PLASTIC WAVE PROPAGATION WITH COMBINED STRESSES Thesis for the Degree of Ph. D. MICHfGAN STATE UNNERSITY RAM PARKASH GOEL 1963 «mmmmmmnmnnmnm; I rm... mm 3 ”93 ““2854 LIBRARY Michigan State University This is to certify that the thesis entitled Plastic Wave Propagation with Combined Stesses presented by Ram Parkash Goel has been accepted towards fulfillment of the requirements for Ph. D. degree in Mechanism Dept. of Metallurgy, Mechanics & Materials Science x», ,/'"4 t "7 ’ 5 ‘ 1 i ’_\ I . H , (K ' i n .I , Av > ' ‘ - r '\ A ' _ ~ r‘ . .3, , \ \. .1' '\’4__ . L L. ‘l ' ' V» J\_’ v 2' Major professor Date August 15, 1969 0-169 MAGIC 2 JAN 1 1 1998 ABSTRACT PLASTIC WAVE PROPAGATION WITH COMBINED STRESSES By Ram Parkash Goel This investigation analyzes combined-stress wave propagation for loading beyond the elastic limit in two geometries: a thin—walled semi—infinite tube subjected to simultaneous axial and torsional impact. and a half space under compressive and shear impact loadings. The major part of the results presented are for the thin- walled tube case. where the effect of different harden- ing assumptions has been investigated. For the half space. isotropic hardening was assumed throughout the study. One type of simple—wave solution not previ- ously investigated is presented and also one example of a numerical solution for a nonsimple wave. For the thin-walled tube analysis the consti- ‘tutive equations were obtained by combining the two most widely used hardening postulates. namely isotropic hard- ening and kinematic hardening. A hardening parameter m 1 Ram Parkash Goel (varying from m = O for purely kinematic hardening to m = 1 for purely isotropic hardening) was used to define the fraction of total hardening attributable to isotropic hard- ening. The boundary value problems of a thin—walled tube were solved for different values of the hardening parameter m and with different initial stress states and different tension-torsion impact loadings on the boundary x = O of the tube. such that simple wave solutions occurred. For some choices of initial and boundary conditions. both qual- itative and quantitative differences were observed in the particle stress histories between the predictions of iso- tropic hardening and the predictions of kinematic or com— bined kinematic and isotropic hardening. In some cases a discontinuity in shear stress occurs. propagating at the elastic shear wave speed c followed by a slow plastic 2. simple wave. according to the kinematic hardening or the combined kinematic and isotropic hardening assumptions. when no such discontinuity is predicted by isotropic hard- ening. It is considerably more difficult to obtain simple wave solutions for the kinematic hardening or the combined kinematic and isotropic hardening case than it is for the isotropic hardening case. because a family of simpleawave 2 Ram Parkash Goel stress trajectories must be constructed to obtain one par- ticular stress trajectory for a prescribed initial and boundary state of the tube. For the isotropic—hardening half space loaded with step function impacts in compression and two shears T and 2 T each simple—wave solution implies a radial loading pro- 3! jection in the T T -plane. But the radial loading T3/T2 2'3 ratio for the fast wave may be different from the ratio for the slow wave. The transition is accompanied by jumps both in 12 and T3 traveling at the elastic shear-wave speed c2. A surprising result was obtained for the one example of numerical solution for a nonsimple wave in a half space. The plotted level lines of stress and velocity turned out to be straight even though it was not a simple wave. More- over the compressive and shear waves appeared to be un- coupled. No general conclusions can be drawn from these limited results. but the possibility exists that closed- form solutions could be obtained in future studies of non- simple waves. for some boundary conditions. PLASTIC WAVE PROPAGATION WITH COMBINED STRESSES BY Ram Parkash Goel A THESIS Submitted to Midhigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mechanics Department of Metallurgy. Mechanics and Materials Science August 1969 To my parents ACKNOWLEDGMENTS I wish to express my gratitude to Professor Lawrence E. Malvern for his invaluable guidance and counsel throughout this research. I shall always be indebted to him for the frequent discussions we had during the period of this research. I express my thanks to Professors William A. Bradley. James L. Lubkin. Robert W. Little. Norman L. Hills. and Charles J. Martin who served on my guidance committee. Also thanks are due to Professor Donald J. Montgomery. chairmanLOf the department. for his encour- agement and counsel. The project was supported by the National Science Foundation under Grant No. 6-1396-x. My sincere appreciation and thanks are due to my dear friend and relative Rajinder Pal Singla who did a perfect job of drawing the numerous graphs. iii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF SYMBOLS . . . . . . . . . . . . . . . . Chapter 1. INTRODUCTION. . . . . . . . . . . . 1.1 PURPOSE . . . . . . . . . 1.2 BACKGROUND HISTORY. . waves 0 O O O O O 1.3 SCOPE OF THE STUDY. . . . . . . 2. WORK HARDENING ASSUMPTIONS. . . . . . . 2.1 ”INTRODUCTION. .-. . . . . . . . 2.2 ISOTROPIC HARDENING . . . . . . . 2.3 PRAGER'S KINEMATIC HARDENING. . 2.4 COMBINED ISOTROPIC AND KINEMATIC HAmENING I O O C O O O O O 0 iv 1.2.1 Theory of Plasticity. . 1.2.2 Plastic Wave Propagation. 1.2.3 Combined-Stress Plastic Page iii vii w 11 15 19 19 24 28 33 TABLE OF CONTENTS (cont.) Chapter 3. 4. 5. THIN-WALLED TUBE. 3.1 INTRODUCTION. . . . . . . . . . . . 3.2 EQUATIONS GOVERNING THE MOTION. . . 3.3 SOLUTION OF GOVERNING EQUATIONS 3.3.1 Wave Speeds . . . . . . . 3.3.2 Characteristic Conditions 3.3.3 Simple Waves. . . . . . . 3.4 PROPERTIES AND SLOPES OF STRESS TRAJECTORIES FOR SIMPLE WAVES . . PLANE WAVES IN A HALF SPACE 4.1 INTRODUCTION. . . . . . . . . . 4.2 EQUATIONS GOVERNING THE MOTION. . . 4.3 SOLUTION OF THE GOVERNING EQUATIONS 4.3.1 Wave Speeds . . . . . . . 4.3.2 Characteristic Conditions 4.3.3 Simple Waves, , . . . . . 4.4 NUMERICAL SOLUTION FOR NONSIMPLE WAVES . . . . . . . . . . . . . . APPLICATIONS AND RESULTS. . . . . . . . . . 5.1 SIMPLE WAVE SOLUTIONS FOR THIN- WALLED TUBE . . . . . . . . . 5.2 EXAMPLES OF COMBINED TENSION- TORSION IMPACTS ON THE THIN- WALLED TUBE . . . . . . . . . V Page 41 41 41 49 49 51 52 58 68 68 69 77 77 82 85 90 96 96 101 TABLE or CONTENTS (cont.) Chapter 5.3 SIMPLE WAVE SOLUTIONS FOR HALF SPACE . . . . . . . . . . 5.4 NONSIMPLE WAVE SOLUTION FOR A HALF SPACE . . . . . . . . . . . 5.5 DISCUSSION OF RESULTS . . . . . . . 5.5.1 Comparison of Different Hardening Assumption Results for Tension- Torsion Impact on Thin-Walled Tube. . . . 5.5.2 Simple Waves in Isotropic Hardening Half Space. . 5.5.3 Nonsimple Waves in Iso- tropic Hardening Half Space . . . . . . . . . 5.6 'CONCLUSIONS AND RECOMMENDATIONS FOR 5.7 Appendix FURTHER RESEARCH. . . . GRAPHICAL DISPLAY OF RESULTS. 1. THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. A1.1 METHOD OF CHARACTERISTICS. . . . . A1.2 SIMPLE WAVE SOLUTION . . .'. . . . 2. LAGRANGIAN FORMULATION OF EQUATIONS OF MOTION IN MATERIAL COORDINATES. . BIBLIOGRAPHY. vi Page 107 112 115 115 118 120 122 125 155 155 160 162 170 23’ In! D.. 1] (DZU go (If) * as 0 d8 LIST OF SYMBOLS coefficient matrix. multiplying yd: slope of the shear stress and shear velocity level lines coefficient matrix. multiplying Ex slope of the normal stress and velocity level lines; proportionaltiy scalar function in ob- taining the translation of the yield surface characteristic wave speed elastic dilatational wave speed elastic shear wave speed fast wave speed elastic bar-wave speed slow wave speed components of the rate-of—deformation tensor submatrix of partitioned matrix A' elastic part of R adjoint of 2? surface element in the undeformed configuration surface element in the deformed configuration vii LIST or SYMBOLS (Cont.) ds1 ds2 ds (to 2 23 M motion of the local yield surface element due to isotropic hardening motion of the local yield surface element due to kinematic hardening total motion of the local yield surface element Ybung's modulus tangent modulus unit vector in stress space function of effective plastic strain faEP yield function shear modulus slope of effective stress-—effective plastic strain curve unit matrix of order j second invariant of plastic strain increments second invariant of deviatoric stresses size parameter of the yield surface (yield stress in shear for isotropic hardenihg)’ initial yield stress in shear = MN - $2 eigenvector transpoSe of,£ submatrix of B = l'+ 4H7}2 G viii LIST OF SYMBOLS (Cont.) 1 2 = _. + M G 4H1] m hardening parameter -i ’2 N - E+H(€) fir unit normal to yield surface P grid point at time (t + At) Q grid point at time t R radius of yield surface in all direction r radius vector of the yield surface in stress space §. submatrix of A e . g‘ elastic part of S s = 2Hg’n 2 s 3 (01-02) T? first Piola-Kirchoff stress tensor (nonsymmetric) To. elements of To J1 ~ t time U displacement in the longitudinal direction u longitudinal particle velocity V displacement in the tangential direction v particle velocity in the i direction v particle velocity in the tangential direction solution vector and its partial derivatives ix LIST OF SYMBOLS (Cont.) 6.. 13 2m material Cartesian coordinates (Lagrangian coordinates) in Appendix 2 spatial Cartesian coordinates (Eulerian coord- inates) in Appendix 2 particle coordinate in the undeformed state. in the text uniaxial yield stress uniaxial yield stress with reversed loading initial yield stress position vector of the center of the yield surface in the stress space components of the position vector g'in Chap. 3; coefficient of jth element in the ith equation in Section 4.4 2(1+v) 1-2v engineering shear strain in Chapter 3; ratio of shear stresses (13/12) in Chapters 4 and 5 plastic angle change (plastic shear strain) elastic angle change (elastic shear strain) ratio of boundary shear stresses ratio of prestress shear stresses Kronecker delta. unity when i=j. otherwise 0 total strain vector longitudinal unit extension elastic longitudinal unit extension LIST OF SYMBOLS (Cont.) 20 0.. 1:1 0* plastic longitudinal unit extension effective plastic strain increment distance in the characteristic direction in the x.t-plane. in Appendix 1 = T-T* n coordinate of a point on the fast-wave stress path just before the jump scalar function used in the plastic potential theory Poissonis ratio n 20 ‘I Q 0 .2. 3 O P 0* (deviatoric part of e) mass density stress vector in nine-dimensional stress space components of 2; also Cartesian tensor Cauchy stress components O-coordinate of a point on the fast-wave stress path just before jump deViatoric stress vector in nine space yield stress in tension (or compression) normal stress coordinate where the slow-wave stress path starts O-coordinate of the yield surface center xi TABLE OF CONTENTS (Cont.) effective stress increment shear stress component; resultant shear stress of 12 and 73 shear stress in y-direction value of T 2 on the boundary shear stress in z-direction value of T3 on the boundary T-coordinate of the yield surface center initial prestress in shear boundary shear stress shear stress coordinate where the slow-wave stress path starts distance in the direction normal to character— istic lines in the x.t—plane in Appendix 1: an element of an eigenvector in Chaps. 3 and 4 dO/dr; an element of an eigenvector xii CHAPTER 1 INTRODUCTION 1.1. PURPOSE The general purpose of this study is to investi— gate the dynamic response of metal structural elements subjected to dynamic loads leading to combined-stress de- formation beyond the elastic limit. with different mater- ial behavior assumptions. To accomplish this purpose. specific boundary value problems will be analyzed. deal- ing with a tension-torsion impact on a thin-walled tube and with plane waves from combined compressive and shear impact loading of a half space. These two cases are con- sidered because they can be checked experimentally. (The transient response of the half space is the same as that of one of two impacting plates in a "plate-slap" experi— ment. until release waves arrive from the edges or reflec- tions occur from the face opposite to the impacted face.) Understanding the dynamic response of structural elements subjected to impact loads beyond the elastic 1 limit has practical value. Moreover the analysis under dif- ferent material assumptions may. when experimental evidence becomes available. contribute to a better understanding of the material response and to the construction of better continuum theories of plasticity. The plastic behavior of metals is not adequately accounted for by any of the present theories. especially the hardening behavior during combined-stress deformation. Several different idealizations have been proposed. but only two of them. namely isotropic hardening and kinematic hardenihg (see Chapter 2). have received much attention in the literature. Actual material behavior is probably intermediate between the isotropic hardening and kinematic hardening idealizations. although some experimental evi- dence indicates that sometimes a corner develops on the yield surface in stress space at the active point. a re- sult not predicted by either isotropic hardening or kine- matic hardening. The thin-walled tube analyses in this study assume a combination of isotrOpic hardening and kinematic hardening. The half Space analyses assume iso- tropic hardening. The hardening assumptions and the constitutive equations of plasticity are presented in Chapter 2 after the background history on plasticity and on plastic wave propagation is reviewed in Section 1.2. and the scope of the present study outlined in Section 1.3. 1 . 2 BACKGROUND HISTORY 1.2.1. Theory of Plasticity When a part of a metallic body is deformed beyond its elastic limit. the part is said to become plastic. Constitutive equations. i.e.. the mathematical descrip— tions of physical relations. in the plastic state. are still not well established. although the plasticity of metals has been studied extensively since Tresca.* in 1864. published a preliminary account of experiments on punching and extrusion. which led him to formulate the yield condition now given his name. Tresca proposed that a metal yields plastically (i.e.. exceeds the elastic limit) when the maximum shear stress attains a critical *For references to the early papers on plasticity men- .tioned in this section. see Hill (1950). Dates given Tin parentheses refer to books and papers referenced in .the bibliography at the end of this thesis. value. In 1871. Lévy proposed a three-dimensional rela- tion between stress and plastic strain. Von Mises in 1913 also independently proposed equations similar to those given by Levy. Prandtl in 1924 for the plane prob- lem. and Reuse in 1930 and R. Schmidt in 1932 for three- dimensional problems generalized the Levy-Mises theory-- bringing elastic strains and work-hardening within the framework of the Levy-Mises theory. The yield condition usually called the Mises yield condition was prOposed by Von Mises in 1913 and in- dependently by Huber in 1904. The Mises yield condition. which stipulates that yield occurs when the second invar- iant of the deviatoric stress reaches a critical value. is the most widely used yield condition. It gives good agreement with experimental observations for the initial yield of most polycrystalline metals. when the crystallite orientation is reasonably random and the grain size is small compared to the dimensions of the yielding region. When plastic deformation leads to residual stresses and preferred orientation of the crystallites. neither the Mises yield condition nor any other isotropic yield con- dition is really accurate for a subsequent loading. This is because the hardening is not isotropic. In a series of papers beginning in 1950. Drucker showed that in a stable work-hardening plastic material. the yield function (a function of the stresses) is also a plastic potential function. such that the plastic natural strain rates (plastic rate—of—deformation com- ponents) are in the same proportion as the partial deriv— atives of the yield function with respect to the stresses; See Drucker (1951). Thus. if the yield condition is f(opq) = constant. where f is a function of the nine stress components 0 . then é? =i§§ 13 50.. 1] (1.1-1) where the égj are the plastic natural strain rates. and X is a scalar function to be determined by the hardening behavior. If the equation f = constant is visualized as a hypersurface in a nine-dimensional space where the nine Oi. are Cartesian coordinates. then Equation (1.1-l) im- plies that the vector whose nine components are éij is parallel to the normal to the yield surface at the active point (the outer normal for the usual definitions of f and X). When the yield function f is the second invariant of the deviatoric stress components. then 3% = Oij . ij and the "plastic potential theory" of Equation (1.1-l) implies éP. = i of.‘. (1.1-2) 13 13 where O’ ‘= o — 1-5 O ij ij 3 ij kk is the deviatoric stress. Equation (1.1—2) is precisely the Levy-Mises stress-strain relation mentioned above. Thus plastic potential theory shows that the Mises yield condition implies the Levy-Mises equations. The Levy-Mises equations and the Mises yield con- dition. suitably modified for the kinematic hardening ‘cases. will be used throughout the present study. The Levy-Mises equations are the most widely used incrementalgplasticity theory. so called because it gives the increment of plastic strain defj = d1 Oij instead of the total strain. Hencky in 1924 proposed a small-strain plastic theory. alternatively known as "total-strain or deforma- tion theory." which gives the total strains. It has the advantage of simplicity. and it agrees with the predic- tions of the Levy—Mises theory when the loading path is a straight line radially outward from the origin in stress space. For a more detailed review of the early histori— cal background of the theory of plasticity. see: Hill (1950). World War II. the advent of high speed computers and development in experimental techniques have stimulated work in the theory of plasticity. and extensive studies of non-elastic metal behavior Ender dynamic loads have been going on in the United States and in the Soviet Union for the last three decades. 1.2.2 Plastic Wave Propagation The entire field of dynamic plasticity is of com— paratively recent origin. L. H. Donnell (1930) analyzed the problem of a uniaxial stress pulse propagating in the longitudinal direction of a nonlinear elastic thin bar. He regarded the transient loading pulse as a superposition of stress increments. each traveling at its appropriate 8 . where E is the tangent modulus and speed cp = (Et/p) t p is the mass density of the material of the bar. The idea was an extension of the linear elastic bar speed co = (E/P)8. Donnell showed that such a wave suffers a change in form as it moves along the bar. For the load- ing part of the pulse the nonlinear elastic wave does not differ from a plastic wave in a rate-independent plastic material. More than a decade later Von Karman in the United States. Taylor in Great Britain. and Rakhmatulin in the Soviet Union. working independently of each other. de- rived the partial differential equations for plane longie tudinal plastic waves in thin bars. In treating the prob— lem. Von Karman and Duwez (1946) and Rakhmatulin (1945) used Lagrangian coordinates and derived a single second order nonlinear wave equation for the axial displacement. Taylor (1942) developed the basic theory of one-dimensional finite-amplitude plastic wave propagation using Eulerian coordinates and observed that the governing equations were formally identical with those for the propagation of one-dimensional finite—amplitude waves in an ideal compressible fluid (the pressure being a function only of the density). He noted that. when suitable transformations of the coordinates are made. his theory was similar to that of Von Karman (the essential equivalence was estab- lished by Bohnenblust and others*). A series of longitudinal impact tests were per- formed By Duwez. Clark and others* to check the validity of the Von Harman theory. The results were in fairly good agreement with the theory. but there were some dis- crepancies. In particular. in some of the early tests it appeared that the force-time variation at the fixed end of the bar showed higher stress than the theory pre- dicted. Which indicated that stress increments were propa- gated at a higher wave speed than predicted. It was sug- gested that these discrepancies might be due to strain- rate effect. To explain the discrepancies exhibited by Von Karman theory. many authors have proposed various consti- tutive equations for materials that exhibit a rate effect. For a work-hardening material Malvern (1949) solved the bar problem on the basis of a constitutive relation of the form *The reference to these papers can be found in a survey article by H. G. Hopkins (1966). 10 Eé = O + 9(O.e) where g is an arbitrary function expressing the strain rate dependence. He obtained numerical results only for g of the form g[O-f(€)]. where O = f(€) is the static stress-strain relation. Sokolovskii (1948) had already considered a special case of this constitutive relation for non—workhardening material. More recent experimental studies have shown that the rate—independent theory gives a good account of fin— ite amplitude waves in bars of copper. steel. and annealed aluminum alloys. if a single dynamic stress-strain curve is used. not explicitly containing rate effects. See. for example. Bell (1965). Some impact experiments in one-dimensional strain ("plate-slap") [Barker. Lundergan. and Herrmann (1964). Butcher and Karnes (1966)] have indi- cated that no single dynamic curve can correlate their results. while a Malvern-type rate-effect law can. There is also some evidence that the rate-dependent theory gives a better account of the propagation of the leading edge of an incremental plastic wave traveling into a prestressed plastic region. See the discussion by Williams and Mal- vern (1969). In the present study the effect of rate de- pendence on the constitutive equations will not be included. 11 For additional details of the history of plastic wave propagation. see: Hopkins (1966). Cristescu (1967). and Craggs (1961). Most of the papers cited so far have been connected with waves of uniaxial stress. Craggs does present a deri- vation of the wave speeds for combined-stress plane waves in an unbounded medium (based on a paper published by Craggs in 1957). and Cristescu includes some treatment of Combined-Stress waves in his book. 1.2.3 Combined-Stress Plastic Waves The study of stress wave propagation in metals is not always simple even when there is perfect elastic be- havior. When the metal behavior is nonelastic. it becomes quite complex even when there is only one stress component acting. The complexity is compounded for combined-stress plastic waves. Craggs (1957) derived the wave speeds for combined-stress plane waves in an unbounded medium (for small strain and rotation). He found that quite generally there are two wave Speeds. a fast wave speed cf and a slow wave speed cs. such that 12 I c ‘5 c '5 c s 2 '5 C f 1 where c is the elastic dilatational wave speed and c 1 2 is the elastic shear wave Speed. See also Craggs (1961). Rakhmatulin (1958) studied analytically the prob— lem of combined-stress loading in the edge impact of two plates with the velocities of impact directed in the planes of the plates and oblique to the impacting faces. assuming zero normal strain in the direction parallel to the impacting faces and the plane of the plates. and using total-deformation theory. Rakhmatulin assumed that in the case of combined-stress in a plastic body. resulting from an impact. there propagates at first a plastic uniaxial stress state II followed by a constant state III. then a combined-stress state propagating as a wave of strong dis- continuity )t= bt followed by a constant state IV. See Fig. 1-1- ,Cristescu (1959) considered another possible solution of the same equations established by Rakhmatulin. and postulated that combined dynamic stress is transmitted in a body only by combined stress waves. either continuous waves or waves of strong discontinuity. For an instantan- eous loading. he asserted. one will have four regions: 13 Fig. 1.1 the region I is not deformed. the region II is an elastic region. and the regions III and Iv are plastic regions. which may be separated from each other or from region II by a combined-stress plastic wave of strong discontinuity. See Fig. 1.1. Bleich and Nelson (1965) presented a closed form solution for the case of a uniformly distributed step load of pressure and shear on a half space for an elastic. perfectly plastic material. Clifton (1966) presented the solution of a thin- walled cylindrical tube with end impacts in torsional shear and axial compression for an elastic-plastic iso- tropic hardening material. He concurred with Cristescu's conclusion that there could be a plastic simple wave of 14 combined stress. but he disagreed with Cristescu in that. according to Clifton. (1) a wave of strong discontinuity occurs only in elastic regions. and (2) a leading uniaxial plastic wave exists for a step load of normal stress and shear. A part of the present study considers the same problem formulated by Clifton for the thin-walled tube. but the material behavior assumed is a combination of isotropic hardening and kinematic hardening. Ting and Nan (1968) generalized the half space prOblem treated by Bleich and Nelson to a general elastic- plastic isotrOpic hardening material. Ting (Dec. 1968) gave a formulation of the partial differential equations for a general plane wave in a half space. In another paper (May 1968) Ting also solved a boundary value problem for a half space with two shear loadings and no compressive loading. A part of the present study uses a formulation similar to Ting's formulation for the general plane wave in a half Space. but treats boundary loadings not treated by Ting or Ting and Nan. Nan (1968) obtained numerical solutions for a half space subjected to two impact loads in shear or one compression and one shear. 15 1.3 SCOPE OF THE STUDY The study presently reported was undertaken to (1) understand the effect of a combination of iso— tropic and kinematic material hardening assump- tions. (2) carry out some simple wave solutions for a general plane wave in.a half-space, (3) carry out nonsimple wave solutions of a plane wave in a half-space. For (1). examples of a semi-infinite thin-walled cylin- drical tube subjected to normal and shear step loads at the boundary have been solved. while for (2). a half space of isotropic hardening material with uniformly distributed pressure and two independent shear loads on the boundary was considered. For the nonsimple wave solutions only one shear load and a compressive load were considered to be acting on the boundary of the half space. in such a combination that simple waves do not result. 16 The stress-strain curve for simple tension was assumed to be concave toward the strain axis. and inde- pendent of the rate of strain. The curve was also assumed smooth at yield. It was assumed that the mater- ial satisfies Mises yield condition (suitably modified for the kinematic hardening cases). that the elastic and plastic strain rates are additive. and that the plastic strain rate is related to the yield function by plastic potential theory. In other words. the constitutive theory is the same as the quasi-static elastic-plastic theory. The analysis differs only in that the inertia terms are included. Chapter 2 gives the formulation of the consti- tutive equations for an elastic. plastic strain-hardening material governed by the Mises yield condition under the assumptions of isotropic hardening. kinematic hardening. and combined isotropic and kinematic hardening. In Chap- ter 3. simple wave solutions for a thin-walled tube under tension-torsion impact are obtained for the different hardening assumptions. The characteristic conditions that would be used for a numerical nonsimple wave solu- tion are also given in Chapter 3. although no such numer- ical solutions for the tube case are included in this study. 17 In Chapter 4. simple wave solutions for loading of a half Space of isotropic hardening material by com- bined compression and two independent shear loadings are given. A formulation for nonsimple wave numerical solu- tions in a half space is also given in Chapter 4; the numerical scheme used is an adaptation of the one devel- oped by Vitiello and Clifton (1967) for the thin-walled tube. In Chapter 5.the specific problems solved are further described and the results and conclusions pre- sented. These results show both quantitative and quali- tative differences between the simple-wave solutions for isotropic hardening obtained previously by Clifton (1966) and the solutions for kinematic hardening or combined kinematic and isotropic hardening. For example. with some boundary conditions the solutions assuming kinematic or combined kinematic and isotropic hardening Show a dis- continuity in shear stress propagating at the elastic Shear wave Speed c . while isotropic hardening predicts 2 no discontinuity for those particular boundary conditions. The half-space simple-wave solution for step- function loading by compression and two shears reduces 18 to the result for loading by compression and a single shear. previously obtained by Ting and Nan (1968). when there is no initial prestress beyond yield. since the simple—wave equations require the ratio of the two shear stresses to be constant. But when the half space is pre— stressed statically beyond yield by shear in one direc- tion there may occur a fast simple wave with a constant ratio of the two shear stress components followed by a slow simple wave with a different constant ratio of the two shear stress components. The transition from the fast wave to the slow wave involves a discontinuity in both shear stress components propagating at the elastic shear wave speed c This discontinuity wave either 2. follows immediately after the fast simple wave. or in some cases it may be separated from the fast wave by a constant state region in the x.t-plane. One specific example of a numerical solution is given for nonsimple waves in a half Space loaded by com- pression and shear boundary tractions that increase linearly with time after the initial impact in such a way that simple waves are not produced. CHAPTER 2 WORK HARDENING ASSUMPTIONS 2.1 INTRODUCTION The question of just how the yield function changes during plastic deformation under combined stresses is a very important one. Several hardening hypotheses have been proposed for incorporation into phenomenolog- ical theories of plasticity. all failing to account for all the phenomena observed. Only two of the hypotheses have received much attention. The oldest and most widely used assumption is that of isotropic hardeninglwhich assumes that the yield surface in stress space maintains its shape. center. and orientation. while its size increase is defined by a single parameter depending on the plastic deformation or on the plastic work done. Hill (1950) attributes the earliest formulations of isotropic hardening for com~ bined stresses (with elastic Strains neglected) to Taylor and Quinney in 1931. Schmidt in 1932. and Odquist in 1933. The isotropic-hardening assumption has been fairly 19 20 successful in correlating different kinds of radial paths in stress space (paths. such that the stress components maintain their ratios as they increase). See. for example. Lubahn and Felgar (1961). But the isotropic-hardening assumption has not been so successful when the loading is not radial and especially When unloading and reversed loading occur. For such reversed loadings. most metals exhibit a Bauschinger effect. such that the yield stress for the reversed loading is smaller in magnitude than it would be for continued loading in the original direction and may even be smaller than the yield stress would have been for an initial loading in the reverse direction. See. for example. Lubahn and Felgar (1961). The Baushinger effect is illustrated schematically for a uniaxial stress load- ing in Fig. 2.1 where Y is the yield stress after re-' R versal while Y is the yield stress for continued loading in the original direction. and Y0 is the initial yield stress. Isotropic hardening would require YR = -Y and therefore implies no Bauschinger effect. The Simplest hypothesis incorporating a Bauschin- ger effect is Prager's kinematic-hardening assumption. where the yield surface does not change its size. shape. 21 Fig. 2.l.--Bauschinger effect for uniaxial stress (schematic) or orientation but merely translates in stress space in such a way that its center always moves in a direction parallel to the normal to the yield surface at the active point. Hodge (1957) and Ke-chzhi (1958) have introduced combinations of kinematic hardening with isotropic harden- ing. The present study will examine the effect on com- bined-stress waves in a thin—walled tube of varying the 22 mixture of the two hardening assumptions. The basis for the combination will be discussed in connection with the normal component of the motion of the local surface ele- ment of the yield surface in stress space. V 011 Pig. 2.2.--Yie1d surface in stress space (schematic). Figure 2.2 is a schematic illustration of the yield surface in stress space; it should be understood to represent a hypersurface in a nine-dimensional 23 Euclidean space in which the nine stress components Oij are Cartesian coordinates. The vector g'in this space represents the state of stress. while the vector 3’ is the position vector of the center of the yield surface and £'= g’- e, For purely isotropic hardening we have E.’ o and 2.5 E} and the origin 0 coincides with the center C. Then. for any infinitesimal incremental loading d2, the yield surface simply expands. and the normal component ds = dr’o fi’of the motion of the surface element is equal to dg'- E, Here dr denotes the vector from A to D; it would coincide with d2 if the new loading were radially outward from the previous center to the subsequent yield surface. For purely kinematic hardening. we have again ds = dg'- fi/ but this time the motion of the surface ele- ment is entirely in the normal direction. and it is .caused by a translation of the whole surface rather than by an enlargement. For combined kinematic and isotropic hardening. the normal motion ds will be taken to be the sum of a part produced by enlargement and a part produced by 24 translation. This combined hardening will be discussed in Section 2.4. First we shall consider isotropic hard- ening and kinematic hardening separately in Sections 2.2 and 2.3. 2.2 ISOTROPIC HARDENING This simple assumption is the only one that has received extensive application. According to the iso- tropic hardening assumption the yield surface merely en- larges. with its size governed by a single parameter. as was pointed out in Section 2.1. What remains to discuss is how the size parameter depends upon the deformation and how this is related to the normal component ds of the motion of the local surface element. The magnitude of this single parameter is usually assumed to depend on the plastic deformation by one of the following two schemes. which are equivalent when used with the Mises yield condition. According to the first scheme. a 222? versal plastic stress-strain curve is assumed to relate two scalar quantities. the effective stressZE (measuring the size of the yield surface) and the integral of the 25 effective plastic strain increment EEP. while the second scheme assumes that E is a single—valued function of the total plastic work Wp = foij dEEj‘ When the Mises yield ' EEP condition is used. 0 and are defined as follows. See. for example. Hill (1950). -= lab-1’ ’% O (3J2) - (2 Oij Cid) (2.2-l) —p_g 83.2. B p32 de (3 12) (3 deij deij) (2.2-2) Here the Oij are the components of the stress tensor for i.j = 1.2.3 while the primes denote the deviatoric stress components. Repeated letter subscripts imply summation. J’ and I are the second invariants of the deviatoric 2 2 stress tensor and the plastic strain increment tensor respectively. The numerical factors in Equations (2.2—l) and (2.292) are so chosen that. for a uniaxial stress HEP reduce to O and dep respectively. state. O and 11 11 Hence the assumption of a.universal stress strain_curve E = .mazP] (2.2-3) permits the determination of the function F. in principle. by a single stress—strain curve in simple tension. The 26 assumption that one universal stress-strain curve governs all possible combined-stress loadings of a given material is a very strong one. but it has been fairly successful in correlating radial paths in stress space. How does the assumption of Equation (2.2-3) re- late to the motion of the yield surface element in the stress Space? Fig. 2.2 is reproduced in Fig. 2.3 for the isotropic hardening case where g’a 0 and g a g. f N :o QI Fig. 2.3.--Isotropic hardening case (schematic) For isotropic hardening all radii increase in proportion. Thus 27 (ii = (dz/mg. (2.2—4) and d2? ds . d}; ° :1 = (dB/Mg-TEpT (2.2-5) since d2? is parallel to the normal at the active point of the yield surface. by plastic potential theory. if the degj coordinate directions are superposed on the oij directions of the stress space. Let 9 denote the slope of the universal stress-strain curve of Equation (2.2-3). Then. since |d£P| ..Jr§:EZP. and g3? - g. Equa- tion (2.2-S) yields ~ ds =J 35 {1,3 - dep. (2.2-6) For the Mises yield condition. the plastic work increment is Earp = g - dip. (2.2—7) whence ds =.J g- g HEP = %g |d£P| . (2.2-8) The last result is a form convenient to work with later on. 28 Attempts to construct a more complicated iso- tropic hardening theory than the one based on the Mises yield condition have met with little success because of the inherent inaccuracy of isotropic hardening. which always neglects the Bauschinger effect. The simplest assumption incorporating a Bauschinger effect is Prager's kinematic hardening. 2.3 PRAGER'S KINEMATIC HARDENING Prager (1956) proposed an alternative assumption. which he called kinematic hardening. In kinematic hard- ening the yield surface in stress space does not change its size. shape or orientation. but merely translates in the direction of its normal at the active point. Thus an initial yield surface f(Oij) = 0 is changed into fu’ij - aij) = f(€ij) '5 0 . (2.3-1) where eij and aij are the nine components of‘g and g. respectively. defined in Section 2.1. Prager assumed that the plastic strain history determines the aij history as follows. The time deriva- tives dij and éij are assumed to be related by 29 _ -P ij - b eij . (2.3—2) where b is a scalar function,whose dependence on the de- formation history is to be specified. Hodge (1957) pointed out that the concept of kinematic hardening must be applied in the nine-dimensional stress space even when the loading is such that some of the stress components are zero. Apparently all applications up to the present have assumed linear kinematic hardening in which the parameter b does not change during the deformation. In order to make the hardening prediction reduce in uniaxial tension to a stress—strain curve with a nonlinear plastic range. it will be assumed in the present study that b is the same function of the accumulated effective strain deP in a combined-stress deformation as it is in uniaxial tension. There is of course no fundamental justification for making this assumption (or for assuming kinematic hardening). But it leads to a stress-strain history which coincides with the predictions of isotropic hard- ening when the loading is radial. and isotropic harden- ing has been fairly successful in radial loadings. In a uniaxial tension loading 0 the nonzero 11 components of the unit normal §,to the yield surface are proportional to the nonzero deviatoric stresses. 3O , 2 °11 3 O11 ’ ’ 1 (2.3-3) “22 = °33 ‘ "3 011 Hence. for uniaxial 011' A _ 2 A l: A l; A n ”J? 311 "J? 322 J? 9.33 ”'3‘“ where each Eij is a dimensionless unit vector in the di- rection of the Oij axis in the stress space. Since dd = d . we have G11 211 _ _ 3_ (2.3-5) d8 d2 - g ~/-6- doll Also. in general. purely kinematic hardening implies ds = ldgj = b IdEP| . (2.3—6) Hence. for uniaxial tension 011' kinematic hardening gives 2. 9 ds / z'b dell . (2.3 7) From equating the two expressions for ds given in Equa- tions (2.3-5) and (2.3-7) it follows that. in uniaxial tension. - 2. p - 2. _ b - 3 (doll/dell) — 3 g . (2.3 8) where g is the slope of the curve of stress versus plastic strain. In a combined-stress deformation with kinematic 31 hardening. it will be assumed that b is two-thirds of the slope of the uniaxial curve at the point on the curve P 11 is equal to the value that faEp has in the com- where e bined-stress case. Since the Mises yield condition will be used. the normal to the yield surface will have no component in the direction of the hydrostatic line 011 = 022 - 033. Hence the normal component of the surface-element motion will be parallel to the deviatoric hyperplane all + 022 + 033 = 0 . (2.3-9) Thus dg f3 = dg’ Q = ds . (2.3—10) and di.’ g = drf - §'- ldrfil - ds. (2.3—11) where d2: is the deviatoric stress increment with compon- ents doij and drf is the deviator of dr, (For the Mises yield condition. the normal is parallel to grand hence to drf.) For any radial loading under combined stresses. both E and d2: will be parallel to g: and the center moves on the line from the origin to the deviatoric stress point 32 at the initial yield. so that éjis also parallel to g Then for radial loading ds = dg . g = dg’ fig = [dg’] . (2.3-12) Also ds = bldgpl = — g ldepl . (2.3-13) whence for radial loading . I 2 p ldg’I -'§'g‘|de | (2.3-l4) or 3 Thus for combined-stress radial loading this kind of kine- matic hardening gives a? = g 52" (2.3-16) where g is the same function of [52p as it is in uniaxial tension. in agreement with the predictions of isotropic hardening based on the same uniaxial curve. There is some evidence (see. for example. Naghdi (1960)) that the kinematic hardening assumption in some cases predicts phenomena better than does isotropic 33 hardening. although such a simple assumption cannot give an accurate account of the extremely complex actual ma— terial behavior. It is expected that a combination of the above two assumptions might offer an improvement over the separate ones. 2.4 COMBINED ISOTROPIC AND KINEMATIC HARDENING Hodge (1957) has used combined isotropic and kinematic hardening in a general approach to the problem of piecewise linear plasticity. He assumed Tresca's yield condition for the initial yield. A similar formu- lation was given by Ke-chzhi (1958). In the present in- vestigation. a combination of isotropic and kinematic hardening is also used. but with the Mises yield condi- tion and without any restriction to linear hardening. It is assumed that the total hardening is a combination of expansion and translation of the yield surface in the stress space. In Fig. 2.2 the solid curve gives schematically the instantaneous position of the yield surface. while the dashed curve represents a portion of the new yield 34 surface after an infinitesimal stress increment d2, The normal component ds fi'of the motion of the local surface element at A will be considered to be the sum of a motion produced by expansion of the surface from its current center C (as in isotropic hardening from the center 0) and a translation of the surface in the direction of the normal fi'at A. The portion ds of ds caused by the ex- 1 pansion can be calculated as in Section 2.2. if'O is re- placed everywhere by R. Here ,,15 3 R ' (3 eij eij) is the radius of the current yield surface in a direction , *5 axis. just as'O = (g-aij aij) was when the center remained at the origin 0. From Equation parallel to the all (2.2-8). the result is (181 -% : l dip I . (2.4-1) de where dR/d'emp would be equal to g. the slope of the uni- axial 0 versus 6p curve at the point where e 11 11 the current value of dep. if the incremental hardening P 11 equals were all associated with expansion of the yield surface from the current center C. The translation of the yield 35 surface causes additional motion. Let this translation be denoted by dsz; it can be written as ds2 - b | d5? | (2.4-2) as explained in Section 2.3. If there were no incre— mental expansion. we would have b - g-g. and dR/d'é"p - 0. When expansion and translation occur simultan- eously. we let m denote the fraction of g accounted for by the expansion term and l-m the fraction accounted for by the translation. Then dR/dtP = mg (2.4-3) b = gii—m)g (2.4-4) dsl . %-mg|d£P| (2.4—5) as2 - %ii-m)g|dg?| . (2.4-6) and the total ds is (2.4-7) 36 The fraction m can also evidently be interpreted as the fraction of the total ds accounted for by the ex- pansion. i.e. dsl strate. Thus d2 §=ds=fidsl=%g|depl=~/ggdzp. (2.4-8) whence d3 ' '(Vf/l-Vfl) =~/_%—"9 agp . (2.4-9) where Vf denotes the gradient of the yield function f. For the Mises yield function. -_1. I I 2- f - 2 gij gij — k - 0 . (2.4-10) we have af/aaij = af/agij = eij and (2.4-11) w _ I . I k = |Vf| - (gij gin.) Jik. Hence do - Vf = ’ da . (2.4-12) - m ds. as the foregoing equations demon- 37 and Equation (2.4-9) furnishes the expression ._p =./§ , de d0 (2.4-l3) 29k aPq pq which will be useful in transforming the plastic strain increment constitutive equation. The fraction m is still to be assigned. It could be supposed to vary during the deformation. but in this investigation it will be assumed constant during the de- formation. and the effect of assuming different constant values for m between zero and unity will be examined. The plastic potential flow rule determines the plastic strain increments as follows de?. a digT. . (2.4-l4) 1] 13 where d1 is a scalar multiplier to be determined. Squar— ing and adding the nine component equations gives dep d6? - ((11)2 ET. 5? .. . . (2.4-15) 13 13 13 13 or g-(EEP)2 = (di)2 0% R2) . (2.4-l6) 38 whence d1 . g-(dEP/R) (2.4-17) and Equation (2.4-l4) becomes "P 9 2.2g. , _ This equation can be transformed by use of Equa- tion (2.4-13) and R =«f3 k to the form p . 2 I I deij [3/49k ] [épq dopq] éij (2.4-19) [qu dom] 5ii I where H = 3/4gk2 . (2.4-20) The last Equation (2.4—19) is the form that will be used in Chapter 3. It should be recalled that g is the slope of the uniaxial curve of 011 versus Gil at the point where 651 equals the current value of dep in the com- bined-stress deformation. and that k2 = k 57. g’. is not 13 in a constant except in the case of purely kinematic harden- ing. 39 It should also be noted that in the usual formulation of combined-stress plasticity theory the strain increments deij are the natural strain increments. such that ds.. 5v. 5v. dt ij 2 Oxj Oxi where the Dij are the components of the rate of deforma- tion tensor. and the xk are spatial or Eulerian coordin- ates. The natural-strain increments are approximately equal to the increments of Lagrangian small strain in a finite body when the displacement gradient components are everywhere small compared to unity. In the combined- stress wave propagation problems to be considered. the displacement gradients will be sufficiently small that the eij may be interpreted as Lagrangian small strains. The stress components of the combined-stress plas- ticity theory are also the usual Cauchy stress components Oij' defined relative to the deformed configuration. In Appendix 2 it is shown that. for the small strains con- . _ o _ o O o Sidered. a11 T11. and 012 T12. where T11 and T12 are the only components of the nonsymmetric first Piola-Kirchoff 40 stress tensor that appear (for plane waves in the x- direction) in the equations of motion written in mater- ial coordinates (Lagrangian formulation). For the half space case. the equality is not limited to small strains. This completes the formulation of the plastic part of the constitutive equations. In Chapters 3 and 4 these equations will be combined with the usual elastic Hooke's law and applied to combined tension and torsion waves in a thin-walled tube and compression and shear waves in a half space. CHAPTER 3 THIN-WALLED TUBE 3 . 1, INTRgUCTION In the following Section 3.2. the equations gov- erning the motion of a loading wave of combined normal and shear stresses (with nonelastic deformations) in a thin-walled cylindrical tube are given and solved. These equations are a generalization of the equations originally derived by Von Karman for a thin unstretched wire sub— jected to an impact load in tension. The generalized equations for the tube case were solved by Clifton (1966) under the assumption of isotropic hardening. Character- istics and characteristic conditions for the problem. and the particularly simple forms to be integrated for simple wave solutions are given in Section 3.3. 3.2 EQUATIONS GOVERNING THE MOTION The system of governing equations for the propa- gation of combined longitudinal normal stress and 41 42 torsional shear stress in a thin-walled cylindrical tube given below is obtained by assuming that plane sections remain plane and that the stress is uniform across each section. Lateral inertia effects are assumed to be negli- gible. Thus all the stresses and particle velocities are functions of x. the initial coordinate along the axis of the tube. and the time t. Material (Lagrangian) coordin- ates will be used. In Appendix 2. it is shown that. for small strains and for the geometry considered. the assump- tion that stress and particle velocity are independent of the coordinates other than x implies that the Cauchy stress component a is equal to the first Piola-Kirchoff tensor ll component T11 for the same particle. Their common value will be denoted by 0. Similarly a = T° . which will 12 12 be denoted by T. The other components of the two tensors are not all equal. but only these two appear in the equa- tions of motion. To facilitate writing. the following symbols are adopted I x 8 initial coordinate of the section. measured along the axis of the tube 0* 1* 43 time period elapsed after the boundary x = 0 is loaded axial stress shear stress a——-coordinate of the yield surface center T~—-coordinate of the yield surface center 0-0* 2 _. _ * 30 O T - 1* longitudinal particle velocity tangential particle velocity axial unit extension engineering Shear strain (angle change) mass density of the material of the tube in the reference state Young's modulus shear modulus 44 Subscripts x and t denote the partial derivatives with re- spect to x and t respectively. Superscripts e and p denote the elastic and plastic parts respectively. The equations of motion. for no body forces. are given in the reference state by '%E = p‘%% (3.2-1a) 3:: a p gLVt. (3.2-lb) See Appendix 2 for a demonstration that the equations of motion take this form. Let U(x,t) and V(x,t) respectively be the longi- tudinal and tangential displacements at time t of a cross- section initially at a distance x. Loading occurs at x = 0. Then 6 = U* (3.2—2a) u = Ut (3.2-2b) y =‘Vx . (3.2-3a) v = Vt (3.2-3b) The Sign convention to be used for compressive impact is that a and 6 represent compression. while positive 45 displacement U and velocity u represent motion in the negative x-direction. For tensile impacts these are re— versed; then the same Equations (3.2-l) and (3.2-2) can be used for either compressive or tensile impact. Differentiating Equation (3.2-2a) with respect to t and Equation (3.2-2b) with respect to x yields the following continuity equation 6 = u . . (3.2-4a) Similarly Equation (3.2-3) yields another continuity equation given by yt = vx . (3.2-4b) It is assumed that the total strain rate is separable into elastic and plastic parts. Thus e P = + . .— et 6t 6t (3 2 5a) and - - ‘Yt 'yt + y: . (3.2 5b) where the elastic parts are e =% a (3.2-6a) and (3.2—6b) .3 n ll QIH d (If 46 while the plastic strain rates. given by the plastic po- tential theory. are p - . a_f - . ’ and p - . -a_f . . - 7t - X 5n X (2n) . (3.2 7b) Here f is the yield function and l is a scalar function. both defined in Chapter 2. The assumption of additivity of the elastic and plastic Lagrangian strain rates (in- stead of rate-of—deformation components) limits the ap- plicability of the theory to small strains (less than about 0.05). as is explained in Appendix 2. It is assumed that the material yields according to the Mises yield condition given by %'(€’)2 + n2 - k2 = O . .(3.2-8) For the case of only two nonvanishing stress components d1 is much simplified: from Equation (2.4-19). X is given by i . H [g’ at + 2n Tt] . (3.2-9) 47 Thus Equation (3.2-7) can be written as e: = i g’ = H (g’ot + 2n 1t) g’ (3.2-10a) vi - 1(2n) = H (g’ot + 2n Tt)(2n), (3.2-10b) Substituting Equations (3.2-5) through (3.2-10) into Equation (3.2-4) gives 1 , _ + E a H (g a t + Zth)§ = 9x (3.2-11a) t l , G Tt + H (g at + Zth) (2n) - vx . (3.2-11b) Then Equations (3.2—l) and (3.2-11) form a complete sys— tem of first order partial differential equations. which can be written in the matrix form A'w + B = o . (3.2—12) w «t ~x where w is the vector 2€. n a < o a and A and B are the 4x4 square matrices given on the fol- lowing page. 48 ’— —1 p 0 0 0 1 I 2 a 0 [E + H(€ ) ] 0 2H€ n i a 0 0 p 0 o 2Hg’n o [ 81- + 411.12] r— ‘1 0 -l 0 0 -1 0 0 0 E = 0 0 0 -1 Q 0 -l 0 ._ .1 Note that A and g are square symmetric matrices. In the elastic region H = 0; therefore. A becomes p 0 0 0 1 O E 0 0 A6 = "’ o o p o 1 0 0 o G while B remains the same. 49 The solution of Equation (3.2-12) will be carried out in Section 3.3 by the method of characteristics. 3.3 SOLUTION OF GOVERNING EQUATIONS The system (3.2-12) is a quasilinear. symmetric. hyperbolic system of partial differential equations. From the theory of partial differential equations wave speeds. characteristic conditions and the simple wave solutions for this system of equations will be obtained in the following subsections. 3.3.1 Wave Speeds The characteristic velocities c are the roots of the determinantal equation |cA7§J = 0 (see Appendix 1). Expansion of this determinant gives the quadratic equa- tion. L(pc2)2 - (M + N) pc2 + l = o . (3.3-1) where 2 1 2 - -S =- L MN M + 4H n .2 ’ (3.3-2) N=%+H(€) 3.2235,. 50 Solution of Equation (3.3-1) yields 2”_ ' _ 2 2 35 pc pm.) i [(BZ‘ILN) +45 1 (3.34, The positive Sign in the numerator gives the fast wave speed c while the negative sign gives the slow wave f speed cs. Both cf and c8 depend upon the state of stress and plastic deformation through Equations (3.3-2). For the elastic case both these wave speeds reduce to elas- tic bar velocities: c f reduces to co =I¢E7 . while cs reduces to c2 uIJG7p. By rearranging Equation (3.3-l). we can write it as D(C) H (t’)2 (CZ/cg-lwc2 + 4Hn2(c2/c:-1)pc2 + (cz/ci-l) (cZ/cg-l) = 0 . (3.3-4) whence it is seen. since 0 < c2 < CD that D(co) ) 0 D(c2) g 0 D(0) > 0 . (3.3-5) Therefore. the roots cf and c8 satisfy the inequalities 0.$ c ‘$ c s '$ c .g c . (3.3-6) 51 3L3.2 Characteristic Conditions The characteristic condition along the charac- teristic dx = c dt is given (see Appendix 1) by if Q d = 0 . (3.3-7) (V where A? = [21. 22. 23. 24] denotes the transpose of a left eigenvector of (CA - g) . Because of the symmetry of A and B. ’9; is also a right eigenvector of (CA - B) . Thus 2"” (cg -§) = (CA - E) 3: = o. (3.3-8) Expansion of Equation (3.3-8) yields tlpc + £2 = 0 . (3.3-9a) 9. 9. 9. = , , - 1 + Nc 2 + SC‘4 0 (3 3 9b) pc23 + 24 = 0 . (3.3-9C) SC22 + 9.3 + Mc24 = 0 (3.3-9d) Solution of any three of the above four equations gives (for that value of c) the eigenvector. which is determined only up to an arbitrary constant. Solution of Equations (3.3-9a. b. c) yields (3.3-10a) N c2 — 1 If we arbitrarily choose 2. = Sc. we get 1 3“ = [Sc. -Spc2. -Nc + fie. Npcz-l] (3.3-10b) Instead. solving Equations (3.3-9a. C. d). we get an al- ternative choice * 2T ~‘l ‘g—c" - MCI MPCZ-lo Co 'SPCZ] o (3.3-10C) if we arbitrarily choose 23 = Sc. The elements a: are re- lated to the elements 2i by the equation 2: = -Nc£i. as can be verified by using Equation (3.3-l). A left eigenvector f; also determines a simple- wave solution. as follows. 3.3.3 Simple Waves A Simple wave solution is defined as the particu- lar solution of Equation (3.2-12) in which the vectorjw is a constant along each characteristic of a family of the characteristic lines. Since c is dependent on the stress 53 state. in a simple wave region. the characteristic lines belonging to the family are straight. Also from the theory of characteristics (see Appendix 1). in a simple wave region. as we go from one straight characteristic to a neighboring one. dw is given by (CA - B) dw = 0 . (3.3-ll) From Equations (3.3-8) and (3.3-ll). we see that g and d! are proportional. Therefore. with 3? given by Equa- tion (3.3-10b). it; a da 3 d3; ___ d1 (3 3_12) Sc ' 2 l_ 2 -spc pc NC Npc 1 From Equation (3.3-12) it immediately follows that d1 .1 (3 13a) '*" - N 2 pc The alternative form of £?. given by Equation (3.3-10c). yields 1 .2 - M 91—9 -‘E‘=‘——- . (3.3-13b) d1 S Other equalities of Equation (3.3-12) yield 54 dv du l arar-az- (3'34“ Let Y denote the function giving the value of 3% . We will consider the history of an individual particle. so that the path of integration from one characteristic to another will be along the line x = constant of the x.t- plane. It will be convenient to take the shear stress T or the axial stress a as the independent variable along this line. instead of the time t. The system of equa- tions to be integrated then takes the form d0 d1 Y (3.3-15) du .1. 1 d1 pc Y (3.3-16a) 51y. d1 (3.3-16b) ‘DIH O to determine a. u. and v as functions of T. P P The plastic strains 6 and y can be obtained by integrating the constitutive equations. Equations (3.2-11). Dividing Equation (3.2-11a) by at and Equation (3.2—11b) by Tt gives 55 1 I 2 I dT d€ E + H(§ ) ZHRE da da (3.3 17a) 1 2 Ida d7 G + 4H1] + 2Hn§ _d't = —dT (3.3'17b) along the line x = constant. since ux = 6t and vx = Vt by Equations (3.2—4). If we substitute Equation (3.3-13b) in Equation (3.3-17a) we obtain or. in terms of T as the independent variable. de 1 d1 - 2 Y. (3.3-18a) pc Similarly substituting for92 in Equation (3.3-17b) yields dT d7 1 __ .. __ (3.3—18b) d1 pc2 along the line x . constant in a simple wave region. The translation of the center of the yield surface as the loading continues is obtained as follows. Since da* = b dep . (3.3-19a) p dT* = b 511— . (3.3-19b) 2 56 we Obtain p I da* 2de 2d1 .g -' - . - = . 3.3-20 dT dVP dk(2n) n ( ) Also from Equations (3.3—18b) and (3.3-19b) -2 e,1o.___1 a: dT* ' 2(d7‘d7 ) 2( 2 d1 - G ) pc Hence d1* b l 1 d1 2 ‘ 2 ' a) (3°3'21) pc If we substitute b = g-(l-m)g. H in Equation (3.3-21). 49k2 d * l 1 '5:- =—"E' (1-m)(—2- '- 4Hk pc From Equations (3.3-20) and (3. d * . l a? "—2' <14“) (‘7 '- 4Hk pc Equations (3.3-15). (3.3-16). a complete set of simultaneous. ferential equations. which can by Equation (2.4-4). and we obtain 1 _ G) (3.3-22a) 3-22a) .1 .S’ _ G) n (3.3 22b) (3.3—18). and (3.3-22) form first order. ordinary dif- be integrated numerically 57 for an assigned value of m. with c set equal to the ex- pression for c in the fast simple wave or to the expres- f sion for c8 in the slow simple wave region. Results for various values of m and particular choices of the material parameters and loading conditions are given in Chapter 5 and the stress trajectory plots of Fig. 1 to Fig. 12.* If the stress-strain curve has a continuous tan- gent at the yield stress. H will be zero at a point on the initial yield surface. and the right sides of Equa- tions (3.3-22) are indeterminate. In Section 3.4 it is shown by L’Hospital's rule that the limiting values of da*/dr and dT*/dT as H 4.0 are given for a slow simple wave (cs-q c2) by 2 ‘ * gill-1:0 g_: - (1_m) 1L2. (3.3-23a) k lim 60* I qu d? = (1m) 3% (3.3-23b) k *For Figures 1 to 31 see Section 5.7. 58 3.4 PROPERTIES AND SLOPES OF STRESS TRAJECTORIES FOR SIMPLE WAVES In this section first of all some of the general properties of the stress trajectories for simple wave so- lutions will be discussed. Then. in order that numerical integration of the equations obtained in Section 3.2 can be carried out. the initial slopes of the stress trajec- tories at the starting points of the integration proceSS will be established. because in some cases there are cer- tain curves. e.g. the initial yield surface. where some of the parameters may become unbounded or indeterminate and thus not able to be handled numerically. When the tube is initially unstressed. the plastic simple wave stress trajectory in the 0.1 plane begins at a point (00.10) on the initial yield curve (assumed in this study to be the Mises yield ellipse a2+3r2 = 3k2). If the impact loading is assumed to be a boundary stress (ab.Tb) applied instantaneously at x = 0 at time t = O and then maintained constant. there is an elastic wave ahead of the plastic wave in which axial stress a0 propa- gates as a shock wave at a speed Co. This is followed by an elastic shear wave at speed c propagating as a 2 59 jump in T to the value To. unless To = 0. If T0 = 0. a fast plastic simple wave of uniaxial stress a follows the elastic wave. The stress trajectory for this fast Simple wave begins at (00.0) and moves along the a-axis. If To * 0. there is no fast plastic simple wave for the case of no prestress; instead the elastic shear wave is fol- lowed by a slow plastic simple wave whose trajectory be- gins at (0°.To) on the initial yield surface and leads to the final stress (ab.Tb). When there is a uniaxial fast plastic wave (case To = O. mentioned above) the fast wave may be followed immediately by a slow plastic wave of combined stress. whose trajectory leads from a point say (a1.0) on the a-axis (01“ ac. where 0C is the criti- cal point). Several slow wave trajectories for no ini- tial prestress are shown in Fig. l to Fig. 3 . emanat- ing either from a point (OO.TO) on the initial yield sur- face. or from a point on the a-axis to the right of the ini- tial yield surface. When (ab.Tb) is given. the initial point (0°.To) or (01.0) for the slow wave stress trajec- tory leading to (ab.Tb) is not known at first. The appro— priate initial point for a given (ab.Tb) can be located approximately by interpolating between two of the previ- ously calculated trajectories. 60 When there is initial shear stress Tl beyond the elastic limit. there is no initial elastic shock wave but there is a fast plastic simple wave of combined stress leading from (0.Tl) . Several examples of such fast plastic waves are shown in Fig. 5 to Fig. 11. The fast wave may be followed immediately by a slow wave. whose stress trajectory begins at the end of the fast-wave tra- jectory. as in Fig. 6. or in some cases the fast wave may be separated from the slow wave by a jump in shear stress. so that the slow wave stress trajectory emanates from a point different from the end of the fast wave trajectory. as in Fig. 7. These cases involving a shock wave separ- ating the fast wave from the slow wave will be discussed further in the results and discussion of Chapter 5. Here we consider only the case that the slow—wave trajectory begins at the point where the fast-wave trajectory ends. and Show that at such a point the slow-wave trajectory is orthogonal to the fast-wave trajectory. as was demon- strated by Clifton (1966) for the case of isotropic hard- ening. Equations (3.3-l3) give the slopes of the stress trajectories for the simple wave solutions. For the slow 61 Simple wave the slope (da/dT) given by Equation (3.3-13b) can be written as 1 CZ-M p8 da ( E ) := I (3.4-l) Cs S where M and S are given by Equation (3.3—2) and _i2 can pc s be obtained by rearranging the quadratic Equation (3.3—l) into a quadratic for -l3-as the variable as fol- pc lows. Dividing Equation (3.3-l) by (pcz)2 gives 1 1 2 2 - (M+N) 2 + L - O . (3.4—2) (PC ) PC Solution of the quadratic Equation (3.4-2) yields t 4-2- = ‘i' (mu): [(M—N)2 + 452] . (3.4-3) pc 'where the positive Sign corresponds to the slow wave speed cs while the negative Sign yields the fast wave speed cf. Therefore. at a point where a trajectory changes from a fast wave to a slow wave. 1 2 " " ""12 + (“+1“) - (3.4-4) pc8 Pcf 62 Substituting Equation (3.4-4) into Equation (3.4-1) im- mediately yields N- — ——2-1 pc ( -—d° > = ———f (3 4-5) 'dT cs S ' ' Now for the fast simple wave. the slope of the stress trajectory can be written by Equation (3.3-13a) as (93) - s . (3.4-6) d" c T— f '-:3 - N p f Multiplying Equations (3.4-5) and (3.4-6) gives gg (d1) 19. _ ' ( dT ) '- '1 o (3.4-7) cs Of which implies that the simple wave stress trajectories for the fast waves and slow waves are orthogonal to each other. when they intersect for a particular stress history. In order that the numerical integration can be started. some of the limits evaluating the initial slopes of stress tra- jectories and other associated slopes will be evaluated next. 63 From Equation (3.3-2) we see that. since 8 = 2Hg’n vanishes whenever H = 0. g’ = 0. or n = 0. the slow wave speed for these Special cases is given by f if M-N > 0 pc (3.4-8) i=4 if M-N < O ZHP‘ Ska The case H = 0 occurs for a point on the initial yield surface. if the uniaxial stress—strain curve has a con- tinuous Slope at yield. For the case H = 0. we have M --c1:and N .l . so that (M-N) > 0 and pc: =i'? For E the case g’ = 0 . we have M-N = é’+ 4Hn2 - é-and pc: =:% . Finally. for the case n = 0. we have M-N = é’- %'- H(€’)2 and (M—N) > 0 for H(g’)2 < i-- fi'. while M-N < 0 for H(€’)2 > é'- i". We see that there is a critical point on the n = 0 axis where H(§’)2 = é‘- fi'. From Equation (3.2-11a). with q = 0 and ux = at. we see that 1 I2 _ [E + H(g ):lat — 6t . (3.4-9) Hence. if this critical point is reached by uniaxial loading. it represents the point on the uniaxial curve where de/da = é-or where the slope of the uniaxial 64 tension curve equals G. This critical point is very im- portant for the construction of simple wave trajectories. as the following discussion shows. The quantity Y given by Equations (3.3-13) as Y = S (3.4-10a) 8 m L-N c2 p 8 takes the form w = 3— (3.4-10b) for M-N > 0 in the special cases H = 0. g’ = 0. or q = 0. Hence. since S = O in these cases. we have Y8 = 0 in these special cases as long as (M-N) is positive. At the critical point mentioned above M-N = 0. and the expression for Ys is indeterminate. For larger values of g (on n=0). we have MPN < 0 so that pc: ='%'and the expression Y - appears to be indeterminate at all points S s _l_,_ N pc: on the g axis to the right of the critical point. By using the alternative form for 3% in Equation (3.3-13). we see that .1. gig—.5... g .._3_ (3.4-11) 65 to the right of the critical point. This shows that a: do = 0 there. This im- $-= 0 at these points and that plies that the only slow wave trajectory that can orig- inate on the 5 axis to the right of the critical point is a wave of uniaxial tensile or compressive stress. Any combined-stress trajectory originating on the g-axis must originate to the left of the critical point or at the critical point. Trajectories originating to the left of the critical point offer no difficulty. For them the limiting value of T as n—pO is zero. Hence trajectories departing from the n - 0 axis at points to the left of the critical point have a vertical tangent. It may. how— ever. be noted that c -. c f 2 as r|-.0. Therefore. the fast simple-wave, combined—stress trajectories in the a. T plane end at a point corresponding to n = 0. This point in the a.T plane does not correspond to T = 0 ex— cept for the case of m - 1. At the critical point the slope of the trajectory is indeterminate. An infinite number of trajectories de- part from the critical point with slopes anywhere between zero and infinity. Several trajectories will be computed by assuming different values of the initial slope. with particular choices of the material parameters. 66 Finally. before we start constructing these stress * trajectories. we notice that 3% given by Equation (3.3-22a) takes the indeterminate form % for H = 0 (point on the ini- tial yield surface) and thus requires the limiting value of the function to be determined. By L'Hospital's rule. this limit is evaluated as follows. By Equation (3.3-22a) dT* (3;) = (l-m) C 12(J§--Gl—). (3.4-12) 4m: 8 pcs Substituting in Equation (3.4-12) the value of pci given by Equation (3.3-3) we obtain an dT ).——5H {(G+4HT})+(E+H(€)) ( 4k 2 {ewe - am“) ‘5 + 4(2nng’)2] }- Differentiating with respect to H the numerator and the (3.4—13) mhw denominator of the expression on the right in Equation (3.4-l3). we obtain 67 dT* 1- 2 1 8k , Li? - i) + 43.2 - H + 411112 - 30332 + 16132112 (632:1 as H 4.0. Hence the limiting value is .___ _ , 2)- ::T) '37? " :3“ 4n 2+(g’)2+(G E<)-(-'l: _) (E ) G E or Lim dT* 2 H...o d? = (l-m) 33 . (3.4-14a) k” From Equations (3.4-14a) and (3.3-20) . we obtain a * I m 9.0 _ (1-..) n _E._ H-vO dT 2 . (3.4-14b) k Having evaluated the value of the slopes for crit- ical points. we are now in a position to construct the stress trajectories. For various initial conditions and specific material properties the stress trajectories are plotted in Fig. 1 to Fig. 12. CHAPTER 4 PLANE WAVES IN A HALF SPACE 4.1 INTRODUCTION In the following Section 4.2 the equations that govern the motion of a loading plane wave in a half space of isotropic hardening material obeying the Mises yield condition are given. Similar equations were originally given by Bleich and Nelson (1965) for an elastic. ideally plastic material and later by Ting (Dec. 1968) for an isotropic hardening elastic. plastic material with a gen- eral yield condition f(aij) - k2. Ting and Nan (1968) considered combined pressure and shear loadings leading to simple waves and gave some simple—wave stress trajec- tories based on the Mises yield condition. In Sections 4.3 and 4.4 closed form solutions for the stress trajec- tories of simple waves. and a numerical solution for non- simple waves are obtained. These are applied to some boundary loadings not previously treated. including a case of compression and two independent shear loadings. 68 69 4.2 EQUATIONS GOVERNING THE MOTION The basic equations governing wave propagation in a general elastic. plastic material consist of a yield condition defining the transition to the plastic state. a set of constitutive equations relating stress. strain and/or their rates. and the equations of motion. For the most general plane wave propagating in a half space of isotropic-hardening material governed by the Mises yield condition. bounded by the plane x - 0. the above mentioned basic equations are given below. Material coordinates will be used (Lagrangian formulation) in the equations of motion. Thus x is the initial coordinate of the particle. Let v1(x.t). v2(x.t) and v3(x.t) be the x.y and z compon- ents of the velocity of a particle. which are assumed to be functions of x and t only. Let al(x.t) be the normal stress component a in the x-direction at any section x. 11 and let 02(x.t) and a3(x.t) be the lateral normal stresses 0 and 0 Also let T2(x.t) and T3(x.t) be the shear 22 33' stresses C12 and 013 at section x. It is assumed that the normal strains 622 and C33 in the lateral direction at the section x are zero. The above assumption leads to the result 70 02 (XIt) - 03 (XIt) I (4.2-l) as follows. The elastic strain increments are given by the classical theory of elasticity as .e 1+v . v .. ...-_5 I .- 61] E 013 E 1Japp (4 2 2) where v is Poisson's ratio and Oij are the time rates of the stress components oij’ By plastic potential theory with the Mises yield condition. éP. - i 0.. (4.2-3) where l is a scalar function. and the prime on aij denotes the deviatoric part of the stress component. Since the deij will be interpreted as increments of Lagrangian strain. rather than as the increments of natural strain. the assumption of additivity of elastic and plastic parts of the strain limits this treatment to small strains (less than about 0.05). Equations (4.2-2) and (4.2-3) then yield é.. 6.. '.. _'2. ° I 13 13 13 E oij E 6ijopp + Aoij (4.2-4) 71 For all and €22 this gives 6 1 . . ~ ° I 22 = 0 = E [622-v(all+a33)] + 1022 (4.2-5) é - o = -1- [d — (d +5 )1 + io’ (4 2-6) 33 E 33 V 11 22 33 ° Subtracting Equations (4.2-5) and (4.2-6) yields l+v ° I I _ E (622-633) + 1(022-033) - 0. (4.2-7) As long as the material remains elastic. we have 1 - 0 and hence 022 = 633. If the material is initially stress free. it follows that at yield 022 - 033. whence 022-033 = 0 at yield. Hence Equation (4.2-7) shows that 022—033 remains equal to zero during the deformation. In the notation defined at the beginning of this section. the equations of motion in material coordinates can be written as 01x = pvlt (4.2-8a) sz = pv2t (4.2-8b) 13x : pv3t (4.2-8C) 72 where the letter subscripts x and t denote partial deriv- atives. (See Appendix 2 for a demonstration that in this problem the Lagrangian equations of motion take this form.) Under the assumption of isotropic hardening. the Mises yield condition f(oij) = k2 can be written as (° '° )2 2 2 1 2 2 3 + T2 ‘l‘ T3 - k 0 (4.2-9) where k. the yield stress in pure shear. is the single parameter defining the hardening. For isotropic harden- ing. the scalar function X of Equation (4.2-3) is i = Hf . (4.2-10) where H(k) is a function of the accumulated effective plastic strain or of the plastic work done. which may be expressed as a function of k during loading. and f = o of . j at Chapter 2. that See Equations (2.4-l7) to (2.4-20). Recall from for loading [f-- k2. and f 2 0] 110:) 8 2 49k (4.2-11a) H = O for unloading [f~$ k2. and/or f < O] (4.2-11b) 73 where g = dEVEEP for isotropic hardening. The slope g of the universal stress-strain curve is really a function g of the accumulated effective plastic strain deP. but during loadingaJdk - 3 = F[fdzp]. Hence 9 may be con- sidered a function of k during loading. For the half space problem under consideration. Equations (4.2-4) represent four constitutive equations. Note that €22 - e - 0 (4.2-12) and that the continuity requirements give le I Ellt (4.2-13a) sz ' 2e12t ' 7121: (4'2'13b) v3x I 2613t = 713t' (4.2—13c) The constitutive equations. Equations (4.2-4). take the following form when i is given by Equation (4.2-10) and the strain rates are replaced by velocity gradients by using the continuity equations. Equations (4.2-13). MI? 1 E alt - OZt (4.2-14a) + H [salt-302t+21 T2t+ZT T3t](S) = le 2 3 74 ~2v 2gl-v} E alt + E °2t (4.2ql4b) + H[salt-s02t+21212t+21313t1 (-s) a 0 T v (4.2-14c) 3T3t](12) 2x QIH T + 2H[so -so t+21 2t 1t 2 t+ZT 2 2 aha T + 2H[so ~30 +21 1 +2137 3t 1t 2t 2 2t v (4.2-14d) 3t](T3) 3x where .Z s = 3 (01-02) . (4.2-15) Note that 01 = 8. while 02 = 03 tions (4.2-l4) form with the three equations of motion. = -'% s. The four Equa- Equations (4.2-8). a set of seven first order partial dif- IVIVI ferential equations for the seven functions v1 2 3 o o T and T for the plane wave propagating into a 1' 2' 2 3 half space. These equations differ from those given by Ting (Dec. 1968) for the general plane wave only in that the specific choice of the yield condition as the Mises yield condition has furnished the explicit form for i The seven equations may be written in matrix form as = 0 (4.2-16) 2: +12. w w ~t ~x 75 where w is the vector given by T g=[v.v.v.o O T (4.2—l7) 1 2 3 1' 2' 2’ T3] ' and g and E are the 7 X 7 square matrices given below. p1 fl = (4.2-18) 9. 20 2V! where ’25 is a 3 X 3 unit matrix and § is the 4 X 4 matrix: 3 ._J: 2 :_2_\_._ _ 2 [E + as] [E as] ZH'rzs 2H13s [.31 - Hsfl [gsé—'l)+ H82 'ZHT s -2H'r s ...J. —_ E 2 3 s = 3. 2 ,(4.2-19) 2H'rzs -2H'rzs L5 + 4H1“; 41-11213 2H 2H 8 4HT '1‘ + 4HT2 T38 ' T3 2‘3 G 3 L__ ...J The matrix 2 is 51 at" (4.2-20) up I 238 20 76 where §,= 0 0 l O . (4.2-21) Alternatively the matrix § can be written as §’= se + H2;(v§)T (4.2—22) ~ where §f is obtained by setting H = 0 in g and thus cor— responds to the elastic case. and Y5 is the gradient of the yield function f. while (Y§)T represents its trans- pose. (v§)T = [s. -s. 21 . 213] (4.2—23) 2 The solution of the system of Equations (4.2—16) will be carried out by the method of characteristics in Section 4.3. 77 4.3 SOLUTIONOF THE GOVERNING EQUATIONS In the following subsections the simple wave so- lutions of the system of equations given by Equation (4.2—16) will be obtained. Subsection (4.3.1) will de- termine the wave speeds. while Subsection (4.3.2) will discuss the characteristic conditions. Simple wave so- lutions will be obtained in Subsection 4.3.2. 4.3.1 wave Speeds In this subsection the solution for the wave speeds for the system of equations (4.2-16) will be car- ried out by the method of characteristics (see Appendix 1). The wave velocities c are the roots of the 7 X 7 determinantal equation lcgng = 0. The order of the de- terminant can be lowered as follows. The matrix (céfg) can be written as a product of two partitioned matrices 9915.3 E P993 9. £3 (l/pCMI. (ch-E) =- = ‘ (4.3-1) '1' 'r M cS M g g (l/pc)I4 ~ ~ ~ 78 where g and S are the matrices defined in Equations (4.2—19) and (4.2-21). 243 is a 3 X 3 unit matrix. and Q’is given by Q'= pc2 §.- 53% . (4.3-2) The determinant of the product of two square ma- trices is the product of the two determinants of the indi- vidual matrices. and the determinant of a triangular square matrix is the product of its diagonal elements. Therefore. Equation (4.3-l) yields the characteristic equation 1 lei-El =3; IRI = 0 . (4-3-3) where IQJ is a 4 X 4 determinant. The determinant can be further simplified by fol- lowing a procedure suggested by Ting (Dec. 1968). Combin- ing Equations (4.2-22) and (4.3-2) yields 2 = pc2[§e + H v; (37;)1'] - ET 5 (4.3-4a) or g = If + Hpc2 z; (ng (4.3—4b) where De = pczse - 5T 5 . (4.3-5) ~ ~ 79 Ting (Dec. 1968) showed that if E’is an r’x E matrix. 9 and h are column vectors with r elements. and "a" is a scalar. then the determinant of the matrix E.+ a 2.2? is equal to the determinant of E plus a h?§?g/ where {f is the transpose of the adjoint of EX i.e. the element sz in {f is the cofactor of the element Pij in a; For the problem of interest the adjoint matrix is symmetric. Thus M = lgel + pc2 H(Vj)T(ge)*(VLf) (4.3-6) N where (De)* is the adjoint of 2?. The 4 X 4 matrix Q? has only six nonzero elements: e 3 9c _ e 2fl-v2 2 D11 E 1 D22 E 9° (4.3-7) e ch e e -2v 2 ”33"‘344' G '1 D12'1321= E pc The adjoint matrix (2?)* has the following six nonzero elements 80 2 2 (De)* - £1.15!)— pc2(‘E—;- _:D 2 2 11 E (”6’22 = (3:?— '17(%' ‘1) 2 F 2 2 e * = e * a .Qc _ 2c l-2v _ _ 29c (D) (D )44' (G 17L; 2 (1 ”J 33 E e * _ e * _ 2v 29c2 2 (D )12 ' (D )21 - E—'pc G - l . (4.3-8) Hence Equations (4.3-3) and (4.3—6) yield _l_ IDI = C(‘ch _ J) 2 EC2 _ 1) pc2 1-2v _ (l-v) pc ~' ' '.G E' G G 2 2 ecz 3S1-2v2 2 2 ecz l-2v +Hs< -]) pc-1+8HT G E G 2 .. (l—v) " O (4.3-9) 2 2 2 . where T = 72 + 13. EVidently c = 0 and c = t c2 are roots of the equation. where c I‘JG/p is the elastic 2 shear wave speed. The remaining roots are given by the equation ob- . . . . ch . tained by dividing through by Hc ( G -l). The resulting equation is denoted as D(c) = 0 below. It has been trans— formed by substituting G = pcg. noting that 81 2 c l = 2(l-v -—2 ——)-l_2v (4.3-10) ‘2 where c1 is the dilational elastic wave speed. and by eliminating Poisson's ratio v from the resulting equa- tion by introducing the parameter 5 defined by Bleich and Nelson (1965) as . 2§1+v) _ B l-2v . (4.3 11) After some algebraic manipulation. the reduced equation below is obtained. 2 D(c)=l£2__1 £3_1 32+9-T2 23.23-:l '53 2 6 c2 C2 c2 2 C2 2 2 2 2 2 2 c 3 1 c c 1 _ +3+1 HE cz'l c:2"cz ”0 (4'3”) 2 2 2 Thisis the same equation as was obtained by Ting and Nan (1968) for compression and a single shear T with the Mises yield condition (except for a minor difference in notation for the parameter H). 82 As Ting and Nan have pointed out. it can be eas- ily seen from Equation (4.3-12) that D(cl) > 0 D(cz) \< 0 D(o) > 0 . (4.3-13) whence the roots 1 cf and 1 c8 of Equation (4.3-12) sat- isfy the inequalities 0 g c ‘< c s 2 (4.3-14) ‘< c < c f 1 4.3.2 Characteristic Conditions The characteristic condition along the character- istic of slope c is given (see Appendix 1) by f 5 d = o . (4.3-15) ~ where 2 is a column vector obtained from gr (cg—g) = o . (4.3-16) If 21. are the elements of the column vector 220...27 39 Equation (4.3-16) gives the set of seven homogeneous simultaneous equations. cm.1 + £4 = cpz2 + 26 = cp23 + £7 = l 1 2 2v ._ + ._ _ ... cml (E +Hs ”'4 (E 2v 2 2gl-v) ’ _ + (E H5 )14 + ( E 1 C22 + Zs‘rZHfi.4r — 28T2H£5 l _£.+ .. E c 3 2513H£4 2313H 5 83 0 (4.3-17) 0 (4.3-18) 0 (4.3-19) + H52)£ + 251 Hz + 251 H2 = O 5 2 6 3 7 (4.3-20) + H52)£ - 251 Hi - 251 Hi - O 5 2 6 3 7" (4.3-21) l 2 + (G + 412H)£6 + 41213H£7 - 0 (4.3-22) l 2 + 41213H26 + (G + 413ml7 - 0 . (4.3-23) and 3 can be obtained up to an arbitrary constant by solving any six of the above seven equations. we arbi- trarily drop Equation (4.3-20) from the set. Then solv- ing the remaining set of six equations yields (up to an arbitrary constant) the T following vector 3. g, = (Y. 1. 13/12. - pcw. ¢. + pc. - pc13/12] . (4.3-24) 84 where 2 2 12 c1 sg 9+4 c2 \1/ = — 3';— 7 - 4T 1 + 2 l - "'3 (4.3-'25) 2 c2 2 BBGHs c 2 ’ (:2 ¢ ='§§2- §:- pc + Eféé' l - -§:£—§ l - -§'(4.3—26) 2 BBGHs c and 2 2 2 1 I 12 + 13 (4.3-27) By means of Equation (4.3-12) and Equations (4-3-25) through (4.3-27) Y and ¢ can alternatively be written as 2 2 s c 7 c2 Y =';- 2 2 (4.3-28) 2 c — c l 2 2 2 gps (c - c2) (c - Bcz) ¢ (4.3-29) 21 c (c - c2) 2 1 Substituting Equations (4.3-24) through (4.3-29) in Equa- tions (4.3-15) yields the characteristic conditions along each characteristic of slope c given by Equation (4.3-9). After some algebraic manipulations the resulting equation takes the following form. 85 2 5c 3 2 . .39 ,-2 2 E12(de1+dv2) + E13dv3 2 EH5 + 2(B+1) ‘EZE'EHS c2 .. 12.-:2. .2. 2 2 5+1 8 2 c‘l’12 + 2EHc125 + 2EHc13s dcl 2c Bcz 2 3 2 3g 2 2 @+4 c + - + - ———| — -——- 2 EH8 2(5+1) 2c2 EH3 5+1 6 2c2 CWT2 2 2 2 c2 + 2EHc125 + 2EHc13 s dc:2 — (B-EC—2 ) EH1zsc‘l/ 2 c + (§-+ 4EH12 + 4EH12)c d1 — (3--8—a ) EH1 ch G 2 3 2 c2 3 E 2 2 + (G + 4EH12 + 4EH13)c d13 - 0 (4.3—30) This is the characteristic condition. an interior differ- ential equation governing along each characteristic when the appropriate c for that characteristic is used. A left, eigenvector £ also determines a simple wave solution of the system of equations (4.2-16). as given below in Subsection (4.3.3). 4-313 Simple waves In Appendix 1 it is shown that in a simple wave region. if we go from one straight characteristic to a 86 neighboring one. the elements of the increment dw in the solution are proportional to the elements of 3, Thus dvl dv2 dv3 dol do2 - d13 1 _ -pc13/12 ' (4.3—31) Y - -ch ' with the understanding that if any denominator in Equation (4.3-31) vanishes. so does the corresponding numerator. In particular if we put 13 = O and 12 = 1. Equation (4.3-31) reduces to the form given by Ting and Nan (1968). Note that the last equality in Equation (4.3-31) implies d12/12 = d13/13 . (4.3-32) Integration of Equation (4.3-32) yields 13 = 712 where 7 is a constant. Hence any plastic simple wave involves radial loading from the‘origin . in the 1 _plane. 2'13 This would seem to imply that there is no solution based on simple waves which could not be reduced by a rotation of the coordinate axes to the results of Ting and Nan for loading by compression and a single shear. We shall see. however. in Section 5.3 that the constant 7 associated with the fast simple wave need not be the same as that 87 associated with the slow simple wave. and that the tran- sition from one 12.13-plane to another involves discon- tinuities in 12 and 13 propagating at the elastic shear wave speed c2. Equation (4.3-31) also implies d(ol-02) d1 = P—g—‘gfl . (4.3-33) 2 Substituting for Y and ¢ from Equations (4.3-28) and (4.3-29) into Equation (4.3-33) furnishes the following ex ression where s =-3 (o —o ) p I 3 l 2 I 2 2 2 2 1c -c2) (3c —Bc2) ___ a (4.3-34a) d12 312 c2(c2-ci) ) . (4.3-34b) We shall see that. as a consequence of Equation (4.3-37) below. Equations (4.3-34a) and (4.3-34b). respectively. can be written as (c2 - c2) (3c2 - ficz) 29.-.; 2 2 d1 ' 31 2 2 2 (4'3‘34c) c (c —c ) 1 or alternatively ds _ '4': _ 1 d1 - - 3 s BGH51 (4°3-34d) Equation (4.3-31) gives dol s cz-cg a;-" :f’ 2 2 . (4.3-35a) 2 2 c -c1 which. by virtue of Equation (4.3-37) below. can be written alternatively as 2 2 dol s c -c2 afif'- ;' 2 2 . (4.3-35b) c -c1 Since 1 = y12 in a simple wave. by Equation (4.3—32). 3 1 -.J 12+1§ . 1+72 T2 (4'3'36) for 12 > 0. Hence = -—'= -'. (4.3-37) 89 Equations (4.3-34) and (4.3-35a) can thus be written in the alternative forms of Equations (4.3-34c). (4.3-34d). and (4.3-35b) given above. Equations (4.3-34d) and (4.3- 35b) are identical to the equations given by Ting and Nan for the single shear compression loading. Since the right side of Equation (4.3-34d) is a function of s and 1 only (since c depends only on s and 1). that equation is un- coupled and can be integrated independently to give a one- parameter family of 5-1 curves. as Ting and Nan showed. When the resulting s as a function of 1 is substituted into Equation (4.3-35b). that equation can be integrated by a numerical quadrature. with an additive constant of integration. The additivity of the integration constant shows that for any one curve in the 3.1-plane the family of curves in the O .1-plane can be obtained by translating 1 one member of the family in the 01-direction. Integration of Equations (4.3-34) and (4.3-35) gives the stress history experienced by a particle in the simple wave region. Numerical integration results for several such particles with various initial and boundary conditions are shown in Figures 13 to 14. and discussed in Section 5.3. Additional properties of simple 9O wave solutions in a half space have been discussed by Ting and Nan (1968). When the half space is initially prestressed. the simple wave solution for such a case is different. as is discussed in detail in Section 5.3. 4.4 numsagggp SOLUTIONS FOR NONSIMPLE WAVES In the preceding sections. the characteristics and characteristic conditions for the system of equations (4.2- 16) have been obtained. Closed form solutions (i.e. simple wave solutions by numerical quadrature) can be obtained for suitable boundary conditions. In this section. the charac- teristic conditions will be integrated for a boundary con- dition which does not give a simple wave solution. Closed form solutions are not possible and thus the characteristic conditions given in the incremental form will be integrated numerically. Equation (4.3-15) represents the increment dw of vector w along a characteristic with wave speed c. where Equation (4.3-9) gives the values of c. For sim- plicity. numerical solution is sought here only for cases where one compressive load and one shear load act on the 91 boundary x = 0 of the half space under consideration. Therefore. by setting 13 = 0 and 12 = 1 in Equation (4.3-30) the incremental relations. after some algebraic manipulations. can be written as follows. 2 2 _ 38c 2_ 2 EH5 _ QSQ-Z) 2 Ec‘l’ dv1 + Ec dv2 {[2(5+1) + (3c Bcz) ——2 2(B+1) c2]? 2 2 2 3§c 2 2 EH5 gg§¢4z 2 + — ———— - + 2EHc 1s dol +{l:2(8+1) (3c Bcz) 2 ”3+” CJ‘Y + 2EHc21s do2 - (3c2-Bc:) EH1sY + (§-+ 4EH12)c2 d1=0. (4.4-l) And. after rearrangement of terms. Equation (4.3-9) or the reduced Equation (4.3-12) can be written as 2 2 2 2 4c 2 ‘3 2 41 3 g_.‘_ §+3 2 l .1_ c[é s + B + EH(B+1):] (c2) c 6 s + c2 5 2 2 2 3 c2 _E:_7_ s. 2 __.___ _1 - _ + (B+1)EH] ‘32 + C 3 + (B+l)EH c2 0 ' ”'4 2’ 2 2 c s O is evidently a root of Equation (4.4-2); the other four roots are 92 which can be obtained by solving the quadratic equation left after cancelling c from Equation (4.4-2). Then for the five values of c. Equation (4.4-l) yields a set of five simultaneous equations for the increments of the five elements of the vector y, A systematic scheme to compute these increments will be discussed here. Vitiello and Clifton (1967) developed an explicit difference scheme for the case of a thin—walled tube prob- lem. In their scheme the solution of a system of differ— ence equations of the type of Equation (4.4-l) at a time t + At is obtained as an explicit function of the solution at time t. Here At is the time increment for the differ- ence grid (see Fig. 20). Equation (4.4-l) may be abbre- viated as a..dw. = 0 for i = l to S (4.4-3) 13 3 along the characteristic whose wave speed is ci. (In Fig. 20. c - 0 gives the vertical line QP). The coeffi- 5 cients'aij depend on the state of stress and the wave speed. Following Vitiello and Clifton (1967). we approx- imate all the coefficients dij for the difference scheme by their values at Q. i.e. by the values calculated using 93 the known stress values at Q. and obtain the values wi at points Qi by linear interpolation along the horizontal grid line through mesh points 0’. Q. Q” after locating Q . Q2. Q3. Q4 by approximating the characteristic seg- 1 ments 0i? by straight lines of slope dx/dt = ci. where the ci are evaluated at Q. If A denotes the mesh ratio At/Ax. then the interpolation gives wj(Qi) = wj (Q) +>~ci(Q) [wj (0’) -wj (0)] for i = 1.2 (4.4-4a) wjmi) = wj (Q)+>~ci(Q) [wj (Q) -wj (Q )] for i = 3.4 (4.4-4b) and wj(Qi) - wj(Q) for i = 5 . (4.4-4c) Substituting Equations (4.4-4) into Equations (4.4-3) gives Gij(Q)[wj(P)-Wj(0)] Xci(Q)aij(Q)[wj(Q )-wj(Q)] for i = 1:2 (4.4-5a) aij (Q) [W]. (P)'-.'Wj (0)] ' >~<31((2).Clij (Q) [wj (Q) -Wj (Q. )] for i ' 3:4 (4.4—5b) ll 0 aij (Q) [wj (P)-—wj (Q)] for i = 5 . (4.4-5c) 94 Solution at p is found by solving Equations (4.4-5). a set of five simultaneous linear algebraic equations for the five unknown values wi(P). A more sophisticated iterative procedure could for the nth iteration be used with the coefficients oi?) approximated (see Nan (1968)) by a(n)= 1,[a(n—l) (n-l) ij 2 (Q.)+-a ij i ij (9)] . or by some other average value instead of using all coef- ficients evaluated at Q. No such sophisticated procedure was attempted. According to Vitiello and Clifton (1967). theorems of the theory of numerical analysis imply that the exact difference solution converges to the exact so- lution w'of the differential equation in the maximum norm with an error that is 0(Ax). provided the functions in- volved are Lipschitz continuous and that the mesh ratio Ax/At z'cmax' This mesh ratio restriction is also that required as a necessary condition that the domain of de— pendence of the difference equations must include the do- main of dependence of the differential equations. Vitiello and Clifton (1967) pointed out that the convergence for plastic wave propagation problems is. however. not always 95 assured because discontinuities at wave fronts and bound- aries between elastic and plastic regions violate the smoothness assumptions. But in some numerical experi- ments they found good agreement between the solutions and the numerical quadratures of simple wave solutions. CHAPTER 5 APPLICATIONS AND RESULTS 5.1 SIMPLE WAVE SOLUTIONS FOR THIN-WALLED TUBE In Clifton's (1966) solutions for the simple wave stress trajectories for a thin-walled tube of isotropic hardening material. loaded by combined tension-torsion impact idealized as an assumed instantaneous application of uniform stress (Ob.1b) at the boundary x = 0 (applied at t = O and held constant thereafter). the slow simple wave trajectory of 0 versus 1. leading to the final values (0b.1b) for any particle. could be constructed by backward integration of Equation (3.3-15) from the final values (ob.1b). since the right-hand side I is a function only of O and 1. See Fig. 3* for several such trajectories. If this trajectory leads back to a point on the initial yield curve of 0 versus 1. then the impact on an initially stress-free tube produces only the slow simple wave fol- lowing the elastic jumps in<3(traveling at the elastic *Figures 1 to 31 are given at the end of this chapter. 96 97 bar-wave speed co) and in 1 (traveling at the elastic shear-wave speed c2). When the backward-integrated slow- wave trajectory leads back to a point on the U-axis out- side the initial yield locus. then a fast simple wave of uniaxial stress a precedes the slow simple wave. When the bar is prestressed in shear into the plastic range before impact. there is also a fast wave of combined stress. which is constructed by forward integration from the initial condition (0.11) point until it intersects the slow wave leading to (Ob.1b) at a point (00.10) out- side the initial yield locus. In this case the slow-wave O trajectory initiates at this intersection point (00.1 ) instead of on the initial locus. but from the intersec- tion point to the final point (0b.1b) the slow-wave tra- jectory is the same as it would have been without the prestress. when the hardening is purely isotropic. For kinematic hardening or combined kinematic and isotropic hardening the solution is more complicated. Because the function Y on the right-hand side of Equation (3.3-15) depends in these cases on the location of the center of the yield surface and on the accumulated effec— tive plastic strain. the slow-wave 0.1-trajectory cannot 98 be constructed by backward integration from (Ob.1b). It is necessary to construct it by forward integration. so that the motion of the yield—surface center and the accum- ulated effective strain can be calculated by following the history of the deformation. For a given choice of the hardening parameter m other than the isotropic hard- ening choice m = 1. it is not too difficult to construct a slow wave trajectory by forward integration beginning at some point on the initial yield surface (no plastic prestress). but it is not known in advance what starting point should be used for the trajectory leading to arbi— trary prescribed constant boundary stresses (ob.1b) at the impact end x = 0. It is therefore necessary to con- struct a family of slow-wave trajectories starting from a sequence of points on the initial yield locus. Then the member of that family leading to the prescribed (Ob.1b) point is selected. Since in practice only a finite number of such slow-wave trajectories will be constructed. it may be necessary to interpolate between them. or to use an interative construction to converge on the desired trajectory leading to arbitrary prescribed (ob.1b). The complexity is compounded when there is 99 initial prestress beyond yield. since because of the de— formation in the combined-stress fast simple wave. the plastic strain and the yield-surface center location at the intersection of the fast wave trajectory with a pre- viously constructed slow—wave trajectory (leading to (ob.1b) from a point on the initial yield locus) are not in general the same as the plastic strain and the yield- surface center of the previously constructed slow simple wave at the intersection point. Thus the slow simple wave leading to (ob.1b) will be different for different prestress conditions when the hardening is not isotropic. We shall consider first some examples where the tube is initially stress free and then some cases where the tube is initially stressed beyond yield. Numerical integration of the simultaneous first- order ordinary differential Equations (3.3-15). (3.3-16). (3.3-18). and (3.3-22) can be carried out by any of the standard methods. A fourth order RungeéKutta method was used here. The standard computer program to carry out the integration was obtained from the Michigan State Uni- versity Computer Laboratory. Program Documentation No. 00000035. March 64. The Runge—Kutta method has the 100 advantages that it is self starting and stable and pros vides good accuracy. It keeps no account of the error estimate. but changing the step size to different values and then comparing the results gives an estimate of the accuracy of the method. To illustrate the procedure outlined in Chapters 2 and 3. specific examples with prescribed initial and boundary conditions that result in simple-wave solutions for the thin—walled tube will be discussed for different values of the hardening parameter m. The material. for which the simple wave stress trajectories are shown in Figures 1 to 12. was assumed to have the following stress-strain curve in simple tension. the same stress- strain curve as was used by Clifton (1966). 8 1.732 ep = 0.403 x 10’ (0 - qY) (5.1-1) where 0y = 1750 psi. is the initial yield stress in ten— sion. Hence. in uniaxial tension. do ——— = 78410 . (5.1-2) dep (ep)0.4226 101 For a uniaxial tension test. dep = HEP and thus ep=dep. Therefore. for a general three-dimensional deformation state. Equation (5.1-2) implies 1 78410 10.4226 (5.1-3) 9 fde and g is a monotonically decreasing function of dep. The value of Poisson's ratio v and Young's modulus E are assumed to be 0.33 and 107 psi.. respectively. 5.2 EXAMPLES OF COMBINED TENSION- TORSION IMPACTS ON THE THIN- WALgD TUBE Consider a semi-infinite thin-walled tube ini- tially at rest and unstressed. which is given simultan— eously instantaneous compressive and torsion impacts causing uniformly distributed boundary normal and shear stresses ab and 1b respectively on the end x - 0. which remain constant for t > 0. A family of slow-wave trajec- tories is constructed for each of the choices m = 0. m = 0.5. and m = l of the hardening parameter. The choice m = 0. corresponding to the purely kinematic 102 hardening case. gives the trajectories of Fig. 2. The choice m = 1 corresponding to the purely isotropic hard- ening case previously solved by Clifton (1966) gives the trajectories of Fig. 1. while the choice m = 0.5 is a combination of both isotropic and kinematic hardening. Fig. 3 shows the network of simple wave stress trajec- tories for m = 0.5. Figure 4. shows examples of the relative differ- ence in the stress history experienced by a particle at x = constant. due to the different choices of material hardening parameter m. This figure is constructed by choosing an appropriate stress trajectory from each of the networks for m = 0. m = 0.5. and m l and then plotting on a single graph. Now consider the case of a thin—walled tube ini- tially stressed with static shear load causing positive shear stress 11 greater than the yield stress of the tube in shear. which is then given simultaneously instantan- eous compressive and torsion impacts causing uniformly distributed boundary normal and shear stresses ab and 1b. respectively. constant for t > 0 at the and x = 0. As was noted earlier. complexity of constructing simple wave 103 solutions is greater when there is an initial prestress beyond yield. For each initial condition of prestress a fast wave stress trajectory is obtained for each of the choices m = 0. m = 0.5. and m = l of the hardening parameter. From a series of points on any one of these fast-wave trajectories a series of slow—wave trajectories is constructed by forward integration with the initial conditions of stress. yield—surface center and accumu— lated effective plastic strain for each slow wave of the series given by the values of those quantities at the starting point of that slow-wave trajectory on the fast- wave trajectory. All of the slow wave trajectories starting from any one fast-wave trajectory are constructed with the same value of the hardening parameter m as was used for that fast wave. Figures 5 to 10 show several such fast-wave stress trajectories and a family of slow- wave stress trajectories originating from each fast-wave stress trajectory. For the particular choice of m = 1. i.e. corresponding to pure isotropic hardening. a family of slow-wave stress trajectories originating from each fast-wave stress trajectory is not necessary because for that case the stress trajectories in the 0.1—plane are 104 explicit functions of o and 1 only and thus the slow simple wave stress trajectories need only be constructed originating from the initial yield surface. As was pointed out earlier in Section 3.4. each member of the family of fast-wave trajectories in the 0.1-plane will end at a point corresponding to n = 0. and c = c 2. For positive initial stress 11. the point in the 0.1 plane corresponding to q = 0. and c = c2 will be for some positive value of 1. say 1 = 1d. except in the case m = l for which 1d = 0. The slow-wave trajec- tory from the end of the fast-wave trajectory is a hori— zontal Straight line in the 0.1-p1ane. with 1 = 1d. as is shown in Fig. 9 . If the boundary condition 1b is such that 1b < 1d. then there is no slow-wave trajectory leading from a point on the fast—wave trajectory to b b). (0 .1 Instead there will be a jump in shear stress from a point on the fast-wave trajectory to a point on the slow-wave trajectory; this jump propagates at the elastic shear wave speed c The point on the fast-wave 2. trajectory from which the jump occurs is not known in advance when arbitrary boundary conditions (0b.1b) are prescribed with 1b < 1d. Just as it was necessary to 105 construct a family of slow—wave trajectories emanating from a series of points on the fast-wave trajectory and to interpolate between them to find the slow—wave trajec- tory leading to a given point (0b.1b) with 1b > 1d. it is now necessary to construct a family of trajectories for which the slow—wave trajectory is separated from the fast—wave trajectory by a jump in 1. The value of this jump in 1 is determined by the yield condition just before the jump. where k has the current value determined by the hardening up to the jump. In the €.n-plane the jump is represented as a vertical line from a point on the upper boundary of the ellipse 52 + 3n2 = 3k2 to a point on the lower boundary. a jump in n by the amount -2nf. if qf is the value of n in the fast-wave just before the jump. Since the jump across the interior of the yield curve represents an elastic change. there is no hardening associated with the jump. Hence the yield curve center (0*.1*) and the faEP are unchanged. This means that the jump in 1 in the 0.1— plane is also of amount -2qf from the point (of.1f) on the fast-wave trajectory to (of.1f—2nf) at the beginning of the slow-wave trajectory. A slow-wave trajectory 106 starting from (of.1f-2nf) is then constructed for a series of points (of.1f) on the fast-wave trajectory. with initial values of 0*.1* and faEP for each such slow- wave given by the values of these parameters at (of.1f). Figures 7 to 8 show several examples of such trajec- tories involving a jump in 1. When the hardening is isotropic. no jump occurs in 1 when both 11 and 1b are positive; only when 1b is negative (for positive 11) does the jump followed by additional plastic deformation occur with isotropic hard- ening. since then 1b < 1d = 0. Also when the boundary 0b)2 + 3(1b)2~$ 3k2. the jump loading is elastic. i.e. ( in shear occurs but then for this elastic boundary load- ing there is no combined—stress slow wave incorporating any additional plastic strain. A case of elastic bound- ary loading demonstrating the jump in shear for the iso- tropic hardening case is shown in Fig. 11. Ting and Nan (1968) discussed the possibility of such jumps for plane waves in a half space in the case of isotropic hardening. Because of the jump. there is a qualitative difference between the predictions of kinematic or combined kine- matic and isotropic hardening and the predictions of iso- tropic hardening. for 0'< 1b < 1d. 107 5.3 SIMPLE;WAVE'SOLUTIONS'FOR HALF SPACE Construction of stress trajectories for plane plastic simple-wave regions in a half space of isotrOpic hardening material loaded by one compression and two shears reduces to the case of one compression and one shear previously obtained by Ting and Nan (1968) when the half space is initially unstressed. But the con- struction procedure changes when the half-space is pre- stressed statically with a shear component in one direc— tion while the shear part of the impact with two shear stress components 1: and 1? has a resultant shear stress 1b in a different direction. Such simple wave solutions will be considered in the later part of this section. The stress trajectories for plane waves in a half space are four-dimensional trajectories in 0 o 1' 2' 12. 13-space. But. for a simple wave. we have 13 = y12. and it is possible to discuss each simple wave in the 2 2 three-dimensional 0 0 1-subspace. where 12 = 12+13 = 1' 2' (l + yz)1:. as was pointed out in Section 4.3. This I C three—dimensional stress trajectory in 01 2. 1-5pace is conveniently represented by its projections onto two nonparallel planes. for example onto the 0 .1—plane and 1 108 onto the (Cl-Osz-plane as in Figures 13 to 14. The ma- terial properties assumed for these examples are the same as those assumed by Ting and Nan (1968) . The hardening parameter H is assumed during loading to be related to the current value of the shear yield stress k by the equation The initial shear yield stress ko.and the elastic modulus E and Poisson's ratio v are assumed to be k0 = 1000 psi.. E - 107 psi.. v . 0.25 (s - 51 Ting and Nan (1968) obtained similar stress paths for B = 5 for one compression and one shear. They also gave solu— tions for B - 2.25 and observed that the character of the solutions changes somewhat at B - 3 (v = 0.125). Consider the case of a half space with no initial prestress. subjected to uniformly distributed step-function impact loads in compression 0b and two shear stress com— 1 ponents 12 and 1: such that the deformations are non- elastic. In this case the two shear stress components 109 5 will reduce to a single shear stress 1b = [}1:)2+ (1§)€] The admissible stress paths in the 0 .1-plane and (cl-02). 1 1-plane for such a case would either be like odgb’ or like odeb” in Figures 13 and 14. For the path odeb” there is only a slow simple wave following the elastic jumps in 0 and s (traveling at the dilatational-wave-speed 1 cl) and in 1 (traveling at the elastic shear—wave speed c2). For the path odgb’. a fast simple wave of zero shear stress precedes the slow simple wave. Figures 16 and 17 show the simple wave solutions in the x.t—plane for the two cases considered above. Now consider the case when the half space is sta— tically prestressed beyond yield with two shear stress components 1; and 1;” which have resultant shear stress 11, and then uniform step—function impact loads of one com-~ and two shear stress components 1b pressive stress 0b 2 l and 1:. with resultant shear stress 1b. are applied on the boundary x = 0 of the half space. If 7 denotes the . b _ b b ratio of shear stress components. then let 7 - 13/12 denote the ratio of boundary shear stress components. while the initial value of 71 = 1g/1i. 110 First consider the case when yl = 7b. This is again equivalent to the case considered by Ting and Nan (1968) of one compression load and one shear load. As was pointed out in Subsection 4.3.3. the stress trajec- tories for slow simple waves in the 0 .1 plane may be 1 given an arbitrary rigid body displacement in the 01- direction. corresponding to the additive integration constant. The need for this is apparent from Figures 13 and 14. because the intersections of the fast—wave stress trajectory pqr and the calculated slow-wave stress- trajectory eb"c does not have the same shear ordinate in both the figures. which are supposed to be projections of the same stress trajectory in the 0 . O .1-space. There- 1 2 fore. for the case of initial static prestress of shear 11. each slow-wave stress trajectory in the 0 .1—plane 1 must be given a horizontal displacement (i.e. parallel to the ol-axis) by an amount such that the intersection of the slow wave stress trajectory. with the fast-wave stress trajectory has the same shear stress ordinate in both the projection planes. For different values of pre- stress 11. the amount of translation of the slow-wave stress trajectory in the 0 .1-p1ane will be different. 1 111 Fig; 15.5hows\the modified network of stress trajectories in the 01.1-plane for one such prestress case in shear. Now consider the case when yb*yl. In this case too the slow-wave trajectories in the 0 .1-plane need modification as described for when vb = 71. But Equation 1 (4.3-32) implies radial loading in 1 -plane in simple 2"‘3 wave regions (50 that 7 remains constant during plastic loading). However. it is conceivable that an elastic jump may occur in 7. traveling at elastic shear wave speed c Such a jump is only possible at the end of a 2. fast simple wave or at the end of a constant—state region following the fast simple wave. Therefore. y = yl re- mains constant in the fast-wave and constant-state re- gions. and 7 - vb remains constant in the constant-state region following the jump and in the slow simple—wave re- gion. A jump in 7 implies a jump in both 12 and 13. Fig. 18 shows one such example in the x.t-plane. The corresponding stress trajectory could be obtained from ’Figures 14 and 15. 112 5.4 NONSIMPLE WAVE SOLUTION FOR A HALF SPACE Only one example of numerical integration of the characteristic conditions obtained in Section 4.4 has been worked out. for a half space. subjected to compres- sion and one shear loading. For simplicity. the boundary and initial conditions were chosen to avoid discontinu— ities in the stresses. except at the leading wave front. The half-space was assumed statically prestressed in shear to a value 10 below yield. Then at time t - 0 the bound— ary x - 0 was assumed loaded suddenly in compression to a value (01)o such that yield is just reached at (Ol)o. 10. For t > 0 the boundary tractions both increase lin— early with time. Specific numerical values of 10 - 200 psi. and (01)o - 2575 psi. were chosen for the illustra— tion. With this type of loading the leading wave is an elastic shock wave. a jump in 01 from zero to (Ol)o. propagating at the elastic dilatational wave speed c1. Immediately above this line in the x.t—plane the stress state is uniform with 01 = (ol)o. 1 = 10. The plastic wave region above the line x - clt has the following initial conditions 113 vl(x. X/cl)=‘—(Gl)o/pcl' v2(x.x/c1) = 0.0 forx>/0 01(x.x/cl) = (Ol)o. 1(x.x/cl) = 10. The boundary conditions on x = 0 in the half space are 9 01(0.t) — (01)O + 10 t 1(O.t) = 10 + 0.25 X 109t for stress in psi. and time in seconds. The increment in the time step is given by At = Ax/cl. For the above example. solutions were obtained with different values of grid size Ax = 0.05 in.. 0.025 in.. and 0.0125 in. in order to see the effect of change in grid size. Figures 21 and 22 demonstrate the stress profiles at t .5. 2.5 microseconds and the stress histories exper- ienced by particles originally at x = 0.3 in. and x = 0.7 in. For the example considered. c 2.174 X 105 in./sec.. l and therefore. with the final value of Ax = 0.0125 in. the final time-increment size was At = Ax/cl R50.57 micro seconds. The integration process was terminated after a time period equal to the time that would be required for 114 an elastic dilatational wave traveling at speed c to pass 1 through a 3/2 in. thick plate. This choice was made ar- bitrarily. For the mesh size of Ax % 0.0125 in.. there were 7140 grid points to calculate. and the total compu- tation time on the CDC 3600 computer was 2 min. 30 sec. The numerical results obtained are presented in Figures 23 to 28 as velocity and stress histories at dif— ferent sections. x-constant. and as velocity and stress profiles at different time intervals. Projections of stress trajectories in.the 0.1'and (01-0 1-planes are l 2)' shown in Fig. 29 for particles initially located at x = 0.5 in.. and x = 1.0 in. Fig. 30 shows level lines for v . 1. and v in the x.t- various constant values of 01. l 2 plane. Despite the fact that this is not a simple wave it is observed that the level lines are all straight. Each level line of Cl is also a level line of VI. and each level line of 1 is a level line of v2. Further comment on this unanticipated result is given in Section 5.5.3. The slopes dx/dt of the level lines of 01 and v1 are de- noted by 2_while the slopes of the level lines of 1 and v are denoted by b, Fig. 31 shows a plot of a_versus 2 Cl and b versus 1. 115 5.5 DISCUSSION OF RESULTS 5.5.1 Comparison of Different Hardening Assumption Results for Tension—Torsion Impact on Thin-Walled Tube The relative differences. caused by varying the hardening parameter m. in the simple-wave stress histories experienced by a particle in a thin-walled tube depend on . b b . - the ratio a /1 of the impact boundary normal and shear stresses and on the initial state of the tube. For no . . . b b initial prestress and for 0 /1 less than about one. the relative differences in the stress history for a given boundary condition are not significant. as is evidenced by Fig. 4. These differences still remain small when the tube is prestressed with a static shear stress beyond yield. if the ratio of impact boundary normal and shear stresses remains small. Significant differences begin b b . to occur when a /1 is large. For the case of initial static prestress beyond yield. the differences may even become qualitative as well as quantitative. Fig. 12 demonstrates these dif- ferences in the stress history for certain boundary and 116 initial conditions. In the case shown in Fig. 12. for m = 1 or m = 0.5 the simple wave solution in the x.t- plane consists of a fast simple wave. a constant state region and a slow simple wave region. while for m =() (purely kinematic hardening). following the constant state region a jump occurs in shear stress. traveling at the elastic shear speed c Such a jump can also 2. occur for m = 0.5. but for conditions with positive shear stress such a jump followed by a plastic loading is never predicted by the isotropic hardening assumption. This qualitative difference in the predictions by the kinematic hardening assumption (or by the combined kine- matic and isotropic hardening assumption) from the pre— dictions by the isotropic hardening assumption could be an important factor. if experimental studies confirm or fail to confirm the existence of such a jump in shear stress for suitable impacts on a tube prestressed in shear. When there is no prestress. such qualitative differences do not appear. but the quantitative differ— ence becomes noticeable when the impact (Sb/1b increases beyond unity. In Fig. 4 the quantitative differences in 117 the stress histories when the tube is initially stress free and the ratio of the impact boundary normal and shear stresses is large. are shown. When the tube is prestressed with normal stress beyond the critical point. and then loaded with normal and shear impact loads. the phenomenon of unloading predicted by the isotropic hardening assumption is repeated by kinematic hardening and combined kinematic and isotropic harden- ing. but again the process of constructing simple wave solutions is more difficult than it is in the isotropic hardening cases. This unloading phenomenon has been pre- viously predicted by Clifton for isotropic hardening. as follows. As Clifton (1966) showed. when an isotropic— hardening tube is prestressed in uniaxial normal stress beyond the critical point (the point where the slope dO/de of the uniaxial stress-strain curve equals the elastic shear modulus G) and then loaded with normal and shear impact loads. an elastic wave precedes the combined-stress plastic wave. This elastic wave con— sists of a jump decrease in 0 traveling at speed co fol- lowed by a jump increase in 1 traveling at speed c2. 118 After these two elastic waves have passed, the stress point is at the proper (00.10) value to begin a slow- wave trajectory leading to (0b.1b) from a point on the (hardened) yield locus that passes through the prestress point (01.0). When the hardening is kinematic hardening or combined kinematic and isotropic hardening. a similar unloading phenomenon occurs. but the analysis is more complex because of the greater history dependence. No examples for such a case have been worked out here. It would be necessary to construct a family of slow-wave trajectOries: leading from various points (00.10) on the hardened yield locus. For isotropic hardening case the slow-wave trajectory could be constructed by backward integration from (Ob.1b) to determine the beginning point (00.10) on the hardened yield locus. but this is not pos- sible with the other hardening assumptions. 5.5.2 Simple Waves in Isotropic Hardeninngalf Space The simple wave solutions with one compression and two shear impact step-function loadings on the bound- ary of a half space of isotropic hardening material 119 reduce to the case of one compression and one shear im— pact step—function loadings when 71 = vb. Then 13 = 712. and all plastic simple-wave stress histories are radial in the 1 . -plane. A rotation of the coordinate axes 23 then reduces the problem to the case of loading by com- pression and a single shear previously treated by Ting and Nan (1968). 1 b . . . For the cases when 7 *7 . an elastic jump in y traveling at the elastic shear wave speed c occurs. In 2 these cases the stress history is not radial with a single 13/12 ratio. but the fast simple wave is radial with one 13/12 ratio while the slow simple wave is radial with an- other 13/12 ratio after the jump in y. (There are also. as a rule. constant state regions in the x.t-plane pre- ceding and following the x - c t wave front.) These 2 cases cannot be reduced by a rotation of the coordinate axes to the cases treated by Ting and Nan. But the stress-trajectory projections onto the 0 .1—plane and l 2 2 2 . the (01-02).1-p1ane (1 - 12 + 13). belong to the fami- lies of trajectories computed for compression and one shear. Therefore. all simple wave solutions for one compression and two shear loadings for a half space of 120 isotropic hardening material can be constructed from stress trajectories like those already obtained by Ting and Nan (1968) for the case of compression loading and one shear loading. The simple-wave solutions reveal the possibility that elastic unloading regions can occur in the interior even when the dynamic boundary tractions are increasing. so that the final (0b.1b) point lies outside the (hard- ened) yield locus. Similarly plastic loading regions may occur evernwhen the boundary loading is elastic. This suggests that unforeseen elastic and plastic load— ings and possible discontinuities may also occur with nonsimple waves. When they do occur they will make the numerical integration procedures difficult. 5.5.3 Nonsimple Waves in Isotropic Hardening Half Space Stress histories and stress profiles obtained for different values of Ax show that a further reduction in the mesh size is needed to obtain a more accurate so- lution. But the difference method used to approximate the system of partial differential equations itself needs 121 improvements. Therefore. it was felt that reducing the grid size to still a smaller value would waste computer time without contributing much to the accuracy of the solution. The surprising result that the level lines of 01. v1. 1. and v2 from the numerical solution are all straight raises some interesting questions about the possibility of finding analytic solutions. or at least solutions obtained by quadratures instead of by the step- by-step numerical integration of the hyperbolic system of equations. Since the surprising result was only dis- covered during the final stage of preparing this manu- script. no investigation of these possibilities has yet been made. Because only one numerical example has been worked out. it is not possible to make any generaliza- tions about what other boundary conditions would lead to straight level lines. This is not a simple wave solution. since both 01 and 1 are not constant along the same level line. and the level lines are not in general characteristics of the governing hyperbolic system. But the compressional wave and the shear wave appear as though they were uncoupled. 122 since the level line of 01 is also a level line of VI. and the level line of 1 is also a level line of v2. Such uncoupling was not expected because the governing constitutive equations are coupled. Nonsimple wave solutions for several different boundary conditions on a half space have been reported by Nan (1968). but he did not show any level lines. 5.6 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH The present study has been carried out with a view of predicting material behaviors under combined stresses. offering to experimentalists the possibility of a qualitative difference in the results due to dif- ferent material behaviors and different boundary condi- tions. For some loadings the kinematic hardening and combined kinematic and isotropic hardening assumptions lead to predictions qualitatively different from the predictions of isotropic hardening. This opens the pos- sibility of experimental verifications. 123 The simple wave solutions with kinematic harden- ing and combined kinematic and isotropic hardening are more complex to construct than with the isotropic harden- ing assumption. The difficulty is a result of the history- dependence of the hardening even during continuous loading. This difficulty will likely persist with any other history- dependent hardening theories. Nonsimple wave solutions might not be much more difficult with history-dependent hardening assumptions. since the numerical integration follows the history any- way. But the numerical integration procedures should be used with caution. since unforeseen discontinuities and elastic regions may occur. Not enough examples of non- simple waves have been worked out to permit generaliza- tions. Although simple waves in an isotropic hardening half space loaded by compression and two independent shears cannot always be reduced by a rotation of axes to the compression and one shear case treated by Ting and Nan (1968). the stress trajectories they obtained can be adapted to give the solution for the more general case. 124 The nonsimple wave solution obtained for the par- ticular example in this study offers a possibility of closed—form solutions because the level lines of the so- lution are straight and the compressional and shear waves appear to be uncoupled. But it should be noted that not enough has been done to make any definite conclusions. This possiblity of closed-form nonsimple wave solutions should certainly be a subject for further research. Many details of error analysis and convergence in the numerical procedure also remain to be investigated. as well as techniques for handling unloading waves and reflections from the opposite face when the half space is replaced by a plate of finite thickness. This subject of dynamic plasticity with combined stresses has drawn the attention of research workers only very recently. A few simple examples. such as the thin- walled tube and the half space. have been solved for certain simple boundary conditions and material behaviors. The phenomena predicted by these examples need to be con— firmed or contradicted by experimental studies. Rapid progress in this direction may not occur because of the difficulties of the combined-stress experiments. Only 125 one experimental investigation has been reported so far. by Lipkin and Clifton (1968). who were able to confirm the existence of fast waves and slow waves in tests on tubes. In the meantime. solutions for more boundary value problems need to be obtained. and. if possible. procedures worked out to handle the cases where unfore— seen unloading occurs. 5.7 GRAPHICAL DISPLAY OF RESULTS Figures 1 to 31 are given on the following pages in this section to display the results obtained for the boundary value problems of the thin-walled tube and the half space. Ao.H u Ev wflsy ommmmuumoumsoz 0 now mnumm mmmuum m>mz.mamEHm BonII.H.mfim mom use 05% mauasH Aummv o mmmmem uezmoz m we . a . . I owoo comm owow omen OOON coma o 126 o ¥.m.N.H fl H 82 S m r w I! 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F L r b 0 N> 00m L .0 [Cm 00001 0000 S H> u m m 1 .m. 00001 000 H0 00001 1000 00004 T000 0 (333/°NI) ZA 'TA meooqu 0 1 0000 0000 9 TL 0m S Hu 1 0000 1 0000 0 000.00 152 mmHH :03 0 0000. 0000 0 — ouumnmue mmmuum m>mB mamEHmcozII.mm .00m 0 000m 000m _ .CH 0.0 ll X 00m 000 1000 N 00000 A 01000 00mm 00mm oowa 0000 .100N 000 000 153 00:00 Hm>mq >u00000> 0cm mmmuamll.0m .000 0200 x 00200000 00.0 0.0 00H0 0.0 00.0 0 (DEISOHDIW) HWLI. 154 p mamum> mmcfia Hm>ma N> wnm P mo 9 macaw can an msmum> mmcaa Hm>wq H> 6cm HO MD M wmoamll.am .mflm AHmmv Ho ooo.NH ooo.oa ooom 0000 000% ooom o it P _ $ _ _ x? ‘1 0.0 Imm.o Q [.m.o S n1 0 d 3 S m .mb.o _ _ 4 _ i _ OO.H coma coca com com 00¢ com o Gummy P 4f APPENDIX 1 THEORY OF PARTIAL DIFFERENTIAL EQUATIONS* rs , A1.1 METHOD OF CHARACTERISTICS ._7_ The method of characteristics for solving quasi- linear. hyperbolic system of first order partial differ— ential equations involves choosing appropriate directions along which the system of equations is considerably sim- plified. The method is very useful when there are only two independent variables. Letthe general system of partial differential equations be A'w + 2' = O (A1.1-l) ~t Yo: where the coefficient matrices g and E depend on x.t and E but do not depend on the derivatives of w” There may exist a wave front ¢(x.t) = 0 across which there may Lr’ *For a detailed study of the theory of partial differen- tial equations see Jeffrey and Taniuti (1964). 155 156 occur discontinuities in the normal derivatives of the solu- tion. If we introduce new variables §(x.t) and ¢(x.t) where g is the arc length along the curve ¢(x.t) = 0. then Equation (A1.1-l) becomes ¢(X,t) okx,t\g O §(x.t) 5w ow _ +§¢X)_: + (.1ng t+§gx)_~ - 0 , (A1.1-2) (54> 54> 5g t In order that the jump discontinuities. if any. across these characteristic lines be permitted. set the coeffi— dete rminant cientAof the normal derivatives across these lines equal to zero. i.e.. [Apt + B¢ | = O (A1.1-3a) or. alternatively. |c§ - 21 = o (A1.1-3b) 157 where -¢ c=E-:-:- =-:—: =g—fi. (A1.1-3c) This means (1) that the characteristic curves are curves across which the normal derivatives are not determined by the equations and the values of the functions on the curves. and it will also be shown (2) that the character- istic curves are curves along which the differential equa- tions can be transformed into interior differential equa- tions. containing only derivatives with respect to a par- ameter varying along the curves. The first property permits discontinuities in the normal derivatives of an acceptable solution across a characteristic. while the second property furnishes a con- venient procedure of solution. We now seek to transform the system (A1.1-2) into a set of interior differential equations by premultiply- ing it by a vector g?. which is to be determined. We obtain T 53 T 531’. z. (Q‘PJE‘DX 5 + £ (ggt + ggx S? = o. (A1.1-4) ~ 158 If this system is not to contain derivatives aw/a¢ then the coefficient matrix of ow/o¢ should vanish. Then E? (§¢t + §¢X) = 0 (A1.1-5) If we choose the elements of £? as the roots of the sys- tem of algebraic Equations (A1.1-5). then the surviving terms in the system of Equations (Al.l—4) will contain only derivatives with reSpect to g. and it will therefore be the system of interior differential equations that we are seeking. The system (Al.l-S) of linear algebraic equations is homogeneous. The necessary and sufficient condition for the existence of a solution is that the determinant of coefficients vanishes. i.e.. + = - lg wt 3 ¢X[ 0 (A1.1 6a) or. alternatively. [c A - Bl = 0 (A1.1-6b) where c = -——'=‘- ='—- . (A1.1-6c) 159 Equations (A1.1—6) are the same as Equations (A1.1-3). which shows the truth of the earlier assertion that the characteristic directions are the directions in which the equations can be transformed into interior differen- tial equations. The lepes c = dx/dt of the character- istics are the eigenvalues of the matrix CETEJ and the 2 required for the transformation is a left eigenvector of the matrix. such that T g’ (cg - E) = O. (A1.1—7) . . . T . With this chOice of c and 3’. the system of Equations (A1.1-4) becomes ow T “3t + E g)???” = o (A1.1-8) 2 . l . or. since B = cg, while gx =‘E gt. Equation (A1.1-8) be- comes T 5% Q. — = O A1.1-9 2 ~ 5 5g gt ( ) along the characteristic curve, Since ¢ is constant along the curve. 3;: t = as - (A1.1-10) 160 From Equations (A1.1-9) and (A1.1-10). the interior dif- ferential equation becomes 3.?" g dw = o (A1.1-ll) along the characteristic curve if £’is a left eigenvector of cA - B. associated with the eigenvalue c. A1.2 SIMPLE WAVE SOLUTION The set of Equations (Al.l-l) is a hyperbolic sys— tem of partial differential equations. A simple wave solu- tion of this system is a particular solution of this sys- tem of equations. in which the vector w'is a constant vector along a characteristic line 3% = c. c is the wave speed,which is function of the elements of w; Since‘w is defined constant along c. thus the characteristic lines are straight in the simple wave region. w'is con- stant implies the following 53' dx -—- -—-+ w = O or cw + 5x = . .2- dt N. x ‘1. 0 (A1 1) Elimination of wt from Equations (A1.1-l) and (A1.2-l). yields 161 (of: - §)wx= O (Al.2-2) and elimination of wX yields (cg - §)Y¢ = O. (A1.2-3) Equations (A1.2—2) and (A1.2-3) imply that (cg — §)dw = O (A1.2—4) for the change dw associated with any change dx.dt in- volving going from one straight characteristic of the simple wave to another one. [Equation (A1.2-4) is of course trivially satisfied for a change dx = cdt. where c is the wave speed of the straight characteristic. since then dw’= 0.] Iflg is a left eigenvector of (CA - E) and A and g are symmetric. then (cf: - g) g = o . (A1.2-5) Equations (A1.2-4) and (A1.2—5) imply that. as we go from one straight characteristic of a simple wave to a neigh- boring one. dw is proportional to g. APPENDIX 2 LAGRANGIAN FORMULATION OF EQUATIONS OF MOTION IN MATERIAL COORDINATES The combined-stress plastic-potential flow theory is formulated in terms of the rates of deformation (time rates of the natural or logarithmic strain components) and the Cauchy stress components. both referred to the deformed configuration and using the spatial (Eulerian) coordinates xk of the deformed position of the particle whose initial position coordinates are the material co- ordinates XK of the particle. Only in this appendix is the notation of capital X.K for material coordinates and lower-case xk for spatial coordinates. Throughout the body of the thesis lower—case x denotes the material co- ordinate. Both sets of coordinates will be supposed measured from the same set of rectangular Cartesian axes. Then they are related as follows 162 163 where U1. U2. and U3 are the displacement components. which are considered as functions of the XK in the La- grangian formulation. The combined-stress yield condi— tion is also stated in terms of the Cauchy stresses. For comparison with experimental measurements of wave propagation it is convenient to have the solu- tions in terms of material coordinates; for example. a strain gage is located by giving its initial position. It also tUrns out that the equations and simple wave so- lutions take an especially simple form when material co- ordinates are used. In the cases of uniaxial stress waves in a bar and of uniaxial strain in a half space the re— sulting simplified equations are not limited to small strains. since the combined-stress plasticity theory is not used. Even when ell = oUl/Bxl is not small. the en- gineering stress-strain curve of the bar can be used to determine the dependence of the nominal stress Til on e For combined-stress waves. however. the rates of 11' deformation Dij can be approximated as time rates of the small strain components only when the strains are small compared to unity (less than about 0.05). Then. for example 164 when the displacements are functions only of the one ma- terial coordinate X1 and the time. as is assumed in the problems treated in this study. This replacement of the Dij in the constitutive equations by the corresponding rates of small strain is the only approximation limiting the half-space plane-wave analysis to small strains. since we shall see that the equations of motion are exact for the half-space case. Additional approximations are implied by the neglect of the variation of the displace- ment components through the thickness in the tube case. The equations of motion in terms of material co- ordinates take an especially simple form when the nonsym- metric first Piola-Kirchoff stress tensor Tim is used. This tensor gives the force vector on a deformed surface element per unit area of the undeformed element. Thus 0 . ' = _ = . . , 2-2 NK TKl dso dpl nj 031 d8 (A ) where the first equality expresses the components dPi in terms of the T21. the undeformed area dSo and its unit um:- .w. 165 normal components NK' while the second equality is the usual equation in terms of the Cauchy stresses. the de- formed area dS. and its unit normal components nj; see. for example. Malvern (1969). or Eringen (1967). The two stress tensors are related by 2. 5x. 0 0.. = -1-T . (AZ-3) 31 p0 BXJ Ji where p and p0 are the current and initial density at the particle. Both sides of Equations (AZ-2) and.(A2-3) are to be understood as expressed in terms of the XM and t. by using the equations xk = xk(X1. x2. x3. t) (AZ-4) defining the motion. The equations of motion in terms of the material coordinates then take the following form (for negligible body forces) ngi ovi 8x = po57§E (AZ-5) J where the vi(X1. X2. X3. t) are the particle velocity components vi = oUi/ot. 166 For the two cases considered in this study. the stresses are assumed independent of X and X . Hence the 2 3 equations of motion reduce to o BT11 _ 6Y1 ax " po' 37-: (AZ-6) l The only stresses appearing in these equations of motion 0 O O are T11: T12: 13 be zero and neglect the lateral inertia term Ipo Bv3/Bt). Ti3. (In the tube case we assume T to These are related to the Cauchy stresses 0 . a . a 11 12 13 as follows. by Equations (AZ-3). ‘ O = B 3:]; To + Efl. To + .6:— To 11 p0 8X1 11 5X2 21 5X3 31 Bx 5x , Bx ‘.g .__I o + __1_ o +‘__1 o _ °12. 9 5x T12 ax T22 ax T32 ? (A2 7) o 1 2 3 13 p 6X 13 BX 23 BX 33 o 1 2 3 J Since x a x +U . and U is independent of x2 and x . 1 l 1 1 3 Equations (AZ-7) reduce to where tinuity equation in material coordinates =.B 011 po -.B 012 p0 =.E 013 p0 501 e =-—- 11 axl‘ (AZ-8) The density ratio can be obtained from the con- 32 p Q. 4% = J = determinant Ixk M| (AZ-9a) where J is the Jacobian determinant of the transformation. [See Malvern (1969)]. twfo o a: (AZ-9b) 168 For the case of plane waves in a half space all the dis- placements are independent of X and X . so that this 2 3 reduces to po p - l + 611 . (AZ—9c) F whence we have 0 5 C11 ‘ Tii ’ F a = To (AZ-10) 12 12 ' o 013 ‘ T13 in this case without any assumption of small strains. For the thin-walled tube case. however. if the X3-direction is taken as the thickness direction (radial direction) at a point. while X is tangential to the cir- 2 cumferential direction. we would not expect 6 2 = BUZ/BX2 2 and 633 = SUB/5X3 to vanish because of the Poisson effect. If we assume that plane sections remain plane. we would have Po ‘ 'E- = (1+6 ) (1+6 (1+6 (AZ—11) ll 22) 33) 169 and o 11 22) T r, o 33) 11 g (1+6 (1+e T° (AZ-12) 12 U 22) 12 = (1+e (1+6 33) . o 0 Thus in the tube case. replacement of T11 by all and T12 by 012 requires restriction to small strains. In either case the other stress components are 0 0 not equal. For example. T21 * 021. but T21 does not enter the equations of motion. Barker. 3811! J. BlEiCho BIBLIOGRAPHY L.M.. Lundergan. C.D.. Herrmann. W.. "Dynamic Response of Aluminum." J. App. Phys.. Vol. 35. pp. 1203-1212. 1964. F.. ”The Dynamic Plasticity of Metals at High Strain Rates: .An Experimental Generalization." Behavior of Materials Under Dynamic-Loading.- edited by N. J. Huffington. Jr.. New Ybrk: ASME. pp. 19—41 (1965). H. H. and Nelson. 1.. "Plane Waves in an Elastic- Plastic Half Space Due to Combined Surface Pres- sure and Shear." J. App. Mech.. Vol. 33. Trans. ASME Vol. 88. Series E. pp. 149—158. 1966. Butcher. B. 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"Harmonic Dispersion Analysis of Incremental Waves in Uniaxially Pre- stressed Plastic and Viscoplastic Bars. Plates. and Unbounded Media." J. App. Mech.. Vol. 36. Series E.. pp. 59-64. 1969. b ‘a~ Department of Applied Mechanics J *4 Division of Applied Mathematics "I1111111111111111113