Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates 3031170725, 9783031170720

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Advanced Structured Materials

Maxim Yu. Orlov Visakh P. M.   Editors

Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates

Advanced Structured Materials Volume 176

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.

Maxim Yu. Orlov · Visakh P. M. Editors

Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates

Editors Maxim Yu. Orlov National Research Tomsk State University Tomsk, Russia

Visakh P. M. Tomsk State University Controls System and Radioelectronics Tomsk, Russia

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-031-17072-0 ISBN 978-3-031-17073-7 (eBook) https://doi.org/10.1007/978-3-031-17073-7 © 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

This collective monograph presents 18 chapters devoted to the study of the behavior of materials and individual structures at high pressures, strain rates, as well as shock and explosive loading. The reader is offered a list of numerous sources to get acquainted in detail with any of the topics discussed in the book. The authors tried to adhere to the university style in presenting the material, since most of them teach courses at various Russian universities, including national research ones. Therefore, with a certain degree of confidence, we can say that the presented results reflect the current achievements of Russian science. The research work of the authors is supported by various scientific foundations, including the Russian Science Foundation. The book contains the most famous studies, some of which were reported at well-known international scientific events: conference on Mechanics and Materials and Design (MDM), conference on Protective Structure (ICPS), conference on Integrity–Reliability–Failure (IRF), DYMAT, etc. Thematic symposiums were organized specifically to discuss some of the actively developing scientific topics. The authors are known in the Russian Federation for their regular presentations at scientific conferences and publications in Russian scientific journals. Some of the obtained results were discussed in various scientific communities concerning the topic of chapters. As the tool it used commercial software packages, experimental equipment in shared use centers, as well as Russian non-commercial software packages and computer codes developed in the author’s laboratories and departments. So, for example, several chapters are devoted to this. We hope that the algorithms will be developed and improved to simulate the considered class of scientific problems. Each book has its own background. The prehistory of this book is inextricably linked with the annual scientific conference on the continuum mechanics and celestial mechanics, organized by Tomsk State University (www.cimcm.tsu.ru). For almost 10 years of the event’s history, 20 volumes of conference Proceedings have been published in Russian and International publishing houses. Two years ago, a collection of selected articles was published in the Serbian scientific journal “Thermal Sciences”. The total number of participants reached 2000 people from Russia, Kazakhstan, Ukraine, Belarus, Poland, France, Belgium, Germany, Canada, China, Czech Republic and other countries. One of the priorities of this event is the creation of v

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a single information space, the exchange of experience in the field of continuum mechanics, computational mechanics and the design of advanced materials. Some of the chapters in the book are based on plenary reports presented at the conference by authors at different times. To be honest, in some cases this motivated them. Therefore, we would like to thank the program committee for the opportunity. We hope that the second edition of the book will be released in a certain period of time, in which the scientific results would be supplemented with additional materials, including computer animation of computational experiments, high-speed shooting of experiments, infographics, etc. In our opinion, this can clarify some points related to numerical modeling and will be very useful at the stage of testing mathematical models and calculation algorithms. The collective monograph is of scientific and methodological interest for a wide range of specialists in mechanics, physicists, materials scientists working in various fields of modern science and technology, as well as for students, graduate students and early-experienced researchers from universities and technical universities. This book was supported by the Tomsk State University Development Program. Tomsk, Russia July 2022

Maxim Yu. Orlov Visakh P. M.

Introduction

The book presents results of research on materials under impact, explosion load, high pressures and dynamic strain rates. A well-known fact is materials and structures present a complex response when subjected to highly intense dynamic loading. This is due to effects like inertia and material nonlinear response. To understand their behavior, it is necessary for an integrated approach comprising experimental, analytic and numerical analysis. In the book is collected such research and their applications. The monograph aims at providing to the scientific community a synthesis of the research activities performed at the author’s laboratories in the field of the characterization of material constitutive behavior and damage under dynamic loadings. Purely Russian developments for solving modern problems of penetration and perforation are presented in Chaps. 1, 6 and 10. The chapters show the results of numerical experiments, and preliminary test calculations were carried out to establish the reliability of the results obtained. Chapter 1 not only describes the architecture of the programming environment, but also proposes a technology that provides visual representation of complex algorithms based on visual programming of functional object diagrams. Chapter 6 describes the REACTOR 3D software package for solving problems of penetration and perforation. A comparison is made with experimental results in terms of the penetration depth when a long rod penetrates a massive aluminum alloy target. When modeling high-speed collision of compact elements with an aluminum target, new scientific results were obtained. The performed simulations made it possible to modify the analytical formula for calculating the hole diameter in the velocity range from 2 to 5 km/s. Chapter 10 represents the noncommercial software package Impact 2D for simulating modern problems of penetration and perforation. The software package solver is constituted of a numerical Lagrangian method with algorithms for splitting computational nodes and erosion triangular elements. The method has been extensively tested, and comparisons with the analytical solution and experimental data are given in the chapter. The authors attempt to explain the effect of layers in a multilayer metal design on its impact

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resistance against three types of strikers (ogival, conical and blunt ones). A series of numerical experiments was carried out at an initial velocity above the ballistic limits of all targets. Chapter 2 presents combined theoretical and experimental studies of the resistance of glass-reinforced plastic (GRP) and carbon-fiber-reinforced plastic (CFRP) specimens during contact underwater explosions. Research objects were four groups of samples made from various reinforcing materials. The structure of reinforcement ensured quasi-isotropic properties of the material. The research objects were tested in the explosive tank of the Krylov State Research Center. The chapter contains some photographs of the experimental setup, including an explosion chamber, etc. The results of the experiments are shown as photographs of samples after the blast with some graphical dependencies. Computational models have been developed and verified using LS-DYNA and AUTODYN software for the experimental conditions. Based on computer modeling, the details of the stress-strained state in GRP and CFRP are considered by the instant of damage initiation (rupture of fibers). Chapter 3 proposes a hypothesis for the physics of the jet formation process due to plastic deformation of the cumulative lining material. The section along the generatrix of the axisymmetric cumulative lining has a mesh grid, which makes it possible to determine the movement of selected cells in the process of cumulative jet-forming and, consequently, the final location of this cell, i.e. the selected volume of cumulative lining material, in the pestle or the cumulative jet. Knowing which volume of cumulative lining material is used to form the pestle and cumulative jet will enable the design of optimum combined material lining. The chapter presents the results of a numerical experiment on the spatial distribution of the metal of the jetforming layer. The numerical experiment was carried out in the Ansys AUTODYN numerical simulation environment for rapid dynamic processes. The authors of this chapter determined the spatial distribution of the cumulative lining material both in the cumulative jet and in the pestle. The calculations determined the unevenness of the jet-forming process, namely that the deformation of the cumulative lining material proceeds in a wave-like manner, i.e. alternating. The results of the presented materials determined a complicated flow path of the cumulative liner material during product formation of cumulative jet and pestle. Mechanical behavior on a mesoscopic scale and effective mechanical properties of metal matrix composite materials under shock-wave loading were investigated in Chap. 4 by means of the computer simulation method. Shock-wave propagation in the representative mesoscopic volume of metal matrix composite, deformation processes, nucleation and growth of damages and evolution of structure of composites consisted of an aluminum matrix, and randomly distributed ceramic inclusions were numerically simulated. The mechanical behavior of the matrix was described by the elastic–plastic model. The model of the damaged brittle solids was used for inclusions. The problem was solved with the application of the finite-difference method. It was shown that the deformation of composites in the shock-wave front is accompanied by a change in the initial orientation of the structural elements. In this process, it is possible to form a dissipative structure from volumetric blocks, including a

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certain number of inclusions, which are displaced as a whole. The computer simulation predicts the occurrence of cracked ceramic particles, local damages in metal matrix and growth of meso-scale and macro-scale cracks in composite materials under shock-wave loading. The results of numerical calculations were used for the numerical evaluation of effective elastic and strength properties (elastic moduli and elastic limits) of composites with different values of volume concentration and various shapes of ceramic inclusions. The numerical values of the effective mechanical characteristics of investigated materials were obtained, and the character of the dependence of the effective elastic and strength properties on the structure parameters of composites was determined. In Chap. 5, some experimental data are reported on the microstructure, phase composition and microhardness of the titanium surface layers after their modification with a flow of nitrogen, carbon and boron nanoparticles accelerated by a shaped charge explosion. The applied surface modification method has enabled to improve functional properties and may be deployed as an alternative to other ones, based on processing with laser, electron or ion beams, as well as plasma flows. In comparison to them, the key advantages of the shaped charge synthesis are the possibility to control the particle flow rate, obtain a multi-phase composition of the flow, and average particle velocities in the flow even when using powder mixtures of materials characterized by significantly different densities. These capabilities are unattainable within the framework of other above-mentioned methods. The results of both currently reported and some previous studies on the shaped charge synthesis in associated flows have confirmed its perspective for the formation of advanced surface layers with improved functional properties. It should be noted that the early experiments have been carried out using mechanically mixed powders of titanium, tungsten, as well as their carbides and borides. As a result, such accelerated particles have penetrated into the molten surface of titanium targets without synthesis, forming a coarse-grained composite coating based on the matrix material of the using targets. Its thickness has been determined by the penetration depth of the particles. In this study, an attempt had been made to use a novel mixture of specially prepared nitrogencontaining salts, boron powders and multi-walled carbon nanotubes as sources of corresponding atoms. It has been found that initial compounds were chemically decomposed in the cumulative jet, followed by the synthesis of new nanosized ones on the target’s surface from released compound atoms and atoms of target material due to the simultaneous impact of high speeds, temperatures and pressures. During the shaped charge synthesis, such newly formed nanoparticles of titanium nitrides, carbonitrides and borides have formed a strong composite coating with nanoscale inclusions on the titanium surface, possessing a tenfold increase in microhardness. The obtained results have shown that the implemented method opens up wide opportunities for the formation of unique low-dimensional surface layers via the interaction of accelerated nanoparticles containing nitrogen, carbon and boron atoms with titanium substrates. The results of numerical experiments based on the author’s model developed to determine the thermodynamic parameters of high-energy action on heterogeneous

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materials were presented in Chap. 7. The chapter demonstrated the results of computational experiments of shock-wave loading of alloys containing gold as a component. The verification of the obtained results with the available data, which were determined on the basis of the experiment, had been carried out. The thermodynamic equilibrium components of author’s model TEC was based on a low-parameter equation of state and took into account the interaction of components. The preliminary test calculations were carried out to establish the reliability of the results obtained for pure gold and germanium materials. The model allowed us to take into account the polymorphic phase transition of components under the shock-wave loading of the materials, which models based on the additive approximation couldn’t do. New scientific results were obtained by modeling high-energy loading on the investigating alloys. The author specified the scope of the model. This model makes it possible to determine the compositions of heterogeneous materials and the ratios of their components, which make it possible to obtain the necessary pressure and temperature values during shock-wave loading of solid and porous alloys and mixtures. The results of numerical simulation of two-dimensional flows that arose after the dam failure are presented in Chap. 8. At the beginning of the chapter, the results of analytical studies of the problem of the breakup of a special discontinuity necessary for the set of a numerical experiment are presented. For the first time, for discontinuous initial conditions, a solution of a two-dimensional problem is constructed using nonstationary self-similar variables in the form of converging series in physical space. When calculating the flow fields, these series were used to set approximate initial conditions at the moment following the destruction. The boundaries of the computational domain were contact and sound characteristics. Analytical methods have been used to find the laws of motion of the boundaries and the values of the parameters of the medium on them. The obtained formulas were used to set approximate boundary conditions. Since the boundaries of the computational domain are movable, new variables were introduced into the system of equations, in which the computational domain became rectangular. Then the numerical simulation of the flow fields was performed using an explicit difference scheme. The aim of the numerical experiment was to obtain flow fields over a long time interval and identify the features arising in them. Calculations have shown that the use of approximate analytical initial and boundary conditions makes it possible to construct flows over a sufficiently large time interval. We also found the moments of time when there are features on the contact surface. In Chap. 9, estimates were obtained for the protective properties of permeable barriers made of granular material under the influence of strong shock waves. Such barriers are promising elements that protect critical structures from impulse and explosive effects. Similar processes have been described under the assumption that the granular layers are not deformable. A distinctive feature of this work was the rejection of this assumption, since sufficiently intense shock waves cause significant deformations of the granular layers. For the first time in computational practice, the processes of interaction of gaseous media with deformable layers were modeled on the basis of nonlinear equations of the dynamics of two interpenetrating continuums. As interfacial forces, the resistance forces in the flow of gas around spherical particles

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and the Stokes friction force were taken into account. The numerical solution of the equations was carried out according to the modified scheme of S. K. Godunov, adapted to the problems of the dynamics of interpenetrating media. The contact surfaces of the pure gas with the porous granular layer and the pore gas were the discontinuity surface of porosity and permeability. At these boundaries, the laws of conservation of mass, momentum and energy were fulfilled as on a moving jump in porosity. Numerical implementation of the contact conditions was made on the basis of solving the nonlinear problem of disintegration of the discontinuity, taking into account the abrupt change in porosity. When solving the problem, the data obtained earlier by the authors on the dependence of the permeability of the barrier on the degree of its compression were used. These data were taken from the solution of three-dimensional problems of compression of fragments of granular layers. New results were obtained for the impact of intense plane shock waves up to 0.3 MPa with a flat deformable granular barrier of lead balls. The features of the ongoing nonlinear processes, the parameters of shock waves reflected and passing through the barrier were analyzed. It was shown that the parameters of transmitted waves depended significantly on the degree of compression of the layer. Therefore, the evaluation of the protective properties of permeable barriers under the influence of strong shock waves should be carried out taking into account changes in their permeability due to deformation. Developments of load analysis during high-speed impact deformation mechanical equivalents on-board equipment military and civil aircraft are presented in Chap. 11. The chapter demonstrates the results of numerical experiments where highly porous aluminum alloy samples impact a rigid wall and a movable finite mass barrier. The barrier mobility influence on the loading history is revealed. The comparison with experimental results in terms of the length of the deformed sample is carried out. Numerical experiments were conducted in the ANSYS LS-DYNA software package using the 3D Lagrangian method. A direct geometric assignment of porosity was used in the issue. The results were compared with a solution based on Riera’s approach with iterative load correction on barrier movement. Additionally, the author’s onedimensional model based on a deformable highly porous rod was proposed. The behavior of a highly porous medium in a one-dimensional model is described by a modified Carroll-Holt model. Obtained results made it possible to evaluate the accuracy of simplified approaches (Riera’s and one-dimensional model) at impact velocity from 100 m/s to 250 m/s for aluminum alloy samples with an average density from 576 kg/m3 to 1287 kg/m3 . Chapter 12 presents a computational and experimental approach to study the motion of supercavitating projectiles in an aqueous medium. The motion is accompanied by numerous associated processes, such as acceleration of the projectile assembly in a bore, separation of the sabot on an air path, entry and motion of supercavitating projectiles in an aqueous medium, and interaction of the projectiles with a set of targets to hit. It is impossible to solve the whole complex of problems by using only theoretical or experimental methods. In this regard, we propose a comprehensive theoretical and experimental approach to study the supercavitating projectile motion in an aqueous medium which comprises the following branches of

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science: hydrodynamics, mechanics of deformable solid bodies, materials science, internal ballistics and armored ballistics. The study considers the high-speed interaction of a group of bodies in an aqueous medium and the interaction with different underwater obstacles in conditions of various depths and distances in the speed range up to 1500 m/s. The experimental part of the study is performed by using a unique hydroballistic track capable of registering the fundamental ballistic characteristics, the trajectory of the projectiles, the body state in the air and water areas, and the interaction between projectiles and the required set of targets. For each area of the projectile motion, unique software packages have been developed and tested, which provide high-accuracy descriptions of the accompanying processes starting from the projectile acceleration in an accelerator channel and ending with the projectile interaction with obstacles. Some results obtained using the developed computational and experimental approach for a set of supercavitating projectiles in an aqueous medium are presented. Chapter 13 discusses the present issues with slow deformation wave generation and their connection to contemporary geodynamic deformation processes. Slow dynamic deformation disturbances in geomaterials are mostly caused by natural processes in the Earth’s crust and lithosphere and manifest themselves in changes in seismic activity and geophysical fields. A model of autowave deformation processes is put forward in the chapter, and it involves a relay race and consecutive restarts of deformation activity from fault to fault or from one activated part of the fault to another. The model is based on dynamic equations of solid mechanics, constitutive equations of elastoplasticity for geomaterials and cellular automaton conception. For its numerical implementation, both the finite difference method and the cellular automaton methodology were employed. The cellular automaton algorithm for transmitting slow perturbations can be implemented in a variety of ways. Von Neumann and Moore were the two variations used to set the neighborhood in two-dimensional cellular automaton models. The impact of the cellular automata algorithm’s implementation variant on the spread of slow deformation waves was examined. The findings of an innovative investigation on the influence of loading type and fault orientation on wave front form are presented below. The computations took into account both uniaxial tension and compression loading and varied the angle between the fault and the axis of load application. It was discovered that under tension, if the angle is obtuse, jagged fronts first emerge at the fault’s margins before spreading throughout the fault. The wave front of inelastic deformation propagates from the fault’s margins, if the angle is acute, although not along the fault itself, but rather along the boundaries of the calculation zone perpendicular to the direction of loading. Under compression, if the angle is obtuse, triangular-shaped plastically deformed areas are generated along the spreading front. The situation in the case of an acute angle is similar to that in tension from the outset. Later on, however, weakly curved wave fronts of inelastic deformation spread out from the intersection sites, resulting in triangular regions of a medium that has undergone plastic deformation, the regions being elongated along the fault rather than the region’s boundary. It is crucial to understand slow deformation waves since they occur not only in geodynamics but also in other nonlinear active media.

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Chapter 14 presents a method for solving integral equations of dynamic problems for extended media containing flat cracks at the interface subjected to vibrations, including those at high frequencies. The authors proposed a modification of the fictitious absorption method, which makes it possible to solve spatial problems under any conditions of contact between stamps and the medium, as well as in the presence of flat cavities, cracks and inclusions. The use of more complex basis functions allowed us to obtain a solution in a simpler form. In the problems of oscillations for a semi-infinite medium caused by a load on the edges of internal cracks or on the plates of the surface, the kernel of the integral operator is a generalized function. By carrying out a regularization, we arrived at an integral equation with a classical kernel function. The approach developed in the chapter makes it possible to overcome a number of problems that arise when solving the integral equations of dynamic problems in the theory of elasticity under high-frequency effects. The fictitious absorption method, being a semi-analytical method, made it possible to investigate the features of the IE solutions and take them into account during the development of numerical algorithms. Chapter 15 presents the results of the development of a model of a mixed multicomponent polydisperse mechanochemically reacting sintered deformable body, compacted under conditions of stationary and dynamic thermomechanical action. A model of the state of a solid deformed dispersed body with components capable of mechanochemical transformations and sintering has been constructed in a wide range of temperatures and dynamic loading. The developed approach made it possible to obtain a substantiation of technological regimes for the creation of new low-temperature co-sintered composite materials (LTCC) by mechanochemical synthesis and sintering methods. The new complex model takes into account the possibility of phase transitions of components, mechanical activation and chemical transformations of reacting components, processes of heat and mass transfer, shrinkage, sintering of components, and formation of a skeleton of individual fractions of interacting particles of refractory components at various structural levels. The authors developed a method for estimating residual stresses in the matrix of sintered LTCC, taking into account the refractory products of thermal degradation of the binder. A series of computational experiments showed that the possibility of forming a refractory skeleton of any fraction of refractory ceramic components of the mixture makes a decisive contribution to the formation of the structure of sintered LTCC, and the nature of the distribution of zones with the formed skeleton of refractory particles is determined by the parameters of the initial dispersion. Chapter 17 presents the results of experimental and theoretical studies of a force actuator capable of providing shock-free opening of large-sized transformable space structures. The proposed force actuator uses an active element made of titanium nickelide material with a shape memory effect (SME). A spring element was used to simulate the working force developed by the active element in the process of experimental research. To assess the stability of the parameters of the force actuator, original experimental setups were created for deforming active elements and determining their main characteristics. It has been established that force actuators with active elements made of titanium nickelide material with SME have the advantages of generating significant forces, low weight and low power consumption. As a

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result of the research, temperature dependences of shape memory deformation and deformation–force dependences were obtained for active elements of a force actuator that operates under uniaxial compression conditions. To ensure an approximate similarity of the processes during ground tests and in orbit, a model of the operation of the actuator was proposed, which made it possible to determine the voltage of the power supply in space by knowing the voltage of the power supply during ground tests. The model allows calculations with different estimated heat transfer coefficients and also the calculations can take into account changes in the resistance and heat capacity of active elements during their heating. Experiments have confirmed that the actuation time of the actuator decreases significantly with an increase in the voltage of the power supply. At the same time, the working stroke of the active element (a decrease in its relative elongation) remains almost constant despite the difference in the actuation time of the actuator. In Chaps. 16 and 18, ice was the object of research. The chapters have an emphasis on experimental studies that were carried out in the author’s laboratories and ice experimental basins. In Chap. 1, the bending behavior of ice reinforced with polypropylene fibers was studied. Experimental curves are obtained for various schemes of amplifying samples with polypropylene pipes. The behavior of the curves and the formation of cracks in the ice cover is compared with photo and video recording of the samples. It has been experimentally proven that the most efficient scheme was a scheme with four pipes with a cross section. In addition, it was found that polypropylene as a reinforcing material increases the plasticity of ice samples. Chapter 2 is devoted to measuring the ice resistance of the ship’s hull. Numerous towing tests of a typical ship model in continuous flat ice are presented. On the basis of the data obtained, the measurement errors in the assessment of the main ice properties and ice resistance of ships were investigated, and possible ways to reduce them were considered.

Contents

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“Algozit” Programming Environment for Continuum Mechanics Problem-Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valerii O. Kaledin, Anna E. Paulzen, Sergey V. Belov, and Sergey V. Ponomarev 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 “Algozit” Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Example of Algorithm Implementation . . . . . . . . . . . . . . . . . . . . . . 1.4 Application for Experimental Problem Solution . . . . . . . . . . . . . . . 1.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Modeling Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Software Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Estimation of Deformation and Heat Radiation Parameters for Fabric Sample with Damaging Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Correlation of Dynamic Temperature Fields with Experiment Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Functionality of “Algozit” Programming Environment for Calculation of Cooled Structures . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative Assessment of Underwater Explosion Resistance for GRP and CFRP Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrey I. Dulnev and Ekaterina A. Nekliudova 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Materials and Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Computer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Numerical Study of the Geometric Distribution of Metal of Cumulative Lining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ekaterina M. Grif, Anatoliy V. Guskov, Valeriya A. Kiryukhina, Konstantin E. Milevsky, and Alena A. Nesterova 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Statement of the Problem of Metal Distribution of Local Zones of the Inner Surface of the Cumulative Lining . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Statement of the Problem of the Distribution of the Metal of the Inner Surface of the Cumulative Lining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . About Mechanical Behavior and Effective Properties of Metal Matrix Composites Under Shock Wave Loading . . . . . . . . . . . . . . . . . Valerii V. Karakulov and Vladimir A. Skripnyak 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simulation of the Mechanical Behavior of Metal Matrix Composite with Reinforcing Ceramic Inclusions Under Plane Shock Wave Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Investigation of the Mechanical Behavior of Metal Matrix Composites with Reinforcing Ceramic Inclusions Under Plane Shock Wave Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results of Computer Simulation of Damage to Metal Matrix Composites with Reinforcing Ceramic Inclusions Under Loading by Shock Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Investigation of Effective Mechanical Properties of Metal Matrix Composites with Different Concentrations of Reinforcing Ceramic Inclusions Under Shock Wave Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Investigation of the Effective Elastic and Strength Properties of Metal Matrix Composites with Reinforcing Ceramic Inclusions of Different Shapes Under Shock Wave Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaped-Charge Treatment Effects Accompanying the Formation of Hard Structure and New Phase States in Coatings on Titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gulzara Kanzamanova, Sergey A. Kinelovskii, and Alexander A. Kozulin 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

36 37 40

44

47 48 51 53 54

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5.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

8

9

Reactor 3D Software Performance on Penetration and Perforation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aleksandr E. Kraus, Evgeny I. Kraus, and Ivan I. Shabalin 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Taylor Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Non-monotonic Dependence of Penetration Depth on Impact Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Modeling the Crater Formation Process . . . . . . . . . . . . . . . . . . . . . . 6.5 Evaluation of the Ballistic Resistance of Ceramics . . . . . . . . . . . . 6.6 High-Speed Impact on Thin Targets . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation of Thermodynamic Parameters for Gold Alloys Under Shock-Wave Loading . . . . . . . . . . . . . . . . . . . . . K. K. Maevskii 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Calculation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Modeling Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical and Analytical Modeling of Two-Dimensional Water Flows Arising After the Dam Failure . . . . . . . . . . . . . . . . . . . . . . Sergey L. Deryabin and Alexey V. Mezentsev 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Statement of the Problem for Numerical Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation of the Interaction of a Shock Wave with a Permeable Granulated Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anatoliy V. Kochetkov and Ivan A. Modin 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Experimental Studies of the Deformation Properties of a Porous Granular Layer Under Static and Dynamic Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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71 73 80 81 83 83 85 87 90 92 93 98 98 103 103 104 106 109 110 113 114 115 119 120 123 126 127 129 130 131

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9.4 Statement of the Problem of Numerical Modeling . . . . . . . . . . . . . 136 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10 Research of the Behaviour of Multi-layered Steel Targets Impacted by High-Velocity Projectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxim Yu Orlov and Talgat V. Fazylov 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Verification of Numerical Results . . . . . . . . . . . . . . . . . . . 10.3 Numerical Simulation Results of Perforation of Targets with High-Velocity Projectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Problem’s Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Projectiles and Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Perforation of Targets with Ogival Projectile . . . . . . . . . . 10.3.4 Perforating Targets with a Conical Projectile . . . . . . . . . . 10.3.5 Perforating Targets with a Rod . . . . . . . . . . . . . . . . . . . . . . 10.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Approaches to Determining the Load on a Free Body of Finite Mass upon Impact of a Highly Porous Cylinder . . . . . . . . . . . . . . . . . . Yulian V. Popov, Georgy V. Belov, Vladimir A. Markov, Vladimir I. Pusev, and Viktor V. Selivanov 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem’s Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Results of Numerical Calculations and Discussion . . . . . . . . . . . . 11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Comprehensive Experimental and Theoretical Study of High-Speed Entry into Water and Movement of Supercavitating Strikers at Gunfire Start . . . . . . . . . . . . . . . . . . . . . . Aleksandr N. Ishchenko, Viktor V. Burkin, Aleksey S. D’yachkovskiy, Konstantin S. Rogaev, Ivan S. Bondarchuk, Anton Yu. Sammel’, Alexey D. Sidorov, Evgeniy Yu. Stepanov, Andrey V. Chupashev, Vladimir Z. Kasimov, and Leonid V. Korol’kov 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Vacuum Silencer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Air Section of the Track . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146 147 147 148 148 149 149 150 151 155 159 164 165 166 169

170 172 174 175 178 180

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12.2.3 Water Section of the Track . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Intraballistic Studies of the Acceleration of Supercavitating Strikers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 External Trajectorystudies in the Air . . . . . . . . . . . . . . . . . . . . . . . . 12.5 External Trajectory Studies in the Water . . . . . . . . . . . . . . . . . . . . . 12.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Study of the Slow Deformation Wave Propagation from the Faults Having Different Inclinations to the Loading Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aigerim A. Kazakbaeva and Igor Yu. Smolin 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 The Calculations for Different Types of the Neighborhood in the Cellular Automata Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Tension and Compression of a Region with Different Inclinations of the Faults . . . . . . . . . . . . . . 13.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Fictitious Absorption Method in a Dynamic Problem for a Layer Weakened by a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ilya S. Telyatnikov and Alla V. Pavlova 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Fictitious Absorption Method for Solving Integral Equations with an Oscillating Kernel Symbol . . . . . . . . . . . . . . . . 14.4 Transformation of the Integral Equation for the Crack . . . . . . . . . 14.5 Construction of Auxiliary Solutions for IE in Case of Damping Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Solutions of the Integral Equation for an Axisymmetric Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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189 189 192 193 194 194

197 198 199 202

203 203 208 209 211 212 212 214 215 218 221 226 228 229

15 Computer Simulation of Related Problems of Sintering Low-Temperature Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Aleksandr O. Tovpinets, Vladimir N. Leitsin, Maria A. Dmitrieva, and Anastasia V. Puzatova 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

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15.2 Simulation of Sintering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Experimental Investigation of Ice Sample Behaviour When Reinforced with Polypropylene Void Rods . . . . . . . . . . . . . . . . . . . . . . . Alexey S. Vasilyev, Vitaliy L. Zemlyak, and Victor M. Kozin 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Mathematical Modeling of Deployment Dynamics of Large Transformable Space Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir N. Zimin, Alexey V. Krylov, Georgy N. Kuvyrkin, and Artur O. Shakhverdov 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Mathematical Model of the Transformable Configuration Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Model of the Actuator’s Operating . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Estimation of Uncertainty for Measurement of Ship Ice Resistance in Ice Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aleksei A. Dobrodeev 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Fundamental Principles of the Search for Uncertainty in Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Procedures of Ship Model Tests in the Ice Basin . . . . . . . . . . . . . . 18.4 Procedure for Calculation of Uncertainties for Values Measured in Ice Basin Model Tests . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 Measurement Error For Physical and Mechanical Ice Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.1 Flexural Strength of Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.2 Ice Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Measurement Error of Ice Resistance . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

232 235 243 243 245 246 247 249 253 256 259

260 261 265 272 274 276 276 279 279 280 282 283 284 286 288 288 292 293

Chapter 1

“Algozit” Programming Environment for Continuum Mechanics Problem-Solving Valerii O. Kaledin, Anna E. Paulzen, Sergey V. Belov, and Sergey V. Ponomarev Abstract The specific features of the “Algozit” programming environment for functional-object implementation of numerical experiment algorithms in continuum mechanics are investigated. Not only “Algozit” architecture is described, but also the technology providing visibility of complex algorithm representations with convenient program debugging, being based on visual programming of functional-object schemes, is proposed. Implementation of a mathematical model for multi-layer fabric package behavior under rigid element impact is described. Numerical modeling results and the correlation with physical experiment data are presented. Keywords Programming environment · Continuum mechanics · Numerical experiment

1.1 Introduction Software implementation of “grid methods”, especially the finite-element method, is considered to be an overwhelming task for one developer and demands enormous joint efforts of a team, the main reason of which is the existing universally expensive software systems.

V. O. Kaledin · A. E. Paulzen Kuzbass Humanitarian Pedagogical Institute of Kemerovo State University, 23 Tsiolkovsky Str., 654041 Novokuznetsk, Russia e-mail: [email protected] S. V. Belov (B) · S. V. Ponomarev National Research Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, Russia e-mail: [email protected] S. V. Ponomarev e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Y. Orlov and Visakh P. M. (eds.), Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates, Advanced Structured Materials 176, https://doi.org/10.1007/978-3-031-17073-7_1

1

2

V. O. Kaledin et al.

The impossibility for a user to formulate a fully comprehensive view of the numerical methods implemented within these software systems is triggered by the delusion of fully calculated data reliability and excludes the application of investigated methods and algorithms. Moreover, this fact triggers user deskilling in the application domain. Frequently, unfeasible elaborated calculation methods to solve relatively simple problems decrease the work efficiency of computing engineers. When developing and implementing updated algorithms into tailored software programs, software application engineers are faced with the visualization problem of executing algorithms, their modification simplicity, and debugging. In this case, software program applications should be a development framework, including a universal language programming system as only one of the subsystems. The proposed technology, implemented into the functional-object “Algozit” programming environment, provides clear visualization of complex algorithms and possible convenient debugging of programs based on visual programming of functional-object schemes [1, 2]. The application area of this technology is the design of complex algorithms for engineering calculations, and in particular, implementation of the finite-element method on this platform, which could be applied in analyzing the stress–strain state of any object in heavy engineering, complex solid ores, etc. and this technology could be applied in designing other algorithms as well. The present paper describes the application of a software package in simulating the shock processes in fabric obstacles [3–5]. Commercial software packages (LSDYNA, ANSYS, NASTRAN, and others) are applied in the numerical calculation of such problems [6–9]. Besides, some author tools are also applied, which only implement specific behavior models for dynamic deformation and destruction of impactors and targets. World-leading institutes and companies have developed a series of numerical calculation methods for high-speed processes, including obstacle penetration [10, 11]. Nowadays, the experimentally well-proven software program LS-DYNA [12– 14] is being applied in enterprises involved in the design and production of polymer composite and fabric items. As described in the publications [15, 16], the LS-DYNA package has been applied in finite-element modeling of flexible fabric materials, based on the continuum mechanics model where the modeling is restricted to the one-layer fabric material. Finite-element analysis ANSYS software program is one of the most widespread systems for problem-solving in physics and mechanics. ANSYS system has made it possible to plot geometrical models through home embedded applications or imported from external sources. ANSYS Autodyn package has been specially designed and developed for modeling high-speed non-linear dynamics in solid bodies, fluids, and gases. ANSYS Autodyn makes it possible to simulate such high-speed processes as detonation and deflagration explosions, response behavior of structures under intensive dynamic impact outside the range of load-carrying capacity, destruction, fragmentation, material penetration, etc.

1 “Algozit” Programming Environment for Continuum Mechanics …

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Existing commercial program packages are rather expensive and involve restricted program code access. This, in its turn, either hampers the process extension of solved problem classes or makes it impossible. Functional-object programming environment “Algozit” includes simple-to-learn interface components and partially open code, which enables the user to modify the set of defined solutions for specialized problems. Algorithm elaboration is accomplished by visual programming tools, which significantly simplifies the interpretation of complex algorithm structures. The paper is structured as follows: (1) “Algozit” architecture; (2) an example of environment algorithm implementation; (3) an applied problem for stimulated deformation of multi-layer fabric material under rigid element impact; (4) “Algozit” advantages, including an applied problem solution.

1.2 “Algozit” Architecture Figure 1.1 illustrates the diagram of the “Algozit” architecture. The module “Designer” is targeted at designing functional-object schemes by visual programming tools. Software implementation within the “Algozit” environment involves both the encoding of function object classes to C++ language and the visual designing of algorithms as functional-object schemes. Due to a fairly wide range of preprogrammed functional classes, it is possible to dispense complement writing-specific classes in many cases of program development. Functional-object classes are C++ classes determined within dynamic-link library “Object factories.” They are base class descendants, within which specific overlaying and selecting methods are defined: execution, argument initialization, input reading

Fig. 1.1 “Algozit” architecture

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from a file, recording a file, etc. All these methods or any part of them can be overlapped within class-descendants to execute the logic process of one specific algorithm. Base class includes the static methods, within which the behavior of the object as a finite automaton is determined. The specific subprogram of the whole algorithm is realized by every functional class instance being applied in the functional-object scheme. The result is step-by-step execution of the algorithm making it possible to check the transient data through examination of the object memory content or file in combination with the transient data without applying the programming environment. When developing and introducing new algorithms, the usage of the above-mentioned process is simultaneously active. Functional object, a class instance of C++, reveals a behavior distinctly restricted by well-defined rules; it differs from the more conventionally established concept “functional object.” In this case, it is defined as “algomat.” In terms of implementation details, algomat is a class instance; in terms of implementation of the algorithm, as a whole part, algomat is an abstraction level independent of the implementation environment. Algomat provides functionality which computes the values and is stored in memory. Algomat sets, connected by functional relations, generate oriented loop-free graphs (network), representing a functional-object scheme. Graph nodes depict algomats, while the graph edges, their relations. Besides the configuration nodes representing executing objects, there are structure-forming algomats. They regulate the instruction execution: developing loops, branching, scheme extension, and referencing to algomats on other pages. The availability of such tools provides the possible building of advanced command sequences for function blocks and recurring subgraph compression into one or several elements in the initial scheme. Such a scheme is the source representation part of the algorithm itself. It is designed by the module “Designer” tools and is transformed through “Interpreter” as command sequences, according to the construction loops, branching, and other structural units. Command operands are the algomats. The commands are divided into types according to their function: dereferentiation of a reference, computation of algomat, transmission of data from one algomat to another, start of loop, etc. It should be noted that the conversion of the functional-object schemes, included in the command list, is completely automated, and user involvement is not required. Obtained commands are executed through the module “Interpreter”. In the case of downloading compiled program for each algomat, a corresponding class instance is created and initialized. All classes are implemented into a separate dynamically linked library “Objects factory”. These classes have open-source code. As commands are executed, functional classes generate temporary and output data, which, in turn, are written in files and interpreted through the service modules “Data export” and “Graphs.” Service modules are convenient for analyzing output data.

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1.3 Example of Algorithm Implementation One implementation step of the finite-element method in the “Algozit” programming environment is illustrated. An example of the functional-object scheme to compute local loads is illustrated in Fig. 1.2. The functional objects and their relations in building local load vectors for the majority of finite element types are demonstrated on the page. The page includes the object “Iterator of FE” for circular computing. All objects within the iterator are calculated for each finite element. After each successive step of the calculation, the results are accumulated in the algomat memory “Total load”. After the cycle is performed, the object data are downloaded into the text file. In Fig. 1.2, the blueframe linked objects are off-page references. In order to translate the page to the queue commands, the object type “Result” should appear. The translation starts from the node “Result;” further, its dependent arguments are analyzed, and all references are resolved.

Fig. 1.2 Design window of functional-object scheme

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1.4 Application for Experimental Problem Solution 1.4.1 Problem Statement Deformation of multi-layer fabric material under rigid element impact is simulated. Physical testing of multi-layer fabric samples on impact is the main method to obtain data on strength of projected obstacles. However, this is pertinent to high expenses. Numerical modeling of the rigid element shock processes in fabric casing [17] has become the topic of intensive research, and, it is this, that makes it possible to investigate structure behavior in the setting of different design parameters [18, 19]. However, excluding the advantages, applied commercial software packages are not only rather expensive, but also designed to solve only specific problems, whereas their restricted program code access excludes either possible upgrading of existing objects and extending solved problem classes or this process becomes time-consuming and labor intensive. Actually, either high-end computers (HPCs) are required to calculate multi-layer fabric packages under impact [16, 20–23] or simplified models should be established [24–27]. The balance is achieved by applying indirect estimation of absorbed energy by dynamic temperature field registration and numerical model adjustment according to the experiment data. This requires the integration of the heat emission process into the model which directly implies complexity and flexible modification of the algorithm calculations.

1.4.2 Modeling Object The material object, exposed to high-speed impact, is represented as a multi-layer fabric sample. Fabric layers are arranged freely without any intertwining. Each fabric layer is an orthogonal overlapping of two fiber families. Fibers could be arbitrary angel packing relative to the edge. Three structure heterogeneity are considered (Fig. 1.3): integral package layer, layer level, and micro level, constituting layer fibers—warp and weft which deform under conditions of slipping to the layer as a continuum environment. Under impact conditions, the sample center contacts with the impactor which has preassigned mass M and initial velocity V o . These values are the variables (parameters) specifying this impact. During the exposure process, the impactor is decelerated, while its kinetic energy transforms into the work of fiber friction force, structural changing and material fiber damage, irreversible work of stresses at the base, and heating of the impactor and fiber samples. In this respect, state variables, i.e. output parameters are displacement U, velocity V, internal stress Σ, and temperature T, change in time.

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Fig. 1.3 Three-level package structure of fabric layers: a multi-layer packing; b fabric layer; c fiber in fabric layer

Model parameters are both geometrical dimensions, physico-mechanical and thermo-physical material constants. The model includes the following simultaneous processes: • fabric layers’ motion coupled with damaging element followed by deceleration due to external viscous base reaction forces and internal forces—stresses in fabric fibers and fiber friction force during relative displacement; • fiber distortion and stretching, resulting in both reversible and irreversible stress within the fibers; • fiber relative displacement in the layer, as well as layer displacement within the package. • friction force transition to thermal energy; • transition of partial work of stress on irreversible deformation into thermal energy. There are two impact stages in the model: wave and shell type. Both model stages include material governing equations, displacement compatibility conditions, kinematic relations, motion equations, and initial and boundary conditions for displacement and stress. The wave stage is restricted by fabric consolidation time involving initial porosity elimination at the longitudinal wave passing from the surface of impact. Simultaneously, fibers break in the fabric layers which are close to the contact area with the damaging element. Velocity and displacement in the wave stage are considered to be the initial conditions in calculating the shell-type stage where penetration into the fabric and deceleration of the impactor occur.

1.4.3 Software Implementation Successive software implementation of the above-mentioned problem is as follows. Importing initial data of the topolo-geometrical model initiates the execution of the algorithm. Further, the cycle is introduced where the iteration loop repeats a preassigned number of time steps. The overall impact time and the number of single

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stepping are specified in the source file “options.txt”, while time step τ is calculated as the ratio of these variables. Discrete motion equations are integrated according to the specific algorithm over time. Each time step is split into two stages—Eulerian and Lagrangian. The first stage involves the successive calculation of local stiffness matrices, viscosity and mass of multi-layer package elements, and local load matrices. These matrices depend on the displacement and velocity of transient and relative motion at step start. Then, the equation global coefficient matrices are assembled and transfer displacements and velocities are determined by solving the simultaneous linear algebraic equations. The second stage involves the calculation of local fabric layer element matrices which depend on the relative displacement at step start and determined transfer displacements. The global matrices are assembled and relative displacements at the step end are determined by the solution of simultaneous linear equations. Consequently, the stress in fabric layers, absorbed energy, and adiabatic temperatures in quadrature element points in all layers are calculated. The numerical solution algorithm for multi-layer package deformation under the impact, including damaging elements, could be illustrated in the following blockdiagram (Fig. 1.4). The functional scheme for an algorithm is illustrated in Fig. 1.5. The above-mentioned diagram illustrates each algomat as a single application. The output is calculated by data uploading into the test file. The deformation model in the shell-type stage involves two types of finite elements: quadrilateral membrane element—to calculate relative motion parameters, and quadrilateral shell element—to calculate transient motion parameters. A separate application is designed to calculate local matrices for every stage, whereas only one and the same application is used for assembly and simultaneous linear equation solutions. Executed applications are enclosed in the iterator “Cycle over time”; its index object is the panel “Number of steps” where data is downloaded from the text file with options (Fig. 1.5). Exception is the “Initial state”, application where output results are downloaded into the fiber state file, considering accepted initial conditions after the wave stage. According to the diagram (Fig. 1.5), application execution results, involved in each of the stages, are the simultaneous linear equation solutions as (

){ }t+1 { }t M + τ C + τ 2 K U˙ = M U˙ − τ K {U }t + τ {F}

(1.1)

whereas, to determine nodal displacements, the following expression should be applied: { }t+1 {U }t+1 = {U }t + τ U˙

(1.2)

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Fig. 1.4 Algorithm block-diagram including process-steps

Determining left-side expression (1.1) involves the successive execution of the application line: “Local matrices”—“Renumbering”—“Local loads”—“Assembly.” Based on the execution of these applications, mass, viscosity, and stiffness matrices are established. Mass and stiffness matrices are also applied in determining the right side of the simultaneous linear algebraic equations. In this case, the local matrices are assembled on the basis of shell element topology. The applications “Renumbering” and “Assembly” are included in the standard application “Algozit” for the finiteelement method, and are not changed. After each time, energy dispersion and adiabatic temperature increments are determined. Changing the mathematical model and numerical scheme could be required to maintain the modeling adequacy while changing full-scale testing conditions. To improve the program controllability and its adjustment to sudden modifications, the functional-object schemes are divided into pages (related object aggregates) according to the equations of defined models: objects of governing equations for materials (stress under non-linear elastic deformation, viscous friction, and others), topology, element geometry, objects of basic functions and interpolation, objects of quadrature formulae, and other specific library pages of finite elements. Thus, if one

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Fig. 1.5 Functional scheme page illustrating algorithm implementation including process-steps

equation group in the mathematical model or approximation method change, then only the pages which include the objects relative to these changing aspects in the model itself should be changed.

1.4.4 Estimation of Deformation and Heat Radiation Parameters for Fabric Sample with Damaging Element One-layer fabric sample will be considered to examine the behavior adequacy of the fabric obstacle under impact with the damaging element in the shell-type stage of deformation. This one layer is an orthogonal intersection of two fiber families at zero fiber angle lengthwise to the sample edge. A square sample on a side of 0.06 m is examined (Fig. 1.6). Time step τ = 2 ms, whereas all calculations involve 300 steps. Figure 1.10 illustrates nodal displacement in the fabric sample at time points 0.1 m/s, 0.3 m/s, and 0.6 m/s (50, 150, and 300 steps, respectively) at damaging element velocity of 100 m\s and impactor radius −0.00635 m. The results showed that before the time point 0.1 m/s, a formed dome with maximum displacement was observed in the central section of the sample. Thinning fiber packing was revealed in the cross-shaped zones centered lengthwise towards the edges. Increasing impactor and fabric contact time further increase displacement in the impact point, while in the course of fiber movement, more and more fabric surface is involved.

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Fig. 1.6 Fabric sample displacement at time points: a 0.1 m/s, b 0.3 m/s, and c 0.6 m/s at impactor velocity 100 m/s, m

Figure 1.7 illustrates the distribution of the displacement for one and the same fabric obstacle, but the impactor velocity, in this case, equals 30 m/s and the impactor radius is 0.0127 m. As we can see in Fig. 1.7, at increasing damaging element and fabric contact time, only the cross-shaped zones from the center to the edges are involved during this process, while angular parts of the fabric sample revealed insignificant deformation. The central part dimension, involving the most significant displacement throughout the impact process, practically does not change. Obtained results on deformed dome formation correlated with the experiment results [28, 29]. A 24-layer fabric obstacle model was used to analyze the influence of initial damaging element velocity. Warp and weft fibers showed zero reinforcing angles relative to the edge. Two fiber families had different initial and final elastic modulus values: 12.7 GPa and 14.2 GPa, respectively. Figure 1.8 illustrates the calculating results of adiabatic temperature for facing fabric layer at different initial impactor velocities. At a velocity less than 500 m/s, maximal actual stress does not exceed the ultimate stress. In this case, it is possible to conclude that the cross-shaped zones with increased temperature are formed as a result of energy absorption due to fiber friction (Fig. 1.8a– c). At an impactor velocity of 500 m/s, maximal actual stress exceeds the ultimate stress. In this case, the zones with the highest increased temperature revealed energy absorption as a result of the destructive energy on the fibers (Fig. 1.8d).

Fig. 1.7 Displacement of fabric sample in the following time: a 0.1 m/s, b 0.3 m/s, and c 0.6 m/s at impactor velocity 30 m/s, m

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Fig. 1.8 Temperature changes on the package surface at the following initial impactor velocities: a 100 m/s, b 200 m/s, c 400 m/s, and d 500 m/s, C°

1.4.5 Correlation of Dynamic Temperature Fields with Experiment Data Full-scale experiments were conducted by means of an impact-induced steel ball on the multi-layer fabric obstacle at initial velocity variations of the impactor. The surface impact itself was registered on an infrared imaging system, Fig. 1.9 [19]. The experiment unit includes a multi-layer fabric sample, placed between the base surface and shot firing unit; registering unit for the impactor velocity (3); infrared imaging system (4); data-processing system (10); and a sensor for motion start (6). Infrared imaging system (4) is positioned in such a way such that the field-of-view could embrace the location of the impact. Unit (6) input (entrance) is connected to unit (3) output (exit), while unit (6) output (exit) is connected to the infrared imaging system (4) input (entrance). System (4) output (exit) is connected to system (10) input (entrance). RUSAR aramide fiber-based multi-layer fabric package, involving different weave-structured types and total package surface density, was examined. The experiment included steel ball shot firing of the layered package. The steel balls were 6.3 mm in diameter and 1.05 g. Temperature field was registered by the infrared imaging system IRTIS 2000. Figure 1.10 illustrates the recorded thermogram fabric material images (in C°) after the interaction with the impactor, as well as the calculated temperatures. The thermograms (Fig. 1.10) registered significantly increased temperature within the damaging element and fabric obstacle contact zone, compared to the initial temperature. However, the thermograms indicated that the temperature distribution

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Fig. 1.9 Experiment unit diagram: 1—investigated multi-layer fabric package; 2—plasticine block (if available); 3—unit to register impactor velocity; 4—infrared imaging system IRTIS 2000; 5— shot firing mechanism; 6—motion start sensor for damaging element; 7—infrared imaging system field-of-view; 8—impact location; 9—velocity vector of damaging element; 10—data-processing system

Fig. 1.10 Thermograms of the sample at damaging element velocity: a 300 m/s; and b 500 m/s, C°

pattern is different. This could be explained by the fact that in the first experiment— there was no penetration into the fabric while, in the second experiment—there was a breakthrough (disruption). Figure 1.11 illustrates the deformed shapes of the layered fabric shell under the impact with the damaging element, whereas the colors indicate the temperatures in the nodes of the finite-element model. The most significant displacement was observed in the cross-shaped zones from the center to the edges, directionally to the fiber packing. Consequently, increasing temperature could be observed within these zones

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Fig. 1.11 Simulation experiment results under conditions of the following damaging element velocity: a 300 m/s and b 0 500 m/s: nodal displacement in finite-element model of layered shell, m

(Fig. 1.10a). From a different angle, another result was observed (Fig. 1.10b)—the zone with the most significant displacement and increased temperature on the fabric sample surface localized in the impact point with the damaging element. Unlimited displacement increases in the impact point (Fig. 1.10b) revealed the possibility of penetration. To confirm this assumption, actual (effective) deformation and stress results are required. In the case under consideration, the ultimate deformation value is exceeded. Thus, the mathematical simulation (modeling) results will be in good agreement with the experiment data only after selecting the model setup parameters.

1.4.6 Functionality of “Algozit” Programming Environment for Calculation of Cooled Structures Calculation of temperature deformation of cooled structure elements, including electrical and electromechanical units for which temperature distribution in material volumes is established, could be executed in the “Algozit” programming environment. The algorithm of temperature-induced stress and deformation in cooled structures are introduced into the functional-object system as a separate component. This, in its turn, is also applied in estimating static and durability under conditions of mechanical actions. In view of this, the same finite elements which are applied in analyzing deformation under mechanical loads could be included in the computation grid. Additional material characteristics, such as thermal expansion coefficients, are introduced and stored with other physico-mechanical characteristics. Consequently, not only static linear thermos-elasticity, but also stability problems under conditions of correlated mechanical and temperature interaction could be solved.

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1.5 Conclusion “Algozit” programming environment is a flexible tool for developing and debugging new calculation algorithms. Open library “Object factories” provides the user to complement its personal functional classes in order to expand the problem-solving classes. Applying visual programming to functional-object schemes proves the visualization of developed algorithms. Step-by-step execution of a high-level program gains the check-out of intermediary results of each element in the scheme. The software package is intensively applied in technical computing within numerous engineering domains. An example of adapting the “Algozit” programming environment system to implement a mathematical deformation model of a multi-layered fabric sample under impact with a rigid active object and heat liberation within the material is described. For the considered problem, an integrated investigation of dynamic temperature fields in multi-layered polymer samples under impact conditions of a rigid spherical element is conducted. It was proved that the experimentally measured temperatures, both qualitatively and quantitatively, are in agreement with the mathematical modeling results. Acknowledgements The work was carried out within the framework of the State mission of the Ministry of Science and Higher Education of the Russian Federation (theme 0721-2020-0036).

References 1. Kaledin VO, Gileva AE (2017) Functional-object programming mathematical modeling algorithms. In: Materials XXI international science and research conference (in memory of the Mikhail Fedorovich Reshetnev, General Constructor of space vehicles and rocket systems) “Reshetnev Reading”, Reshetnev Siberian State University of Science and Technology, Krasnoyarsk, 8–11 November 2017 2. Kaledin VO, Paulzen AE, Ulyanov AD (2020) Automated visualization of calculation results in the Algozit programming environment. IOP Conf Ser Mater Sci Eng 865 012019. https:// doi.org/10.1088/1757-899X/865/1/012019 3. Batoz JL, Ben Tahar M (1982) Evaluation of a new quadrilateral thin plate bending element. Int J Numer Meth Engng 18:1655–1677. https://doi.org/10.1002/NME.1620181106 4. Bazhenov SL (1997) Dissipation of energy by bulletproof aramid fabric. J Mater Sci 32:4167– 4173. https://doi.org/10.1023/A:1018674528993 5. Mamivand M, Liaghat GH (2010) A model for ballistic impact on multilayer fabric targets. Int J Impact Eng 37(7):1056–1071 6. Blankenhorn G (2003) Improved numerical investigations of a projectile impact on a textile structure. In: 4th European LS-DYNA Users conference: proceedings of the European users conference, Ulm, 23–24 May 2003 7. Zheng D, Cheng J, Binienda W K (2006) Numerical modeling of friction effects on the ballistic impact response of single-ply tri-axial braided fabric. In: 9-th international LS-DYNA users conference, Dearborn, 4–6 June 2006 8. Wang Y, Sun X (2001) Digital-element simulation of textile processes. Compos Sci Technol 61:311–319

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9. Zhou G, Sun X, Wang Y (2004) Multi-chain digital element analysis in textile mechanics. Comps Sci Technol 64:239–244 10. Ha-Minh C, Imad A, Kanit T, Boussu F (2013) Numerical analysis of a ballistic impact on textile fabric. Inter J Mech Sci 69:32–39 11. Mossakovsky PA, Antonov FK, Kolotnikov ME, Kostyreva LA et al (2012) Experimental investigation and FE analysis of fiber woven layered composites under dynamic loading. In: Proceedings of the 12th international LS-DYNA users conference, Dearborn, 3–5 June 2012 12. Dolganina NY, IgnatovA AV, Shabley AA, Sapozhnikov SB (2019) Ballistic resistance modeling of aramid fabric with surface treatment. Commun Comput Inf Sci 965:185–194 13. Hallquist JO LS-DYNA Theoretical Manual (1998) Livermore software technology corporation. Livermore 14. LS-DYNA. Keyword User’s manual. Volume II material models version R7. Livermore software technology corporation, Livermore 15. Tabiei A, Ivanov I (2002) Computational micro-mechanical model of flexible woven fabric for finite element impact simulation. Int J Numer Meth Engng 53:1259–1276. https://doi.org/10. 1002/NME.321 16. Shahkarami A, Vaziri R (2006) An efficient shell element based approach to modeling the impact response of fabrics. In: 9th international LS-DYNA users conference: proceedings of the international users conference, Dearborn, 4–6 June 2006 17. Reddy JN (2003) Mechanics of laminated composite plates and shells: theory and analysis. 2nd Ed. Boca Raton 18. Kaledin VO, Budadin ON, Gilyova A Ye, Kozelskaya SO (2017) Modeling of thermomechanical processes in woven composite material at blow by the striking element.: IOP Conf Ser J Phys 894:012019. https://doi.org/10.1088/1742-6596/894/1/012019 19. Kaledin VO, Gileva AE, Budadin ON, Kozel’skaya SO (2018) Quality control of armor fabric by modeling thermomechanical processes under projectile impact. Russ J Nondest Test 54(5):363– 371 20. Setoodeh Sh (2005) Optimal design of variable-stiffness fiber-reinforced composites using cellular automata (Master Thesis), Shiraz University 21. Sevost’yanov PA, Monakhov VI, Samoilova TA, Dasyuk PE (2016) Modeling fabric sample elongation and breaking dynamics, taking account of random variations and changes in fabric structure and interaction of yarns. Fibre Chem 47(6):501–504 22. Smolin A, Shilko EV, Buyakova SP, Psakhie S et al (2015) Modeling mechanical behaviors of composites with various ratios of matrix–inclusion properties using movable cellular automaton method. Def Technol 11:8–34 23. Sueki S, Soranakom C, Mobasher B, Peled A (2007) Pullout–slip response of fabrics embedded in a cement paste matrix. J Mater Civ Eng 19(9):718–727 24. Gu B (2003) Analytical modeling for the ballistic perforation of planar plain-woven fabric target by projectile. Compos B: Eng 34:361–371 25. Guiberteau F, Padture NP, Lawn BR (1994) Effect of grain size on hertzian contact damage in alumina. J Am Cerum Soc 77:1825–1831 26. Ha-Minh C, Imad A, Boussu F, Kanit T (2013) On analytical modelling to predict of the ballistic impact behaviour of textile multi-layer woven fabric. Compos Struct 99:462–476 27. Jacobs MJN, Van Dingenen JLJ (2001) Ballistic protection mechanisms in personal armor J Mater Sci 36:3137–3142 28. Kobylkin IF, Selivanov VV (2014) Materialy i struktury legkoj bronezashchity (Materials and structures of light armor). Bauman MSTU, Moscow 29. Orlov MYu, Glazyrin VP, Orlov YuN (2020) Research of the projectile’s layout for penetration capability through metal targets. J Phys Conf Ser 1709 012001. https://doi.org/10.1088/17426596/1709/1/012001

Chapter 2

Comparative Assessment of Underwater Explosion Resistance for GRP and CFRP Specimens Andrey I. Dulnev and Ekaterina A. Nekliudova

Abstract The results of experimental and computational studies on the resistance of glass reinforced plastic (GRP) and carbon fiber reinforced plastic (CFRP) specimens under proximity underwater explosions are given. Four groups of specimens made with different reinforcement materials were tested. The structure of reinforcement ensured quasi-isotropic properties of the material. Based on experimental data, the patterns of damage accumulation and failure for these materials are considered. The damage to material is assessed in terms of three criteria: damage to binder, rupture of individual fibers, and through-type fracture. Experiments have established that the resistance to the explosion of GRP specimens is much higher than that of the CFRP specimens under consideration in terms of individual fiber ruptures. Computer models have been developed and verified using LS-DYNA and AUTODYN software for the experimental conditions. Based on computer modeling, the details of the stress-strained state in GRP and CFRP are considered by the instant of damage initiation (rupture of fibers). Keywords Underwater explosion · Glass reinforced plastic · Carbon fiber reinforced plastic · Experiment

2.1 Introduction Ship hull structures in operating conditions, apart from static and quasi-static loads, may be subject to the loads excited by unsteady dynamic effects like an explosion or high-speed shock, which they have to sustain and remain functional. The performance of hull structures largely depends on the load resistance of the materials used for their manufacture. In recent years, composite materials are increasingly used in manufacturing ship hulls. This type of material not only features high specific strength, A. I. Dulnev (B) · E. A. Nekliudova Krylov State Research Centre, Moskovskoe shosse 44, 196158 St. Petersburg, Russia e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Y. Orlov and Visakh P. M. (eds.), Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates, Advanced Structured Materials 176, https://doi.org/10.1007/978-3-031-17073-7_2

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good corrosive resistance, low conductivity, and non-magnetic performance but also provides a wide range of capabilities to make structures with effective resistance to any type of load. Resistance of composite materials depends on the reinforcement material and binders, type of reinforcement, manufacturing processes, etc. These factors have broadly been studied with respect to static loads and to a lesser degree with respect to unsteady dynamic loading, while such investigations are relevant for the selection of advanced composite materials for hull structures. This study considers the resistance of composite materials under proximity underwater explosion. Various aspects of underwater explosion effects were considered in [1–13]. The fundamental role in the analysis of material dynamic strength belongs to test data. In this connection, the studies pay much attention to the technology used to obtain experimental results. In [3, 5, 6], the loading of specimens was done in a conical shock tube where a micro-charge of explosive generated a plane underwater shock wave. In [4, 8, 9], tests were conducted using a cylinder shock tube. In this case, the underwater shock wave is formed by an impact on the water of a heavy piston accelerated by a gas gun. In [10–12], the tests were conducted in an explosion tank where specimens were fitted in a special setup and exposed to blasts of explosives of various masses. In [1, 7], the specimens were tested in an open-water range. Research [13] gives some comparisons of explosion resistance for carbon fiber reinforced plastics and glass reinforced plastics. Along with tests for analysis of deformation dynamics in composite material specimens [2–6, 8–13], analytical methods and computer models were developed for the analysis of underwater shock wave effects on specimens. This paper is using test results [10–12] to consider the explosion effects on the amount and type of damage to GRP and CFRP specimens. The paper is structured as follows. Section 2.2 describes the mechanical characteristics of tested composite materials and geometric parameters of test specimens. Section 2.3 describes the test conditions (explosive charges, recorded parameters, etc.), as well as criteria to assess the test results. Experimental results are discussed in Sect. 2.4, energy capacity is analyzed and explosion resistance of tested composite material types is compared. Section 2.5 considers computer modeling of the stress-strained state of GRP and CFRP specimens in given test conditions. Section 2.6 summarizes the conclusions drawn from the studies.

2.2 Materials and Specimens This section describes the composite materials used to make the test specimens (versions of reinforcement fabric, types of layup, and mechanical characteristics). The geometric characteristics of test specimens are given. Four types of materials made with different reinforcement fabrics were tested: • type 1—twill weave carbon fabric (layup 0°/90°) and biaxial diagonal carbon fabric (layup +45°/−45°);

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Table 2.1 Main parameters of specimens Material type

Number of specimens

Thickness, δ, mm

Density, ρ, kg/m3

Surface mass, m, kg/m2

Number of plies

1

4

5.85–5.94

1503–1526

8.93

14 + 6

4

5.16–5.21

1519–1532

7.91

12 + 6

2

4

5.73–5.76

1493–1500

8.60

7+6

3

4

5.28–5.37

1551–1577

8.33

7

4

6

7.91–8.75

1740–1860

14.71–15.23

10

5

7.36–7.40

1960–1990

14.49–14.69

10

5

5.18–5.20

1970–2000

10.20–10.40

7

5

3.66–3.75

1990

7.28–7.46

5

layup in stack [(0°/90°)2 /(+45°/−45°)1 /…/(+45°/−45°)1 /(0°/90°)2 ]; • type 2—biaxial carbon fabric (layup 0°/90°) and biaxial diagonal carbon fabric (layup +45°/−45°); layup in stack [(0°/90°)2 /(+45°/−45°)1 /…/(+45°/−45°)1 /(0°/90°)2 ]; • type 3—quadraxial carbon fabric; layup [0°/ + 45°/90°/−45°]; • type 4—quadraxial glass fabric; layup [0°/ + 45°/90°/−45°]. All specimens were made of the above materials by vacuum molding (infusion method) using a vinyl ester binder. Parameters of specimens are given in Table 2.1. The main mechanical characteristics of the materials under considerations obtained in static tests are given in Table 2.2. Specimens of each type have approximately the same characteristics. Layup in stacks provided quasi-isotropic qualities of materials in the reinforcement plane. Specimens were round plates with a radius of 300…400 mm. Each plate had 18 holes for bolts. The installation radius of bolts was 250 mm. Specimens of less than 7 mm thickness were made with ring molding in way of bolted joints. It let us ensure the maximum similarity in patterns of damage for the specimens, namely inflict damage in the middle part of specimens (rupture of fibers, through fracture) and prevent damage in supports or bolted joints, which are accidental and significantly depend on the way how the specimens are fixed.

2.3 Experimental Setup This section considers the explosion tests of GRP and CFRP specimens and parameters registered during these experiments. Criteria used to assess the test results are given. The specimens were tested in the explosion tank of the Krylov State Research Centre. The inner diameter of the test portion is about 3600 mm (access hatch is

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Table 2.2 Main mechanical characteristics of materials Characteristic

Orientation, deg

Type 1

Type 2

Type 3

Type 4

Modulus of elasticity, GPa

0

50.8

50.0

42.4

21.8

45

33.2

37.5

45.6

20.8

90

50.0

55.3

47

22.2

Shear modulus in reinforcement plane, GPa

0

14.0

11.5

9.25

8.0

45

7.83

6.85

9.59

8.9

90

14.90

11.2

–

8.4

Interlaminar shear moduli, GPa

0

2.77

3.03

4.49

4.2

90

2.70

3.13

4.17

4.5

Poisson’s ratio in plane 12

–

0.27

0.27

0.32

0.29

Ultimate tensile elongation, %

0

1.09

1.98

1.92

2.6

45

1.96

1.85

1.74

–

90

1.22

1.94

1.90

2.8

Ultimate tensile/compression strength, MPa

0

568/350

993/372

778/372

394/371

45

574/315

619/359

752/372

334/368

90

607/278

1076/414

866/387

410/357

Ultimate interlaminar shear strength, MPa

0

44.1

40.0

55.2

–

45

37.6

48.0

55.8

–

90

38.0

42.0

57.2

–

1150 mm), and the height is 4000 mm. Inside of the tank is clad with a special antireflection coating. Figure 2.1 gives a photo of a test specimen setup being loaded into the explosion tank and a post-test photo of a specimen setup. The tests were conducted according to the scheme “water−air”, i.e. there was an air space on the side opposite to explosion. Each specimen was exposed to one explosion. In all tests, the charge was placed in front of the specimen center at a fixed stand-off distance (300 mm). The charge mass Q was varied (from 8 to 90 g). An explosive plastic of cylinder shape was used with a height-to-diameter ratio equal to unity. Composition of explosive: hexogen—80%, phlegmatizing agent—20%. Density of the explosive is 1350 kg/m3 , velocity of detonation is 7200 m/s, and TNT equivalent for specific energy of blast is ~1. In accordance with the relations contained in [14], the design parameters of shock wave acting on a specimen in tests were as follows: maximum front pressure pm = 35…85 MPa, and exponential decay constant θ = (2.0…3.1)·10–2 ms. The test specimen was fixed between the tubing and the pressure ring bar using bolts (Fig. 2.1). The diameter of the specimen’s effective area when it was fixed to the tubing was 400 mm. In the process of the tests, strains opposite the blast (back) surface of the specimen as well as blast pressure in open water were recorded. Strain gauges for measuring

2 Comparative Assessment of Underwater Explosion Resistance …

21

Fig. 2.1 Loading of a specimen in the explosion tank and a specimen setup after tests

strains in radial and circular directions were placed in the specimen’s center and at various points of the circle with a radius of r = 110–125 mm. Pressure records made it possible to control the completeness of charge detonation and TNT equivalent as well as make sure that wave reflections from walls of the explosion tank have no effect on specimen strains. As a result of tests, the explosive charge mass corresponding to the various levels of specimen damage was determined. The rate of damage was assessed using three signs (criteria): 1—damage to binder (for GRP specimens this was visually recorded as a change in color (bleaching) of specimens in the middle part after test); 2— rupture of individual fibers; 3—through-type fracture (breach). In accordance with these signs, three levels of explosion resistance were established. To generalize the test results for specimens with different surface mass, the relative mass of explosive β = m/mex was used, where mex, kg/m2 is the mass of explosive charge per unit of specimen’s effective area; m, kg/m2 is the surface mass of the specimen. Thus, the relative charge mass giving rise to one or other type of damage is the measure of specific explosion resistance of specimens.

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2.4 Test Results This section presents the results of a test series performed to study the damage patterns of GRP and CFRP specimens under exposure to underwater explosions of different power and to obtain comparative estimates of explosion resistance of these materials. Figures 2.2 and 2.3 show photos of characteristic damage patterns. The test results show that as the power of explosive impact (mass of charge) grows, the first damage is suffered by the binder in the specimen’s central part and in way of the specimen support, the damaged patch is then gradually spreading over the entire effective area of the specimen. Next, individual fibers are broken in the middle part, starting from the specimen back ply. The rupture of fibers at this stage is chaotic as a rule. Gradually, the number of plies with broken middle fibers grows. Finally, a through fracture occurs. Table 2.3 contains the relative masses of explosive charges for different levels of specimen explosion resistance obtained based on experimental data (test data for type 1 CFRP specimens are grouped together). The specific resistance to explosion characterized by through damage is equal for all groups of GRP specimens (with less than 15% variations). A typical pattern of through damage is two crossing main fractures/ruptures directed at about ±45º to the warp of reinforcement material. It should be noted that the resistance to an explosion of GRP specimens in terms of fiber rupture somewhat increases with a reduction in thickness, which is apparently caused by a lower level of bending strains. However, this difference is small, when the thickness is doubled the value of β2 changes, by not more than 20%. The energy capacity of GRP specimens is determined largely by the binder damage prior to fiber rupture. In general, if from the beginning of binder damage to through fracture the specific explosion resistance of specimens increased by more than 5 times, during the through fracture development (from the beginning of fiber rupture)

а) Binder damage, back side view (Q=8 g, β=0.44 %, δ=7.32 mm, m=14.49 kg/m2)

b) Rupture of fibers in three plies, back side view (Q=70 g, β=3.83 %, δ=7.40 mm, m=14.50 kg/m2)

Fig. 2.2 Characteristic patterns of damage for GRP specimens

c) Through fracture, face side view (Q=55 g, β=4.20 %, δ=5.20 mm, m=10.40 kg/m2)

2 Comparative Assessment of Underwater Explosion Resistance …

23

а) Type 1, rupture of fibers in two plies from back side (Q=30 g, β=3.01 %, δ=5.21 mm, m=7.91 kg/m2)

b) Type 1, through fracture, face side view (Q=40 g, β=3.55 %, δ=5.88 mm, m=8.93 kg/m2)

c) Type 2, rupture of fibers in several plies from back side (Q=45 g, β=4.15 %, δ=5.76 mm, m=8.60 kg/m2)

d) Type 2, through fracture, back side view (Q=50 g, β=4.61 %, δ=5.73 mm, m=8.60 kg/m2)

e) Type 3, rupture of fibers in several plies from back side (Q=40 g, β=3.81 %, δ=5.29 mm, m=8.33 kg/m2)

f) Type 3, damage in way of molding, view from face side (Q=50 g, β=4.76 %, δ=5.37 mm, m=8.33 kg/m2)

Fig. 2.3 Characteristic patterns of damage for CFRP specimens Table 2.3 Explosion resistance of material types tested Relative mass of charge β for different levels of explosion resistance, %

Material type

Surface mass, kg/m2

β1

β2

β3

CFRP (type 1)

8.93 7.91

–

≤1.78

≈3.29

CFRP (type 2)

8.60

–

≈1.84

≈4.34

CFRP (type 3)

8.33

–

≤1.90

>4.76 ≈4.38

GRP (type 4)

14.71–15.23

≈0.52

≈3.13

14.49–14.69

≤0.44

≈3.27

>3.83

10.20–10.40

≤0.62

≈3.50

≈3.85

7.28–7.46

≤0.87

≈3.77

>4.26

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A. I. Dulnev and E. A. Nekliudova

this increase was about 10–40%. Two comparative tests with specimens of 7.36– 7.40 mm thickness were carried out to confirm that the binder damage has an effect on the energy capacity of specimens. In one case, the specimen was first exposed to blast with a charge of 8 g (β = 0.44%). The binder was damaged (at the specimen’s back side practically over the entire effective area, at the specimen’s face side in way of support contour) without fiber rupture. Then, the same specimen was loaded by a charge explosion of 70 g (β = 3.83%). A through fracture was inflicted. In the second case, a similar specimen was loaded by a single explosion of 70 g charge (β = 3.83%). In this case, no through fracture was broken, while fibers were ruptured at the back, in the specimen center. Thus, preliminary loading of the specimen, which caused the binder damage, led to a drastic reduction of its explosion resistance by the second and third criteria. For CFRP specimens, a significant contribution to the energy absorption process in specimen damage prior to through fracture is made by fiber rupture. As compared to GRP specimens, the relative power of the explosion is increased by about 2– 2.5 times between the beginning of fiber rupture and formation of through fracture. The type of through fracture in CFRP specimens depends on the reinforcement material and layup schedule. Generally, the through damage of these specimens can be characterized as a “breach”. In general, the following conclusions can be drawn from the test results: • CFRP specimens made of quadraxial carbon fabric (type 3) have the highest explosion resistance in terms of through fracture criterion. However, the explosion resistance of these specimens is only slightly (by about 10–15%) better than that of GRP specimens (type 4). Specimens made of twill weave carbon fabric and biaxial diagonal carbon fabric (type 1) have the lowest explosion resistance in terms of through fracture criterion. • GRP specimens have significantly higher explosion resistance (about 2 times) in terms of fiber rupture criterion than all CFRP specimens. This difference is primarily related to the ultimate tensile elongation of CFRP and GRP considered. For GRP, the relative elongation is ε1 = 2.6%, ε2 = 2.8%, for CFRP: type 1—ε1 = 1.09%, ε2 = 1.22%, ε12 = 1.96%; type 2—ε1 = 1.98%, ε2 = 1.94%, ε12 = 1.85%; type 3—ε1 = 1.92%, ε2 = 1.90%, ε12 = 1.74%. • CFRP specimens made of different materials have approximately equal explosion resistance until fibers start to rupture. At the same time, the explosion resistance of quadraxial fabric (type 3) is significantly higher before through fracture than that of the type 2 specimens (see Table 2.2). However, the ultimate value of interlaminar shear of the type 3 specimens is significantly higher. Therefore, the explosion resistance of CFRP specimens is related not only to ultimate strain, as was mentioned above, but also to this material characteristic. Figure 2.4 shows a typical example of a pressure record in the explosion tank during experiments. Analysis of pressure records indicated that there is no influence of wave reflections from the explosion tank walls on the parameters of specimen dynamic loading. The experimental curve of pressure was compared with the calculated explosion parameters in unlimited fluid for a TNT charge. Calculation was

2 Comparative Assessment of Underwater Explosion Resistance …

25

done following the empirical relations [14, 15]. As it is seen from a comparison of experimental and calculated data, there was full charge detonation in the tests, and explosion parameters were in good agreement with the empirical relations for TNT charges. This circumstance allowed us to employ the Johns-Wilkins-Lee (JWL) equation of state along with empirical TNT parameters for explosive charges used in the tests for further computer modeling of test conditions. Figure 2.5 gives characteristic strain records of GRP specimens at different levels of exposure: β: 0.44% (Q = 8 g, δ = 7.32 mm); 1.04% (Q = 20 g, δ = 8.75 mm); 1.64% (Q = 15 g, δ = 3.66 mm); 3.27% (Q = 60 g, δ = 7.36 mm). For each test (value of β), the figure shows the records of 2–3 strain gauges located at different points of the circle r = 125 mm. Figure 2.6 gives similar records for CFRP specimens at β = 1.78% (Q = 20 g, δ = 5.94 mm), β = 1.84% (Q = 20 g, δ = 5.75 mm), β = 1.90% (Q = 20 g, δ = 5.28 mm). 20 Experiment Empirical dependence

1

Pressure, MPa

15 10

2

3

.

5 0 -

5

0

0.5

1

1.5

2

2.5

3

3.5

Time, ms

Fig. 2.4 Experimental and calculated explosion pressure (Q = 45 g, distance to sensor r = 1.04 m): 1—shock wave front, 2—wave reflection from specimen, 3—wave reflection from tank walls

Fig. 2.5 Time history of strain for GRP specimens

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Fig. 2.6 Time history of strain for CFRP specimens

Analysis of strain measurements indicates the following. Time of transient specimen deformation is 2–4 ms. It depends both on the frequency responses of specimens and on the level of explosion effect. The transient process time is reduced at higher frequencies and levels of explosion effects. The latter is related to increased influence of membrane radial strains. Maximum value of radial strains in measurement points was increased by about 2–2.5 times above the circumferential strains. At the explosion, initiating the fiber rupture in GRP specimens (β = 3.27%), the maximum radial strains in the circle r = 125 mm reached 2.8%. In the specimen center where maximum strains occur, this value is even higher and above the ultimate tensile strain of GRP under static tests. Unlike GRP, the CFRP specimens at the explosion initiating the fiber rupture (β ≈ 1.78–1.90%) had the maximum radial strains in measurement points of not more than 1–1.5%, and in the center of specimens, their values correspond to the ultimate tensile strain of these materials under static tests. The measurements of strains were used for the verification of computer models.

2.5 Computer Modeling The purpose of computer modeling is to develop finite element (FE) computer models for adequate prediction of specimen deformation when exposed to underwater explosion, as well as application of these models for a more detailed analysis of the stressstrained state of specimens in tests. Computer models are required, among other things, because of limited experimental information that could be obtained about the deformation process in tests. In particular, there is no possibility to record strains on the face side of specimens during the proximity explosion and, in the center of specimens, even at the back side.

2 Comparative Assessment of Underwater Explosion Resistance …

27

Fig. 2.7 Computational domain in test modeling

Computer modeling of underwater blasts and their effects on obstacles have been investigated in many studies. Various aspects of such modeling using commercial and in-house software have been considered in Ref. [2–6, 8–13, 16–20] and other works. Test conditions were modeled with two programs: LS-DYNA and AUTODYN. In both cases, problems were formulated in a three-dimensional statement. In view of the symmetry, the quarter of the computational domain was considered as a segment of fluid (water) with a radius of 660–900 mm and a height of 810–1050 mm. The upper part was cut along the outline. The cut-away dimensions were the same as the overall tubing dimensions (Fig. 2.7). The specimen was placed inside the cutaway with air area over it corresponding to the free volume in the tubing behind the specimen. The design diameter of the specimen was assumed as equal to the installation diameter of the bolt joint, which was 500 mm. In modeling, LS-DYNA and AUTODYN used the same boundary conditions and the same types of finite elements. The no-fluid-loss condition was set on the boundary of the air domain, along the contour simulating tubing was set absolutely rigid boundary with the no-fluid-loss condition, and condition of free flow was set on the external water domain boundary. The boundary condition for the specimen was no displacements normal to its plane within a circle of 50 mm wide, which ensured the same effective field of specimen in calculation and test. For modeling all objects, including the specimen, air, water, and explosives, the SOLID-type finite elements were used. For modeling air, water, and explosives the Eulerian mesh was used, while the specimen was modeled using the Lagrangian mesh. Mesh parameters (dimensions, height-to-width ratio, and congestion) were chosen to have a small influence on the results of calculations. Specifically, the characteristic size of the Eulerian mesh for water and air near the specimen was about 4–5 mm, the Lagrangian mesh size in the specimen plane was −3.4 mm, and the number of elements across the specimen thickness was 6. For water, the polynomial equation of state was used. It was assumed that pressure in liquid could not take negative values. Air was modeled using the perfect gas equation. For modeling explosion products, the JWL equation of state was used. The coefficients and parameters of Chapman-Jouguet were assumed as for TNT [21].

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Specimen material was considered as quasi-isotropic with linear elastic characteristics. For correct comparison with the tests, the specimen characteristics (thickness, density, moduli of elasticity, etc.) in calculations were assumed as in the tests under consideration. The effects of the underwater explosion were modeled in two steps. In the first phase, the explosion was considered in free unlimited fluid in one-dimensional (AUTODYN) and two-dimensional (LS-DYNA) formulations. Calculations in this formulation were performed until the shock wave approached the specimen. Then, the obtained solution was exported into a 3D model containing the specimen, and the second phase of calculations was performed. Verification of computer models was done by experimental records of strains in tests for different types of specimens. Charge masses in tests were Q = 8, 10, 15, 45, and 60 g for GRP and 20 g for CFRP. Figures 2.8 and 2.9 by way of example compare calculation relations for radial and circumferential strains versus time with similar relations obtained experimentally for GRP and CFRP (type 3). Quantitatively, an agreement between the calculation results and the experiment was estimated based on the approach suggested and validated in [22]. Following this approach, the assessment is done by the complex coefficient RC, which takes into account the discrepancy between two unsteady processes in terms of amplitude as well as phase. Reference [23] suggests the following ratings for the complex coefficient RC: if RC ≤ 0.15, the convergence of two processes is excellent, if 0.15 < RC ≤ 0.28 it is acceptable, at RC > 0.28 it is poor. Fig. 2.8 Comparison of calculation results with experiment for GRP (Q = 10 g, δ = 8.75 mm)

2 Comparative Assessment of Underwater Explosion Resistance …

29

Fig. 2.9 Comparison of calculation results with experiment for CFRP, type 3 (Q = 20 g, δ = 5.28 mm)

Figure 2.10 gives the coefficient RC calculated for radial and circumferential strains in 9 tests with GRP specimens (triangles refer to the LS-DYNA calculations, circles refer to the AUTODYN calculations). As seen from this figure, as well as Figs. 2.8 and 2.9, calculation results have good agreement with the experiment. For GRP specimens, the coefficient RC is larger than 0.28 only for 4 cases of LS-DYNA calculations and 2 cases of AUTODYN calculations. Some discrepancies between calculations and experimental data, as well as between solutions obtained by LSDYNA and AUTODYN, can be associated with the following conditions: errors in the determination of the actual specimen thickness, discrepancies between coordinates of points for strain determination and locations of strain gauges, errors in modeling specimen’s boundary conditions, different accuracy of defining the explosion parameters in the first phase of the problem solution using LS-DYNA and AUTODYN, and failure to accurately synchronize the starting instant (initiation of specimen deformation) between calculations and experiments. The membrane and bending strains as well as strain intensity (von Mises equivalent strain) were assessed in the specimen center and the specimen support for a more detailed analysis of the stress-strained state of specimens. Figure 2.11 shows the strain rate in the center, on the back side of specimens, and in the specimen support on the face side. Analysis of calculation results showed the following. At the power of explosion tentatively starting to rupture individual fibers, there is practically the same level of bending and membrane strains in the specimen center.

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Fig. 2.10 Complex coefficient of agreement between calculations and experiment for GRP specimens

Note that GRP specimens have about 2% of membrane strains, while CFRP specimens have about 0.5–0.6% of membrane strains. In the support, the bending strains are prevalent, with their levels significantly, more than 4 times, higher than the level of membrane strains. Maximum radial strains (sum of bending and membrane strains) in the support are about 1.4–1.5 higher than in the specimen center. In spite of the fact that there is a higher level of radial strains, the damage (rupture of fibers) begins in the specimen center in line with the results of tests. In this connection, because the material of specimens could be considered as a quasi-isotropic material it was interesting to compare the level of strain intensity in the center and in the support of specimens. This comparison indicated that both for GRP and CFRP specimens the maximum strain intensity in the center of specimens is higher than in the support (for GRP specimens, it is 1.5–2 times higher, and for CFRP specimens it is 10–20% higher). By the time the fibers start to rupture, the strain intensity in the GRP specimens reaches about 5%, which is more than the ultimate tensile strain of this material in static tests, while for the CFRP specimens (type 3) it is 1.7%, which is somewhat less than the similar ultimate tensile strain for this material. Based on this result, it can be presumed that the resistance to an explosion of composite materials, whose structure ensures quasi-isotropic behavior, can be assessed (criterion of individual fiber rupture) by the strain intensity whose limiting value is obtained from a combined analysis of experimental data and numerical modeling results.

2 Comparative Assessment of Underwater Explosion Resistance …

31

Fig. 2.11 Computational relations for von Mises equivalent strain (a GRP, Q = 60 g, δ = 8.75 mm, b CFRP (type 3), Q = 20 g, δ = 5.28 mm)

Further, it would be interesting to consider the accumulation of material damage (in particular, binder damage) in the specimen deformation process for the estimation of strain intensity limit.

2.6 Conclusion Results of experimental and computational studies on the resistance of GRP and CFRP specimens to the underwater explosion are presented. An approach based on three criteria is suggested: damage to binder, rupture of individual fibers, and through-type fracture (breach). The explosion effect corresponding to one or the other criteria is measured in terms of the relative explosive charge mass (charge mass per unit effective area of specimen related to specimen surface mass). It was established experimentally that the resistance of GRP specimens when fibers start to rupture is significantly (about 2 times) higher than the explosion resistance of all types of CFRP specimens under consideration. CFRP specimens made of quadraxial carbon fabric (type 3) have the highest explosion resistance in terms

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of through fracture. However, the explosion resistance of these specimens is only slightly (about 10–15%) higher than that of GRP specimens (type 4). Based on computer modeling, the details of GRP and CFRP stress-strained states are considered by the instant when the fibers start to rupture. In this case, the von Mises equivalent strain can be an ultimate state characteristic for the materials under consideration. The obtained results can be used to validate the choice of composite material for hull structures exposed to blast loads. The approach implemented for assessing the explosion resistance of GRP and CFRP specimens is based on a combined analysis of test data and computer modeling results. It was found effective and revealed specific details of test specimens’ deformation and damage. This approach can be recommended for assessing the explosion resistance of other composite materials made of various reinforcement materials and binders.

References 1. Mouritz AP (1995) The effect of underwater explosion shock loading on the fatigue behaviour of GRP laminates. Composites 26(1):3–9 2. Batra RC, Hassan NM (2007) Response of fiber reinforced composites to underwater explosive loads. Compos B 38:448–468 3. LeBlanc J, Shukla A (2010) Dynamic response and damage evolution in composite materials subjected to underwater explosive loading: an experimental and computational study. Compos Struct 92(10):2421–2430 4. Avachat S, Zhou M (2011) Effect of facesheet thickness on dynamic response of composite sandwich plates to underwater impulsive loading. Paper presented at the 18th international conference on composite materials (ICCM18), Jeju, South Korea, 21–26 August 2011 5. LeBlanc J, Shukla A (2011) Response of E-glass/vinyl ester composite panels to underwater explosive loading: effects of laminate modifications. Int J Impact Eng 38:796–803 6. LeBlanc J, Shukla A (2011) Dynamic response of curved composite panels to underwater explosive loading: experimental and computational comparisons. Compos Struct 93(11):3072– 3081 7. Arora H, Hooper PA, Dear JP (2013) Blast loading of sandwich structures and composite tubes. In: Abrate S, Castanie B, Rajapakse DS (eds) Dynamic failure of composite and sandwich structures. Springer, Dordrecht, pp 47–92 8. Schiffer A, Tagarielli VL (2014) The dynamic response of composite plates to underwater blast: theoretical and numerical modelling. Int J Impact Eng 70:1–13 9. Schiffer A, Tagarielli VL (2015) The response of circular composite plates to underwater blast: experiments and Modelling. J Fluids Struct 52:130–144 10. Dulnev A, Nekliudova E (2017) Exsperimentalno-raschetnaya ocenka vzryvosoprotivlyaemosti obrazcov iz stecloplastica (Experimental and computational blast resistance assessment of fiberglass samples). Tomsk State Univ J Math Mech 47:51–62 11. Dulnev AI, Nekliudova EA (2017) Resistance of GRP samples to non-contact underwater explosion. J Phys Conf Ser. https://doi.org/10.1088/1742-6596/919/1/012003 12. Dulnev AI, Nekliudova EA (2018) Influence of reinforcement materials on explosion resistance of carbon fiber reinforced plastic. Paper presented at the 18th European conference on composite materials (ECCM18), Athens, Greece, 24–28 June 2018 13. Mouritz A, Rajapakse Y (eds) (2017) Explosion blast response of composites. Woodhead Publishing, Cambridge

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14. Cole R (1948) Underwater explosions. Princeton University Press, Princeton 15. Zamyshlyaev BV, Yakovlev YuS (1967) Dinamicheskie nagruzki pri podvodnom vzryve (Dynamic loads in underwater explosion). Sudostroenie, Leningrad 16. Kormilitsin YuN., Melnikov SYu, Tomashevsky VT (2006) Podvodnyi vzryv i ego vzaimodeistvie so sredami i pregradami (Underwater explosion and its interaction with media and obstacles). Nauka, St. Petersburg 17. Dombrovsky SE, Palmin SA (2003) Metody rascheta polya davleniya v blizhnei zone podvodnogo vzryva (Methods for calculation of pressure field in near zone of underwater explosion). In: Abstracts of the VII international conference “Zababakhin Scientific Talks”, Snezhinsk, Russia, 8–12 September 18. Yanilkin YuV et al (2014) Kod EGIDA dlya modelirovaniya dvuhmernyh i trehmernyh zadach mehaniki sploshnoi sredy (EGIDA code for modeling two-dimensional and threedimensional problems in continuum mechanics). In: Abstracts of the XII international conference “Zababakhin Scientific Talks”, Snezhinsk, Russia, 2–6 June 2014 19. Glazyrin VP, Orlov YuN, Orlov MYu (2012) Modelirovanie processa vzaimodeistviya udarnika napolnennogo VV s pregradami (Modeling of explosive-filled striker interaction with obstacles). Izvestia vyshikh uchebnykh zavedeniy Physics 55(7–2):61–64 20. Orlov MY, Orlova YN (2021) Research of the destruction of ice under shock and explosive loads. In: Altenbach H, Eremeyev VA, Igumnov LA (eds) Multiscale solid mechanics. Advanced structured materials, vol 141. Springer, Cham. https://doi.org/10.1007/978-3-03054928-2_27 21. Orlenko LP (ed) (2002) Fizika vzryva, tom 1 (Explosion physics, vol 1), 3rd edn. Phizmatlit, Moscow 22. Russell DM (1997) Error measures for comparing transient data: part I Development of a comprehensive error measure Part II Error measures case study. Paper presented at the 68th shock and vibration symposium, Hunt Valley, Maryland, 3–7 Nov 1997 23. Russell DM (1998) DDG53 Shock trial simulation acceptance criteria. Paper presented at the 69th shock and vibration symposium, Minneapolis St. Paul, 12–16 Oct 1998

Chapter 3

Numerical Study of the Geometric Distribution of Metal of Cumulative Lining Ekaterina M. Grif, Anatoliy V. Guskov, Valeriya A. Kiryukhina, Konstantin E. Milevsky, and Alena A. Nesterova Abstract Based on the review of scientific and technical literature, the well-known facts about the course of the cumulation phenomenon and its features are presented, including elements of the hydrodynamic theory of cumulation. The known information on the geometric distribution of the metal of a cumulative lining during the operation of the cumulative charge is analyzed. The need to clarify the existing knowledge about the geometric distribution of the material in the jet-forming process is explained. The hypothesis of the jet-forming process by plastic deformation of lining material is proposed. A numerical experiment on the functioning of an axisymmetric cumulative charge is performed, and the data obtained on the geometric distribution of the lining metal are analyzed. Keywords Cumulative jet · Hydrodynamic theory of cumulation · Metal distribution of cumulative lining · Plastic deformations

E. M. Grif Faculty of Applied Mathematics and Informatics, Novosibirsk State Technical University, Novosibirsk, Russia A. V. Guskov (B) · V. A. Kiryukhina · K. E. Milevsky · A. A. Nesterova Faculty of Aircraft, Novosibirsk State Technical University, Novosibirsk, Russia e-mail: [email protected] K. E. Milevsky e-mail: [email protected] A. A. Nesterova e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Y. Orlov and Visakh P. M. (eds.), Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates, Advanced Structured Materials 176, https://doi.org/10.1007/978-3-031-17073-7_3

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3.1 Introduction The cumulative effect is the phenomenon of the concentration of energy in a certain place or direction [1, 2]. Cumulation is understood as a phenomenon during an explosion, consisting in an increased effect of the explosion, which is achieved by the direction of the masses set in motion by the explosion toward the common axis. The cumulative effect of an explosive is carried out by means of a notch made in the base of the explosive charge [3–5]. The process of formation of cumulative jets during the oblique impact of plates thrown by detonation products or during explosive compression of axisymmetric metal facings was first explained by the hydrodynamic theory of cumulation, which was based on the model of an ideal incompressible fluid [6, 7]. In general, the hydrodynamic theory of the cumulative effect remains a simple and universal approximation [8]. The scientific and technical community [9] identifies the following reasons for refining the hydrodynamic theory: – existing models of jet stretching and fracture, and the conclusions obtained by different authors are not consistent with each other, although a comparison of theoretical dependencies with experimental data is generally satisfactory; – in the experiments, the dependence of the limiting elongation of the jet on the structural-mechanical characteristics of the metal (grain size, hardening, texture) is manifested, despite the fact that the hydrodynamic model does not predict the behavior of the metal under the conditions of a cumulative jet [10]; – the reasons for the possible anomalous elongation of the material of the shaped jet before destruction by 1000%–2000%, as well as the dependence of this ultimate elongation on the initial diameter and density of the jet, are unknown; – the reasons for the appearance of initial perturbations causing the destruction of the jet into many fragments are unknown; – the dependence of the ultimate elongation and the nature of the destruction of the jet on the chemical (phase) composition and degree of purity of the metal. That is, according to experimental studies, the course of the process of stretching and destruction of cumulative jets largely depends on the physical and mechanical characteristics of the real metal of the cumulative lining: despite the fact that in classical hydrodynamic theory the lining material is taken to be an inviscid liquid, during jet formation and stretching it exhibits properties characteristic of crystallites [9, 11–13]. Based on this fact, there is a question of identifying the features of the real spatial distribution of the material of a cumulative lining in the jet and pestle. According to the formed theoretical basis, in the process of jet formation, the lining material is conditionally divided into two parts by a separating surface lying between the inner and outer parts of the initial cone. The metal of the outer part of the lining forms a cumulative pestle that moves along the axis with relatively low velocity

3 Numerical Study of the Geometric Distribution …

37

(500…1000 m/s), and the metal of the inner part of the lining forms a cumulative jet that moves along the axis with very high velocity (2000…10,000 m/s) [14–18]. The authors of [19] propose a technology of a multilayer cumulative lining design aimed at ensuring the possibility of increasing the density of the cumulative jet material while maintaining the optimum mass of the cumulative lining. This is achieved by using a high-density material for the inner layer from which the jet is formed, and for the outer layer, a material with a lower density, such as aluminum and copper [20–22]. The results of mathematical modeling performed by the authors showed that the maximum efficiency of the products is observed at the coating thickness of 0.2…0.3 mm, which corresponds to the depth of occurrence of the jet-forming layer. However, the analysis of the given graphical data of the results of the numerical experiment revealed that, indeed, the formed cumulative jet consists of the material of the jet-forming layer, but this material also fills part of the pestle, which reveals inconsistency of the experimental results with the accepted existing model of the distribution of the material of the cumulative lining in the jet and the pestle. Thus, the aim of the paper is to investigate the spatial distribution of the metal of the jet-forming lining during the cumulative process, based not on determining the kinematic parameters of lining metal points, but on determining the parameters of the metal volumes that make up the cumulative lining. The paper is structured as follows: the first section shows the model and methods used in which the cumulative process is treated as some unit volumes of metal based on the properties characteristic of crystals. The problem statement for the numerical experiment in the Ansys AUTODYN numerical simulation environment is given in Sect. 3.2. Section 3.3 presents an investigation of the spatial metal distribution of the jet-forming layer of the cumulative lining. Section 3.1 provides a numerical calculation of the metal distribution of the jet-forming layer of the cumulative lining in space in the Ansys AUTODYN software, and Sect. 3.2 provides a test calculation which gives a more complete picture of metal movement over the entire cumulative lining surface due to the distribution of sensors along the entire length of the cumulative lining. Section 3.4 presents the conclusions drawn from the results of the study.

3.2 Model and Methods The above-mentioned inconsistencies are also revealed by a study conducted earlier in the numerical simulation in the environment [23] of the functioning of an axisymmetric cumulative charge. Gauges were installed in a special way on the metal of the inner part of the cumulative lining: localization was chosen in the area of the apex at a depth of 0–500 μm (gauges 1–25) and also in several sections along the length of the cumulative lining at the same depth (gauges 26–35). The location of the gauges was conditioned by the depth of occurrence of the jet-forming layer (for a

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given charge of the order of h ≈ 300 μm), i.e. in the study it was planned to monitor the geometrical distribution of metal located up to the interface and behind it. The computational pictures of the conducted numerical experiment are shown in Fig. 3.1. According to the computational picture, a part of the metal inside the cumulative lining does form a cumulative jet (gauges 20, 23, 30, 32). However, a significant part of it is spent on the formation of a cumulative pestle, as in the previous example. This fact reveals a discrepancy with the known data on the process of cumulative jet formation: if the calculated cell size h = 250 μm is smaller (or equal) to the thickness of the jet-forming layer, the entire released metal volume of the inner surface of the cumulative lining should have gone into the cumulative jet. The inconsistency of the observed phenomenon may be explained both by a combination of features of the geometry of the cumulative funnel and the features of the formulation of the numerical experiment, and, as noted earlier, by possible preconditions for refining the jet-forming theory.

a)

b) Fig. 3.1 The computational pictures. a at the initial moment of time t = 0, b at the moment of time t = 1.65 μm

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As noted above, the material of lining in the actual cumulation process exhibits properties characteristic of crystallites. Then, based on this fact, it is logical to conduct a study based not only on determining the kinematic parameters of the lining metal points, but also on determining the parameters of the metal volumes constituting the cumulative lining: it is the metal volume having the surface area that experiences the pressure that controls the course of the plastic deformation process. It is assumed that the jet-forming process proceeds as follows as shown in Fig. 3.2. The element of the cumulative lining, at the initial moment of time is a ring (a) with certain parameters: diameter, thickness, length and cone angle; in the process of compression, force (P) undergoes a gradual change in these parameters and, accordingly, in shape (intermediate states b and c), eventually forming a cylindrical body (d) (the shape of a truncated cone). The plastic deformation process cannot change the physical and mechanical characteristics of the metal. The cumulative charge pressure creates a region of all-round compression in the plastic deformation area in which the material exhibits superplasticity properties [24, 25]. A scheme is proposed as shown in Fig. 3.3, where the cumulative charge velocity vector V is related in the direction and module of three vectors: deformation velocity Vm , deformation velocity Vε and cumulative charge material displacement Vk . The choice of these characteristics is caused by the fact that, considering the material of a cumulative lining as a real metal, it is logical to represent the cumulative charge as a “tool-material” system, where the deforming tool is an explosive, and

Fig. 3.2 Transformation of a single metal volume during the formation of a cumulative jet

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Fig. 3.3 The kinematic scheme of the movement of a single volume of cumulative jet

the preform of the future product—the cumulative jet—the cumulative lining whose metal has certain physico-mechanical properties. The numerical experiment was carried out in the environment of numerical simulation of high-speed dynamic processes Ansys AUTODYN.

3.3 Problem Statement The operation of an axisymmetric cumulative charge in 2-D space is considered and the Lagrangian-Euler solver is used, with the Lagrangian method describing the metal nodes of the cumulative lining. The material of cumulative lining is copper, the explosive is Composition B (hexogen-trotyl 60/40), the body material is steel grade 1006 and the surrounding space is air [26, 27]. Figure 3.4 shows the cumulative charge on the basis of which the study is based. The Jones-Wilkins-Lee equation of state (JWL) is used to describe the behavior of an explosive as formula (3.1):     ω ω ωE0 −R1 V 1 − 1 − e e−R2 V + , + B1 Pe = A1 R1 V R2 V V where Pe —pressure, A1 , B1 , R1 , R2 and ω—constants, V—specific volume, E0 —specific internal energy per unit mass. The specific volume V is calculated according to formula (3.2):

(3.1)

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Fig. 3.4 The cumulative charge. 1—a cumulative lining; 2—an explosive, 3—a detonator, 4—a body

V =

1 , ρB

(3.2)

where ρ B —the density of the explosive. The equations of state for metal of cumulative lining are based on the impact model (Shock). There is an empirical linear relationship between the detonation velocity Us U S and mass particle velocity up u p , , which holds for most solids and liquids over a wide range. In Autodyn code, this relationship is defined as (3.3) Us = C 0 + S · u p ,

(3.3)

where S—a constant reflecting the slope of the dependence Us (u p ), C0 —the speed of sound in the substance. Then it is convenient to present in the form of Mi-Gruneisen an equation based on the Hugoniot shock dependence, according to the formula (3.4): P = PH +  · ρ(E − E H ),

(3.4)

where PH —Hugoniot pressure, —Gruneisen coefficient, and EH EH —Hugoniot energy,  · ρ = 0 · ρ0 = const. The Gruneisen coefficient G is calculated by the formula (3.5): =

B0 , 1−μ

where B0 —constant, μ—compressibility, calculated according to the formula (3.6):

(3.5)

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μ=

ρ , ρ0

(3.6)

where ρ0 —reference density. The Hugoniot pressure PH is calculated according to formula (3.7): PH =

ρ0 C20 μ(1 + μ) . [1 − (S − 1)μ]2

(3.7)

The Hugoniot energy EH is calculated according to formula (3.8):   μ 1 PH . EH = 2 ρ0 μ + 1

(3.8)

The equations of state of the steel shell are also subject to the shock model of shock. The steel sheath strengthening model is described by the Johnson–Cook equation, which determines the yield strength Y by the formula (3.9):     Y = A + Bεnp 1 + Clogε∗p 1 − THm ,

(3.9)

where A,B,C, n and m—constants, εp —the effective rate of plastic deformation, ε∗p —the relative effective plastic deformation rate, THm —the homologous temperature. The relative effective plastic deformation rate ε∗p is calculated by the formula (3.10): ε∗p =

εp , ε0

(3.10)

where ε0 —the normalized effective rate of plastic deformation, computed by the formula (3.11): ε0 = 1S−1 .

(3.11)

The homologous temperature THm is calculated by the formula (3.12): THm =

T − Troom , Tmelt − Troom

(3.12)

where Tmelt Tmelt —the melting temperature, Troom —the room temperature. The distortions of the computational grid during numerical simulation are presented in Fig. 3.5. The results of the calculation lead to the following conclusions:

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t = 4 µm Fig. 3.5 The result of calculating the functioning of the cumulative charge

– firstly, it is true that as the element moves, its shape changes in a similar way to that shown in Fig. 3.3—the short ring becomes a “tube”; – secondly, at time t = 4 μm, the deformation velocity is equal to the detonation velocity of the explosive VD = 7.98 km/s, the velocity of the deformation Vε = 2.9 105 s−1 and the velocity of the material’s movement Vm = 1.35 km/s. Since, of the previously reviewed studies which found that a significant part of the metal in the cumulative lining goes to the formation of the cumulative pestle, it was decided to research the spatial distribution of the metal in the jet-forming layer during the cumulation process. Research into the spatial distribution of the metal of the jet-forming layer during the cumulation process. The study of the metal of the jet-forming layer of the cumulative lining assumes a more detailed design of the cumulative charge presented in Fig. 3.6 than in the previous study. Based on the purpose of the study, the conical shape of the cumulative lining is chosen: this ensures the formation of a high-speed and high-gradient continuous monolithic jet. The choice of the solution angle of the lining of 2α = 45◦ is conditioned by the fact that the optimal range is 2α = 40..50 in order to obtain maximum penetration. The mass of the cumulative lining was Mkl = 44 g. As it is known, the penetrative effect of the explosive charge is influenced by the detonation and energy characteristics of the explosive used that are determined by the explosive density ρ E H , detonation velocity D and a number of other parameters. Therefore, hexogen is assigned as an explosive charge material, because, firstly, it refers to powerful explosives, which have a positive impact on the penetrating effect of the cumulative charge, and, secondly, it is equipped with a press, which expands the possibility of ensuring possible accuracy of charge formation, which contributes to

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Fig. 3.6 The cumulative charge design

increasing the stability of cumulative jet formation and, therefore, positively affects the performance of the cumulative charge. The shape of the explosive charge also has a direct influence on the penetrating effect of the cumulative charge, and its choice is based on the principle of optimal decrease of the explosive mass without reducing the effectiveness of the device. Therefore, the shape of the charge is cylindro-conical, with the height of the cylindrical part being at least half the height of the cumulative lining: thus, the so-called “active” mass of the explosive, whose energy is used in the cumulative lining compression process, is retained, while the explosive mass is reduced without affecting the kinematic characteristics of the cumulative jet. The explosive charge mass was MEH = 172 g.

3.3.1 Statement of the Problem of Metal Distribution of Local Zones of the Inner Surface of the Cumulative Lining It is proposed to perform a numerical calculation of the metal distribution of the jet-forming layer in the space of the ANSYS AUTODYN program in order to further analyze the results by comparing them with existing knowledge of the cumulation process. The model is axisymmetric, in a 2-D design space, with boundaries open except for the axis, with the condition of flowing materials outside the design domain (in the Autodyn code, the boundary condition is Flow out). The Euler method is used to solve

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the basic equations. The computational step of the grid is chosen as the optimal one based on the purpose of the experiment, ratio of reasonable computational accuracy, computational costs and peculiarities of the simulated process, and is set to 0.25 μm. The Jones-Wilkins-Lee equation of state (JWL) is used to describe the behavior of an explosive as formula (3.13):     ω ω ωE 0 −R1 V e e−R2 V + , PE = A1 1 − + B1 1 − R1 V R2 V V

(3.13)

where PE —pressure, A1 , R1 B1 , R2 and ω—constants, V —specific volume, E 0 —specific internal energy per unit mass. The specific volume V is calculated according to formula (3.14): V =

1 , ρB

(3.14)

where ρ B —the density of the explosive, The equations of state for metal of cumulative lining are based on the impact model (Shock). There is an empirical linear relationship between the detonation velocity Us and mass particle velocity up , which holds for most solids and liquids over a wide range. In Autodyn code, this relationship is defined as (3.15) Us = C 0 + S · u p ,

(3.15)

where S—a constant reflecting the slope of the dependence Us (u p ), C0 —the speed of sound in the substance. Then it is convenient to present in the form of Mi-Gruneisen an equation based on the Hugoniot shock dependence, according to the formula (3.16): P = PH +  · ρ(E − E H ),

(3.16)

where PH —Hugoniot pressure, —Gruneisen coefficient, and EH —Hugoniot energy,  · ρ = 0 · ρ0 = const. The Gruneisen coefficient G is calculated by the formula (3.17): = where B0 —constant,

B0 , 1−μ

(3.17)

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μ—compressibility, calculated according to the formula (3.18): μ=

ρ , ρ0

(3.18)

where ρ0 —reference density. The Hugoniot pressure PH is calculated according to formula (3.19): PH =

ρ0 C20 μ(1 + μ) . [1 − (S − 1)μ]2

(3.19)

The Hugoniot energy EH is calculated according to formula (3.20): EH =

  μ 1 PH . 2 ρ0 μ + 1

(3.20)

Gauges are locally installed at the depth of the jet-forming layer according to the computational scheme shown in Fig. 3.7. The trajectory of the gauges installed in the local area adjacent to the top of the cumulative lining is obtained, as shown in Fig. 3.8. Although in the ideal situation all of the allocated metal volumes should have gone into the jet, most of it remained in the pestle. Consider the distribution of the coordinates of the end points of the trajectory (t = 18 μm) and the maximum velocity of the selected volumes of metal along the length of the lining, as shown in Fig. 3.9.

Fig. 3.7 The computational scheme

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Fig. 3.8 The calculated trajectory of the gauges’ movement

Firstly, according to the presented graphs, the metal of the cumulative lining is distributed along both the jet and the pestle. Secondly, the uneven distribution of metal is expressed in the formation of the head part of the jet from the first quarter of the length of the cumulative lining, and not only from the area adjacent to its apex (region 1). This fact is also supported by the fact that the highest velocities have the metal of local areas 2–4, despite the fact that the highest velocity values should have been in local area 1.

3.3.2 Statement of the Problem of the Distribution of the Metal of the Inner Surface of the Cumulative Lining The next step is to carry out a test calculation similar to the previous one, but giving a more complete picture of metal movement across the entire surface of the cumulative lining. To do this, the gauges indicated in Fig. 3.10 are evenly spaced along the entire length of the cumulative lining, while locally condensing in between zones 2–4: Consider the distribution of the coordinates of the end points of the trajectory (t = 18 μm) and the maximum velocity of the selected volumes of metal along the length of the cumulation lining, as shown in Fig. 3.11. The above dependencies also demonstrate the non-uniformity of the jet-forming process at the initial stages. The obtained calculation results can be explained both by peculiarities of the numerical simulation problem statement and peculiarities of the

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E. M. Grif et al. xt=18 µm (xt=0 µm) 14.00 12.00 xt=18 µm, sm

10.00 8.00 6.00 4.00 2.00 0.00 0.00

1.00

2.00

3.00

4.00 x, sm

a) v max, t=18 µm (xt=0 µm) 8.00 vmax, t=18 µm, sm/ µm

7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0.00

1.00

2.00

3.00

4.00 x, sm

b)

Fig. 3.9 Calculation results at t = 18 μm: distribution for the selected volumes of metal along the length of the lining. a Coordinates of the end points of the trajectory; b a maximum velocity

cumulative lining apex geometry and inaccuracy of the existence of the jet-forming model.

3.4 Conclusion Firstly, it can be assumed that the collapse of the cumulative lining proceeds in a wavy manner: along the length of the jet-forming layer, local zones of the collapse of the material arise, in which the elements of the cumulative lining bordering on its inner surface go into the jet, alternating with zones, during the collapse of which the

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Fig. 3.10 The computational scheme

elements of the cumulative lining bordering on its inner surface go already into the pest. Secondly, it can be assumed that the disturbances in the spatial distribution of the material of the cumulative lining during the cumulation process are introduced by a conditional “middle” layer of the lining, which is a thin layer of metal lying on both sides of the interface in its classical representation; and the peculiarity of the proposed version of the process is that already at the interface (in the “middle” layer), local volumes are distributed in waves, alternating the movement of the volume and, possibly, the metal adjacent to it either in the pestle or in the cumulative jet. In this case, it is advisable to represent the interface not as a straight line, but as a wave-like one, and then it can be assumed that the results of numerical modeling obtained within the subsection are explained by the fact that with the Eulerian formulation of the problem and the accuracy of the computational grid set within this problem, the gauges located in the elements of the computational grid, at the moment of collapse of which the bend of the interface is directed downward, go into the pest, because the thickness is too small and, consequently, the mass of the real metal tends to flow into the cumulative jet, which explains the large percentage of gauges from the material of the inner layer of the metal lining that has gone into the pestle. A possible explanation for these phenomena is collapse, carried out by a mechanism whose basic principle is that the inner layer of the cumulative cladding does not behave as a perfect liquid, but as a combination of annular volumes of metal, transformed under pressure into tubular-shaped elements. The elements, shrinking like a telescopic effect (overlapped), form zones of unstable jet formation. Determining the flow of metal in these zones is one of the target objectives of a study on the spatial distribution of cumulative lining material during cumulative jet-forming.

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xt=18 µm (xt=0 µm)

xt=18 µm, sm

11.00 9.00 7.00 5.00 3.00 0.00

1.00

2.00

vmax, t=18 µm, sm/ µm

a) 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0.00

1.00

v max, t=18 µm (xt=0 µm)

2.00 b)

3.00

3.00

4.00 x, sm

4.00 x, sm

Fig. 3.11 Calculation results at t = 18 μm: distribution for the selected volumes of metal along the length of the lining. a Coordinates of the end points of the trajectory; b a maximum velocity

The above objectives can be expanded, proved or disproved in further experimental studies: first, a series of numerical experiments with a smaller step value of the computational grid of the order of 75–125 microns is required in order to study the continuity of the jet-forming layer: the research task is to obtain a computational picture where all gauges located on the inner boundary of the cumulative lining go into the jet while preserving the features of the problem statement (material models, the main type of solver equations, boundary conditions, geometry of the computational domain); secondly, a series of physical experiments is required to investigate the behavior of the metal both in the jet-forming and middle layers, as well as in the layer that forms the pest according to the hydrodynamic theory of cumulation;

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the experimental study of the spatial distribution of the cumulative lining material between the cumulative jet and the pestle; an experimental study of the spatial distribution of the cumulative lining material between the head, middle and tail parts of the cumulative jet and the head, middle and tail parts of the cumulative pestle.

References 1. Monroe CE (1888) On certain phenomena produced by the detonation of gun cotton, Report № 6. Newport Natural History Society, Proceedings 1883–1888 2. GOST 34000–2016. Shaped charge. Test methods for function and safety [Electronic resource]. https://meganorm.ru/Data2/1/4293764/4293764696.htm. Accessed 1 Sep 2017 3. Minin IV, Minin OV (2003) The world history of the development of cumulative ammunition. In: IV All-Russian scientific and technical conference “SCIENCE. INDUSTRY. DEFENSE”, Novosibirsk, April 23–25, 2003. Novosibirsk State Technical University, Novosibirsk, pp 51–52 4. Zlatin NA (ed) (1971) Physics of fast-flowing processes, vol 2. Mir, Moscow 5. Heinz Feiwald G (1941) The history of hollow charge effect of high explosive charges. German Academy of aviation research, German 6. Dave N, Mallery Ml (2007) Historical development of linear shaped charge. 43rd AIAA/ASME/SAE/ASEE joint propulsion conference & Ex-hibit, July 8–11, 2007. Cincinnati, OH 7. Clark GB (1948) Secret of the shaped charge. Ordnance Mag 1948:49–51 8. Voevoda A, Witkowski T (2014) Modelling of jet formation in linear shaped charges. Combustion, explosion, and shock waves, pp 130–136 9. Fomin VM, Zvegintsev VI, Braguntsov EY (2019) Phenomenon of nonideal cumulative action. In: Zababakhinsky scientific readings: collection of materials 14 international conference, Snezhinsk, March 18–22, 2019. Publishing House of RFNC—VNIITF, Snezhinsk, pp 17–18 10. Guskov AV, Milevsky KE, Grif EM (2019) Influence of the metal microstructure on the jet formation process. Bull South Ural State Univ Ser Mech Eng Industry 19(4):28–38. https:// doi.org/10.14529/engin190404 11. Trishin UA (2005) Physics of cumulative processes: Monograph. Publishing house of Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk 12. Lavrentyev MA (1957) (1957) The cumulative charge and principles of its operation. UMN 12(4):41–52 13. Minin VF, Minin OV, Minin IV (2016) Technology of anisotropic cumulative charge cladding manufacture. Bull Siberian State Uni Geosyst Technol 36(4):237–242 14. Mishnev VI, Guskov AV, Milevsky KE, Trishin YA (2003) Cumulative jet formation: laboratory practical work for the 4th year of FLA (specialties 171400, 330500, 120400) full-time form of training. Novosibirsk State Technical University, Novosibirsk 15. Orlenko LP (ed) (2002) Physics of explosion, vol 2. Fizmatlit, Moscow 16. Babkin AV, Vildanov VA, Gryaznov EF (eds) (2008) Means of destruction and ammunition: Textbook. Publishing House of the Bauman Moscow State Technical University, Moscow 17. Orlenko LP (ed) (2004) Physics of explosion, vol 2. FIZMATLIT, Moscow 18. Szuck MJ (2011) Simulation of micro-shaped charge experiments and analysis of shock states for a hyper-elastic solid. Urbana, Illinois 19. Kalashnikov VV, Demoretsky DA, Trokhin OV (2011) Technology of manufacturing of shaped charge liners with increased piercing ability. Izvestia Samara No 1–2:373–376 20. Yi J, Wang Z, Yin J, Zhang Z (2019) Simulation study on expansive jet formation characteristics of polymer liner. Materials 12(5):744. https://doi.org/10.3390/ma12050744

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21. Tregubov VI (ed) (1997) Rotary extraction by thinning the walls of cylindrical parts from pipes on specialized equipment. Tula State University, Tula 22. Ulyanitsky VU (2001) Physical foundations of detonation spraying. Dissertation, The Lavrentiev Institute of Hydrodynamics SB RAS, Novosibirsk 23. Grif EM (2020) Features of geometrical distribution of metal in the process of cumulative jet formation. In: The collection of scientific publications “Science. Technologies. Innovations”, part 9, 30 November – 4 December, 2020. Publishing house of NSTU, Novosibirsk, pp 127–130 24. Minin IV, Minin OV (1999) Diffraction quasi-optics and its applications. SibAGS, Novosibirsk 25. Guskov AV, Milevsky KE (2017) Technological processes of metal processing in the production of shells: textbook, manual, part 2. NSTU Publishing House, Novosibirsk 26. Monroe CE (1888) Modern explosives. Scribner’s Magazine III:563–576 27. Monroe CE (1900) The applications of explosives. Popular Sci Monthly 300:455

Chapter 4

About Mechanical Behavior and Effective Properties of Metal Matrix Composites Under Shock Wave Loading Valerii V. Karakulov and Vladimir A. Skripnyak

Abstract In this work, the results of studying the mechanical behavior at the mesoscopic scale and effective properties of composites with aluminum matrix and reinforcing ceramic inclusions are submitted. Computer simulation of mechanical reaction of a representative volume of composite material, considered as an ensemble of the interacting structural elements (ceramic particles and metal matrix), is used for studying mechanisms of deformation and processes of the nucleation and growth of damage in metal–ceramic composites at the mesoscopic scale under the loading by shock waves. The mechanical behavior of aluminum matrix is described by the model of the damaged elastic–plastic medium. The model of the damaged brittle solid is used for ceramics. The problem is solved in the 2D statement with the application of finite-difference method. Results of numerical simulation have shown the formation of non-stationary and essentially non-uniform fields of stresses and strains at the mesoscopic scale. Generation of a dissipative structure at the mesoscopic scale in the composites under shock wave loading was revealed in the simulations. Cracks in ceramic particles, cracks between particles and matrix, and damage to the matrix can appear in composites under shock wave loading. The values of effective mechanical characteristics of composites were defined in this work. Keywords Computer simulation · Mechanical behavior · Shock-wave loading · Composite materials · Structure · Effective properties

V. V. Karakulov (B) · V. A. Skripnyak National Research Tomsk State University, 36, Lenin Avenue, Tomsk 634050, Russia e-mail: [email protected] V. A. Skripnyak e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Y. Orlov and Visakh P. M. (eds.), Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates, Advanced Structured Materials 176, https://doi.org/10.1007/978-3-031-17073-7_4

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4.1 Introduction Metal–ceramic composites are widely used now in various fields of industry and are used under extreme operating conditions [1, 2]. In this connection, it is necessary to adequately predict the mechanical behavior of these materials under intensive energy impacts. The experimental results show that the mechanical behavior of composites at high strain rates qualitatively differs from the behavior of its components under the same conditions [3–8]. This specificity of the mechanical behavior of composites can be caused by processes on meso-scale level. In the simulations of mechanical behavior, the composites are often taken as homogeneous [9, 10], but these materials represent a complex of components with different physical and mechanical properties. The components interconnected on inner contact surfaces form the structure of composites. The structure and its evolution during deformation can have a significant influence on the mechanical behavior of composite materials. The present work aims at the numerical simulation of the mechanical behavior of stochastic metal–ceramic composites under loading by shock waves and study mechanisms of deformation on meso-scale level of composites with different structure parameters. This paper is structured as follows: the physical–mathematical model and the method of determining the effective parameters are described in Sect. 4.2; the results of numerical investigation of mechanical behavior of metal matrix composites under plane shock wave loading are presented in Sect. 4.3; Sect. 4.4 analyzed the results of computer simulation of damage to metal matrix composites under loading by shock impulses; the results of numerical investigation of effective mechanical properties of metal matrix composites with different concentrations of ceramic inclusions under shock wave loading are discussed in Sect. 4.5; Sect. 4.6 presented the results of numerical evaluation of the effective elastic and strength properties of metal matrix composites with ceramic inclusions of different shapes under shock wave loading; Sect. 4.7 presented the conclusions derived from the results of the study.

4.2 Simulation of the Mechanical Behavior of Metal Matrix Composite with Reinforcing Ceramic Inclusions Under Plane Shock Wave Loading In this section, the physical–mathematical model of the behavior of composite material under shock wave loading and the method of determining the effective parameters of mechanical state are described. The physical–mathematical model of the two-phase condensed heterogeneous medium with an explicit description of its structure is used to describe the mechanical behavior of the composite on mesoscopic scale level under the considered loading conditions.

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The used model represents the heterogeneous medium as a complex of interconnected structural elements—matrix and inclusions. Inclusions have different shapes and are randomly distributed in the matrix. Within interfaces of each structural element, the medium is taken as homogeneous and isotropic while transition through the interface mechanical properties of the medium change abruptly. The Johnson–Holmquist model of the damaged elastic–brittle medium is used for the mechanical behavior of ceramic inclusions and the Johnson–Cook model of the elastic–viscoplastic medium for the metal matrix. Dimensions of the simulated area and the number of structural elements are chosen in such a way as to determine effective values of parameters of the mechanical state of the medium. Effective mechanical parameters of the heterogeneous medium loaded by a plane shock wave are determined by volume averaging of local values of the state parameters in thin flat layers perpendicular to the shock front direction. The physical–mathematical model and the method of determining the effective parameters are presented in [11–13]. The loading by a shock wave of a plate of stochastic metal–ceramic composite material consisting of a metal matrix and reinforcing ceramic inclusions is considered in this work. The numerical simulation is performed on a rectangular fragment of the plane section of the plate along the direction of the shock wave. The problem was solved in the 2D statement with an application finitedifference scheme after Wilkins. The fragments of simulated areas of the two-phase heterogeneous medium with different concentrations of inclusions are shown in Fig. 4.1. The model of the behavior of the composite material under shock wave loading and method for determining effective parameters of the mechanical state are adopted here for metal-ceramic composites Al-B4 C, Al-SiC, and Al-Al2 O3 with different concentrations of inclusions.

Fig. 4.1 Simulated areas of the two-phase heterogeneous medium with the model structure of the composite composed of the matrix (light region) and arbitrary-shaped inclusions (dark regions). The characteristic size of inclusions is 5 μm, the volume concentration is: a −25%, b −75%

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4.3 Investigation of the Mechanical Behavior of Metal Matrix Composites with Reinforcing Ceramic Inclusions Under Plane Shock Wave Loading In this section, the results of the numerical investigation of the mechanical behavior of metal matrix composites at the mesoscopic scale under plane shock wave loading are presented. Propagating in the composite material, the shock wave interacts with the inner boundaries of structural elements. These processes cause the distribution of local values of parameters of the mechanical state at the meso-scale level. The results of computer simulation show that there is a strong variation of stresses and strains at the meso-scale level in metal–ceramic composites under shock wave loading. The distribution of stresses depends on meso-scale structure of composites but has no essential dependence on the shock wave amplitude. Figure 4.2 shows the distribution of pressure and strains in the composite Al-75%B4 C. Deformation of composites in the shock wave front is accompanied by a change in the initial orientation of the structural elements. In this process, it is possible to form a dissipative structure from volumetric blocks, including a certain number of inclusions, which are displaced as a whole. The simulation results reveal the formation of the dissipative structure in the shock wave front at the meso-scale level of the metal–ceramic composites. Figure 4.3 shows the formation of the dissipative structure at the meso-scale level of the composite Al-75%B4 C. Simulation results indicate that the scale of the dissipative structure depends on the amplitude of the shock wave. Formation of the dissipative structure is accompanied by the formation of the bimodal distribution of velocities of material particles in the shock wave at the meso-scale level of the composite. The simulation results reveal the formation of the bimodal distribution of velocities of material particles in the shock wave at the

Fig. 4.2 The distributions of local values of pressure (a) and strain (b) at the meso-scale level in the metal–ceramic composite Al-75%B4 C under shock wave loading

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Fig. 4.3 Calculated vector field of particle velocity at the meso-scale level in the metal–ceramic composite Al-75%B4 C under shock wave loading. Nucleation of the dissipative structure under intensive shock compression of the composite

meso-scale level of the composite Al-75%B4 C, as one can see in Fig. 4.4. Distribution function of particle velocities behind the front of shock wave is similar to logarithmically normal distribution function.

4.4 Results of Computer Simulation of Damage to Metal Matrix Composites with Reinforcing Ceramic Inclusions Under Loading by Shock Impulses In this section, the results of computer simulation of damage to metal matrix composites under loading by shock impulses are shown and analyzed. The results of experimental investigations show that the actual strength of the metal–ceramic composites under shock loading is lower than the theoretical strength estimates. The results of investigations of microstructure of the composites after loading show the fractured ceramic particles and cracks between particles and matrix. This specificity of the mechanical behavior of composites can be caused by processes on meso-scale level. Propagating in a composite material, the shock impulse interacts with the inner boundaries of structural elements. These processes cause the distribution of local values of the parameters of the mechanical state on meso-scale level. The results of calculations forecast that due to the strong variation of local values of stresses at the meso-scale level in metal–ceramic composites under shock impulse loading, the local tensile stresses may appear in unloading wave. The strength of

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Fig. 4.4 The distribution of particle velocity in the shock wave at the meso-scale in the metalceramic composite Al-75%B4 C. Formation of bimodal distribution of particle velocity

ceramics is very different in compression and in tension, for this reason, some local damages appear in ceramic particles after passing of unloading wave. Figure 4.5a shows that local tensile stresses may appear in the unloading wave in the composite Al-75% B4 C and the Fig. 4.5b shows that this may be the reason for the appearance of damages in ceramic particles. There are more significant damages to the metal–ceramic composite in the region where two opposing unloading waves interact (spall zone). The calculation predicts the existence of cracked ceramic particles, local damages in the metal matrix, and the existence of meso-scale and macro-scale cracks. The spall zone in metal–ceramic composite has larger dimensions than in metals and ceramics. There is the effect of bridging meso-cracks in composite materials under loading by shock pulses. The efficiency of strengthening metal–ceramic composites depends on not only the concentration of ceramics, but also on the meso-structure of the composites. Figure 4.6 shows the formation of the spall zone in composite Al-75% B4 C under tensile stresses. The volume of the damaged composite material catastrophically increases in the spall zone. There are three kinds of damages in the spall zone: cracks in ceramic particles; cracks between ceramic particles and matrix; damages in the aluminum matrix. Cracks between ceramic particles and matrix and damages in aluminum matrix may appear under tensile stresses when two opposing unloading waves interact. Cracks in ceramic particles may appear under local tensile stresses in unloading waves.

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Fig. 4.5 a Structure of shock pulse in metal–ceramic composite Al-75%B4 C: the upper curve on the graph Pmax (t) is the maximum pressure in the shock pulse, the middle curve

(t) is the effective pressure in the shock pulse and the lower curve Pmin (t) is the minimum pressure in the shock pulse; b Damages in ceramic particles after the passage of a shock pulse due to local tensile stresses in the unloading wave

Fig. 4.6 a Structure of shock pulse under spallation in metal–ceramic composite Al-75%B4 C, (1) effective pressure

, (2) minimum pressure Pmin , (3) maximum pressure Pmax , (4) specific number of damaged cells, (5) specific number of damaged ceramic mesh cells, (6) specific number of damaged aluminum matrix mesh cells; b structure of spall zone in metal–ceramic composite Al-75%B4 C: (1) damage to ceramic particles, (2) damage to aluminum matrix

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4.5 Investigation of Effective Mechanical Properties of Metal Matrix Composites with Different Concentrations of Reinforcing Ceramic Inclusions Under Shock Wave Loading In this section, the results of the numerical investigation of effective mechanical properties of metal matrix composites with different concentrations of ceramic inclusions under shock wave loading are discussed. In the simulation of high-rate deformation of constructional elements made of composite materials under intensive dynamic loading, composites are often taken as homogeneous or quasi-homogeneous. However, these materials present a complex of components with different physical and mechanical properties. The components interacting by distinct internal contact surfaces (interfaces between components) form the structure of composites. The structure and its evolution during highrate deformation can have a significant influence on the mechanical behavior and properties of composite materials. The model of mechanical behavior of the stochastic composite under shock wave loading and method for determining effective parameters of the state are employed to predict effective mechanical properties of stochastic metal–ceramic composites Al-B4 C, Al-SiC, Al-Al2 O3 under loading by plane shock waves with the amplitudes from 0.5 to 30 GPa [14]. Composites contain 25%, 50%, 65%, and 75% ceramic inclusions by volume. In Fig. 4.7, the simulation results for the propagation of plane shock waves in metal–ceramic composites are drawn on to plot dependencies of effective shock wave velocity on effective mass velocity . The plotted dependences in the considered range of shock compression are linear = CB + λ , which agrees well with the experimental and theoretical data. The results prove that the concentration of ceramic inclusions affects bulk sound velocity CB and proportionality factor λ. The simulation allows for determining effective values of longitudinal, bulk, and shear sound velocities of the investigated materials. The calculated dependences of effective sound velocities on volume concentration of ceramic inclusions are presented in Fig. 4.8. The simulation data on effective bulk, longitudinal, and shear sound velocities make it possible to determine effective values of elastic moduli of composites under normal conditions. The dependences of values of the effective bulk modulus on volume concentration of ceramic inclusions are presented in Fig. 4.9. The dependences of values of the effective shear modulus on volume concentration of ceramic inclusions are presented in Fig. 4.10. The dependences of values of the effective Young’s modulus on volume concentration of ceramic inclusions are presented in Fig. 4.11. The simulation results show that values of effective mechanical properties of the studied composites under shock wave loading depend strongly on the volume

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Fig. 4.7 Calculated effective shock wave velocity versus effective mass velocity in metal–ceramic composites: a Al-B4 C; b Al-SiC

Fig. 4.8 Calculated values of effective longitudinal and shear sound velocities versus volume concentration of inclusions in composites, solid lines exhibit the square approximation of the simulation results: a effective longitudinal sound velocity; b effective shear sound velocity

concentration of ceramic inclusions. The dependence of values of effective bulk, longitudinal, and shear sound velocities and elastic moduli of metal–ceramic composites on volume concentration of reinforcing ceramic inclusions is nonlinear and monotonically increasing. The influence of the concentration of ceramic inclusions on values of the effective Hugoniot elastic limit is also studied. Values of the effective Hugoniot elastic limit in dependence on volume concentration of ceramic inclusions are presented in Fig. 4.12.

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Fig. 4.9 Values of effective bulk modulus versus concentration of inclusions in composites, solid lines exhibit the square approximation of the simulation results, dotted lines exhibit linear approximation of component’s properties of investigated composites: a Al-B4 C; b Al-SiC

Fig. 4.10 Values of effective shear modulus versus concentration of inclusions in composite materials, solid lines exhibit the square approximation of the simulation results, dotted lines are linear approximation of component’s properties of investigated composites: a Al-B4 C; b Al-SiC

The effective Hugoniot elastic limit increases non-uniformly with the growing concentration of inclusions, and its increment is maximal at high concentrations of inclusions. The effective Hugoniot elastic limit demonstrates a weak dependence on the concentration of reinforcing inclusions up to 70%. This is explained by the fact that inelastic deformation develops at the shock front due to plastic flow of the aluminum matrix. At the inclusion concentration above 70%, composites demonstrate a framework structure characterized by a negligible thickness of matrix layers or direct contact between particles. In this case, plastic deformation of the matrix

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Fig. 4.11 Values of effective Young’s modulus versus concentration of inclusions in composites, solid lines exhibit the square approximation of the simulation results, dotted lines are linear approximation of component’s properties of investigated composites: a Al-B4 C; b Al-SiC

Fig. 4.12 Calculated values of the effective Hugoniot elastic limit of composites as a function of volume concentration of ceramic inclusions, solid lines exhibit the approximation of the simulation results, dotted lines exhibit linear approximation of component’s properties of investigated composites: a Al-B4 C; b Al-SiC

is unable to ensure relaxation of increasing shear stresses. This causes the effective Hugoniot elastic limit to increase significantly, but not higher than the elastic limit in the corresponding polycrystalline ceramics.

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4.6 Investigation of the Effective Elastic and Strength Properties of Metal Matrix Composites with Reinforcing Ceramic Inclusions of Different Shapes Under Shock Wave Loading In this section, the results of numerical evaluation of the effective elastic and strength properties of metal matrix composites with ceramic inclusions of different shapes under shock wave loading are presented and analyzed. The model of the behavior of the composite material under shock wave loading and method for determining effective parameters of the mechanical state are used for numerical evaluation of effective elastic and strength characteristics of stochastic metal–ceramic composites Al-50%B4 C, Al-50%SiC, and Al-50%Al2 O3 with the reinforcing ceramic inclusions of different shapes [15]. The fragments of simulated areas of the two-phase heterogeneous medium with inclusions of different shapes are shown in Fig. 4.13. Dimensions of the simulated area and the number of structural elements are chosen in such a way as to determine effective values of parameters of the mechanical state of the medium. The numerical simulation of the loading by a plane shock wave of composites on mesoscopic scale level makes it possible to determine values of effective longitudinal, bulk, and shear sound velocities and Hugoniot elastic limits for investigated materials. Values of the effective sound velocities are used for determining values of effective bulk, shear, and Young’s modulus of composites under normal conditions. The results are shown in Tables 4.1, 4.2 and 4.3.

Fig. 4.13 Simulated areas of the two-phase heterogeneous medium with the model structure composed of the matrix (light region) and reinforcing inclusions (dark regions) of different shapes: a arbitrary shape; b spherical shape; c shape of short fibers. The average size of inclusions is: a the characteristic dimension is 5 μm; b the diameter of spheres is 5 μm; c the diameter of fibers is 1 μm and their length is 10 μm. The volume concentration of inclusions is 50%

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Table 4.1 Calculated values of the effective mechanical characteristics of metal–ceramic composite material Al-50%B4 C with reinforcing inclusions of different shapes Composite with Composite with Effective mechanical Composite with inclusions of arbitrary inclusions of spherical inclusions in the shape properties of shape shape of short fibers composite material Mass Density ρ0 , g/cm3

2.64

2.64

2.64

Longitudinal Sound Velocity Cl , km/s

8.8

8.8

8.9

Bulk Sound Velocity Cb , km/s

7.25

7.26

7.23

Shear Sound Velocity Cs , km/s

4.32

4.31

4.4

Bulk Modulus K, GPa

138.8

139.1

138.1

Shear Modulus μ, GPa

49.3

49.1

51.1

132.3

131.7

136.4

Young’s Modulus E, GPa Hugoniot Elastic Limit σHEL , GPa

0.54

0.54

0.55

The simulation results showed that values of the effective mechanical characteristics of stochastic metal–ceramic composites Al-50%B4 C, Al-50%SiC, and Al50%Al2 O3 weakly depend on the shape of reinforcing inclusions and mainly are defined by their volume concentration. Calculated values of effective mechanical characteristics of composites with inclusions of spherical shape and composites with inclusions of arbitrary shape coincide with an accuracy of 1%. For composites with reinforcing inclusions in the shape of short fibers, with fiber diameter to length ratio equal to 1/10, the determined values of longitudinal sound velocities and Hugoniot elastic limits are higher. This is due to the effect of orientation of fibers, and that the speed of propagation of elastic waves in all considered ceramic compounds is much higher than the corresponding values for the aluminum matrix.

4.7 Conclusions The simulation results have shown that: • there is a strong variation of local values of stresses and strains at the meso-scale level in metal-ceramic composites under shock wave loading and the distribution

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Table 4.2 Calculated values of the effective mechanical characteristics of metal–ceramic composite material Al-50%SiC with reinforcing inclusions of different shapes Composite with Composite with Effective mechanical Composite with inclusions of arbitrary inclusions of spherical inclusions in the shape properties of shape shape of short fibers composite material Mass Density ρ0 , g/cm3

2.97

2.97

2.97

Longitudinal Sound Velocity Cl , km/s

8.05

8.02

8.07

Bulk Sound Velocity Cb , km/s

6.54

6.56

6.53

Shear Sound Velocity Cs , km/s

4.06

4.03

4.11

Bulk Modulus K, GPa

126.9

126.8

126.5

Shear Modulus μ, GPa

49.1

47.8

50.1

130.2

129.4

132.5

Young’s Modulus E, GPa Hugoniot Elastic Limit σHEL , GPa

•

•

•

• •

0.548

0.547

0.551

of stresses depends on meso-scale structure of composites, but has no essential dependence on the shock wave amplitude; the deformation of composites in the shock wave front is accompanied by a change in the initial orientation of the structural elements, in this process, it is possible to form a dissipative structure from volumetric blocks, including a certain number of inclusions, which are displaced as a whole, the scale of the dissipative structure depends on the amplitude of the shock wave; the formation of the dissipative structure leads to the formation of the bimodal distribution of velocities of material particles in the shock wave at the mesoscale level of the composite, and the distribution function of particle velocities behind the front of the shock wave is similar to logarithmically normal distribution function; due to the strong variation of local values of stresses at the meso-scale level in metal–ceramic composites under shock pulse, loading the local tensile stresses may appear in unloading wave and may be the reason of the appearance of damages in ceramic inclusions; the spall zone in composites has larger dimensions than in metals and ceramics, there are cracks in ceramic particles, cracks between particles and matrix, and damages in the matrix in the spall zone; the efficiency of metal–ceramic composites depends not only on the concentration of ceramics, but also on the meso-structure of the composites;

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Table 4.3 Calculated values of the effective mechanical characteristics of metal–ceramic composite material Al-50%Al2 O3 with reinforcing inclusions of different shapes Composite with Composite with Effective mechanical Composite with inclusions of arbitrary inclusions of spherical inclusions in the shape properties of shape shape of short fibers composite material Mass Density ρ0 , g/cm3

3.24

3.24

3.24

Longitudinal Sound Velocity Cl , km/s

7.01

7.01

7.03

Bulk Sound Velocity Cb , km/s

5.94

5.94

5.93

Shear Sound Velocity Cs , km/s

3.22

3.21

3.26

Bulk Modulus K, GPa

114.1

114.1

113.8

Shear Modulus μ, GPa

33.5

33.3

34.3

Young’s Modulus E, GPa

92.1

91.1

93.7

Hugoniot Elastic Limit σHEL , GPa

0.6

0.6

0.61

• the dependence of values of effective bulk, longitudinal, and shear sound velocities and elastic moduli of metal–ceramic composites on volume concentration of reinforcing ceramic inclusions is nonlinear and monotonically increasing; • the effective Hugoniot elastic limit demonstrates a weak dependence on the concentration of reinforcing inclusions up to 70%; • at the inclusion concentration above 70%, the effective Hugoniot elastic limit increases significantly, but not higher than the elastic limit in the corresponding polycrystalline ceramics; • values of the effective mechanical characteristics of stochastic metal matrix composites with reinforcing ceramic inclusions Al-50%B4 C, Al-50%SiC, and Al-50%Al2 O3 weakly depend on the shape of reinforcing inclusions and mainly are defined by their volume concentration. Acknowledgements The work was carried out with the support of the Tomsk State University Competitiveness Improvement Program and partly within the framework of the Fundamental Research Program of the State Academy of Sciences for 2013–2020, line of research III.23.

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References 1. Robinson JH, Nolen AM (1995) An investigation of metal matrix composites as shields for hypervelocity orbital debris impact. Int J Impact Eng 17(1–6):685–696 2. Silvestrov VV, Plastinin AV, Pay VV et al (1999) Issledovaniye zashchitnykh svoystv ekranov iz kompozita keramika/alyuminiy pri vysokoskorostnom udare (Investigation of the protective properties of screens made of ceramic/aluminum composite under high-speed impact). Fizika Goreniya i Vzriva 35(3):126–132 3. Grady DE (1994) Hydrodynamic compressibility of silicon carbide through shock compression of metal-ceramic mixtures. J Appl Phys 75(1):197–202 4. Gray III GT, Hixson RS, Johnson JN (1996) Dynamic deformation and fracture response of Al 6061-T6–50 vol.% Al2 O3 continuous reinforced composite. In: Proceedings of the international conference on shock waves in condensed matter, pp 547–550 5. Vaidya RU, Song SG, Zurek AK et al (1996) The effect of structural defects in SiC particles on the static and dynamic mechanical response of a 15 volume percent SiC/6061-A1 matrix composite. In: Proceedings of the international conference on shock waves in condensed matter, pp 643–646 6. Batkov YuV, Novikov SA, Fishman ND et al (1999) Sdvigovaya prochnost’ alyuminiyevogo kompozita v udarno-szhatom sostoyanii (Shear strength of an aluminum composite in a shockcompressed state). Khim Fiz 18(11):47–49 7. Johnson JN, Hixson RS, Gray GT III (1994) Shock-wave compression and release of aluminum/ceramic composites. J Appl Phys 76(10):5706–5718 8. Blumenthal WR, Gray III GT (1989) Structure–property characterization of shock loaded B4 CAL. In: Proceedings of the international conference on mechanical properties of matter at high rates of strain, pp 363–370 9. Altman BS, Nemat-Nasser S, Vecchio KS et al (1991) Homogeneous deformation of a particle reinforced metal-matrix composite. In: Schmidt SC, Dick RD, Forbes JW et al (eds) Proceedings of the international conference on shock compression of condensed matter, pp 543–546 10. Anderson CA, O’Donoghue PE, Skerhut D (1990) A mixture theory approach for the shock response of composite materials. J Compos Mater 24(10):1159–1178 11. Platova TM, Skripnyak VA, Karakulov VV (1995) Ob osobennostyakh rasprostraneniya udarnykh voln v geterogennykh sredakh s prochnost’yu (On the features of shock wave propagation in heterogeneous media with strength). Vychislitel’niye Tekhnologii 4(1):200–210 12. Skripnyak VA, Karakulov VV (2004) Lokalizatsiya deformatsii pri vysokoskorostnom nagruzhenii metallokeramicheskikh materialov (Strain localization under high-speed loading of cermet materials). Fizicheskaya Mezomekhanika 7(spec iss, part 1):329–331 13. Karakulov VV, Smolin IYu, Skripnyak VA (2013) Chislennaya metodika prognozirovaniya effektivnykh mekhanicheskikh svoystv stokhasticheskikh kompozitov pri udarno-volnovom nagruzhenii s uchotom evolyutsii struktury (Numerical method for predicting the effective mechanical properties of stochastic composites under shock-wave loading, taking into account the evolution of the structure). Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika 4:70–77 14. Karakulov VV, Smolin IYu, Skripnyak VA (2014) Numerical simulation of effective mechanical properties of stochastic composites with consideration for structural evolution under intensive dynamic loading. AIP Conf Proc 1623:237–240. https://doi.org/10.1063/1.4898926 15. Karakulov VV, Smolin IYu, Skripnyak VA (2016) Numerical investigation of effective mechanical properties of metal-ceramic composites with reinforcing inclusions of different shapes under intensive dynamic impacts. AIP Conf Proc 1783:020081-1-020081-4. https://doi.org/ 10.1063/1.4966374

Chapter 5

Shaped-Charge Treatment Effects Accompanying the Formation of Hard Structure and New Phase States in Coatings on Titanium Gulzara Kanzamanova, Sergey A. Kinelovskii, and Alexander A. Kozulin Abstract This paper presents the results of research on superhard coatings obtained by the interaction of high-speed powder clouds with a titanium substrate. High-speed powder clouds were obtained under conditions of a shaped-charge explosion. Shapedcharge synthesis was conducted using specially prepared mixtures containing atoms of nitrogen, carbon, and boron. The interaction of the initial mixture components and the substrate material yielded nitride, carbonitride, and boride phases that formed superhard coatings. The relationship between the mechanical properties of the investigated coatings and their structural state was established in this study. The specific features of the obtained layers were determined by the fact that they are multiphase coatings built into the crystalline structure of the titanium substrate. The microhardness of these layers varies in the range of 11–21 GPa. Keywords Shaped-charge synthesis · X-ray diffraction analysis · Superhard coatings · Microhardness

Acronym CE CSR DC

Container with explosive compound Coherent scattering region Detonating charge

G. Kanzamanova · A. A. Kozulin (B) Department of Deformable Solids Mechanics, National Research Tomsk State University, Tomsk, Russia e-mail: [email protected] S. A. Kinelovskii Lavrentyev Institute of Hydrodynamics Siberian Branch Russian Academy of Sciences, Novosibirsk, Russia e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Y. Orlov and Visakh P. M. (eds.), Behavior of Materials under Impact, Explosion, High Pressures and Dynamic Strain Rates, Advanced Structured Materials 176, https://doi.org/10.1007/978-3-031-17073-7_5

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EC F FCC HCP MWCNT PL RC TS XRD XRF

G. Kanzamanova et al.

Explosive compound Distance to the target Face-Centered Cubic lattices Hexagonal close-packed lattices Multiwalled carbon nanotubes Conical liner with powder Reaction chamber components Titanium sample X-ray diffraction X-ray fluorescence

5.1 Introduction The line of research in this paper is in a trending and promising field of modern materials science and solid state physics, since it enables the creation of novel composite coatings on the surfaces of construction alloys with advanced thermophysical and physicomechanical properties [1, 2]. These composite materials attract attention due to their outstanding mechanical properties: high strength, hardness, coefficients of friction, wear resistance, and corrosion resistance [3]. Extreme treatment of solids is an efficient way to create new compositions and structures and new phase states. Further, the use of the energy of explosion is a promising synthesis technique. The problem of extreme treatment can be solved by colliding high-speed jets of reacting particles with metal targets, thus achieving greater (by orders of magnitude) parameters of interaction (pressure, temperature). Such high-speed jets of reacting particles can be created by using the products of explosion to propel the powders, and this can be attained by using specially prepared shape charges with powder linings. For the purpose of shaped-charge synthesis, it is possible to use a wide array of chemical elements and compositions, including the concurrent use of heavy and light elements, which is not possible with known dynamic synthesis methods, much less static synthesis. It is known from the literature [4–6] that if boron, carbon, or tungsten is included in the powder mixture, then in the course of shaped-charge synthesis, boride and carbide phases of titanium and tungsten will emerge on the surface of the titanium sample. This increases the microhardness of the alloy surface from 1.7 to 30 GPa, thus setting a record value. One of the latest fields in shaped-charge synthesis involves the use of mixtures of complex salts [7] that contain atoms of initial substances. It was deduced from the theoretical evaluations that during exposure to high-energy treatment, the chemical elements of the salts should interact more efficiently and synthesis with the elements of the substrate material at the atomic level should take place [8, 9]. It ensues from practical use that the synthesis of nitrogen-containing salts (potassium ferricyanide K3 (Fe(CN)6 + potassium ferrocyanide K4 (Fe(CN)6 ) yields negative results, even though theoretical speculation suggests the formation of nitride

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phases. In the experiments described in [10], mixtures for the purpose of synthesis were produced on a base of ammonium nitrate (NH4 NO3 ) and specially prepared complex salts ((Co(NH3 )6 )(WO4 )NO3 ) [11]. In these cases, traces of titanium and tungsten nitrides and carbon nitrides were detected. In this paper, we suggest using ammonium salt oxalic acid–ammonium oxalate ((NH4 )2 C2 O4 ). It is assumed that the decomposition of the salt at greater temperatures and pressures will free up nitrogen, which will be followed by the synthesis of nitride-containing phases. The purpose of this research was to investigate the peculiarities in the formation of nitride-containing coatings by shaped-charge synthesis on titanium substrates using mixtures of different salts comprising light chemical elements. The article was structured as follows. Details of the experiment on obtaining superhard coatings on titanium substrates using high-energy cumulative synthesis and a description of the original experimental reactor, composition of the initial mixtures, and experimental conditions are stated in Sect. 5.2; Sect. 5.3 presents the results of an experiment on the cumulative synthesis of coatings with an assessment of the chemical and phase composition, data of electron microscopy, study of the microhardness of the nearboundary regions of treated surfaces, correlation of the structural and phase states of the surface material with its high microhardness values were carried out; Sect. 5.4 presented the main conclusions derived from the results of the study.

5.2 Materials and Methods To achieve the goal of the present study, experiments were conducted to produce and enable the investigation of superhard composite coatings based on mixtures of salts and powders using shaped-charge synthesis. The target treatment experiment was conducted on the original experimental equipment for cumulative synthesis (Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences in Novosibirsk, Russia). A schematic illustration of the reactor for synthesis and the development of the process of formation and movement of a particle cloud are presented in Fig. 5.1. Usually, the cumulative charge is employed in the base of destructive explosive technology, which ensures the formation of a compact cumulative jet that produces penetrating or cutting action upon an obstacle [12–14]. If the lining of the cumulative charge is made of a porous material, a loose flow of the powder (rather than a compact cumulative jet) can be formed under certain conditions [15]. In this case, the cloud of particles flows with high velocity from the shapedcharge lining through the reactor and interacts with the titanium target surface under high pressure and temperature without destroying it. In the present research, a series of shaped-charge syntheses on substrates (disks 7 mm thick and 15 mm in diameter) from commercially pure titanium alloy (Russian grade VT1-0) were conducted. The disk-shaped substrates were obtained by mechanical cutting from titanium rods and cleaned in an ultrasonic bath with ethanol for 10 min immediately before being loaded into the reactor. The main characteristics

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Fig. 5.1 Schematic illustration of experimental synthesis equipment and shaped-charge reaction at different times: a state before detonation; b cloud of particles flowing from an axisymmetric shaped-charge liner after explosion; and c high-energy interaction of particles with titanium target surface. AIR, air environment; DC, detonating charge; CE, container with explosive compound; EC, explosive compound; PL, conical liner with powder; RC, reaction chamber components; TS, titanium sample

of the shaped-charge devices and the experimental conditions were similar to those described in [4]. After thorough mechanical stirring, the mixture was poured into the tight gap between 2 cones made of thin filter paper, one inserted into the other. In the experiments, shaped charges with conic lining and apex angle α = 30° were used, and the distance to the target (F) was 300 mm (Fig. 5.1). The substrate surfaces were subjected to treatment once (samples N1 and N2) and 3 times (sample N3) to evaluate the effects of multiple treatments. To achieve the main goals of this research, after the treatment, batches of samples were selected from different reactor parts. For the purpose of comparison, a sample that had not been subjected to shaped-charge treatment (sample N0) was studied, which was from the same batch of substrates used as targets for the shaped-charge synthesis. The following components containing atoms of nitrogen, carbon, and boron were included in the initial powder mixtures for lining formation: (NH4 )2 C2 O4 (an oxalate salt with ammonium) + multiwalled carbon nanotubes (MWCNTs) + boron powders in a 2/1/1 volume proportion. Having studied the shaped-charge synthesis of functional coatings, it was found that MWCNTs proved to be a better alternative to carbon powders [16] for the purpose of synthesizing carbides. Additionally, due to their characteristic sparse structure, MWCNTs allow for homogenizing mixtures after thorough mechanical activation, which prevents agglomeration and the tendency to cake [17]. Boron powders in the nanostructured state have exhibited greater efficiency in terms of boride composition yields when subjected to shaped-charge treatment. In previous experimental studies,

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where boride powders were used in lining mixtures [18], the contribution of borides to the aggregate picture of X-ray diffraction analysis was determined. Multiple research methods to study the peculiarities of the formation of coatings were used after conducting shaped-charge synthesis on the surfaces of titanium substrates: 1. A scanning electron microscope (Vega II LMU, Tescan, Czech Republic) was used to determine the microstructure state of the coating. 2. Quantitative X-ray fluorescence (XRF) methods were used to determine the chemical composition of the coating material on a sequential X-ray fluorescence spectrometer (Lab Center XRF-1800, Japan). 3. The X-ray diffraction (XRD) method was used to determine the structure and phase composition of the substrate and coating materials with an X-ray diffractometer (XRD-6000, Shimadzu, Japan) with filtered CuKα radiation. Full-profile analysis of the XRD diagrams was conducted using Powder Cell 2.4 PC programs [19] and PDF-2 material identification electronic database [20]. The lattice parameters were determined using reflections in an angular range of 25° < 2θ < 80°. 4. A method of coating and substrate microhardness testing [21] was used with a Vickers indenter and an automatic micro Vickers hardness tester (HMV G21ST Shimadzu, Japan). To exclude the possible influence of impurities on the research results, the samples were subjected to purification in an ultrasonic bath for 30 min.

5.3 Results and Discussion During the shaped-charge synthesis of coatings, a characteristic heterogeneous surface landscape was formed. The landscape had micron surface roughness with occasional smooth molten areas and microcracks. Visually, the treated surfaces had a golden tint, which is peculiar to nitride titanium compositions. Typical micrographs of the treated surface are shown in Fig. 5.2. The appearance of treated surfaces with nonuniform relief for all samples is absolutely identical. In the central part, one can see smooth areas with an almost complete melting of the matrix material and separate islands of the molten mechanocomposite. On the peripheral areas of the coating there are traces of spreading gas jets in the form of bulk bundles of complex structure and solidified melt of the substrate in the form of spherical droplets. With higher magnification, a certain number of craters that formed when the agglomerates of the mixture collided with the material of the molten surface are visible. Their formation was due to the impact of refractory agglomerates of sprayed composites, as evidenced by the presence of craters filled with particles of the initial mixture.

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Fig. 5.2 SEM images of surface structure in different parts of sample N1. Markers indicate the locations of investigated areas

Powder particles, both spherical and shapeless in the form of scales, are visible on the surface of the coatings and inside the craters. On the surface of the samples, there is a grid of microscopic cracks (Fig. 3d). Microcracks indicate the process of rapid crystallization of high-temperature melt and the occurrence of large internal stresses. Microscopic studies in the orthogonal plane to the treated surface indicated that there was no clear boundary between the coating and the substrate. We can only talk about the transition zone, where the grains of the titanium substrate are mixed with the coating material. Similar areas were found over a considerable part of the border. Fig. 5.3 SEM images of characteristic areas of the treated surface: a craters, b pore, c molten mechanocomposite, d cracks

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The mixing of materials of the substrate and coating led to an assumption about the high adhesive properties of the coating. The results of the chemical analysis using XRF on the initial substrates before and after treatment are provided in Table 5.1. The chemical elements in the table are provided in weight proportions. Titanium and oxygen were determined to be the main chemical elements for the initial alloy and the rest of the elements were determined to be impurities, with their quantity corresponding to the respective alloy specification. The peculiarities of the XRF analysis lie in the fact that the elemental composition is determined at the surface of the sample, i.e., the measurement depth is shallow. Therefore, the presence of large amounts of oxygen is attributed to the formation of a thin oxide film on the surface. An XRD diagram of the initial substrate obtained by using full-profile analysis is provided in Fig. 5.4. In the diagram, the initial titanium substrate (defined as VT1-0 titanium alloy by the specification) is represented by main peaks that correspond to the hexagonal close-packed (HCP) lattices of alpha-titanium (α-Ti), and no other phases were detected. Table 5.1 Results of element analysis on investigated sample surfaces using XRF Sample number

Ti

N0

85.25

N1

40.04

N – 25.39

O

Al

Na

Si

Fe

C

Others

13.28

0.49

0.31

0.13

0.09

0.07

0, which means the discontinuity of the gas density (Fig. 8.1a). At the point of time t = 0, the impenetrable wall G is immediately destroyed, and the gas movement starts (Fig. 8.1a). In the formed flow, a RW adjoins the vacuum through the free boundary G02 and is separated from the area of the resting gas with the line G12 that is the sound characteristic of these flows (Fig. 8.1b). The weak discontinuity takes place on the characteristic G12 . It is assumed that the density of medium on the boundary G02 between water and air equals zero at all times. The latest assumption makes the used model an approximate one. It is required to find the laws of boundaries’ movement G12 , G02 and construct a flow in the area of the RW.

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Fig. 8.1 Schematic illustration of the breakup of a special discontinuity at different times: a state before damage at t = 0; b state after damage at t = t 0 > 0

As a mathematical model, a system of equations of gas dynamics with a polytropic index γ = 7.02 is used. The system of equations takes into account the action of gravity. It is assumed that this model adequately describes two-dimensional water flows [9, 10]. The system of equations describing the isentropic flows of an ideal polytropic gas under the action of gravity has the form [19]: γ −1 c(u x + wz ) = 0, 2 2 ut + u x u + uz w + ccx = 0, γ −1 2 ccz = −g. wt + wx u + wz w + γ −1

ct + cx u + cz w +

(8.1)

Unknown functions in it are c—the sound speed of gas; u, w—cartesian coordinates of the gas velocity vector, independent variables—t, x, z; and g—acceleration of gravity. If we put u = w = 0 in the system (8.1), then the first two equations are fulfilled identically, and in the third equation, we get 2 ccz = −g. γ −1 Integrating this equation, in [17] it is obtained as follows: c = c0 (z) =

/

2 c00 − (γ − 1)gz

—distribution of the sound speed of a stationary gas. Here c00 is the sound speed of a resting gas at z = 0. Next, the RW is constructed for the values of z from the interval: 0≤z