Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations 1527593398, 9781527593398

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Problems of Impact and Non-Stationary Interaction in ElasticPlastic Formulations

Problems of Impact and Non-Stationary Interaction in ElasticPlastic Formulations By

Vladislav Bogdanov

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations By Vladislav Bogdanov This book first published 2023 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2023 by Vladislav Bogdanov All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-9339-8 ISBN (13): 978-1-5275-9339-8

CONTENTS

Foreword ................................................................................................... ix Introduction ................................................................................................ 1 Chapter 1 .................................................................................................. 12 Formulation of elastic and elastic-plastic problems §1.1. Impact of fine elastic cylindrical shells of Kirchhoff–Love and S.P. Tymoshenko types on an elastic half-space .................... 14 §1.2. Impact of fine elastic spherical shells of Kirchhoff–Love and S.P. Tymoshenko types on an elastic half-space .................... 21 §1.3. Impact of hard cylinders on an elastic layer ............................... 28 §1.4. Impact of fine elastic cylindrical shells on the elastic layer ....... 31 §1.5. Quasi-static elastic-plastic formulation – three-dimensional problem ......................................................................................... 33 §1.6. Dynamic elastic-plastic formulation .......................................... 36 §1.6.1. Plane stress state ................................................................ 38 §1.6.2. Plane strain state ................................................................ 40 §1.6.3. Three-dimension stress-strain state ................................... 42 Chapter 2 .................................................................................................. 46 Algorithms for solving mixed non-stationary and dynamic boundary value problems §2.1. Impact of fine elastic cylindrical shells on an elastic half-space .. 46 §2.2. Impact of fine elastic spherical shells on an elastic half-space ..... 70 §2.3. Impact of hard cylinders on an elastic layer ............................... 92 §2.4. Impact of fine elastic cylindrical shells on an elastic layer ........ 98 §2.5. Quasi-static elastic-plastic formulation – three-dimensional problem ....................................................................................... 101 §2.6. Dynamic elastic-plastic formulation ........................................ 104 §2.6.1. The case of plane stress state ........................................... 104 §2.6.2. The case of plane strain state ........................................... 107 §2.6.3. The case of three-dimension stress-strain state ............... 111

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Chapter 3 ................................................................................................ 114 Numerical implementation of the solution algorithm §3.1. Impact of fine elastic cylindrical shells on an elastic halfspace ............................................................................................ 116 §3.2. Impact of fine elastic spherical shells on an elastic half-space ... 121 §3.3. Impact of a hard cylinder on an elastic layer ........................... 126 §3.4. Impact of fine elastic cylindrical shells on an elastic layer ...... 127 §3.5. Three-dimensional problem – quasi-static elastic-plastic formulation .................................................................................. 128 §3.6. Dynamic elastic-plastic formulation ........................................ 128 §3.6.1. Problems of plane stress and strain states ........................ 128 §3.6.2. Three-dimension problem of stress-strain state ............... 129 §3.7. Crack growth in a dynamic elastic-plastic formulation............ 129 §3.7.1. In the problems of plane stress and strain states according to the criteria A ...................................................................... 129 §3.7.2. In the problems of plane stress and strain states according to the criteria B ...................................................................... 129 §3.7.3. In the three-dimension problem of the stress-strain state according to the criteria A ..................................................... 130 §3.7.4. In the three-dimension problem of the stress-strain state according to the criteria B...................................................... 130 §3.8. Considering the process of unloading of the material in problems in dynamic elastic-plastic formulation ........................................ 131 §3.8.1. Problems of plane stress and strain states ........................ 131 §3.8.2. Three-dimension problem of stress-strain state ............... 131 Chapter 4 ................................................................................................ 132 Numerical results analysis §4.1. Impact of fine elastic cylindrical shells on an elastic half-space .................................................................................... 132 §4.2. Impact of fine elastic spherical shells on an elastic half-space ... 151 §4.3. Impact of a hard cylinder on an elastic layer ........................... 159 §4.4. Impact of fine elastic cylindrical shells on an elastic layer ...... 167 §4.5. Three-dimensional problem – quasi-static elastic-plastic formulation .................................................................................. 170 §4.6. Problems in dynamic elastic-plastic formulation ..................... 173 §4.6.1. Plane stress state .............................................................. 173 §4.6.2. Plane strain state .............................................................. 176 §4.6.3. Three-dimension stress-strain state ................................. 182

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§4.7. Problems in dynamic elastic-plastic formulation considering the process of unloading of the material...................................... 192 §4.7.1. Plane stress state .............................................................. 192 §4.7.2. Plane strain state .............................................................. 198 §4.7.3. Three-dimension stress-strain state ................................. 202 §4.8. Problems in dynamic elastic-plastic formulation considering the process of crack growth provided that the maximum breaking stresses are provided directly on the continuation of the crack tip ............................................................................. 204 §4.8.1. Plane stress state .............................................................. 204 §4.8.2. Plane strain state .............................................................. 206 §4.8.3. Three-dimension stress-strain state ................................. 212 §4.9. Problems in dynamic elastic-plastic formulation considering the processes of crack growth provided that the maximum breaking stresses are provided directly on the continuation of the crack tip and unloading of the material ...................................................... 215 §4.9.1. Plane stress and strain states ............................................ 215 §4.9.2. Three-dimension problem of stress-strain state ............... 218 §4.10. Problems in dynamic elastic-plastic formulation considering the process of crack growth according to the generalized local

VTT criterion of brittle fracture

................................................. 219

§4.10.1. Plane stress state ............................................................ 220 §4.10.2. Plane strain state ............................................................ 225 §4.10.3. Three-dimension stress-strain state ............................... 230 §4.11. Problems in dynamic elastic-plastic formulation considering the process of crack growth according to the generalized local

VTT criterion of brittle fracture and unloading of the material ... 232 §4.11.1. Plane stress state ............................................................ 232 §4.11.2. Plane strain state ............................................................ 234 §4.11.3. Three-dimension stress-strain state ............................... 236 Chapter 5 ................................................................................................ 238 Determination of material fracture toughness based on problem solving in dynamic and quasi-static elastic-plastic formulations §5.1. Plane problems ......................................................................... 240 §5.1.1. Dynamic elastic-plastic formulation of the plane stress state problem – numerical results analysis ............................ 240 §5.1.2. Dynamic elastic-plastic formulation of the plane strain state problem – numerical results analysis ............................ 242

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§5.2. Three-dimension problem of stress-strain state – solution algorithm and numerical implementation .................................... 244 §5.2.1. Quasi-static elastic-plastic formulation – numerical results analysis .................................................................................. 245 §5.2.2. Dynamic elastic-plastic formulation – numerical results analysis .................................................................................. 247 Chapter 6 ................................................................................................ 250 Non-stationary interaction on reinforced composite materials in dynamic elastic-plastic formulation §6.1. Plane strain state of reinforced composite glass materials ....... 251 §6.1.1. Problem formulation and solution algorithm................... 251 §6.1.2. Numerical solution .......................................................... 253 Conclusions ............................................................................................ 261 References .............................................................................................. 263

FOREWORD

This book is dedicated to my mother, Lyudmila Antonovna Popel, who was my first listener, and without her support I would not have been able to achieve and get everything I have. The book considers and compares three dynamic mathematical models: elastic, quasi-static and elastic-plastic. The problems of impact and non-stationary interaction of absolutely hard bodies and fine elastic shells are solved. The results are presented in the form of figures and tables. It is shown that the elastic model of impact is convenient to use to calibrate the numerical process in solving impact problems and non-stationary interaction in elastic-plastic formulation. This was achieved using a new methodology of solving dynamic contact problems in elastic-plastic dynamic mathematical formulation. Specifically, for elastic-plastic dynamic mathematical modelling, a new methodology of destruction toughness K Ic (T ) determination has been developed. The book includes the results of calculations of destruction toughness for irradiated RPV reactor steel. There is a comparison of the results of the impact of solid hard bodies and fine elastic shells and non-linear contact interaction developed using two approaches: 1) an elastic mathematical model and 2) an elastic-plastic mathematical model on the stage of elastic deformation. This book considers and compares three dynamic mathematical models: elastic, quasi-static and elastic-plastic. Solutions for contact problems are important for the determination of resource of the strength of the materials, crack resistance, plastic deformation and strain-stress states of construction such as aeroplanes, rockets, ships, trains, bearings, magistral gas- and oil-pipelines, large-scale metal constructions and constructions that have cylindric and spheric panels. New methodologies and approaches described in this book are useful for the precise solution of the problems of shock, thrust and impact and for the reliable simulation of dynamic contact processes. The newly developed methodology and approach of solving contact problems in dynamic elastic-plastic formulation offer the ability to design new composite reinforced and armed materials, such as the composite two-layer reinforced glass base material proposed in this book.

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Foreword

This book should prove of interest to scientists, students, post-graduate students and engineers. V.R. Bogdanov

INTRODUCTION

It is necessary to determine the resource of the strength of the materials and crack resistance of constructions such as aeroplanes, rockets, ships, trains, bearings, magistral gas- and oil-pipelines, large-scale metal constructions and constructions that have cylindric and spheric panels. A new methodology of solving dynamic contact problems, more precisely elastic-plastic dynamic mathematical formulation, is developed. On this basis it is possible to simulate more reliable and suitable nonlinear, plastic processes in the material or, specifically, to calculate plastic deformation of the material. Constructions made of any material can have cracks. Constructions with cracks can survive for a hundred years; on the other hand, ones without cracks can collapse at short notice. The question is: does a material have brittle characteristics or not? The main parameter that describes the brittle characteristics of a material is destruction toughness or critical intensity factor KIc (T ) . This book sets out a more precise elastic-plastic dynamic mathematical model in which a new methodology of destruction toughness KIc (T ) determination has been developed [21, 24, 28, 29, 31, 39, 54]. In the book are the results of calculations of destruction toughness for irradiated RPV reactor steel. There is a comparison of the results of the impact of solid hard bodies and fine elastic shells and non-linear contact interaction, developed using two approaches: 1) an elastic mathematical model and 2) an elastic-plastic mathematical model on the stage of elastic deformation. Problems related to the non-stationary interaction of deformable and absolutely rigid bodies with the environment are of great practical and theoretical interest. The progress and development of modern technology leads to the need to study non-stationary and dynamic processes in various designs. Such processes are important in shipbuilding, aviation and rocketry. They occur, as a rule, in explosions, blows, shocks and impacts. The main elements of most designs are shells, plates and rods. Therefore, the study of the dynamic processes in such objects is of great interest. Shock processes are encountered when solving a variety of problems. Their successful solution is associated with the harmonious interaction of various sciences: aero- and hydrodynamics, the theory of elasticity and plasticity, solids mechanics, destruction mechanics, the theory of shells

2

Introduction

and plates, applied and computational mathematics and others. The difficulty of solving problems of this kind is that it is necessary to jointly integrate systems of equations describing the motion of the body and the environment, when setting boundary conditions on unknown (moving) curved surfaces of the section. The position of these surfaces is determined in the process of solving. Therefore, precise solutions in this field of solids mechanics have mostly applied only to idealized, absolutely rigid objects. Given the complexity of constructing analytical solutions to the problem of the penetration of deformable structures into the fluid (due to a significant change in the shape of contact and free surfaces, the emergence and development of cavitation zones in the liquid and elastic-plastic deformations in the structure material), both numerical-analytical and numerical methods are considered. The problem of the penetration of elastic shell structures into an elastic medium, when the boundary of the contact area lags behind the front of the waves that occur during impact, is also rarely studied. The issues related to the non-stationary interaction of bodies and structures with a continuous medium are set out in a number of monographs: V.M. Alexandrov [1, 2], A.S. Volmir [64, 65], Sh.U. Galiev [72, 73], L.A. Galin [74, 172], W. Goldsmith [79, 80], V.T. Grinchenko [89], A.M. Guz’ and V.T. Golovchan [90], A.M. Guz’ and V.D. Kubenko [77, 91 – 93], R.M. Davis [94], K. Johnson [95], B.T. Diduh [96], J.A. Zukas, T. Nicholas and H.F. Swift [102], N.A. Kilchevsky [108], Y.V. Kolesnikov and E.M. Morozov [111], A.V. Kolodyazhny and V.I. Sevryukov [112], V.D. Kubenko [116, 120], E.N. Mnev and A.K. Pertsev [152], H.A. Metsaveer [149], G.I. Petrashenya [160], V.Z. Parton and P.I. Pearl [158], G.Ya. Popov [162], V.B. Poruchikov [170], S. Prasad [234], A.Ya. Sagomonyan [177, 178], L.I. Slepyan [182], L.I. Slepyan and Yu.S. Yakovlev [183], O.Y. Zhariy and A.F. Ulitko [100] and others, as well as in works by E.F. Afanasyev [8], A.E. Babayev [10, 11], A.G. Bagdoev [12], N.M. Belyaev [14], A.K. Efremov [13], F.M. Borodich [57–59], O.G. Goman [81], A.G. Gorshkov [60–62, 82–86], B.V. Kostrov [115], V.B. Poruchikov [169, 170] and D.V. Tarlakovsky [185, 186]. G. Kirchhoff was one of the first to consider problems relating to the unstable rectilinear motion of a rigid ball in an acoustic medium at a given speed in his 1876 monograph [211]. On the basis of the solution of the wave equation in a spherical coordinate system, A. Love [225] investigated the translational motion of a sphere in an acoustic medium in the presence of elastic force and a given velocity. The same solution is given in the monograph of G. Lamb [215]. Determining the loads at a given law of motion of a rigid body is the first step in solving the problem of the

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interaction of a moving obstacle within the environment. This problem, based on the Laplace transform over time, is considered in the monographs of B.V. Zamyshlyaev and Y.S. Yakovlev [101], E.I. Grigolyuk and A.G. Gorshkova [88], A.M. Guz’, V.D. Kubenko and M.A. Cherevko [91], F. Muna [154], and W. Pao and S. Mou [233]. The problems of impact and penetration of absolutely rigid and elastic bodies into compressible and incompressible fluid are quite well studied. Solutions for the plane and axisymmetric problems of impact and penetration of stamps and elastic bodies into a compressible fluid were given in the works of V.D. Kubenko and his students [66–71, 84, 116– 122, 124, 128–130]. The impact and penetration of stamps of different profiles into the elastic half-space were studied by V.D. Kubenko and S.N. Popov in [121, 133–135, 163–166]. The impact and penetration of elastic cylindrical and spherical shells into an elastic half-space were studied by V.D. Kubenko, S.N. Popov and V.R. Bogdanov in [15–17, 125–127, 134, 167, 168]. Analysis of the current state of research on the penetration of bodies into an elastic medium makes it possible to conclude that most of the work relates to the study of the penetration of non-deformable (absolutely solid) bodies. The penetration models used do not take into account a number of fundamental features of the dynamic interaction of the penetrating body and the elastic half-space. First of all, the possibility of deformation of the penetrating body is not taken into account, although taking it into account is extremely important when calculating penetration into the environment of thin-walled structures. The problems of impact and penetration of elastic shells into the elastic half-space have been little studied and the formulation, taking into account the influence of the rate of penetration of the shell and the corresponding rise of the medium due to it, has not been studied, despite these problems being of great scientific and practical interest. M.A. Lavrentyev, dealing with the impact of a projectile on armour, approached the elastic half-space as incompressible fluid. D.V. Tarlakovsky studied the impact of elastic shells on an elastic halfspace at the superseismic stage, when the rate of change of the boundary of the contact area is greater than the speed of longitudinal waves occurring in the half-space, and solved the unmixed boundary problem. There are two major categories of numerical methods for solving partial differential equations: direct and indirect. Direct methods [217, p.3], which are known as strong-form methods, include well-known methods of finite differences [212], collocations with regularization [210, 217, 218], smoothing of hydrodynamic particles (smoothed particle

4

Introduction

hydrodynamics) [207, 219, 220, 226, 246], the gradient smoothing method [222, 223, 224, 244, 245], the discretisation method and the analytical method. Indirect methods, known as weak-form methods, include finite element methods (MFE, mash dependent) [98, 208, 221, 222, 247]. The typical and most widely used weak form is the Galerkin weak form. Weak form equations are usually formulated in integral form, which implies the need to satisfy them only in the integral (average) sense, which is a weak requirement. The formulation of weak form methods is more general, and often these methods are more effective for less precise practice and engineering tasks. There are two types of stability: spatial (existing in space) and temporal. The spatial stability of the method includes only the formulation of the method based on the finite spatial sampling model. When a method provides a stable solution for static problems, it is said that the method is spatially stable. The time-stable method provides a stable solution for dynamic problems and, therefore, includes precise formulation based on spatial and temporal discretization. However, when a stable temporal integrated circuit is used, a spatially stable method will not necessarily be stable when solving dynamic problems. Catastrophic loss of accuracy occurs when the element mesh is significantly distorted. Therefore, the standard finite element method requires strict adhesion. In an isoperimetric element for which the socalled Jacobian matrix is defined, when the shape of the element is violated, the Jacobian matrix becomes poorly conditioned. The idea of smoothed finite element methods (S-FEM) [221, p.8] is to modify the combined deformation fields or to construct deformation fields using only displacement, hoping that Galerkin’s model will give some good qualities. Such modification/construction of the deformation field can be carried out inside the element and more often outside the element, bringing information from neighbouring elements. Deformation fields satisfy certain conditions, and Galerkin’s standard weak form must be modified accordingly to ensure stability and convergence. The question remains about the sufficiency of smoothing procedures to ensure a stable numerical solution. According to the method of execution and formulation [180, p.9–10] of the basic equations of MFE or equations for individual finite elements, there are four main types of MFE: direct, variational, residual and energy balance. The direct method is similar to the deformation method in the calculations of linear bearings. It is used to solve relatively simple

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problems and is convenient for clear geometric and mechanical values of individual approximation steps. The variational method is based on the principle of stationarity of the functional. In the problem of solid mechanics, the functional is usually the potential energy of the system (Hellinger–Rayon, Hu-Vashizi). Unlike the direct method, which can only be applied to elements of a very simple type, the variational method is used with equal success on elements of both simple and complex types. The residual method (weight residual method) is a generalized type of MFE approximation based on the differential equations of the problem under consideration. This method is used to solve problems for which it is difficult to formulate the functional or problems that do not have such a functional. The method of energy balance is based on the balance of different types of energy; it is used in thermostatic and thermodynamic analysis of a continuous medium. In the mechanics of solid deformed bodies of the above types of MFE, a special place belongs to the methods of variation and residual, which in the area under discussion are two complementary methods of equal accuracy. The variational method is widely used, as expressions in the functional usually have a lower number of derivatives compared to the derivative in the corresponding differential equation of the problem, which allows one to choose interpolation functions from a wide palette of simple functions. The variational form of MFE is derived from the classical Ritza method and the residual method from the classical Bubnov–Galerkin method. In principle, from other variational methods, as well as from the residual method, it is also possible to derive the appropriate types of FEM. However, they are used much less often. Unlike classical variational methods, in which the choice of interpolation functions depends on the configuration of the problem under consideration, in FEM this does not happen, as interpolation functions are defined exclusively within the framework of exclusive finite elements. Interpolation functions – a family of independent functions that describe an element, have zero values everywhere for all elements except the elements to which they relate. This is the main difference between FEM and the classical Rayleigh–Ritza and Bubnov–Galerkin methods, in which interpolation functions are determined for the whole domain. The errors in FEM [180, p.119] by their nature can be twofold: sampling errors, which are the difference between the real geometry of the body and its approximation by a system of finite elements, and errors of interpolation functions, which appear due to the difference between the real field of unknown functions and their approximation by polynomials.

6

Introduction

The type of finite element is determined by the degrees of freedom (and other parameters) attributed to its nodes [159, p.155]. By nodes we mean the geometric point of the area occupied by the finite element and in which is concentrated one or more degrees of freedom of this finite element. The corresponding class of finite elements could be called a class of finite elements with solid nodes (or finite elements of high precision) – an absolutely solid body of infinitesimal size. Finite elements with nonsolid nodes are widely used. Elements of this type (for example, a triangular element with six degrees of freedom at the Cowper–Kosko– Lindberg–Olson node [203]) use as generalized degrees of freedom not only the values of the displacement function at nodal points and the values of its first derivatives in coordinates but also the values of the second derivatives of the same function. The motivation for introducing additional degrees of freedom is usually the desire to eliminate gaps in the displacement fields (eliminate incompatibilities) by increasing the rank of approximation of the quest functions, which is useful for improving asymptotic estimates of convergence. However, this can lead to complications and loss of accuracy in the linear transformation of the coordinates (the transition from the local coordinate system to the global system and vice versa). The finite element methods, being mesh dependent, do not allow for the consideration of large deformations, unlike mesh-free methods of finite differences. It should be noted that the method of finite differences imposes the weakest requirements on the desired functions – they must be piecewise differentiable. The success and rapid development of finite element methods does not deny or diminish the importance of finite difference methods. The use of these methods in the problems of dynamics is strictly justified; the relations for estimating the error of the method are proved [103]. The impossibility of obtaining a guaranteed stable and convergent solution by the finite element method stimulates the further development and generalization of the finite difference method. A method [238] of generalized differences is being developed, which is a modification of the well-known and reliable method of finite differences with a variable partition step. The study of impact in elastic staging continues to develop. Thus, in [148] the plane problem of the interaction of an absolutely solid drummer and an elastic isotropic homogeneous half-space at the supersonic stage within the framework of the theory of elasticity under conditions of rigid adhesion of contact surfaces is considered. The contact zone can be a multi-connected area. The problem of the dynamics of the interaction of

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the drummer with the half-space is reduced to the initial Cauchy problem for a system of quasi-linear differential equations. In the dynamic problems of non-linear mechanics, the theory of plastic flow stands out among the existing models of the theory of plasticity. Among the most widely used fluidity conditions are two conditions: Treska–Sen–Venan and Maxwell–Huber–Mises–Genky, of which the second condition is more accurate, as shown by G. Genky at the First International Congress of Technical Mechanics. L.M. Kachanov made a great contribution to the development of the theory of plasticity and fracture [106, 107]. It is necessary to highlight problems with fast loading (such loading can be attributed to explosion). The contact process can be formalized by a rigid-plastic model. To date, studies on the dynamics of rigid-plastic structures cover a wide range of issues [156] and are detailed in a number of monographs [78, 97, 99, 173, 174] and reports [142, 175, 195, 209, 213, 216]. Since the impulse load is used in stamping products, the static and dynamic problems for rectangular, circular, annular plates and membranes with different shapes of the load impulse are well studied [156]. The condition of plasticity of Treska was mainly used. Regarding the bending of plane obstacles, most of the research concerns the problems of the axisymmetric deformation of circular and annular plates. A.A. Gvozdev [75] made modifications to generate an approximate solution for rectangular plates. The work of V.N. Mazalov [141] is devoted to the study of the dynamic behaviour of annular plates with fixed contours loaded with a evenly distributed high-intensity explosive load. V.N. Mazalov and Yu.V. Nemirovsky [229] constructed a complete solution for the problem of dynamic bending of a rigid plastic annular plate with a free inner contour loaded with an evenly distributed explosive load. According to a single scheme, the problem is analysed for any method of fastening the outer circuit – from hinged resistance to clamping. In [231], D. Nipostin and A. Stanchuk constructed a complete solution for the problem of dynamic bending of an annular plate, similar to that studied by V.N. Mazalov [141], but for a load given by an arbitrary integrable time function and using the approximate Johansen plasticity condition. K.L. Komarov and Y.V. Nemirovsky [109, 110] considered the dynamic behaviour of rigid-plastic rectangular plates, taking into account their own weight and rectangular plates under the action of a rectangular moving impulse. To simplify the calculations, it was assumed that the linear hinges move uniformly within each interval of movement.

8

Introduction

In the work of S. Bak and M. Muzhynsky [196], the dynamics of a hinged rectangular plate made of a rigid plastic material with the Huber– Mises–Genki plasticity condition were studied. The study of the dynamic behaviour of rigid plastic bodies in a resistant medium was carried out by G.S. Shapiro [190] for an infinite beam and by A.A. Amandosov [4], A.A. Amandosov and K.M. Stamgaziev [5], A.A. Amandosov and A.R. Uskombaev [6], M.M. Aliyev and G.S. Shapiro [3] and A. Kumar [214] for a circular hinged plate. In these works, the assumption of G.S. Shapiro was confirmed [190] that if the hinge lines are non-stationary, then the dynamics equations can be integrated in the case when the resistance of the medium depends on the speed of movement of the points. For plates, it is also concluded that the distribution of moments is independent of the resistance to the foundation. In the monograph of V.A. Smirnov [181], an elastic-plastic solution to the distribution of plastic deformations both on the plane and on the thickness of square plates with mixed boundary conditions under a single applied, monotonically increasing static load is considered, and the load values corresponding to the limit state are obtained. In monograph [156], a technique is developed based on the model of an ideal rigid plastic body that makes it possible to calculate the dynamic deformation of various shapes and fastenings of sheet metal structures from a single position. The results of the monograph are based on the authors’ study of the dynamics of various single- and double-connected plates with arbitrary contour shape with different ways of fixing it. The influence of viscoelastic resistance to the foundation, as well as the variable thickness of plates and rigid inserts, is considered. Many important application problems have been solved. The results of the dynamic analysis are presented in a simple analytical form, convenient for further use. Throughout the work, comparisons are made of results calculated based on exact and approximate solutions. Publications [235–237, 243] develop an approach to study the dynamic development of cracks in experimental samples based on the Rayleigh method, which approximates the dynamic processes in the beam using the so-called single degree of freedom (SDOF). This makes it possible to replace the dynamic model with a quasi-static one. In [235–237], the motion of the beam is described as a superposition of vibrational modes. To achieve greater accuracy of the model, the curvature of the drummer and supports is also considered. This method determines the dynamic stress intensity factor (DSIF) for the destruction process for 1.8 microseconds. In [243], an experimental-computational method for determining the dynamic stress intensity factor (DSIF) K I (t ) was

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proposed. The load and failure time of short compact specimens were determined experimentally. The single signal-response was calculated separately by the finite element method. DSIF was defined according to linear theory as a convolution of load and a single signal-response, while the critical value of DSIF corresponded to the moment of failure. The total period during which the destruction process was studied was 40 microseconds. Attempts are being made to build a suitable model describing the motion of a fractured medium. The theory of asymmetric (moment stress and couple stress) elasticity was developed by E. Cosserat and F. Cosserat [202]. According to this theory, not only ordinary stresses but also moments must be determined in an infinitesimal neighbourhood. Another difference from the classical elasticity is that the deformation of the body is described by two vectors: the displacement vector u and the rotation vector I . Cosserat’s modern theory is developed for linearized two- and three-dimensional cases [191, 205, 230, 232, 239]. In bound moment theory, the vectors of rotation and displacement are not free but are related by a relation I (curl u) / 2 . To distinguish between the cases of bound and unbound moment theories, Ehringen [205] proposed calling the unbound theory micropolar elasticity. In [192], an analysis of two crack models in an unbounded two-dimensional micropolar medium is carried out. The models relate to the problems of plane and antiplane deformation, respectively. In both cases, one of the three modes (forms) is not connected and the other two are connected. For a semi-infinite crack, the problem is reduced to a scalar and a second-rank Riemann–Hilbert vector with a Hermitian matrix of coefficients (couple stress is moment stress). In [138, 147, 155], an approach was developed to use surface influence functions to solve non-stationary problems. In [155], non-stationary surface influence functions for an elastic-porous half-plane were determined and the problem of propagation of non-stationary axisymmetric perturbations from the circle surface of a Cosserat medium was investigated. In the dissertation work [147], the peculiarities of solving the problem of diffraction of acoustic waves on unfixed elastic and deformable structures and non-stationary problems of diffraction of elastic and acoustic waves on inhomogeneous transversally isotropic inclusions of spherical shape, as well as a number of non-stationary contact problems based on smooth apparatus of surface influence functions, were researched. In [18–52, 54, 201], the authors examine the approach of tracking the surface of plastic flow described by the Maxwell–Huber–Mises–Genki

10

Introduction

condition, based on the numerical solution of dynamic problems using the method of finite differences with variable pitch. The problems of the impact of rigid bodies on deformable bodies and their collision remain relevant and are studied in various formulations. One of the most important areas of such research is to identify the features of the destruction of cut-notched beam specimens when they are destroyed on a three-point bend with a hammer. Relevant experiments make it possible to determine the much-needed fracture mechanics characteristic of the material – the fracture toughness or critical stress intensity factor associated with the stress intensity factor at the crack tip. Because the process is dynamic and can be accompanied by significant plastic deformations, its study is a complex and multifaceted task that requires analysis of the impact on the test body, the dynamic interaction of the body with bearings and the process of destruction and its development. This topic is extremely important for the avionics, rocket and shipbuilding industries when examining the resource of the strength of the materials. [20, 23–52, 54] explore the problems of non-stationary shock, an impact interaction of an absolutely rigid plane drummer with a specimen notchcut in the middle section in a dynamic elastic-plastic setting. In [37], the three-dimensional quasi-static problem corresponding to [23] in the elastic-plastic formulation was solved, and it was found that the stresses differ significantly from the stresses obtained from the solution of a similar problem in the dynamic elastic-plastic formulation. The publication [27] solves the problem of determining the stresses and the limit state in the plane strain state from the three-point bending of the beam specimen with a middle notch. In [25], a similar problem is explored of plane stress state under the criterion condition of the beginning of crack growth at the moment of moving the calculated maximum stress from the place of direct extension of the crack tip to a certain distance from it in order to ensure the existence of the maximum directly at the crack tip. In publications [26, 30], plane problems of stress and strain states with a crack are solved, the growth of which is controlled by a generalized local

VTT

criterion of brittle fracture. In publications [28, 31], the fracture

toughness of the material was determined based on the study of solutions of the plane strain state and spatial problems under the assumption that the crack is stationary, respectively. The proposed models made it possible in their development to significantly increase the level of adequacy of the obtained theoretical approaches. In [146], it was found that the quantitative characteristics of the necessary conditions for the formation of cold (brittle) cracks in the welding of low-alloy high-strength steels are quite clear when using a probabilistic model of brittle fracture with the Weibull

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

11

distribution function, whose parameters generally depend on material microstructure and the percentage of diffuse hydrogen contained in the metal. There is no mention of the mathematical model and the problem from which the stresses used in the Weibull distribution relations are determined. This indicates that the probabilistic approach is quite universal and productive, but it is clear that if one uses a more accurate dynamic elastic-plastic formulation, the result will be more reliable. This is why in [43, 44] the three-dimensional process of crack growth with a straight front under the condition of shifting the maximum stresses from the crack tip and the local criterion of brittle fracture, respectively, were studied. In [131], the effect of non-stationary loading on the end surface of an elastic half-strip was investigated. In [132], a supersonic impact is solved. The impact of a rigid cylinder is interesting as a limiting case of impact of elastic shells [11]. This book uses the approach in [17, 22, 120, 121, 123, 126, 132, 133, 136, 164, 165], which is based on a reducing of the initial equations of the dynamics of the shell-layer system to an infinite system of Volterra integral equations of the second kind. This makes it possible to effectively determine the numerical solution of the problems and reliably calculate the dynamic and kinematic quantitative characteristics that describe the collision process, depending on the magnitude of the initial impact velocity and the parameters of the shell and layer.

CHAPTER 1 FORMULATION OF ELASTIC AND ELASTICPLASTIC PROBLEMS

A thin elastic shell moving perpendicular to the surface of an elastic half-space z t 0 collides with an elastic half-space at a time t 0 . The shell begins to penetrate into the elastic medium at a rate vT (t ) , and the initial rate of penetration V0 vT (0) . The shell’s thickness h is much smaller than the radius R of the middle surface of the shell ( h / R d 0.05 ). Let us denote the undisturbed surface of the half-space by the sign Ƚ. Let us denote the part of the surface Ƚ that has not collided with the penetrating body *1 (t ) ; the rest of the surface Ƚ, through which penetration takes place * 2 (t ) , will be denoted and it will be called the contact area. Denote the surface bounding the penetrating body by F (t ) . The area of the surface

F (t )

that has not come into contact with the half-

space is denoted F1 (t ) , the rest is denoted by – F2 (t ) and we call it the contact surface. Equalities are valid for these surfaces:

*(t ) *1 (t )  * 2 (t ); F (t )

F1 (t )  F2 (t ).

The deformed free surface of the half-space is denoted by *1 (t ) . We assume that there is no friction or just a slip condition between the elastic half-space and the penetrating body. Boundary conditions will be as follows:

uz

* 2 (t )

 u0 *

2 (t )

§ x wT (t )  f ¨ © r & & 1  ( n , m) 2 ,

· & & ¸  w0 * 2 (t ) (n , m)  ¹

(1.1)

Formulation of elastic and elastic-plastic problems

ux r

w0 *

2 (t )

& & 1  (n, m)2  u0 *

2 (t )

&

0, – slip condition

zx

§ x · f¨ ¸ © r ¹

(1.2)

(1.3)

1

zr

& & (n, m),

*2 (t )

V zz * (t ) 0,

V

13

(1.4)

* (t )

­ f ( x), if plane problem ® ¯ f (r ), if axisymmetric problem

&

where m and n are normal orts to the surfaces of the shell and halfspace, respectively. Since the law of motion of the shell is not known in advance, it is necessary to add the equation of motion of this shell to equations (1.1)– (1.4) and equations describing the dynamics of the elastic shell: d 2 wT & & M V ( t , s )( n , m)dF2 . (1.5) zz dt 2 F2 (t ) The refined model of fine shells of the S.P. Tymoshenko type makes it possible to consider the displacement and inertia of rotation of the crosssection of the shell. Perturbations spread in S.P. Tymoshenko-type shells at a finite speed. Therefore, the study of the dynamics of the propagation of wave processes in thin shells of the S.P. Tymoshenko type is an important aspect, just as it is important to study the wave processes of impact in the elastic medium into which the striker penetrates. The method of reducing the solution of dynamics problems to the solution of the second type of Volterra’s infinite system of integral equations (ISIE) and the convergence of this solution are well studied. This approach has been successfully used for the study of problems of the impact of solids [133– 135, 143, 164–166] and elastic thin shells of the Kirchhoff–Love type [15– 17, 125–127, 136, 167, 168] on the elastic half-space and problems of the impact of solids [123, 132, 200] and elastic thin shells of Kirchhoff–Love type [22] on the elastic layer. An attempt was made [198, 199] to solve plane and axisymmetric problems of the impact of elastic thin cylindrical

³³

14

Chapter 1

and spherical shells of the S.P. Tymoshenko type on the elastic half-space by the method of reducing the problems of dynamics to the solution of the second type of Volterra’s ISIE. It is shown that this approach is unacceptable for the plane and axisymmetric problems of the impact of fine elastic shells on an elastic half-space studied in this book. Discretization using Gregory’s methods for numerical integration and Adams for solving the Cauchy problem for the second type of Volterra’s ISIE leads to the solution of a poorly defined system of linear algebraic equations: as the size of reduction increases, the determinant of such a system extends to infinity. This technique does not allow one to solve plane and axisymmetric dynamics problems for thin shells of the S.P. Tymoshenko type and elastic bodies [198, 199]. This shows the limitations of this approach [15–17, 22, 66–71, 116–136, 163–168, 200] and explains the need to develop other mathematical approaches and models [19–21, 23–54, 143–146, 227]. It should be noted that to calibrate the computational process in the dynamic elastic-plastic formulation at the elastic stage, it is convenient and appropriate to use the technique of summarizing the problems of dynamics to solve the second type of Volterra’s ISIE [15–17, 22, 66–71, 116–136, 163–168, 200]. The solution of problems for elastic shells [63, 140, 150, 187], elastic half-space [9, 104, 151], elastic layer [137] and elastic rod [206, 241] was developed using the method of influence functions [87]. In [187], the process of nonstationary interaction of an elastic cylindrical shell with an elastic halfspace at the so-called ‘supersonic’ stage (when the velocity of penetration is higher than the speed of the impact waves in the elastic half-space) of interaction is studied. This stage is characterized by exceeding the rate of expansion of the region of contact interaction by the rate of propagation of tensile-compressive waves in the elastic half-space. The solution was developed using influence functions corresponding to the concentrated force or kinematic effects for the elastic isotropic half-space, which were found and studied in [87].

§1.1. Impact of fine elastic cylindrical shells of Kirchhoff– Love and S.P. Tymoshenko types on an elastic half-space A thin elastic cylindrical shell comes into contact with the elastic halfspace z t 0 of its lateral surface along the generatrix of the cylinder at the time t 0 . As shown in Figure 1, we associate with the shell a moving cylindrical coordinate system: rT y c : T is the polar angle, which is postponed from

Formulation of elastic and elastic-plastic problems

15

the positive direction of the axis Oz. The axis O cy c coincides with the axis of the cylinder. Denote by

w0 (t,T ) ,

u0 (t ,T ) ,

p (t , T )

and

q (t , T ) the tangential and normal displacements of the points of the middle surface of the shell and the radial and tangential components of the distributed external load acting on the shell, respectively. We connect a fixed Cartesian coordinate system xyz with a halfspace, so that the Oz axis is Fig. 1.1 Calculation scheme directed inwards, the Ox axis is on the surface of the half-space and the Oy axis is parallel to the cylinder generator. The physical properties of the half-space material are characterized by elastic constants: the modulus of volume expansion K, the shear modulus P and the density U. The elastic medium with constants K, P and U will correspond to the hypothetical acoustic medium with the same constants K and U ; thus, P

0 . By C p , CS and C0 we mean the speed of

longitudinal and transverse waves in an elastic half-space, as well as the speed of sound in a hypothetical acoustic medium corresponding to the considered half-space. Let into the notation: 2 2 P 4P · 2 CS 2 Cp § E , D ¨1  ¸, 2 2 C0 K C0 © 3K ¹ E2 3P K (1.1.1) , b2 . C02 2 U 3K  4 P

D

We introduce dimensionless variables:

Chapter 1

16

C0t ui u0 x z , xc , zc , uic , u0c , R R R R R V ij w0 vT wT w0c , V ijc , vTc , wTc , R K C0 R

tc

p , qc KR

pc where

q , Mc KR

u (ux , u y , uz ) is

M

U R2

, (i, j

(1.1.2)

x, y, z ),

the displacement vector of points of the

environment, Vzz, Vxz are the non-zero components of the stress tensor of the medium, M is the linear weight of the fine shell and vT (t ) , wT (t ) are the velocity and displacement of the shell as a solid. In the future we will use only dimensionless quantities, so we omit the dash. The elastic half-space and the shell are in a state of plane deformation. Given (1.1.2), from the system of basic equations of dynamics of arbitrary thin elastic shells based on Kirchhoff–Love’s hypotheses, we obtain the equations of motion of a thin elastic cylindrical shell written with respect to dimensionless variables. w 2u0 w w0 w 3w0 w 2u0 (1  a1)   a E  E 2 q, (1.1.3) 1 1 2 3 2 wT

wT

w u0 w 3u0 w 4 w0   a1  w0  a1 wT wT 3 wT 4 a1

h2 12 R

Q 0 , E0 , U0

, E1 2

wt

wT

 E1

C02 (1 Q 02 ) U0 , E2 E0

w 2 w0

 E 2 p, w t2 R U (1 Q 02 ) K , h U0 E0

are Poisson’s ratio, Young’s modulus of elasticity and the

density of the shell material, and p and q are the radial and tangential components of the distributed load acting on the shell, respectively. Differential equations (of the S.P. Tymoshenko type), which describe the dynamics of cylindrical shells and consider the shear and inertia of rotation of the cross section, by virtue of (1.1.2) have the following form [178, p.87]:

Formulation of elastic and elastic-plastic problems

17

w 2u0 w 2u0 w w0  (1  )  a4)  a4u0  E3q, a 4 wT w t2 wT 2 2 2 wu 2 w w0 w w0 w ) K0   (1  a3 ) 0  a3w0  E 4 p, 2 2 wT wT wt wT w w0 w 2) w 2) J 02 2   a2)  a2u0 , a 2 wT wt wT 2 J 02

(1.1.4)

where

J 02 a2 a4

C02

2 C02

,

K02

C02

2 C01

6(1 Q 0 )b12 R 2 h

2

1 , E3 a3

,

E0

2 C01

, b12

(1 Q 02 ) U0 5 , a3 6

(1 Q 02 ) K 2 R E02 h

, E4

,

2 C02

2 (1 Q 0 )b12

,

2(1 Q 0 ) K 2 R b12 E02 h

b12 E0 , 2(1 Q 0 )

,

where Ɏ is the angle of rotation of the normal section to the middle 2 surface and b1 is a factor that considers the distribution of tangential forces in the cross section of the shell. The motion of an elastic medium is described by scalar potentials of equations M and the non-zero component of vector potential \ satisfying the wave equations:

'M

w 2M , '\ D 2w t 2

w 2\ w2 w2 , ' {  . E 2w t 2 w x2 w y2

(1.1.5)

Physical quantities are expressed in terms of wave potentials as follows:

Chapter 1

18

ux

wM w\  , uz wx wz

V zz

(1  2b 2 )

V xz

2E 2

wM w\  , uy wz wx

0,

§ 2 w 2M w 2\ · 2 w M E  2  ¨ ¸, ¨ w z 2 w xw z ¸ w t2 © ¹

w 2M w 2\ w 2\ , V xy  2  2E 2 w xw z w t w x2

4 V zz  V xx

w 2M 2(1  b ) 2 , V xx wt 2

V yz

4  V zz .

0, (1.1.6)

In a plane problem, it is natural to express normal displacements as

wM w\ . However, expressions (1.1.6) were used  wz wx further in this book, because in this case the transition function Fn (t ) in ux

wM w\ , uz  wx wz

(2.1.44) is the same as in the axisymmetric problem (§2.2) and, has a simpler form and is easier to calculate. If the shear modulus P 0 , then the equations of motion of the elastic medium will be the equations of acoustics. Physical quantities will then be expressed in terms of displacement potential as follows: 2 2

Vx

w M , V w xw t z

w M , V 0, w zw t y

if i z j ­0, ° V ij ® (1.1.7) 4 w 2M , if , ( , , , ) p i j i j x y z  ° D 2w t 2 2 ¯ where Vx , Vy , Vz are components of the velocity vector of the points of the medium V (Vx ,Vy ,Vz ) . Consider the initial stage of the process of impact of elastic shells on the surface of the elastic half-space, when there is no plastic deformation and the amount of deepening of the shell into the environment is small. The problem of interaction of elastic shells with an elastic half-space is solved in a linear formulation; therefore, we linearize boundary conditions [120]: boundary conditions from the perturbed surface are demolished to undisturbed surfaces of deformable bodies.

Formulation of elastic and elastic-plastic problems

19

As can be seen from Figure 1.1, the projection of the functions

u0 , w0 , p and q on the axis Ocr and Oczc will be equal: prZ c w0 (t , T ) w0 (t ,T ) cos T , prZ cu0 (t ,T ) u0 (t ,T )sin T , prZ c p(t ,T ) p(t ,T ) cos T , prZ cq(t ,T ) q(t ,T )sin T , prr w0 (t ,T )  w0 (t ,T ) sin T , prr u0 (t ,T ) u0 (t ,T ) cos T , prr p(t ,T )  p(t ,T ) sin T , prr q(t ,T ) q(t ,T ) cos T . Then, in the coordinate system zOx, movement uz , ux

V zz

and

V zx

and stresses

at the surface points of the contact area will be written in

the form:

u z (t , x, 0)

wT (t )  f ( x)  w0 (t ,T ) cos T  u0 (t ,T ) sin T , (1.1.13)

u x (t , x, 0)

 w0 (t , T ) sin T  u0 (t ,T ) cos T ,

V zz (t , x, 0)  p(t ,T ) cos T  q(t ,T ) sin T , V xz (t , x, 0)  p(t ,T ) sin T  q(t ,T ) cos T ,

(1.1.14) (1.1.15) (1.1.16)

wT (t ) is moving the shell as a solid; the function f(x) describes the profile of the shell. On the other hand, the radial and tangential components of the distributed load acting on the shell are expressed through normal and tangential stresses arising on the surface of the half-space in the contact zone. p (t , T )

V zz (t , x, 0) cos T  V xz (t , x, 0) sin T , T  T * , (1.1.17)

q (t , T )

V zz (t , x , 0) sin T  V xz (t , x , 0) cos T , T  T * , (1.1.18)

where 2T * , as can be seen from Figure 1.1, is the value of the sector of the shell that has contact with the half-space.

Chapter 1

20

The kinematic condition that determines the half-size of the contact area x* (t ) will be written as follows:

wT (t )  f ( x)  u z (t , x, 0)  w0 (t , T ) cos T  u0 (t , T ) sin T

x d x (t )

­°0, if ® °¯H  0, if

(1.1.19)

x ! x (t )

In this case, we assume that the contact region is simply connected, and this statement is equivalent to the fact that normal to the contact area stresses are compressive.

V zz

z 0

 0,

x  x (t ).

(1.1.20)

Mathematically, we have a non-stationary mixed boundary problem of the theory of elasticity when displacements are given in the contact region and the rest of the half-space boundary is free of stresses. We will require compliance with the condition of complete slippage. V zx 0, x  f, V zx 0, x  f, (1.1.21) z 0

z 0

Based on (1.1.8), the boundary conditions in the absence of friction in the contact zone can be formulated as follows:

w w0 (t ,T ) w uz vT (t )  cos T  wt z 0 wt w u (t ,T )  0 sin T , x  x (t ), wt V zz 0; x ! x (t ), V zx z 0 0, x  f.

(1.1.22)

The initial conditions for the potentials M and \ are zero:

Mt

0

wM wt

0, \ t 0

t 0

w\ wt

0.

(1.1.23)

t 0

For the problem of the impact of the elastic shell on the elastic halfspace, the velocity and displacement of the impact body is found from the equation of motion by its integration. The equation of motion of the shell with mass M for the problem of impact with initial velocity V0 has the form:

Formulation of elastic and elastic-plastic problems

d 2 wT (t )

M

 P(t ),

dt 2

vT (t ) t

(1.1.24)

V0 , wT (t ) t

0

21

0

0,

(1.1.25)

x (t )

P(t ) 2

³

V zz (t , x, 0)dx.

(1.1.26)

0 In addition, we have the condition of no perturbations in front of the front of the longitudinal waves and the condition of attenuation of perturbations at infinity.

M U !D t C 1

where

U1

D

0, \ U !D t C 1 D

x 2  z 2 , CD

0,

(1.1.27)

const.

M U of o 0, \

o 0. (1.2.28) U1of Thus, we obtained the initial boundary problem for equations i) (1.1.3), (1.1.5) and (1.1.19) with mixed boundary conditions (1.1.17) and initial conditions (1.1.18) and (1.1.20); ii) (1.1.4), (1.1.5) and (1.1.19) with mixed boundary conditions (1.1.17) and initial conditions (1.1.18) and (1.1.20). Solving the formulated problem, it is necessary to investigate the stress-strain state of the shell and half-space, which arises as a result of their collision. To do this, find the normal stresses V zz , normal 1

displacements

uz , displacements of the middle surface of the shell w0

and other characteristics.

§1.2. Impact of fine elastic spherical shells of Kirchhoff– Love and S.P. Tymoshenko types on an elastic half-space A thin elastic spherical shell reaches a surface of the elastic half-space z t 0 at a time t 0 . As shown in Figure 1.2, we connect to the shell a moving spherical coordinate system r cM cT , where M c is the longitude of the radius vector and T is the polar angle. The shell penetrates the elastic medium with a velocity vT  t , (0 d t d Ɍ ) , the initial penetration rate

V0

vT (0) and T is the time during which the shell interacts with the

half-space.

Chapter 1

22

Denote by u0 (t , T ) , w0 (t , T ) , p (t , T ) and q (t , T ) the tangential and normal displacements of the points of the middle surface of the shell and the radial and tangential components of the distributed external load acting on the shell, respectively. We connect a fixed cylindrical coordinate system r M z with half-space, the Oz axis is directed inwards, M is the polar angle. The angle T is plotted from the positive direction of the Oz axis. The physical properties of the half-space material are characterized by elastic constants: the modulus of expansion K, the shear modulus P and the density U. An elastic medium with constants K, P and U will be matched by a hypothetical acoustic medium with the same constants K and

U(P

0 ). By C p , CS

and C0 , we mean the speed of longitudinal and

transverse waves in an elastic half-space, as well as the speed of sound in this hypothetical acoustic environment. Coefficients C p , CS , C0 , D, E and b are defined in the same way as in (1.1.1). We introduce dimensionless variables: C0t ui u0 r z , rc , zc , uic , u0c , tc R R R R R (1.2.1) V ij w0 vT wT , V ijc , vTc , wTc , w0c R K C0 R pc

p KR 2

, qc

q KR 2

, Mc

M

U R3

, (i, j

r , M , z ).

Here

u (ur , uM , uz )

is the

displacement vector of the moving points of the environment; V zz , V rz are the non-zero components of the medium stress tensor; M is the linear weight of the shell and vT (t ), wT (t )

Fig. 1.2. Calculation scheme

are velocity and displacement of the shell as a solid, respectively. In

Formulation of elastic and elastic-plastic problems

23

future we will figure out only dimensionless values, so we omit the dash. The elastic half-space and the shell are in a state of axisymmetric deformation. Taking into account (1.2.1), from the system of basic equations of dynamics of arbitrary thin elastic shells based on Kirchhoff–Love’s hypotheses, we obtain the equations of motion of a thin elastic spherical shell written with respect to dimensionless variables. w 2u0 wu ww  ctgT 0  (Q 0  ctg 2T )u0  (1 Q 0 ) 0  2

wT

wT

wT

ª w 3 w0 w 2u0 § w 2 w0 w u0 ·  a1 «  +ctgT ¨  ¸ 3 ¨ wT 2 wT ¸¹ wT 2 «¬ wT © §ww ·º (Q 0  ctg 2T ) ¨ 0  u0 ¸» © wT ¹¼

E5

w 2u0  E 6 q, w t2

ª w 4 w0 w 3u0 ªw u º (1  Q 0 ) « 0  ctgT u0  2 w0 »  a1 «   4 ¬ wT ¼ wT 3 «¬ wT § w 3 w0 w 2u0 · § w 2 w0 w u0 · 2 Q T (1 ctg )      2ctgT ¨ ¸ ¨ ¸ 0 2 ¸ ¨ wT 3 ¨ wT 2 ¸ wT wT © ¹ © ¹ §ww ·º  ctgT (2 Q 0  ctg 2T ) ¨ 0  u0 ¸» © wT ¹¼

E5

w 2 w0  E 6 p, w t2 (1.2.2)

where a1

h

2

(1 Q 02 ) U0 K

(1 Q 02 ) KR ,

, E5 , E6 E0 U E0h 12R2 Q 0 , E0 , U0 are Poisson’s ratios, Young’s modulus of elasticity and

the density of the shell material, and p and q are the radial and tangential components of the distributed load acting on the shell, respectively. Differential equations (of the S.P. Tymoshenko type), which describe the dynamics of spherical non-sloping shells and take into account the shear and inertia of rotation of the cross section, by virtue of (1.2.1) take the following form [178, p.87]:

Chapter 1

24

w 2u0

1

1 Q 02 wT 2 



ctgT w u0

1 Q 02 wT

Q 0  (1 Q 0 ) cos 2 T (1 Q 02 ) sin 2 T

u0 



2(1 Q 0 )ks  1 Q 0 w w0 2(1 Q 02 )ks

) 2(1  Q 0 )k s

J 02

wT w 2u0 w t2



 q,

w 2 w0 w w0 1 w u0 ctgT    2(1 Q 0 )k s wT 2 1 Q 0 wT 2(1  Q 0 )k s wT 1 w ) ctgT 2 u0  w0    2(1  Q 0 )k s wT 1 Q 0 1 Q 0 1



w 2 w0 ctgT ) J 02  p, 2(1  Q 0 )k s w t2

E0 hR 2 w w0 w 2) w) T    ctg wT 2(1 Q 0 )ks D wT wT 2 

(1 Q 0 )ks D(2Q 0  (1 Q 0 ) sin 2T )  E0 hR 2 sin 2 T 2(1 Q 0 )ks D sin 2 T

w 2) , w t2 U0k1C02 , K02 E0

)

(1.2.3)

K02

J 02

kr 1 

3h2 20R

, D 2

U0h3C02kr 12D

E0h3

12(1 Q 02 )

, k1 1 

, ks

h2 12R2

,

5 , 6

where Ɏ is angle of rotation of the normal section to the middle surface,

ks

is the shear coefficient and D is the cylindrical stiffness.

The motion of an elastic medium is described by scalar potential M and the non-zero component \ of vector potential that satisfy the wave equations:

Formulation of elastic and elastic-plastic problems

'M

w 2M , '\ D 2w t 2

25

w 2\ w2 w w2 , ' {   . (1.2.4) E 2w t 2 w r 2 rw r w z 2

Physical quantities are expressed in terms of wave potentials as follows:

ur

wM w\  , uz wr wz

wM w\ \   , uM wz wr r

0,

§ 2 w 2M w 2\ w\ · 2 w M  2   E ¨ 2 ¸, ¨wz ¸ r z r z w w w w t2 © ¹

V zz

(1  2b 2 )

V rz

w 2M w 2\ w 2\  2  2E 2 2E , V rM w rw z w t w z2

V rr

(1  2b 2 )

2

VM z

(1.2.5)

0,

§ 2 w 2M w 2\ · 2 w M   2 E ¨ 2 ¸. ¨wr ¸ r z w w w t2 © ¹ modulus P 0 is set to zero, then

the equations of If the shear motion of the elastic medium will be the equations of acoustics. Physical quantities will then be expressed in terms of displacement potential as follows: w 2M w 2M Vr , Vz , VM 0, (1.2.6)

w rw t

w zw t

if i z j, ­0, ° V ij ® w 2M p , if i j, (i, j r , M , z )  ° D 2w t 2 ¯ where Vr , VM , Vz are components of the velocity vector of the points of the medium

V (Vr ,VM ,Vz ) .

Consider the initial stage of the process of the impact of elastic shells on the surface of the elastic half-space, when there is no plastic deformation and the amount of deepening of the shell into the environment is small. The problem of interaction of elastic shells with an elastic halfspace is solved in a linear formulation; therefore, we linearize boundary conditions [120]: boundary conditions from the perturbed surface are demolished to the undisturbed surface of deformable bodies.

Chapter 1

26

As can be seen from Figure 1.2, the projections of the functions

u0 ,

w0 , p and q on the axis O cz c and Or will be equal: prZ c w0 (t ,T ) w0 (t , T ) cos T , prZ cu0 (t , T ) u0 (t , T ) sin T , prZ c p(t ,T ) p (t , T ) cos T , prZ c q(t , T ) q (t , T ) sin T , prr w0 (t ,T )  w0 (t ,T ) sin T , prr u0 (t ,T ) u0 (t ,T ) cos T , prr p(t ,T )  p(t ,T ) sin T , prr q(t ,T ) q(t ,T ) cos T . Then, in the coordinate system zOr, displacements uz, ur and stresses

V zz

and

V rz

at points on the surface of the contact area will be written

as:

u z (t , r , 0) wT (t )  f ( r )  w0 (t , T ) cos T  u0 (t , T ) sin T , ur (t , r , 0)  w0 (t ,T )sin T  u0 (t ,T ) cos T , V zz (t , r , 0)  p (t ,T ) cos T  q (t ,T ) sin T , V rz (t , r , 0)  p (t , T ) sin T  q (t , T ) cos T ,

(1.2.7) (1.2.8) (1.2.9) (1.2.10)

p (t , T )

V zz (t , r , 0) cos T  V rz (t , r , 0) sin T , T  T * , (1.2.11)

q (t , T )

V zz (t , r , 0) sin T  V rz (t , r , 0) cos T , T  T * , (1.2.12)

wT (t ) is the movement of the shell as a solid, function f® describes the shell profile and 2T * , as can be seen from Figure 1.2, is the magnitude of the sector of the shell in contact with the half-space. The kinematic condition that determines the half-size of the contact

area r (t ) will be written as follows:

wT (t )  f (r )  uz (t , r,0)  w0 (t ,T )cosT  u0 (t ,T )sin T

Formulation of elastic and elastic-plastic problems

­°0, if r d r (t ) ®

°¯H  0, if r ! r (t )

27

(1.2.13)

In this case, we assume that the contact area is simply connected, and this statement is equivalent to the fact that normal to the contact area stresses are compressive:

V zz

z 0

 0, r  r (t ).

(1.2.14)

Mathematically, we have a non-stationary mixed boundary problem of the theory of elasticity when displacements are given in the contact region and the rest of the boundary is free of stresses. We will require compliance with the condition of complete slippage at the contact zone. V zr 0, r ! 0. (1.2.15) z 0

Based on (1.2.7), the boundary conditions in the absence of friction in the contact zone can be formulated as follows:

w uz wt V zz

vT (t ) 

z 0 z 0

w w0 (t ,T ) w u (t ,T ) cos T  0 sin T , r  r (t ), wt wt

0, r ! r (t ), V zr

z 0

0, r ! 0. (1.2.16)

Initial conditions for potentials M and \ are zero:

wM wt

M

0

0, \ t 0

w\ wt

0. (1.2.17) t 0 t 0 For the problem of the impact of the elastic shell on the elastic halfspace, the velocity and displacement of the impact body is found from the equation of motion by its integration. The equation of motion of the shell with mass M for the problem of impact with initial velocity V0 has the form:

Mt

d 2 wT (t ) dt 2

vT (t ) t

0

 P(t ),

V0 , wT (t ) t

(1.2.18) 0

0,

(1.2.19)

Chapter 1

28

r (t )

P(t ) 2S

³

rV zz (t , r ,0)dr.

(1.2.20)

0 In addition, we have the condition of no perturbations in front of the front of the longitudinal waves and the condition of attenuation of perturbations at infinity.

M U !D t C 1

where

U1

D

0, \ U !D t C 1 D

r 2  z 2 , CD

0,

(1.2.21)

const.

M U of o 0, \

o 0. (1.2.22) U1of Thus, we obtained the initial boundary value problem for equations i) (1.2.2), (1.2.4) and (1.2.18) with mixed conditions at the boundary (1.2.16) and initial conditions (1.2.17) and (1.2.19); ii.) (1.2.3), (1.2.4) and (1.2.18) with mixed boundary conditions (1.2.16) and initial conditions (1.2.17) and (1.2.19). Solving the formulated problem, it is necessary to investigate the stress-strain state of the shell and half-space, which arises as a result of their collision. To do this, find the normal stresses V zz , normal 1

displacements uz , displacements of the middle surface of the shell and other characteristics.

w0

§1.3. Impact of hard cylinders on an elastic layer Perpendicular to the surface of the elastic layer, an absolutely rigid cylinder [53] with a flat surface | x |d d parallel to the surface of the layer moves and in time t 0 reaches this surface. Contact occurs along the line ^| x |d d ; z 0` , or on the generatrix of the cylinder, when d 0. The cylinder is connected to a moving cylindrical coordinate system rT z c , the O cy c axis coincides with the axis of the cylinder as shown in Figure 1.3 and the layer is connected to the fixed Cartesian coordinate system ɯɭz. Layer thickness is h. The cylinder penetrates into the elastic medium with a speed

VT (t ) , (0 d t d T )

and the initial penetration rate V0

VT (0) , T is the

time during which the cylinder interacts with the layer. We introduce dimensionless variables as in §1.1. In this case U, P, Ʉ, CP and

CS

are the

Formulation of elastic and elastic-plastic problems

29

density, shear modulus, volume deformation modulus and velocity of wave propagation in the elastic medium;

E0 , v0 , U0 , M

are Young’s

modulus, Poisson’s ratio, and the density and linear mass of the shell, respectively. We omit the dash everywhere below. The equations of motion of the elastic layer are written in the form [22, 53, 200] (1.1.5). If the shear modulus ȝ is set to zero, then the equations of motion of the elastic medium will be the equations of acoustics. Physical dimensionless quantities are expressed in terms of wave potentials, as described in §1.1. Denote by u x , u y , u z the components of the displacement vector, and

Fig. 1.3 Scheme of stamp-layer system.

by

V zz , V zx

the

components of the stress tensor. The solution of the problem uses the approach in [15–17, 22, 53, 117, 124–136, 200], which allows one at the initial stage of penetration to identify linear coordinates along the surface of the layer and the body; therefore, the approximate relations will be valid (2.1.46). In the contact area, considering (2.1.46), there is a relationship between displacement

u z (t , x, 0) *

uz

and pressure p.

wT (t )  H (| x |  d ) §¨1  1  (| x |  d ) 2 ·¸ , © ¹ (1.3.1)

t

x  x , wT (t )

³ VT (W )dW , 0

p (t , x )

V zz (t , x, 0),

x  x* .

The linearized boundary conditions are as follows:

(1.3.2)

Chapter 1

30

w uz wt

V zz

z 0 z 0

V zx

{ V (t , x) vT (t ),

0, 0,

z 0

x  x (t ),

(1.3.3)

x ! x (t ),

(1.3.4)

x  f.

(1.3.5)

For the interaction time 0 d t d T , we select the rectangle {| x |d l; 0 d z d h) occupied by the environment, and the problem of hitting the layer can be considered as the problem of hitting the rectangle. The width of the rectangle l is chosen so that the perturbations do not reach its boundary:

§ dx* * ¨ | x | l l ! D (T  t0 )  x (t0 ), ¨¨ dt t ©

t0

· D¸. ¸¸ ¹

For specificity, we choose the zero initial conditions of the problem and on the side surface of the rectangle the conditions of sliding foundation (1.3.6) ux 0, V zx 0, x l

x l

wI wt

0

0, \

w\ wt

0. (1.3.7) t 0 t 0 The motion of the shell as a body is determined by Newton’s second law

It

t 0

T (t )  F (t ), VT (0) V0 , wT (0) 0, Mw where F (t ) is the reaction force of the elastic layer,

(1.3.8)

determined considering (1.3.2) and (1.3.4) as an integral of the pressure in the contact area. x (t )

F (t ) 2

³

p(t , x)dx.

(1.3.9)

0

The boundary of the contact area x* considering the rise of the medium and the slowing of the penetration of the shell into the elastic medium is determined from the condition

Formulation of elastic and elastic-plastic problems

31

wT (t )  H (| x* |  d ) §¨1  1  (| x* |  d ) 2 ·¸  u z (t , x* , 0) © ¹ ­°0, ® °¯H  0,

(1.3.10)

x d x (t ) x ! x (t )

§1.4. Impact of fine elastic cylindrical shells on the elastic layer A thin elastic cylindrical shell [22] moving perpendicular to the surface of the elastic layer z = 0 at time t = 0 reaches this surface. Contact occurs on the generatrix of the cylinder. The shell is connected to the moving cylindrical coordinate system rT zc , the Ocyc axis coincides with the axis of the cylinder shell and the layer is connected to the fixed Cartesian coordinate system xyz . Layer thickness is h (see Figure 1.4). The shell penetrates into the elastic medium with a velocity penetration rate is

V0 VT (0) .

VT (t ) , (0 d t d T ) ,

and the initial

T is the time during which the shell

interacts with the layer. The thickness h1 of the shell is much less than the radius R of the middle surface of the shell (h1 / R d 0,05) . We introduce dimensionless variables as in §1.1. In this case,

U, P, Ʉ, CP and CS are density, shear modulus, volume deformation modulus, and velocities of waves propagation in an elastic medium;

E0 , v0 , U0 , M

Fig. 1.4 ɋɯɟɦɚ ɫɢɫɬɟɦɢ ɨɛɨɥɨɧɤɚ-ɲɚɪ.

are

Young’s modulus, Poisson’s ratio, density and linear mass of the shell. We omit the dash everywhere below.

Chapter 1

32

The motion of a cylindrical shell is described by a system of basic equations of dynamics of thin elastic shells based on Kirchhoff–Love’s hypotheses (1.1.3). The equations of motion of the elastic layer are written in the form [22, 53, 200] (1.1.5). In the solution of the problem, as well as for rigid cylinders, the approach [15–17, 22, 53, 117, 124–136, 200] is used, due to which the approximate relations (2.1.46) will be valid. In the contact area, taking into account (2.1.46), there is a relationship between displacements

u z (t , x, 0)

uz , w0

and pressure p.

wT (t )  1  1  x 2  w0 (t , x),

x  x* , wT (t )

t

(1.4.1)

³ VT (W )dW , 0

p (t , x )

V zz (t , x, 0),

x  x* .

(1.4.2)

The linearized boundary conditions are as follows:

w uz wt

z 0

{ V (t , x) vT (t ) 

V zz

z 0

V zx

z 0

w w0 (t , x) , wt

x  x (t ), (1.4.3)

0,

x ! x (t ),

(1.4.4)

0,

x  f.

(1.4.5)

For the interaction time 0 d t d T , we select the rectangle {| x |d l; 0 d z d h) occupied by the environment, and the problem of hitting the layer can be considered as the problem of hitting the rectangle. The width of the rectangle l is chosen so that the perturbations do not reach its boundary

§ dx* | x | l ¨ l ! D (T  t0 )  x* (t0 ), ¨¨ dt t ©

t0

· D¸. ¸¸ ¹

Let us choose the initial conditions of the problem (1.1.18). The motion of the shell as a body is determined from Newton’s second law as in (1.3.8) with the reaction force of the elastic layer as in (1.3.9).

Formulation of elastic and elastic-plastic problems

33

The boundary of the contact area x* considering the rise of the medium and the slowing of the penetration of the shell into the elastic medium is determined from the condition

wT (t )  1  1  x*2  u z (t , x* , 0)  w0 (t , x* ) ­°0, ® °¯H  0,

x d x (t )

(1.4.6)

x ! x (t )

§1.5. Quasi-static elastic-plastic formulation – three-dimensional problem The Odquist parameter, effective and main stresses are determined from the numerical solution of the three-dimensional bending problem of small-scale compact specimens, using a well-known technique [18, 20, 37, 39, 144–146]. Consider the deformation of an isotropic beam { L / 2 d x d L / 2; 0 d y d B; 0 d z d H } having a rectangular shape in the cross-sectional plane (Figure 1.5 – length L, width B, thickness H). The beam has a crack-notch with length l along the rectangular area

^x

0; 0 d y d l ; 0 d z d H } and is in contact with

two fixed supports in the area {L* d x d L*  a; y 0; 0 d z d H } . From above on a body the absolutely rigid drummer contacting a bar in area falls { x d A; y B ; 0 d z d H } . Its action is replaced in the contact zone by an evenly distributed normal stress  P , which changes over time as a linear function

P

Fig. 1.5. Geometric scheme of the problem

p01  p02t . Due to

the symmetry of the deformation process relative to the plane x 0 , only the right part of the beam (Figure 1.5) is considered below, when x t 0 . It is believed that

Chapter 1

34

the contact area remains unchanged throughout the interaction period. Known methods of research of plane quasi-static elastic-plastic [18, 20, 37, 39, 144–146] models are used in the calculations. The equations of the spatial quasi-static theory for which the components of the displacement vector u (ux , u y , uz ) are related to the components of the strain tensor by Cauchy relations are considered

H xx H yy

H zz

1 § wu x wu y ·  ¨ ¸, 2 © wy wx ¹ wu y 1 § wu z wu y · , H yz  ¨ ¸ 2 © wy wz ¹ wy

wu x , H xy wx

wu z , H xz wz

(1.5.1)

1 § wu x wu z · .  wx ¸¹ 2 ¨© wz

Equilibrium equations take the form:

wV xx wV xy wV xz 0,   wx wy wz wV yx wV yy wV yz 0,   wx wy wz wV zx wV zy wV zz   0. wx wy wz

(1.5.2)

Figure 1.5 shows a diagram of the calculation area. We assume that under impact loading the material is elastic-plastically strengthened, and the calculation of stress fields, deformations and their increments, p

including increments of plastic intensity d H i as well as the Odquist d H ip will be performed on the basis of the numerical parameter N solution of the quasi-static elastic-plastic problem. The boundary conditions of the problem will be written as follows:

³

x 0, 0  y  l , 0  z  H , V xx x 0, l  y  B, 0  z  H , ux

0, V xy 0, V xy

x L / 2, 0  y  B, 0  z  H , V xx

0, V xz

0,

0, V xz

0,

0, V xy

0, V xz

0,

Formulation of elastic and elastic-plastic problems

y 0, 0  x  L*, 0  z  H , V yy

0, V xy

y 0, L*  x  L*  a, 0  z  H , u y y

B, 0  x  A, 0  z  H , V yy

y

B, A  x  L / 2, 0  z  H , V yy H , 0  x  L / 2, 0  y  B, V zz

0, V yz

0,

0,

0, V yz

0, V zy

0, V xz

0,

0, V yz

0, V yz

0, V xy

0, V zx

0, (1.5.3)

0, V yz

0, V xy

 P, V xy

z 0, 0  x  L / 2, 0  y  B, uz z

0, V yz

0, V xy

y 0, L*  a  x  L / 2,0  z  H , V yy

35

0,

0, 0.

The determinant relations of the mechanical model are based on the theory of non-isothermal plastic flow of the medium with hardening under the condition of Huber–Mises fluidity. The effects of creep and thermal expansion are neglected. Then, considering the components of the strain tensor by the sum of its elastic and plastic components [7, 106, 179], we obtain for them

H ij

H ije  H ijp , H ije

1 sij  K V  M , d H ijp 2G

sij d O . (1.5.4)

where sij V ij  G ijV is the stress tensor deviator, G ij is the Kronecker symbol, ȿ is the modulus of elasticity (Young’s modulus), G is the shear

3K1 is the volumetric compression modulus, which binds in the ratio H K V  I volumetric expansion 3H (thermal expansion I { 0 ), V (V xx  V yy  V zz ) 3

modulus,

K1 (1  2Q ) / (3E) ,

K

is the mean stress and d O is a scalar function, which is determined by the shape of the load surface and quadratically depends on the stress deviator sij [106, 179].

Chapter 1

36

­° 3d H ip 2 2 0 ( f V V ( T ) 0); (f {   ® i S 2 V i ¯° ( f ! 0  impossible); dO

2§ p p  d H yy d H xx ¨ 3 ©



d H ip 



p d H yy



2





2 V xy

where

2  V xz

V i is

  dH xxp  dH zzp

§ p  6 ¨ d H xy ©



1 § ¨ V xx  V yy 2©

Vi 6

 d H zzp

2

2  V yz



2

2



p d H xz

2

2



2

½° 0) ¾ ¿°

0, df



p d H yz



1/2 (1.5.5) 2 ··

,

¸¸ ¹¹



 V xx  V zz  V yy  V zz



2



1

2 .

the stress intensity and

H ip is the intensity of plastic

deformations. We assume that due to plastic deformation the material is strengthened with strengthening coefficient

K*

at this temperature ratio [18, 20, 23–52,

144–146, 200]:

K*

§ N (T ) · V S (T ) V 02 (T0 ) ¨1  ¸ , H0 H0 ¹ © where T is temperature,

N

V 02 (T0 ) E

is the Odquist parameter, T0

the hardening coefficient and

V S (T )

,

(1.5.6)

20$ C , K * is

is the yield strength after hardening

of the material at temperature T.

§1.6. Dynamic elastic-plastic formulation The approach [116–123] for solving dynamic problems, which was developed by V.D. Kubenko, makes it possible to determine the stressstrain state only on the surface of the medium into which the drummer penetrates. Furthermore, this approach does not allow one to study the impact of elastic shells of the S.P. Tymoshenko type. The equations describing the dynamics of the shell are Laplace transformed and

Formulation of elastic and elastic-plastic problems

37

developed into trigonometric Fourier series with parameter n. After returning to the space of the originals and using the convolution theorem in integral expressions of the components of the series of normal and tangential displacements of the middle surface of shells of the S.P. Tymoshenko type, some kernels will have asymptotic

exp  O  n

on

the parameter n. Therefore, with increasing size of the reduced system of Volterra integral equations of the second kind, the determinant of the system of linear algebraic equations obtained as a result of discretization will increase indefinitely – the matrix of this system will be poorly conditioned. However, if we use Kirchhoff–Lowe-type shells [15–17, 22, 125–127, 136], the convergence of the solution will be guaranteed when solving impact problems. This led to the expediency of developing other mathematical approaches and models. In [20, 23–52, 200, 201] a new approach to solving problems of impact and non-stationary interaction in elastic-plastic mathematical formulation was developed. In this book, the results of solving two plane problems of an impact of a circular rigid cylinder with plane surface on the elastic layer, namely i) impact within a purely elastic model and ii) non-stationary interactions in the elastic-plastic formulation are compared. The circular cylinder initially contacts the surface of the layer along the plane surface. Due to the impact load, we assume that the material is elastic-plastic with strengthening, and the calculation of stress, strain fields and their p

increments, including plastic deformations intensity increments d H i , as well as the Odquist parameter

N

p

³ dHi

will be based on numerical

solution of the corresponding dynamic elastic-plastic problem. During the calculations we use the known methods of research of the quasi-static elastic-plastic [37, 39, 144–146] model, which take into account the non-stationary load and apply numerical integration implemented in the calculation of the dynamic elastic model [15–17, 22, 53, 125–127, 136, 200]. When calculating the dynamic fields of stresses and strains, we do not take into account the interaction of wave fields, the reflection from the body boundary and the possible contact interaction between the edges of the notch.

38

Chapter 1

§1.6.1. Plane stress state The deformation [20, 25, 26, 32, 36, 38, 47] of a beam specimen in the shape of a rectangle 6 L u B (  L / 2 d x d L / 2 ; 0 d y d B ) with a notch-crack of initial length

0 d y d l0 },

l l0

along the segment { x

0;

which is in contact with two fixed bearings along

{ L* d x d L*  a ; y 0 }. The thickness w of the specimen is considered so small that it is possible to adjust the dependencies of the plane stress state ( V zz 0, V xz 0, V yz 0 ). An absolutely rigid drummer falls on top of the body, making contact along the segment

x d A; y B }. Its

{

action on the body is replaced by an evenly distributed stress  P in the contact region, which changes over time as a linear function P p01  p02t . Given the symmetry of the deformation Fig. 1.6. Geometric scheme of the problem (a) and break grid near the tip of the crack (b)

process relative to the line x 0 , only the right part of the cross section of the body is considered below (Figure 1.6a). The equations of the plane dynamic theory for which the components of the displacement vector u (ux , u y , uz ) are related to the components of the strain tensor by Cauchy relations are considered (1.5.1). The equations of motion of the medium have the form:

Formulation of elastic and elastic-plastic problems

wV xx wV xy  wx wy wV xy wx



wV yy wy

U U

w 2u x wt 2

39

,

w 2u y wt 2

(1.6.1)

,

where U is the material density. The boundary conditions of the problem, which take into account the change in crack length but are based on the assumption of invariance of the reaction area and the location of supports, as well as determination of support reactions using static methods, are written as follows:

x

0,

0 yl:

V xx

0, V xy

x

0,

l  y  B:

ux

x

L / 2, 0  y  B :

V xx

0, V xy

0,

y

0,

0  x  L* :

V yy

0, V xy

0,

y

0,

L*  x  L*  a :

uy

y

0,

L*  a  x  L / 2 : V yy

0, V xy

y

B,

0  x  A:

V yy

 P, V xy

y

B,

A x  L / 2:

V yy

0, V xy

0, V xy

0, V xy

0, 0,

(1.6.2)

0, 0, 0, 0.

The initial conditions are as follows:

ux t u y

0

0, u y

t 0

0, u z t

0, u z t 0, l t

0

0, u x t l0 .

0

0, (1.6.3)

0 0 t 0 The determinant relations of the mechanical model are based on the theory of non-isothermal plastic flow of the medium with hardening under the condition of Huber–Mises fluidity. The effects of creep and thermal expansion are neglected. Then, considering the components of the strain tensor by the sum of its elastic and plastic components [7, 106, 179], we obtain for them expression (1.5.4), where d O is a scalar function (1.5.5) determined by the condition of plasticity (shape of the load surface) and

quadratically depends on the components of the stress deviator sij [106,

Chapter 1

40

179], in the expression of which the stress intensity and the differential of the intensity of plastic deformations has the form: 2§ p p 2 p p 2 d H ip ¨ d H xx  d H yy  d H xx  d H zz 

3 ©





Vi

p d H yy



 d H zzp





2

6

1 § ¨ V xx  V yy 2©





p d H xy



2





1 2· 2

¸ ¹

2

,

(1.6.4)



 V xx  V yy

2

1 2 · 2  6V xy ¸ .

¹

The material is strengthened with a strengthening coefficient

K*

[18,

20, 23–52, 144–146, 200] as in (1.5.6). Rewrite (1.5.4) in expanded form:

d H xx d H yy d H xy

§ V V ·  KV ¸  (V xx  V )d O , d ¨ xx 2 G © ¹ § V yy  V ·  KV ¸  (V yy  V )d O , d¨ © 2G ¹ § V xy · d¨ ¸  V xy d O. © 2G ¹

(1.6.5)

§1.6.2. Plane strain state Deformations and their increments [27, 28, 30, 34, 41, 42, 45, 46, 48, 50], the Odquist parameter, effective and principal stresses are obtained from the numerical solution of the dynamic elastic-plastic bending problem of infinite beam {  L / 2 d x d L / 2 ; 0 d y d B ; f d z d f} in the plane of its cross section in the form of specimens of the Charpy type. It is assumed that the stress-strain state in each cross-section of the cylinder is the same, close to the plane deformation, and therefore it is necessary to solve the equation for only one section in the form of a rectangle 6 L u B with a notchcrack with length l along the segment {x 0 ; 0 d y d l} and contacts two fixed supports along { L* d x d L*  a ;

y

0 }.

Formulation of elastic and elastic-plastic problems

41

From above on a body the absolutely rigid drummer contacting along a segment { x d A;

y B}. Its action is replaced by an evenly distributed

stress  P in the contact region, which changes over time as a linear function P p01  p02t . Given the symmetry of the deformation process relative to the line x 0 , only the right part of the cross-section is considered below (Figure 1.6a). The calculations use known methods for studying the quasi-static elastic-plastic [18, 20, 37, 39, 144–146] model, taking into account the non-stationarity of the load and using numerical integration implemented in the calculation of the dynamic elastic model [27, 28, 30, 34, 41, 42, 45, 46, 48, 50]. The equations of the plane dynamic theory are considered, for which the components of the displacement vector u (u x , u y ) are related to the components of the strain tensor by Cauchy relations (1.5.1), and the equations of motion of the medium have the form (1.6.1). The boundary and initial conditions of the problem have the same form as in (1.6.2) and (1.6.3), respectively. The determinant relations of the mechanical model are based on the theory of non-isothermal plastic flow of the medium with hardening under the condition of Huber–Mises fluidity. The effects of creep and thermal expansion are neglected. Then, considering the components of the strain tensor by the sum of its elastic and plastic components [106, 179], we obtain expressions for them (1.5.4). The material is strengthened with a hardening factor

K*

[18, 20, 23–

52, 144–146, 200] (1.5.6). Rewrite (1.5.4) in expanded form:

d H xx d H yy d H zz

d H xy

§ V V · d ¨ xx  KV ¸  (V xx  V )d O , © 2G ¹ § V yy  V · d¨  KV ¸  (V yy  V ) d O , © 2G ¹ § V V · d ¨ zz  KV ¸  (V zz  V )d O , © 2G ¹

§ V xy d¨ © 2G

· ¸  V xy d O , ¹

(1.6.6)

Chapter 1

42

d O is determined by the shape of the load surface and is quadratically dependent on the stress deviator sij [106, 179], and A scalar function

the stress intensity and plastic deformation intensity differential have the forms of (1.5.5) and (1.6.4), respectively and considering that in case of plane strain state: 2 1 § 2 Vi ¨ V xx  V yy  V xx  V zz 

2©







 V yy  V zz In

contrast



2





1 2 · 2  6V xy ¸ .

¹

to

the

traditional

flat

deformation,

when

'H zz ( x, y ) const , for a refined description of the deformation of the specimen, taking into account the possible increase in longitudinal elongation

'H zz , we present its form [27, 28, 30, 34, 41, 42, 45, 46, 48,

50, 56, 144]: 0 'H zz ( x, y) 'H zz  'F x x  'F y y,

where unknown

'F x

and

'F y

(1.6.7)

describe the bending of the prismatic

body (which simulates the plane strain state in the solid mechanics) in the 0 Ozx and Ozy planes, respectively, and ' H zz are the increments according to the detected deformation bending along the fibres x y 0 .

§1.6.3. Three-dimension stress-strain state Deformations and their [23, 31, 40, 43, 44, 49, 51, 200] increments, the dHip ; d H ip are increments of intensity of Odquist parameter (N plastic deformations), effective and main stresses are obtained from the solution of the dynamic elastic-plastic problem of bending beams

³

{ L / 2 d

x d L / 2 ; 0 d y d B ; 0 d z d H ` in the plane of its cross-

section in the form of Charpy-type specimens. The beam under consideration 6 L u B u H has a notch-crack with length l along the

Formulation of elastic and elastic-plastic problems

43

strip { x

0 ; 0 d y d l ; 0 d z d H } and contacts two fixed supports along {L* d x d L*  a ; y 0 ; 0 d z d H }. An absolutely rigid drummer falls on top of the body, making contact

y B ; 0 d z d H }. Its action is replaced by an evenly distributed stress  P in the contact region, which changes over time as a linear function P p01  p02t . Given the symmetry of the deformation process relative to the plane x 0 , only the right part of the

along the strip { x d A;

cross-section is considered below (Figure 1.7a). The calculations use

Fig. 1.7. Geometric scheme of problem (a) and the partition grid near the crack tip (b)

known methods for studying the quasi-static elastic-plastic [18, 20, 37, 39, 144–146] model, which take into account the non-stationarity of the load and use numerical integration implemented in the calculation of the dynamic elastic model [23, 31, 40, 43, 44, 49, 51, 200]. Boundary and initial conditions of the problem will be written as follows:

x

V xx

0,

0  y  l, 0, V xy 0, V xz

x 0, l  y  B, u x 0, V xy 0, V xz

0 zH : 0, 0 z H : 0,

Chapter 1

44

x

L / 2, 0  y  B, V xx 0, V xy 0, V xz 0,

y

0, V xy 0, 0,

y

0, V yz

0, V xy B,

V yy

0, V yz

0, V yz

B, A  x  L / 2, V yy 0, V xy 0, V yz

V zz

0,

0 zH : 0, 0,

0  x  A, 0 z H :  P, V xy 0, V yz 0,

y

z

0,

L*  a  x  L / 2, 0  z  H :

V yy y

0 zH :

L*  x  L*  a,

0, V xy

uy

0,

0  x  L* ,

V yy y

0 zH :

0  x  L / 2, 0, V xz 0, V yz

H , 0  x  L / 2, V zz 0, V xz 0, V yz

0 zH : 0, 0 y  B: 0, 0 y  B:

z

t

0 : ux

0, u y

(1.6.8)

0,

0, u z

0.

The equations of the spatial dynamic theory are considered, for which the components of the displacement vector u (u x , u y , u z ) are related to the components of the strain tensor by Cauchy relations (1.5.1). The equations of motion of the medium have the form: wV xx wV xy wV xz w 2u x   U 2 , wx wy wz wt (1.6.9) wV xy wV yy wV yz w 2u y U 2 ,   wx wy wz

wt

Formulation of elastic and elastic-plastic problems

wV xz wV yz wV zz   wx wy wz

U

w 2u z wt 2

45

.

The determinant relations of the mechanical model are based on the theory of non-isothermal plastic flow of the medium with hardening under the condition of Huber–Mises fluidity. The effects of creepage and thermal expansion are neglected. Then, considering the components of the strain tensor by the sum of its elastic and plastic components, we obtain expressions for them (1.6.10). The material is strengthened with a hardening coefficient K* [18, 20, 23–52, 144–146, 200] (1.5.6). Rewrite (1.5.4) in expanded form::

§ V ij 1 · · §  G ij ¨ K  d¨ ¸V ¸  2G ¹ ¹ © © 2G  (V ij  G ijV )d O , (i, j x, y, z ),

d H ij

(1.6.10)

d O is determined by the shape of the load surface and is quadratically dependent on the stress deviator sij [106, 179], and A scalar function

the stress intensity and the intensity differential of plastic deformations have the form as in (1.5.5).

CHAPTER 2 ALGORITHM FOR SOLVING MIXED NON-STATIONARY AND DYNAMIC BOUNDARY PROBLEMS

When solving the problem of the impact of elastic shells with elastic half-space, the Laplace transform in time coordinate is used. The Laplace transform for a function f (t ) that is integrable on any interval is expressed by the formula: f L f ( s) f (t )e st dt , (2.1)

³ 0

f (t )

1 2S i

J if

³

f L ( s )e st ds,

Re J ! 0.

(2.2)

J if where the interval is taken along the path lying to the right of the singularities of the integrated function.

§2.1. Impact of fine elastic cylindrical shells on an elastic half-space Since the impact process is short-lived, the perturbation region at each time t is finite. Limiting the finite time interval of the interaction  0 d t d T , we can select a half-space region that by the time T covers the entire perturbation zone. From this point of view, for the times 0 d t d T elastic half-space can be replaced by elastic half-strip

 x d l;

z t 0 ; perturbations do not reach the limits of time T. l DT  x (T ).

(2.1.1)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

47

Thus, for the time being 0 d t d T and the problem under consideration is reduced to a non-stationary problem for a half-strip under mixed boundary conditions at its end. To represent the displacement vector in the form (2.1.2) u grad M  rot\ , div\ 0, on the side surface of the semi-strip, we choose, for example, the conditions of sliding: (2.1.3) ux 0, V zx 0, x l

x l

or

uz

x l

0, V xx

x l

0.

(2.1.4)

Consider the initial boundary value problem (1.1.3), (1.1.5), (1.1.17) and (1.1.20). We present the normal w0 (t , T ) and tangential u0 (t , T ) displacements of the points of the middle surface of the shell and the radial p (t ,T ) and tangential q (t , T ) components of the distributed external load acting on the shell in the form of trigonometric Fourier series:

w0 (t , T )

u0 (t , T ) p (t , T ) q (t , T )

f

¦ w0n (t ) cos(nT ),

(2.1.5)

¦ u0n (t ) sin(nT ),

(2.1.6)

¦ pn (t ) cos(nT ),

(2.1.7)

n 0 f

n 1 f

n 0 f

¦ qn (t ) sin(nT ).

(2.1.8)

n 1

If in the initial boundary value problem (1.1.3), (1.1.5), (1.1.17) and (1.1.20) equations (1.1.4) are considered instead of equations (1.1.3), then in the form of trigonometric Fourier series we have to present another angle of rotation of the normal ) ( t , T ) :

) (t , T )

f

¦ ) n (t ) sin(nT ).

n 1

(2.1.9)

Chapter 2

48

In space of the Laplace transformant with the parameter s transformants of functions ) , w0 , u0 , p , q will, by virtue of (2.1.5)– (2.1.9), have the form: f w0L ( s, T ) w0Ln ( s ) cos(nT ), n 0

¦

u0L ( s, T ) p L ( s, T ) q L ( s, T ) ) L ( s, T )

(2.1.10)

f

¦ u0Ln (s) sin(nT ),

(2.1.11)

¦ pnL (s) cos(nT ),

(2.1.12)

¦ qnL (s) sin(nT ),

(2.1.13)

¦ ) nL (s) sin(nT ).

(2.1.14)

n 1 f

n 0 f

n 1 f

n 1

Next, apply to the system of equations (1.1.3) the Laplace transform (2.1) on the variable t with the parameter s and substitute equations (2.1.10)–(2.1.13). We are equating the coefficients at the same cos(nT ) and

sin(nT )

, and we obtain the relations that connect the components of L

L

the transformants of the functions w0 , u0 , p L w0,0 (s)

L

L

and q .

E 2 p0L ( s ) , 1  E1s 2

(2.1.15)

 E2 (E1s2  (1  a1)n2 ) pnL (s)   nE 2 (1  a1n 2 )qnL ( s)  E12 s 4  L w0, n ( s)

(2.1.16)



((1  a1 )n 2  a1n 4  1) E1s 2  a1n 2 (n 2  1) 2 ,

Algorithms for solving mixed non-stationary and dynamic boundary value problems

 nE2 (1  a1n2 ) pnL (s)  E2 (E1s2   a1n 4  1)qnL ( s )  E12 s 4  ((1  a1 )n 2  a1n 4  1) E1s 2  a1n 2 (n 2  1)2 , (n 1, f)

49

L u0, n (s)

(2.1.17)

Then, applying to (2.1.15)–(2.1.17) the inverse Laplace transform (2.2), by the convolution theorem of the originals of the two functions we obtain: t E2 t W  w0,0 (t ) p0 (W ) cos dW , (2.1.18) E1 0 ( E1)1/2

³

t

w 0,n (t )

³ pn (W )G11 (n, t  W )dW  0

(2.1.19)

t

 ³ qn (W )G12 (n, t  W ) dW , 0

t

u0,n (t )

³ pn (W )G21(n, t  W )dW  0

(2.1.20)

t

 ³ qn (W )G22 (n, t  W )dW , (n 1, f) 0

where

ª§ (1  a )n 2 · 1 G11 (n, t )  X1 ¸ cos X1 t  «¨ ¸ E1 ( X 2  X1 ) «¬¨© E1 ¹ º § (1  a1 )n 2 · ¨  X 2 ¸ cos X 2 t » , ¨ ¸ E1 »¼ © ¹

E2

Chapter 2

50

E 2 n(1  a1n2 ) (cos X1t  cos X 2 t ), E12 ( X 2  X1 )

G12 (n, t )

G21 (n, t ) G12 (n, t ), ª§ 1  a n 4 · 1  X1 ¸ cos X1 t  «¨ ¸ E1 ( X 2  X1 ) «¬©¨ E1 ¹ º § 1  a1n 4 · ¨  X 2 ¸ cos X 2 t » , ¨ E1 ¸ »¼ © ¹

E2

G22 (n, t )

X1

b § b 2 ¨© 2



2

1/2

· C¸ ¹

, X2

b ((1  a1 )n2  a1n4  1) E1, c

b § b 2 ¨© 2



2

1/2

· C¸ ¹

,

a1n2 (n2  1)2 ,

t

w0,0 (t )

E2 t W ( )sin W p dW , 0 ³ ( E1 )1/2 0 ( E1 )1/2

(2.1.21)

t

w0,n (t )

³ pn (W )G11 (n, t  W )dW  0

(2.1.22)

t

 ³ qn (W )G12 (n, t  W ) dW , 0

t

u0,n (t )

³ pn (W )G 21(n, t  W )dW  0

(2.1.23)

t

 ³ qn (W )G 22 (n, t  W )dW , (n 1, f) 0

where

G11(n, t )

ª§ (1  a )n2 · sin X1t 1  X1 ¸  «¨ ¸ E1( X 2  X1) «¬©¨ E1 X1 ¹

E2

Algorithms for solving mixed non-stationary and dynamic boundary value problems

51

§ (1  a1 )n2 · sin X 2 t º ¨  X2 ¸ » , ¨ ¸ E1 X 2 »¼ © ¹ E 2 n(1  a1n 2 ) § sin X1 t sin X 2 t ·  G12 (n, t )  ¨ ¸, X1 X 2 ¸¹ E12 ( X 2  X1 ) ¨© G 21 (n, t ) G12 (n, t ), G 22 (n, t )

ª§ 1  a n 4 · sin X1 t 1  X1 ¸  «¨ ¸ E1 ( X 2  X1 ) «¬©¨ E1 X 1 ¹

E2

§ 1  a1n 4 · sin X 2 t º ¨  X2 ¸ » . ¨ E1 ¸ X »¼ 2 © ¹ We now apply the Laplace transform (2.1) to the variable t (s is the transformation parameter) to the system of equations (1.1.4) and substitute there (2.1.10)–(2.1.14). Equating the coefficients at the same cos nT and sin nT we obtain the ratios connecting the components of the L

L

L

development of functions ) , w0 , u0 , p

L

and q

L

in trigonometric

series [199]:

L w0,0 (s)

E4 p0L (s) , K02s2  a3

(2.1.24)

L w0, n (s)

L L Q11 ( n , s ) pnL ( s )  Q12 ( n, s ) qnL ( s ),

(2.1.25)

L u0, n (s)

L L Q21 ( n, s ) pnL ( s )  Q22 ( n, s ) qnL ( s ),

(2.1.26)

) nL ( s )

L L Q31 ( n, s ) pnL ( s )  Q32 ( n, s ) qnL ( s ),

(2.1.27) where QijL ( n, s )

' 21  n, s

' ij ( s ) '(s)

, (i

1, 2, 3; j

1, 2; n

1, f ),

 E 4 n ª a2  (1  a4 )(J 02 s 2  n 2 ) º , ¬ ¼

Chapter 2

52

E3 ª(K02 s 2  n2 )(J 02 s 2  n2 ) 

' 22  n, s

¬

,

 (K02 a2  J 02 a3 ) s 2  a3n2  a2 a3 º , ¼ '11  n, s E 4 ª (J 02 s 2  n 2 )(J 02 s 2  n 2 )  a2  a4 º , ¬ ¼ 2 2 2 '12 ( n, s )  E3n ª (1  a3 )(J 0 s  n )  a2 a3 º , ¬ ¼ 2 2 2 ' 31 ( n, s ) E 4 a2 ª n(J 0 s  n )  1º , ¬ ¼ ' 32 (n, s ) E3a2 ªK02 s 2  a3 (1  n 2 ) º , ¬ ¼ 2 4ª 6 4 2 ' ( s ) K0 J 0 s  Aa s  Bb s  Cc º , ¬ ¼ 2 2 2 2 2 Aa ((2K0  J 0 )n  a3J 0  a4K0 ) / (K02J 02 ), ((K02  2J 02 )n4  ((a3  2)J 02  a4K02 )n2 

Bb

 a2 (K02J 02  a3J 02  a4K02 )  J 02 ) / (K02J 04 ), ( n 6  2n 4  ( a2K02  1)n 2  a3a4K02 ) / (K02J 04 ).

Cc

Then, applying to (2.1.24)–(2.1.27) the inverse Laplace transform (2.2), by the convolution theorem of the originals of the two functions we have [199]: t E4 t W w 0,0 (t ) p0 (W ) cos dW , (2.1.28) 2 K0 0 K0 (a4 )1/2

³

t

w 0,n (t )

³ pn (W )Q11 (n, t  W )dW  0

t

 ³ qn (W )Q12 (n, t  W ) dW , 0

(2.1.29)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

53

t

u0,n (t )

³ pn (W )Q21(n, t  W )dW  0

(2.1.30)

t

 ³ qn (W )Q22 (n, t  W )dW , 0

t

 (t ) ) n

³ pn (W )Q31(n, t  W )dW  0

(2.1.31)

t

 ³ qn (W )Q32 (n, t  W )dW , (n 1, f) 0

where

Qij (n, t )

4 ª¬ (' r Rij  ' i Iij )ch(r0t ) cos(V 0t ) 

 '2r  'i2  +2' ij ( n, s12 )  H( s12 )ch( s1t )  H(  s12 )cos( s1t ) 'c( s12 ) ,

 (' i Rij  ' r I ij )sh(r0t ) sin(V 0t ) º¼ where ɇ(x) is the Heaviside step function, r0 (r 2  V 2 )1/4 cos(M / 2), V 0

(r 2  V 2 )1/4 sin(M / 2),

M arctg(V / r ), r (( A  B) / 2  Aa / 3), V

3( A  B ) / 2, s12

A  B  Aa / 3,

A (qc / 2  Q1/2 )1/3 , B (qc / 2  Q1/2 )1/3 , Q ( pc / 3)3  (qc / 2)2 , pc  Aa2 / 3  Bb , qc R11 I11

2( Aa / 3)3  Aa Bb / 3  Cc , r1

r 2  V 2 , V1

2rV ,

E 4 ªJ 04 r1  J 02 (2n 2  a2  a4 )r  n 2 (n 2  a2  a4 ) º ,

¬ E 4 ªJ 04V 1  J 02 (2n 2  a2  a4 )V º , ¬ ¼

¼

Chapter 2

54

R22

E3 ªK02J 02 r1  ((J 02  K02 )n2  K02 a2  J 02 a3 )r  ¬

 n 2 (n2  a3 )  a2 a3 º , ¼ I 22 E3 ªK02J 02V1  ((J 02  K02 )n 2  K02 a2  J 02 a3 )V º , ¬ ¼ 2 2 R12  E3n ª(1  a3 )(J 0 r  n )  a2 a3 º , ¬ ¼ R21

 E 4 n ª(1  a4 )(J 02 r  n2 )  a2 º , ¬ ¼

I12

E3n(1  a3 )J 02V , 'i K02J 04 > 6V1  4 AaV @ ,

E4n(1  a4 )J 02V , ' r K02J 04 > 6r1  4 Aa r  2Bb @ , 'c( s) K02J 04 ª6s 4  4 Aa s 2  2 Bb º , ¬ ¼ I 21

w0,0 (t )

E 4 ( a4 )1/2 K0

t

³ p0 (W ) sin K 0

t W 1/2 0 ( a4 )

dW ,

(2.1.32)

t

w0,n (t )

³ pn (W )Q11 (n, t  W )dW  0

t

(2.1.33)

 ³ qn (W )Q12 (n, t  W ) dW , 0

t

u0,n (t )

³ pn (W )Q 21(n, t  W )dW  0

t

 ³ qn (W )Q 22 (n, t  W )dW , 0

(2.1.34)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

55

t

) n (t )

³ pn (W )Q31(n, t  W )dW  0

(2.1.35)

t

 ³ qn (W )Q 32 (n, t  W )dW , (n 1, f) 0

where

Q ij ( n, t )

4 ª¬ (G r Rij  G i I ij )sh( r0t ) cos(V 0t ) 

(G i Rij  G r Iij )ch(r0t ) sin(V 0t ) º¼

G r2  Gi2 



+2'ij (n, s12 ) H( s12 )sh( s1t )  H( s12 )sin( s1t )

Gr

r0 ' r  V 0 'i , G i



( s1'c( s12 )) ,

V 0 ' r  r0 'i .

We apply to the system of equations (1.1.5) the Laplace transform (2.1) on the variable t (s is the transformation parameter) and the Fourier method of separation of variables, taking into account the even of x potential M and odd potential \, and using conditions (1.1. 27)–(1.1.28). Then, in the space of the Laplace transform we obtain the following expressions for wave potentials [77]: f I L (s, x, z ) An (s) exp( z (s 2 / D 2  On2 )1/2 ) cos On x, n 0 (2.1.36) f \ L (s, x, z ) Bn ( s) exp( z (s 2 / E 2  On2 )1/2 )sin On x. n 0

¦

¦

where On are the eigenvalues of the problem; in this case

On nS / l, meets the condition of sliding sealing (2.1.3) and On

( n  1 / 2)S / l , ( n 1, f )

(2.1.37) (2.1.38)

meets the condition (2.1.4). In (2.1.36),

An (s)

and

Bn (s)

are determined from the conditions at

the boundary. From expressions (2.1.36) and relations (1.1.6), it follows that the required functions on the surface of the half-space are represented

Chapter 2

56

in the form of series according to the system of eigenfunctions of the problem. f

uz (t, x,0)

ux (t, x,0)

¦ uzn (t)cos On x,

n 0 f

¦ uxn (t)sin On x,

(2.1.39)

n 1 f

¦ V zn (t)cos On x,

V zz (t, x,0)

n 0 f

¦V zxn (t)sin On x.

V zx (t, x,0)

n 1 Assume that the vertical component of velocity is known on the surface of the half-space still:

V (t, x) . The conditions on the boundary are

w uz wt

V (t , x), V zx z 0 0. (2.1.40) z 0 Applying to the last equations of the Laplace transform (2.1) on t and considering (2.1.39), we obtain the conditions for the transformant harmonics of the development of the corresponding functions on the surface of the half-space into trigonometric series: L L (2.1.41) su zn ( s ) V nL ( s ), V zxn ( s ) 0, ( n 0, f ). From (2.1.36), (2.1.41), using (1.1.6), we obtain ( s 2  2E 2On2 )VnL ( s) An ( s )  , 3 2 2 2 s s / D  On 2E 2On2 L Bn ( s ) Vn ( s ). 3

(2.1.42)

s

We consider (2.1.36), (2.1.42) in (1.1.6) and obtain such a connection between the transformants of the harmonics of the functions

(n 0, f) :

V zz

and V

Algorithms for solving mixed non-stationary and dynamic boundary value problems

L V zn (s)

§

DVnL ( s) ¨1 

s 2  D 2On2  s

¨ ©

s

2

 D 2On2

57



§

E2 2¨ 1 4  On ¨ s2 / D 2  O 2 D n © ·§ 2 ·· 1 2 On ¸ ¸  ¨ E 3 ¸ . ¨ s ¸¹ ¸ s 2 / E 2  On2 ¸¹ © s ¹ 1

(2.1.43)

Applying to (2.1.43) the inverse Laplace transform (2.2) and using the convolution theorem, we find the relationship between the harmonics of the vertical component of velocity and normal stresses on the surface of the half-space: t § · V zn (t ) D ¨ Vn (t )  Vn (W ) F (t  W )dW ¸ , (2.1.44)

¨ ©

³

¸ ¹

0

where

Fn (t )

^

DOn J1 (DOnt )  2bEOn E 2 On2t 2 ( J 0 (DOnt ) 

 J 0 ( EOnt )  J1 (DOnt )  J1 ( EOnt ))   EOn t (bJ 0 (DOnt )   J 0 ( EOnt ))  (2  b 2 ) J 0 (DOnt )  J 0 ( EOnt )` , where

J 0 (t ), J1 (t )

are the Bessel functions of the first kind of zero and

first shape, respectively, and the function J 0 ( t ) is defined as follows: t

J 0 (t )

³ J 0 (W ) dW . 0

Further, mixed boundary conditions (1.1.17) satisfice. From (1.1.17), (2.1.44), using ɇ(x), which is a Heaviside step function, we obtain the following expression for the vertical component of the velocity on the surface of the half-space:

Chapter 2

58

f

¦ Vn (t ) cos On x

H( x  x )  vT (t ) 

n 0

 w 0 (t , T ) cos T  u0 (t , T ) sin T  t

f

(2.1.45)

 H( x  x ) ¦ cos On x ³ Vn (W ) Fn (t  W )dW .

n 0 0 For solving the problem, an approach is used that assumes at an early stage of penetration the identification of linear coordinates along the surfaces of the half-space and the body; therefore, the approximate relations will be valid [120]: (2.1.46) x | sin T , ctg T | 1 / T . Substituting (2.1.10) and (2.1.11) into (2.1.45) taking into account (2.1.46) and representing both parts (2.1.45) in the form of series on

cos On x ,

we obtain an infinite system of integral equations (ISIE) of

Volterra of the second kind relatively unknown velocity harmonics on the surface of the half-space (n 0, f) : t f f (1) (2)  Vn (t )  D mn ( x ) Vm (W ) Fm (t  W )dW  ªD mn ( x )w0m (t )  ¬ m 0 m 0 0 t (3)  D mn ( x )u0m (t ) º Vm (W ) Fm (t  W )dW Cn ( x )vT (t ), (2.1.47) ¼ 0 where l 1 (1) cos Om x cos On xdx, D mn ( x ) N n2 x

¦

³

¦

³

³

(2) (x ) D mn

1

x

³

N n2 0

1  x 2 D1m ( x) cos On xdx,

Algorithms for solving mixed non-stationary and dynamic boundary value problems

x

1

(3) D mn (x )

³ xB1m ( x) cos On xdx,

N n2 1

Cn ( x )

59

x

0

2 ³ cos On xdx, N n

N n2 0

l

³ cos

2

On xdx,

0

D1m ( x) cos(mS / 2)Tm ( x)  sin(mS / 2)U m ( x), B1m ( x) sin(mS / 2)Tm ( x)  cos(mS / 2)U m ( x), where

Tm ( x)

and

U m ( x)

are Chebyshev polynomials of the first

and second kind.

w 0m (t ) , u0m (t )

Functions

are determined from the relations

(2.1.18)–(2.1.20), but they include unknown functions

pn (t )

and

qn (t ) . Let us exclude them; for this purpose we use conditions (1.1.12) and (1.1.13), which can be rewritten using (2.1.44) in the form: f

¦

pn (t ) cos nT

n 0

f

D H(T  T ) cos T ¦ cos(On sin T ) u n 0

t

§ · u ¨ Vn (t )  ³ Vn (W ) Fn (t  W ) dW ¸ , ¨ ¸ 0 © ¹ f

¦ qn (t ) sin nT

n 0

(2.1.48)

f

D H(T  T ) sin T ¦ cos(On sin T ) u n 0

t § · u ¨ Vn (t )  ³ Vn (W ) Fn (t  W ) dW ¸ . ¨ ¸ 0 © ¹

(2.1.49)

Using the orthogonality of the functions cosnT and sinnT , we obtain the relations that establish the relationship between the harmonics of development in a series of functions p, q and V: t f § · (1) ¨ pn (t ) J mn (T ) Vm (t )  Vm (W ) Fm (t  W )dW ¸ , (2.1.50) ¨ ¸ m 0 0 © ¹

¦

³

Chapter 2

60

qn (t )

f

¦

§

(2) ¨ J mn (T ) Vm (t ) 

¨ ©

m 0

t

· ¸ , (2.1.51) V ( W ) F ( t W ) d W  m m ³ ¸ 0 ¹

where (1) J mn (T )

D Nn2

D

(2) J mn (T )

N n2

S

T

³ cosT cos nT cos(Om sin T )dT , 0

T

³ N n2 0

sin T sin nT cos(Om sin T )dT ,

2

2 ³ cos nT dT , N n

S

³ sin

2

nT dT .

0 0 Thus, the final form of Volterra ISIE of the second kind will be as follows: t f (1) Vn (t )  D mn ( x ) Vm (W ) Fm (t  W )dW  m 0 0 t f f (1) (2)  D mn (x ) J km (T (W )) u m 0 k 00 W § ·

¦

¦

³

¦³

u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G11 (m, t  W )dW  ¨ ¸ 0 © ¹ f

¦

m 0

(2) D mn (x )

f t

(2) (T (W )) u ¦ ³ J km

k 00

W § · u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G12 (m, t  W )dW  ¨ ¸ 0 © ¹

Algorithms for solving mixed non-stationary and dynamic boundary value problems

f

f t

m 0

k 00

61

(1) (3)  ¦ D mn ( x ) ¦ ³ J km (T (W )) u

W § · u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G21 (m, t  W )dW  ¨ ¸ 0 © ¹ f

f t

m 0

k 00

(2.1.52)

(2) (3) ( x ) ¦ ³ J km (T (W )) u  ¦ D mn

W § · ¨ u Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G22 (m, t  W )dW ¨ ¸ 0 © ¹

Cn ( x )vT (t ), ( n 0, f). To solve the problem, when the rate of penetration of the shell

vT (t )

is a predetermined function, it is sufficient to numerically implement equation (2.1.52). The expression for the reaction force of the elastic half-space (1.1.26) using (2.1.44) is rewritten as: X (t ) P(t ) 2 V zz (t , x, 0)dx 2D vT (t ) x (t )  0 (2.1.53)

t f ½ sin On x ° Vn (W ) Fn (t  W )dW ¾ .  On 0 °¿ n 0 The equation of motion of the shell (1.1.24) with the initial conditions takes the form: t f ­° ½° sin On x dv (t ) 2D ®vT (t ) x (t )  M T Vn (W ) Fn (t  W )dW ¾ . dt On 0 n 0 ¯° ¿° (2.1.54)

^

³

¦

³

¦

³

To solve the problem of impact with the initial velocity

V0 , the system

of equations (2.1.52) must be supplemented by the equation of motion (2.1.54).

Chapter 2

62

The contact area is determined taking into account the rise of the medium from the condition: t

f

t

G1 j vT t  G 2 j ³ vT (W )dW  f ( x )  ¦ cos On x ³ Vn (W ) dW  0

 1  x 2

n 0

0

f

¦ cos(n arcsin x ) u

n 0 f t

(1) (arcsin x (W ))G11 ( n, t  W )  u ¦ ³ ªJ mn ¬ m 00

(2) (arcsin x (W ))G12 (n, t  W ) º u  J mn ¼

W § · ¨ u Vm (W )  ³ Vm ([ ) Fm (W  [ )d [ ¸ dW  ¨ ¸ 0 © ¹ f

 x ¦ sin( n arcsin x ) u n 0 f t

(1) (arcsin x (W ))G 21 ( n, t  W )  u ¦ ³ ªJ mn ¬ m 00

(2) (arcsin x (W ))G 22 (n, t  W ) º u  J mn ¼

W § · u ¨ Vm (W )  ³ Vm ([ ) Fm (W  [ )d [ ¸ dW ¨ ¸ 0 © ¹ ­° 0, if x  x (t ), (2.1.55) ®

H x x t 0, if ( ),  ! °¯ where G ij {0, if i z j ; 1, if i j} is the Kronecker symbol. Index

j = 1 corresponds to the case when the body enters the environment at a rate that varies according to a predetermined law (statement 1); if the velocity of the penetrating body is known only at the initial time t = 0, and

Algorithms for solving mixed non-stationary and dynamic boundary value problems

63

in the following moments is determined from the equation of motion (statement 2), then j = 2. If we exclude the fourth addition in relation (2.1.55), we obtain the condition from which the boundary of the contact area is determined without taking into account the rise of the medium. Assume that normal stresses are known on the surface of the half-space

V zz (t , x, 0) . The conditions at the border are as follows: V zz

z 0

V z ( t , x ), V zx

z 0

0.

(2.1.56)

Applying to the last equations the Laplace transform (2.1) on t and taking into account (2.1.39), we obtain the conditions for the transformant harmonics of the development of the corresponding functions in the series: L L L V zzn ( s ) V zn ( s ) V zxn ( s ) 0.

(2.1.57)

From (2.1.36) and (2.1.57), using (1.1.6), we have: L An ( s ) §¨ 2 E 2On2 s 2 / D 2  On2 ·¸ V zn ( s ) Cn ( s ) , © ¹

Bn ( s )

 s2  2E 2On2 V znL (s) Cn (s) ,

Cn ( s )

4E 4On2 s 2 / D 2  On2 s 2 / E 2  On2 

(2.1.58)

( s 2  2 E 2On2 ) 2 . Given (2.1.36) and (2.1.58) in (1.1.6), we can write:

VnL ( s )

L s3 s 2 / D 2  On2 V zn ( s ) Cn ( s).

(2.1.59)

Applying to (2.1.59) the inverse Laplace transform (2.2) and using the convolution theorem, we obtain the following relationship between the harmonics of normal stresses and the vertical velocity component on the surface of the half-space: t

Vn (t )

V (t )  zn  ³ V zn (W )Gn (t  W ) dW , D 0

where t

Gn (t )

On J1 (DOnt )  ³  J 0 ( EOnW ) f1 (n, t  W )  0

 J 0 (DOnW ) f 2 (n, t  W ) @ dW ,

(2.1.60)

Chapter 2

64

f j ( n, t )

4 ª¬(' r R j  'i I j ) ch r0t cos V 0t 

('i R j  ' r I j )shr0t sin V 0t º¼ (' 2r  'i2 )  2 N j (n, s1 )(H( s12 )chs1t  H( s12 ) cos s1t ) / 'c( s1 ), ( j 1; 2) where ɇ(x) is the Heaviside step function,

r0

(r 2  V 2 )1/4 cos(M / 2), V 0

M

arctg(V / r ), r

V

(r 2  V 2 )1/4 sin(M / 2),

(( A  B ) / 2  a / 3), a 8E 2On2 ,

3( A  B ) / 2, s12

A  B  8E 2On2 / 3,

A 2 E 2On2 ((17  45b 2 ) / 27  ((2 / 9  4b 2 / 3)3   (17  45b 2 )2 )1/2 )1/3 , B

2 E 2On2 ((17  45b 2 ) / 27  ((2 / 9  4b 2 / 3)3 

 (17  45b 2 )2 )1/2 )1/3 ,

R1 4E 2b2On3 ªr 2  V 2  (D 2  E 2 )On2r  D 2E 2On4 º , ¬ ¼ I1 4E 2b2On3 ª2rV  (D 2  E 2 )On2V º , ¬ ¼ 3 On ª 2 R2 (D  4 E 2 )(r 2  V 2  4 E 2On2 (D 2  ¬ D 5E 2  4 E 2b 2 )r  4 E 4On4 (D 2  4 E 2 (1  b 2 )) º , ¼

On3 ª 2 (D  4 E 2 )2rV  4 E 2 On2 (D 2  5E 2  4 E 2b 2 )V º , ¼ D ¬ N1(n, s) 4E 2b2On3s ª s4  (D 2  E 2 )On2s2  D 2 E 2On4 º , ¬ ¼ 3 On s ª 2 (D  4 E 2 ) s 4  4 E 2 On2 (D 2  5E 2  N 2 ( n, s ) ¬ D 2 2 2 4 E b ) s  4 E 4On4 (D 2  4 E 2 (1  b 2 )) º , ¼ I2

Algorithms for solving mixed non-stationary and dynamic boundary value problems

'r

6(r 2  V 2 )  32E 2On2 r  16E 4On4 (3  2b2 ),

'i

12rV  32E 2On2V ,

'c( s )

65

s (6 s 4  32 E 2On2 s 2  16 E 4On4 (3  2b 2 )).

Similarly, given (2.1.36), (2.1.58) and (2.1.41) in (1.1.6), we find the relationship between the transformants of the harmonics of the functions uz and Vzz: L L uzn (s) s 2 s 2 / D 2  On2 V zn (s) Cn ( s).

(2.1.61)

Applying to (2.1.61) the inverse Laplace transform, we can obtain the following relationship between the harmonics of normal displacements and normal stresses on the surface of the half-space: t

u zn (t )

³ V zn (W )G n (t  W ) dW ,

(2.1.62)

0

where

G n (t )

t

On J 0 (DOnt )  ³ > J 0 ( EOnW ) F1 ( n, t  W )  0

 J 0 (DOnW ) F2 (n, t  W ) @ dW ,

F j (n, t )

4 ª¬(G r R j  G i I j )shr0t cos V 0t 

(G i R j  G r I j )chr0t sin V 0t (G r2  G i2 )  2 N j (n, s1 )(H( s12 )shs1t  H( s12 ) sin s1t ) / 'c( s1 ), ( j 1; 2).

Gr

r0'r  V 0'i , Gi

G 0 (t )

0, G n (0)

r0'i  V 0'r ,

0, G 0 (t )

1 / D .

Now mixed boundary conditions (1.1.12) satisfice. From (2.1.60), (1.1.12), using H(x), a Heaviside step function, we obtain the following expressions for normal stresses on the surface of the half-space:

Chapter 2

66

^

V z (t ) D H( x  x ) vT (t )  1  x 2

f

¦ w 0n (t ) D1n ( x) 

n 0 t

½°  x ¦ u0n (t ) B1n ( x)  ¦ cos On x ³ V zn (W )Gn (t  W ) dW ¾ . °¿ n 0 n 0 0 f

f

Representing both parts (2.1.63) in the form of series on

(2.1.63)

cos On x , we

obtain ISIE Volterra of the second kind concerning unknown harmonics of normal stresses on a half-space surface: t f (1) V zn (t )  E mn ( x ) V zm (W )Gm (t  W )dW  m 0 0 f (2)  (3)  ª E mn  ( x ) w0m (t )  E mn ( x )u0m (t ) º (2.1.64) ¬ ¼ m 0 Dn ( x )vT (t ), (n 0, f) where x D (1) ( x )  2 cos Om x cos On xdx, E mn Nn 0

¦

³

¦

³

D

(2) (x ) E mn

³

N n2 0 x

D

(3) (x ) E mn

Dn ( x )

x

N n2 

D N n2

1  x 2 D1m ( x) cos On xdx,

³ xB1m ( x) cos On xdx, 0

x

³ cos On xdx.

0 Equations (1.1.12) and (1.1.13) after substitution using (2.1.7), (2.1.8) and (2.1.39), will be rewritten as:

Algorithms for solving mixed non-stationary and dynamic boundary value problems

f

f

n 0 f

n 0 f

67

¦ pn (t )cos nx  H(T  T )cosT ¦ V zn (t)cos(On x), (2.1.65) ¦ qn (t )sin nx

 H(T  T )sin T ¦ V zn (t )cos(On x). (2.1.66)

n 0 n 0 Using the orthogonality of the functions cosnT and sinnT, we rewrite the last two equalities in the form: f (1) pn (t ) [mn (T )V zm (t ), (2.1.67) m 0 f (2) qn (t ) [mn (T )V zm (t ), (2.1.68) m 0

¦

¦

(1) where [ mn (T )

1

D

(1) (2) (T ), [ mn (T ) J mn

1

D

(2) (T ). J mn

Substitute (2.1.18)–(2.1.20) taking into account (2.1.67) and (2.1.68) in (2.1.64) and obtain ISIE of Volterra of the second kind in the final form: t f (1) V zn (t )  E mn ( x ) V zm (W )Gm (t  W ) dW  m 0 0 f f t (1) (2)  (T (W ))V zk (W )G11 (m, t  W )dW  E mn ( x ) [ km m 0 k 00 f f t (2) (2)  (T (W ))V zk (W )G12 (m, t  W )dW  E mn ( x ) [ km m 0 k 00 f t f (1) (3)  (T (W ))V zk (W )G21 (m, t  W )dW  E mn ( x ) [ km k 00 m 0 f f t (2) (3)  (x ) (T (W ))V zk (W )G22 (m, t  W )dW  E mn [ km m 0 k 00

¦

³

¦

¦³

¦

¦³

¦

¦³

¦

¦³

Chapter 2

68

Dn ( x )vT (t ).

(2.1.69)

The system (2.1.69) is solvable in the case of shell penetration at a given speed vT (t) . The expression for the reaction force (1.1.26) can be rewritten as:

X (t )

P(t ) 2

³

f

V zz (t, x,0)dx  2 ¦

sin On x

On

V zn (t ).

(2.1.70)

n 0 0 The equation of motion of the shell (1.1.24) with the initial conditions takes the form: f sin On x dv (t ) (2.1.71) M T 2 V zn (t ). dt O n n 0

¦

To solve the problem of impact with the initial velocity of the shell V0 with mass M, the system of equations (2.1.69) must be supplemented by the equation of motion (2.1.71). The contact area is determined considering the rise of the medium from the condition: t t f

G1 j vT t  G 2 j vT (W )dW  f ( x )  cos On x V zn (W )G n (t  W )dW  n 0 0 0 t f f (1) ª[ mn  1  x 2 cos(n arcsin x ) (arcsin x (W ))G11 (n, t  W )  ¬ n 0 m 00 (2)  [ mn (arcsin x (W ))G12 (n, t  W ) º V zn (W )dW  ¼

¦

³

¦

x

f

³

¦³

f t

(1) (arcsin x (W ))G 21 ( n, t  W )  ¦ sin(n arcsin x ) ¦ ³ ª¬[mn

n 0

m 00

(2) (arcsin x (W ))G 22 (n, t  W ) º V zm (W )dW  [ mn ¼ ­° 0, if x  x (t ), (2.1.72) ®

°¯H  0, if x ! x (t ),

Algorithms for solving mixed non-stationary and dynamic boundary value problems

69

where G ij is the Kronecker symbol. In the case of statement 1, the index j = 1; in the case of statement 2, j = 2. Excluding in relation (2.1.72) the fourth addition, we obtain the condition from which the boundary of the contact area is determined without considering the rise of the medium. and u0m are determined from relations (2.1.28)–(2.1.30) and (2.1.32)–(2.1.34), then ISIE of Volterra of the second kind (2.1.52) and (2.1.69) and equations (2.1.55) and (2.1.72) will take a similar form, only instead of kernel functions G ij and G ij there If

w 0m , u0m , w0m

will be functions Q ij and Q ij with (2.1.28)(2.1.35). However, we note that with increasing parameter n some kernel functions are asymptotic [199]:

Q11 (n) exp  O(n) , Q11 (n) O 1 n exp  O(n) , (2.1.73) Q22 (n) exp  O(n) , Q 22 (n) O 1 n exp  O(n) . This means that if in a plane problem is to use shells of the S.P. Tymoshenko type, the use of kernel functions Q ij and Q ij will not ensure the convergence of the numerical solution of ISIE of Volterra. After reduction of ISIE and the subsequent discretization, the received system of linear algebraic equations will be badly defined: with an increase of the size of this system, its determinant will increase indefinitely. To solve the problem, we obtain two solvable ISIE of Volterra of the second kind: (2.1.52), (2.1.54) and (2.1.69), (2.1.71). The first system is written relative to the unknown harmonics of the vertical velocity component on the surface of the half-space; the second is relative to the unknown harmonics of normal stresses occurring at the surface points of the elastic medium. In the numerical implementation of the impact problem, one may use either of the two solvable systems. The first system (2.1.52), (2.1.54) is more complex than the second (2.1.69), (2.1.71); however, in the second system the function

Gn (t )

is

more cumbersome and less convenient in numerical solution than the function

Fn (t) .

Therefore, the system (2.1.52), (2.1.54) was chosen for

the numerical implementation, and all further considerations will be made in relation to it. Taking into account (2.1.46), (1.1.8)–(1.1.13) ISIE (2.1.52) and equation (2.1.55) will be rewritten as:

Chapter 2

70

Vn (t ) 

t

f

(1) ( x ) ³ Vm (W ) Fm (t  W ) dW  ¦ D mn

m 0

f

¦

(2) D mn (x )

m 0

(2.1.74)

0

f t

¦³

k 00

uG11 (m, t  W )dW

§

(1) J km (T (W )) ¨ Vk (W )  ¨

©

W

· ¸u  V ( [ ) F ( W [ ) d [ ³ k k ¸ 0 ¹

Cn ( x )vT (t ).

t

G1 j vT t  G 2 j ³ vT (W )dW  f ( x )  0

f

t

 ¦ cos On x ³ Vn (W ) dW  n 0

 1 x

0

2

f

f t

(1) (arcsin x (W )) u ¦ cos(n arcsin x ) ¦ ³ ª¬J mn

n 0

m 00

(2.1.75)

(2) uG11 (n, t  W )  J mn (arcsin x (W ))G12 (n, t  W ) º u ¼

W § · u ¨ Vm (W )  ³ Vm ([ ) Fm (W  [ )d [ ¸ dW ¨ ¸ 0 © ¹

­° 0, if x  x (t ) ®

°¯H  0, if x ! x (t ) §2.2. Impact of fine elastic spherical shells on an elastic half-space Since the impact process is short-lived, the perturbation region at each moment of time t is finite. Limiting the finite time interval of the interaction  0 d t d T , we can distinguish the region of the half-space,

Algorithms for solving mixed non-stationary and dynamic boundary value problems

71

which at time T covers the entire perturbation zone. From this point of view, for moments of time 0 d t d T , the elastic half-space can be replaced by an elastic half-cylinder  r d l ; z t 0 , to the boundaries of which perturbations do not reach at time T.

l D T  r (T ). (2.2.1) Thus, for the time being, where 0 d t d T , the considered problem is reduced to a non-stationary problem for a half-cylinder under mixed boundary conditions at its end. To represent the displacement vector in the form (2.1.2) on the side surface of the half-cylinder, choose, for example, the conditions of sliding sealing: (2.2.2) ur 0, V zr 0, r l

r l

or

uz

r l

0, V rr

r l

0.

(2.2.3)

Consider the initial boundary value problem (1.2.2), (1.2.4), (1.2.13) and (1.2.16)–(1.2.20). We present the normal w0 (t,T ) and tangential

u0 (t,T )

displacements of the points of the middle surface of the shell

and the radial p(t,T ) and tangential q(t ,T ) components of the distributed external load acting on the shell in the form of trigonometric Fourier series: f w0 (t,T ) w0n (t )Pn (cosT ). (2.2.4) n 0 f u0 (t,T ) u0n (t )Pn (sin T ). (2.2.5) n 1 f p(t ,T ) pn (t ) Pn (cosT ). (2.2.6) n 0 f q(t ,T ) qn (t ) Pn (sin T ). (2.2.7) n 1 If in the initial boundary problem (1.2.2), (1.2.4), (1.2.13) and (1.2.16)–(1.2.20) instead of equations (1.2.2) we consider equations

¦

¦

¦

¦

Chapter 2

72

(1.2.3), then in the form of series of derivatives of Legendre polynomials it is necessary to present another angle of rotation of the normal

)(t ,T )

)(t,T ) :

f

¦ )n (t ) Pn1(cosT ).

(2.2.8) n 1 In space, the Laplace transformant with the parameter s transformants of functions

), w0 , u0 , p , q

will, by virtue of (2.2.4)–(2.2.8), have

the form:

w0L (s,T ) u0L (s,T ) pL (s,T ) qL (s,T ) )L (s,T )

f

¦ w0Ln (s)Pn (cosT ).

(2.2.9)

n 0 f

¦ u0Ln (s)Pn1(cosT ).

(2.2.10)

¦ pnL (s)Pn (cosT ).

(2.2.11)

¦ qnL (s)Pn1(cosT ).

(2.2.12)

n 1 f

n 0 f

n 1 f

¦ )nL (s)Pn1(cosT ).

(2.2.13) n 1 Apply to the system of equations (1.2.2) the Laplace transform (2.1) on the variable t with the parameter s and substitute equations (2.2.9)– (2.2.12). Equating the coefficients at the same Pn (cosT ) and

Pn1 (cosT ) , obtain the ratios connecting the components of the L

L

decomposition of the functions w0 , u0 , p E 2 p0L ( s ) L w0,0 (s) , E1s 2  2(1  Q 0 )

L

L

and q . (2.2.14)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

L w0, n (s)

E 6 (( E5 s 2  J 1 ) pnL ( s )  J 3qnL ( s )) , E52 s 4  (J 1  J 4 ) E5 s 2  J 1J 4  J 2J 3

E6 (J 2 pnL ( s)  ( E5 s 2  J 4 )qnL ( s))

L u0, n ( s)

E52 s 4  (J1  J 4 ) E5 s 2  J1J 4  J 2J 3

73

(2.2.15)

,(n 1, f) (2.2.16)

where

J1 (1  a1 )(n2  n Q 0  1), J 2 a1(n2  n)  1  (1  a1 )Q 0  a1, J 3 a1n(n  1)(n(n  1)  1 Q 0  a1 (1 Q 0 )), J 4 2(1 Q 0 )  a1n(n  1)(n(n  1)  1 Q 0 ). Then, applying to (2.2.14)–(2.2.16) the inverse Laplace transform (2.2), by the convolution theorem of the originals of the two functions we obtain: t § 2(1 Q 0 ) · E6 w 0,0 (t ) p0 (W ) cos ¨ (t  W ) ¸dW , (2.2.17) ¨ ¸ E5 0 E1 © ¹

³

t

w 0,n (t )

t

³ 0

pn (W )G11 ( n, t  W ) dW  ³ qn (W )G12 ( n, t  W ) dW , (2.2.18) 0

t

u0,n (t )

t

³ pn (W )G21(n, t  W )dW  ³ qn (W )G22 (n, t  W )dW , (2.2.19) 0

0

(n 1, f) where

2 Nij (n, is1 )

Gij (n, t )

'c(is1 )

cos( s1t ) 

2 Nij (n, is2 ) 'c(is2 )

cos( s2t ),

E 6 n( E5 s 2  J 1 ) s, N12 (n, s) E6J 3s, N 21 (n, s) E6J 2 s, N 22 ( n, s ) E 6 ( E 5 s 2  J 4 ) s ,

N11 (n, s )

J1  J 4 ,B 2 E5

s1

| A  B |, A

s2

| A  B |, 'c( s )

(J 1  J 4 ) 2  4J 2J 3 2 E5

,

2 E5 s (2 E5 s 2  J 1  J 4 ), ( n 1, f).

Chapter 2

74

t

§ 2(1 Q 0 ) ·  ( W )sin ( W ) p t dW , ¨ ¸ 0 ¨ ¸ (2.2.20 E 2E5 (1 Q 0 ) ³0 5 © ¹

E6

w0,0 (t )

) t

t

0 t

0 t

0

0

³ pn (W )G11 ( n, t  W ) dW  ³ qn (W )G12 (n, t  W ) dW , (2.2.21)

w0,n (t )

u0,n (t )

³ pn (W )G 21(n, t  W )dW  ³ qn (W )G 22 (n, t  W )dW , 2.2.22)

( n 1, f) where

Gij (n, t )

2 Nij (n, is1) s1'c(is1)

sin(s1t ) 

2 Nij (n, is2 ) s2'c(is2 )

sin(s2t ).

We now apply the Laplace transform (2.1) over the variable t (s is the transformation parameter) to the system of differential equations (1.2.4) and substitute there (2.2.9)–(2.2.13). Equating the coefficients at the same

Pn (cosT )

and

Pn1 (cosT ) obtain the ratios that connect the L

L

L

components of the decomposition of functions ) , w0 , u0 , p

L

and

q L into series by polynomials and connected Legendre polynomials [198, 199]: L w0,0 (s)

where

p0L ( s )

J 02 s 2  2 / (1 Q 0 )

,

(2.2.23)

L w0, n (s)

L L Q11 ( n, s ) pnL ( s )  Q12 ( n, s ) qnL ( s ),

(2.2.24)

L u0, n (s)

L L Q21 ( n, s ) pnL ( s )  Q22 ( n , s ) qnL ( s ),

(2.2.25)

L L ) nL ( s ) Q31 (n, s ) pnL ( s )  Q32 (n, s ) qnL ( s ),

(2.2.26)

QijL (n, s)

'ij (s) '( s )

, ( i 1,2,3 ; j=1,2; n 1, f ),

Algorithms for solving mixed non-stationary and dynamic boundary value problems

'11  n, s

75

§ n(n  1) · 1 2 2   s J ¨ ¸u 0 ¨ 1 Q 2 1  Q 0 ¸ 0 © ¹





u n(n  1)  1 Q 0  RR  K02 s 2 ,

'12 (n, s)

n(n 1) n(n 1) 1Q 0  RR K02s2 , 1Q 0





§ § 2(1 Q 0 )ks · ' 21  n, s = ¨ RR  ¨  1¸  n(n  1)  1 Q 0  RR  ¨ © 1 Q 0 ¹ © K02 s 2

 2(1 Q 0 )Dks ,

' 22  n, s

· n(n  1) RR § n(n  1) 2 ¨   J 02 s 2 ¸ u 2(1 Q 0 )ks © 2(1 Q 0 )ks 1 Q 0 ¹







u n(n  1)  1 Q 0  RR  K02 s 2 , ' 31 ( n, s )

· RR § 1  J 02 s 2 ¸ , ¨  n( n  1)  h © 1 Q 0 ¹

'32 (n, s) 

n(n  1) RR (1 Q 0 )h



, RR

R 2 E0 h , 2(1  Q 0 ) Dk s



'(s) K02J 04 s6  Aa s 4  Bb s 2  Cc , 1 § n( n  1) n( n  1) 1 2 ·    ¨ ¸ J 02 ¨© 1 Q 02 2(1  Q 0 ) k s 1  Q 0 1 Q 0 ¸¹ R 2 E0 h · 1 §  2 ¨ n( n  1)  1  Q 0  ¸, 2(1  Q 0 ) Dk s ¸¹ K0 ¨© Aa

Chapter 2

76

Bb

1 § 2 § n( n  1) 1 · § n( n  1) 2 · ¨K0 ¨   ¸ ¨ ¸ K02J 04 ¨© ¨© 1 Q 02 1  Q 0 ¸¹ © 2(1 Q 0 )ks 1 Q 0 ¹

§ n(n  1) n( n  1) 1 2 · J 02 ¨    ¸u ¨ 1 Q 2 ¸    2(1 ) k 1 1 Q Q Q s 0 0 0 0 © ¹ 2 § R E0 h n( n  1) 2 · u ¨ n( n  1)  1  Q 0   u ¸ ¨ ¸ 2(1  Q 0 ) Dk s k 2(1 ) 1 Q Q   s 0 0 © ¹ § K02 J 02 R 2 E0 h · · 2(1 ) 1 u¨     k Q Q ¸¸,  0 s 0 ¨ (1 Q ) 2 ¸¸  Dk 2(1 ) Q s 0 0 © ¹¹ 1 § § n(n  1) 1 · § § n(n  1) 2 · ¨¨ Cc   ¸ ¨ ¨ ¸u K02J 04 ¨© ¨© 1 Q 02 1 Q 0 ¸¹ ¨© © 2(1 Q 0 )ks 1 Q 0 ¹

§ R 2 E0 h · n(n  1) R 2 E0 h · u ¨ n(n  1)  1 Q 0  ¸ ¸ ¨ ¸ 4(1 Q )2 Dk 2 ¸ Q 2(1 ) Dk  0 s 0 s ¹ © ¹ 2 § 2(1  Q 0 ) k s · n( n  1) § R E0 h  ¨  1¸ u ¨ 2 ¨ 2(1 Q 0 ) k s © 2(1  Q 0 ) Dk s © 1 Q 0 ¹ § R 2 E0 h u ¨ n( n  1)  1  Q 0  ¨ 2(1  Q 0 ) Dk s ©

··· ¸ ¸ ¸. ¸¸ ¹ ¹ ¹¸

Then, applying to (2.2.23)–(2.2.26) the inverse Laplace transform (2.2), by the convolution theorem of the originals of the two functions we have [198, 199]: t § · 1 t W w 0,0 (t ) p ( W ) cos (2.2.27) ¨ ¸dW , 0 ¨ J (1 Q ) / 2 ¸ J 02 0 0 © 0 ¹

³

t

w 0,n (t )

t

³ pn (W )Q11 ( n, t  W ) dW  ³ qn (W )Q12 ( n, t  W ) dW , (2.2.28) 0

0

Algorithms for solving mixed non-stationary and dynamic boundary value problems

t

u0,n (t )

³ 0

t

pn (W )Q21 ( n, t  W ) dW  ³ qn (W )Q22 ( n, t  W ) dW , (2.2.29) 0

t

 (t ) ) n

77

t

³ pn (W )Q31(n, t  W )dW  ³ qn (W )Q32 (n, t  W )dW ,(n 0

1, f)

0

(2.2.30) where

Qij (n, t )

4 ª¬( ' r Rij  ' i I ij )ch( r0t ) cos(V 0t ) 

 '2r  'i2 + +2'ij (n, s12 )  H( s12 )ch( s1t )  H( s12 )cos( s1t ) 'c( s12 ) ,

 ('i Rij  ' r I ij )sh(r0t ) sin(V 0t ) º¼

where ɇ(x) is the Heaviside step function;

r0

( r 2  V 2 )1/4 cos(M / 2), V 0

M

arctg(V / r ), r

(( A  B ) / 2  Aa / 3),

3( A  B ) / 2, s12 1/2 1/3

V

B (qc / 2  Q

(r 2  V 2 )1/4 sin(M / 2),

A  B  Aa / 3, A (  qc / 2  Q1/2 )1/3 ,

) , Q ( pc / 3)3  (qc / 2)2 ,

pc  Aa2 / 3  Bb , qc 2( Aa / 3)3  Aa Bb / 3  Cc , r1

r 2  V 2 , V1

2rV ,

§ § n(n  1) 1 · R11 K02J 02 r1  ¨K02 ¨  ¸ ¨ ¨ 1  Q 2 1 Q 0 ¸ 0 ¹ © ©



 J 02 (n(n  1)  1 Q 0  RR ) r  § n(n  1) 1 · ¨  ¸ (n(n  1)  1 Q 0  RR ), ¨ 1 Q 2 1 Q 0 ¸ 0 © ¹

Chapter 2

78

§ § n(n  1) 1 · I11 K02J 02V1  ¨K02 ¨  ¸ ¨ ¨ 1 Q 2 1  Q 0 ¸ 0 ¹ © ©



 J 02 (n(n  1)  1 Q 0  RR ) V ,

R12 R21

n(n 1) n(n 1) (n(n 1) 1Q 0  RR K02r), I12 K02 V, 1Q 0 1Q 0 § § 2(1 Q 0 )ks ·  1¸ u ¨¨  RR  ¨ 2(1 Q 0 ) Dk s © © 1 Q 0 ¹ 1



u(n(n  1)  1 Q 0  RR  K02 r ) , § 1 · 1  I 21 K 02 ¨ ¸V , © 1  Q 0 2(1  Q 0 ) k s ¹ § § n(n  1) 2 · R22 K02J 02 r1  ¨K02 ¨  ¸ ¨ © © 2(1 Q 0 )ks 1 Q 0 ¹

§ n(n  1) 2 · J 02 (n(n  1)  1 Q 0  RR ) r  ¨  ¸u © 2(1 Q 0 )ks 1 Q 0 ¹ n(n  1) RR u( n(n  1)  1  Q 0  RR )  , 2(1  Q 0 ) k s



§ § n(n  1) 2 ·  I 22 K02J 02V1  ¨K02 ¨ ¸ ¨ © © 2(1  Q 0 )ks 1 Q 0 ¹



 J 02 (n(n  1)  1 Q 0  RR ) V , · RR § 1 R  J 02 r ¸ , I31 J 02 R V , ¨  n( n  1)  h 1 Q 0 h © ¹ R R32  R n(n  1), I32 0, ' r K02J 04 ª¬6r1  4 Aa r  2 Bb º¼ , h

R31

Algorithms for solving mixed non-stationary and dynamic boundary value problems

'i

79

K02J 04 ª¬6V1  4 AaV º¼ , 'c( s) K02J 04 ª6s 4  4 Aa s 2  2 Bb º . ¬ ¼ t § · 1 t W w0,0 (t ) p ( W )sin (2.2.31) ¨ ¸dW , ¨ J (1 Q ) / 2 ¸ J 0 ³0 0 0 © 0 ¹ t

w0,n (t )

t

³

0 t

u0,n (t )

) n (t )

pn (W )Q11 ( n, t  W ) dW  ³ qn (W )Q12 ( n, t  W ) dW , (2.2.32) 0 t

³ pn (W )Q 21 (n, t  W ) dW  ³ qn (W )Q 22 (n, t  W ) dW , (2.2.33)

0 t

0 t

0

0

³ pn (W )Q31 (n, t  W )dW  ³ qn (W )Q32 (n, t  W )dW , (2.2.34)

( n 1, f ) where



Qij (n, t ) 4 (G r Rij  G i Iij )sh(r0t ) cos(V 0t )  (G i Rij  G r Iij )ch(r0t )sin(V 0t ) Q ij (n, t )

G r2  Gi2 



4 (G r Rij  G i I ij )sh(r0t ) cos(V 0t ) 

(G i Rij  G r Iij )ch(r0t )sin(V 0t )

G r2  Gi2 



+2'ij (n, s12 ) H( s12 )sh( s1t )  H( s12 )sin( s1t )

Gr

r0 ' r  V 0 'i , G i



( s1'c( s12 )),

V 0 ' r  r0 'i ,

We apply to the wave equations (1.2.4) the Laplace transform (2.1) for the variable t (s is the transformation parameter) and the Fourier method of separation of variables. We will demand satisfaction of the condition (1.2.21), (1.2.22). Then, in the space of the Laplace transforms we obtain the following expressions for wave potentials [153]:

Chapter 2

80 f

I L ( s, r , z )

¦ An ( s ) exp(  z ( s 2 / D 2  On2 )1/2 ) J 0 (On r ),

n 0 f

\ L ( s, r , z )

(2.2.35)

¦ Bn ( s ) exp(  z ( s 2 / E 2  On2 )1/2 ) J1 (On r ).

n 0

where

On

are the eigenvalues of the problem, which are determined

from conditions (2.2.2), taking into account (1.2.5), and are the roots of the equation:

J1 (On l ) In (2.2.35),

0, ( n

An (s)

0, f ).

and

Bn (s)

(2.2.36) are determined from the boundary

conditions. From expressions (2.2.35) and relations (1.2.5), it follows that the required functions on the surface of the half-space are represented in the form of series according to the system of eigenfunctions of the problem. f

u z (t , r , 0) ur (t , r , 0)

V zz (t , r , 0) V zr (t , x, 0)

¦ uzn (t ) J 0 (On r ),

n 0 f

¦ urn (t ) J1(On r ),

n 1 f

(2.2.37)

¦ V zn (t ) J 0 (On r ),

n 0 f

¦ V zrn (t ) J1(On r ).

n 1 Assume that the vertical component of velocity is known on the

surface of the half-space follows:

w uz wt

V (t, r) .

The boundary conditions are as

V (t , r ), V zr z 0 0. (2.2.38) z 0 Applying the Laplace transform (2.1) by t to the last equations and taking into account (2.2.37), we obtain the conditions for the transformant

Algorithms for solving mixed non-stationary and dynamic boundary value problems

81

harmonics of the decompositions of the corresponding functions on the surface of the half-space into trigonometric series: L L (2.2.39) su zn ( s ) V nL ( s ), V zrn ( s ) 0, ( n 0, f ). We obtain from (2.2.35), (2.2.39), using (1.2.5): ( s 2  2E 2On2 )VnL ( s) 2E 2On2 L An ( s)  Vn ( s). (2.2.40) , Bn ( s) 3 3 2 2 2 s s s / D  On Given (2.2.35), (2.2.40) in (1.2.5), we can obtain the following relationship between the transformants of the harmonics of the vertical component of velocity Vn and normal stresses

V zn (n 0, f) :

§

s 2  D 2On2  s

¨ ©

s 2  D 2On2

L V zn ( s ) DVnL ( s ) ¨1 

§ 1 1 u¨  2 ¨ 2 2 s 2 / E 2  On2 © s / D  On

4

E2 2 O u D n

·§ 2 ·· ¸ ¨ 1  E 2 On ¸ ¸ . ¸¨ s s3 ¸¹ ¸ ¹© ¹

(2.2.41)

Applying the inverse Laplace transform (2.2) to (2.2.41) and using the convolution theorem, we find the relationship between the harmonics of the vertical component of velocity and normal stresses on the surface of the half-space: t § · V zn (t ) D ¨ Vn (t )  Vn (W ) Fn (t  W )dW ¸ . (2.2.42)

³ ¨ ¸ 0 © ¹ where the function Fn (t ) is the same as the function in (2.1.44).

Next, we satisfy the mixed boundary conditions (1.2.16). From (1.2.16), (2.2.42), using H(x), a Heaviside step function, we obtain the following expression for the vertical component of velocity on the surface of the half-space: f Vn (t ) J 0 (On r ) H(r  r ) vT (t )  w 0 (t ,T ) cos T  (2.2.43) n 0 t f u0 (t , T )sin T  H(r  r ) J 0 (On r ) Vn (W ) Fn (t  W )dW . n 0 0



¦



¦

³

Chapter 2

82

In the following solving of the problem, we assume that at an early stage of the penetration the approximate relations will be valid [120]: r | sinT , ctgT | 1/ T. (2.2.44) Substituting (2.2.9) and (2.2.10) into (2.2.43), taking into account (2.2.44) and representing both parts (2.2.43) in the form of series on

J 0 (On r ) ,

we obtain an infinite system of integral equations (ISIE) of

Volterra of the second kind relatively unknown velocity harmonics on the surface of the half-space (n

Vn (t ) 

0, f) :

f

t

f

m 0

0

m 0

(4) (5) (r )³ Vm (W ) Fm (t  W )dW  ¦ ªD mn (r ) w 0m (t )  ¦ Dmn

¬

t

(6) (r )u0m (t ) º ³ Vm (W ) Fm (t  W )dW Cn (r )vT (t ),  D mn ¼ 0

(2.2.45) where (4) D mn (r )

(5) D mn (r )

(6) (r ) D mn

Cn (r ) Functions

l

1 N n2

³ rJ 0 (Om r ) J 0 (On r )dr ,

r

r

1

1  r 2 Pm §¨ 1  r 2 ·¸ J 0 (On r )dr , © ¹

³r

Nn2 0

r

1

³r

N n2 0 1

2

1 r2

r

w Pm §¨ 1  r 2 ·¸ J 0 (On r )dr , wr © ¹

2 ³ rJ 0 (On r )dr, N n

N n2 0

w 0m (t ) , u0m (t )

l

³ r  J 0 (Onr )

2

dr.

0

are determined from the relations

and qn (t) . To find them, we use conditions (1.2.11), (1.2.12), which can be rewritten using (2.2.42) in the form: (2.2.17)–(2.2.19), but they include unknown functions

pn ( t )

Algorithms for solving mixed non-stationary and dynamic boundary value problems

f

83

f

D H(T  T ) cos T ¦ J 0 (On sin T ) u

¦ pn (t )Pn (cos T )

n 0

n 0

(2.2.46)

t § · u ¨ Vn (t )  ³ Vn (W ) Fn (t  W ) dW ¸ , ¨ ¸ 0 © ¹ f

f

¦ qn (t )Pn1 (sin T ) D H(T  T ) sin T ¦ J 0 (On sin T ) u

n 0

n 0

(2.2.47)

t

§ · u ¨ Vn (t )  ³ Vn (W ) Fn (t  W ) dW ¸ . ¨ ¸ 0 © ¹

We use the orthogonality of Legendre polynomials and attached Legendre polynomials and obtain the following relations that establish a relationship between the components of the decompositions of functions p, q ɿ V in a series: t f § · (3) ¨ pn (t ) J mn (T ) Vm (t )  Vm (W ) Fm (t  W )dW ¸ , (2.2.48) ¨ ¸ m 0 0 © ¹

¦

³

f

§

m 0

¨ ©

t

·

0

¸ ¹

(4) ¨ (T ) Vm (t )  ³ Vm (W ) Fm (t  W )dW ¸ , ¦ J mn

qn (t )

(2.2.49)

where (3) J mn (T )

(4) (T ) J mn

N n2

D N n2

D

T

³ cosT sin T Pn (cosT ) J 0 (Om sin T )dT ,

0 T

³ N n2 0

sin 2 T Pn1 (cos T ) J 0  Om sin T dT ,

S

³ sin T  Pn (cos T ) 0

2

dT , N n2

S



1 ³ sin T Pn (cos T ) 0



2

dT .

Thus, the final solvable form of ISIE of Volterra of the second kind will be as follows:

Chapter 2

84

Vn (t ) 

f

t

m 0

0

(4) (r ) ³ Vm (W ) Fm (t  W )dW  ¦ D mn

f

f t

m 0

k 00

(3) (5)  ¦ D mn (r ) ¦ ³ J km (T (W )) u

W § · u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G11 (m, t  W )dW  ¨ ¸ 0 © ¹ f

f t

m 0

k 00

(4) (5) (r ) ¦ ³ J km (T (W )) u  ¦ D mn

W § · ¨ u Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G12 (m, t  W )dW  ¨ ¸ 0 © ¹ f

¦

(6) D mn (r )

m 0

f t

(3) (T (W )) u ¦ ³ J km

k 00

W § · u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G21 (m, t  W )dW  ¨ ¸ 0 © ¹

(2.2.50)

f t

f

(4) (6) (r ) ¦ ³ J km (T (W )) u  ¦ D mn m 0

k 00

W

§ · u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d [ ¸ G22 (m, t  W )dW ¨ ¸ 0 © ¹ Cn (r )vT (t ), (n

0, f).

To solve the problem when the penetration rate of the shell is a predetermined function, it is enough to numerically implement equation (2.2.50). The expression for the reaction force of the elastic half-space (1.2.20) using (2.2.42) is rewritten as:

Algorithms for solving mixed non-stationary and dynamic boundary value problems

r (t )

P (t )

2S

³

85

rV zz (t , r , 0) dr DS r * (t ) u

0



­ f J1 On r °

u ®vT (t )r (t )  2 ¦ On ° n 0 ¯



½ ° ³ Vn (W ) Fn (t  W )dW ¾ . ° 0 ¿ t

(2.2.51)

The equation of motion of the shell (1.2.23) with the initial conditions will take the form:

M

dvT (t ) dt

DS r (t ) u



­ f J n On r °

u ®vT (t )r (t )  2 ¦ On ° n 0 ¯

t V (W ) F (t  W )dW ½°¾. ³

n

n

(2.2.52)

° ¿

0

To solve the problem of the impact with the initial velocity

V0

(statement 2), the system of equations (2.2.50) must be supplemented by the equation of motion (2.2.52). The contact area is determined considering the rise of the medium from the condition: t t f G1 j vT t  G 2 j vT (W )dW  f (r )  J 0 On r Vn (W )dW  n 0 0 0 f f t (3) ªJ mn  1  r 2 Pn §¨ 1  r 2 ·¸ (arcsin r (W )) u ¬ © ¹m 0 n 0 0 (4) uG11 (n, t  W )   J mn (arcsin r (W ))G12 (n, t  W ) º u ¼

¦

³

¦



³

¦³

W § · u ¨ Vm (W )  ³ Vm ([ ) Fm (W  [ )d[ ¸ dW  ¨ ¸ 0 © ¹

r

1 r

2

f

¦

n 0

Pn1 §¨ ©

1 r

2

f t

(3)

· ¸ ¦ ³ ª¬J mn (arcsin r (W )) u ¹m 0 0

Chapter 2

86

(4) G 21 (n, t  W )  J mn (arcsin r (W ))G12 (n, t  W ) º u ¼

W § · u ¨ Vm (W )  ³ Vm ([ ) Fm (W  [ )d[ ¸ dW  ¨ ¸ 0 © ¹

r

1 r

2

f

¦

n 0

Pn1 §¨ ©

1 r

2

f t

(3)

· ¸ ¦ ³ ª¬J mn (arcsin r (W )) u ¹m 0 0

(4) uG 21 (n, t  W )  J mn (arcsin r (W ))G 22 (n, t  W ) º u ¼ W § · ¨ u Vm (W )  ³ Vm ([ ) Fm (W  [ )d[ ¸ dW ¨ ¸ 0 © ¹

­° 0, if r  r (t ) . (2.2.53) ®

°¯H  0, if r ! r (t ) where G ij {0, if i z j ; 1, if i j} is the Kronecker symbol. Index

j = 1 in the case of statement 1; j = 2 in the case of statement 2. If we exclude the fourth addition in relation (2.2.53), we obtain the condition from which the boundary of the contact area is determined without taking into account the rise of the medium. Assume that normal stresses V zz (t, r,0) are known on the surface of the half-space. The boundary conditions are as follows: L L L V zzn ( s ) V zn ( s ), V zrn ( s ) 0.

(2.2.54)

Applying the Laplace transform (2.1) by t to the last equalities and taking into account (2.2.37), we obtain the conditions for transformant harmonics of the decompositions of the corresponding functions into series: L L L V zzn ( s ) V zn ( s ), V zrn ( s ) 0.

(2.2.55)

From (2.2.35), (2.2.55), using (1.2.5), determine the coefficients An (s) and Bn (s) :

An ( s )

L  ( s 2  2 E 2On2 )V zn ( s ) Cn ( s ) ,

Algorithms for solving mixed non-stationary and dynamic boundary value problems

L Bn ( s) 2E 2On2 s 2 / D 2  On2 V zn ( s) Cn ( s).

87

(2.2.56)

Given (2.2.35), (2.2.56) in (1.2.5), we can obtain an expression that establishes the relationship between the transformants of the components of normal stresses and the vertical velocity components on the surface of the half-space:

VnL ( s )

L s3 s 2 / D 2  On2 V zn ( s ) Cn ( s).

(2.2.57)

Applying the inverse Laplace transform (2.2) to (2.2.57) and using the convolution theorem, we obtain the following relationship between the harmonics of normal stresses and the vertical velocity component on the surface of the half-space: t

Vn (t )

V (t )  zn  ³ V zn (W )Gn (t  W ) dW , D 0

where kernel function

Gn (t )

(2.2.58)

is determined in (2.1.60).

Similarly, taking into account (2.2.35), (2.2.39), (2.2.56) and (1.2.5), applying the inverse Laplace transform (2.2) and the convolution theorem, we can obtain the relationship between the components of normal displacements and normal stresses on the half-space surface. t

u zn (t )

³ V zn (W )G n (t  W )dW .

(2.2.59)

0

where kernel function G n (t ) is determined in (2.1.62). Let us now satisfy the mixed boundary conditions (1.2.16). From (2.2.58), (1.2.16) we obtain the following representation for normal stresses in the contact zone and on the free surface of the half-space: f

¦ V zn (t ) J 0  On r

V z (t , r ) D H (r *  r ) u

n 0

§ u ¨ vT (t )  1  r 2 w 0 §¨ t , 1  r 2 ·¸  © ¹ © t f ·  ru0 ¨§ t , 1  r 2 ¸·  ¦ J 0  On r ³ V zn (W )Gn (t  W )dW ¸ , ¸ © ¹ n 0 0 ¹

where ɇ(x) is the Heaviside step function.

(2.2.60)

Chapter 2

88

Representing both parts (2.2.60) in the form of series on J 0  On r , we obtain ISIE of Volterra of the second kind concerning unknown harmonics of normal stresses on the surface of the half-space: t f (4) V zn (t )  E mn (r ) V zm (W )Gm (t  W )dW  m 0 0 f (5)  (6)  ª E mn  (r ) w0m (t )  E mn (r )u0m (t ) º (2.2.61) ¬ ¼ m 0 Dn (r )vT (t ), (n 0, f), where

¦

³

¦

(4) E mn (r ) (5) E mn (r )

D

x

 Om r J 0  On r dr , Dn (r )

rJ 0 N n2 0 (5) DD mn (r ), 

³

D Cn (r )

(6) (6) E mn (r ) DD mn (r ).

Using conditions (2.2.6), (2.2.7) and (2.2.37) we will rewrite (1.2.11) and (1.2.12) in the form: f f pn (t )Pn (cosT )  H(T  T )cosT V zn (t ) J0 Onr , (2.2.62 n 0 n 0 ) f f qn (t )Pn1(cosT )  H(T  T )sin T V zn (t ) J0 Onr . (2.2.63 n 0 n 0 )

¦

¦





¦

¦





Using the orthogonality of the functions we rewrite the last two equations in the form: f (3) pn (t ) [mn (T )V zm (t ), m 0 f (4) qn (t ) [mn (T )V zm (t ), m 0

Pn (cosT )

and Pn1 (cos T ) ,

¦

(2.2.64)

¦

(2.2.65)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

(3) where [ mn (T )

1

D

(3) (4) (T ), [ mn (T ) J mn

1

D

89

(4) (T ). J mn

We substitute (2.2.17)–(2.2.19) considering (2.2.64) and (2.2.65) in (2.2.61) and obtain ISIE of Volterra of the second kind in its final form: t f (4) V zn (t )  E mn (r ) V zm (W )Gm (t  W )dW  m 0 0 f f t (3) (5)  (r ) (T (W ))V zk (W )G11 ( m, t  W )dW  E mn [ km m 0 k 00 f f t (4) (5)  (T (W ))V zk (W )G12 ( m, t  W )dW  E mn (r ) [ km (2.2.66) m 0 k 00 f t f (3) (6)  (T (W ))V zk (W )G21 ( m, t  W )dW  E mn (r ) [ km k 00 m 0 f f t (4) (6)  E mn (r ) [ km (T (W ))V zk (W )G22 ( m, t  W )dW  m 0 k 00 Dn (r )vT (t ), (n 0, f).

¦

³

¦

¦³

¦

¦³

¦

¦³

¦

¦³

The system (2.2.66) gives the solution of the problem in the case of penetration of the shell at a given speed vT (t ) (case of statement 1). The expression for the reaction force (1.2.25) can be rewritten as: r (t )

P(t )

2S

³

rV zz (t , r , 0)dr

0

f

2S ¦



r (t ) J1 On r (t )

On

V

(2.2.67) zn (t ).

n 0 The equation of motion of the shell (1.2.18) with the initial conditions will take the form:

Chapter 2

90

dv (t ) M T dt

*



f

J1 On r (t )

n 0

On

2S r (t ) ¦

V

zn (t ).

(2.2.68)

To solve the problem of impact with the initial velocity V0 of the shell of mass M (case of statement 2), the system of equations (2.2.66) must be supplemented by the equation of motion (2.2.68). The contact area is determined considering the rise of the medium from the condition: t G1 j vT t  G 2 j vT (W )dW  f (r )  0 t f  J 0 On r V zn (W )G n (t  W )dW  n 0 0 f f t

2

2 · § ª[ (3) (arcsin r (W )) u  1 r Pn ¨ 1  r ¸ ¬ mn © ¹ n 0 m 00 (4) (arcsin r (W ))G12 (n, t  W ) º V zn (W )dW  uG11 (n, t  W )   [ mn ¼

³

¦



³

¦

r

1 r

2

¦³

f

¦

n 0

Pn1 §¨ ©

1 r

2

f t

(3) · ¸ ¦ ³ ª¬[ mn (arcsin r (W )) u ¹m 0 0

(4) uG 21 ( n, t  W )  [ mn (arcsin r (W ))G 22 (n, t  W ) º V zm (W )dW ¼

­° 0, if r  r (t ) . (2.2.69) ®

0, if r r ( t )  ! H °¯ where Gij is the Kronecker symbol. In the case of statement 1, the index j

= 1; in the case of statement 2, j = 2. If we exclude the fourth addition in relation (2.2.69), we obtain the condition for formulations 1 and 2, from which the boundary of the contact area is determined without taking into account the rise of the medium. Note that with increasing parameter n, kernel functions Q ij and Qij are asymptotic [198, 199]:

Algorithms for solving mixed non-stationary and dynamic boundary value problems

91

Q11 (n) exp  O(n) , Q11 (n) O 1 n exp  O(n) , (2.2.70) Q22 (n) exp  O(n) , Q 22 (n) O 1 n exp  O(n) . This means that if in the axisymmetric problem we use shells of the S.P. Tymoshenko type then the numerical solution of ISIE of Volterra with kernel functions Q ij and Qij will not convergence. As in the case of the plane problem, after the reduction of the ISIE and subsequent discretization, the resulting system of linear algebraic equations will be poorly defined: with increasing size of this system, its determinant will increase indefinitely. To solve the problem, we obtained two ISIE of Volterra of the second kind, which give the solution (2.2.50), (2.2.52) and (2.2.66), (2.2.68). The first system is written for unknown harmonics of the vertical component of velocity on the surface of the half-space, the second for unknown harmonics of normal stresses occurring at surface points of the elastic medium. In the numerical implementation of the impact problem, it is possible to use any of these two systems. The first system (2.2.50), (2.2.52) is more complex than the second (2.2.66), (2.2.68); however, in the second the function

Gn (t )

is more

cumbersome and less convenient in the numerical solution than the function

Fn (t ) . Therefore, the system (2.2.50), (2.2.52) was chosen for

the numerical implementation and in future all considerations will be made in relation to it. Considering (2.2.44), (1.2.12)–(1.2.17) ISIE (2.2.50) and equation (2.2.53) will be rewritten as: t f (4) Vn (t )  D mn (r ) Vm (W ) Fm (t  W )dW  m 0 0 f f t (3) (5)  D mn (r ) J km (T (W )) u (2.2.70) m 0 k 00 W § ·

¦

¦

³

¦³

u ¨ Vk (W )  ³ Vk ([ ) Fk (W  [ )d[ ¸ G11 (m, t  W )dW ¨ ¸ 0 © ¹ Cn (r )vT (t ), (n

0, f).

Chapter 2

92

t

f

0

n 0



G1 j vT t  G 2 j ³ vT (W )dW  f ( r )  ¦ J 0 On r

t

³ Vn (W )dW  0

f t

f

(3)  ¦ Pn ¨§ 1  r 2 ¸· ¦ ³ ªJ mn (arcsin r (W ))G11 (n, t  W )  ¬ © ¹m 0 n 0 0

(4)  J mn (arcsin r (W ))G12 (n, t  W ) º u ¼

W § · u ¨ Vm (W )  ³ Vm ([ ) Fm (W  [ ) d [ ¸ dW ¨ ¸ 0 © ¹ ­° 0, if r  r (t ) . ®

0, if r r ( t ) H  ! °¯

(2.2.71)

§2.3. Impact of hard cylinders on an elastic layer We apply the Laplace transform of the variable t with the parameter s to equations (1.1.5) and the Fourier method of separation of variables [22, 52, 200], and considering the conditions of attenuation at infinity their general solution is written in the form L

I ( s , x, z )

f

¦ An (s) exp §¨©  z

n 0

s 2 / D 2  On2 ·¸ cos On x  ¹

f

 ¦ Bn ( s ) exp §¨ z s 2 / D 2  On2 ·¸ cos On x, © ¹ n 0

\ L ( s , x, z )

f

¦ Cn (s) exp §¨©  z

n 0

s 2 / E 2  On2 ·¸ sin On x  ¹

Algorithms for solving mixed non-stationary and dynamic boundary value problems

93

f

 ¦ Dn ( s) exp §¨ z s 2 / E 2  On2 ·¸ sin On x, © ¹ n 0 where

On

nS / l , ( n

(2.3.1)

0, f ) are the eigenvalues of the problem,

which are determined from the conditions on the side surfaces. Functions V , uz , V zz , V zx on the surface of the medium are written in the form of series according to the system of eigenfunctions, and the function p – in the form of trigonometric series: f

¦ Vn (t ) cos On x,

V (t , x , o )

n 0 f

¦ u zn (t ) cos On x,

u z (t , x , o )

V zz (t , x, o)

V zx (t , x, o)

(2.3.2)

n 0 f

¦ V zn (t ) cos On x,

n 0 f

¦ V zxn (t )sin On x,

n 1

p(t , x)

(2.3.3)

f

¦ pn (t ) cos(nx).

n 0

Next, the auxiliary problem for equations (1.1.5) with boundary conditions is solved.

w uz wt

z 0

V (t , x), V zx

z 0

0, uz

z h

0, ux

z h

0. (2.3.4)

Satisfying (2.3.4) and considering (2.3.1) and (2.3.2), applying the inverse Laplace transform and the convolution theorem, we obtain the equality that establishes the relationship between the components of

Chapter 2

94

normal stresses and the vertical velocity component on the top surface of the layer:

V zn (t )

t § · D ¨ Vn (t )  ³ Vn (W ) Fn (t  W ) dW ¸ , ¨ ¸ 0 © ¹

(2.3.5)

where t

Fn (t )

Fn (t )  I1 ( n, t )  ³  J 0 ( EOn[ )I2 ( n, t  [ )  0

 J 0 (DOn[ )I3 (n, t  [ ) d [ , 4

I j (n, t ) J j (n, D , E , h)  ¦ H j (n, si , h) cos Ei t , i 2

Ei

Im si , (i

2;3; 4), ( j 1; 2;3), H j (n, si , h)

J j (n, D , E , h) 

B j D j Fj · 1 § ¨ G 3 j b0  2  2  2 ¸ , a0 ¨© a1 b1 c1 ¸¹

2 N j (n, si ) '( si )

,

Algorithms for solving mixed non-stationary and dynamic boundary value problems

95

'( s) a0 s 2 (9s 6  7(a1  b1  c1 ) s 4  5(a1b1  a1c1  b1c1 ) s 2  3a1b1c1 ), a0 c1

3

b 4 h6 , a1 108

§ 2 D 2 h2On2 E 

¨ h2 ¨©

3

E 2On2



6D 2 h

2

, b1

· ¸, ¸ ¹

N1 (n, s) a01  a11s 2  a21s 4  a31s 6 , N 2 (n, s) a02  a12 s 2  a22 s 4  a32 s 6 , N3 (n, s) a03  a13s 2  a23s 4  a33s 6  a43s8 , a01

2hE 9bOn2 (1  b 2 )(1  h2On2  2h2On4 / 3),

a11

E 5b( E 2  2h2On2 ( E 2 (1  b2 )(1  h2On2 / 3) 

b2 (2E 2 (1  On4 / 3)  h2On2 (1  b2 )2 / 6))) / h,

§

E 2 ¨ On2  ©

6 · ¸, h2 ¹

Chapter 2

96

a21

E 5bh((1  b 2 )(1  h 2On2 / 3)  h 2On2 (1  2b 2 

3b 4 ) / 3)) / 2, a31

E 3bh3 (1  6b 2  b 4 ) / 24,

a02

2 E 10bOn4 (2b 2  h 2On2 (2(1  b 2 )  13h 2On2 / 20) / 3),

a32

E 4b3On2 (1  2b 2  b 4 / 5) / 6, a12

2 E 8bOn2 (6b 2 

 (2  b 2 (5  b 2 ))h 2On2  (7 / 20  b 2 (7 / 5  2b 2 ))h 4On4 ) / 3, a22

2E 6bh 2On2 (b 2 (1  b 2 / 3)  (4 / 5  b 2 (4  b 2 (10 

b 2 / 5)))h 2On2 ), a03

2E 10On4 (2  (19 / 3  b 2 )h 4On4 / 20 

2(1  b 2 / 3)h 2On2 ), a23

 E 6 (2  (2  10b 2 / 3 

4b 4 / 3)h 2On2  (7 / 12  19b 2 / 20  8b 4 /15  E 2b 2 (1  2b 2  b 4 / 5) / 3)h 4On4 ) / 2, a33

 E 4 (h 2 (b 2 (1  b 2 / 3) 

 (8b 4 / 5  3b 2  4 / 5)h 2On2 /12)  (1  2b 2  b 4 / 5) E 2 h 2 u uOn2b 2 / 3), a43

 E 4 h 4b 2 (1  2b 2  b 4 / 5) / 24, b0

B j ; D j ; F j ( a , b, c )

B; D; F (b2 j , b1 j , b0 j ), B

 c  a1b)(c1  b1 ) / ' 0 , D

b2k

a12 (c1  b1 )  b12 (a1  c1 ) 

a0 k  a3k m2 / a0 , b1k

a2 k  a3k k2 / a0 , (k 1; 2), b03

 a43k2 / a0 ) / a0 , b13 / a0 , b23

(a12 a 

(b12 a  c  b1b)(a1  c1 ) / ' 0 , F

(c12 a  c  c1b)(b1  a1 ) / ' 0 , ' 0  c12 (b1  a1 ), b0k

2a43 / a0 ,

a1k  a3k l2 / a0 ,

a03  m2 (a33 

a13  a43m2 / a0  l2 (a33  a43k2 / a0 ) /

a23  a43l2 / a0  k2 (a33  a43k2 / a0 ) / a0 , k2

(a1  b1  c1 )a0 , l2

(a1b1  a1c1  b1c1 )a0 , m2

iD 6 / h 2  On2 , s3

s1

0; s2

s2

i 3E 2 / h 2  D 2On2 .

i E 6 / h 2  On2 ,

a1b1c1a0 ,

Algorithms for solving mixed non-stationary and dynamic boundary value problems

97

In this case, the exponents in (2.3.1) are decomposed into power series in which the first six members are retained. Here,

J 0 (t ), J1(t )

are the

Bessel functions of the first kind of zero and first order, respectively, and t J 0 (W )dW . the function J 0 (t ) is defined as follows: J 0 (t )

³

0 It is easy to check that when the thickness of the layer extends to

infinity lim Ii ( n, t ) h of

0, (i 1; 2;3) , the functions Ii are zero and

equality (2.1.5) coincides with the corresponding equality for the halfspace. Using the last equation and (1.3.5) when satisfying mixed boundary conditions (1.1.17), making redevelopments into series by eigenfunctions and equalling the coefficients for the same

cos On x , we obtain an ISIE of

Volterra of the second kind with respect to unknown components of the vertical velocity components: t f (1) Vn (t )  D mn ( x ) Vm (W ) Fm (t  W )dW  m 0 0 f (2) (3) ªD mn  ( x ) w 0m (t )  D mn ( x )u0m (t ) º u (2.3.6) ¬ ¼ m 0 t u Vm (W ) Fm (t  W )dW Cn ( x )vt (t ), 0

¦

³

¦

³

where

(i ) D mn ( x ), (i 1; 2;3), Cn ( x )

are as in (2.1.47).

The equation of motion of the body (1.1.19) taking into account (2.1.50), (2.3.3) will be:

Chapter 2

98

dvT (t ) dt



2D vT (t ) x (t )  M

^

f

sin On x (t )

n 0

On

¦

t

½  (t  W )dW °¾ . ( ) V F W n n ³ °¿ 0

(2.3.7)

§2.4. Impact of fine elastic cylindrical shells on the elastic layer Applying the Laplace transform on the variable t with the parameter s [22] to equations (1.1.5) and the Fourier method of separation of variables, considering the conditions of attenuation at infinity, their general solution is written in the form [153] (2.3.1). Functions

V , uz , V zz , V zx

on the surface of the medium are

written in the form of series according to the system of eigenfunctions as in (2.4.1), and functions u0 , f

V (t , x, 0)

u z (t , x, 0)

V zz (t , x, 0) V zx (t , x, 0)

w0 , p

– in the form of trigonometric series:

¦ Vn (t ) cos On x,

n 0 f

¦ u zn (t ) cos On x,

n 0 f

¦ V zn (t ) cos On x,

n 0 f

¦ V zxn (t ) sin On x,

n 1

(2.4.1)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

99

f

¦ w0n (t ) cos(nx),

w0 (t , x)

n 0 f

¦ u0n (t ) sin(nx),

u0 (t , x)

(2.4.2)

n 1 f

¦ pn (t ) cos(nx).

p (t , x)

n 0 In equations (1.1.3), considering (2.4.2) and applying consistently the direct and inverse Laplace transform and the convolution theorem, we

obtain the following expressions for the harmonics of functions

w0

u0

and

and their time derivatives by analogy with (2.1.18)–(2.1. 20) and

(2.1.21)–(2.1.23). t

³ pn (W )G1(n, t  W )dW ,

w0,n (t )

0 t

³ pn (W )G 2 (n, t  W )dW ,

u0,n (t )

0 t

w 0,n (t )

³ pn (W )G1(n, t  W )dW , 0

t

u0,n (t )

³ pn (W )G2 (n, t  W )dW ,

(n 1, f),

0

t

w0,0 (t )

E2 t W p (W ) sin dW , 1/2 ³ 0 ( E1 ) 0 ( E1 )1/2 t

w 0,0 (t ) where

E2 t W p0 (W ) cos dW , ³ E1 0 ( E1 )1/2

(2.4.3)

Chapter 2

100

G1 (n, t ) G11 (n, t ), G2 (n, t ) G22 (n, t ), G1 (n, t ) G11 (n, t ), G 2 (n, t ) G 22 (n, t ). In (1.1.26), considering (2.4.1) and (2.4.2) and using the orthogonality of trigonometric functions, we obtain the expression for n-th pressure harmonics S f

pn (t )

 ¦ J mn ( x )V zz ,m (t ), N n2 m 0

J mn ( x )

D N n2

³ cos

2

nxdx,

0

(2.4.4)

T

³ cos nx cos(Om x)dx. 0

The equality that establishes the relationship between the components of normal stresses and the vertical velocity components on the top surface of the layer is used as in (2.1.5). Using equation (2.1.5) when satisfying mixed boundary conditions (1.4.4), (1.4.5), making redevelopments into series by eigenfunctions and equalizing the coefficients for the same

cos On x ,

we obtain an ISIE of

Volterra of the second kind with respect to unknown components of the vertical component speed: t f (1) Vn (t )  D mn ( x ) Vm (W ) Fm (t  W )dW  m 0 0

¦

³

f

(2)  (3)   ¦ ªD mn ( x ) w0m (t )  D mn ( x )u0m (t ) º u ¬ ¼

(2.4.5)

m 0 t

u³ Vm (W ) Fm (t  W )dW Cn ( x )vt (t ), 0

where

(i ) D mn ( x ), (i 1; 2;3), Cn ( x )

are as in (2.1.47).

The equation of motion of the body (1.1.19) considering (2.1.50), (2.4.2), (2.4.4) will take the form as in (2.3.7).

Algorithms for solving mixed non-stationary and dynamic boundary value problems

101

§2.5. Quasi-static elastic-plastic formulation – three-dimensional problem We use, as in [18, 20, 37, 39, 144–146], the method of sequential (step-by-step) tracking of the development of elastic-plastic deformations, when at each step of increase load stress-strain state is found considering the previous solution steps. To do this, the entire period of growth of the load is divided into separate small intervals. Consider some interval of change in load, determined by the time is the current time and t0 is the moment of time preceding the current one. We integrate equations with respect to the variable t, using the theorem of the mean V xx  V , V yy  V , interval

't

t1  t0 , where t1



V zz  V

and the mean values of V xy ,

V xz , V



y z replace with the

corresponding values at the time t1 with precision O('t) . Due to the short duration of the process of destruction of small-scale specimens, increments ') can be neglected as an infinitesimal value and ignored. Increments of deformations will be written down in the form:

'H xx \ (V xx (t1 )  V (t1 ))  KV (t1 )  bxx (t0 ), 'H yy \ (V yy (t1 )  V (t1 ))  KV (t1 )  byy (t0 ), 'H zz \ (V zz (t1 )  V (t1 ))  KV (t1 )  bzz (t0 ), 'H xy \V xy (t1 )  bxy (t0 ),

(2.5.1)

'H xz \V xz (t1 )  bxz (t0 ), 'H yz \V yz (t1 )  byz (t0 ). where

bij ( t 0 )

V ij ( t 0 ) 2G

§ 1 ·  G ij ¨  K ¸ V ( t 0 ), © 2G ¹

V (t0 ) (V xx (t0 )  V yy (t0 )  V zz (t0 )) / 3, (i, j x, y, z). 1  'O , ' O – differences equivalent to the differential 2G d O and for the stress state at the time t1 is determined by the yield Let

\

condition, which taking into account (2.5.1) can be rewritten as follows:

Chapter 2

102

­ 1 if f  0; ° 2G , ° f !0 ® p ° 1  3'Hi , if f 0; °¯ 2G 2V i

\

2ª p p 'H xx  'H yy « 3 ¬



'H ip 



p 'H yy

 'H zzp



2

2

  'H xxp  'H zzp

§ p  6 ¨ 'H xy ©



inadmissible

2



p 'H xz

2



2

(2.5.2)



p 'H yz



1 2 ·º 2

¸» ¹¼

,

1 1 · § V ii  ¨ K  ¸V , 2G 2G ¹ © 1 V (V xx  V yy  V zz ) / 3, 'H ije V ij , (i z j ), 2G (i, j x; y; z ). 'H ijp

'H ij  'H ije , 'H iie

For considering the physical non-linearity contained in conditions (2.5.2), we use the method of successive approximations, which allows the solution of a non-linear problem to be reduced to the solution of a set of successive solutions of linear problems.

\ ( n1)

­ ( n) 1 p , if V i( n )  V S (T )  Q, °\ p  2G ° ° ( n) if  Q  V i( n )  V S (T )  Q, (2.5.3) ®\ , ° (n) °\ ( n ) V i , if V i( n )  V S (T ) ! Q, ° V S (T ) ¯

where Q is the value of the largest deviation of the stress intensity

Vi(n) in

step n from the strengthened yield strength, n is the approximation number and empirical constant 0 d p d 1 is determined for different types of materials. From system (2.5.1), we find expressions for calculating stresses:

Algorithms for solving mixed non-stationary and dynamic boundary value problems

V xx

A1'H xx  A2 'H yy  A2 'H zz  Yxx ,

V yy

A2 'H xx  A1'H yy  A2 'H zz  Yyy ,

V zz

A2 'H xx  A2 'H yy  A1'H zz  Yzz ,

V xy

A3'H xy  Yxy ,

V xz

A3'H xz  Yxz ,

V yz

A3'H yz  Yyz ,

103

(2.5.4)

where

Yxx

A1bxx  A2byy  A2bzz , Yxy

A3bxy ,

Yyy

A2bxx  A1byy  A2bzz , Yxz

A3bxz ,

Yzz

A2bxx  A2byy  A1bzz , Yyz

A3byz ,

A1

1 \  2K \ K , A2 , A3 . 3\ K 3\ K \

The increments of the components of the displacement vector ' u are related to the components of the deformation increments by the following relations:

'H xx

wu , 'H xy wx

1 § wu wv · ¨  ¸, 2 © wy wx ¹

'H yy

wv , 'H xz wy

1 § wu ww · ¨  ¸, 2 © wz wx ¹

'H zz

ww , 'H yz wz

1 § wv ww · ¨  ¸, 2 © wz wy ¹

where u

'u x , v

'u y , w

(2.5.5)

'u z .

The calculated value of the stress intensity factor

KI

(SIF) near the

crack in the static problem for an elastic deformed compact specimen was chosen as the main independent parameter for studying the change in the studied values. SIF at each current moment of time was determined according to the approximate ratio [176, §1.2] for a static solution

Chapter 2

104

2 § l §l · ¨ KI 1,93  3, 07  14,53 ¨ ¸  ¨ B ©B¹ © , 3 4· §l · §l · 25,11¨ ¸  25,8 ¨ ¸ ¸ , ©B¹ © B ¹ ¸¹

l 12 F BH

(2.5.6)

where F 2 A H P is the contact power and L 4 B is length between supports. The independent parameter that characterizes the loading process is the time tk k 't , and hence the force of contact interaction of the drummer with the specimen F 2 AP corresponding to this moment of time. Since in fracture mechanics fracture toughness (crack resistance) is mainly obtained in quasi-static experiments and comparing this with the limit value of the SIF

K1

obtained from the elastic solution, then to describe

the change of individual characteristics in many dependencies in the role of an independent parameter (variable) we choose the approximate value of SIF

KI

(hereafter referred to as elastic SIF) for the elastic problem of a

three-point bending beam with notch-crack.

§2.6. Dynamic elastic-plastic formulation §2.6.1. The case of plane stress state Let the non-stationary interaction [20, 25, 26, 32, 36, 38, 47] occur in a time interval

t  [0, t* ].

Then, for each moment of time t from this

interval: e H xx

H eyy e H xy

V xx  V 2G

V yy  V 2G

V xy 2G

,

p d H xx dt

V xx  V ddtO ,

p d H yy  KV , dt

V yy  V ddtO ,

 KV ,

p d H xy dt

V xy d O , dt

(2.6.1)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

e  H e ), H p  Q (H xx yy zz 1 Q

e H zz

105

p H p . H xx yy

For numerical integration over time, Gregory’s quadrature formula [189] of order

m1

3

with coefficients

Dn

was used. After even

discretization in time with nodes tk k 't  [0, t* ] (k 0, .) for each value k we write down the corresponding node values of deformation increments:

'H xx,k

B1V xx,k  B2V yy,k  bxx ,

'H yy,k

B2V xx,k  B1V yy,k  byy ,

'H xy,k

B3V xy,k  bxy , bzz

'H zz,k

B4 (V xx,k  V yy,k )  'H xx,k  'H yy,k  bzz ,

B1 B2

B4 (V xx,k 1  V yy,k 1),

(2.6.2)

 1 K  1  D 'O , 3 2G

1 K  1  2D 'O , 0 k G 3 0

k



B3

1  D 'O , B 0 k 4 2G

bij

1 V  G K  1 V k 1  2G ij,k 1 ij 2G

m1 1







1  2Q 2K  1 , 3(1 Q ) 2G





 ¦ Dn V ij,k n  G ijV k n 'Ok n (i, j x, y). n 1

The function \ 1 (2G)  'O characterizing the yield condition, considering (1.6.6), (2.6.1) is

Chapter 2

106

p °­ 1 1  3'H i ( f ( 0), f  ® 2G 2G 2V i °¯ (f ! 0  inadmissible ),

\

'H ip



2 § 'H p  'H p 2  'H p  'H p 2  xx yy xx zz 3 ¨©



p  'H p  'H yy zz p 'H xx p 'H xy

e H xx e H xy

°½ 0) ¾ °¿

0, df





2



1



p  6 'H xy



e , 'H p 'H xx  'H xx yy

2· 2

¸ ¹

(2.6.3)

,

'H yy  'H eyy ,

e , 'H p 'H p  'H p , 'H xy  'H xy zz xx yy 1 V  K  1 V, He 1 1 yy 2G V yy  K  2G V , 2G xx 2G 1 V , V (V  V ) / 3. xx yy 2G xy









To consider the physical non-linearity contained in the dependencies (2.6.3), the method of successive approximations is used, which allows us to reduce the non-linear problem to a sequence of linear problems [18, 20, 23–52, 53, 144–146] as in (2.5.3). The solution of the system [25, 26, 32, 36, 38, 47] gives expressions for the components of the stress tensor at each step:

V xx,k

A1'H xx,k  A2'H yy,k  Yxx , Yxx

A1bxx  A2byy ,

V yy,k

A2'H xx,k  A1'H yy,k  Yyy , Yyy

A2bxx  A1byy ,

V xy,k

A3'H xy,k  Yxy , Yxy

A1

B1

 B12  B22 ,

A2

A3bxy , A3 1 B3 ,

 B2

(2.6.4)

 B12  B22 .

The increments ' u of the displacement vector are related to the increments of the deformations by such relations:

Algorithms for solving mixed non-stationary and dynamic boundary value problems

w'u y

'H xx

w'u x , 'H yy wx

'H xy

1 § w'u x w'u y  ¨ 2 © wy wx

wy

, 'H zz

w'u z , wz

· ¸. ¹

107

(2.6.5)

The intensities of stresses and strains used above were determined for each unit cell of the numerical solution. The independent parameter that characterizes the loading process at the time tk k 't , and hence the force of contact interaction of the drummer with the specimen F 2 A P corresponding to this moment of time is SIF

K1e

KI

from (2.5.6).

§2.6.2. The case of plane strain state Let the non-stationary interaction [27, 29, 30, 34, 41, 42, 45, 46, 48, 50] occur in a time interval t  [0, t* ] . Then, for every moment of time t: e H xx

H eyy e H zz e H xy

V xx  V 2G

V yy  V 2G

V zz  V 2G

 KV ,  KV ,

2G

p d H yy

dt

d H zzp  KV , dt

p V xy d H xy

,

p d H xx dt

dt

V xy

dO

V xx  V dt ,

V yy  V ddtO , dO V zz  V , dt

(2.6.6)

dO . dt

For numerical integration over time, Gregory’s quadrature formula [189] of order m1 3 with coefficients Dn was used. After discretization in time with nodes tk k 't  [0, t* ] ( k 0, K ) for each value k we write down the corresponding node values of deformation increments:

Chapter 2

108

D 2 (D1  D 2 ) , D1

'H yy ,k

B2V xx,k  B1V yy ,k  E yy , B2

'H zz ,k

D1V zz ,k  D 2 (V xx,k  V yy,k )  bzz , 1  D0 'Ok , 2G bxx  D 2 (bzz  'H zz ) / D1, B3V xy ,k  bxy , B3

'H xy ,k

E xx E yy

byy  D 2 (bzz  'H zz ) / D1, E zz

 (bzz  'H zz ) / D1,

1§ 1 1§ 1 · ·  D0 'Ok ¸ , ¨ K   2 D0 'Ok ¸ , D 2 ¨K  3© 3© 2G G ¹ ¹ 1 1 · § bij V ij ,k 1  G ij ¨ K  ¸ V k 1  2G 2G ¹ ©

D1

m1

(2.6.7)





 ¦ Dn V ij ,k  n  G ijV k  n 'Ok  n (i, j n 1

x, y, z ).

Function \ 1 (2G)  'O , which characterizes the yield condition, taking into account (1.6.13), (2.6.6), (1.6.14) is: p ­° 1 ½° 1 3'H i  \ ® ( f  0), ( f 0, df 0) ¾ 2G 2V i ° 2G ° (2.6.8)

¯ ( f ! 0  inadmissible),

¿

Algorithms for solving mixed non-stationary and dynamic boundary value problems

'H ip

2§ p p 'H xx  'H yy ¨ 3 ©



2



p  'H yy  'H zzp





2

  'H xxp  'H zzp

p  6 'H xy



2

109



2·

¸, ¹ 1 1 · § p e e , 'H xx 'H xx 'H xx  'H xx V xx,k  ¨ K  ¸V k , 2G 2G ¹ © 1 1 · § p 'H yy 'H yy  'H eyy , 'H eyy V yy,k  ¨ K  ¸V k , 2G 2G ¹ © 1 p e e , 'H xy 'H xy 'H xy  'H xy V xy,k , 2G 1 1 · § e e V zz ,k  ¨ K  , 'H zz 'H zzp 'H zz  'H zz ¸V k , 2G 2G ¹ © V k (V xx,k  V yy ,k  V zz ,k ) 3. Consider when calculating the value

'Hzzp

that its impact is so small

that without reducing the accuracy of calculations it can be considered that

'H zzp

0.

To take into account the physical non-linearity contained in conditions (2.6.8) the method of successive approximations is used, which makes it possible to reduce a non-linear problem to a sequence of linear problems [18, 20, 23–53, 144–146] (2.5.3). The solution of the system (2.6.7) gives expressions for the components of the stress tensor at each step:

V xx,k

A1'H xx,k  A2 'H yy ,k  Yxx , Yxx

A1E xx  A2 E yy ,

V yy,k

A2 'H xx,k  A1'H yy,k  Yyy , Yyy

A2 E xx  A1E yy ,

V xy,k

A3'H xy,k  Yxy , Yxy

A3bxy , A3 1 B3 ,

Chapter 2

110

D 2 (V xx,k  V yy ,k ) / D1  E zz ,

V zz ,k A1

B1

 B12  B22 , A2

Unknown [57]

 B12  B22 .

 B2

(2.6.9)

0

'F x , ' F y and 'Hzz in (1.6.14) are determined from

the conditions of equilibrium of even with respect to x normal stresses V zz .

³³ V zz (x, y)Udxdy

MU , (U 1, x, y),

(2.6.10)

6

when M 1 where

Mx

0,

My

M1 is the projection on the axis Oz of the main vector of

contact stresses, and M x , M y are the corresponding projections of the main moment of the forces acting on the resistance (no torsion, as noted). Given the symmetry of the problem and V zz ( x, y) V zz ( x, y) , this equation in case of U x is satisfied automatically. If we substitute (1.6.14) and (2.6.9) into (2.6.10), taking into account the symmetry of the integration domain with respect to x and the even of functions V xx , k , V yy , k , b zz , we have 'F x 0 . A system of linear algebraic equations is obtained for the calculation of 0 'H zz LU1  'F y LU y

MU

³³ 6

LU r

³³



M U , (U



D 2 V xx  V yy  bzz D1

0 ' H zz , 'F y :

1, y ),

U rdxdy,

U rdxdy , (r , U 1, x, y ). D1

(2.6.11)

6 The increments ' u of the displacement vector are related to the increments of the deformations by the relations (2.6.5). The intensities of stresses and strains used above were determined for each unit cell from the numerical solution. The stress intensity factor (SIF)

Algorithms for solving mixed non-stationary and dynamic boundary value problems

KIe

at each point in time

tk

111

k 't was determined from the ratio

(2.5.6).

§2.6.3. The case of three-dimension stress-strain state Let the non-stationary interaction [23, 31, 39, 43, 44, 49, 51, 52, 54] occur in a time interval t  [0, t* ] . Then, for each point in time t:

H ije d H ijp dt

V ij

1 · §  G ij ¨ K  ¸V , 2G 2G ¹ ©

V ij  GijV ddtO , (i, j

(2.6.12)

x, y, z ).

Like for plane problems for numerical integration over time, Gregory’s quadrature formula [189] of order m1 3 with coefficients Dn was used. After sampling in time with nodes tk k 't  [0, t* ] (k 0, K ) for each value k we write down the corresponding node values of deformation increments:

Chapter 2

112

'H xx,k

B1V xx,k  B2 (V yy ,k  V zz ,k )  bxx ,

'H yy ,k

B1V yy ,k  B2 (V xx,k  V zz ,k )  byy ,

'H zz ,k

B1V zz ,k  B2 (V xx,k  V yy ,k )  bzz ,

'H xy ,k

B3V xy ,k  bxy , 'H xz ,k

B3V xz ,k  bxz ,

1§ 1 · ¨ K   2 D0 'Ok ¸ , 3© G ¹ (2.6.13) 1§ 1 1 ·  D0 'Ok ¸ , B3  D0 'Ok , ¨K  3© 2G 2G ¹ 1 1 · § V ij ,k 1  G ij ¨ K  ¸ V k 1  2G 2G ¹ © B3V yz ,k  byz , B1

'H yz ,k B2 bij m1





 ¦ Dn V ij ,k n  G ijV k n 'Ok n . n 1

Function \ 1 (2G)  'O , which characterizes the yield condition, taking into account (1.6.19), (2.6.12) is equal to: p ­° 1 ½° 1 3'H i (f  0); (f 0, df 0) ¾ \ ®  2G 2V i ° 2G ° (2.6.14)

¯ ( f ! 0  inadmissible),

2ª p p 'H xx  'H yy « 3 ¬



'H ip 



p 'H yy

'H ijp

¿

 'H zzp



2

2

  'H xxp  'H zzp

§ p  6 ¨ 'H xy ©



'H ij  'H ije , (i, j

2



p 'H xz

2



2



p 'H yz



1 2 ·º 2

¸» ¹¼

,

x, y, z ).

To take into account the physical non-linearity contained in conditions (2.6.14) the method of successive approximations is used, which makes it possible to reduce a non-linear problem to a sequence of linear problems [19, 20, 23–53, 144–146] (2.5.3).

Algorithms for solving mixed non-stationary and dynamic boundary value problems

113

The solution of system (2.6.13) gives expressions for the components of the stress tensor at each step:

V xx,k

A1'H xx,k  A2 'H yy ,k  A2 'H zz ,k  Yxx ,

V yy ,k

A2 'H xx,k  A1'H yy ,k  A2 'H zz ,k  Yyy ,

V zz ,k

A2 'H xx,k  A2 'H yy ,k  A1'H zz ,k  Yzz ,

V xy ,k

A3'H xy ,k  Yxy , V xz ,k

V yz ,k

A3'H yz ,k  Yyz ,

(2.6.15)

A3'H xz ,k  Yxz ,

where

Yxx

A1bxx  A2byy  A2bzz , Yxy

A3bxy ,

Yyy

A2bxx  A1byy  A2bzz , Yxz

A3bxz ,

Yzz

A2bxx  A2byy  A1bzz , Yyz

A3byz ,

A1

( B1  B2 ) (( B1  B2 )( B1  2 B2 )) ,

A2

 B2 (( B1  B2 )( B1  2 B2 )), A3 1 B3 .

The increments ' u of the displacement vector are related to the increments of the deformations by such relations:

'H xx 'H yy 'H zz

w'u x , 'H xy wx

1 § w'u x w'u y  ¨ 2 © wy wx

· ¸, ¹

w'u y

, 'H xz

1 § w'u x w'u z  wx 2 ¨© wz

· ¸, ¹

w'u z , 'H yz wz

1 § w'u y w'u z  ¨ 2 © wz wy

· ¸. ¹

wy

(2.6.16)

The intensities of stresses and strains used above were determined for each unit cell from the numerical solution. The stress intensity factor K I at each time point tk

k 't was determined from the relation (2.5.6).

CHAPTER 3 NUMERICAL IMPLEMENTATION OF THE SOLUTION ALGORITHM

To justify the application of the method of reduction, it is necessary, strictly speaking, to prove that V nN Ÿ V nN ( t ), n o f , t  [0, f ]. Here Ÿ indicates the uniformity of convergence. At present, in the mechanics of a continuous medium, many static and stationary dynamic problems are reduced to solving infinite systems of algebraic equations (ISAE), which in most cases are solved by reduction. However, there are theorems that state the fact that the reduction process coincides if certain conditions are imposed on the coefficients of the matrix of an algebraic system [105]. ISIE of Volterra of the second type (2.1.74), (2.2.70) were studied in [114]. It is also noted here that systems of the type (2.1.74), (2.2.70) are found when solving boundary problems with mixed boundary conditions for equations of the hyperbolic type by the Fourier method. This paper proved the theorem that establishes the conditions for the coefficients of the system, under which the reduction process coincides. However, it should be noted that the substantiation of the convergence of the reduction method for ISIE (2.1.74), (2.2.70) with the coefficients (3.1.1)–(3.1.4) and (3.2.1)–(3.2.4) is not the subject of this research and will not be conducted below. The order of reduction N will be chosen for reasons of practical convergence. To smooth the oscillations that occur when summing a finite number of members of a series, as well as Gibbs phenomena near a weak mathematical break, we used the averaging operation defined in [139], which consists, in the case of the sum of a finite number of members of trigonometric series on V n of the Lancosh multipliers [139]:

Vn

if n 0, ­1, ° ® sin(nS /N ) °¯ nS /N , if n z 0,

(3.1)

Numerical implementation of the solution algorithm

115

To calculate the integrals, the method of mechanical quadratures was used, in particular the symmetric quadrature Gregory formula for m equidistant nodes [189] (0 d t d T ) . The coefficients Gi 1 are obtained by disclosing the differences up to m1  1 rank in Gregory’s formula [189]. Formulas up to m1 4 rank were used in the calculations. Rewrite the differential equations of motion with initial conditions (2.1.54) and (2.2.52) in the form:

dvT (t ) dt



P (t ) , M

vT (t ) t 0

V0 .

(3.2)

The Cauchy problem for differential equation (3.2) was solved by the Adams method (closed-type formulas) [113] of m 1 rank with local truncation error O ( ' t m1 1 ) . The general scheme of the method is reduced to the formulas: m 1 't 1 vT ,k vT ,k 1  Di Pk -i  O('t m11 ). (3.3) M i 0 The coefficients D i are obtained by disclosing the differences up to

¦

rank

m1  1 in the Adams formula [113]. Formulas up to rank m1

4

were used in the calculations. The use of Gregory’s formulas [189] and (3.3) makes sense if the initial nodes are calculated fairly accurately. For this purpose, the beginning of the solution was calculated with step 't1 't / 16 , followed by doubling the step and applying these formulas of a higher rank

m1 . The inaccuracy in the detection of the first

nodes is not higher than O ( ' t 2 / 16 2 ) . Moreover, for subsequent nodes the error did not change, because higher-rank formulas were used.  0 m , u0 m , w0 m and u 0 m are determined from the relations If w (2.2.27)–(2.2.29), (2.2.31)–(2.2.33), then ISIE of Volterra of the second kind (2.2.53) and (2.2.69) will take a similar form, only instead of kernel

 there will be functions Q and Q from (2.2.27)– functions G ij and G ij ij ij (2.2.32).

116

Chapter 3

§3.1. Impact of fine elastic cylindrical shells on an elastic half-space Numerical calculations will be performed for eigenvalues (2.1.37), in this case F ( t ) { 0 , and from (2.1.44) it follows that F ( t ) { 0 . The coefficients in (2.1.74) take the form [171]:

­

(1) D mn (x )

lx , ° ° l ° sin(O x ) m ° , Oml ° °

° 2sin(On x ) , ° Onl ° ® §

· °1 ¨ l  x  sin(2On x ) ¸ , ¸ °l ¨ 2On ¹ ° ©

° sin((O  O ) x ) m n °  (Om  On )l ° °

° sin((Om  On ) x ) , ° (Om  On )l ¯

if m

n

if m z 0, n

0,

0,

if m 0, n z 0, if m

n, m z 0, n z 0,

if m z n, m z 0, n z 0,

(3.1.1)

Numerical implementation of the solution algorithm

­ x ° , ° l ° sin(mx ) ° , ° ml ° 2sin(O x ) n ° (2) , D mn ( x ) ® Onl ° ° sin((m  O ) x ) n °  ° (m  On )l °

° sin((m  On ) x ) , ° (m  On )l ¯

Cn ( x )

­ x if ° , ° l ®

° 2sin(On x ) , if ° Onl ¯

n

117

if m n 0, if m z 0, n 0, if m 0, n z 0,

(3.1.2)

if m z 0, n z 0,

0, (3.1.3)

n z 0,

­ x ° , °S ° sin(O x ) m ° , ° OmS ° ° 2sin(nx ) (1) J mn ( x ) D ® , ° nS ° sin((O  n) x ) m °  ° (Om  n)S °

° sin((Om  n) x ) , ° (Om  n)S ¯

if m n 0, if m z 0, n 0, if m 0, n z 0, (3.1.4)

if m z 0, n z 0,

Chapter 3

118

To solve the system (2.1.74) we will use the method of reduction (truncation). We write the truncated system (2.1.74): t N (1) VnN (t )  D mn ( x ) VmN (W ) Fm (t  W )dW  m 0 0 t N N (1) (2)  D mn (x ) J km (T (W )) VkN (W )  (3.1.5) m 0 k 00 W ·  VkN ([ ) Fk (W  [ )d[ ¸ G11 (m, t  W )dW ¸ 0 ¹

Cn ( x )vT (t ). In the section below we will understand the solution VnN (t ) of

¦

³

¦



¦³

³

reduced systems, and the index N in (3.1.5) will be omitted. We rewrite the reduced system (3.1.5) in nodes using Gregory’s quadrature formulas for equidistant nodes [189], considering that Fn (0) { 0 . N

k 1

Vnk  't ¦

(1) D mnk

N

k

N

i 0

k1 0

m 0

't ¦

m 0

(2) D mnk

¦ giVmi Fm (k  i) 

i 0

¦ giG11 (m, k  i) ¦ J k(1)mi Vk i  1

i 1 ·  't ¦ g jVk1 j Fk1 (i  j ) ¸ Cnk vT ,k , ( n ¸ j 0 ¹

1

(3.1.6)

0, N ).

We write expression (2.1.53) with respect to the reaction force of the elastic half-space:

Pk



2D vT ,k xk*  N

 't ¦ V m m 0

sin Om xk* k 1

Om

· ¦ giVmi Fm (k  i) ¸¸ . i 0 ¹

(3.1.7)

Numerical implementation of the solution algorithm

119

The equation of motion (2.1.54) in discretized form is solving relatively vTk and has the form:

§ 2D't 2 D0 u ¨ vT ,k 1  M M  2D'tD0 xk* ¨© M

vT ,k N

u ¦ Vm

sin Om xk* k 1

m 0

Om

't ¦ giVmi Fm (k  i)  M i 0

m1 1

¦

i 0

· Di Pk i ¸ . ¸ ¹

(3.1.8)

The condition for determining the boundary of the contact area (2.1.75) in truncated and discretized form takes the form: k xk* 1  G1 j k 'tV0  G 2 j 't gi vTi  i 0 k N 't V n gi Vni cosOn xk*  i 0 n 0 k 1 N N (3.1.9) 't gi V n cos( nxk* )G11 ( n, k  i ) J k(1)ni Vk1i  1 i 0 n 0 k1 0

 

¦

¦

¦

¦ ¦

¦

i -1 ·· 't ¦ giVk1 j Fk1 (i  j ) ¸ ¸ ¸¸ j 0 ¹¹



1 2· 2

¸ ¸ ¸ ¹

.

The system (3.1.6) can be rewritten as: N § k 1 (1) Vnk Cnk vT ,k  't giVmi Fm (k  i )  ¨¨ D mnk m 0© i 0 N k 1 (2) J (1) gi G11 (m, k  i ) u D mnk jmk j 0 i 0

¦

¦

¦

¦

i 1 § ·· ¨ u V ji  't ¦ gi1V ji1 F j (i  i1 ) ¸ ¸  ¨ ¸¸ i1 0 © ¹¹

Chapter 3

120

'tg k

N

N

m 0

j 0

(2) G11 ( m, 0) ¦ J (1) ¦ D mnk jmk V jk 

(3.1.10)

§ k 1 ·  ¨ 't ¦ giV ji F j ( k  i ) ¸ . ¨ ¸ © i 0 ¹

Equation (3.1.9) and system (3.1.10) are solved by iterations. Determining the boundary of the contact area xk from equation (3.1.9),

Vnk from the system of equations (3.1.10). Then, using the found harmonics Vnk , we refine the values xk by

we determine the harmonics

solving equation (3.1.9) again and find the refined values of the harmonics of the vertical velocity component Vnk and so on until the conditions are met:

xk*(i )  xk*(i 1)  H ,





(i ) (i 1) max Vnk  Vnk  H,

0dnd N

(3.1.11)

where i is the iteration number; when performing calculations H was chosen equal 10-6. Next, we perform calculations for the time (k  1)'t . Taking as a *(0) xk* , V n(,0k) 1 V n k , n 0, N by formula zero approximation xk 1 (3.1.9), we find

(1) xk*(1) 1 , and from (3.1.10) we find V n , k  1

Thus, we will take

x1*(0)

x10 ,

v T( 0,1)

for n

0, N .

V 0 . For the effective

application of the method of mechanical squaring, the asymptotics of functions Fn (t ) were considered: lim Fn ( t ) 2 E b O n (1  b 2 ), and no f

their oscillating nature. As the number n increases, the frequency of oscillations of the function Fn (t ) increases. For the effective application of quadrature formulas, it is necessary that the half-wave of oscillation of the function with the largest number FN (t ) had several points of division of the time interval. Since the first oscillation of function

J1 (t ) has the

Numerical implementation of the solution algorithm

121

largest amplitude, the last condition is approximately written in the form [184]: (3.1.12) DOn k L 't 3.8317, where 3.8317 is first (non-zero) root of

J1 (t ) with an accuracy of 10-4

kL is the number of breakpoints per half-wave of oscillation. For eigenvalues (2.1.37), condition (3.1.12) has the form: (3.1.13) 't 3.8317l (D N S k L ).

and

In addition, condition (2.1.1) must be considered. For efficient operation of the algorithm, it is enough to set kL equal to 2, ( kL t 0 ). As test examples, the problems of impact of an absolutely rigid cylinder with mass M = 0.25 with an initial velocity V0 0.001 on a steel half-space P / K 0.45 and the problem of impact of a steel shell with mass M = 1.0 and thickness h / R 0.01 with a velocity V0 0.05 on the water

surface were selected. The results of the calculations coincide with the results in [68, 165]. The above problems are special cases of the problem of impact of an elastic cylindrical shell on an elastic half-space. To obtain the problem of the impact of an absolutely rigid cylinder on an elastic medium, the functions Gij ( n , t ) { 0 in (2.1.18)–(2.1.20) and G ij ( n, t ) { 0,  i , j 1; 2 in (2.1.21)–(2.1.23) must be set equal to zero; to obtain the problem of impact of the steel shell on the water surface, it is necessary to take the speed of sound in the medium equal C 02 modulus equal to zero, P 0 .

12.2 and set the shear

§3.2. Impact of fine elastic spherical shells on an elastic half-space Numerical calculations were performed for eigenvalues determined from (2.2.36). The coefficients of the system (2.2.70) take the form:

Chapter 3

122

­§ ·2 °¨ r ¸ , if °°¨ l ¸ Cn (r ) ®© ¹ ° 2r J1 (On r ) , if ° 2 2 ( ) l J l O O  °¯ n 1 n ­ § ·2 °1  ¨ r ¸ , ° ¨ l ¸ ° © ¹ ° 2(lJ (O l )  r J (O r )) 1 m 1 m , ° 2 Oml ° ° (4) (r ) ® 2(lJ1 (Onl )  r J1 (On r )) D mn , ° 2 2 l J l ( ) O O  ° n 1 n ° 2 ° § r J1 (On r ) · ¸ , °1  ¨¨ ° © lJ1 (Onl ) ¸¹ ° ¯0,

n 0, (3.2.1)

n z 0,

if m

n 0,

if m z 0, n 0, if m 0, n z 0,

if m

n, m z 0, n z 0,

if m z n, m z 0, n z 0, (3.2.2)

(5) * D mn (r )

*

Cn (r ),

2 ­ § · °(2n  1) ¨ r ¸ , if ¨ 2 ¸ °° (3) © ¹ J mn (r ) D ® ° (2n  1)r J1 (Om r ) , if ° 2Om °¯

(3.2.3)

m

0, (3.2.4)

m z 0,

To solve system (2.2.70), we will use the method of reduction (truncation). Let us write down the truncated system (2.2.70):

Numerical implementation of the solution algorithm

N

t

m 0

0

VnN (t ) 

123

(4) (r ) ³ VmN (W ) Fm (t  W )dW  ¦ D mn

N

N t

m 0

k 00

(3) (5)  ¦ D mn (r ) ¦ ³ J km (T (W )) u

(3.2.5)

W § · N u ¨ Vk (W )  ³ VkN ([ ) Fk (W  [ )d[ ¸ G11 (m, t  W )dW ¨ ¸ 0 © ¹

Cn (r )vT (t ), ( n

0, N ).

In this section, as well as for the system (3.1.5), the index N will be omitted in the future. Also, as in the previous paragraph, V n are the Lancosh multipliers [139] (3.1) used to smooth oscillations and Gibbs phenomena near weak break points when summing up a finite number of series members. The method of mechanical quadratures was used to calculate the integrals, in particular the symmetric Gregory quadrature formula for equidistant nodes [189]. The Cauchy problem for the differential equation (2.2.52) was solved using the Adams method [113] (closed-type formulas) of rank m1 with local truncation error

O ( ' t m 1  1 ). The general scheme of the method was reduced to formulas (3.3). The beginning of the solution was calculated with a step 't1 't /16 , followed by a doubling of the step and the application of Gregory’s and (3.3) formulas of a higher rank

m1 .

We rewrite the reduced system (3.2.5) in nodes using Gregory’s quadrature formulas [189], given that Fn (0) { 0, F0 (t ) { 0 . N

k 1

(4) Vnk  't ¦ D mnk ¦ giVmi Fm (k  i)  m 1

N

't ¦

m 0

(5) D mnk

i 0

k

N

i 0

k1 0

¦ giG11(m, k  i) ¦ J k(3)mi Vk i  1

1

Chapter 3

124

i 1 ·  't ¦ g jVk1 j Fk1 (i  j ) ¸ Cnk vT ,k , (n 0, N ). ¸ j 0 ¹

(3.2.6)

Write the expression for the reaction force of the elastic half-space (2.2.51) in discretized and reduced form:

Pk

N

§ ¨ ©

DS rk ¨ vT , k rk  2 ' t ¦ V n

J1 ( O n rk ) k 1

n 0

On

·

¦ g iVni Fn ( k  i ) ¸¸ .

i 0

¹

(3.2.7) The equation of motion of a spherical shell (2.2.52) in discrete form is solving relatively vT , k and has the form:

§ 2DS't 2 D0 rk  u v ¨ T ,k 1 M M  DS'tD0 rk 2 ¨© M

vT ,k N

u¦ Vm m 1

J1 (Om rk ) k 1

Om

't ¦ giVmi Fm (k  i)  M i 0

m1 1

¦

i 0

· Di Pk i ¸ . ¸ ¹

(3.2.8)

The condition for determining the boundary of the contact area (2.2.71) in truncated and discretized form takes the form: k rk 1  G1 j k 'tV0  G 2 j 't gi vTi  i 0 k N 't gi V nVni J 0 (On rk )  i 0 n 0 k 1 N N (3.2.9) 't gi V n (n  0,5) Pn (rk ) J k(3)ni u 1 i 0 n 0 k1 0

 

¦

¦ ¦

¦ ¦

¦

j 1 § · u ¨ Vk1i  't ¦ giVk1 j Fk1 (i  j ) ¸ G11 (n, k  i ) ¨ ¸ j 0 © ¹

The system (3.2.6) can be rewritten as:



2·

¸ ¸ ¹

1

2

.

Numerical implementation of the solution algorithm

125

2

§ rk · 2 't ¨ ¸ vT ,k  2 u ¨ l ¸ l © ¹

V0 k

(3.2.10)

lJ1 (Oml )  rk J1 (Om rk ) k 1 giVmi Fm ( k u O m m 1 i 0 2 k N § r · J ji u D ¨ k ¸ 't gi ¨ l ¸ i 0 j 0 © ¹ N

¦

¦

 i) 

¦ ¦

i 1 § · N u ¨ V ji  't ¦ gi1V ji1 F j (i  i1 ) ¸ ¦ ( m  0,5)G11 (m, k  i ), ¨ ¸m 0 i1 0 © ¹

where

J ki

Vnk

­ ( ri ) 2 , if k 0; ° ° 2 ®

° ri J1 (Ok ri ) , if k z 0; ° Ok ¯

2rk J1 (On rk )

On (lJ1 (Onl )) 2

vT ,k 

2· §§ § rk J1 (On rk ) · ¸ k 1 ¨ ¨ 't 1  ¨ g V F (k  i)  ¨ ¨¨ ¨ lJ (O l ) ¸¸ ¸¸ ¦ i ni n 1 n ¨ ¹ ¹i 0 ©© ©



2D rk J1 (On rk )

On (lJ1 (Onl )) 2

k

N

i 0

j 0



¦ gi ¦ J ji V ji 

· N  't ¦ gi1V ji1 F j (i  i1 ) ¸ ¦ ( m  0.5) u ¸m 0 i1 0 ¹ u G11 (m, k  i ) , (n 1, N ). i 1

Chapter 3

126

Equation (3.2.9) and system (3.2.10) are solved by the method of iterations according to the algorithm described in paragraph §3.1, until the following conditions are met: (i ) (i 1) (3.2.11) rk (i )  rk (i 1)  H , max Vnk  Vnk H. 0d n d N





where i is the number of iterations and H is equal to 10-6. When performing calculations for the moment of time (k  1)'t , we *(0) rk* , V n(,0k) 1 V n k , n 0, N , and take the zero approximation rk 1

*(1) by formula (3.2.9) we find rk 1 . From (3.2.10), we calculate V (1) for n , k 1

n

0, N and so on until condition (3.2.11) is satisfied in step k  1 .

Conditions (2.2.1) and (3.1.13) were considered for the effective application of the method of mechanical squaring and efficient operation of the algorithm. As test examples, the problems of impact of an absolutely rigid sphere with mass M = 0.25 with an initial velocity V0 0.01C0 on an aluminium half-space for which P / K 0.33 and the problem of impact of a steel spherical shell with mass M = 0.25 and thickness h / R 0.01 with an initial velocity V0 0.05 on the water surface were selected. The results of the calculations coincide with the results in [130, 166]. The above problems are special cases of the problem of impact of an elastic spherical shell on an elastic half-space. To obtain the problem of the impact of an absolutely rigid sphere on an elastic medium, the functions G ij ( n , t ) { 0 in (2.2.17)–(2.2.19) and G ij (n, t ) { 0,  i, j 1; 2 in (2.2.20)–(2.2.22) must be set equal to zero; to obtain the problem of impact of the steel spherical shell on the water surface, it is necessary to take the speed of sound in the medium equal C 02 to zero, P 0 .

12.2 and set the shear modulus equal

§3.3. Impact of a hard cylinder on an elastic layer The numerical implementation of the solution system [52, 200] of equations (2.1.52), (2.1.54) is based on the combined application of quadrature and reduction methods. The integrals in (2.1.52), (2.1.54) were

Numerical implementation of the solution algorithm

127

calculated by the symmetric quadrature Gregory fifth-rank formula for equidistant nodes [189]. The Cauchy problem for the differential equation (2.1.54) was solved by the fourth-rank Adams method with local truncation error O ( ' t 6 ) [113], where ' t is the length of the partial intervals into which we divide the segment [0, T ] . The beginning of the solution was calculated in steps ' t / 1 6 . The rank of reduction N was chosen for reasons of practical convergence. To smooth the oscillations that occur when adding a finite number of members of a series, as well as Gibbs’ phenomena, we used the averaging operation [139], which reduces the sum of a finite number of members of a trigonometric series to the product of members of a finite sum by V n , being the Lancosh multipliers (3.1).

§3.4. Impact of fine elastic cylindrical shells on an elastic layer The numerical implementation [22] of the solution system of equations (2.2.50), (2.2.52) is based on the combined application of quadrature and reduction methods. The integrals in (2.2.50), (2.2.52) were calculated by the symmetric quadrature Gregory fifth-rank formula for equidistant nodes [189]. The Cauchy problem for the differential equation (2.2.52) was solved by the fourth-rank Adams method with local truncation error

O (' t 6 ) [113], where 't is the length of the partial intervals into which we divide the segment [0, T ] . The beginning of the solution was calculated in steps 't /16. As well as cylindrical dies, the order of reduction N was chosen and V n , being the Lancosh multipliers, were used to smooth the oscillations [139] (3.1). Giving the shear modulus ȝ zero value, we have, as a partial case, the problem of the shell hitting the surface of the fluid layer. If functions and

Gi

G i , (i 1; 2) are equal to zero, as another partial case, we have the

problem of hitting the cylindrical stamp on the elastic layer.

Chapter 3

128

§3.5. Three-dimensional problem – quasi-static elasticplastic formulation The finite difference method [189] with variable steps of partitioning along the axes Ox (N elements), Oy (M elements) and Oz (K elements) were used to calculate compact specimens from RPV reactor steel 15H2NMFA. The step between the break points was the smallest near the crack tip and at the specimen boundaries. The division over time is uniform. The characteristic cell size near the crack tip is equal to the average grain size of the test metal (0.05 mm). The intensities of stresses and strains were determined for each cell from the solution of the threedimensional problem and were compared with the solution of the problems of plane stress and plane strain states. The calculation scheme was compared with the known theoretical results calculated for the cell centre on the basis of classical one-membered asymptotic dependencies [176, §1.2].

§3.6. Dynamic elastic-plastic formulation The finite difference method [189] with a variable splitting step was used for the calculations. The characteristic size

Uuc

(index ‘uc’ from the

English – utilized cell [227]) of the cell of the breakdown grid around the crack tip (cells within a radius of 1–2 mm) was taken equal to the average grain size of the test metal and was not more than 0.05 mm. The division over time is uniform. The step between the points of split by coordinates was the smallest in the vicinity of the crack tip and at the boundaries of the specimen.

§3.6.1. Problems of plane stress and strain states Calculated strain fields were used to determine the plastic and elastic components of deformation energy [56].

U ije

1 H ijeV ij dxdy, ³³ 2

U ijp

1 H ijpV ij dxdy (i, j ³³ 2

6

(3.6.1)

x, y ).

6 The use of finite differences [189] with variable partition steps for wave equations is justified in [103], and the accuracy of calculations with

Numerical implementation of the solution algorithm





an error of no more than O ( ' x ) 2  ( ' y ) 2  ( ' t ) 2 , where

'y

129

'x ,

and ' t are increments of spatial variables x and y and time variable

t.

§3.6.2. Three-dimension problem of stress-strain state The use of finite differences [189] with a variable partition step for wave equations is justified in [103], and provides the accuracy of calculations with an error of no more than





O ( ' x ) 2  ( ' y ) 2  ( ' z ) 2  ( ' t ) 2 , where ' x ,

'y , ' z

and

' t are increments of spatial variables x, y and z and time variable t.

§3.7. Crack growth in a dynamic elastic-plastic formulation We define the criteria that the maximum breaking stresses are provided directly on the continuation of the crack tip as criteria A and the generalized local

VTT

criteria for brittle fracture as criteria B.

§3.7.1. In the problems of plane stress and strain states according to the criteria A The following algorithm was used to determine the actual crack length l at each time point. If the largest values of any stresses did not appear in cell 1 (figure 1.6b) but in cell 2 (figure 1.6b), then the crack moved up one cell. Cell 2 became cell 1. All stresses were recalculated for the increased crack. This was done until all the greatest stresses were in cell 1.

§3.7.2. In the problems of plane stress and strain states according to the criteria B The crack length l at each time point was determined by the local criterion of brittle fracture. If in cell 1 (figure 1.6b) the highest main stress

V1

has reached or exceeded the level of critical stresses of brittle

destruction SC (N )

1 2

¬ªC1  C2 exp( Ad N ) ¼º

, provided that the

Chapter 3

130

effective

stress

V eff

V i  V 02

(temperature-dependent

V 02 (T ) a  c(T  273)  b exp( h(T  273)) ) is not negative, it is assumed that this cell collapses [143, 228], the length of the crack increases to the height of this cell and the grid is rearranged so that near the tip of the crack was again cell 1. Parameters [143] a, c, b, h, and

Ad

C1, C2 ,

of these dependencies characterize the properties of the studied

polycrystalline material.

§3.7.3. In the three-dimension problem of the stress-strain state according to the criteria A The following algorithm was used to determine the actual crack length l at each moment of time. If the highest values of normal tensile stresses

V xx

are found not in the cells of row 1 (figure 1.7b) but of row 2, and

when this condition of failure corresponds to the thickness of the row cells, which together make at least 65 percent of the thickness of specimen H, then the crack moved one row of cells up. The cells of row 2 became the cells of row 1, the breakdown grid was changed, the cells were renumbered and all stresses were recalculated for the enlarged crack. This was done until the above-mentioned maximum stresses were recorded throughout the cells of row 1.

§3.7.4. In the three-dimension problem of stress-strain state according to the criteria B A local criterion of brittle fracture was used to determine the actual crack length l at each moment of time. If in any cell of row 1 (figure 1.7b) the main stress V1 has reached or exceeded the level of critical stresses of brittle destruction [143, 228] SC (N ) provided that the effective

1 2

, ª¬C1  C2 exp( Ad N ) º¼ stress V eff V i  V 02 ,

V 02 (T ) a  c(T  273)  b exp( h(T  273))

is not negative, it

is considered that this cell is destroyed. When this condition of brittle fracture is met by cells of row 1 in thickness, which together are not less than 65 percent of the thickness of the specimen H, then the crack moved one row of cells up. The cells of row 2 became the cells of row 1, the

Numerical implementation of the solution algorithm

131

breakdown grid was changed, the cells were renumbered and all stresses were recalculated for the enlarged crack. This was done until the abovementioned maximum stresses were recorded throughout the cells of row 1.

§3.8. Considering the process of unloading of the material in problems in dynamic elastic-plastic formulation §3.8.1. Problems of plane stress and strain states The process of unloading of the material took place according to the following algorithm. If in any cell the absolute value of stress became less than the maximum value, then the plastic deformations stop increasing and the hardening of the material stops. Again, plastic deformations begin to increase, and the hardening of the material continues when the absolute value of stresses exceeds the maximum values.

§3.8.2. Three-dimension problem of stress-strain state To consider the process of unloading of the material in the case of the spatial problem of the stress-strain state, the same algorithm was used as for plane problems in §3.8.1.

CHAPTER 4 NUMERICAL RESULTS ANALYSIS

§4.1. Impact of fine elastic cylindrical shells on an elastic half-space a

Throughout this paragraph the calculations consider the penetration of steel cylindrical shell with the following parameters:

E0

20 u 1011 dyn / cm 2 ,

U0

7.7 g / cm3. Below are the results of calculations of the impact of

K 0 17 u1011 dyn / cm 2 ,

Q0

0.28 ,

steel thin shells of different thickness and mass with an elastic half-space: a) aluminium with elastic characteristics: E

K

7.5 u 1011 dyn / cm 2 , Q

0.34 , U

7 u1011 dyn / cm 2 ,

2.7 g / cm3 . The results are

illustrated in Figures 4.1–4.34. b) steel with elastic characteristics the same as the shell; the results are illustrated in Figures 4.35–4.55. Figures 4.1–4.7 show the time dependencies of the normal stresses V zz (t,0,0) in the half-space at the point of initial contact (Figure 4.1), the reaction force P(t) of the elastic half-space (Figure 4.2), the normal displacement of the initial touch points in the half-space u z (t , 0, 0) (Figure 4.3), the speed of penetration of the shell into the elastic medium

vT (t) (Figure 4.4), the movement of the middle surface of the shell w0 (t,0) at the frontal point (Figures 4.5 and 4.6) and the radius of the contact area of the shell with the half-space (Figure 4.7).

Numerical results analysis

Fig. 4.1 Stresses

133

V zz (t ,0,0)

Fig. 4.2 Reaction power

P(t )

Fig. 4.3 Displacement uz (t,0,0)

134

Chapter 4

Fig. 4.4 Velocity VT (t )

Fig. 4.5 Displacement w0 (t,0)

Fig. 4.6 Displacement w0 (t,0)

Numerical results analysis

135

Fig. 4.7 Contact area

In these figures, curves 1, 2, 3, 4 correspond to the cases of penetration of the steel shell with an initial impact velocity V0 0.001 , relative thickness h / R 0.01 and mass M equal to 0.01, 0.03, 0.05 and 0.1 respectively. The figures show that the lighter the body, the less normal stresses and the faster they reach a maximum. The same is true for normal displacements

uz , reaction forces P and displacements w0 . As the mass

of the shell increases, the boundary of the contact area increases, and the penetration rate of the drummer vT decreases more slowly.

Fig. 4.8 Stresses

V zz (t, x,0)

Fig. 4.9 Displacement uz (t , x,0)

Fig. 4.10 Displacement w0 (t ,T )

Fig. 4.11 Stresses V zz (t , x, 0)

Chapter 4

136

Fig. 4.12 Displacement uz (t, x,0)

Fig. 4.14 Stresses

V zz (t, x,0)

Fig. 4.13 Displacement w0 (t,T )

Fig. 4.16 Displacement w0 (t,T )

Fig. 4.15 Displacement uz (t , x,0)

Fig. 4.17 Stresses V zz (t , x,0)

Fig. 4.18 Displacement u z (t , x,0)

Numerical results analysis

137

Fig. 4.19 Displacement w0 (t,T )

Figures 4.8–4.19 show the normal displacements normal stresses

V zz (t , x, 0)

uz (t , x,0)

and

arising at the points of the half-space

surface in the contact zone of the shell and the elastic medium, and the displacements

w0 (t,T )

of the points of the middle surface of the shell at

fixed points in time. The initial impact speed is V0 0.001 . In Figures 4.8–4.16, curves 1, 2 and 3 correspond to the moments of time t1 0.3, t2 0.6, t3 0.9 , respectively. In Figures 4.17–4.19, curves 1, 2 and 3 correspond to the moments of time t1 0.2, t2 0.4, t3 0.6 , respectively. The results shown in Figures 4.8–4.10 correspond to a shell mass M 0.001 , those in Figures 4.11–4.13 to a shell mass M 0.03 , in Figures 4.14–4.16 a shell mass M 0.05 and in Figures 4.17–4.19 a shell mass M 0.1 . From Figures 4.8, 4.11, 4.14 and 4.17 it can be seen that the normal stresses are different from zero at the area of contact and take the maximum value at the point of initial contact at each moment of time. From Figures 4.9, 4.12, 4.15 and 4.18, it can be seen that the normal displacements u z (t , x, 0) decrease from the centre of the contact area to its boundary, not reaching zero. The normal displacement of the half-space * u z (t , x* , 0) at the point of the boundary of the contact area x is the amount of sediment (rise) of the medium. From Figures 4.10, 4.13, 4.16 and 4.19 it can be seen that with increasing mass of the shell, the displacements w0 (t,T ) of the middle surface of the shell increase.

138

Chapter 4

Fig. 4.20 Stresses

V zz (t,0,0)

Fig. 4.21 Reaction power P(t )

Fig. 4.22 Displacement uz (t,0,0)

Numerical results analysis

139

Fig. 4.23 Velocity VT (t )

Fig. 4.24 Displacement w0 (t,0)

Fig. 4.25 Contact area

Figures 4.20–4.25 illustrate normal stresses V zz (t,0,0) (Figure 4.20), reaction power P(t) of the half-space (Figure 4.21), normal displacements (Figure 4.22), penetration rate vT (t) (Figure 4.23), displacement of the shell w0 (t,0) at the frontal point (Figure 4.24) and radius of contact areas

x* (t ) (Figure 4.25) that change over time and occur as a result of impact of the steel shell with a relative thickness h/R = 0.02 with an initial impact velocity V0 0.001 .

140

Chapter 4

Fig. 4.26 Stresses

Fig. 4.27 Displacement uz (t, x,0)

V zz (t, x,0)

Fig. 4.28 Displacement w0 (t ,T )

Fig. 4.29 Displacement uz (t, x,0)

Numerical results analysis

Fig. 4.30 Displacement uz (t, x,0)

Fig. 4.32 Stresses

Fig. 4.33 Displacement uz (t, x,0)

141

Fig. 4.31 Displacement w0 (t,T )

V zz (t, x,0)

Fig. 4.34 Displacement w0 (t,T )

In these figures, curves 1, 2 and 3 correspond to the variants of the impact of the shell with a mass M equal to 0.1, 0.2 and 0.5, respectively. The analysis of the figures shows that the normal stresses, as well as for

Chapter 4

142

impact variants, the results of which are shown in Figures 4.1–4.7, are smaller, the easier the penetrating body; however, in contrast to the above options, normal stresses V zz (t,0,0) reach one maximum instead of two. The same is true for normal displacements u z (t , 0, 0) , reaction forces P and displacements w0 (t,T ) . As the mass of the shell increases, the boundary of the contact area increases and the rate of penetration of the shell vT (t) decreases more slowly. In Figures 4.26–4.34, curves 1, 2 and 3 correspond to the moments of time t1 0.3, t2 0.6, t3 0.9. These figures show graphics of normal stresses V zz (t , x, 0) and displacements w0 (t,T ) at fixed moments of time, while the variables x * x  [0, x ( t )[, T  [0, S / 4[ .

and

T

change

Fig. 4.35 Stresses V zz (t ,0,0)

Fig. 4.36 Reaction power P(t )

in

intervals

Numerical results analysis

Fig. 4.37 Displacement uz (t ,0,0)

Fig. 4.38 Velocity VT (t )

143

144

Chapter 4

Fig. 4.39 Displacement w0 (t,0)

Fig. 4.40 Displacement w0 (t,0)

Fig. 4.41 Contact area

Numerical results analysis

145

Fig. 4.42 Stresses V zz (t , x,0)

Fig. 4.43 Displacement w0 (t,T )

Analysing the graphs presented in Figures 4.26, 4.27, 4.29, 4.30, 4.32 and 4.33, we can see that with increasing stiffness of the shell at the contact area the normal displacements remain compressive, and normal displacements uz reach a maximum at the point of initial contact, decreasing when approaching the boundaries of the contact area.

146

Chapter 4

Fig. 4.44 Stresses V zz (t , x,0)

Fig. 4.45 Displacement w0 (t ,T )

Fig. 4.46 Stresses V zz (t , x,0)

Numerical results analysis

147

Fig. 4.47 Displacement w0 (t ,T )

Figures 4.35–4.47 show graphs of changes of stress V zz (t , 0, 0) , reaction force P(t), displacement u z (t , 0, 0) , velocity vT (t) and contact area radius

x*(t ) over time, as well as normal stresses V zz (t , x, 0) distributed along the contact area and displacement w0 (t,T ) of the middle shell surface points at fixed moments of time t1 0.3, t2 0.6, t3 0.9 , arising from the impact of steel shells with an initial impact velocity V0 0.001 , relative thickness h/R = 0.01 and different mass M. In Figures 4.35–4.41, curves 1, 2 and 3 correspond to the variants of the impact of the shell with a mass M equal to 0.03, 0.05 and 0.1, respectively. In Figures 4.42–4.47, curves 1, 2 and 3 correspond to the moment of time t1 0.3 ; 2 – t2 0.6 ; 3 – t3 0.9 . It should be noted that the radius of the contact area x*(t) depending on the time reaches two maxima.

148

Chapter 4

Fig. 4.48 Stresses V zz (t ,0, 0)

Fig. 4.49 Reaction power P(t )

Fig. 4.50 Displacement uz (t ,0,0)

Numerical results analysis

Fig. 4.51 Velocity VT (t )

Fig. 4.52 Displacement w0 (t,0)

Fig. 4.53 Contact area

Fig. 4.54 Stresses

V zz (t, x,0)

149

Chapter 4

150

Fig. 4.55 Displacement

w0 (t,T )

Figures 4.48–4.55 show graphs of changes in the parameters

V zz (t ,0,0) , u z (t , 0, 0) , P(t ), vT (t) and w0 (t,0) over time and the values of functions V zz (t , x, 0) , w0 (t,T ) , at fixed points of time t1 0.3, t2 0.6, t3 0.9 . Curves 1, 2 and 3 in these figures correspond to the moments of time

t1 , t2 , t3 . In addition, Figures 4.68 and 4.69 show

for comparison the values of the corresponding parameters for the impact of an absolutely rigid cylinder with a mass M 0.03 with an initial impact velocity V0 0.001 and the results with respect to the impact of a steel shell with relative thickness h / R 0.02 and mass M 0.1 with an initial impact velocity V0 0.001 on the compressible fluid surface. In the following, we will call the results of the study of the impact of shells on the elastic half-space, shown in Figures 4.1–4.55, case A; elastic shell with mass M = 0.1 on the acoustic half-space, case B; and solid body with mass M = 0.03, case C. From Figures 4.1, 4.8, 4.11, 4.14, 4.17, 4.20, 4.26, 4.29, 4.32, 4.35, 4.42, 4.44, 4.46, 4.48, 4.54, 4.68a, and 4.69a, it follows that the maximum stresses

V zz

for the solving plane problem occupy an intermediate

position between cases B and C; the maximum normal displacements of the half-space

uz

below the frontal point of the shell in case A are much

smaller than in cases B and C (this can be seen from Figures 4.3, 4.9, 4.12, 4.15, 4.18, 4.22, 4.27, 4.30, 4.33, 4.37, 4.50, 4.68c and 4.69b). The same is stated for the reaction power of the medium P (Figures 4.2, 4.21, 4.36, 4.49, 1.3.68g and 4.69c). The rate of penetration of the shell into the elastic medium

vT

changes more smoothly than in cases B and C

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151

(Figures 4.4, 4.23, 4.38, 4.51, 4.68b and 4.69d). From the curves shown in Figures 4.5, 4.6, 4.10, 4.13, 4.16, 4.19, 4.24, 4.28, 4.31, 4.34, 4.39, 4.40, 4.43, 4.45, 4.47, 4.52, 4.55 and 4.69d, it follows that the deflection of the shell in the front point when hitting an elastic medium significantly exceeds the case of B.

§4.2. Impact of fine elastic spherical shells on an elastic half-space Throughout the calculations below, the penetration of a steel spherical shell with the following elastic parameters was considered:

20 u 1011 dyn / cm 2 ,

E0

K0

17 u 1011 dyn / cm 2 ,

Q0

0.28 ,

U0 7.7 g/cm3. Figures 4.56–4.67 illustrate the results of calculations of the impact process of steel thin shells of different thickness and mass with the aluminium half-space. Elastic characteristics of the aluminium halfspace

K

are

as

7.5 u 1011 dyn / cm 2 , Q

7 u1011 dyn / cm 2 ,

follows:

E

0.34 , U

2.7 gɝ/cm3 .

Fig. 4.56 Stresses V zz (t ,0,0)

152

Chapter 4

Fig. 4.57 Displacement u z (t ,0,0)

Fig. 4.58 Reaction power P(t )

Numerical results analysis

Fig. 4.59 Velocity VT (t )

Fig. 4.60 Displacement w0 (t,T )

Fig. 4.61 Contact area

153

154

Chapter 4

Fig. 4.62 Stresses

V zz (t, r,0)

Fig. 4.63 Displacement uz (t, r,0)

Fig. 4.64 Stresses

V zz (t, r,0)

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155

Fig. 4.65 Displacement u z (t , r , 0)

Fig. 4.66 Displacement w0 (t,T )

Fig. 4.67 Displacement w0 (t,T )

Figures 4.56–4.61 show the time dependencies of normal stresses

V zz (t,0,0) (Figure 4.56) and normal displacements u z (t , 0, 0) (Figure 4.57) at the point of initial contact, the reaction force P(t) of the elastic half-space (Figure 4.58), the velocity of shell penetration into the elastic medium vT (t ) (Figure 4.59), the movement of the middle surface of the shell w0 (t ,0) at the frontal point (Figure 4.60) and the radius r*(t ) of the area of contact of the shell with the half-space (Figure 4.61). In these figures, curves 1 and 2 correspond to the following cases of penetration of the steel shell:

156

1) initial impact velocity V0 and mass M 0.03 ; 2) initial impact velocity V0

Chapter 4

0.001 , relative thickness h / R

0.01

0.0005 , relative thickness h / R

0.02

and mass M 0.1 . The graphs show that the higher the speed the larger normal stresses appear in the medium and the faster they reach their maximum over time; the same can be said for normal displacements uz (t ,0, 0) , reaction force P(t), displacement w0 (t , 0) and radius of contact area r*(t) . Figures 4.62–4.67 show the normal stresses (option a) V zz (t , r ,0) and normal displacements u z (t , r , 0) that occur at the points of the half-space surface in the contact zone of the shell and the elastic medium, and the displacement w0 (t,T ) of the points of the middle surface of the shell at fixed moments of time. In figures 4.62, 4.63 and 4.66, curves 1, 2 and 3 correspond to the moments of time t1 0.3, t2 0.6, t3 0.9 .

Fig. 4.68

Numerical results analysis

Fig. 4.69

157

158

Chapter 4

Fig. 4.70

Numerical results analysis

159

In figures 4.64, 4.65 and 4.67 (option b), curves 1 and 2 correspond to the moments of time t1 0.6, t2 1.5 . The graphs shown in Figures 4.62– 4.65 show that the normal stresses V zz (t, r,0) decrease from the centre of the contact area towards its boundary and the normal displacements u z (t , r , 0) in the case of compressive normal stresses decrease from the centre to the contact zone boundary. Figure 4.70 shows the numerical results for the case of impact of a steel shell with mass M 0.03 with an initial velocity V0 0.0005 on the surface of the corresponding acoustic medium: normal stresses (Figure 4.70a) and displacement (Figure 4.70b), the reaction force of the acoustic half-space P (Figure 4.70c), the rate of penetration of the shell VT (Figure 4.70d) and radial movements of the shell at the frontal point (Figure 4.70e). We name the results of the study of the impact of shells on the elastic half-space, as shown in Figures 4.56–4.57, case A, and elastic shell mass M 0,03 on the acoustic half-space as case B. It follows from figures 4.56, 4.62, 4.64 and 4.70a that the maximum stresses V zz for the solving axisymmetric problem (case A) are greater than in case B; the maximum normal displacements of the half-space

uz

below the frontal point of the shell in case A are much smaller than in case B (this can be seen from Figures 4.57, 4.63, 4.65 and 4.70b). The same is stated with respect to the reaction strength of the medium P (Figures 4.58 and 4.70c); the rate of penetration of the shell into the elastic medium vT in case A changes more smoothly than in case B (Figures 4.59 and 4.70d). From the curves presented in Figures 4.60, 4.66, 4.67 and 4.70e, it follows that the deflection of the shell at the frontal point when hitting an elastic medium significantly exceeds the case of B.

§4.3. Impact of a hard cylinder on an elastic layer An aluminium layer was chosen as an example [52, 200]. Figures 4.1.1 and 4.1.2 show the results of calculations when V0 0.0002 , M 0.001 ,

l 0.6 , T 0.05 , h 0.4 , d 0.02 and 't 4.166667E–5. If we give the shear modulus ȝ a value of zero, we obtain, as a partial case, the problem of the impact of the shell on the surface of the fluid layer.

Chapter 4

160

In Figure 4.71, the dashed line is the time dependence of the component of the displacement vector

uz

in the impact problem of a rigid

cylinder with a flat platform in an elastic formulation (the first problem). In the elastic-plastic model, the axis Oy coincides with the axis O z c . In this figure, the solid line corresponds to the component of the displacement vector

uz , which is denoted here as uz , for the problem of

non-stationary interaction of the stamp and the layer in elastic-plastic formulation (second problem) with the following values: material hardening factor K * 0.05; L 600 mm ; h 400 mm ; d 2 m m ; M 80 ; N 101 ; p01 10.1 MPa ; p02 4 .04 MPa .

Fig. 4.71. Displacement vector component uz

The smallest step of the partition grid was near the upper surface and was equal to 0.01 mm ( 'xmin 0.01 mm ; 'ymin 0.01 mm (only the first three layers of partition)), T 50 $ C . The components of the displacement vector uz calculated in the centre of the contact zone at the point (0, 0) for the first problem and the point (0.01, 399.99) for the second problem are compared. The percentage deviation of the values of the displacement vector uz obtained for the first and second tasks are shown in Figure 4.72. The time interval at which this deviation does not exceed 8% is found. The results of solving plane impact problems of rigid cylinders with a flat area in an elastic formulation and non-stationary interaction in an

Numerical results analysis

161

elastic-plastic mathematical formulation at the elastic stage coincide well. The use of the elastic-plastic mathematical formulation allows us to: 1. Determine the stress-strain state at the points defined by the partition grid of the computational domain and not only on the surface of this domain. 2. Obtain a reliable description of the development of plastic deformations. The stage corresponding to plasticity is a continuation of the elastic stage. 3. Authentically determine the fracture toughness KIc . 4. Check and calibrate the first steps in the time of solving problems in elastic-plastic dynamic formulation; when the deformation process is elastic, it is convenient to use the solution of the corresponding elastic problem.

Fig. 4.72. Deviation of results Below are the results of the problems of impact of a cylindrical stamp without a flat area, that is, when d 0 , on an elastic layer. The numerical implementation of the solution system of equations (2.3.6) and (2.3.7) is based on the combined application of quadrature and reduction methods. The integrals in (2.3.5)–(2.3.7) were calculated by the symmetric quadratic Gregory formula for equidistant fifth-rank nodes [189]. The Cauchy problem for differential equation (2.3.7) was solved by the fourth-rank Adams method with local truncation error O('t 6 ) [113], where ' t is the length of the partial intervals into which we divide the segment [0, T ] . The beginning of the solution was calculated in steps ' t / 1 6 . The rank of reduction N was chosen for reasons of practical convergence. To

162

Chapter 4

smooth the oscillations that occur when adding a finite number of members of a series, as well as Gibbs phenomena, we used the averaging operation [139], which reduces the sum of a finite number of members of a trigonometric series to the product of members of a finite sum by V n , the Lancosh factors (3.1). The first example was the impact over time T 0.6 of an absolutely rigid cylinder of radius R 1 0 0 , mass M 0.03 with an initial velocity V0 0.003 on an elastic layer of reactor steel (15X2NMFA, 10GN2MFA, 2Cr-Ni-Mo-V) with thickness h 2 . In Figures 4.73–4.76, the results corresponding to the case of reactor steel P 0.5357 K , l 1 0 are given by solid lines; the dotted lines correspond to the case of the aluminium layer P 0.3582K , l 1 0 .

Fig.4.73. Contact area

x*

Fig.4.74. Normal stresses V zz

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163

Fig.4.75. Displacement u zz

Fig.4.76. Velocity

vT

Figures 4.73–4.76 show the time dependencies of the contact zone x*, normal stresses V zz and normal displacements uz at the point of initial contact, and the rate vT of penetration of the body into the environment. As a second example, the problem of impact of an absolutely rigid cylinder on the surface of an aluminium elastic layer was solved in comparison with the impact on an elastic aluminium half-space, when V0 0.003 , P 0.3582K , M 0.03 , l 10 , T 0.6 and h 2 . Solid lines in Figures 4.77–4.80 show the time dependencies of normal stresses V zz , normal displacements uz at the point of initial contact of the cylinder, the reaction force of the elastic layer P and the rate of penetration of the body into the medium vT .

164

Chapter 4

Fig. 4.77. Normal stresses V zz For comparison, dashed lines show results from a similar problem of hitting a rigid cylinder on an aluminium half-space.

Fig. 4.78. Displacement uz

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165

Fig. 4.79. Reaction power P

Fig. 4.80. Velocity vT

At the initial stage of the impact (at a dimensionless time of less than 0.07 – this is the time during which the shock wave CS does not reach the lower limit of the layer), the solutions for the strip and half-space coincide. Since in (2.3.1) in space the Laplace transform was decomposed into power series, the rather stable time process of impact was actually modelled, but the obtained results differ significantly from the case of impact on the half-space surface, when there are no waves reflected from the lower boundary. It is also worth noting that at less than 30% of normal stresses V zz , the movement uz at the point of initial contact in the case of an impact on half the space is five times less. Figures 4.81–4.84 show the results of calculations of the impact of a rigid cylinder with a plane area d 0.05 (Fig. 1.3), the mass M 0.25 on the surface of the elastic aluminium P 0.3582K layer with a thickness

Chapter 4

166

h 2 with speed V0

0.003 . Figures 4.81–4.84 show the normal

stresses Vzz (Fig. 4.81) and normal displacements uz (Fig. 4.82) at the central point of the contact area, the reaction force of the layer P (Fig. 4.83) and the penetration rate of the impact V into the elastic medium (Fig. 4.84).

Fig.4.81. Normal stresses

V zz

Fig. 4.82. Displacement uz

Fig. 4.83. Reaction power P

Numerical results analysis

167

Fig. 4.84. Velocity vT

The solid line corresponds to the case of impact of a rigid cylinder with a plane area with length 2d on the surface of the elastic layer (Case 1). For comparison, dashed and dotted lines are shown corresponding to the problems of impact of the cylinder with a length 2d of the plane area on the elastic half-space (Case 2) and a rigid cylinder without a plane area on the surface of the elastic layer (Case 3), respectively. Maximum normal stresses occur in Case 3. In Case 2, the normal stresses are much lower than in Cases 1 and 3, and in Case 3 the normal stresses are higher than in Case 1 on 30%.

§4.4. Impact of fine elastic cylindrical shells on the elastic layer A steel shell [22] and an aluminium layer were chosen as examples. Figures 4.85–4.89 show the results corresponding to the case V0 0.003 ,

P 0.3582K , h1 / R 0.01 , M 0.03 , l 10, T 0.6 and h 2. As in the case of the impact of the cylindrical stamp on the elastic layer in (2.4.1) in the space of the Laplace transformant, decomposition into step rows was performed and, in fact, a rather time-stable impact process was modelled, because the decomposition procedure was made in the space of the Laplace transformant near neighbourhood point t 0 .

168

Chapter 4

Fig.4.85 Normal stresses V zz

Fig. 4.86. Displacement uz

Numerical results analysis

Fig. 4.87. Reaction power P

Fig. 4.88. Velocity vT

169

170

Chapter 4

Fig. 4.89. Displacement w0

Lancosh multipliers were taken in the form (3.1). Figures 4.85–4.89 show the time dependencies of the normal stresses V zz , normal displacements uz at the point of initial contact, the reaction force of the elastic layer P, the rate of penetration of the body into the environment vT and the movement w 0 of the middle surface of the shell at the frontal point.

§4.5. Three-dimensional problem – quasi-static elasticplastic formulation Numerical studies [37, 39] were performed for bars with a compact profile made of 15H2NMFA reactor steel. Figures 3.90–3.93 show the results of calculations for bars 60 mm long, 10 mm wide, 50 mm thick and with a depth of notch in the middle of 3 mm and a hardening factor of the material K * 0.05 . Distance between supports was 40 mm and coefficients were p01 8 MPa , p02 10 MPa and temperature T 50 $C. Figures 4.90–4.93 show graphs of the distribution of normal stresses in MPa that occur around the top of the notch-crack of a three-dimensional beam with a compact profile in a plane that is 8.7 mm away from the side surface of the beam. Figure 4.93 shows graphs of the values of the Odquist parameter N in MPa m. The solid and dashed lines in the figures refer to the solution of problems in quasi-static and dynamic formulations. The

Numerical results analysis

171

Odquist parameters are calculated in the plane z 41,3 around the top of the crack of the spatial beam with a compact profile. Vx x

Vyy

1200

1200

800

800

400

400

0

2, 4

32, 8

63, 2

Fig.4.90. Stresses

93, 6

124, 0

0

KI

2, 4

Vzz

N 0, 3

800

0, 2

400

0, 1 2, 4

32, 8

63, 2

Fig.4.92. Stresses

V zz

63, 2

Fig.4.91 Stresses

V xx

1200

0

32, 8

93, 6

124, 0

KI

0

2, 4

32, 8

63, 2

93, 6

KI

124, 0

Vyy

93, 6

124, 0

Fig.4.93 Odquist parameter

KI

N

As can be seen from Figures 4.90–4.93, the stress values and Odquist parameters differ significantly from each other. In the quasi-static formulation, the stresses increase rapidly when the SIFs K I  45 MPa m and do not change after that. In addition, when

K I ! 120 MPa m , in a quasi-static setting, it is necessary to significantly increase the number of iterations (the process begins to coincide poorly). In the dynamic setting, starting with K I 93.5 MPa m , the stress values fluctuate markedly, which indicates the instability of the deformation process and the need to take into account the growth of cracks and unloading of the material. The quasi-static model cannot be used to adequately model the processes of concentration of plastic deformations and stresses and the destruction of specimens (compact, Charpy, etc.). The solution of the three-dimensional problem [23, 31] of the stressstrain state with respect to a material with a rectangular cross-section with a notch-crack in the middle (compact profile) for the determination of the destruction toughness in three-point bending in a dynamic elastic-plastic setting allows us to determine the fields of plastic deformation and stresses more precisely than the solution of a quasi-static elastic-plastic spatial

172

Chapter 4

problem.

Fig. 4.94. Mean stresses

V

Fig. 4.95. Odquist parameter

N

Figures 4.94 and 4.95 show (solid lines) the temperature T dependencies of the mean stress V and Odquist parameter N in the region of the crack tip in the plane z 41.3 mm , when the SIF value was equal to K I 72.3 MPa m ; dashed and dashed with a triangle lines correspond to the cases of dynamic elastic-plastic formulation of plane stress and strain states, respectively [21, 28]. The difference between the values of the mean stresses V and the Odquist parameter N in the spatial quasi-static formulation and the plane problem of the stress state in the dynamic formulation exceeds 200 per cent. However, despite such a large quantitative difference, the qualitative aspect of the problem of the process of material destruction is captured well by the quasi-static formulation. With increasing temperature, we observe an obvious increase in plastic deformation.

Numerical results analysis

173

§4.6. Problems in dynamic elastic-plastic formulation Compact specimens of 15H2NMFA reactor steel (Young’s module

E 2.15u1010 kg m2

Poisson’s

ratio

Q 0.272727 ,

density

3

U 7700 kg m ) were selected for calculations. For all calculations below was realised such procedure of precision check. The mean stresses V in cells near the crack tip on the stage when there are no plastic deformations were compared using classical monomial asymptotic dependencies [176, §1.2] and expression (2.5.6) in the cells of the discretized problem and the solution of the problem is purely linearly elastic, were compared with those mean stresses V calculated for the cell centre based on classical monomial asymptotic dependencies [176, §1.2] using expression (2.5.6) for the SIF of the elastic solution. For cells 1 and 6 (see Figure 1.6b), when x 0.01 mm, y 3 r 0.04 mm , the difference did not exceed 0.3%. In all cases below, variable partitioning steps along the Ox (M elements), Oy (N elements) and Oz (K elements) axes were used in the finite differences method to calculate the compact specimen of two- and three-dimensional models.

§4.6.1. Plane stress state The four figures below show the results [21] of calculations of Charpy specimens with a length of 60 mm, a width of 10 mm, a thickness of 50 mm and a depth of cut in the centre of the specimen 3 mm, with a coefficient strengthening K * 0.05 . The length between the bearing points is 40 mm. The time step was taken equal to 0.0005 s. The length of the contact zone was 5 mm.

Fig.4.96. Stresses near the crack’s top

Fig.4.97. Odquist parameter

174

Chapter 4

Figure 4.96 shows the graphs of stresses occurring in the crack apex of the two-dimensional Charpy specimen. Solid, solid with a triangle, solid with a cross and dotted lines correspond to the stresses V xx , V yy , yield strength after the material strengthening at temperature T VS and stress intensity Vi , respectively. Figure 4.97 shows how the Odquist parameter changes. Plastic deformations pulsate over time. The following figures show the results at the time when the value of the stress intensity factor K I 57.1 MPa m .

Fig.4.98. Main normal stress V1

Numerical results analysis

175

Fig.4.99. Main tangential stress W1

Figures 4.98 and 4.99 show the fields of the main normal and tangential stresses V1 and W1 , respectively. The zone of the largest plastic deformations is located in a radius of about 1 mm from the crack tip. The largest plastic deformations occur in the area of maximum main stresses.

Fig. 4.100. Mean stresses

V

Fig. 4.101. Odquist parameter

Figures 4.100 and 4.101 show the temperature dependencies of the mean stress V and the Odquist parameter N at the crack tip area, respectively. With increasing temperature, there is an increase in plastic deformations.

Chapter 4

176

§4.6.2. Plane strain state Figures 4.102–4.108 show the results of calculations of Charpy specimens with a hardening factor of the material K *

0.05 . Calculations

were made at the following parameter values: temperature T

50 $C ;

4 L 60 mm ; B 10 mm ; l 3 mm ; 't 5 ˜ 10 s ; a 5 mm ; p01 8 MPa ; p02 10 MPa ; M 60 ; N 77 . The smallest splitting step was 0.02 mm and the largest 2.6 mm ( 'xmin 0.02 mm;

'ymin

0.04 mm

'ymax

0.6 mm ).

(only

the

first

layer);

'xmax

2.6 mm ;

Figure 4.102 shows the graphs of stresses (MPa) that occur in the crack tip area of a two-dimensional compact specimen: lines 1–5 correspond to stresses V xx , V yy , V zz , yield strength after the material strengthening at temperature T

VS

and stress intensity

Vi . When the stress intensity factor

takes the value K I 60.1 MPa m , the maximum stresses occur in the cell, which is located above the crack line at a distance of one cell from the crack tip. At this point in the cell of the top of the crack, the stresses begin to fluctuate. When the value of the stress intensity K I K I* 93.5 MPa m , these oscillations become significant and the stress intensity, as shown in Figure 4.102, also begins to fluctuate.

Fig. 4.102. Stresses in cell 1 (Fig. 1.6b) on the continuation of the crack axis

Numerical results analysis

177

Fig. 4.103. Maximal stresses on the continuation of the crack axis

The study of the K I dependence of the maximum in absolute value of the stresses occurring on the extension x 0, y ! l , the crack axis (Figure 4.103) suggests that when the stress intensity becomes significant K I K I* , the maximum stresses occur in the second from the tip of cell 2. At this point, in the first cell affecting the tip the stresses begin to experience fluctuations. Most likely, these oscillations indicate a loss of stability of the deformation process in the area of the crack tip (cells 1, 4–6 in Figure 1.6b) and the probable beginning of its movement. Solid, solid with a triangle, dotted, solid with a square and solid with a cross lines refer to the stresses V xx , V y y , V x y , V zz , and yield strength after the material strengthening at temperature T V S , respectively. Figures 4.104 and 4.105 show graphs of normal stresses V xx and V yy occurring in the crack vertex region (line 1) in comparison with similar stresses occurring at the plane stress state (line 2).

178

Chapter 4

Fig. 4.104. Stresses

V xx

Fig. 4.105. Stresses

V yy

Calculation of the SIF K I dependencies in cell 1 of the Odquist parameter N (Figure 4.106), temperature dependencies of the mean stress V (Figure 4.107) and the Odquist parameter N (Figure 4.108) when K I K I0 57.1 MPa m  K I* for cases of plane strain state (solid lines) and plane stress state (dashed lines) shows that in all cases of plane stress state the level of accumulated plastic deformations is higher. The parameter N and the accumulated plastic deformations with increasing temperature grow monotonically with the development of the deformation process (see Figures 4.106 and 4.108). In case of plane strain state, at higher mean stress the plastic deformations accumulate less than in the case of plane stress state.

Numerical results analysis

179

With increasing temperature at a given subcritical load level, the level of mean stresses in cell 1 generally decreases.

Fig. 4.106. SIF K I dependence of the Odquist parameter 1.6b)

Fig. 4.107. Temperature dependence of mean stresses

V

N

in cell 1 (Fig.

in cell 1 (Fig. 1.6b)

The results of the three-dimensional elastic-plastic problem [23] in the planes z 32.3 mm and z 41.3 mm are presented by lines 2 and 3, respectively.

180

Chapter 4

Fig. 4.108. Temperature dependence of Odquist parameter

Fig. 4.109. Main normal stress

V1

N

in cell 1 (Fig. 1.6b)

near the tip of the crack

A study of the distribution of the main normal

V1

(figure 4.109) and

the main tangential stresses W1 (figure 4.110) in the vicinity of the crack tip length l 3 mm when K I K I0 found that the region of greatest normal and tangential stresses are concentrated in the area in front of the tip, and their extremums approximately coincide and are on the axis of the crack at a distance of 0.1 mm from its tip.

Numerical results analysis

181

Fig. 4.110. Main tangential stress W1 near the tip of the crack

p p Figures 4.111a–c show the fields of plastic deformations H xx , H yy p and H xy , respectively, which arise in the area of the crack tip at the value

of SIF K I 90.5 MPa m . The zone of the largest plastic deformations, as shown in Figure 4.111, is located within a radius of approximately 0.1 mm from the crack tip and much less than the corresponding area in the case of plane stress state. The largest plastic deformations occur in the region of maximal main stresses.

Fig. 4.111. Plastic deformations fields

The analysis of the results obtained above and their comparison with [21 and 23] shows that the values of the solution of the spatial problem

Chapter 4

182

[23] occupy an intermediate place between the values of the solutions of the problems of plane stress state [21] and plane strain state. The solution of the problem of plane strain state for a material with a rectangular cross-section with a crack in the middle (compact profile) to determine the fracture toughness in three-point bending in a dynamic elastic-plastic setting makes it possible to determine much more accurately the fields of plastic deformations and stresses than in solving the quasistatic elastic-plastic problem of a plane strain state.

§4.6.3. Three-dimension stress-strain state Figures 4.112 and 4.113 [44] show the results of calculations of compact specimens with a material hardening coefficient of K * 0.05 . Calculations were made at the following parameter values: L 60 mm ; 2 A H P ; ' t 5 ˜ 10  4 s ; B 10 mm ; H 50 mm ; l0 3 mm ; F A 2.5 mm ; p01 8 MPa ; p02 10 MPa ; M=22; N=22; K=21. The smallest splitting step was 0.02 mm and the largest was 2.6 mm ( ' xmin 0.02 mm ; (only the first layer); 'ymin 0.04 mm 'xmax 2.6 mm ; 'ymax 0.6 mm ); T 50 q C . Graphs of the calculated the SIF KIe dependencies of stresses on the extension of the crack axis near its tip in the plane z 41.3 mm (cell 1 in Figure 1.7b) of a three-dimensional model of a compact specimen (Figure 4.112) show that with the development of deformation of the specimen e when SIF KIe reaches level K Ie K I* 99.6 MPa m , some stresses begin to decrease, and in the case of exceeding the elastic SIF KIe level K Ie

e K I**

120.8 MPa m , the stresses become compressive.

Lines 1–5 correspond to stresses

V xx , V

yy ,

V zz ,

yield strength

after the material strengthening at temperature T V S and intensity Vi , respectively. In Figure 4.113, the solid and dashed lines correspond to the cases of crack increase according to the criteria of local brittle fracture and the absence of maximum stresses, respectively. The calculation showed that at the specimen temperature T 50$ C the crack began to increase when the elastic SIF level K Ie K Iec { 66.2 MPa m was exceeded.

Numerical results analysis

183

Fig. 4.112. Stresses in cell 1 (Fig. 1.7 b) on the extension of the crack axis

The study of the K Ie dependence of the crack length (Figure 4.113a) makes it possible to state that when the stress intensity factor is less than e , the solid and dashed lines correlate well. the value KIe  KI**

When the elastic SIF reaches KIe

e the stress values KI** V xx ,

V

yy

and V zz sharply decrease (see Figure 4.112) and become compressive (negative) and the crack length increases sharply and reaches a value of l 7.9 mm . At this point, the Odquist parameter (Figure 4.113b) in cell 1 decreases to zero. From this point, most likely, the refraction of the specimen begins.

Fig. 4.113. Dependencies: ɚ). Crack length, b). Odquist parameter in cell 1 (Fig. 1.7b)

The calculation of the KIe dependence of the Odquist parameter N , which characterizes the accumulated plastic deformation in cell 1 (immediately before the crack tip), is shown in Figure 4.113b. While the

Chapter 4

184

deformation is elastic N 0 . Then, in cell 1, plastic deformations begin to accumulate monotonically and at the moment when the crack makes the first jump, the location of cell 1 changes to a zone with smaller values of parameter N (the value N decreases abruptly) and the parameter N accumulation process begins again. Therefore, when the elastic SIF exceeds the value KIec , the value of the N changes, and at the same time the magnitude of the plastic deformation accumulated in cell 1 has an oscillating (non-monotonic) nature. In the e , the amplitude of the oscillations of the parameter N case when KIe KI* e decreases to zero. increases and, as mentioned above, when KIe KI** A variable splitting step was used along the Ox (N elements), Oy (M elements) and Oz (K elements) axes. Figures 4.114–4.121 [31] show the results of calculations of Charpy specimens with a material hardening coefficient K * 0.05 . Calculations were made at the following parameter values: L 60 mm ; B 10 mm ; 2 A H P ; 't 5 ˜ 10 4 s ; A 2.5 mm ; H 50 mm ; l 3 mm ; F p01 8 MPa ; p02 10 MPa ; M=60; N=77; K=22. The smallest splitting

step was 0.02 mm and the largest was 2.6 mm ( 'xmin (only

'ymin

0.04 mm

'xmax

2.6 mm ; 'ymax

the

first

0.6 mm ; 'zmax

layer);

'zmin

0.9 mm ); T

0.02 mm ; 0.02 mm ; 50 q C .

Graphs of the calculated SIF K I dependence of stresses that occur in the middle cell of the crack continuation of the three-dimensional model of the Charpy specimen (Figure 4.114) show that with the development of the loading process in the case of exceeding the level of stress intensity factor K I K I* 75.3 MPa m , the stresses at this point change the monotonic nature of the increase to oscillating. In Figures 4.114 and 4.115 solid, solid with triangle, solid with round, solid with plus, solid with star, dotted with triangle, dotted and solid with cross lines correspond to stresses V xx , V y y , V zz , V xy , V xz , V y z , yield strength after the material strengthening V S and stress intensity Vi , respectively. Figure 4.114 shows the graphs of stresses that occur in the crack apex of the three-dimensional Charpy specimen in the plane z 41.3 mm . Figure 4.115 shows the maximum absolute values of the stresses occurring on the line x 0 . When the stress intensity becomes significant

Numerical results analysis

KI

185

75.3 MPa m , the maximum stresses occur in the cell located at the

top at a distance of one cell from the top of the crack. The values of the highest stresses in the three-dimensional model are less than similar stresses in the case of the plane strain model by 8–10 per cent. Stress values in the case of a spatial problem occupy an intermediate position between similar stresses that occur in the case of models of planar stress and planar deformation. The study of the K I dependence of the maximum in absolute value of the stresses occurring on the extension x 0, y ! l of the crack plane (Figure 4.111) suggests that when the stress intensity becomes significant K I K I* , the maximum stresses occur in the second cell from the tip line. At this point, stress fluctuations begin to be observed in the first cell touching the tip. Most likely, these oscillations indicate a loss of stability of the deformation process in the area of the crack tip and the probable beginning of its movement.

Fig.4.114. Stress around the crack tip

Figures 4.116 and 4.117 show graphs of stress distribution over the thickness of a three-dimensional Charpy specimen in the crack vertex region when the stress intensity values K I are 26.7 and 93.5 MPa m , respectively. The used model assumes that the crack plane does not change. Considering such a change requires special research.

186

Chapter 4

Fig.4.115. The maximum stress in the crack in the area of the crack tip

Fig.4.116. Stresses when K I

Fig.4.117. Stresses when K I

26.7 MPa m

93.5 MPa m

Numerical results analysis

187

The following figures show the results at the time when the value of the stress intensity factor K I 57.1 MPa m .

Fig. 4.118. Stresses V1 in planes

z

27.3, 48.3, 49.88 and 49.98 mm

Figure 4.118 shows the main normal stresses

z

27.3, 48.3, 49.88 and 49.98 mm, respectively.

V1

in the planes

188

Chapter 4

Fig. 4.119. Stresses W1 in planes

z

27.3, 48.3, 49.88 and 49.98 mm

Figures 4.118 and 4.119 show the main normal V1 and tangential W1

stresses in the planes z 27.3, 48.3, 49.88 and 49.98 mm, respectively. The zone of the largest plastic deformations is in a radius of approximately 5 × 10-5 m around the crack tip. The largest plastic deformations occur in the region of maximum principal stresses. The values of the highest principal stresses occur at a distance of 5 × 10-4 m from the side surfaces of the specimen and decrease towards the free side surfaces. Figures 4.120 and 4.121 show the temperature dependencies of the mean stress and Odquist parameter in the crack vertex region, respectively. In these figures, the solid with a triangle, solid and solid with a round correspond to the values in the planes of z = 32.3, 41.3 and 48.3 mm, respectively. These dependencies are compared with the case of a plain strain state, which corresponds to the dashed lines. With increasing temperature, there is an increase in plastic deformations.

Numerical results analysis

Fig. 4.120. Mean stresses

189

V

Fig. 4.121. Odquist parameter

The values of the modulus of elasticity, Poisson’s ratio, density and the fraction of shear modulus and volumetric compression GK G / K of the materials under study are given in Table 1. Figures 4.122 and 4.123 show [53] the results of calculations for specimens with length L 60 mm, height B 10 mm , width H 50 mm , depth of cut in the centre of the specimen l 3 mm and hardening factor K * 0.05 . The distance between the bearing points was 40 mm .

Metal GK E (kg/m2) Ȟ ȡ (kg/m3)

Reactor RPV steel 0.535714 2.15ǜ1010 0.272727 7700

Titan

Aluminium

Table 1 Silver

0.409091 1.12ǜ1010 0.32 4505

0.358209 0.7ǜ1010 0.34 2688.9

0.284672 0.8ǜ1010 0.37 1050

Chapter 4

190

A

Time step 't 0.0005 s . Half of the length of the contact zone was 2.5 mm , N1 22, N 2 22, N 3 21 , coefficients p01 8 MPa ,

p02

10 MPa , temperature T

50$ C .

In [243] it was recorded that the specimens were destroyed 21–23 ms after the collision of bodies. To verify the approach proposed here, the process of destruction of specimens of the same material as in [243], size and with the same contact load is modelled in a dynamic elastic-plastic setting taking into account material unloading and crack growth according to the local criterion of brittle fracture (5.2.6). Calculations showed that the specimens were completely destroyed after 23 ms. This to some extent confirms the correctness of the problem and the adequacy of the developed model. Figures 4.122 and 4.123 show the results of calculations for metals, considering the possible unloading of the material. The unloading of material was considered according to the following algorithm. If in any cell the absolute value of stresses became less than the maximum value reached there earlier, it was considered that plastic deformations cease to increase, the material ceases to strengthen and the linear dependence of stresses on deformations is restored. Plastic deformations began to increase again, and the hardening of the material continued when the absolute value of stresses exceeded the previously reached maximum value. Figure 4.122 shows graphs of average stresses V (MPa) occurring in the crack apex of a three-dimensional compact specimen in the plane z 41.3 mm (inside cell 1 in Figure 1.7b). Solid, solid with square marker, dashed with triangular marker and dashed lines correspond to the calculations for reactor steel, titanium, aluminium and silver, respectively. As long as the ‘elastic SIF’ does not reach the value of K I K I* 72.3 MPa m , the average calculated stresses V are greater when the coefficient GK is smaller, and the dependence on K I is practically monotonic. However, when the SIF K I exceeds the value of K I* , the monotonic nature of the change stops and the average stress V begins to oscillate.

Numerical results analysis

Fig. 4.122. Mean stresses

191

V

Graphic dependencies are not given for the cross section of z 49.88 mm ; however, we note that the oscillations in it begin when the elastic SIF reaches the value of K I 54 MPa m , and the level of average stresses V is 23% less than in the plane of z 41.3 mm . Figure 4.123 shows graphs of the Odquist parameter, which characterizes the plastic deformation accumulated in cell 1 in front of the crack front in the case of reactor steel, titanium, aluminium and silver (solid, solid with square marker, solid with triangular marker and dotted lines, respectively). Analysing figure 4.123, it can be seen that the smaller the ratio of shear modulus and volumetric compression GK , the greater the Odquist parameter and, hence, the accumulated plastic deformation in front of the crack front.

Fig. 4.123. Odquist parameter

Chapter 4

192

Comparing the plastic deformations in the cross sections z 41,3 mm (Figure 4.123a) and z 49.88 mm (Figure 4.123b), it can be seen that in the sections closer to the side surfaces of the specimen, the plastic deformations are greater than in its middle. While the elastic SIF K I takes values from the range of reactor steel [84.4–102.6], titanium [84.4–105.7], aluminium [84.4–108.7] and silver [84.4–87.4], the value of the scatter of the values of the Odquist parameter in the studied cross sections z 41.3 mm and z 49.88 mm is not more than 6%. Then, until the end of the load process, the value of their largest deviation reaches values of 35%, 33%, 42% and 25%, respectively.

§4.7. Problems in dynamic elastic-plastic formulation considering the process of unloading of the material For calculations of the mathematical model of the compact specimen, RPV steel 15ɏ2ɇɆɎȺ is chosen as in §4.6.

§4.7.1. Plane stress state The results of calculations of mean stresses V in cells near the crack tip on the stage, when loads are relatively small and when there are no plastic deformations in the cells of the discretised problem and the solution of the problem is purely linearly elastic, were compared with those mean stresses V calculated for the cell centre based on classical monomial asymptotic dependencies [176, §1.2] using expression (2.5.6) for the SIF of the elastic solution. Figures 4.124 and 4.125 [32, 38] show the results for the following parameter values: K * 0.05 ; L 60 mm ; B 10 mm ; l 3 mm ; 't 0, 0005 s ; A 2.5 mm ; p01 8 MPa ; p02 10 MPa ;

M 60 ; N 77; T

50$ C .

Numerical results analysis

193

Fig. 4.124. K1e dependencies of the stresses at cell 1 (Fig. 1.6b) on the continuation of the crack axis

Graphs of the calculated stress dependencies near the crack tip (cell 1, Figure 1.6b) of the two-dimensional compact specimen (Figure 4.124) show that with the development of the specimen deformation process in the case where the elastic SIF K1e level is exceeded K1e

e K1* 90.5 MPa m , the stress at this point changes to oscillating. Solid, solid with a triangle, solid with a cross and solid with a circle lines correspond to stresses V xx , V yy , yield strength V S and stress

intensities Vi , respectively. Dashed, dotted with triangle, dotted with cross and dotted with circle lines apply to the same stresses, but obtained from the solution of the problem without considering the process of unloading of the material. Figure 4.125 compares the value of the Odquist parameter of the problem under consideration against the values of the corresponding problem of the plane stress state without taking into account the process of unloading the material. The calculations show (Figure 4.125) that the level of plastic deformations is higher, considering the unloading of the material and when the elastic SIF does not exceed the level K1e 129.9 MPa m .

194

Chapter 4

Fig. 4.125. SIF K1e dependence of Odquist parameter N in cell 1 (Fig. 1.6b)

Figures 4.126 and 4.127 show graphs of temperature T dependencies, of the mean stress V and the Odquist parameter N , respectively, in cell 1 when the SIF is equal to K1e 57.1 MPa m . The solid line in these figures corresponds to the case of the plane stress state problem taking into account the process of unloading of the material in comparison with the results of the corresponding problem without taking into account the unloading (dotted lines).

Fig. 4.126. Temperature dependence of mean stress

V

in cell 1 (Fig. 1.6b)

Numerical results analysis

195

Fig. 4. 127. Temperature dependence of Odquist parameter

N

in cell 1 (Fig. 1.6b)

When the temperature T does not exceed  50$ C , then the mean stresses V and the Odquist parameters N coincide. Figures 4.128–4.130 [47] show the results of calculations of compact specimens with length, width, thickness, depth of cut in the centre, hardening factor of the material and distance between the bearing points as in previous examples. Calculations were performed for the following parameter values: ' t 0.0005 s ; A 2.5 mm ; p01 8 MPa ; p02

10 MPa ; M 60 ; N

77 ; temperature T

50 $ C .

Fig. 4.128. Odquist parameter

N

196

Chapter 4

As stated in [243], during the experiments the compact specimens were destroyed in 21–23 ms. The process of destruction of compact specimens of material, size and contact load as in [243] was modelled in a dynamic elastic-plastic setting, taking into account the unloading of material and crack growth according to the local criterion of brittle fracture. The specimens were destroyed in 23 ms. This confirms the correctness and adequacy of the developed formulation and model. Figure 4.128 shows the Odquist parameter. As can be seen from the figure, plastic deformation is greater when the value of the GK parameter is smaller. Figures 4.129 and 4.130 show the areas of plastic deformations for specimens made of reactor steel and aluminium, respectively, which occur in the area of the crack tip at the time when K I 123.9 MPa m . Figures a), b) and c) correspond to plastic deformations H xxp , H yyp and H xyp . The zone of plastic deformations is located at an angle of approximately 45° to the right and left of the crack tip; the maximum plastic deformations occur directly near its tip. In the case of reactor steel (the harder material), the values of plastic deformations are about four times less than for aluminium, while the width and height of the plasticity regions in both cases are approximately the same. As in the problem of plane deformation, plastic deformations H yyp are the smallest in value, and plastic deformations rank.

H xxp

and

H xyp have the same

Numerical results analysis

197

Fig. 4.129. Plastic deformations around the crack tip of the reactor steel specimen: a)

H xxp ; b) H yyp ;

c)

H xyp

Fig. 4.130. Plastic deformations in the area of the crack tip of the aluminium specimen: a) H xxp ; b) H yyp ; c) H xyp

Based on the solution of the plane stress state problem for a material with a rectangular cross-section with a cut-crack in the middle in a dynamic elastic-plastic setting, taking into account the unloading process, it is shown that when the value of fracture of shear and volumetric compression ratios is smaller, there are more plastic deformations.

Chapter 4

198

§4.7.2. Plane strain state Figures 4.131 and 4.132 [46] show the results of the calculation of some important mechanics of fracture quantities for the following parameter values: the hardening factor of the material K * 0.05 ; L 60 mm ; B 10 mm ; l0 3 mm ; 't 0.0005 s ; A 2.5 mm ;

p01 8 MPa ; p02 10 MPa ; M 60 ; N 77 ; temperature T

50$ C .

The smallest step of partitioning was 0.02 mm and the largest 2.6 mm ( ' xmin 0.02 mm; (only the first layer); ' ymin 0.04 mm

'xmax

2.6 mm ; 'ymax

0.6 mm ).

In Figures 4.131 and 4.132, solid lines correspond to this problem, and dotted lines show the results of the problem of plane strain state with a fixed crack without considering the process of unloading of the material. Solid and dotted, solid and dotted with a triangle, solid and dotted with a quadrilateral, solid and dotted with a cross, and solid and dotted with a circle lines relate to the stresses V xx , V yy , V zz , yield strength V S and stress intensity Vi of the respective problems.

Fig. 4.131. KIe dependencies of stresses in cell 1 (Fig. 1.6b) on the continuation of the crack axis from

Numerical results analysis

199

Graphs of the calculated SIF K Ie dependencies of the stresses on the extension of the notch axis near its tip (cell 1 in Figure 1.6b) of the twodimensional model of the compact specimen (Figure 4.131) show that with the development of the deformation process of the specimen in the case of exceeding the elastic SIF KIe level K Ie

63.2 MPa m , only the stresses

V xx differ significantly. K Ie

The calculations showed that after the elastic SIF reaches value 124.3 MPa m , the values of the stresses V xx and V xy begin to

decrease (see Figure 4.131) and their oscillations increase. The calculation of the dependence of the Odquist parameter N in cell 1, which characterizes the plastic deformation accumulated in cell 1 (all the time directly in front of the crack tip), is shown in Figure 4.132. While the deformation is elastic, N 0. As can be seen, the level of plastic deformation in the case of considering the unloading process is less than without taking it into account.

Fig. 4.132. SIF K1e Odquist parameter

N

dependence in cell 1 (Fig. 1.6b)

Figures 4.133–4.136 [50] show the results of calculations of compact specimens with a length of 60 mm, width of 10 mm, thickness of 50 mm and depth of cut in the centre of 3 mm, with a coefficient of hardening of the material K * 0.05 . The distance between the bearing points was 40 mm. Calculations are carried out using the following parameter values:

Chapter 4

200

't N

2.5 mm ; p01 8 MPa ; p02 10 MPa ; M 60 ;

0, 0005 s ; A 77 ; temperature T

50$ C .

Fig. 4.133. Odquist parameter

N

Table 1 shows the values of the ratio of shear modulus and volumetric compression. In [133, 165] it was shown that the rise of the medium on impact depends on the GK parameter. Figure 4.133 shows the Odquist parameter for different metals. As can be seen from Table 1 and Figure 4.133, when the ratio of shear modulus and volumetric compression GK is smaller, the plastic deformations are greater. p, p Figures 4.134–4.136 show the areas of plastic deformations H xx H yy , Hxyp , respectively, that occur in the area of the crack tip at the time when

K I 108.7 MPa m . Figure a) applies to the case of specimens of aluminium, figure b) to specimens of reactor RPV steel. The zone of plastic deformations is located at an angle of approximately 45° to the right and left of the crack tip; the maximum plastic deformations occur directly near its tip.

Numerical results analysis

201

p in zone near the crack tip: a) aluminium b) Fig. 4.134. Plastic deformations H xx reactor steel

p in zone near the crack tip: a) aluminium b) Fig. 4.135. Plastic deformations H yy reactor steel

p in zone near the crack tip: a) aluminium b) Fig. 4.136. Plastic deformations H xy

reactor steel

202

Chapter 4

In the case of reactor steel (the harder material), the value of plastic deformations is about four times less than for aluminium, while the width of the plastic region in case a) is 25% greater than in case b), and the p are the smallest in height is almost the same. Plastic deformations H yy p and p have the same degree. size; plastic deformations H xx H xy

Based on the solution of the plane deformed state problem for a material with a rectangular cross-section with a cut-crack in the middle in a dynamic elastic-plastic setting, taking into account the unloading process of the material, it is shown that when the ratio of shear modulus and volumetric compression is smaller, the plastic deformations are greater.

§4.7.3. Three-dimension stress-strain state Figures 4.137 and 4.138 [51, 54] show the results of calculations for bars with length L 60 mm, height B 10 mm and width H 50 mm , with the initial depth of notch-cut in the centre of the specimen l 3 mm and the hardening factor of the material K * 0.05 . The distance between the bearing points was 40 mm . Time step was 't 0.0005 s . Half the length of the contact zone was A 2.5 mm ; N1 22; N2 22; N3 21 ; coefficients of p01 8 MPa ; p02 10 MPa ; temperature T 50$ C . The material was unloaded according to the following algorithm. If in any cell the absolute value of stress became less than the maximum value already reached there, then it was considered that plastic deformations stop increasing, the material ceases to strengthen and the linear dependence of stresses on deformation is restored. Plastic deformations began to increase again and the hardening of the material continued when the absolute value of stresses exceeded the previously reached maximum value. Figure 4.137 shows the graphs of the mean stresses V of the threedimensional specimen in the plane of z 41.3 mm . Solid, solid with a square, dashed with a triangle and dashed lines correspond to reactor steel, titanium, aluminium, and silver, respectively. As long as the elastic SIF e has not reached the value of K Ie KI* 72.3 MPa m , the mean stress V is greater when the coefficient GK is smaller. When the elastic SIF e , the mean stress begins to oscillate. exceeds the value K I*

Numerical results analysis

Fig. 4.137. Mean stresses

In the case when z

203

V

49.88 mm , oscillations begin when the elastic

SIF reaches the value of K Ie 54 MPa m , and the mean stress V is 23% less than in the plane of z 41.3 mm . Figure 4.138 shows graphs of the Odquist parameter, which characterizes the accumulated plastic deformation in the area in front of the crack tip in the case of reactor steel, titanium, aluminium, and silver (solid, solid with square, solid with triangle and dotted lines, respectively). If the parameter GK (see Table 1) is smaller, the Odquist parameter is greater; hence the plastic deformation. Comparing Figures 4.138a and 4.138b, it can be seen that the plastic deformations are greater in the cross sections of z 49.88 mm and z 0.12 mm , located closer to the side surfaces of the compact specimen.

Fig. 4.138. Odquist parameter in the area of the crack tip in plane: ɚ). z 41.3 mm , b). z 49.88 mm

Chapter 4

204

Based on the solution of the spatial problem of plastic deformation accumulation in a compact specimen in a dynamic elastic-plastic setting considering the material unloading process, it is shown that when the ratio of shear modulus and volumetric compression is smaller, plastic deformations are bigger, and plastic deformations are greater in the crosssections that are located directly under lateral surfaces of a compact specimen.

§4.8. Problems in dynamic elastic-plastic formulation considering the process of crack growth provided that the maximum breaking stresses are provided directly on the continuation of the crack tip Solving spatial and planar problems for a compact specimen to determine the fracture toughness of three-point bending in a dynamic elastic-plastic formulation considering crack growth in the absence of maximum stresses at the crack tip makes it possible to determine fields of plastic deformations and stresses more accurately than solving quasi-static elastic-plastic similar problems and also allows one to adequately model the process of crack propagation. 15X2NMFA steel was chosen for the calculations of the mathematical model of the compact specimen.

§4.8.1. Plane stress state For comparison with [144], the calculations were performed under the assumption of the absence of plastic deformations H zzp . Figures 4.139 and 4.140 show the results of calculating the values for the following parameters: L 60 mm ; B 10 mm ; K* 0.05 ; l0 3 mm ;

't

0.0005 s ; A

2.5 mm ; p01 8 MPa ; p02 10 MPa ; M 60 ;

N

77 ; temperature T 50$ C.

Graphs [25] of the calculated dependence of stresses near the crack tip (cell 1, Figure 1.6b) show (Figure 4.139) that with the development of the deformation process of the specimen in the case of exceeding the elastic e SIF K1e level K1e K1* 90.5 MPa m , stresses at this point change the monotonic nature of the increase to oscillating. Solid, solid with a triangle, dotted, solid with a cross and solid with a circle lines relate to stresses V xx , V y y , V xy , yield strength V S , and

Numerical results analysis

205

stress intensity Vi , respectively. Calculations revealed (Figure 4.140a) that the crack began to increase in the case of exceeding the elastic SIF level K1e K1ec { 51 M Pa m .

Fig. 4.139. SIF K1e dependencies of stresses in cell 1 (Fig. 1.6b) on the continuation of the crack axis

After the elastic SIF reaches the value of K1e 148.2 MPa m , the Odquist parameter (Figure 4.140b) in cell 1 (Figure 1.6b) decreases to zero, and the crack length reaches l 8.35 mm .

Fig. 4.140. SIF

K1e dependencies ɚ). Crack length l; b). Odquist parameter N cell 1 (Fig. 1.6b)

in

Chapter 4

206

This means that at such a crack length in front of its tip, the material is not plastically deformed and further refraction of the specimen is practically elastic.

§4.8.2. Plane strain state Figures 4.141–4.148 [41] show the results of calculation of some important mechanics of fracture characteristics for the following parameter values: material hardening coefficient K* 0.05 ; L 60 mm ; B

p02

0.0005 s ; A 2.5 mm ; p01 8 MPa ; 10 MPa ; M 60 ; N 77 ; temperature T 50$ C . The smallest

10 mm ; l

3 mm ; 't

step of partitioning was 0.02 mm and the largest 2.6 mm ( 'xmin 0.02 mm ; 'ymin 0.04 mm (only the first layer); 'xmax 2.6 mm ; ' ymax 0.6 mm ). Graphs of the calculated dependence on the mean stress SIF KIe on the extended notch axes near its tip (cell 1 in Figure 1.6b) of the twodimensional model of the compact specimen (Figure 4.141) show that with the development of the specimen deformation process in the case when e

e SIF KI exceeds the value K Ie K I* 78.3 MPa m , at this point some stresses (for example V y y ) change the monotonic nature of the increase

to oscillating, and in the case of exceeding the elastic SIF KIe level e K Ie K I** 111.7 MPa m , stress fluctuations become more intense. Solid, solid with a triangle, dotted, solid with a cross and solid with a circle lines refer to the stresses Vxx , V yy , Vxy , yield strength V S and stress intensity V i , respectively. Calculations showed that at the temperature T 50$ C the crack began to increase in the case of exceeding the elastic SIF level K Ie K Iec { 60.1 MPa m . The study of the dependence of crack length on

KIe (Figure 4.142) gives the opportunity to confirm that when the stress e , the process of increasing crack intensity becomes significant KIe KI* length becomes more intense (with much larger length increments) with increasing K Ie .

Numerical results analysis

207

Fig. 4.141. SIF K Ie dependence of stresses in cell 1 (Fig. 1.6b) on the continuation of the crack axis

After reaching the elastic SIF value K Ie

K Ied { 169.4 M Pa m , the

stresses values Vxx and Vyy begin to decrease (see Figure 4.142). At this point, the Odquist parameter (Figure 4.143) in cell 1 is oscillating in nature with a damped amplitude, and the crack length reaches l 5.94 mm . From this point, most likely, further refraction of the specimen begins. Calculation of the dependence of the Odquist parameter N characterizes the accumulated plastic deformation in cell 1 (Figure 1.6b) from the SIF K Ie (Figure 4.143). While the deformation is elastic N 0, then plastic deformations begin to accumulate monotonically in cell 1 and at the moment when the crack makes the first jump, the location of cell 1 changes to a zone with a smaller parameter N value (the value N decreases by a jump) and the accumulation process of N begins again. Because of this, when the elastic SIF exceeds the value KIec , dependence of N , and at the same time the magnitude of the plastic deformation accumulated in cell 1 are oscillating (non-monotonic). In the case when KIe

N

e , the amplitude of the oscillations of the parameter KI*

increases and, as noted above, when KIe

KIed its growth slows down.

208

Chapter 4

Figures 4.144 and 4.145 show, respectively, the temperature dependencies of the mean stress V and the Odquist parameter N in cell e 1, obtained at a fixed value K Ie K I0 75.3 MPa m K Ieɫ in the case of crack development (solid lines) when the load reaches this level, and the assumption that the crack does not change its original size [28] (dashed lines). For comparison, the corresponding results of the problem of plane stress state (lines with a triangle) are given.

Fig. 4.142. SIF K Ie dependencies of crack Length l

Fig. 4.143. SIF KIe dependencies of Odquist parameter

N

in cell 1 (Fig. 1.6b)

Numerical results analysis

209

In cases where the temperature level allows an increase in the crack e (including when size at a given KIe KI0 T 50$ C ), a plane stress state of the specimen with a fixed crack in the area before the crack tip is characterized by a higher level of mean stresses and accumulated plastic deformations. If at a certain temperature (in this case T   100 $ C ) the stress level for the crack is insufficient, then both approaches give the same results.

Fig. 4.144. mean stress (Fig. 1.6b)

V

in cell 1

Fig. 4. 145. Odquist parameter in cell 1 (Fig. 1.6b)

N

When the crack is stationary and the stress intensity becomes significant

KIe

60.1 MPa m

KIeɫ

(see

Figure

4.146),

plastic

p , H p and H p occur in front of the crack tip. deformations H xx yy xy Solid, solid with a triangle and dotted lines (Figure 4.147) relate to the p and p, p , respectively. Solid, energies of plastic deformations U xx U yy U xy

solid with triangle and dotted lines (Figure 4.148) show the energy of e, e , respectively. elastic deformations Uxx U eyy and U xy

210

Chapter 4

Numerical results analysis

211

p (ɚ, d, g, Fig. 4.146. Diagrams of distribution of maximal plastic deformations H xx

p (b, e, h, k) ɿ p (c, f, i, l) in crack zone j), Hyy Hxy

The study of the distribution of plastic energies U ijp , (i, j

x, y )

(Figure 4.147) from the moment the elastic stress intensity factor exceeds e indicates their oscillatory nature. Elastic energies the level K I* Uije , (i, j

x, y) (Figure 4.148) increase monotonically, with jumps

occurring most likely through a discrete computational scheme. When SIF

KIe ! 123.9 MPa m , darker and hatched areas on the diagrams correspond to higher values of deformation. It is noticeable that the zones of maximum plastic deformations (most likely) are adjacent to the free surface of the crack, and its tip is located in a less plastically deformed material (as if trying to break into such a zone). If the increase in crack length is large enough, a number of such zones of increased plastic deformation adjoin the crack edges. It is clear that reducing the size of the partition grid will somewhat smooth the images at the bottom of the distribution diagram, but most likely such uneven plastic deformations may be due to the dynamic propagation of the crack.

Chapter 4

212

Fig. 4.147. Energies of plastic deformations U p ij

Fig. 4.148. Energies of elastic deformations U ije

When the crack begins to move and the stress intensity becomes the value of

K Ieɫ  K Ie

90.5 MPa m

e , the maximum plastic K I*

p , H p and H p are located outside the tip of the crack. deformations H xx yy xy

§4.8.3. Three-dimension stress-strain state To calculate the models of compact specimens, the partition steps along the axes Ox (M elements), Oy (N elements) and Oz (K elements) were variable.

Numerical results analysis

213

Figures 4.149 and 4.150 [43] show the results of calculations of compact specimens with a coefficient of hardening of the material K* 0.05 for the following parameter values: L 60 mm; B 10 mm ; H p02

50 mm ; l0

3 mm ; ' t

10 MPa ; M 22 ; N

0.0005 s ; a

p01 8 MPa ;

22 ; K 21 . The smallest splitting step was

0.02 mm and the largest 2.6 mm ( ' xmin (only the first layer); ' xmax

5 mm ;

0, 02 mm ; ' ymin

2.6 mm ; 'ymax

0.04 mm

0.6 mm ).

Fig. 4.149. Stresses in cell 1 on the continuation of the crack axis

Graphs of the calculated SIF KIe dependencies of stresses that occur in the plane z 41,3 mm in the cell of the crack extension (row of cells 1 in Figure 1.7b) of the three-dimensional model of a compact specimen (Figure 4.149) show that with the development of the loading process e when elastic SIF exceeds the value K Ie K I* 90.5 MPa m , then the stresses at this point change the monotonic nature of the increase to chaotic oscillations. Lines 1–5 relate to stresses V xx , V y y , V zz , stress intensities Vi and yield strength V S , respectively. At the value of SIF K Ie 108.7 MPa m , the oscillatory nature of the stresses increases, and at the value of SIF K Ie 114.8 MPa m , the calculated stresses V yy become compressive. Calculations show that the crack begins to move when the elastic SIF KIe reaches a value K Ie 60.1 MPa m . In Figure 4.150, lines 1–3 refer to the spatial problem and the plane problems of strain and stress states,

214

Chapter 4

respectively, when the crack moves in accordance with the local criterion of brittle fracture. The dependence of the crack length on the elastic SIF in the three-dimensional case with the straight crack front concurs quite well with the results obtained in the case of a plane strain state, as long as the SIF K Ie does not exceed K Ie 117.8 MPa m . Figure 4.151 shows the graphs of the dependencies of the Odquist parameter on the crack tip (cell 1, Figure 1.6b) in the plane z 41.3 mm .

Fig. 4.150. Crack length l

Fig. 4.151. Odquist parameter

N

SIF KI dependence in cell 1

Lines 1–5 concern the cases of analysis of problems in elastic-plastic formulation of the spatial dynamic problem considered here, spatial problems in dynamic and quasi-static formulations with a fixed crack, as well as plane problems of strain and stress states with a crack growing by the local criterion of brittle fracture. Lines 1–3 obtained for a moving crack are oscillating. This is explained by the fact that starting from the

Numerical results analysis

215

beginning of the crack movement, the values of the Odquist parameter are given at different points of the body (cells), which due to the crack propagation reach the tip, becoming cell 1.

§4.9. Problems in dynamic elastic-plastic formulation considering the processes of crack growth provided that the maximum breaking stresses are provided directly on the continuation of the crack tip and unloading of the material The solution of three-dimensional and plane problems of stress-strain state for a material with a cross-section in the form of a rectangle with a cut-crack in the middle (compact profile) in a dynamic elastic-plastic setting taking into account the unloading process give one the ability to determine the fields of plastic deformations and stresses more precisely than when solving a similar problem without taking into account the process of unloading of the material or when using a quasi-static mathematical model. Reactor steel 15H2NMFA was used to calculate the mathematical model of the compact specimen.

§4.9.1. Plane stress and strain states Stress, crack length and the Odquist parameter for plane stress problems with material unloading and without unloading practically coincide. To calculate the mathematical model of the compact specimen, the partition steps along the axes Ox (N elements) and Oy (M elements) were variable. Figures 4.152–4.154 show the results of the calculation of some important mechanics of fracture characteristics for the following parameter values: the coefficient of hardening of the material K* 0.05 ;

L 60 mm; B 10 mm ; l 3 mm ; 't 0.0005 s ; A 2.5 mm ; p01 8 MPa ; p02 10 MPa ; M 60 ; N 77 . The smallest step of partitioning was 0.02 mm, and the largest 2.6 mm ( ' xmin 0.02 mm ; (only the first layer); 'ymin 0.04 mm ' xmax 2.6 mm ; ' ymax

0.6 mm ), 'xmax

2.6 mm ; T

50ɨ ɋ .

Graphs of the calculated SIF K Ie dependence of the mean stress on the extension of the axis of notch near its tip (cell 1 in Figure 1.6b) in the two-

216

Chapter 4

dimensional model of a compact specimen (Figure 4.152) show that the stress for plane strain state problems with unloading of the material (solid lines) and without unloading (dotted lines) differ when the value of the e elastic SIF K Ie exceeds the level KIe KI* 99.6 MPa m . Solid, solid with a triangle, solid with a cross and solid with a circle lines refer to stresses ıxx , ı yy , yield strength ı S and stress intensity

ıi , respectively, for the problem, taking into account the unloading of the material. Dotted, dotted with a triangle, dotted with a cross and dotted with a circle lines apply to the same stresses, but for the problem without unloading the material. Calculations showed that at the specimen temperature T 50 $C the crack began to increase when the elastic SIF level was exceeded K Ie KIec { 60.1 MPa m , and starting (Figure 4.153) with the value of the KIe 117.8 MPa m , crack length for the plain strain state problem taking into account the material unloading process (solid line) grows faster than for the corresponding problem without considering the unloading process.

Fig. 4.152. SIF K Ie dependencies of stresses in cell 1 on extension of the crack axis

The calculation of the dependence of the Odquist parameter N , which characterizes the accumulated plastic deformation in cell 1 (directly in front of the crack tip), taking into account the unloading of the material

Numerical results analysis

217

(solid line) and without unloading (dotted line), is shown in Figure 4.154. While the deformation is elastic, N 0 .

Fig. 4.153. SIF KIe dependencies of crack length l

Fig. 4.154. SIF KIe dependencies of Odquist parameter

N

in cell 1

Then, in cell 1, plastic deformations begin to accumulate monotonously, and at the moment when the crack makes the first jump, the location of cell 1 changes to a zone with smaller parameter N values (the value N of the jump decreases) and the process of the accumulation N begins again. Therefore, when the elastic SIF exceeds the value of KIec the parameter N changes, and at the same time the value of the accumulated plastic deformation in cell 1 is oscillating (non-monotonic).

218

Chapter 4

§4.9.2. Three-dimension stress-strain state Numerical studies were performed for bars with a compact profile made of 15H2NMFA steel. Figures 4.155 and 4.156 [48] show the results of calculations of compact samples 60 mm long, 10 mm wide, 50 mm thick and with a 3 mm deep cut-notch in the centre, with a hardening factor of the material K* 0.05 . The distance between the bearing points was 40 mm. Calculations were made at the following parameter values: 't 0.0005 s ; A 2.5 mm , p01 8 MPa ; p02 10 MPa ; M 22 ;

N 22 ; K 21; temperature T 50ɨ ɋ . Figures 4.155 and 4.156 [43] show, respectively, graphs of the SIF dependencies of crack length and values of the Odquist parameter N in the plane z 41.3 mm . The dotted line refers to the results of calculations of the problem without considering the unloading of the material.

Fig.4.155. Crack length

Fig.4. 156 Odquist parameter

Numerical results analysis

219

Fig. 4.157 Stresses when crack length grows

Calculations show that the crack begins to grow when the SIF becomes the value K I { K I* 63.2 MPa m (in the case without considering the unloading of the material, this value is equal to 66.2 MPa m ). When the elastic SIF reaches a value K I { K I** 129.9 MPa m (in case of excluding material unloading this value is equal to 123.9 MPa m ), the stresses V xx , V y y and V zz decrease sharply and become compressive (negative) and the crack length increases sharply (see Figure 4.155) and reaches l 7.9 mm . However, as can be seen from Figure 4.157, there is no contradiction in this fact. Figure 4.157 shows the process of increasing the crack gradually in iterations. Here solid, solid with a triangle, solid with a circle, solid with a cross and dashed lines correspond to stresses V xx , V y y , V zz , strengthened yield strength V S and stress intensity V i , respectively. At the last moment of time, when the crack has increased to the value of l 7.9 mm , the stresses become compressive.

§4.10. Problems in dynamic elastic-plastic formulation considering the process of crack growth according to the generalized local VTT criterion of brittle fracture Solving spatial and plane problems in dynamic elastic-plastic formulation taking into account the crack increase according to the generalized local VTT criterion of brittle fracture makes it possible to

Chapter 4

220

determine fields of plastic deformations and stresses much more accurately than when solving quasi-static elastic-plastic spatial problems of deformed stress state, and makes it possible to simulate the process of crack propagation more accurately. Reactor steel 15H2NMFA was used to calculate the mathematical model of the compact specimen.

§4.10.1. Plane stress state Figures 4.158–4.165 [26] show the results of calculating some important mechanics of fracture characteristics for the following parameter values: material hardening factor K * 0.05; L 60 mm ; B 10 mm ; l0 3 mm ; 't 0.0005 s ; A 2.5 mm ; p01 8 MPa ; p02 10 MPa ;

M 60 ; N 77 . The smallest splitting step was 0.02 mm and the largest 2.6 mm ( 'xmin 0.02 mm ; 'ymin 0.04 mm (only the first layer); 'xmax a1

2.6 mm ;

867 ɆPɚ ; 2

h 1.04 ˜ 10

'ymax b

0.6 mm

);

975 ɆPɚ ;

K -1 ; C1 1.92 ; C 2 3.04 ; Ad

temperature

T

50$ C ;

c 0.0305 ɆPɚ ˜ K -1 ; 2.92 .

Fig. 4.158. SIF K1e dependencies of stresses in cell 1 (Fig. 1.6b) on the continuation of the crack axis

Graphs of the calculated SIF K1e dependence of the mean stress on the extension of the axis of the notch near its tip (cell 1 in Figure 1.6b) of the two-dimensional model of a compact specimen (Figure 4.158) show that

Numerical results analysis

221

with the development of the process of deformation of the sample when e elastic SIF K1e exceeds the value K1e K1* 90.5 MPa m , the monotonic nature of the increase is changed to oscillating, and in the case of e exceeding the elastic SIF K1e level K1e K1** 166.3 MPa m , stresses fluctuations become larger in amplitude. Solid, solid with a triangle, dotted, solid with a cross and solid with a circle lines relate to stresses V xx , V yy , V xy , yield strength V S and

stress intensity Vi , respectively. Calculations showed that at the sample temperature T 50 $C , the crack began to increase in the case of exceeding the elastic SIF level K1e K1ec { 72.3 MPa m . The study of the dependence of the crack length on K1e (Figure 4.159) provides the ability to declare that when the stress e , the process of increasing intensity factor becomes significant K1e K1* the crack length becomes more intense (with much larger length

increments with the same increase of K1e ). After reaching the elastic SIF value K1e

K1ed { 324.3 MPa m , the

values of stresses Vxx , V xy decrease sharply (see Figure 4.158) and the stresses V xx become compressive. At this point, the Odquist parameter (Figure 4.160) in cell 1 (Figure 1.6b) decreases to almost zero, and the crack length reaches l0 8.35 mm . From this point, the stresses V xx and

V yy begin to fluctuate around zero and this is likely to correspond to the further refraction of the sample. The calculation of the SIF K1e dependence of the Odquist parameter N , which characterizes the accumulated plastic deformation in cell 1 (all the time immediately in front of the crack tip), is shown in Figure 4.160. While the deformation is elastic, N 0 . Then, in cell 1 (Figure 1.6b), plastic deformations begin to accumulate monotonously, and at the moment when the crack makes the first rise, the location of cell 1 changes to a zone with smaller parameter N values (the value of parameter N decreases with each jump) and the Odquist parameter N accumulation process begins again. Because of this, when the elastic SIF is exceeded K1ec , the value of the change of the parameter N , and at the same time the magnitude of the

Chapter 4

222

plastic deformation accumulated in cell 1 is oscillating (non-monotonic). e , the amplitude of the oscillations of the In the case when K1e K1*

N

increases significantly and, as noted above, when K1e it decreases to almost zero and begins to grow slowly. parameter

K1ed

Fig. 4. 159. SIF K1e dependence of crack length l

Fig. 4.160. Odquist parameter

N

SIF K1e dependence in cell 1 (Fig. 1.6b)

Figures 4.161 and 4.162 show the temperature T dependencies of mean stress V and Odquist parameter N in cell 1, respectively, obtained at a e e in the case of considering fixed value K1e K10 78.3 MPa m ! K1c crack development (solid lines) when the load reaches this level, and assuming that the crack does not change its original size ( dashed lines). When the temperature level allows an increase in the crack size at a e (including at T 50 $C ) in the case of plane stress state of given K1e K10 the specimen with a fixed crack in the area before the crack tip, the level of mean stresses and accumulated plastic deformations is higher. If at a certain temperature (in this case T  100 $C ) the stress level for the crack movement is insufficient, then both approaches give the same results.

Numerical results analysis

223

Figure 4.163 shows the distribution diagrams of the maximum p , H p and H p in the area surrounding achieved plastic deformations H xx yy xy the crack tip and adjacent to its fragment, formed as a result of growth.

Fig. 4.161. Mean stress V dependence in cell1 1 (Fig. 1.6b) K1e

Fig. 4.162. Odquist parameter N dependence in cell 1 (Fig. 1.6b)

The upper part corresponds to the value of the elastic SIF when 60.1 MPa m (when the crack is still not growing), and the lower

e , the crack has grown by 3.24 mm and begins the phase of when K1e K1* intensive growth, which precedes the refraction of the specimen. In this case, the darker zones on the diagrams correspond to higher values of deformation. It is noticeable that the zones of maximum plastic deformations (they, as can be seen from the top of the figure, are formed directly around the crack tip and the crack moves into the unloading zone) adjoin the free crack surface, and with sufficient crack development the crack’s tip is located in less plastically deformed materials. If the increase in crack length is large enough, then a number of such zones of increased plastic deformation adjoin the crack edges, that is, the thickness of the near-surface plastically deformed metal is neither constant nor monotonically variable. It is clear that reducing the size of the partition grid will slightly smooth the images at the bottom of the distribution diagram, but most likely such unevenness of plastic deformations may be inherent in the dynamic nature of the crack growth. In Figures 4.164 and 4.165 solid, solid with a triangle and dotted lines p, refer to the change of additive components (3.6.1) energies of plastic U xx

p and p , respectively, (Figure 4.164) and elastic U e , Ue and e U yy U xy U xy xx yy

Chapter 4

224

(Figure 4.165) deformations in cell 1 (Figure 1.6b) in front of the crack’s tip. p Deformation H xx

p Deformation H yy

K1e

Deformation

p H xy

60.1 MPa m

e K1e K1*

Fig. 4.163. Distribution diagrams of the maximum achieved plastic deformations p , p and p in crack area H yy H xx H xy

The study of the distribution of the components of plastic energies U ijp (i , j x , y ) (see Figure 4.164) from the moment the elastic stress

intensity factor exceeds the level KI* indicates that they are generally oscillating.

Numerical results analysis

225

Fig. 4.164. Changing the components of the energy of plastic deformations U p ij

Fig. 4.165. Changing the components of the energy of elastic deformations U ije

The elastic components of energy U ije (i, j x, y ) (see Figure 4.165) increase monotonically, with minor jumps which can be explained by the discreteness of the computational scheme.

§4.10.2. Plane strain state Figures 4.166–4.169 [30] show the results of calculating some important mechanics of fracture characteristics at the following parameter values: material hardening coefficient K * 0.05; L 60 mm ; B 10 mm ; 't 0.0005 s ; A 2.5 mm ; l0 3 mm ; p01 8 MPa ;

Chapter 4

226

p02 10 MPa ; M 60 ; N 77 . The smallest step of partitioning was 0.02 mm and the largest 2.6 mm ( 'xmin (only the first layer); ' xmax T

50$ C ;

h 1.04 ˜ 10

a1 867 ɆPɚ ; 2 -1

0.02 mm ; ' ymin

2.6 mm ; 'ymax

b

975 ɆPɚ ;

K ; C1 1.92 ; C2 3.04 ; Ad

0.04 mm

0.6 mm ); temperature

c

0.0305 ɆPɚ ˜ K -1 ;

2.92 .

Graphs of the calculated SIF K Ie dependencies of the mean stress on the crack axis near its tip (cell 1 in Figure 1.6b) of the two-dimensional model of compact specimen (Figure 4.166) show that with the development of the sample deformation process in case of elastic SIF level e K Ie K I* 129.9 MPa m , for some stresses (for example V yy ) at this point the monotonic nature of the increase is changed to oscillations, and e in the case of exceeding the elastic SIF level K Ie K I** 145.1 MPa m , stresses fluctuations become more intense. Solid, solid with a triangle, solid with a square, dotted, solid with a cross and solid with a circle lines correspond to stresses V xx , V yy , V zz ,

V xy , yield strength VS and stress intensity V i , respectively. Calculations showed that at the sample temperature T 50$ C , the crack began to increase in the case of exceeding the elastic SIF level K Ie K Iec 87.4 MPa m . The study of the dependence of the crack length on KIe (Figure 1.167a) shows that when the stress intensity factor e , the process of increasing the crack length reaches a value KIe KI* becomes more intense (with much larger length increments) with

increasing values of KIe . When the elastic SIF is reached K Ie K Ied 318.2 MPa m , the values of stresses V xx and V yy decrease sharply (see Figure 4.166) and become compressive (negative). At this point, the monotonic increase in the Odquist parameter (Figure 4.167b) in cell 1 slows down, and the crack length reaches l 7.87 mm . From this point, most likely, further fragmentation of the sample begins.

Numerical results analysis

227

Fig. 4.166. Stresses in cell 1 (Fig. 1.6b) on the continuation of the crack axis.

The calculated dependencies of the Odquist parameter N , which characterizes the accumulated plastic deformation in cell 1 (immediately in front of the crack tip), are shown in Figure 4.167b. While the deformation is elastic, N 0 . Then, plastic deformations begin to accumulate monotonically in cell 1 and at the moment when the crack makes the first rise, the location of cell 1 changes to a zone with smaller values of the parameter N (the value of N decreases by a jump) and the accumulation process of N begins again. In view of this, when the elastic SIF exceeds the value of KIec , the change of N , and with it the magnitude of the accumulated plastic deformation in cell 1, is oscillating (non-monotonic). In the case when

KIe

e , the amplitude of the oscillations of the parameter KI*

significantly and, as mentioned above, when

KIe

KIed ,

N

increases

its growth slows

down.

Fig. 4.167. Dependencies: ɚ) Crack length b) Odquist parameter in cell 1 (Fig. 1.6b)

228

Chapter 4

Figure 4.168 shows the temperature dependencies of the mean stress and Odquist parameter N in cell 1, obtained at a fixed value e e K I K I0 87.4 MPa m K Iec in the case of crack development (solid

V

lines) when reaching a load of this level, and assuming that the crack does not change its initial size (dashed lines).

Fig. 4.168. Temperature dependencies in cell 1 (Fig. 1.6b): ɚ) Mean stress b) Odquist parameter

In cases when the temperature allows an increase in the crack size at a e in the case of plane strain state of the specimen with a given KIe KI0 fixed crack in the area in front of the crack tip, the deformation process is characterized by a lower level of mean stresses and a higher level of accumulated plastic deformation. If at a certain temperature (in this case T  50$ C ) the stress level for crack growth is insufficient, then both approaches give the same results. When the crack is stationary and the stress intensity factor reaches the p value K Ie 60.1 MPa m  K Iec (see Figure 1.169), plastic deformations H xx occur in front of the crack tip. At this point in time, the crack is stationary. When the crack begins to move, and the stress intensity factor reaches e , the maximum plastic the value of K Iec  K Ie 90.5 MPa m  K I* p are located behind the crack tip. deformations H xx The darker areas on the diagrams correspond to higher values of deformations. It is noticeable that the zones of maximum plastic deformations (most likely) are adjacent to the free surface of the crack and its tip is located in a less plastically deformed material (as if trying to break into such a zone).

Numerical results analysis p Deformation H xx

p Deformation H yy K Ie

229 p Deformation H xy

60.1 MPa m

K Iec  K Ie

90.5 M P a m

Fig. 4.169. Diagrams of maximum plastic deformations in the area of the crack tip when the crack: ɚ) stationary b) growing

Chapter 4

230

p and p have approximately the Diagrams of plastic deformations H xx H xy

p are maximised. same shape and size; however, plastic deformations H yy

§4.10.3. Three-dimension stress-strain state Figures 4.170 and 4.171 [43] show the results of calculations of compact samples with a coefficient of hardening of the material K * 0.05 . Calculations were made at the following parameter values: L 60 mm ; ' t 0.0005 s ; a 5 mm ; B 10 mm ; H 50 mm ; l0 3 mm ; p01 8 MPa ; p02 10 MPa ; M 22; N 22 ; K 21. The smallest step of partitioning was 0.02 mm and the largest 2.6 mm ( ' xmin 0.02 mm ; ' y min

0.04 mm (only the first layer); ' xmax

temperature T

2.6 mm ; ' ymax

0.6 mm );

50 C . $

Fig. 4.170. Stresses in cell 1 (Fig. 1.7b) on the continuation of the crack axis

The model used assumes that the crack length varies. Graphs of the calculated dependence on the SIF K Ie of stresses that occur in the plane z 41.3 mm in the crack extension cell (cell 1 in Figure 1.7b) threedimensional model of a compact sample (Figure 4.172) show that with the development of the loading process in the case of a stress intensity factor e level K Ie K I* 90.5 MPa m , the monotonic nature of the increase in stresses at this point is changed to oscillating. Lines 1–5 refer to stresses

Numerical results analysis

231

V xx , V yy , V zz , stress intensity V i and yield strain V S , respectively. At the value of SIF K Ie 108.7 MPa m the oscillatory nature of the stresses increases, and at the value of SIF K Ie 114.8 MPa m stresses V yy become compressive. The crack begins to move when the SIF K Ie reaches the value K Ie 60.1 MPa m . In Figure 4.171, lines 1–3 apply to this problem and to the plane problems of the strain and stress states when the crack moves by the local criterion of brittle fracture. The crack lengths are well adjusted for cases of spatial and plane strain state as long as the SIF K Ie does not exceed K Ie 117.8 MPa m .

Fig. 4.171. Crack length l

Figure 4.172 shows the graphs of the dependencies of the Odquist parameter on the crack tip (cell 1, Figure 1.7b) in the plane z = 41.3 mm. Lines 1–5 concern cases in elastic-plastic formulation according to this problem, spatial problems in dynamic and quasi-static with a fixed crack and plane problems of strain and stress states with a crack growing up by the local criterion of brittle fracture. Lines 1–3, which apply to cases with a moving crack, are oscillating. This is due to the fact that from the moment of movement of the crack values are given in different cells, which in the process of moving the crack become cell 1.

Chapter 4

232

Fig. 4.172. SIF K Ie dependence of Odquist parameter

N in cell 1 (Fig. 1.7b)

§4.11. Problems in dynamic elastic-plastic formulation considering the process of crack growth according to the generalized local V TT criterion of brittle fracture and unloading of the material The methodology of solving the problems of three dimension and plane strain and stress states considering the unloading of the material is described. The solution of these problems for a compact specimen in a dynamic elastic-plastic formulation taking into account crack growth and material unloading makes it possible to determine the fields of plastic deformations and stresses much more accurately than when solving a corresponding quasi-static elastic-plastic problem, and also allows one to adequately model the process of crack propagation. Reactor steel 15H2NMFA was used to calculate the mathematical model of the compact sample.

§4.11.1. Plane stress state Figures 4.173 and 4.174 [35] show the calculation results for the following parameter values:

l0 3 mm ;

't

0.0005 s ;

K* 0.05 ; A

L

60 mm ;

2.5 mm ;

p02 10 MPa ; M 60 ; N 77 ; T 50ɨ ɋ .

B 10 mm ; p01 8 MPa ;

Numerical results analysis

233

Graphs of the calculated stress dependencies near the crack tip (cell 1, Figure 1.6b) show (Figure 4.173) that with the development of the specimen deformation process in the case of exceeding the elastic SIF KIe level

K1e

e K1* 90.5 MPa m , stresses at this point change the

monotonic nature of the increase to oscillating.

Fig. 4.173. K Ie dependencies of stresses in the cell 1 (Fig. 1.6b) on the continuation of the crack axis

Solid, solid with a triangle, solid with a cross and solid with a circle lines relate to stresses V xx , V yy , yield strain VS and stress intensity V i , respectively. Dotted [26], dotted with a triangle, dotted with a cross and dotted with a circle lines apply to the same stresses of the problem without unloading. In Figure 4.174, the solid and dotted lines correspond to the results of calculations in accordance with and without [26] considering the unloading of material, respectively. The calculations revealed (Figure 4.174a) that the crack length begins to differ in the case of exceeding the elastic SIF level

K1e K1erazgr 126.9 MPa m . The values of the Odquist parameter

Chapter 4

234

(Figure 4.174b) coincide when the elastic SIF acquires values from the interval 75.3 MPa m d K1e d 120.8 MPa m .

Fig. 4.174. SIF K1e dependence ɚ) Crack length l; b) Odquist parameter cell 1 (Fig. 1.6b)

N

in

§4.11.2. Plane strain state Figures 4.175–4.176 [33, 45] show the results of calculations of some important characteristics for the mechanics of fracture at the following parameter values: material hardening coefficient K* 0.05; L

B 10 mm ;

l0 3 mm ;

't

p01 8 MPa ; p02 10 MPa ; M

0.0005 s ;

60 ; N

A

77 .

60 mm ; 2.5 mm ;

The smallest

step of partitioning was 0.02 mm, and the largest 2.6 mm ( 'xmin 0.02 mm ; 'ymin 0.04 mm (only the first layer);

' xmax

a1 h

2.6 mm ;

867 ɆPɚ ;

'ymax b

0.6 mm );

975 ɆPɚ;

1.04 ˜ 10  2 K -1 ; C1 1.92 ; C2 3.04 ; Ad

temperature

c

T

50$ C ;

0.0305 ɆPɚ ˜ K -1 ;

2.92 .

The graphs of the calculated dependencies on the SIF K Ie of the crack length (Figure 4.175) with and without [30] considering the unloading of the material are almost the same. Graphs of the calculated dependencies on the SIF K Ie of the Odquist parameter taking into account (solid line) and without taking into account (dotted line [30]) unloading of the material on the extension of the crack axis near its tip (cell 1 in Figure 1.6b) of the two-dimensional model of the

Numerical results analysis

235

compact specimen (Figure 4.176) show that the Odquist parameter differs when the elastic SIF is in the range K Ie

63.2 MPa m d K1e d 84.4 MPa m .

Fig. 4.175. Crack length

Fig. 4.176. Odquist parameter in cell 1 (Fig. 1.6b)

The calculation of the SIF K Ie dependence of the Odquist parameter N , which characterizes the accumulated plastic deformation in cell 1 (immediately before the crack tip), is shown in Figure 4.176. While the deformation is elastic, N 0 . Then, in cell 1, plastic deformations begin to accumulate monotonically, and at the moment when the crack makes

Chapter 4

236

the first jump there is, as in [30], a change in the location of cell 1 to the area with smaller parameter N .

§4.11.3. Three-dimension stress-strain state Numerical investigations have been conducted for bars with a compact profile. Figures 4.177 and 4.178 [40] show the results of calculations of compact samples 60 mm long, 10 mm wide, 50 mm thick and with a 3 mm deep cut-notch in the centre, with the hardening factor of the material K* 0.05 . The distance between the bearing points was equal to 40 mm. Calculations are carried out at the following parameter values: 't 0.0005 s ; A 2.5 mm ; p01 8 MPa; p02 10 MPa ; M 22 ;

N 22 ; K

21 ; temperature T

50ɨ C .

Figures 4.177 and 4.178 show, respectively, graphs of stresses and values of the Odquist parameter N from SIF in the plane z = 41.3 mm in the cell of the crack extension (cell 1 in Figure 1.7b) of the threedimensional model of the compact sample (Figure 4.177). Solid, solid with a triangle, solid with a circle, dotted and solid with a cross lines relate to stresses V xx , V yy , V zz , stress intensity

Vi

and yield strain VS ,

respectively. Calculations show that the crack begins to grow when the SIF becomes e equal to KIe KI* 60.1 MPa m , and its length changes in the same way as in the case without taking into account the unloading of the material. e When the elastic SIF reaches KIe KI** 114.8 MPa m , the values of

stresses V xx , V yy and V zz sharply decrease, the stress V yy becomes compressive (negative) and the crack length increases sharply and reaches the value of l 7,9 mm . The dotted line in Figure 4.177 refers to the results of calculations of the problem without taking into account the unloading of material. When the SIF reaches the value KIe 78.3 MPa m , the Odquist parameter N begins to differ from the case when the hardening of the material is not considered.

Numerical results analysis

237

Fig. 4.177. Stresses

Fig. 4.178. Odquist parameter

Solving the three-dimensional problem of stress-strain state for a material with a cross section in the form of a compact sample with threepoint bending in a dynamic elastic-plastic setting taking into account the unloading process allows one to more accurately determine the fields of plastic deformations and stresses than the process of unloading the material, when the oscillations of the main breaking stresses begin. The proposed method makes it possible to improve the approach and calculation methods used in [17, 19–21, 23–25, 30, 37, 38, 145].

CHAPTER 5 DETERMINATION OF MATERIAL FRACTURE TOUGHNESS BASED ON PROBLEM SOLVING IN DYNAMIC AND QUASI-STATIC ELASTICPLASTIC FORMULATIONS

The problems of impact and the shock loading of deformable solids remain relevant and are studied in a variety of settings. In [144], the plane elastic problem of dynamic interaction of an absolutely solid percussion with an elastic isotropic homogeneous half-space at the supersonic stage of interaction under conditions of rigid adhesion of contact surfaces is considered. At the same time, multiconnectivity of a contact zone is allowed. In [131], the effect of non-stationary loading on the end surface of an elastic half-strip was investigated. In [243], an experimentalcomputational method for determining the dynamic stress intensity factor (DSIF) K I was proposed. Critical destructive loads and fracture time of short compact specimens were determined experimentally. DSIF was defined according to linear theory as a convolution of load and a single response signal, which was calculated separately by the finite element method. Comprehensive studies on the dynamics of rigid-plastic structures are given in [156, 157], where static and dynamic problems for rectangular, circular and annular plates, as well as membranes with different shapes of the load impulse, are studied in detail. There is an experimental technique [188] to study the propagation and stopping of cracks in Charpy specimens under impact load and in disk specimens under thermal shock. Numerical modelling of fracture processes makes it possible to optimize the program of real tests of materials, significantly reduce the number of quite expensive experimental studies and accelerate their completion. When modelling the destruction processes, plane [21, 29, 46, 47] and spatial [23, 31] dynamic elastic-plastic formulations are used. In [21, 23, 31], the destruction toughness was determined through numerical simulations using two-dimensional equations of dynamics of elastic-

Determination of material fracture toughness based on problem solving 239 in dynamic and quasi-static elastic-plastic formulations

plastic deformation of a material with a fixed crack. In [161], an interesting attempt was made to develop a theory for solving initial boundary problems for bodies of rotation in an elastic-plastic setting under thermopower loading. However, the limitation of the used formulation of the problem is the geometric binding of the description of the deformation process to the coordinate plane (the calculated scheme of the finite element method by spatial coordinates is used), as a result of which small deformations occur along trajectories of small curvature. In the experimental determination of the destruction toughness of the material, it is necessary to conduct many series of tests for a large number of specimens, preferably in a fairly wide range of temperature changes. Therefore, theoretical approaches that can significantly reduce the number of such tests deserve attention. In [145], a probabilistic approach to determining the destruction toughness using a quasi-static elastic-plastic model is developed. Similarly, [227] develops a probabilistic approach to determine the destruction toughness, which makes it possible to determine it more accurately than when using the method of the master of curves. Tests are performed on a small number of specimens at normal temperature. Weibull distribution parameters are almost temperatureindependent and, determined only once, are used to calculate the dependencies of destruction toughness over a wide temperature range. In [76], an experimental method for determining the destruction toughness of ceramic materials based on tests for three-point bending of beam specimens with a V-shaped notch is being developed. In [21, 29, 31, 53], it is shown that the developed method of solving spatial problems in dynamic elastic-plastic formulation makes it possible to calculate plastic deformations more accurately than in [227], and therefore more adequately determine the destruction toughness. In addition, in [53], the stress-strain state of bars made of reactor RPV steel, titanium, aluminium and silver in the form of a parallelepiped with a fixed flat notch-crack (sample with edge crack to determine the destruction toughness at three-point bending – SENB [193, 194, 204, 240]) is determined on the basis of the solution of the spatial problem formulated in the elastic-plastic dynamic formulation taking into account the possible unloading of the material. This makes it possible to improve the method proposed in [227] for the estimated determination of the destruction toughness of materials, as well as model the process of development of plastic deformations and stress concentrations in the vicinity of the crack front.

Chapter 5

240

The obtained results more accurately describe the destruction process than the solutions of quasi-static elastic-plastic problems in twodimensional and three-dimensional formulations.

§5.1. Plane problems Based on the fact that the destruction in each cell is an independent event, for the values of T0 and KI the probability Pf (KI ) of brittle destruction at a given

Pf (KI )

T T0

KI

is calculated by the formula [31, 47, 145]:

ª Y M N n,m 1 exp « ¦ ¦ Vnuc  Vd 0 «¬ Vd m 1n 1



Kº

»»¼ ,

(5.1.1)

n ,m n ,m where V nuc V1n,m  mTn,m mHn,mV eff , Y 2 H Uuc (index ‘nuc’ from English – necessary utilized cell [227]), H is the thickness of the smallscale specimen, and m and n are the indices of the unit cells formed by the partition grids along the Ox and Oy axes, respectively. Moreover, the sums of dependence (5.1.1) take into account only those cells that are destroyed by conditions: n,m n,m (5.1.2) V eff t 0, V 1n ,m t SCn ,m , V nuc t V dn,0m .

§5.1.1. Dynamic elastic-plastic formulation of the plane stress state problem – numerical results analysis When calculating the probability curves, the parameter mT was taken as follows [143, 228]:

mT

V 02 (T0 )  V02 (350$ C) .

(5.1.3)

Probability curves were made for irradiated specimens of 15H2NMFA steel, which are in a brittle state. In the temperature dependence of the yield strength (5.1.3) for steel in the brittle state [143, 227], the parameters were taken equal to:

a

867 ɆPɚ, b

975 ɆPɚ, c

0.0305 ɆPɚ ˜ K -1 ,

(5.1.4) h 1.04 ˜ 102 K -1. In determining the values of the Weibull parameters, the experimental values of the critical stress intensity factor were used from [227]:

Determination of material fracture toughness based on problem solving 241 in dynamic and quasi-static elastic-plastic formulations

K Ic (50$ C)

Pf 0.05

$

K Ic (50 C)

Pf 0.95

53, K Ic (50$ C)

Pf 0.5

88, (5.1.5)

123,

which are the points of the dashed lines at the temperature 50$ C shown in figure 5.1. Determining the minimum of the standard deviation function: 2 2 § min ¨ Pf ( K I 53)  0.05  Pf ( K I 88)  0.5  $ $ T 50 C T 50 C ©











1 2· 2

(5.1.6)  0.95 ¸ , ¹ the calculated values of the Weibull parameters were obtained: Vd 25450 MPa , K 61 and V d 0 1395 MPa . Figure 5.1 shows the

 Pf ( K I

123)

T 50$ C

curves of the coefficient KIc (T ) for embrittled steel for 0.05 m thick specimens. For comparison, the values of the Weibull parameters used in [227] are given; the round results correspond to the experimental tests. Problems for different temperature values were solved to calculate the values of functions Pf ( K I ) in (5.1.1). The temperature T0 took values in the range from 200$ C to 200$ C in steps of 50 degrees. The values of the stress intensity factor KI at the values of the probability of brittle failure 0.05, 0.5 and 0.95 were selected, and the required dependencies of the critical stress intensity factor KIc (T ) were constructed from the obtained points on the plane TOKI . The dashed line in Figure 5.1 corresponds to the results of [227]; the solid lines show the destruction toughness curves KIc (T ) for brittle steel, and expression (5.1.2) was used to determine the effective stresses.

242

Chapter 5

Fig. 5.1. Destruction toughness K Ic (T )

The obtained curves better describe the experimental results. The method of non-destructive control of the strength limits and destruction toughness, which is based on the method developed by the author to calculate the problem of plane stress state in a dynamic elastic-plastic model, makes it possible to improve the methods proposed in [144í146, 227].

§5.1.2. Dynamic elastic-plastic formulation of the plane strain state problem – numerical results analysis Probability curves (solid lines) K Ic (T ) (Figure 5.2) were formulated for irradiated 0.05 m thick specimens of 15X2NMFA reactor steel in the brittle state. When calculating them, the parameter mT is taken in the form of [227] (5.1.3). In the temperature dependence of the yield strength (5.1.3), values of parameters [227] (5.1.4) were selected for the steel in the brittle state. To calculate the Weibull distribution parameters, we used three experimental [227] values of destruction toughness (5.1.5), which are the ordinates of the corresponding points on the dashed lines. Minimization of the standard deviation function (5.1.6) gave the following values of the distribution parameters: Vd 17960 MPa , K 6 and V d 0 1590 MPa . For comparison, the figure shows the values of the Weibull parameters used in [227], and the experimental values of KIc , which were determined

Determination of material fracture toughness based on problem solving 243 in dynamic and quasi-static elastic-plastic formulations

from the three-point bending of small-scale compact specimens, are indicated by round markers. To calculate the values of the functions Pf ( KI ) , dynamic elasticplastic problems were first solved by the finite difference method [21] for different values of temperatures T0 in the range from 200$ C to 200$ C with steps of 50 degrees (effective stresses were calculated by expression (5.1.3)), and then, according to (2.5.6), the values of the stress intensity factor KI were calculated from the values of the probability of brittle failure of 0.05, 0.5 and 0.95, and the final dependencies of the critical stress intensity factor KIc (T ) were calculated from the obtained points on the plane TOKI .

Fig. 5.2. Temperature dependencies of KIc (T ) for different values of probability of destruction

For comparison, the dashed line in figure 5.2 shows the results of [227]. The obtained solid curves are generally more consistent with experimental data. Solving the refined plane strain state problem using the more precise dynamic elastic-plastic formulation makes it possible to determine the fields of plastic deformations and stresses much more accurately than in solving the quasi-static elastic-plastic plane deformation problem. The proposed method of non-destructive determination of the strength limits

Chapter 5

244

and destruction toughness makes it possible to improve the calculation methods proposed in [144–146, 227].

§5.2. Three-dimension problem of stress-strain state – solution algorithm and numerical implementation Using the calculated plastic deformations and stresses, the local criterion of brittle fracture of the polycrystalline material and the Weibull distribution function to describe the strength distribution, we find the temperature dependencies of the destruction toughness KIc . For the unit cell, the local criterion of brittle fracture in the form [21, 29, 31, 47, 227] is adopted: (5.2.1) V 1  mT H (T , N )V eff t V d , (5.2.2)

V1 t SC (N ),

where V1 is the maximal main stress, V eff V i  V 02 are the effective stresses, Vd is the effective strength of carbides or other particles on which microcracks of chipping arise,

V 02 (T ) a  c(T  273)  b exp(h(T  273)) is the yield strength, S C (N )

¬ª C1  C 2 exp(  Ad N ) º¼

1 2

is the critical stress of brittle

destruction and a, c, b, h, C1, C2 and Ad are parameters (characteristics) of the material, which are determined on the basis of the results of the destruction of small-scale specimens in their tests for three-point bending. mTH is the temperature T and the plastic deformation dependent parameter, which can be written as [21, 29, 31, 47, 227]: mTH (T , N ) mT (T )mH (N ) ,

(5.2.3)

where mH (N ) S0 / SC (N ) , mT (T ) m0VYs (T ) , S0 { SC (0) ; m0 is a constant, which is determined experimentally and VYs is the temperaturedependent component of the yield strength. Criterion (5.2.1) will be formulated in the probabilistic formulation, as a result of which the parameter Vd will be considered stochastic with the Weibull distribution function [242]:

§ § V  V ·K · d0 P (V d ) 1  exp ¨  ¨ d ¸ ¸, ¨ © V d ¹ ¹¸ ©

(5.2.4)

Determination of material fracture toughness based on problem solving 245 in dynamic and quasi-static elastic-plastic formulations

where K, V d 0 , V d are the distribution parameters. At a given temperature T0 and stress intensity factor KI , the probability of brittle fracture is calculated under the assumption that the fracture in each cell into which the grid breaks the body is an independent event, according to the formula:

Pf (KI )

T

l ,n ,m where V nuc

ª 1 K M N l ,n,m Kº Y ¦ ¦ V nuc  V d 0 » , (5.2.5) 1 exp « ¦ T0 »¼ ¬« Vd l 1 m 1 n 1 l ,n ,m l ,n ,m l ,n ,m l ,n ,m V1  mT mH V eff , Y 2 hl Uuc ; hl is the





step of the grid of the division of the calculation area along the Oz axis – (index ‘nuc’ from English – necessary utilized cell [227]); H is the thickness of the small-size specimen; l, m and n are the indices of unit cells formed by partition grids along the Ox, Oy and Oz axes, respectively. Moreover, the sums of dependence (5.2.5) take into account only those cells that are destroyed by conditions: l ,n ,m l ,n ,m (5.2.6) V eff t 0, V 1l , n , m t S C , V nuc t V dl ,0n , m .

§5.2.1. Quasi-static elastic-plastic formulation – numerical results analysis To calculate the probability curves, the results of calculations for bars 60 mm long, 10 mm wide, 50 mm thick and with a depth of cut in the middle of 3 mm and a hardening factor of the material K * 0.05 were used. Distance between supports was 40 mm; the coefficients used were p01 8 MPa; p02 10 MPa ; temperature T 50$ C . When calculating the probability curves, the parameter mT was taken as follows [21, 29, 31, 47, 227]: mT m0 (V 02 (T0 )  V 02 (350$ C)), m0 0.1 . When using dynamic elastic-plastic formulation, the use of an empirical parameter m0 is not required at all, as in (5.1.3). Thus, m0 1. Probability curves were calculated for irradiated compact specimens of 15H2NMFA reactor steel, which are in the brittle state. The obtained graphs of temperature dependence of the critical SIF KIc (T ) for brittle steel are shown in figure 5.3 by solid lines. Circles and dashed lines correspond to the experimental and calculated results of the publication, respectively [227]. Solid lines with squares correspond to the calculated results of the publication [29].

Chapter 5

246

To determine the values of the Weibull parameters required for the calculations, only three values of the critical stress intensity factor (destruction toughness) [227] (5.1.5) were used, which are the ordinates of the points of the dashed lines of Figure 5.3 with abscissa (temperature) 50$ C . Minimization of the standard deviation function (5.1.6) made it possible to calculate the values of the Weibull parameters: V d 17120 MPa , K 9 and V d 0 1603 MPa . For comparison, Table 2 shows the Weibull distribution parameters from publications [29, 227].

Fig. 5.3. Temperature dependence of critical SIF KIc (T )

Results from publication current [29] [227]

Table 2. Weibull distribution parameters K Vd0 Vd

17120 MPa 17960 MPa 4103 MPa

9 6 12

1603 MPa 1590 MPa 1804 MPa

To calculate the values of the functions Pf ( K I ) in (5.2.5), we solved problems for different values of temperatures T0 in the range from

200$ C to 200$ C with steps of 50 degrees. The SIF KI values were calculated at brittle fracture probability values of 0.05, 0.5 and 0.95, and the required dependencies of destruction toughness K Ic (T ) were calculated from the data obtained on the plane TOKI .

Determination of material fracture toughness based on problem solving 247 in dynamic and quasi-static elastic-plastic formulations

The obtained Weibull distribution parameters differ significantly from those obtained in [21, 29] (see Table 2). This indicates the sensitivity of the Weibull distribution parameters to the applied mechanical model and the method of determining the destruction toughness. It may also be necessary to take into account in some way the dependence of these parameters on temperature, despite the common belief that such dependence is extremely weak. The method developed by the author makes it possible to clarify the method given in [227]. The solution of the three-dimensional problem in quasi-static elastic-plastic formulation to determine the destruction toughness at three-point bending makes it possible to more accurately determine the fields of plastic deformations and stresses than in [227] and can be used in engineering calculations as well as dynamic formulation.

§5.2.2. Dynamic elastic-plastic formulation – numerical results analysis Probability curves (solid lines) K Ic (T ) (Figure 5.4) were calculated for irradiated 0.05 m thick specimens of 15X2NMFA reactor steel in the brittle state. When calculating them, the parameter mT is taken in the form of [143, 227, 228] (5.1.3). Probability curves were made for irradiated specimens of reactor steel 15H2NMFA, which are in a brittle state. In the temperature dependence of the yield strength (5.2.3) for steel in the brittle state [227], parameters are as in (5.1.4). When determining the values of the Weibull parameters, we used [227] the values of the critical stress intensity factor (5.1.5), which are the points of dashed lines at temperature 50$ C , as shown in [227]. Having determined the minimum of the standard deviation function (5.1.6), the values of the Weibull parameters were calculated: V d 16840 MPa , K 7 , V d 0 1793 MPa , (in [227], they were equal to V d 4103 MPa , K 12 , V d 0 1840 MPa ). For comparison, Figure 5.4 shows the values of the Weibull parameters used in [227].

248

Chapter 5

Fig. 5.4. Temperature dependencies of destruction toughness KIc (T ) for different values of probability of destruction

To calculate the values of the function Pf ( K I ) in (5.2.5), we solved problems for different values of temperature. The temperature T0 took values in the range from 200$ C to 200$ C in steps of 50 degrees. The values of the stress intensity factor KI at the values of the probability of brittle fracture of 0.05, 0.5 and 0.95 were selected, as shown in Figure 5.4, and using ordinates of the obtained points on the plane TOKI the required dependencies of the critical stress intensity factor or destruction toughness KIc (T ) were calculated. Figure 5.4 shows critical SIF KIc (T ) dependencies for 0.05 m thick embrittled specimens (solid lines), calculated using the expression V eff V i  V 02 and (5.2.6). The obtained curves fully cover the experimental results, as shown in Figure 5.4 by round marks. A method for determining the reliable three-dimensional fields of values of plastic deformations and stresses from the numerical solution of threedimensional problems in elastic-plastic dynamic mathematical formulation has been developed. The obtained results describe the destruction process more accurately than when solving a quasi-static elastic-plastic spatial

Determination of material fracture toughness based on problem solving 249 in dynamic and quasi-static elastic-plastic formulations

problem. The developed technique effectively models the process of concentration of plastic deformations and stresses on the crack area. The method developed by the author [21, 24, 28, 29, 31, 39, 54] makes it possible to improve the methods proposed in [145, 227].

CHAPTER 6 NON-STATIONARY INTERACTION ON REINFORCED COMPOSITE MATERIALS IN DYNAMIC ELASTIC-PLASTIC FORMULATION

The design of composite and reinforced or armed materials is a requirement of the modern level of production and life. Many methods of calculation and design of such materials are successfully used. In this chapter, for the design of composite and reinforced materials, a technique for solving dynamic contact problems using a more precise elastic-plastic mathematical formulation is used. To consider the physical non-linearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the non-linear problem to a solution of the sequences of linear problems and which is described in previous chapters. It is very important that composite materials are made on a base of glass. Glass is a non-crystalline, often transparent amorphous solid that has widespread practical and technological use in the modern industry. The most familiar types of manufactured glass are ‘silicate glasses’ based on the chemical compound silica (silicon dioxide, or quartz). Glass has high strength and is not affected by the processes of aging of the material, corrosion or creep. In addition, this material is cheap and widely available. Glass can be strengthened, for example in a melt quenching process. If the cooling is fast enough (relative to the characteristic crystallization time), then crystallization is prevented and, instead, the disordered atomic configuration of the supercooled liquid is frozen into a solid state. This increases the strength properties of the glass. The reinforced composite beam is rigidly linked to an absolutely solid base on which an absolutely solid impactor acts from above in the centre of a small area of initial contact.

Non-stationary interaction on reinforced composite materials in dynamic elastic-plastic formulation

251

§6.1. Plane strain state of reinforced composite glass materials In contrast to the traditional plane strain, when one normal stress is equal to a certain constant value, for a more accurate description of the deformation of the sample, taking into account the possible increase in longitudinal elongation, we present this normal stress as a function that depends on the parameters that describe the bending of a prismatic body that is in a plain strain state (§1.6.2, §2.6.2 and §3.6.1). The problem of a plane strain state of a beam made from the composite reinforced doubleglazed material is solved. The reinforced or armed material consists of two layers: the upper (first) thin layer of solid steel and the lower (second) main layer of glass. Glass is a very strong and very fragile material at the same time. The fragility of glass is due to the fact that there are many microcracks on the surface, and when a load is applied to the glass surface, these microcracks begin to grow and lead to the destruction of glass products. If we glue or immobilize the tops of the microcracks on the surface, we will get a strong reinforced armed material that will be lighter, stronger and not subject to degradation of material properties such as aging, corrosion or creep. The upper reinforcing layer of metal can be applied to the glass surface by sputtering so that the metal atoms of the steel penetrate deeply, fill the microcracks and bind their tops. The top layer can be quite thin. Glass is also convenient in that it can be poured into the frame of the reinforcement and thus can be further strengthened. As reinforcing elements, metal wire, polysilicate, polymer or polycarbon compounds, which can have a fairly small thickness, can be used. The thickness of such reinforcing materials can be equal to the thickness of several atomic layers, such as graphene.

§6.1.1. Problem formulation and solution algorithm In §1.6.2, §2.6.2 and §3.6.1 [197], a new approach to solving the problems of impact and non-stationary interaction in the elastoplastic mathematical formulation was developed. In this section, the action of the striker is replaced by a distributed load in the contact area, which changes according to a linear law. The contact area remains constant. The developed elastoplastic formulation makes it possible to solve impact problems when the dynamic change in the boundary of the contact area is considered, and based on this, the movement of the striker as a solid body with a change in the penetration speed is taken into account. Also, such an

Chapter 6

252

elastoplastic formulation makes it possible to consider the hardening of the material in the process of non-stationary and impact interaction. In this section, we investigate the impact process of a hard body with the plane area of its surface on the top of a composite beam that consists of an initial thin metal layer and a second, main glass layer. The fields of plastic deformations and stresses were determined relative to the size of the area of initial contact. Deformations and their increments [20–24], the Odquist parameter, effective and principal stresses are obtained from the numerical solution of the dynamic elastic-plastic interaction problem of an infinite composite beam { L / 2 d x d L / 2; 0 d y d B; f d z d f} in the plane of its cross-section in the form of a rectangle. It is assumed that the stress-strain state in each cross-section of the cylinder is the same, close to the plane deformation, and, therefore, it is necessary to solve the equation for only one section in the form of a rectangle 6 L u B with two layers: an initial steel layer { L / 2 d x d L / 2; f d z d f; B  h d y d B} and

a

second

glass

layer

{ L / 2 d x d L / 2;

0 d y d B  h; f d z d f} with a notch-crack with length l along the segment and contacted along an absolute hard half-space { y d 0} . We assume that the contact between the lower surface of the first metal layer and the upper surface of the second glass layer is rigid.

Fig. 6.1. Calculation scheme

From above on a body the absolutely rigid drummer is contacting along a segment { x d A;

y

B} . Its action is replaced by an evenly

 P in the contact region, which changes over time as a linear function P p01  p02t . Given the symmetry of the deformation

distributed stress

process relative to the line x

0 , only the right part of the cross-section

Non-stationary interaction on reinforced composite materials in dynamic elastic-plastic formulation

253

is considered below (Fig. 6.1). The calculations use known methods for studying the quasi-static elastic-plastic [23, 25–27] model, considering the non-stationarity of the load and using numerical integration implemented in the calculation of the dynamic elastic model [28–30]. The equations of the plane dynamic theory are considered, for which the components of the displacement vector u (u x , u y ) are related to the components of the strain tensor by Cauchy relations (1.5.1), and the equations of motion of the medium have the form (1.6.1):

x

0, 0  y  B : u x

x

L / 2, 0  y  B : V xx

0, V xy

0,

y

0, 0  x  L / 2 : u y

0, V xy

0,

y

B, 0  x  A : V yy

 P, V xy

y

B, A  x  L / 2 : V yy

ux t u x t

0

0, u y

0

0, u y

0, V xy

t 0

0, u z t

t 0

0, u z t

0,

0, V xy 0 0

(6.1.1)

0, 0.

0, (6.1.2)

0.

The determinant relations of the mechanical model are based on the theory of non-isothermal plastic flow of the medium with hardening under the condition of Huber–Mises fluidity. The effects of creep and thermal expansion are neglected. Considering the components of the strain tensor by the sum of its elastic and plastic components [31, 32], we use expression (1.5.4). The material is strengthened with a hardening factor K* [25–29] as in (1.5.6).

§6.1.2. Numerical solution The use of finite differences [34] with variable partition steps for wave equations is justified in [35], with an accuracy of calculations giving an



2

2

error of no more than O ( 'x)  ('y )  ( 't )

2

, where ' x , 'y

and ' t are increments of spatial variables x and y and time variable t.

Chapter 6

254

Figures 6.2–6.11 show the results of calculations of two-layer specimens [197] with a hardening factor of the material K* 0.05 . The first, upper layer is made from hard steel. The second, lower layer has been made from quartz glass. Contact between the two layers is considered to be optimized. Calculations were made at the following parameter T 50 $C ; L 60 mm ; B 10 mm ; values: temperature h 0.3 mm ; 't 3.21˜108 s ; p01 8 MPa ; p02 10 MPa ;

M 62 ; N 100 . The smallest splitting step was 0.005 mm and the largest 2.6 mm ('xmin 0.005 mm; 'ymin 0.01 mm (only the first layer); 'xmax 2.6 mm ; 'ymax 0.65 mm ). Figure 6.2 shows plots of the Odquist parameter N in the cell of the

first layer, which is located in the centre of the specimen at a depth of 0.25 mm. Solid, dotted and solid with a circle lines correspond to cases where the size of the contact zone was equal to

a a2

0.5 mm

and

a a3

a 2 A a1 0.3 mm ,

0.7 mm , respectively.

Figures 6.3–6.5, 6.6–6.8 and 6.9–6.11 show the fields of the Odquist parameter

N

and

t1 2.57 ˜106 s ,

normal

V xx

stresses

t2 3.82 ˜106 s

and

and

V yy

at

times

t3 5.13 ˜106 s ,

respectively. From Figs. 6.2–6.5, it can be seen that the smaller the contact zone, the bigger the plastic deformations; however, at the end of the process of nonstationary interaction, they are of the same degree. Figures 6.3–6.11 show that the highest stresses occur in the upper layer of the metal and the process of accumulation of plastic deformations is more intense there. Figures 6.6í6.8 show areas where the normal stresses V xx in the lower layer are tensile. In the case when the contact zone

t

t1

a a1 ,

at the moment of time

the area with tensile stresses is located in the middle under the

boundary between the layers. This is due to the fact that compressive stresses arise in the upper layer quickly, and the contact between the layers is ideally rigid.

Non-stationary interaction on reinforced composite materials in dynamic elastic-plastic formulation

Areas where the normal stresses

V yy

255

in the lower layer are tensile are

visible in Fig. 6.11.

Fig. 6.2. Odquist parameter

Fig. 6.3. Odquist parameter

N

when

t

N

when

a a1 , t

t1

t1 , a) a a1 , b) a a2 , c) a a3

Chapter 6

256

Fig. 6.4. Odquist parameter

N

when

t t2 , a) a a1 , b) a a2 , c) a a3

Fig. 6.5. Odquist parameter

N

when

t

t3 , a) a a1 , b) a a2 , c) a a3

Non-stationary interaction on reinforced composite materials in dynamic elastic-plastic formulation

Fig. 6.6. Stress

V xx

when

t

t1 , a) a a1 , b) a a2 , c) a a3

Fig. 6.7. Stress

V xx

when

t t2 , a) a a1 , b) a a2 , c) a a3

257

Chapter 6

258

Fig. 6.8. Stress

V xx

when

t

t3 , a) a a1 , b) a a2 , c) a a3

Fig. 6.9. Stress

V yy

when

t

t1 , a) a a1 , b) a a2 , c) a a3

Non-stationary interaction on reinforced composite materials in dynamic elastic-plastic formulation

Fig. 6.10. Stress

V yy

when

t t2 , a) a a1 , b) a a2 , c) a a3

Fig. 6.11. Stress

V yy

when

t

259

t3 , a) a a1 , b) a a2 , c) a a3

At times t2 and t3 , when the contact zone is a1 , and at times t1 , t2 and t3 , when the contact zone is equal to a1 and stresses

V xx

a2 , areas with tensile

in the lower layer are located near its lower boundary. This

is due to the fact that the deformation process has a wave characteristic and the contact of the lower boundary of the lower layer with an absolutely rigid base is ideally rigid.

Chapter 6

260

This also explains that at the moment of time t3 for all cases of the length of the contact zone, the areas where the normal stresses

V yy

are

tensile are located near the lower boundary y 0 . The developed methodology of solving dynamic contact problems in an elastic-plastic dynamic mathematical formulation makes it possible to adequately model the processes of impact, shock and non-stationary contact interaction with the elastic composite base. In this section, the process of impact on a two-layer base, consisting of an upper thin layer of metal and a lower main layer of glass, is effectively modelled. The fields of summary plastic deformations and normal stresses arising in the base are calculated depending on the size of the area of initial contact between the impactor and the upper surface of the base. The area under the stamp in the glass layer under the metal layer is shown where there are small tensile normal stresses

V xx , which are most likely due to the propagation

of impact waves in the base material. The results obtained make it possible to design new composite reinforced armed materials.

CONCLUSIONS

1. A mathematical approach in elastic formulation was implemented to solve plane and axisymmetric problems of impact of elastic shells, described by a system of differential equations derived from Kirchhoff– Love’s hypotheses, on the surface of elastic half-space and absolutely rigid cylinders with or without flat area and cylindrical Kirchhoff–Love fine shells on the surface of the elastic layer, the lower surface of which is coupled with a rigid half-space. This approach makes it possible to conduct an accurate study of the processes of impact and non-stationary contact interaction at a finite time interval comparable to the duration of the impact. Using the Laplace transformation and the method of development into series according to their eigenfunctions, the problems are reduced to an infinite system of Volterra’s integral equations of the second kind, which is numerically realized by the method of reduction and mechanical squaring. It is shown that such an elastic model does not allow the use of shells described by a system of differential equations, which are derived on the basis of the hypotheses of S.P. Tymoshenko. This, and the ability to obtain the values of unknown functions only at the boundary of the contacting bodies, are the limitations of the chosen elastic model. A numerical-analytical method for solving linear plane and axisymmetric problems of penetration of thin elastic cylindrical and spherical shells into an elastic half-space and layer in the case of a moving boundary of contact area taking into account spherical shells is developed. The dependencies of the main dynamic characteristics of the process on the initial impact velocity, mass and thickness of the elastic shell and the physical properties of the half-space or layer and shell material – contact stresses, normal displacements of the medium, reaction forces of the elastic base and penetration rate – are investigated. 2. A mathematical approach to solving problems of non-stationary interaction and impact in a more accurate elastic-plastic formulation has been developed, which successfully overcomes the difficulties of the elastic model, allowing for the reliable study of the processes of nonstationary interaction and impact at the plastic stage and the determination of plastic deformations, destruction toughness or critical intensity factor

KIc . The approach more accurately and adequately models the processes

262

Conclusions

of crack growth, material unloading and material softening, taking into account the rate of deformation. The solution of three-dimensional problems and problems of plane stress and strain states provides for taking or not taking into account unloading of the material and crack growth for samples with a cross-section in the form of a parallelepiped or rectangle with a cut in the middle (compact profile). These problems more reliably simulate the three-point bending process to determine fracture toughness in a dynamic elastic-plastic setting. This makes it possible to determine the fields of plastic deformations and stresses much more accurately than when solving quasi-static elastic-plastic spatial and plane problems. Numerical solutions were obtained for a material with a longitudinal cross-section in the shape of a rectangle with a notch-cut in the middle with a three-point bend and using the finite difference method. Numerical results are given in the form of figures. It is shown that the elastic model of impact is convenient to use to calibrate the numerical process in solving impact problems and non-stationary interaction in the elastic-plastic formulation on the elastic stage when stresses did not reach yield strain and plastic deformations are absent. The developed methodology of solving non-stationary problems in elastic-plastic mathematical formulation makes it possible to determine a probabilistic approach more precisely: it is not necessary to use experimental parameter m0 at all (in [143, 227, 228] m0 0.1). 3. The developed methodology of solving dynamic contact problems in an elastic-plastic mathematical formulation makes it possible to adequately model the processes of impact, shock and non-stationary contact interaction with the elastic composite base.

REFERENCES

Aleksandrov V.M., Romalis B.L. Contact problems in mechanical engineering. Mashinostroenie. Moscow (1986). p. 176. [in Russian] 2. Aleksandrov E.V., Sokolinsky V.B. Applied theory and calculations of impact systems. Nauka. Moscow (1969). p. 199. [in Russian] 3. Aliev M.M., Shapiro G.S. On deformations of plastic plates moving in a resisting medium. Ɉ ɞɟɮɨɪɦɚɰɢɹɯ ɩɥɚɫɬɢɱɟɫɤɢɯ ɩɥɚɫɬɢɧ, ɞɜɢɠɭɳɢɯɫɹ ɜ ɫɨɩɪɨɬɢɜɥɹɸɳɟɣɫɹ ɫɪɟɞɟ. Plast. deformation of the light and special alloys. Moscow (1978). 1. 140–144. [in Russian] 4. Amandosov A.A. Movement of a rigid plastic plate in a resisting medium. News of the Academy of Sciences of the USSR. Mechanics and engineering (1963). 4. 130–132. [in Russian] 5. Amandosov A.A., Stamgaziev K.M. Dynamics of a freely supported plate in a resisting medium. News of the Academy of Sciences of KazSSR. Series Phys.-Maths (1973). 5. 33– 39. [in Russian] 6. Amandosov A.A., Uskombaev A.R. Movement of rigid-plastic structures in a resisting medium. Dynamics of solid medium. AlmaAta (1982). 107–116. [in Russian] 7. Arkulis G.E., Dorogobid V.G. Theory of plasticity. Metallurgy. Moscow (1987). p. 352. [in Russian] 8. Afanasiev E.F., Cherepanov G.P. Some dynamic problems of the theory of elasticity. Applied Mechanics and Mathematics (1973). 37(4). 618–639. [in Russian] 9. Afanas'eva O.A., Mikhailova E.Yu., Fedotenkov G.V. An arbitrary stage of non-stationary contact interaction of a spherical shell and an elastic half-space. Problems of computational mechanics and structural strength. Dnepropetrovsk (2012). 20. 19-26. [in Russian] 10. Babaev A.E., Kubenko V.D. Mechanics of structures made of composite materials. Interaction of non-stationary waves with shells. Naukova Dumka. Kyiv (1983). 2(20). 431–450. [in Russian] 11. Babaev A.E., Savin V.G. Nonstationary Oscillations of a Sectioned Piezoceramic Cylinder under the Action of an Electric Field. Applied Mechanics. Kyiv (1986). 22(12). 26–31. [in Russian] 12. Bagdoev A.G. Spatial non-stationary motions of a solid medium with shock waves. Publishing House of the Academy of Sciences of the ArmSSR. Yerevan (1961). p. 276. [in Russian] 1.

264

References

13. Batuev G.S., Golubkov Yu.V., Efremov A.K. et al. Engineering methods for studying impact processes. Mashinostroenie. Moscow (1977). p. 239. [in Russian] 14. Belyaev N.M. Proceedings on the theory of elasticity and plasticity. GITTL. Moscow (1957). p. 632. [in Russian] 15. Bogdanov V.R., Popov S.N. Vertical Impact of a Thin Elastic Cylindrical Shell on the Surface of an Elastic Half-Spaceɸ Proceedings of the 16th Scientific Conference of Yong Scientists of the Institute of Mechanics of the Academy of Sciences of Ukraine. Manuscript Deposited in VINITI No. 4260 – ȼ91. Kyiv (12/11/1991). [in Russian] 16. Bogdanov V.R., Popov S.N. Vertical impact of a thin elastic spherical shell on the surface of an elastic half-space. Proceedings of the 17th Scientific Conference of Yong Scientists of the Institute of Mechanics of the Academy of Sciences of Ukraine. Manuscript Deposited in UkrNINTI No. 1022 - Uk92. Kyiv (07/07/1992). 24–30. [in Russian] 17. Bogdanov V.R. The problem of collision of thin-walled shells with an elastic medium. Institute of Mechanics of the Academy of Sciences of Ukraine. Manuscript Deposited in GNTB No. 2388 Uk94. Kyiv (13/12/1994). p. 11. [in Russian] 18. Bogdanov V.R., Levitskaya E.R. The problem of rotation of an absolutely rigid cylinder coupled to a one-layer base. Ɂɚɞɚɱɚ ɨ ɩɨɜɨɪɨɬɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɰɢɥɢɧɞɪɚ, ɫɰɟɩɥɟɧɧɨɝɨ ɫ ɨɞɧɨɫɥɨɣɧɵɦ ɨɫɧɨɜɚɧɢɟɦ. Institute of Mechanics of the Academy of Sciences of Ukraine. Manuscript Deposited in UkrNINTI No. 2150 - Uk94. Kyiv (24/11/1994). p. 26. [in Russian] 19. Bogdanov V.R. Construction of the Weibull distribution when using samples of small cross-sections to determine the fracture toughness of a material // II All-Ukrainian Scientific and Technical Conference of Young Scientists and Specialists. Welding and related technologies. Kyiv (25–27/06/2003). 24. [in Russian] 20. Bogdanov V.R. Determining the fracture toughness of the material using samples of small cross-sections. International Conference. Modern problems of welding and resource of structures. Kyiv (24– 27/11/2003). 93. [in Ukrainian] 21. Bogdanov V.R. Determination of fracture toughness of a material based on numerical modeling of the plane stress state. // Bulletin of Kyiv National University. Series of Physical and Mathematical Sciences. Kyiv (2008). No. 3. 51–56. [in Ukrainian]

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

265

22. Bogdanov V.R., Levitska O.R., Prykhodko T.B., Radzivil O.Yu., Samborska L.S. Investigation of a plane impact of a shell on an elastic layer. Collection of scientific works of NTU. Kyiv (2009). 283 – 292. [in Ukrainian] 23. Bogdanov V.R. Three-dimensional dynamic problem of the concentration of plastic deformations and stresses near the top of crack. Bulletin of Kyiv National University. Series Physical and Mathematical Sciences. Kyiv (2009). No. 2. 51–56. [in Ukrainian] 24. Bogdanov V.R., Sulym G.T. Determination of material parameters on the basis of numerical modelling of the plane deformed state. International Conference Modern problems of mechanics. Lviv (79/12/2009). 193. [in Ukrainian] 25. Bogdanov V.R., Sulym G.T. Dynamic growing of a crack in a compact specimen according to the elastic-plastic model of the plane stress state. Bulletin of the Kyiv National University. Phys.-math. Series Sciences. Kyiv (2010). No. 4. 51–54. [in Ukrainian] 26. Bogdanov V.R., Sulym G.T. Modelling of crack movement based on numerical solution of the plane stress state problem. Bulletin of Lviv University. Phys.-math. Series Sciences. Lviv (2010). No. 73. 192– 204. [in Ukrainian] 27. Bogdanov V.R., Sulym G.T. About the solution of the problem of a plane deformed state of a material, taking into account elastic-plastic deformations under dynamic loading. Theoretical and applied mechanics. Donetsk (2010). 1(47). 59–66. [in Russian] 28. Bogdanov V.R., Sulym G.T. To determine the fracture toughness of a material based on numerical simulation of a three-dimensional dynamic problem. International Conference. Strength of Materials and Structures at Low Temperatures. Institute of Strength Problems. Kyiv (25–27/05/2010). Approbation. [in Ukrainian] 29. Bohdanov V. R., Sulym H. T. Evaluation of crack resistance based on the numerical modelling of the plane strained state. Materials Science. (2011). 46(6). 723 734. DOI: 10.1007/s11003-011-9346-0. [in Ukrainian and English] 30. Bogdanov V.R., Sulym G.T. Modelling of crack growth based on the numerical solution of the plane deformed state problem. Collection of scientific works. Problems of computational mechanics and strength of structures. Dnipropetrovsk (2011). No 15. 33–44. [in Russian] 31. Bogdanov V.R., Sulym G.T. Determining the fracture toughness of a material based on numerical modelling of a three-dimensional dynamic problem. International Scientific and Technical Collection.

266

32.

33.

34.

35.

36.

37. 38.

39.

40.

References

Reliability and durability of machines and equipment. Kyiv (2010). No 33. 153–166. [in Ukrainian] Bogdanov V.R., Sulym G.T. On the solution of the problem of the plane stress state of the material, taking into account the process of unloading and elastic-plastic deformations under dynamic loading. Collection Consonance-2011, Institute of Hydromechanics of the National Academy of Sciences of Ukraine. Kyiv (2011). 44–49. [in Russian] Bogdanov V.R., Sulym G.T. Towards the solution of the problem of the plane stress state of the material taking into account the unloading process and elastic-plastic deformations under dynamic loading. Acoustic Symposium Consonance-2011. Kyiv (27–29/09/2011). 9. [in Ukrainian] Bogdanov V.R., Sulym G.T. Plane deformed state of a compact specimen taking into account the processes of unloading and crack growth. International Collection. Modern problems of engineering mechanics. Lutsk (2011). 24–30. [in Ukrainian] Bogdanov V.R., Sulym G.T. Plane strain state of a compact sample taking into account the processes of unloading and crack growth. II International Symposium. Modern problems of engineering mechanics. Lutsk (01/11/2011). 152. [in Ukrainian] Bogdanov V.R., Sulym G.T. Dynamic development of a crack in a compact specimen according to the elastic–plastic model of the plane stress state with consideration of material unloading. Bulletin of the Kyiv National University. Phys.-Math. Series Sciences. Kyiv (2011). No 3. 53–56. [in Ukrainian] Bogdanov V.R. On the three-dimension deformation of specimen of an elastoplastic material with a compact shape profile. Theoretical and Applied Mechanics. Donetsk (2011). 49(3). 51–58. [in Russian] Bogdanov V.R., Sulym G.T. The plane stress state of the material taking into account the unloading process in the dynamic elasticplastic formulation // Acoustic Bulletin. Institute of Hydromechanics of the National Academy of Sciences of Ukraine. Kyiv (2011). 14(4). 3–8. [in Ukrainian] Bogdanov V.R. Determining the fracture toughness based on solving the three-dimension problem in a quasi-static elastoplastic formulation. Bulletin of the Lviv National University. Phys.-Math. Series Sciences. Lviv (2012). 76. 65–75. [in Ukrainian] Bogdanov V.R., Sulym G.T. Modeling of crack movement based on numerical solution of a three-dimension problem taking into account

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

41.

42.

43.

44.

45.

46.

47.

48.

49.

267

the material unloading process. Bulletin of Kyiv National University. Phys.-Math. Series Sciences. Kyiv (2012). 3. 61–66. [in Ukrainian] Bogdanov V.R., Sulym G.T. Dynamic growing of a crack in a compact specimen according to the elastic-plastic model of the plane strain state. Bulletin of the Faculty of Mechanics and Mathematics of the Kyiv National University. Kyiv (2012). 27. 29–35. [in Ukrainian] Bogdanov V.R., Sulym G.T. Dynamic development of a crack in a compact sample according to the elastic–plastic model of plane deformed and stressed states with unloading of the material. Bulletin of the Faculty of Mechanics and Mathematics of the Kyiv National University. Kyiv (2012). 28. 52–56. [in Ukrainian] Bogdanov V.R., Sulym G.T. Modelling of crack movement in a compact specimen based on the numerical solution of a threedimension problem. Collection of Scientific Works. Methods of solving applied problems of the mechanics of a deformed solid body. Dnipropetrovsk (2012). 13. 60–68. [in Ukrainian] Bogdanov V.R., Sulym G.T. Three-dimension modelling of the crack growth process based on numerical solution. Collection of Scientific Works. Problems of computational mechanics and strength of structures. Dnipropetrovsk (2012). 19. 10–19. [in Ukrainian] Bogdanov V.R. Modelling of crack growth based on the numerical solution of the plane deformed state problem taking into account the unloading process. Collection of Scientific Works. Problems of computational mechanics and strength of structures. Dnipropetrovsk (2012). 18. 12–19. [in Russian] Bogdanov V.R., Sulym G.T. Plane strain state in a compact specimen with nonpropagating crack with regard for the process of unloading. Journal of Mathematical Sciences. (2013). 194(3). 285–292. DOI: 10.1007/s10958-013-1528-x. [in Ukrainian and English] Bogdanov V.R., Sulym G.T. Modelling the growth of plastic deformations during impact based on the numerical solution of the problem of a plane stress state. Bulletin of the MAI. Moscow (2013). 20(3). 195–204. [in Russian] Bogdanov V. R., Sulym G. T. Plain deformation of elastoplastic material with profile shaped as a compact specimen (dynamic loading). Mechanics of Solids. Moscow (3013). 48(3). 329–336. DOI 10.3103/S0025654413030096. [in Russian and English] Bogdanov V.R., Sulym G.T. Growing of a spatial crack in a compact specimen taking into account the material unloading process. Theoretical and Applied Mechanics. Donetsk (2013). 52. 37–44. [in Ukrainian]

268

References

50. Bogdanov V.R. Modeling the accumulation of plastic deformations based on the numerical solution of a plane problem taking into account the material unloading process. Bulletin of the Kyiv National University. Phys.-Math. Series Science. Kyiv (2013). 3. 85–88. [in Ukrainian] 51. Bogdanov V.R., Sulym G.T. About the spatial modelling of the process of accumulation of plastic deformations taking into account the process of material unloading under dynamic loading. Collection Consonance-2013. Institute of Hydromechanics of the National Academy of Sciences of Ukraine. Kyiv (2013). 32–37. [in Russian] 52. Bogdanov V.R., Sulym G.T. To the solution of the problem of spatial modelling of the process of accumulation of plastic deformations taking into account the process of material unloading under dynamic loading. Acoustic Symposium Consonance-2013. Institute of Hydromechanics of the National Academy of Sciences of Ukraine. Kyiv (27–29/09/2013). 36. [in Ukrainian] 53. Bogdanov V.R. A plane problem about the impact of a rigid cylinder on an elastic layer. Bulletin of the Kyiv National University. Mechanics and Mathematics. Kyiv (2015). 34. 42–47. [in Ukrainian] 54. Bogdanov V.R., Sulim G.T. Determination of the material fracture toughness by numerical analysis of 3D elastoplastic dynamic deformation. Mechanics of Solids. 51(2). 206-215. DOI 10.3103/S0025654416020084. [in Russian and English] 55. Bozhidarnyk V.V., Sulym G.T. Elements of the theory of plasticity and strength. SVIT. Lviv (1999). p. 676. [in Ukrainian] 56. Boli B., Weiner J. Theory of temperature stresses. Mir. Moscow (1964). p. 360. [in Russian] 57. Borodich F.M. Similarity in the problem of contact of elastic bodies. Applied Mechanics and Mathematics. Moscow (1983). 47(3). 519– 521. [in Russian] 58. Borodich F.M. On the problems of interaction of blunt bodies with an acoustic medium. Applied Mechanics and Mathematics. Moscow (1988). 52(4). 610–616. [in Russian] 59. Borodich F.M. Dynamic contact of a blunt body with an anisotropic linearly elastic medium. Doklady AN SSSR. Moscow (1990). 310(1). 38 – 42. [in Russian] 60. Vestyak A.V., Gorshkov A.G., Tarlakovsky D.V. Nonstationary interaction of deformable bodies with the environment. Itogi Nauki i Tekhniki. Mechanics of a Deformable Solid Body. VINITI. 15. Moscow (1983). 69–148. [in Russian]

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

269

61. Vestyak A.V., Gorshkov A.G., Tarlakovsky D.V. Non-stationary oscillations of elastic (acoustic) media with spherical interfaces. Interaction of plates and shells with liquid and gas. Publishing house of Moscow University. Moscow (1984). 3–26. [in Russian] 62. Vestyak A.V., Gorshkov A.G., Tarlakovsky D.V. Vertical impact by an absolutely rigid body of rotation on an elastic half-space. VI AllUnion Congress on Theoretical and Applied Mechanics. Fan. Tashkent (1986). 163–164. [in Russian] 63. Vestyak A.V., Igumenov L.A., Tarlakovskiy D.V., Fedotenkov G.V. Effect of unsteady pressure on a thin spherical shell with an elastic filler. Computational Continuum Mechanics. Moscow (2016). 9(4). 443–452. DOI: 10.7242/1999-6691/2016.9.4.37 [in Russian] 64. Volmir F.S. Shells in liquid and gas flow. Problems of aeroelasticity. Nauka. Moscow (1976). p. 416. [in Russian] 65. Volmir F.S. Shells in liquid and gas flow. Problems of hydroelasticity. Nauka. Moscow (1979). p. 320. [in Russian] 66. Gavrilenko V.V. Non-stationary loads during penetration of an ellipsoid of revolution into a compressible fluid. Applied Mechanics. Kyiv (1986). 22(8). 113 – 117. [in Russian] 67. Gavrilenko V.V. Determination of non-stationary loads and stressstrain state when axisymmetric bodies penetrate into a compressible fluid. Abstract of the dissertation of Candidate Phys.-Maths. Sciences. Kyiv (1986). p. 16. [in Russian] 68. Gavrilenko V.V. Determination of the stress–strain state of thin elastic shells penetrating a compressible liquid. Applied Mechanics. Kyiv (1988). 24(9). 30–37. [in Russian] 69. Gavrilenko V.N. Penetration of a parabolic cylinder into a compressible fluid. Applied Mechanics. Kyiv (1984). 20(2). 65–71. [in Russian] 70. Gavrilenko V.N. Investigation of hydrodynamic loads during the penetration of symmetrical bodies into a compressible fluid. Abstract of the dissertation of Candidate Phys.-Maths. Sciences. Kyiv (1984). p. 15. [in Russian] 71. Gavrilenko V.N., Kubenko V.D. Plane problem of penetration of rigid bodies into a compressible fluid. Applied Mechanics. Kyiv (1985). 21(4). 41–49. [in Russian] 72. Galiev Sh.U. Dynamics of interaction of structural elements with a pressure wave in a fluid. Naukova Dumka. Kyiv (1977). p. 172. [in Russian] 73. Galiev Sh.U. Dynamics of hydroelastic-plastic systems. Naukova Dumka. Kyiv (1981). p. 286. [in Russian]

270

References

74. Galin L.A. Contact problems of the theory of elasticity. Gostekhizdat. Moscow-Leningrad (1953). p. 264. [in Russian] 75. Gvozdev A.A. To the calculation of structures for the action of a blast wave. Construction Industry. Moscow (1943). 1 and 2. 18–21. [in Russian] 76. Gogotsi G.A. Study of the crack resistance of ceramics on specimens with a V-shaped notch. Problems of Strength. Kyiv (2000). 1. 120– 127. [in Russian] 77. Golovchan V.T., Kubenko V.D., Guz A.N., Grinchenko V.T. Spatial problems of the theory of elasticity and plasticity. 5. Dynamics of elastic bodies. Naukova Dumka. Kyiv (1986). p. 286. [in Russian] 78. Goldenblat I.I., Nikolaenko H.A. Calculation of structures for the action of seismic and impulse forces. Gosstroyizdat. Moscow (I960). p. 320. [in Russian] 79. Goldsmith V. Impact. Gosstroyizdat. Moscow (1965). p. 447. [in Russian] 80. Goldsmith V. Impact and contact phenomena at medium speeds. Physics of fast processes. Mir. Moscow (1971). 2. 153–203. [in Russian] 81. Goman O.G. On the self-similar problem of indentation of a flat punch into an elastic half-space. Applied Mechanics. Kyiv (1983). 19(10). 55–60. [in Russian] 82. Gorshkov A.G. Non-stationary interaction of plates and shells with continuous media. News of Academy of Sciences of the USSR. Solid mechanics. Moscow (1981). 4. 177–189. [in Russian] 83. Gorshkov A.G., Tarlakovsky D.V. Vertical Impact of a Cylinder on an Elastic Half-Planeɸ Problems of Dynamics of Interaction of Deformable Media. Abstracts of Reports. Publishing House of the Academy of Sciences of ArmSSR. Yerevan (1984). 125 – 131. [in Russian] 84. Gorshkov A.G., Tarlakovsky D.V. Resultant reactions in the spatial problem of impact by a rigid body on an elastic half-space. News of Academy of Sciences of the USSR. Solid mechanics. Moscow (1987). 5. 95 – 98. [in Russian] 85. Gorshkov A.G., Tarlakovsky D.V. Dynamic contact problem for a circular cylindrical shell and an elastic half-space. Strength of plates and shells under combined effects. Moscow Aviation Institute. Moscow (1987). 16 – 25. [in Russian] 86. Gorshkov A.G., Tarlakovsky D.V. Dynamic contact problem for absolutely rigid bodies and elastic half-space. Preprint in Moscow Aviation Institute. Moscow (1989). p. 48. [in Russian]

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

271

87. Gorshkov A.G., Tarlakovsky D.V. Dynamic contact problems with moving boundaries. Nauka. Fizmatlit. Moscow (1995). p. 352. [in Russian] 88. Grigolyuk E.I., Gorshkov A.G. Interaction of elastic structures with fluid (impact and immersion). Shipbuilding. Leningrad (1976). p. 200. [in Russian] 89. Grinchenko V.T., Meleshko V.V. Harmonic oscillations and waves in elastic bodies. Naukova Dumka. Kyiv (1981). p. 283. [in Russian] 90. Guz A.N., Golovchan V.T. Diffraction of elastic waves in multiply connected bodies. Ⱦɢɮɪɚɤɰɢɹ ɭɩɪɭɝɢɯ ɜɨɥɧ ɜ ɦɧɨɝɨɫɜɹɡɧɵɯ ɬɟɥɚɯ. Naukova Dumka. Kyiv (1981). p. 283. [in Russian] 91. Guz A.N., Kubenko V.D., Cherevko M.A. Diffraction of elastic waves. Naukova Dumka. Kyiv (1978). p. 308. [in Russian] 92. Guz A.N., Kubenko V.D. Methods for calculating shells. 5. Theory of non-stationary aerohydroelasticity of shells. Naukova Dumka. Kyiv (1983). p. 400. [in Russian] 93. Guz A.N., Kubenko V.D., Babaev A.E. Hydroelasticity of shell systems. Vyscha Shkola. Kyiv (1984). p. 206. [in Russian] 94. Davis R.M. Stress waves in solids. Publishing House of Foreign Literature. Moscow (1961). p. 104. [in Russian] 95. Johnson K. Mechanics of contact interaction. Mir. Moscow (1989). p. 510. [in Russian] 96. Didukh B.I., Lobysev V.L., Lyakher V.M. et al., Dynamics of Continuous Environments in the Calculations of Hydraulic Structures. Energy. Moscow (1976). p. 391. [in Russian] 97. Dikovich I.L. Dynamics of elastoplastic beams. Sudpromgiz. Leningrad (1962). p. 292. [in Russian] 98. Eremenko S.Yu. Finite element methods in the mechanics of deformable bodies. Osnova. Kharkiv University. Kharkiv (1991). p. 272. [in Russian] 99. Erkhov M.I. Theory of ideally plastic bodies and structures. Nauka. Moscow (1978). p. 352. [in Russian] 100. Zhariy O.Yu., Ulitko A.F. Introduction to the mechanics of nonstationary oscillations and waves. Vyscha Shkola. Kyiv (1989). p. 184. [in Russian] 101. Zamyshlyaev B.V., Yakovlev Yu.S. Dynamic loads during underwater explosion. Shipbuilding. Leningrad (1967). p. 387. [in Russian] 102. Zukas J.A., Nicholas T., Swift H.F. etc. Dynamics of impact. Mir. Moscow (1985). p. 296. [in Russian]

272

References

103. Zukina E.L. Conservative difference schemes on non-uniform grids for a two-dimensional wave equation. Work of N.I. Lobachevski Math. Centre. Kazan (2004). 26. 151–160. [in Russian] 104. Igumenov L.A., Okonechnikov A.S., Tarlakovskii D.V., Fedotenkov G.V. Plane non-stationary problem of the motion of a surface load over an elastic half-space. Mathematical methods and physicalmechanical fields. Lviv (2013). 56(2). 157–163. [in Russian] 105. Kantorovich L.V., Krylov V.I. Approximate methods of higher analysis. 3rd edition. Gostekhizdat. Moscow and Leningrad (1949). p. 695. [in Russian] 106. Kachanov L.M. Fundamentals of the theory of plasticity. Nauka Publishing House. Moscow (1969). p. 420. [in Russian] 107. Kachanov L.M. Fundamentals of fracture mechanics. Nauka Publishing House. Moscow (1974). p. 312. [in Russian] 108. Kilchevsky N.A. Dynamic contact compression of solids. Impact. Naukova Dumka. Kyiv 1(976). p. 319. [in Russian] 109. Komarov K.L., Nemirovskii Yu.V. Dynamics of rigid-plastic structural elements. Nauka. Novosibirsk (1984). p. 232. [in Russian] 110. Komarov K.L., Nemirovskii Yu.V. On the dynamic behavior of rigid-plastic rectangular plates. Applied Mechanics. Kyiv (1985). 21(7). 69–76. [in Russian] 111. Kolesnikov Yu.V., Morozov E.M. Contact fracture mechanics. Nauka. Moscow (1989). p. 220. [in Russian] 112. Kolodyazhny A.V., Sevryukov V.I. Impact and pulse effects on structures and materials. Naukova Dumka. Kyiv (1986). p. 168. [in Russian] 113. Korn G., Korn T. Handbook of mathematics for scientists and engineers. Nauka. Moscow (1984). p. 831. [in Russian] 114. Korobeinik Yu.F. On the convergence of the reduction method in solving countable systems of linear integral equations. Uch. Zamp. Rostov-on-Don university. Rostov-on-Don (1959). 43(6). 21–57. [in Russian] 115. Kostrov B.V. Self-similar mixed dynamic problems on the indentation of a rigid punch into an elastic half-space. News of Academy of Sciences of the USSR. REL. Mechanics and engineering. Moscow (1964). 4. 54–62. [in Russian] 116. Kubenko V.D. On the impact of an elastic shell on a liquid surface. Reports of the Academy of Sciences of the Ukrainian SSRɸ Serie A. Kyiv (1974). 2. 164–167. [in Russian] 117. Kubenko V.D. Non-stationary interaction of structural elements with the environment. Naukova Dumka. Kyiv (1979). p. 183. [in Russian]

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

273

118. Kubenko V.D. Movement of a cylindrical body through the surface of a compressible fluidɸ Reports of the Academy of Sciences of the Ukrainian SSR. Serie A. Kyiv (1981). 8. ɋ. 41–47. [in Russian] 119. Kubenko V.D. Impact of an axisymmetric body on the surface of a compressible fluid. Reports of the Academy of Sciences of the Ukrainian SSR. Serie A. Kyiv (1981). 9. 44–48. [in Russian] 120. Kubenko V.D. Penetration of elastic shells into a compressible fluid. Naukova Dumka. Kyiv (1981). p. 160. [in Russian] 121. Kubenko V.D. On one way to solve the problems of penetration of bodies into an acoustic environment. Problems of the dynamics of interaction of deformable media. Abstracts of reports. Yerevan (1984). 183 –187. [in Russian] 122. Kubenko V.D. On solving problems of diffraction of non-stationary waves on obstacles of cylindrical and spherical shape. Reports of the Academy of Sciences of the Ukrainian SSR. Serie A. Kyiv (1975). 10. 901–906. [in Russian] 123.Kubenko V.D. Axisymmetric problem of non-stationary indentation of a blunt rigid body into the surface of an elastic layer. Reports of the National Academy of Sciences. Kyiv (2008). 1. 58–67. [in Russian] 124. Kubenko V.D., Babaev A.E. Interaction of non-stationary waves with shells. Mechanics of composite materials and structural elements. 2. Mechanics of structural elements. Kyiv (1983). 431–450. [in Russian] 125. Kubenko V.D., Bohdanov V.R. The impact of a thin elastic spherical shell on the surface of an elastic half-space. Reports of the National Academy of Sciences. Kyiv (1995). 8. 56–60. [in Russian] 126. Kubenko V.D., Bogdanov V.R. Planar problem of the impact of a shell on an elastic half-space. International Applied Mechanics, 31, No. 6, 483-490. DOI: 10.1007/BF00846802. [in Russian and English] 127. Kubenko V.D., Bogdanov V.R. Axisymmetric impact of a shell on an elastic halfspace. International Applied Mechanics, 31, No. 10, 829– 835. DOI: 10.1007/BF00846884. [in Russian and English] 128. Kubenko V.D., Havrylenko V.V. Penetration of an axisymmetric solid body into a compressible liquid. Applied Mechanics. Kyiv (1986). 22(4). 93–99. [in Russian] 129. Kubenko V.D., Havrylenko V.V. Axisymmetric problem of the penetration of rigid bodies into a compressible liquid. Applied Mechanics. Kyiv (1987). 23(2). 53–60. [in Russian]

274

References

130. Kubenko V.D., Havrylenko V.V. Axisymmetric problem of penetration of thin elastic shells into a compressible liquid. Applied Mechanics. Kyiv (1988). 24(4). 63–74. [in Russian] 131. Kubenko V.D., Havrylenko V.V., Tarlakovsky D.V. Effect of nonstationary loading on the surface of an elastic band. Reports of the National Academy of Sciences. Kyiv (2008). 1. 59–65. [in Russian] 132. Kubenko V.D., Marchenko T.A. The plane problem of nonstationary indentation of a blunted rigid indenter into the surface of an elastic layer. Applied Mechaniks. Kyiv (2008). 44(3). 55–65. [in Russian] 133. Kubenko V.D., Popov S.N. The plane problem of the impact of a rigid blunt body on the surface of an elastic half-space. Applied Mechanics. Kyiv (1988). 24(7). 69–77. [in Russian] 134. Kubenko V.D., Popov S.N. Axisymmetric problem of the impact of a rigid blunt body on the surface of an elastic half-space. Applied Mechanics. Kyiv (1989). 25(7). 16–24. [in Russian] 135. Kubenko V.D., Popov S.N. Impact of a rigid blunt cylinder of curvilinear cross-section on an elastic half-space. Mixed problems of the mechanics of a deformable body. 4 All-Union. Conference Theses of Reports. Odessa (1989). 181. [in Russian] 136. Kubenko V.D., Popov S.N., Bohdanov V.R. The impact of an elastic cylindrical shell on the surface of an elastic half-space. Reports of the National Academy of Sciences of Ukraine. Kyiv (1995). 7. 40–44. [in Russian] 137. Kuznetsova E.L., Tarlakovsky D.V., Fedotenkov G.V., Medvedsky A.L. Effect of a non-stationary distributed load on the surface of an elastic layer. Trudy MAI. Moscow (2013). 71, 1–21. [in Russian] 138. Lai Thanh Tuan, Tarlakovsky D.V. Propagation of nonstationary axisymmetric perturbations from the surface of a layer filled with Kosser's pseudoelastic medium. Trudy MAI. 53. Moscow (2012). p. 20. [in Russian] 139. Lancosh K. Practical methods of applied analysis. Fizmatgiz. Moscow (1961). p. 524. [in Russian] 140. Lokteva N.A., Serdyuk D.O., Skopintsev P.D., Fedotenkov G.V. Non-stationary stress-strain state of a composite cylindrical shell. Mechanics of Composite Materials and Structures. Moscow (2020). 26(4). 544-559. DOI: 10.33113/mkmk.ras.2020.26.04.544_559.08 [in Russian] 141. Mazalov V.N. Dynamic bending of annular rigid plastic plates with immovable circular contours. Dynamics of continuum medium. Novosibirsk (1973). 14. 57–66. [in Russian]

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

275

142. Mazalov V.N., Nemirovskii Yu.V. Dynamics of thin-walled plastic structures // Problems of dynamics of elastic-plastic media. New in foreign science. Mechanics. Mir. Moscow (1975). 5. 155–247. [in Russian] 143. Margolin B.Z., Shvetsova V.A. Brittle Fracture Criterion. StructuralMechanical Approach. Strength Problems. Kyiv (1992). 2. 3–16. [in Russian] 144. Makhnenko V.I. Calculation methods for studying the kinetics of welding stresses and deformations. Naukova Dumka. Kyiv (1976). p. 320. [in Russian] 145. Makhnenko V.I. Improving methods for assessing the residual life of welded joints in long-term structures. Automatic welding. Kyiv (2003). 10–11. 112–121. [in Russian] 146.Makhnenko V.I., Poznyakov V.D., Velikoivanenko E.A., Rozynka G.F., Pivtorak N.I. Risk of formation of cold cracks during welding of high-strength structural steels. Collection of scientific works. Processing of materials in mechanical engineering. National University of Shipbuilding. Mykolaiv (2009). 3. 5–12. [in Russian] 147. Medvedsky A.L. Study of the features of non-stationary interaction of structurally inhomogeneous media and systems based on the method of surface influence functions. Abstract of the dissertation for the degree of Doctor of Physical and Mathematical Sciences. Saratov (2011). p. 36. [in Russian] 148. Medvedsky A. L., Tarlakovsky D. V. Non-stationary contact of a non-deformable striker with imperfections and an elastic half-plane in a supersonic penetration section. Vestnik MAI. Moscow (2011). 18(6). 125–132. [in Russian] 149. Metsaveer H.A., Veksler N.D., Stulov A.S. Diffraction of acoustic pulses on elastic bodies. Nauka. Moscow (1979). p. 240. [in Russian] 150. Mikhailova E.Yu., Tarlakovsky D.V., Fedotenkov G.V. Generalized linear model of the dynamics of thin elastic shells. Scientific notes of Kazan University. A series of physical-mathematical science. Kazan (2018). 160(3). 561–577. [in Russian] 151. Mikhailova E.Yu., Tarlakovsky D.V., Fedotenkov G.V. Nonstationary contact of a spherical shell and an elastic half-space. Proceedings of the MAI. Moscow (2014). 78, 1–26. [in Russian] 152. Mnev E.N., Pertsev A.K. Hydroelasticity of shell systems. Shipbuilding. Leningrad (1970). p. 365. [in Russian] 153. Mors F.M., Feshbakh G. Methods of theoretical physics. Publishing house of foreign literature. Moscow (1960). 2. p. 886. [in Russian]

276

References

154. Moon F. Impact and wave propagation in composite materials. Composite materials. Mashinostroenie. Moscow (1978). 7. 264–334. [in Russian] 155. Nguyen Ngoc Hoa, Tarlakovsky D.V. Non-stationary surface influence functions for an elastic-porous half-plane. Proceedings of the MAI. Moscow (2012). 53. p. 10. [in Russian] 156. Nemirovsky Yu.V., T.P. Romanova Dynamic resistance of flat plastic barriers. Academic publishing house Geo. Novosibirsk (2009). p. 311. [in Russian] 157. Nemirovsky Yu.V., Romanova T.P. Dynamics of a rigid-plastic curvilinear plate of variable thickness with an arbitrary hole. Applied Mechanics. Kyiv (2010). 46(3). 70–76. [in Russian] 158. Parton V.Z., Perlin P.I. Methods of mathematical theory of elasticity. Nauka. Moscow (1981). p. 688. [in Russian] 159. Perelmuter A.V., Slivker V.I., Calculation models of structures and the possibility of their analysis. Stal. Kyiv (2002). p. 597. [in Russian] 160. Petrashen G.I., Molotkov L.A., Krauklis P.V. Waves in layered homogeneous isotropic elastic media. Nauka. Leningrad (1982). p. 287. [in Russian] 161. Piskun V.V., Shevchenko Yu.N. Dynamic elastoplastic processes of deformation of bodies of revolution under impulsive loadingInstitute of Hydromechanics of the National Academy of Sciences: Collection of scientific papers Actual aspects of physical and mechanical research. Mechanics. Naukova Dumka. Kyiv (2007). 239–251. [in Russian] 162. Popov G.Ya. Concentration of elastic stresses near stamps, slits, thin inclusions and reinforcements. Nauka. Moscow (1982). p. 367. [in Russian] 163. Popov S.N. Penetration of a rigid body of revolution into an elastic half-space. Proceedings of the 12th Scientific Conference of Young Scientists of the Institute of Mechanics of the Academy of Sciences of the Ukrainian SSR. Manuscript deposited at VINITI No 5389– ȼ87. Kyiv (29/07/1987). 395–399. [in Russian] 164. Popov S.N. On the problem of superseismic penetration of a parabolic cylinder into an elastic half-plane. Proceedings of the 14th Scientific Conference of Young Scientists of the Institute of Mechanics of the Academy of Sciences of the Ukrainian SSR. Manuscript deposited at VINITI No 5165–ȼ89. Kyiv (02/08/1989). Kyiv (1989). 299–303. [in Russian]

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

277

165. Popov S.N. Vertical impact of a rigid circular cylinder by the side surface on an elastic half-space. Applied Mechanics. Kyiv (1989). 25(12). 41–47. [in Russian] 166. Popov S.N. Impact of a rigid ball on the surface of an elastic halfspace. Applied Mechanics. Kyiv (1990). 26(3). 42–49. [in Russian] 167. Popov S.N., Bogdanov V.R. Vertical elastic impact of a spherical shell on an elastic half-space. Abstracts of scientific reports of scientific-technical conference Operational and Structural Strength of Ship Structures. Ninth Bubnov Readings. Nizhny Novgorod (1991). 69. [in Russian] 168. Popov S.N., Bogdanov V.R. Vertical impact of a thin elastic cylindrical shell on the surface of an elastic half-space. Proceedings of the 16th Scientific Conference of Young Scientists of the Institute of Mechanics of the Academy of Sciences of the Ukrainian SSR. Manuscript deposited at VINITI No 4260–ȼ91. Kyiv (12/11/1991). Kyiv (1991). 332– 337. [in Russian] 169. Poruchikov V.B. Axisymmetric dynamical problem of a stamp on an elastic half-space. Vestnik Moskovskogo universiteta. Mathematicsɸ Mechanics. Moscow (1966). 6. 114–120. [in Russian] 170. Poruchikov V.B. Methods of dynamic theory of elasticity. Nauka. Moscow (1986). p. 327. [in Russian] 171. Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and series. Elementary functions. Nauka. Moscow (1981). p. 798. [in Russian] 172. Development of the theory of contact problems in the USSR. Under the general editorship of L.A. Galina. Nauka. Moscow (1976). p. 492. [in Russian] 173. Rinehart J.S., Pearson J. Behavior of metals under impulse loads. Publishing house of foreign literature. Moscow (1958). p. 296. [in Russian] 174. Rakhmatulin Kh.A., Demyanov Yu.A. Strength under intense shortterm loads. Fizmatgiz. Moscow (1961). p. 399. [in Russian] 175. Reitman M.I., Shapiro G.S. Dynamic theory of plasticity. Elasticity and plasticity. Results of science. VINITI Academy of Sciences of USSR. Moscow (1968). p. 112. [in Russian] 176. Savruk M.P. Fracture mechanics and strength of materials. Stress intensity factors in bodies with cracks. 2. Naukova Dumka. Kyiv (1988). p. 620. [in Russian] 177. Sagomonyan A.Ya. Penetration. Publishing House of Moscow University. Moscow (1974). p. 300. [in Russian]

278

References

178. Sagomonyan A.Ya. Impact and penetration of bodies into a liquid. Publishing House of Moscow University. Moscow (1986). p. 172. [in Russian] 179. Sbornik. Theory of plasticity. Publishing house of foreign literature. Moscow (1948). p. 460. 180. Sekulovich M. Finite element method. Stroyizdat. Moscow (1993). p. 661. [in Russian] 181. Smirnov V.A. Calculation of plates of complex shape. Stroyizdat. Moscow (1978). p. 300. [in Russian] 182. Slepyan L.I. Non-stationary elastic waves. Shipbuilding. Leningrad (1972). p. 374. [in Russian] 183. Slepyan L.I., Yakovlev Yu.S. Integral transformations in nonstationary problems of mechanics. Shipbuilding. Leningrad (1980). p. 344. [in Russian] 184. Spravochnik. Handbook of special functions with formulas, graphs and tables. Edited by M. Abramovitz and I. Stigan. Nauka. Moscow (1979). p. 832. [in Russian] 185. Tarlakovsky D.V. Application of the superposition principle in an axisymmetric dynamic problem for an elastic half-space. News of Academy of Sciences of the USSR. Solid mechanics. Moscow (1988). 2. 76–85. [in Russian] 186. Tarlakovsky D.V. Vertical impact of an absolutely hard sphere with a filler on an elastic half-space. Strength calculation and optimal design of elements of aircraft structures. Moscow Aviation Institute. Moscow (1988). p. 44–46. [in Russian] 187. Fedotenkov G.V. Impact of a cylindrical shell on an elastic halfplane. MAI. Moscow (2001). p. 100. [in Russian] 188. Kharchenko V.V., Kondryakov E.A., Panasenko A.V. Peculiarities of crack propagation in steels during testing of Charpy specimens and disk specimens. VANT. Kharkiv (2013). 2(84). 31–38. [in Russian] 189. Hemming R.V. Numerical methods. Nauka. Moscow (1972). p. 399. [in Russian] 190. Shapiro G.S. On the motion of rigid-plastic bodies in a resisting medium. Applied Mathematics and Mechanics. Moscow (1962). 2. 344–345. [in Russian] 191. Aero E. L., Kuvskhinskii E. V. Fundamental equations of the theory of elastic media with rotationally interacting particles. Sov. Phys. Solid State 2 (1961) 1272–1281. 192. Antipov Y. A. Weight functions of a crack in a two–dimensional micropolar solid. Oxford: QJMAM. Jan. (2012) 33.

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

279

193. ASTM E399. Standard test method for plane–strain fracture toughness of metallic materials. In: Annual Book of ASTM Standards. V. 03. 01. ASTM, Philadelphia, PA. (1991). 194. ASTM Standard E 1921_97. Standard test method for determination of reference temperature. To, for ferritic steels in the transition range. in: Annual Book of ASTM Standards. V. 03.01. pp. 1068–1084. 195. Augusli G. Rigid–plastic structures subject dynamic loads. Meccanica. (1970). V. 5. N 2. 74–82. 196. Bäk ɋ, Muszynski M. Rozwiazania kinematyczne dla sztywno– plastycznej plyty pro–stokatnej z materialy H–M–H obciazonej impulsowo // Biul. WAT J. Dabrowskiego. (1990) T. 39. N l. 3–14. 197. Bogdanov V. Problem of plane strain state of two-layer body in dynamic elastic-plastic formulation (Part I). Underwater Technologies, Kyiv, (2022) No. 12, 3–14. DOI: 10.32347/uwt.2022.12.1101. 198. Bogdanov V. One approach to the axisymmetric problem of impact of fine shells of the S.P. Timoshenko type on elastic half-space. Transfer of Innovative Technologies. Kyiv, (2021) Vol 4(2), 77–89. DOI: 10.32347/tit2021.42.0302. 199. Bogdanov V. One approach to the problems of impact of fine shells of the S.P. Timoshenko type on elastic half-space. Underwater Technologies, Kyiv, (2021) No. 11, 3–18. DOI: https://doi.org/10.32347/uwt2021.11.1101. 200. Bogdanov V. Impact a circular cylinder with a flat on an elastic layer. Transfer of Innovative Technologies, Kyiv. (2018) 1(2) 68–74. DOI: 10.31493/tit1812.0302. 201. Bogdanov V. Impact of a hard cylinder with flat surface on the elastic layer. Underwater Technologies, (2017) Vol. 5, 8–15. 202. Cosserat E., Cosserat F., Th´eorie des Corps D´eformables. Hermann, Paris. (1909). 203. Cowper G. R., Kosko E, Lindberg G. M., Olson M. D. Static and dynamic applications of a high–precision triangular plate bending element. AIAA Journal. (1969) No. 7. 1957–1965. 204. DIN 51 109. Testing of advanced technical ceramics: determination of fracture toughness KIc. (1991). 205. Eringen A. C. Theory of micropolar elasticity. Fracture, Vol. 2 H. Leibowitz (ed.). Academic Press, New York. (1968). pp. 621–729. 206. 31_A. Fedotenkov G. V., Tarlakovsky D. V., Vahterova Y.A. Identification of non-stationary load upon Timoshenko beam. Lobachevskii Journal of Mathematics (2019) 40 No. 4 439–447.

280

References

207. Gingold R. A., Monaghan J. J. (1977). Smooth particle hydrodynamics: Theory and applications to non–spherical stars. Monthly Notices of the Royal Astronomical Society 181, 375–389. 208. Hughes T. J. R. (1987). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice–Hall, Englewood Cliffs, NJ. 209. Jones N. A. Literature review of the dynamic plastic response of structures. Shock and Vibr. Digest. (1975). V. 7. N. 8. 89–105. 210. Kee B. B. T., Liu G. R., Lu C. (2007). A regularized least–squares radial point collocation method (RLS–RPCM) for adaptive analysis. Computational Mechanics, 40, 837–853. 211. Kirchhoff G. Vorlesungen uber Mathematiche Physik.Mechanik. Leipzig: Verland von B.G.Tubner, (1876) p. 466. 212. Kleiber M., Borkowski A. (1998). Handbook of Computational Solid Mechanics. Springer, Berlin. 213. Krajcinovic D. Dynamic response of rigid ideally plastic structures. Shock and Vibr. Digest. (1973). V. 5. N 2. 2–9. 214. Kumar A. Dynamic plastic response of circular plates in a damping medium. Int. J. Impact Eng. (1987). V. 6. N 4. 285–290. 215. Lamb H. The dynamikal theory of sound. London: Edward Arnold and Co. (1931). p. 307. 216. Lee L. H. N. Dynamic plasticity. Nucl. Eng. and Design. (1974). V. 27. N 3. 386–397. 217. Liu G. R., Gu Y. T. (2005). An Introduction to Meshfree Method Methods and Their Programming. Springer, Berlin. 218. Liu G. R., Kee B. B. T. (2006). A stabilized least–squares radial point collocation method (LS–RPCM) for adaptive analysis. Computer Method in Applied Mechanics and Engineering, 195, 4843–4861. 219. Liu G. R. and Liu M. B. (2003). Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, Hackensack, NJ. 220. Liu M. B., Liu G. R., Zong Z. (2008). An overview on smoothed particle hydrodynamics. International Journal of Computational Methods, 5(1), 135–188. 221. Liu G. R., Nguen Thoi Trung, Finite Element Methods. CRC Press, Taylor & Francis Group, New York. (2010). p. 671. 222. Liu G. R., Quek S. S. (2002). The Finite Element Method: A Practical Course. Butterworth Heinemann, Oxford. 223. Liu G. R., Xu X. G. (2008). A gradient smoothing method (GSM) for fluid dynamics problems. International Journal for Numerical Methods in Fluids, 58, 1101–1133.

Problems of Impact and Non-Stationary Interaction in Elastic-Plastic Formulations

281

224. Liu G. R., Zhang J., Lam K. Y. (2008). A gradient smoothing method (GSM) with directional correction for solid mechanics problems. Computational Mechanics, 41, 457–472. 225. Love A. E. H. Some illustrations of modes of vibrationary motions. Proc. London Math. Soc., Second Ser. (1905) V.2. 88–113. 226. Lucy L. (1977). A numerical approach to testing the fission hypothesis. The Astronomical Journal, 82, 1013–1024. 227. Margolin B. Z., Shvetsova V. A., Gulenko A. G., Ilyin A. V., Nikolaev V. A., Smirnov V. I. Fracture toughness predictions for a reactor pressure vessel steel in the initial and highly embrittled states with the master curve approach and a probabilistic model. Pressure Vessels and Piping. Jan. (2002) 219–231. 228. Margolin B. Z., Shvetsova V. A., Karzov G. P. Brittle fracture of nuclear pressure vessel steels. Part 1: Local criterion for cleavage fracture. Int. J. Pressure Vessels Piping. (1997) 72, 73–87. 229. Mazalow V. N., Nemirovsky Ju. V. Dynamic bending of rigid– plastic annular plates. Int. J. Non–Linear Mech. (1976). V. 11. N 1. 25–40. 230. Mindlin R. D., Tiersten H. F. Effects of couple–stresses in linear elasticity. Arch. Rat. Mech. Anal. 11 (1962) 415–448. 231. Niepostyn D., Stanczyk A. Pulse load of annular plastic plates supported on both edges. Arch. Mech. (1981). V. 33 N 5. 675–689. 232. Pal’mov V. A. Fundamental equations of the theory of asymmetric elasticity. J. Appl. Math. Mech. 28 (1964) 496–505. 233. Pao Y.–H., Mow C.–C. Diffractions of elastic waves and dynamic stress concentrations. New York: Crane, Russak, (1973) p. 694. 234. Prasad C. On vibrations of spherical shells. J. Acoust. Soc. Amer., (1964), 36, 3, 489–494. 235. Rokach I. V. On the numerical evaluation of the anvil force for accurate dynamic stress intensity factor determination. Engineering Fracture Mechanics. 70. (2003) 2059–2074. 236. Rokach I. V. Modal approach for processing one– and three–point bend test data for dsif–time diagram determination. Part I: theory. Fatigue & Fracture of Engineering Materials & Structures. 21. (1998). 1007–1014. 237. Rokach I. V. Influence of contact compliance on dynamic stress intensity factor variation during an impact test. International Journal of Fracture. 126. (2004). 41–46. 238. Ronghua Li, Zhongying Chen, Wei Wu. Generalized difference methods for differential equations. Numerical analysis of finite volume methods. Marcel Dekker, Inc., New York (2000) p. 442.

282

References

239. Truesdell C., Toupin R. A. The Classical Field Theories, Encyclopedia of Physics, Vol. III/1. Springer, Berlin. (1960). 240. US Nuclear Regulatory Commission guide 1.99 (TASK ME 305_4). Radiation embrittlement of reactor vessel materials, Revision 2, (May 1988). 241. Vahterova Y.A., Fedotenkov G.V. The inverse problem of recovering an unsteady linear load for an elastic rod of finite length. Journal of Applied Engineering Science. (2020) 18. No. 4. 687–692. DOI:10.5937/jaes0–28073. 242. Weibull W.A. A statistical theory of the strength of materials. Roy Swed Inst. Eng. Res., (1939). 151. 5–45. 243. Weisbrod G., Rittel D. A method for dynamic fracture toughness determination using short beams. International Journal of Fracture. 104. (2000) 89–103. 244. Xu G. X., Liu G. R., Tani A. (2009). An adaptive gradient smoothing method (GSM) for fluid dynamics problems. International Journal for Numerical Methods in Fluids, 62, 499–529. 245. Xu G. X., Liu G. R., Lee K. H. (2009). Application of gradient smoothing method (GSM) for steady and unsteady incompressible flow problems using irregular triangles. International Journal for Numerical Methods in Fluids; submitted. 246. Zhou C. E., Liu G. R., Lou K. Y. (2007). Three–dimensional penetration simulation using smoothed particle hydrodynamics. International Journal of Computational Methods, 4(4), 671–691. 247. Zienkiewicz O. C., Taylor R. L. (2000). The Finite Element Method, 5th edition. Butterworth Heinemann, Oxford.