another important point is the mechanical properties of the model and of the osteosynthesis screws. The tissue of the human bone system is a non-homogeneous

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Fig. 14.3 Creation of the CAD model, by means of the Scan IP® program. a Creation of masks. b 3D model of the bones of the elbow

and anisotropic material, which makes it difficult to have adequate samples for the determination of its mechanical properties. However, based on literature and research, the values to be used can be seen in Table 14.2 (Strafun et al. 2018). Regarding the osteosynthesis screw, its specifications mentioned in Table 14.2. Titanium 6AL4V material was chosen, due to its mechanical characteristics. However, another important point is its high degree of biocompatibility with the human body and its high mechanical resistance, which makes many doctors implement this type of material to have a better healing in any fracture presented. For the fixation of the screws a pilot hole had to be made in the 3D model of the bone, it is worth mentioning that this hole has the same diameter as the core of the screw, since a hole with a diameter of more than 90% provides an optimal fixation, otherwise if the diameter is less than 90%, it does not provide any advantage and produces greater resistance in the insertion of the screw. For the fixation of the screws, the 3D Builder® computer program was used, which facilitates the design, modification or visualization of a 3D model. Likewise, the fixation of the two different types of screws in the 3D model of the elbow bones was done. Figure 14.4 shows the fixation of the two types of screws, which are cortical and cancellous. In the same way, a new fixation was made, which only involves placing two cancellous screws in the area of the fracture to be treated, positioning of two cancellous screws, the computer program SolidWorks® was used, the first step was the Table 14.2 Mechanical properties

Material Cortical bone Trabecular bone Cartilage Titanium Screw 6AL-4 V

Young’s modulus

Poisson’s ratio

E = 18 GPa

v = 0.31

E = 400 MPa

v = 0.26

E = 1 GPa

v = 0.40

E = 104 GPa

v = 0.33

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(b) Three sponge screw fastening

(a) Fixation of three cortical screws

3D model of elbow Fig. 14.4 Titanium screw fixation in 3D model of the elbow. a Cortical screw fixation. b Cancellous screw fixation

Fig. 14.5 Fastening of the two cancellous screws in SolidWorks®

realization of the two pilot holes that will have the biomodel of the elbow, the separation of both holes is 4 mm. can be seen in Fig. 14.5, it is worth mentioning that its specifications are the same as those previously seen in Table 14.2.

14.2.2 Determination of External Agents and Boundary Conditions Applied in the Analyses Using the ANSYS® computer program, the following steps must be performed; the first step is to select a high order solid (20 nodes) called structural solid 186 for 3D models, one of its main characteristics is that it is composed of 6 degrees of freedom and 20 nodes. According to the literature, for this research a linear, elastic and isotropic behavior is considered where the material behaves homogeneously and continuously. The second step is to perform a discretization or meshing in a freeway, where 862 253 nodes and 496 250 elements were obtained. The last and most important step is the motion conditions and constraints. Figure 14.6, shows the constraints and the load that was applied. It is worth mentioning that in the constraints

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all the DOF were applied: i.e., there is no motion in the axes Ux = Uy = Uz = 0 and rotations Rotx = Roty = Rotz = 0. Initially for this research, the numerical analysis was carried out by applying an external agent that causes a compression effect in the system. This can be produced in a person when he/she is performing a sport activity. For example, this activity occurs when the individual stands on his hands on a flat or irregular surface. In this case, the analysis will focus on the forearm of the human body. To determine the compression force (Fcom ) it is important to consider some aspects presented in the literature. The radius bone transfers 60% of the load, while the ulna bone is 40% (Kapandji 2011). For this, an average person of 70 kg was considered (so each arm will carry half of

Elbow position

(b)

Wrist position

(a) Charging application

Ux = Uy = Uz = 0 Rotx = Roty = Rotz = 0

Fig. 14.6 Boundary conditions. a Load application. b Movement restriction

Compressive Strength

Fig. 14.7 Application of external agent and movement restrictions, with compressive loading

Radius bone Unla bone Y Ux = Uy = Uz = 0 Rotx = Roty = Rotz = 0

Z

X

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the load (35 kg), taking into account the load coefficient f1 = 0.6 and a dynamic coefficient of f2 = 1.2, for the radius. While the ulna has a load coefficient f3 = 0.4 and a dynamic coefficient of f4 = 1. Finally, the acceleration of gravity g = 9.81 m/s2 [1 and 17]. To obtain the total compressive force of both bones is solved as follows (IslasJiménez 2019): F1com = m × g × f1 × f2

(14.1)

F2com = m × g × f3 × f4

(14.2)

By substituting the values mentioned above, the following result is obtained, which will be used later for the numerical analysis. F1com = 21 kg × 9.81 m/s2 × 0.6 × 1.2 = F1com = 148.33 N F2com = 14 kg × 9.81 m/s2 × 0.4 × 1 = F2com = 54.94 N Once the corresponding force is obtained, the pressure required for the study is calculated by means of the following equation (Strafun et al. 2018). σy =

F A

(14.3)

σy1 = 0.00678 MPa y σy2 = 0.00268 MPa

14.3 Results The numerical results obtained from the case study presented show the titanium screws. For the mechanical analysis the compression study was performed, the load was established where the screws are and where the fracture was presented, thus giving results that indicate the von Mises stress 13.24 MPa of the cortical screw fixation in the elbow bone system, as well as a total elongation of 0.274 mm, Fig. 14.8. In this part, the results obtained from the case study presented can be observed. It should be noted that these results are from the three cancellous screws, which are fixed to the biomodel of the elbow. It should be noted that the position of the screws is identical to the aforementioned clinical case. Holes were also drilled in the biomodel of the elbow, which are used to fix the screws, thus providing greater reliability in the analyses performed since it is a model identical to real life. Another fundamental point is the application of the finite element method (FEM), because the numerical analysis of the 3D model of the forearm of the elbow with the fixation

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(a)

211

(b) 0.274 (mm) 0.243 0.213 0.182 0.152 0.121 0.091 0.060 0.030 0

1

1.324 x 10 (MPa) 1.173 x 101 1.030 x 101 8.830 x 100 7.358 x 100 5.886 x 100 4.415 x100 2.943 x 100 1.471 x 100 2.673 x 10-6

Fig. 14.8 Compression study of the three Titanium cortical screws. a Maximum elongation of the bone-plate system. b Total von Mises stress (elastic failure criterion)

of three cancellous screws and the proposal of fixation of two cancellous screws was carried out. Figure 14.9, shows the results of the fixation of the three screws and Fig. 14.10, shows the results of the fixation of the two cancellous screws in the 3D model of the elbow. Table 14.3, shows the results of the types of fixation screws analyzed in this research, showing the values of Total Elongation and von Mises stress. Once comparing the results of the fixation of the three cancellous and cortical screws, it can be said that these two types of screws are clinically and mechanically suitable to treat the aforementioned elbow coronoid fracture, since the results obtained are very similar. On the other hand, the results of the fixation of the two cancellous screws are observed (it is important to emphasize that the position of the case study already seen was changed), these results show a similarity with the other results obtained from the fixations of three screws, so it can be said that the fixation of the

(b)

(a) 0.161 (mm) 0.143 0.125 0.107 0.089 0.072 0.054 0.036 0.018 0

(MPa) 1.548 x 101 1.376 x 101 1.204 x 101 1.032 x 101 8.601 x 100 6.881 x 100 5.160 x100 3.440 x 100 1.720 x 100 2.632 x 10-7

Fig. 14.9 Compression study of the three titanium cancellous screws. a Maximum elongation of the bone-plate system. b Total von Mises stress (elastic failure criterion)

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(b)

(a) 0.186 (mm) 0.165 0.145 0.124 0.103 0.082 0.062 0.042 0.021 0

(MPa) 1.227 x 101 1.091 x 101 9.543 x 100 8.180 x 100 6.816 x 100 5.453 x 100 4.909 x100 2.727 x 100 1.363 x 100 4.398 x 10-10

Fig. 14.10 Compression study of two titanium cancellous screws. a Maximum elongation of the bone-plate system. b Total von Mises stress (elastic failure criterion)

Table 14.3 Results for the types of fastening screws

Total elongation (mm)

Von Mises stress (MPa)

0.274

1.324 × 101

Three sponge screw 0.161 fastening

1.548 × 101

Fastening of two sponge screws

1.227 × 101

Three cortical screw fixation

0.186

two cancellous screws in the biomodel of the elbow is optimal and would not cause any alteration of results to the clinical case mentioned in this research.

14.4 Discussion and Conclusion The methodology of this research was carried out through a clinical case with diagnosis and treatment of a coronoid fracture of the right elbow. A computerized axial tomography (CT) scan was used in the arm and forearm area, which by means of its generated DICOM files, a reconstruction of a 3D model of the elbow (consists of cortical bone, trabecular bone and cartilage) was performed, which closely resembles the bones that make up a human elbow. The orthopedic fixation screws used are of characteristics and design specifications already established by the AO of osteosynthesis. Subsequently, the elbow model underwent two study cases (compression and flexion), one study was done with the elbow model without any malformation or fracture, that is, under normal conditions and another with the fracture already treated in the elbow area. For the results, the finite element method (FEM) was used, which shows the stresses, elongations and deformations that are concentrated in the elbow

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bones and in the fixation screws. In particular, this prototype of the elbow model behaved in a satisfactory way since it is at the levels of elastic resistance of the properties of the materials. The main objective of this research work was the analysis of the coronoid fracture of the elbow, taking into account a clinical case. Likewise, the main materials used are plates, screws, wires, nails, needles, among others. These have been used throughout the history of the pathologies that human beings have had to face. In this research, two types of osteosynthesis screws were used to treat fractures. In order to understand the different healing techniques used throughout the world, it was necessary to identify the different cultures of all regions, all this to know how knowledge has been changing according to the evolution of man. A main objective was to identify the methodology to make the construction of the 3D model of the elbow, which was made based on a computerized axial tomography (CAT), this is able to generate images in axial slices, and with the help of computer programs such as Simpleware Scan Ip® and SolidWorks® the 3D model was recreated similar to the characteristics that the elbow has, The geometry of the model was also obtained, such as the type of mesh and the element, and the mechanical properties of each material that was used for the 3D model of the elbow were also added. In the fixation of three titanium cortical screws to treat a coronoid fracture of the elbow. The position of the screws was determined according to a clinical case already treated. The 3D model of the elbow was used to fix the screws in the elbow model. Three titanium cancellous screws with the same position of the clinical case presented were used for the fixation of these screws. The von Mises stress of 15.48 MPa was obtained in the three cancellous screws of the elbow bone system, with a total elongation of 0.161 mm along the system. Likewise, a different fixation was chosen, which consists of two titanium cancellous screws in different positions and at a determined distance assimilated to the coronoid fracture of the elbow. Thus, the results indicate a von Mises stress of 12.77 MPa in the two cancellous screws with a total elongation of 0.186 mm. These results obtained with the different osteosynthesis screws, to treat the coronoid fracture of the elbow, have been of great help to determine that these screw fixations in the clinical case and the proposed ones are recommendable to achieve an optimal recovery of this fracture and thus to obtain a good quality of life. Acknowledgements The authors gratefully acknowledge the financial support of the Mexican government from the Instituto Politécnico Nacional and the Consejo Nacional de Ciencia y Tecnología (National Council of Science and Technology) for this research.

References Anderson R (1944) Comminuted fractures of the distal end of the radius. Surg Gynecol Obstet 78:434–440 Cognet JM, Altman M, Simon P (2009) Material de osteosíntesis: tornillos y placas. J. EMCTécnicas Quirúrgicas-Ortopedia y Traumatología 1(1):1–10

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Cuadrado A (2013) Análisis mediante ensayos in vitro, in vivo y modelos computacionales, de sistemas de fijación interna basados en placas y tornillos para la reparación de fracturas osteoporóticas. Doctoral disertación de la Universidad de Las Palmas de Gran Canaria, España Domínguez PB (2018) Fracturas del área del codo. Tratamiento asistido por artroscopia. Rev Esp Artrosc Cir Articul 25(2):131–141 Islas-Jiménez DI (2019) Mechanical design and numerical analysis of the application of a titanium plate for distal radio fracture. M. Sc. Thesis Instituto Politécnico Nacional SEPI ESIME Zacatenco, CDMX Kapandji AI (2011) Articular physiology. Editorial Medica Panamericana, México Lucas GL, Cooke FW, Friis EA (1999) A primer of, Biomechanics. Springer, New York Ortiz EF, Blasnik JJ (2004) Osteotomías del primer metatarsiano Estabilidad: fundamentos-fijación interna. Rev Asoc Argent Ortop Traumatol 69(3):263–269 Osorio A, Rodríguez D, Gámez B, Ojeda D (2010) Análisis numérico de una placa para fijación de fracturas de radio distal utilizando el Método de Elementos Finitos. Rev Ing UC 17(1):28–36 Roberts TT, Prummer CM, Papaliodis DN, Uhl RL, Wagner TA (2013) History of the orthopedic screw. Orthopedics 36(1):12–14 Sauerbier S, Schön R, Otten JE, Schmelzeisen R, Gutwald R (2008) The development of plate osteosynthesis for the treatment of fractures of the mandibular body–a literature review. J Cranio Maxill Surg 36(5):251–259 Strafun S, Levadnyi I, Makarov V, Awrejcewicz J (2018) Comparative biomechanical analysis of stress–strain state of the elbow joint after displaced radial head fractures. J Med Biol Eng 38(4):618–624 Taljanovic MS, Omar IM, Hoover KB, Chadaz TS (2019) Musculoskeletal imaging. Ed. Oxford, New York Wang T, Boone C, Behn AW, Ledesma JB, Bishop JA (2016) Cancellous screws are biomechanically superior to cortical screws in metaphyseal bone. Orthopedics 39(5):828–832 Yañez-Santana A (2009) Diseño y análisis teórico-experimental de un nuevo sistema de sujeción de tornillos de osteosíntesis en huesos osteoporóticos cilíndricos largos. Doctoral disertación de la Universidad de Las Palmas de Gran Canaria, España Yuehuei-H A (2002) Internal fixation in osteoporotic, Bone. Thieme Medical Publishers, New York

Chapter 15

Energetic Numerical Analysis of the Effect of Impact Loads into a Human Skull (Frontal and Lateral) Fransciso Carrasco-Hernández, Guillermo Urriolagoitia-Sosa, Beatriz Romero-Ángeles, Diego Ivan Islas-Jiménez, José Luis Reyes-Reyes, Christian Díaz-León, Martha Eugenia Espinosa-Hernández, Iván González-Uribe, and Antonio Hernández-Cerón

F. Carrasco-Hernández (B) Departamento de Mecatrónica, Universidad de Durango, Mantenimiento y Energías Renovables, Carretera Durango-Mezquital, km 4.5 Gabino Santillán, 34308 Durango, México e-mail: [email protected] G. Urriolagoitia-Sosa · B. Romero-Ángeles · D. I. Islas-Jiménez · J. L. Reyes-Reyes · C. Díaz-León · M. E. Espinosa-Hernández · I. González-Uribe · A. Hernández-Cerón Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Sección de Estudios de Posgrado e Investigación, Unidad Profesional Adolfo López Mateos “Zacatenco”, Avenida Instituto Politécnico Nacional s/n, Edificio 5, 2do. Piso, Col. Lindavista, 07320 Ciudad de México, México e-mail: [email protected] B. Romero-Ángeles e-mail: [email protected] D. I. Islas-Jiménez e-mail: [email protected] J. L. Reyes-Reyes e-mail: [email protected] C. Díaz-León e-mail: [email protected] M. E. Espinosa-Hernández e-mail: [email protected] I. González-Uribe e-mail: [email protected] A. Hernández-Cerón e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications V, Advanced Structured Materials 171, https://doi.org/10.1007/978-3-031-26466-5_15

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15.1 Introduction Since the beginning of time, humans have analysed accident results, so as to prevent possible damage and protect themselves. In the case of accidents, it is essential for the victim to establish the circumstances that led to the incident, correct the situation in which it occurred and anticipate possible future scenarios. Today, there are a number of activities that involve a high risk for the human body (daily life activities, development of work procedures, sports actions, diverse accidents, etc.). Therefore, it is necessary to account the biological, physical and mechanical ranges that the human body is capable to reach. In regard with the research and evaluation of biological systems, current knowledge is based on previous studies that require the researcher to get involved with the knowledge from various fields of numerous sciences, so that the necessity can be understood more easily and in a more complete manner. Such is the case of Biomechanics, it can be said that human beings have always had the concern to understand the biological, physical and mechanical behaviour of living beings. This research field has been applied and has progressed since ancient times. For example, there are studies by Galileo (2009), Finocchiaro (2001) and Borelli (2014) that explored the relationship between human and animal physiology. Sometime later, Wolff (1986), Carter and Beaupre (2001) and Roux (1909, 1917) contributed greatly to the understanding of bones and the progress of Biomechanics (Xu and Grande-Allen 2010). In more recent times, various organizations have shown great interest and provide economic resources into the development of the area of Biomechanics. From the 1960s to the 1970s, the interest varied widely, from rehabilitation processes and equipment for implants and/or prosthetics. The research attached great importance to the potential characterization of bio-implants and exploring the mechanical properties of connective tissues. The impulse generated in the field of Biomechanics, lead in the 70s, a large number of associations and groups to be focused on the research and equipment development (Fung 1993; Kassab 2004; Chien et al. 2009). It is well known that fighting armed conflicts have generated a large number of people affected by head trauma (Splavski et al. 2000). But not only war conflicts trigger the growth in head trauma, also the development and growth of transportation have proved to generate a raise of human injuries (Rueda Arreguin et al. 2020, 2021a, b). The risk of impact increased in the common population, which led to the study of human tolerance to forces (Thurman et al. 1999). However, the appearance of CT scans that show cranioencephalic trauma due to impacting loads, facilitate its diagnosis and treatment. The development of new technologies has generated innovations in the treatment of brain-cranial trauma. In 1917, one of the most important studies was carried out, it was applied to Hugh DeHaven (Mackay 2007), he was a young pilot in the Canadian air force, who was involved in a plane crash. Hugh DeHaven after recovering from the impact realized that he had survived thanks to the design and integrity of his cabin, in conjunction with the seat belt, which protected him from the impact and diminished the risk of suffering catastrophic injuries. Later

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in 1942, DeHaven carried out impact studies at considerable flying heights (Haven 2000), having a speed range of the airplane between 60 and 95 km/h and the objects they hit were; wooden roofs, fences, soft earth and in two cases car awnings. With the calculation of impact speeds and braking distances (deformations in objects), decelerations and reactions were estimated to take into account the collision forces received by the body. In a more specific sense of ideas and aimed at the protection of human body systems. The skull refuges one of the most important organs of the human body, the brain; which not only controls vital functions of the organism, but also gives the human the capacity of being rational. An impact to the head (depending on the severity), can cause various magnitudes of damage into the brain. Because there is no extra space within the skull socket, the brain tends to squeeze and go through the hole at the base of the skull, this could lead to the development of pressure that injures vital parts of the organ and sometimes destroys regions that control basic body functions, such as breathing, blood circulation, among others. Carrying out a numerical study based on the finite element method (FEM) allows quantifying the skull’s capacity to withstand strategically located impact loads, in order to observe the total behaviour and energy absorption of the skull (Yoganandan et al. 1995; McColl et al. 2018; Hickling and Wenner 1973; Kang et al. 1997; Kleiven and Hardy 2002; Deck et al. 2004; Horgan and Gilchrist 2004; Sahoo et al. 2016; Ren et al. 2020; Cruz-Jaramillo et al. 2020, 2021). Knowing this, it is possible to determine a safety range in the use of head protection, in areas of industrial safety, sports, or other human activities in which there is a risk of an impact on the skull (Thompson et al. 1996; Ryan 1992; Saat and Barkan 2005). Currently, researchers are conducting studies based on accident reconstruction, to name a few, are; Newman (1998), proposed a detailed methodology in order to evaluate the injuries in collisions of the players during American games applying the finite element method. On the other hand, Willinger and Baumgartner (2003) used the finite element method to reconstruct 13 helmets in motorcycle accidents, to understand an approximation of the mechanism of injury to the skull. A very important study with which numerical analysis are verified is the experimental model of Nahum et al. (1977), they measured intercranial pressure and other dynamic responses under specific load intensities. To mention other specialists in this field of study; Mackay (2005), Ruan and co-authors (Baumgartner and Willinger 2005), Zhang and associates (Newman et al. 1999). In order to determine an extreme point that the human body could reach, it is proposed in this research some work to carry out by numerical analysis of impact to the skull, considering it as a mechanical structure, implementing the biological properties of the skull and performing a specific numerical modelling of the skull as a biological system. An impact study is proposed, since this type of sudden loads occur in an extremely short time, which causes the material subjected to these loads to increase its fragility and collapse more easily (compared to the case of applying quasi-loads). The present work includes a methodology directed at computer programs aimed at the degree of damage to the skull produced by impact.

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15.2 Methodology/Development 15.2.1 Principle of Momentum and Movement Amount From Newton’s second law it is defined the principle of momentum and momentum, which originally indicates; The acceleration with which a body changes its speed is proportional to the net force acting on it (González-López de Guereñu et al. 2010). This relationship of mass and speed can be considered as a measure of the difficulty of bringing the particle to rest, defined by the following equation: F=

d (mv) dt

(15.1)

The mass and velocity product defines the momentum of the particle; The force acting on a particle is equal to the rate of change over time, of the particle’s momentum, so it can be specified that the greater the amount of movement of a body, the greater the effect required to change this amount of movement (Riley 1996): F=

d dq = q˙ (mv) = dt dt

(15.2)

The momentum is defined as the force that acts on a particle in a period of time (Vidaurre et al. 1996): d = Fdt dt

(15.3)

Now for the cases in which the applied force is different from zero, it is possible to integrate the equation for the limits in which the momentum presents a change: (tf I=

Fdt = qf − qi

d = mvf − mvi dt

(15.4)

ti

Considering the basis of movement, quatity and momentum, in principle can be derived from Newton’s second law; The force momentum acting on a particle in the time interval (∆t = tf − ti ) is comparable to the momentum of the particle in the same time interval (Hewitt 2004). From the principle of momentum and moment quantity, it is said that, when an momentum is given to a particle, this is the result of an external force applied to the particle and, in effect, that the momentum is transferred from an external agent to the particle:

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(tf mvi +

Fdt = mvf

(15.5)

ti

15.2.2 Component Behavior Under Dynamic Loads When a component is subjected to dynamic loads, an inelastic response occurs. The external energy is large enough to cause permanent deformation. In most plastic dynamic structural problems, it is not possible to apply a standard static analysis methodology, with dynamic magnification factors; since the study in this area in general is used for the design of systems that absorb energy. To study dynamic behavior, the methodology proposed by Norman Jones must be as follow (Jones 1989; Martínez-Miranda et al. 2020): 1. Postulate a kinetic velocity field, which describes the behavior of the element studied (suggested by the characteristics of the material static collapse profile). 2. Involve the plasticity requirements, to find the appropriate yielding proportion of surface that is associated with the speed field assumed in the previous point. 3. Complete the solution by integrating the differential equations that govern and satisfy the initial and boundary conditions. 4. It is necessary to examine whether there are no yield violations during any moment of the complete response, or for certain parameter values. If it is violated, a new solution must be acquired by generating an alternating speed field, which can be obtained from the nature of the yielded violation. Also, repeating the previous steps until an exact theoretical solution is found. The skull will be considered as a spherical shell and the element is subjected to an axial symmetric pressure pulse with the history of a rectangular shape represented in pressure versus time axes Fig. 15.1a, and is described by the following equations: p = po' 0 ≤ t ≤ t or p = 0 t ≥ t

(15.6)

where; po initial pressure, T response lapse, t total time and t pressure pulse time. For the case of study, it is considered a 3D component. Also, a completely spherical shell is considered and it is presented the theoretical background for three different material conditions (Hodge 1963; Baker 1960; Walters and Jones 1972; Jones 1973; Urriolagoitia-Sosa 2005): • Elastic. • Elastic, perfectly plastic. • Rigid, perfectly plastic. Figure 15.1b, shows the representation of the spherical shell that is subjected to a spherical symmetric response, observing the plane and radial shear forces, as well

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Fig. 15.1 General dynamic loading conditions for the case of study. a Rectangular pressure pulse. b Spherical shell segment subject to a symmetrical internal pressure. c Dynamic response of an elastic spheric shell under rectangular pressure rt = π /2(ψo = Tt = 0.25). d Shell directional reaction variation during dynamic response. e Radial displacement in dimentional in a shell

Bone tissue

Delineate Fig. 15.2 SCAN IP® program showing the bone tissue in the CT scan

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as the bending moments (which will latter are considered as zero as the spherical response is symmetric). The membrane reactions of forces are Oθ = Oφ = O. The element presented in Fig. 15.1b requires: m

d2 2O v− p=0 2 R dt

(15.7)

where m is the mass per unit area of the shell and the shell has homogeneous cross section. It is considered a symmetric response in the system, so in the same way, the biaxial deformations of the membrane in the spherical shell is εθ = εϕ = ε, from where it follows that; ε = −v/R. v represnts the radial displacement and it is negative because it is considered that it is directed inside the shell. In Table 15.1, it is presented a resume of formulation for the shell model under diverse material conditions to define critical zones into the component.

15.2.3 Skull Numerical Model (Jones 1989) A methodology was implemented to develop a numerical model of the skull that is adequate to reality. Therefore, it was necessary to use a CT scan from the skull. The suggested methodology is divided into two groups (Carrasco-Hernández 2012; Beristain-Lima 2014; Marquet-Rivera 2017): 1. Obtaining and processing of images. 1. A patient is selected to perform the study. For the study of this work, a healthy 27-year-old man was selected, 1.78 m in height and 100 kg in weight. 2. The patient is subjected to a CT scan study, with a 3D visualization of the skull, generating images in DICOM format. Generating an image for each millimeter of cut (Marquet-Rivera 2017). 3. DICOM file processing. DICOM files are processed using the SCAN IP® program (to separate images and cut by cut view). As well as, the bone tissue must be delineated. This program generates a 3D stereolithography file, triangular surfaces are considered the best organic geometry (Bártolo 2011). 2. File refinement, surface generation (export and import). 1. Smooth, order, separate the triangular surfaces generated in the STL file. Applying a VISCAM MESH® program, the orientation of the triangular surfaces are corrected, reduced and smoothed. Also, merged surfaces are repaired and separated surfaces are joined.

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Table 15.1 Behaviour development for diverse material conditions (Carrasco-Hernández 2012) Elastic

Elastic; Perfectly plastic

Rigid; Perfectly plastic

First phase 0≤t≤t

First phase; O ≤ Oo

First phase; Oθ = Oϕ = 0

Oo =o B; −v ≤

−EBv (1−s)R

O=

2 2EBv m dtd 2 v − (1−s)R 2 p = po

d2 v + r2 v − d dt2 2EB r2 = m(1−s)R 2;

v=

= −p

d=

po m

d{cos(rt)−1} r2

ROo (1−s) ; EB

v