HI ' J_' .3533 LI-l'v "I A 155'." Iwn‘tjuu' . U,.:..-..m9.n.. sin} avivui' This is to certify that the dissertation entitled THE DEFORMATION AND STABILITY OF A PRESSURIZED CIRCULAR TUBE AND SPHERICAL SHELL IN FINITE ELASTICITY AND FINITE PLASTICITY presented by Dong-Teak Chung has been accepted towards fulfillment of the requirements for Ph . D . Mechanics degree in 6.5% ’°"’ ”(7 Date 7//0 /i@ / f MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES ” RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE DEFORMATION AND STABILITY OF A PRESSURIZED CIRCULAR TUBE AND SPHERICAL SHELL IN FINITE ELASTICITY AND EINITE PLASTICITY by Dong-Teak Chung A DISSERTATION Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in Mechanics Department of Metallurgy, Mechanics and Materials Science 1986 u I\ \\ u . . l. A a 5 i a. . . 1 a! i. o . a ’0 ABSTRACT THE DEFORMATION AND STABILITY OF A PRESSURIZED CIRCULAR TUBE AND SPHERICAL SHELL IN FINITE ELASTICITY AND FINITE PLASTICITY by Dong-Teak Chung This dissertation consists of two parts, both concerned with the investigation of problems for a cylinder and/or a sphere under conditions of finite strain. The problem considered in the first part, is carried out within the theory of finite elasticity while the problem in the second part is set within finite plasticity theory. In part I, finite elastic deformation of hollow circular cylinders and spheres under applied uniform internal pressure is studied. Conditions for the initiation of a localized shear bifurcation are obtained. The location of this bifurcation relative to the pressure maximum is investigated. It is shown that when the ratio of the outer undeformed radius to the inner undeformed radius is larger than a critical value, the shear bifurcation occurs before the pressure maximum is attained, while when this ratio is smaller than this critical value, the converse is true. The analysis is carried out for a particular compressible elastic foam-rubber material (the Blatz-Ko material). The results are obtained in closed analytic form. In part II, we carried out an explicit analysis of'a bifurcation problem for a solid sphere composed of an elastic-plastic material. This problem is concerned with the bifurcation of a solid sphere under symmetric tensile load into a configuration involving an internal cavity. The analysis is carried out within the context of plasticity Dong-Teak Chung theory using both finite strain Jz-flow theory and finite strain Jz-deformation theory. It is also shown that the classical infinitesimal theory of plasticity does not predict such a pheonomenon. This model may be used to describe the nucleation of a void from a pre-existing micro-void. To my parents, for the example and inspiration they have provided throughout my life. ii ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my advisors, Professor Cornelius 0. Horgan and Rohan Abeyaratne for their guide and support throughout this work. I would also like to thank my colleagues for their understanding and generous advice. Grateful thanks are extended to the other members of the guidance committee, Professor Nicholas J. Altiero and David H.Y. Yen. Thanks are also due to Mrs. Arlene Klingbiel for typing the manuscript. While preparing this dissertation, I held teaching assistantship awarded by the Department of Metallurgy, Mechanics and Materials Science and research assistantships supported jointly by the Division of Engineering Research, Michigan State University, by the U.S. Army Research Office under Grant DAAG-29-83-K-0145 (6.0. Horgan and R. Abeyaratne) and by the U.S. National Science Foundation under Grant MEA 78-26071 (6.0. Horgan) and Grant MEA 83-19616 (R. Abeyaratne). The support of these institutions is greatfully acknowledged. A final note of appreciation must go to my wife, Inbok, for her tolerance and encouragement. LIST OF FIGURES PART 1: Chapter 1 Chapter 2 Chapte Chapte APPENDIX A APPENDIX B TABLE OF CONTENTS THE FINITE DEFORMATION OF INTERNALLY PRESSURIZED HOLLOW CYLINDERS AND SPHERES FOR A CLASS OF COMPRESSIBLE ELASTIC MATERIALS INTRODUCTION 0 O O O O O O O O O O O O O O 0 0 THE PRESSURIZED HOLLOH CYLINDER; FORMULATION OF BOUNDARY-VALUE PROBLEM . . . . SOLUTION OF BOUNDARY-VALUE SO] Ution O O O O O O O 0 Discussion . . . . . . . RESULTS AND DISCUSSION Thin Shell: . . . . . Thick Cylinder: . Linearization: . . . . Loss of Ellipticity: . THE PRESSURIZED HOLLOW Formulation of Problem . Solution of Boundary-Value Thin Shell: ThiCk She“: O O O O O Linearization: . . . . Loss of Ellipticity: . LIST OF REFERENCES 0 O O O O O O O O 0 PART II: Chapter 1 Chapter 2 ON A BIFURCATION PROBLEM PROBLEM SPHERE . Problem IN FINITE RELATED TO VOID NUCLEATION PLASTICITY INTRODUCTION 0 O O O O O O O O O O O O O 0 O O FINITE STRAIN FORMULATION AND SOLUTION; J2- FLOH THEORY iv Page vi ON 35 37 Chapter 3 FINITE STRAIN FORMULATION ANO SOLUTION; Jz-DEFORMATION THEORY . . . . . . . . . . . . . Chapter 4 INFINITESIMAL STRAIN FORMULATION AND SOLUTION . Chapter 5 DISCUSSION 0 O O O O O O O O O O O C O O O O O 0 LIST OF REFERENCES 0 O O O O O O O O O O O O C O O O O O O O O Page 41 45 49 52 Figure PART 1: PART II: 5. LIST OF FIGURES THE FINITE DEFORMATION 0F INTERNALLY PRESSURIZED HOLLOH CYLINDERS AND SPHERES FOR A CLASS OF COMPRESSIBLE ELASTIC MATERIALS Applied pressure versus ratio of deformed to undeformed radius for a thin cylindrical She]] O O O O O O O O O O O O O O O O O O 0 Applied pressure versus ratio of deformed to undeformed radius for hollow cylinders for different radius ratio a - b/a. The dotted curve connects the points (pe/u, R(a)/a), where pe denotes the pressure at which elliptiCity is 10$t O O O O I O O O O O O 0 Applied pressure versus ratio of deformed to undeformed radius for a thin spherical shell Applied pressure versus ratio of deformed to undeformed radius for hollow spheres for different radius ratios o = b/a. The dotted curve connects the points (pe/u, R(a)/a), where pe denotes the pressure at which ellipticity is lost . . . . . . . . . . . . ON A BIFURCATION PROBLEM IN FINITE PLASTICITY RELATED TO VOID NUCLEATION Schematic graph showing variation of cavity radius c versus applied load p . . . . . . . . vi Page 24 25 26 27 51 PART I THE FINITE DEFORMATION OF INTERNALLY PRESSURIZED HOLLOH CYLINDERS AND SPHERES FOR A CLASS OF COMPRESSIBLE ELASTIC MATERIALS CHAPTER 1 INTRODUCTION Recently Abeyaratne and Horgan [1] obtained an exact solution to a problem describing finite plane strain deformation of an infinite medium, composed of a certain compressible nonlinearly elastic material, the so-called Blatz-Ko material. The problem considered in [I] concerned an infinite medium with a circular cylindrical cavity under pressure loading conditions. In part I of this thesis we show that the solution technique of [1] may be applied also to the case of pressurized hollow cylinders and spheres with jjgjtg_radii, and we carry out a detailed investigation of the solution to these basic problems of nonlinear elasticity. The material considered in this study is a particular homogeneous, isotropic, compressible elastic material, namely the Blatz-Ko material. The pressurized cylinder and sphere problems in finite elasticity for incompressible materials have been considered previously by many authors (see e.g. [2,3]) and are simpler, since the incompressibility constraint immediately yields an explicit expression for the (axially symmetric) deformation field. Such simplification does not occur for compressible materials.* The "Blatz-Ko material" is a mathematical model characterizing the constitutive behavior of a certain foam rubber-like material and was *The pressurized sphere and cylinder problems for a class of (hypothetical) compressible materials, namely harmonic materials, have been investigated recently [4,5]. See also [3], Chapter 5, and [6]. 1 2 proposed by Blatz and Ko [7] on the basis of experiments carried out by them. An extensive discussion of its properties may be found in [7] and also in the paper of Knowles and Sternberg [8]. One interesting feature of the Blatz-KO material is that the system of partial differential (displacement) equations governing the equilibrium of a body composed of such a material may cease to be elliptic at sufficiently severe strain levels [8]. In the present work, we are interested in examining the implications of this for the pressurized cylinder and sphere problems. It should be noted that bifurcation instabilities for pressurized cylinders and spheres for incompressible elastic and elastic-plastic material response have been studied by many authors (see e.g. [3]) but we shall not be concerned with the analogs of these studies for the compressible material of concern here. In the next Chapter, the problem of a hollow circular cylinder composed of the Blatz-KO material subject to internal pressure is formulated. In Chapter 3 the solution to the resulting boundary value problem is obtained and features of the solution are discussed. In Chpater 4, some illustrative examples are presented. For the case of a thin shell, an explicit relation between the deformed radius and the applied pressure is obtained and plotted in Figure 1. In particular, it is found that as the applied pressure p is increased from zero, the deformed radius increases until p reaches a maximum value. Subsequently, p decreases even though the deformed radius still increases. Such non-monotone pressure versus radius relationships are well-known in finite elasticity, particularly for incompressible 3 materials.* Numerical results pertaining to hollow cylinders of arbitrary thickness are also obtained, with corresponding pressure versus deformed radius curves shown in Figure 2. These curves exhibit a single pressure maximum. It is verified analytically in Appendix A that this occurs always for the Blatz-KO material. It is observed that as the thickness of the cylinder increases, the maximum pressure and the corresponding deformed inner radius increases as might be expected. This is verified analytically in Appendix B. when the applied pressure is small, our results coincide with those of infinitesimal elasticity theory. In Chapter 4, we also examine the loss of ellipticity of the governing displacement equations of equilibrium at the deformation at hand. The value of the applied pressure (and the correSponding deformed inner radius) at which the cylinder first loses ellipticity is obtained. It is shown that when the ratio of the outer undeformed radius to the inner underformed radius is larger than a critical value, loss of ellipticity occurs 23:95; the pressure maximum is attained while when this ratio is smaller than this critical value, the converse is true. In Chapter 5, the corresponding three-dimensional problem for a pressurized hollow sphere is treated. The overall behavior of the solution is similar to the two-dimensional case. *An extensive study of this phenomenon for incompressible materials has been carried out recently by Carroll [9]. It is of interest to note that this behavior does ggt_occur in the cylindrical inflation of Mooney-Rivlin or neO-Hookean (incompressible) materials whereas it does for Spherical inflation of such materials [9]. CHAPTER 2 THE PRESSURIZED HOLLOH CYLINDER; FORMULATION 0F BOUNDARY-VALUE PROBLEM Let the open region Do = {(r,e)|a < r < b, O < 6 < 2n} denote the cross-section of a right circular cylinder with inner radius a, and outer radius b, in its undeformed configuration. The cylinder is subjected to an internal pressure of magnitude p. The resulting deformation is a one-to-one mapping which takes the point with polar coordinates (r,e) in the undeformed region Do to the point (R,e) in the deformed region D. We assume that the deformation is an axisymmetric plane strain one so that R = R(r) > 0, e s e on Do, (2.1) where the positive function R(r) is to be determined. Unless explicitly stated otherwise R(r) is assumed to be twice continuously differentiable on a < r < b. The polar components of the deformation gradient tensor E associated with (2.1) are given by Pr, = R(r), Fee = R(r)/r, Fre . Fe, = O, (2.2) where the dot denotes differentiation with respect to the argument. The Jacobian determinant J = det f is required to be positive and hence one has R(r) > O for a < r < b. (2-3) .The principal stretches associated wnth the radial deformation (2.1) are Ar = R(r). A9 = R(r)/r. (2.4) 5 The right Cauchy-Green deformation tensor is defined as C = [TE and its fundamental scalar invariants can be taken as I . tr 9, J a (det 91/2 (2.5) so that in the present problem I - R2 + (R/r)2, J = RR/r. (2.6) Next we turn to the constitutive relation and suppose that the cylinder is composed of a Blatz-KO material* [7]. This compressible, isotropic, elastic material is characterized, in plane strain, by the elastic potential v(1,J) . u/2(I/J2 + 2J - 4), u > O, (2.7) l' representing the strain energy per unit undeformed volume. Here n denotes the shear modulus of the material at infinitesimal deformations. The true stress tensor I associated with a plane deformation is given by I = (2O-lav/aI) EET + (aw/aa)l. (2.8) On substituting from (2.7) and (2.6) into (2.8) one finds that = (1 ---—J%;--). (2.9a) TRRM " R(r)R (r) ( ) (1 r3 ) (2 9b) 1R9 3 19R 3 0, a < T‘ < b. (2.9C) In the absence of body forces, the equilibrium equations div T = O in A the present case reduce to the single equation d A _ Tbe) ‘An extensive discussion of the stress response of this material to various states of deformation may be found in [8]. 6 This, together with (2.9), yields the following nonlinear second-order ordinary differential equation for R(r): 3rR3R - R3R + r3R4 = O for a < r < b. (2.11) The prescribed boundary conditions are TRR = -p at r = a, TRR = O at r = b, (2.12) which, on using (2.9), can be written as R(a)R3(a) . a(l + p/u)‘1, (2.13a) R(b)R3(b) = b. (2.13b) In the next Chapter we derive an exact solution to the boundary-value problem (2.11), (2.13). CHAPTER 3 SOLUTION OF BOUNDARY-VALUE PROBLEM 3.1 Solution It has been shown recently in [1] that the second-order nonlinear ordinary differential equation (2.11) may be reduced to a first-order equation on making the substitution t(r) = "E i (= xr/xe) > o. (3.1) Equation (2.11) then yields 3rt - t(l-t)(t2+t+4) - O for a < r < b, (3.2) where t a dt/dr. It can be shown that there is no loss of generality in assuming that t(r) is less than unity* for a < r < b. Thus it then follows from (3.2) that t increases monotonically with increasing r and so we have 0 < t < 1, dt/dr > 0 for a < r < b. (3.3) Upon integrating (3.2), one finds that 8 c t6 h(t) r = 7T I? , (3.4) (1-t) (t +t+4) where C > O is a constant of integration and we have set 5 -1 2t+1 h(t) =- ex {— tan (——-)l > O. (3.5) p ITS ITS . On the other hand (3.1) and (3.2) also give ‘The arguments given in [1] to justify this assumption carry over, with obvious modification, to the present problem. Notice that since the applied load is that of internal pressure only, one would expect Ar < 1, he > 1 and so one anticipates that t is indeed less than unity. 7 1 an 3 _— a > 0, 3.6 R t (l-t)(t2+t+4) ( ) which yields 4 . O(t2+t+4) h(t) . R (14:)2 (3.7) Again, 0 > 0 is a constant of integration. Observe from (3.3), (3.6) that the undeformed and deformed radial coordinates (r,R) vary monotonically with t. Equations (3.4), (3.5), (3.7) provide a parametric solution to the differential equation (2.11). The range of the parameter t is ta < t < tb, (3.8) where ta > 0 is the value of t corresponding to r = a and is to be determined from (3.4) and tb < 1 is determined in an analogous fashion. The components of true stress TRR: Tee may also be expressed in terms of t on using (2.9), (3.1), (3.4) and (3.5). This leads to 2 1/2 (um I 3 ”[1 ' J: (309) RR h1/2(t)(t2+t+4)3/2 2 2 1/2 t (C/D ) T = u[1 - ], t < t < t . (3.10) ea h1/2(t)(t2+t+4)3/2- a b From the definition of ta, tb, it follows from (3.4) that 6 Ct h(t ) a8 a 2 a a 4 , (3.11) (ta+ta+4)(1-ta) 6 Ct h(t ) b8: 2 b b 4. (3.12) (tb+tb+4)(1-tb) Finally the boundary conditions (2.12), in view of (3.9), may be written as 2 1/2 (C/D ) _p a "[1 - , (3.13) nI/?(ta) (t2+ta+4)3/2 2 1/2 0 = "[1 - C/D ) (3.14) L J. A“2(tb)(t§+tb+4)3‘/f Equations (3.11) -(3.14) consist of four equations for the four unknowns ta, tb, C and D. In the following section we discuss the existence of solutions to these equations. 3.2 Discussion Ne eliminate the integration constant C between equations (3.11) and (3.12) and obtain b89(tb) = 439(ta). (3.15) where g(t) is given by (124.4 111414 g(t) a 6* , O < t < 1. (3.16) t h(t) Also from (3.13) and (3.14), eliminating the constant C/DZ, one has (11.2)? h(tb)(t§+tb+4)3 , (3.17) “ h(ta)(t§+ta+4)3 Thus we now have two equations (3.15), (3.17) for the two unknowns ta and tb. The function g(t) in equation (3.16) tends to infinity as t + 0+, decreases monotonically as t increases and has the value zero when t = 1. Thus for a given ratio of outer undeformed radius b to inner undeformed radius a, one can always express ta in terms of tb and vice versa. He may write tb 8 f(ta,a) (3.18) where a.- b/a and f is an implicit function. Thus, for a given applied pressure p, if (3.17) with tb expressed as in (3.18), can be solved for 10 a number ta such that O < ta < 1, then (3.18) provides a number tb and (3.11)-(3.14) is the desired solution. In order to verify that (3.17) can indeed be solved for an apprOpriate value of ta, we consider the auxiliary function 0(ta) defined by h(f(ta)){f2(ta)+f(ta)+4}3 Q(ta) a 754* 3 , 0 < ta < 1, (3.19) h(ta)(ta+ta+4) which appears on the right hand side of (3.17), when (3.18) is taken into account. For convenience here, we have written f(ta) a f(ta,a). One can readily show that lim Q(ta) = 1, ta+0+ lim dQ(ta) > 0, ta+0+ T—ta (3.20) Tim Q(ta) 3 l, ta+1- lim dQ(ta) < O ta’l- dta It follows that for 0 < ta < 1, there exists a maximum value for Q, with the corresponding value of p given by (3.17) as p = pm. Thus, if 0 < p < pm, there exist at least two solutions for ta. It is shown in Appendix A that Q has only one local maximum and so exactly two solutions for ta exist. Before we proceed to further examination of the foregoing solution, we return to discuss the possible alternative range of values of t(r). Recall that in the preceding discussion we restricted attention to the case when 0 < t(r) < 1 on a < r < b. Note that t(r) E 1 is a solution of (3.2). However (3.1) then yields R(r)=cr (c constant) which does not 11 satisfy the boundary conditions (2.13) unless p = 0. 0n the other hand, if t(r) = 1 at s2mg_point in a < r < b, then (3.2) and a continuity argument shows that t(r) must be unity on the entire range a < r < b. Consequently, the only case remaining to be considered is that for which t(r) > 1 on a < r < b. The preceding analysis up to equation (3.19) continues to be valid (with 1 < tb < t < ta and Obvious minor modifications) and the existence of solution again hinges on the solvability of (3.19) for a root ta, now > 1. It is easily seen that no such number exists (when p > 0) and thus the case when t(r) > 1 need no longer be considered. Finally, we observe that the deformed inner radius R(a), given by (3.7) with t = ta, may be written as 352.1 = ta'3/4(1+p/u)'1/4. (3.21) on using (3.11) and (3.13). CHAPTER 4 RESULTS AND DISCUSSION He now examine some features of the results derived in the previous section and consider some illustrative examples. 4.1 Thin Shell: When the radius ratio a = b/a is very near to unity, it is not difficult to express the relation (3.18) between ta and tb explicitly. Let o.= 1 + e, e = (b-a)/a (<<1), {4-1) and assume tb = ta + e A(ta), (4.2) where A(t) is an, as yet, unknown function. Substituting (4.1), (4.2) into equation (3.15) and neglecting second-order terms in 3 yields A(t) . 1/3(t2+t+4)(1-t)t and so (4.2) then reads tb . ta + (e/3) (taz+ta+4)(1-ta)ta, 0 < ta < 1. (4.3) Upon substituting from (4.3) into equation (3.17), one obtains an explicit relation between ta and the prescribed pressure, which is given, to leading order, by p/u a e (ta-ta3), for O < ta < 1. (4.4) It is clear from (4.4) that p has only one maximum. The deformed cavity radius R(a), is given by (3.21). In the present case, where p is given by (4.4), equation (3.21) yields, to leading order, R(a) a t-3/4 (4.5) 12 13 where the value of ta corresponding to a prescribed pressure p is found from (4.4). A graph of the ratio of the deformed radius to undeformed radius versus pressure, according to (4.4), (4.5), is shown in Figure 1. As the pressure p is increased from zero, the deformed radius increases until p reaches a maximum value of 0.385ue, where R(a) = 1.510a. Subsequently p decreases even though the radius still increases. This phenomenon is well-known in finite elasticity, especially for incompressible materials (see e.g. Ogden [3] pp. 283-287 for the spherical thin shell and the comprehensive recent study by Carroll [9].) 4.2 Thick Cylinder: For a thick-walled cylinder, the relation between ta and tb is implicit and thus computational work is necessary, in general to analyze the behavior of pressure versus radius. It can however be shown analytically that p has only one maximum point and this analysis is carried out in Appendix A. A graph of the ratio of the deformed radius to undeformed radius versus pressure, obtained from solving (3.15), (3.17) numerically and using (3.21), is shown in Figure 2, for different values of the radius ratio a a b/a. As is shown in Figure 2, as the thickness of the cylinder increases, the value of the maximum pressure increases and it occurs at increasingly larger values of R(a)/a, as might be expected. This is verified analytically in Appendix B. The bottom curve in Figure 2, corresponding to a = 1.05, is seen to confirm the thin shell results discussed in section 4.1. On the other hand, as o + a (i.e. a cavity in an infinite medium), the pressure maximum is reached asymptotically as the deformed radius tends to 14 infinity, and has the value pm = 1.51677u. Thus we recover the result Of [1] for this case. 4.3 Linearization: In the particular case when the applied pressure is small (p/u << 1), (3.19) and (3.21) show that ta 2 1, tb z 1 and so t(r) = 1 throughout the body. Let ta = 1 - 5,, tb = 1 -5b, t(r) = 1 - 6(r), (4.6) where the small parameters 5(r),‘63, 5b are unknown. Substituting (4.6) into (3.9), (3.10), (3.13), (3.14) yields, to leading order, Tee = "(5bt5). (4.7) IRR = "(bu-5). (4.8) pl]: 3 6a - 5b. (4.9) Also from (3.4), (3.11), using (4.6), one obtains a2 r Finally the relation between 5a and 6b may be obtained upon substituting (4.6) into (3.15). This leads to 5b = aaarZ, (deb/a). (4.11) The resulting linearized stress fields given by (4.7)-(4.11) yield the well known results of the infinitesimal theory of elasticity (see e.g. Timoshenko and Goodier [10], p. 71): 2 2 a .JEL___ 11. Tee b2_az (1 + F2 )’ (4’12) and 15 2 2 +24 --"-:>- 1..., '3 F 4.4 Loss of Ellipticity: The pressure maxima we have encountered exhibit the unstable behavior of a pressurized tube of the particular compressible material under consideration. It is of interest to examine other possible instabilities for this problem. It is well-known that the displacement equations of equilibrium in finite elastostatics may lose ellipticity at sufficiently severe deformations. In particular, for plane strain deformations of the Blatz-KO material, necessary and sufficient conditions (in terms of the principal stretches) for ellipticity have been obtained by Knowles and Sternberg [8]. In this section, we examine the implications of these results for the pressurized cylinder problem at hand. From equation (4.8) of [8], it follows that ellipticity holds for the Blatz-KO material (2.7) at the axisymmetric solution (2.1) if and only if the principal stretches Ar: 19 introduced in (2.4) are such that 2 - 73' < t < 2 + If, (t =- 1,419). (4.14) Since in the present problem we have 0 < t < 1, it follows that the right hand inequality here always holds. On the other hand, it is clear that ellipticity will be lost whenever the left inequality is violated. In view of the monotonic increasing character of t as r increases (see (3.3)), and recalling that 0 < ta < t < tb < 1, it follows that ellipticity is first lost at r = a and that this occurs when ta = 2-/3. For a given value of radius ratio a = b/a, the corresponding value of the applied pressure, say pe, is found from 16 (3.17), where the value of tb is given by (3.15) with ta = 2 - /3. The corresponding value of R(a)/a then follows from (3.21) with p = pe and ta = 2 - 13. In Figure 2, the pairs of values (pa/u, R(a)/a), for different radius ratios 6, are joined by the dotted curve. There is a critical value of the radius ratio a = at say, at which pe = pm. This may be found numerically from (3.17), (3.18), (3.19) with ta = 2 - f3, and we find “c = 5.1. (4.15) Thus, for a < ac, the load maximum is reached before loss of ellipticity occurs while for a > ac, the converse is true. In either case, after loss of ellipticity, the existence of the smooth solutions obtained here is still ensured. In addition, the possibility exists that non-smooth deformation fields, with discontinuous deformation gradients and stresses, might occur. The argument provided in [1] shows that axisymmetric solutions with such discontinuities do 323 exist in the present problem. Neak solutions, if they exist, must necessarily be non-axisymmetric. There is also the possibility that surface bifurcations might occur, as in [1], but we shall not pursue this issue here. CHAPTER 5 THE PRESSURIZED HOLLOW SPHERE In this Section, we describe briefly how the foregoing considerations can be applied to the analogous problem Of an internally pressurized hollow sphere. 5.1 Formulation of Problem: We are concerned in what follows with the pressure loading of a sphere composed of the Blatz-KO material. Let Do a {(r,6,¢)|a < r < b, 0 < O <2n, O < a < n}, denote the hollow Sphere in its undeformed configuration. The Sphere is subjected to an internal pressure p. The resulting deformation is a one-to-one mapping which takes the point with spherical polar coordinates (r, e, a) to the point (R, e, O) in the deformed region 0. We assume that the deformation is an axisymmetric one so that R - R(r) > O, 9 a e and O = O on Do, (5.1) where R(r) is to be determined. The polar components of the deformation gradient tensor associated with (5.1) are given by 5 . diag (R(r), R(r)/r, R(r)/r). ' (5.2) The principal stretches associated with the radial deformation (5.1) are Ar = R(r), A3 = A¢ = R(r)/r. (503) 17 18 The Blatz-KO material is characterized in the three-dimensional case, by the elastic potential (see e.g. [7]) -2 -2 -2 W(A1,A2,A3) = u/2(A1 +A2 +A3 + 2A1A2A3 - 5). (5.4) The principal components of true stress I are given by J‘1 aw T11 = 414243 311 (no sum on I) (5.5) - '1 72 _ - u(1-J A, ), J - x11213. Substitution from (5.3) and (5.4) into (5.5) yields r2 TRR(P) = U (1 -‘;7;§), (5.58) r4 100(F) = T¢¢(F) = u(1 -'EEE). (5.6b) In the absence of body force, the equilibrium equations div 1 = 0 reduce to d 215 - aF-TRR +-§— (TRR - r - O, for a < r < b. (5-7) 00) which, by virtue of (5.6), yields the nonlinear second-order ordinary differential equation 4 4 2 3“ R = 0, for a < r < b. (5.8) 3r R R - 2rR3R + 2r The prescribed (pressure) boundary conditions are '1' RR 'p at r 8 as (5.93) TRR 0 at r = b. (5.9b) 19 5.2 Solution QE’BOundary-Value Problem: Just as in the two-dimensional case, equation (5.8) may be reduced to a first-order equation on making the substitution t(r) =-£§)§} > o. (5.10) Equation (5.8) then yields 3rt(r) - t(l-t)(2t2+2t+5) = O for a < r < b, (5.11) where t = dt/dr. Again we assume that t(r) is less than unity for a < r < b and so deduce from (5.11) that 0 < t < l, dt/dr > 0 for a < r < b. (5.12) Integration of (5.11) yields 15 c t9d(t) " ’ *2 2 5 . (2t +2t+5) (l-t) (5.13) where C is a positive constant of integration and the function d(t) is defined by d(t) = exp {2 tan-1(££§l)} > 0. (5.14) Also (5.10), (5.11) yield 2 R6 = 013—t +2t+§1d(9), (5.15) (14)? where D > 0 is a constant of integration. The range of the parameter t is (0 <) ta < t < tb (< 1). (5.16) The components of true stress TRR, Tee may also be expressed in terms of t on using (5.6), (5.10), (5.13) and (5.14). This leads to 20 ‘RR u{1 - (DS —)1/6(2t2 +2t+5)'3’2d'1/2(t)}, (5.17) -3/2d-1/2 (t)}, (5.18) T 99 uII - (—§)1/5¥M(2t +2t+5) D for ta < t ac, the converse is true. Finally, after loss of ellipticity, the non-existence of non-smooth axisymmetric solutions may be demonstrated by an obvious modification of the argument given in [1]. The possibility of surface bifurcations also exist, but we shall not pursue this here. t m *7- ll 24 'I Figure 1. 11) 1.5 2.0 2.5 3.0 3.5 R(a)/a Applied pressure versus ratio of deformed to undeformed radius for a thin cylindrical shell. 1“!) y U ' 1u4 142 . 11) * Clai- Ct41~ ONO Figure 2. 25 08100 d 4 4 Applied pressure versus ratio of deformed to undeformed radius for hollow cylinders for different radius ratio a = b/a. The dotted curve connects the points (pa/u, R(a)/a), where pe denotes the pressure at which ellipticity is lost 26 ‘— ’7‘ .24 # t f ‘ j ‘ '— 11) 1.5 2J3 2:5 31) 1l5 Figure 3. Applied pressure versus ratio of deformed to undeformed radius for a thin Spherical shell. : ‘FI'O 2.81’ 2J31' 244 r 242 ’ 210 ( 1.£I’ 1.63’ 1.4»? 1.2 5 1.CD~ ()1! ’ (313 ( Ct4»’ 0L2.’ ()1) ()1) Figure 4. 27 0:11.25 1 ‘ (1:2 \ 081.5 ‘0 =1.1 1.0 2.0 3.0 4.0 5.0 0.0 7.0 3.0 IRLQI ~ 7 Applied pressure versus ratio of deformed to undeformed radius for hollow spheres for different radius ratios a b/a. The dotted curve connects the points (pe/u, R(a)/a), where pe denotes the pressure at which ellipticity is lost. APPENDICES APPENDIX A: Here we present the details of the calculation showing that Q(ta) in equation (3.19) has only one maximum for 0 < ta < 1. We write Q(t)- : Q(t a) , thLt))(f7(tl+f%tl+4l3 (A.1) h(t)(t +t+4)3 using t instead of ta for simplicity. For later purposes, it is also convenient to emphasize here that 0 depends on the radius ratio a = b/a (see 3.18). We recall that (3.20) shows that Q(t,a) has at leaSt one maximum on 0 < t < 1. Now 3 a . . 0g, ) ___6(t 2m4) 22(15 ma)2 11 um,“ a), (M) (NH)2 where H(t) = h(t)(t2+t+4)3, (A.3) and R(t,a) = {(12+t+4)(f+1)%{-- (f2+f+4)(t+1)}. (A.4) Observe from (A.2) that R(t,a) = 0 is equivalent to 13915439-= 0. 3t Thus at a maximum point for Q(t,a), say at t=t(a), it follows that R(E,a) = 0. (A.5) Now we want to show that only one such point t a t(a) exists for which 0 < t < 1. From (A.2) upon differentiating again, we find 28 29 2 a 6 2 2 2 = [2(t +t+4 2t+1 f +f+4 -h-h f R t,a 3:9- 15(2)]7' )( )( ) ( ) ( ) + (t2+t+4)2-2-(f2+f+4)(2f+1)-f'~h-h(f)R(t,a) + (t2+t+4)2(f2+f+4)2-h'-h(f)R(t,a) + (t2+t+4)2(f2+f+4)2-h-h'(f)'f'-R(t,a) (A.6) + (t2+t+4)2(f2+f+4)2-h-n'(f)33§%:393 - {H::)}2(t2+t+4)(t+1)(f2+f+4)2-h-h(f)R(t.a). Using (A.5) we find 2 1.3 a -.- 6 (t2+t+4)2(f2+f+4)2-h'h(f)-aB-§-E-’-E)- “o (A-7) 3t t=t {H(t)} t=t Differentiating (A.4) with respect to t and using (A.5) we find t+1 f+1 2 akgt,a) fl = 4f‘(———-- -—-) + f"(t +t+4)(f+1). (A.8) at t=t(a) f+1 t+1 He now observe from equation (3.15), (3.18) that afgtla) > 0 8t ’ 2 .2_£§£1£Q.< 0, (A.10) at t < f(t,a). (A.11) aR§t,a) 5 < 0 for 0 < t < 1, and so from (A.7), we deduce that at t=t( a) (A.9) Thus 2311 2 < 0 for 0 < E < 1. 31. tif( a) Therefore, we concTude that Q(ta,a) has only one maximum for 0 < ta < 1. APPENDIX B In Appendix A, we have shown that Q(ta,a) in equation (3.19) has only one maximum for 0 < ta < l, and the corresponding value of ta is denoted by t(a). In this Appendix, we analyze the behavior of t(a) as a varies and thereby verify the remark made in Chapter 4.2 regarding the value and location of the pressure maximum pm as the radius ratio a = b/a increases. As we have shown in Appendix A, at the maximum point for Q(ta,a) we have R(E(a).a) = o (3.1) where R(t,a) is defined by (A.4). Differentiating (8.1) with respect to ayields dR - 25.2%. 25. . 1 - 3?. do + 801 0' (8.2) We have shown in Appendix A that -3§ < 0 for 0 < E < 1. at A Observe from (8.2) that 33-:- has the same sign as g—E. Now from (A.4) and (3.18), we have -1 2 2 8R 8 _192 a (l-f) (t +t+4)(8f/8t)g(f)(3f4+3f3+11f2_f_4)’ 7“- 1811(1)(152+1=+4)(ag/01=)2 (8.3) where we use t,f instead of t(a), f(t(a),a) a f(a) for simplicity. The last term on the right hand side of (8.3) vanishes when f = 1//§. Thus from (8.2) we see that 31 g§%§Q-= 0 when a = 5*, (3.4) where a? is the value of a for which f E f(€(3).3) = 1//§. (3.5) Thus recalling (A.9), we conclude that if 0 < f < 1 then %E-> 0, (I * A . ~ _ dt 3 if f - f then do 0, (8.6) * ~ dE andif f 0, do if E =1 then 9f- = 0 (811) do ’ ' . * ~ d“ and if t < t < 1 then 3" > 0. a To complete our analysis, we need the following two auxiliary results. '- meal 1 Let f(x) be a continuously differentiable function on [1,w). Let k be some fixed number. Furthermore, suppose that f and k are such that I f'(x) < 0 whenever f(x) < k. If there exists an xo¢5[1,~) such that f(xo) < k then f(x) < k and f'(x) < 0 on [xo,~). m If f(xo) < k then df a;- x=xo < 0 by hypothesis. Thus f(x) < f(xo) for x0 < x < x0 + e and so f(x) < k for x0 < x < x0 + 6. Thus df d;- xxxo + e < 0‘ Lemma 2 Let f(x) be a continuously differentiable function on [1,»). Let k be some fixed number. Furthermore, suppose that f and k are such that f'(x) > 0 whenever f(x) > k. If there exists an xoe[1,~) such that f(xo) > k then f(x) > k and f'(x) > 0 on [xo,~). 33 The proof of Lemma 2 is similar to that of Lemma 1 and will be omitted. Let * a = 1 + e + 52/2 (3.12) and assume f(t) = t + eA(t) + 223(t)/2, (3.13) where A(t), 8(t) are unknown functions. Proceeding as in the analysis sketched in Chapter 4.1, one can show that E 1//§ - 0.2:, (3.14) ; 1//§ + 0.2:. (3.15) Thus, when the radius ratio a (=b/a) is near to unity we see from (8.14), (8.15) that e * .c 2 0 < t < t, f < f < 1 for a = 1 + e + e /2. (3.16) It would follow from (8.6), (8.11) that d“ d? 2 'a% < 0 and ‘3; > 0 for a = 1 + e + e /2. (3-17) Then, by Lemma 1 and Lemma 2, we can conclude that d? < 0 ~ * s 8 18 To": and 0Oand f O, (2.1)* where (r, 6, ¢) are the current spherical polar coordinates of the point which, in the undeformed configuration, was located at (R, o, o). If the material is assumed to be incompressible, the deformation gradient 5 obeys det E - 1. For the deformation (2.1), this implies rzar/aR = R2, which when integrated gives r = r(R,t) - {R3 + c3(t)}1/3, c(t) > 0, (2.2) where c(t) is to be determined. If it is found that c(t) = O, (2.2) implies that the body remains a solid sphere in the current configuration. On the other hand, if c(t) is found to be positive (i.e. r(O+,t) > 0), there is a cavity of radius c centered at the origin in the current configuration. From (2.1) and (2.2), the non-vanishing components of the Eulerian strain-rate tensor are found to be 0r = -2F/r, 09 - 0, = F/r, (2.3) *Note that the convention used here for undeformed and deformed coordinates is opposite to that employed in Part I. 37 38 where the dot denotes the Lagrangian time derivative. In view of symmetry, and assuming the material to be isotropic, the non-zero components of the (Cauchy) true stress tensor are the radial stress ar(r,t) and the hoop stresses 09(r,t) . a¢(r,t). The prescribed dead load boundary condition on the surface of the sphere requires that °r(a. t) = p(t)(A/a)2. (2.4) where a = r(A,t) a (A3 + c3)“3 represents the deformed outer radius. r The constitutive relation for the elastic-plastic material is taken to be a generalization of dz-flow theory to finite deformations, (see e.g. [3]): kl ’- 0 =-—3 g + A--3 E (o )s (2 5) 2E e 20b P e ~ ’ ' 5‘ Here 5 is the deviatoric Cauchy stress; ac is the effective Cauchy stress; A is a loading coefficient; ep(-) is a given constitutive function representing the effective plastic logarithmic strain. The Jaumann (co-rotational) rate of the Cauchy stress deviator is denoted by g, so that g = S - OS + 59 where g is the spin-tensor. In the case of uni-axial tension, the relation between true stress a and the logarithmic strain a, in monotonic loading, can be obtained from (2.5) as e = 5(0) 5 a/E + ep(o). He assume this relation to be invertible so that we may write the stress-strain relation jn_uni-axial tension as either a = 5(6) or e = 2(0). (2.6) In the present problem, 9 vanishes, and thus g = 5. Also, q9 a 09 - or. Equation (2.5) and (2.6), under conditions of loading (A a 1), yield 39 1 , (2.7) €(Oe) = ZP/T‘, 0e = as "‘ or. On using (2.1), we may integrate (2.7) with respect to the parameter t to obtain c(oe) = 2£n(r/R). Using (2.6) to invert this gives oe = o{2£n(r/R)}. (2.8) Finally, in the absence of body forces, the equilibrium equations reduce to the single equation f; I -335 - 2-32 = O. (2 9) 3r r ' Thus, the problem to bg_solved is the following: We wish to find or(r,t) and c(t) > D such that the field equations (2.2), (2.8), (2.9) fl. and the boundary condition (2.4) hold*. In addition, if c(t) > 0 it is also required that 0r(C.t) = 0. (2.10) This stipulates that when a hole appears at the origin, it must be traction-free. First, it is readily shown that, for all values of p > 0, one solution to the foregoing problem is or(r,t) = p(t), c(t) = O. (2.11) This corresponds to a homogeneous state of deformation r = r(R,t) = R, with resulting stresses “r = 90 = o¢ = p(t). Next we seek a solution for which c(t) > O. Combining (2.2), (2.8) and (2.9), integrating with respect to r, using boundary condition (2.4), and employing a change of variables yields *The remaining physical quantities can be immediately found thereafter from (2.2), (2.9) and 09 = o¢ = °e + °r° 40 ar(r’t) = p(£. 2 - I2£n(l‘/R) 3(6) - 3 3 1/3 2£n(a/A) eon3e/2)-1 d5» R-(r -c ) . (2.12) On enforcing the remaining boundary condition (2.10), one is led to 3(8 _ 3 3 1/3 exp(3e/2)-1 d5: 3 - (A + c ) . (2.13) 2f£n(%) Thus, if for a given value of p, (2.13) can be solved for a positive root c, then this c together with (2.12), provides a solution to the problem at hand. It is readily shown that under usual constitutive conditions, [5(a) = 0(6) as e + 0, 5(5) = 0(5") as e + a, n > O], the integral in (2.13) is bounded for all a > A. Therefore, there does exist a value of pressure p(>O) corresponding to each c > O. A schematic graph of p vs. c is shown in Figure 5. The critical load Pcr at which the cavity is initiated is found by letting c + 0+ in (2.12), i.e. = O 8 per (1; W (16. (2.14) As noted previously, the integral in (2.14) is bounded and so, the cavity is initiated at a finite value of load. One may also study this problem using the finite strain version of Jz-deformation theory (e.g. [4]). This will be done in the next Chapter. CHAPTER 3 FINITE STRAIN FORMULATION AND SOLUTION; Jz-DEFORMATION THEORY In this Chapter, we treat the problem at hand using a finite strain version of Jz-deformation theory (e.g. Hutchinson and Neale [4]). This F3 is simply a Special case of finite elasticity for a particular : incompressible material. Consider a solid sphere of radius A, subjected to a radial tension (dead load), p, applied to its surface R a A. The resulting deformation #5 is a mapping which takes the point (R,e,o) to the point (r,e,¢). He assume that the deformation is axisymmetric one so that e = 0,0 = 0 and r = r(R). In order to avoid interpenetration, it is required that r = r(R) > O on O < R < A, r(O) > O. (3.1) Observe that if r(O) > O, the deformation is not one-to-one at the origin. If the material is assumed to be incompressible, one readily finds that r = r(R) = (R3 + c3)1/3, c > O, (3.2) where the constant c is to be determined. The polar components of the deformation gradient tensor F associated with the radial deformation (3.2) are given by E a diag (($702: 31%)., 1.1-3.1)- (3.3) The corresponding principal stretches are As a A¢ - r(R)/R - x, x, = 19-2 - x-2 (3.4) In view of symmetry, the non-zero components of the true stress tensor 41 42 are the radial stress ar(r) and the hoop stresses 09(r) = a¢(r). The prescribed dead load boundary condition of the surface requires that °r(a) = p(A/a)2. (3.5) where a = r(A) = (A3 + c3)1/3 represents the deformed outer radius. He now turn to the constitutive relation for a general isotropic incompressible elastic material characterized by an elastic potential H(11,12,13) representing the strain energy per unit undeformed volume. The components of the principal true stress a are given by x,- aw . °ii = AIAZAB 3‘1 - p, (no sum on i). (3.6) A For all A > 0 it is convenient to define R(A) by 13(1) w(x‘z,x,x), x > 0. (3.7) From (3.4), (3.6) and (3.7), one can derive the following: - r “' L or - 06 - -212." (R)' (3.8) In the absence of body force, the equilibrium equations div 0 = 0 reduce to -—d +2 -0 39 dror -F(ar-ae) - , c < r < a. ( . ) On using (3.6) and (3.7), (3.8) becomes 1 “1 - -§-w (g) - 0. (3.10) :In Thus, theproblem tgnbg_solved is the following: We wish to find or(r) for r(iEc,a] and c > D such that the field equations (3.2), (3.10) and boundary condition (3.5) hold. In addition, if c > D it is also required that °r(C) = 0 (i.e. arlR=O = O). (3.11) This requires that when a hole appears at the origin, it must be traction-free. 43 First, it is readily shown that, for all values of p > 0, one solution to the foregoing problem is or(r) = p. c = 0. (3.12) This corresponds to a homogeneous state of deformation r = ;(R) = R. Next, we seek a solution for which c > 0. Integrating (3.10) with respect to r, using boundary conditions (3.5) and employing a change of variable 6 = r/R yield .2 A A a A_2 _ H' E = 3_ 3 1/3 or(r) p(a) (r E§§11d£’ R (r c ) . (3.13) 'R On enforcing the remaining boundary condition (3.11) yields p = (g)2 I -E%Lfldfi. a = (A3+ c3)1/3. (3.14) a - 'A Thus, if for a given value of p, (3.14) can be solved for a positive root c, then c together with (3.14), provides a solution to the problem at hand. As we mentioned before, Jz-deformation theory is a special case of finite elasticity, the class of incompressible materials being now characterized by the particular energy function of the form w(11,12,x3) = w(ee), (3.15) where ee = [(2/3){(2011)2+(2012)2+(£013)2}]1/2. (3.16) when 11 = 1'2, 12213-1, as is the case here, (3.16) gives A = exp(ee/2). In the case of uni-axial tension,_the relation between true stress a and the logarithmic strain 5 can be obtained, i.e. 44 0(5) = H'(e)- (3.17) Employing a change of variable, a = Zing, and using (3.17), (3.14) readily becomes p = ($2) —9-(-§-L (is. (3.18) 22n(%) exp(§-€-)-1 Equation (3.18) is seen to be identical to (2.13). CHAPTER 4 INFINITESIMAL STRAIN FORMULATION AND SOLUTION In this Chapter, we consider briefly the infinitesimal strain analog of the problem just treated. In view of symmetry, the r .ywj displacement field has the form UR = U(R.t). Ue = Uo = 0- (4.1) " NM‘A 1.1L ‘-_' |'- The non-vanishing components of the infinitesimal strain tensor are 8 ER = 5%» so = e. = %- “-2) T- h. .- a Incompressibility in the present case requires 8R + $9 + e¢ = 0. (4'3) Thus, (4.2) and (4.3) lead to a simple ordinary differential equation for u whose general solution is u(R,t) - (a(t)-A)A2/R2, (4.4) where a(t) represents the deformed outer radius of the sphere which is to be determined. If it is found that a(t) = A, then u(0+,t) = 0 i.e., the sphere remains solid in the current configuration (in fact, in this case u(R,t) a 0). On the other hand, if it is found that a(t) # A (a(t) > A) then u(0+,t) = a and the body has ruptured at the origin. The non-zero stress components are the radial stress 0R and the hoop stresses 09, o¢ (oe=o¢). The prescribed tensile load boundary condition of the surface of the sphere requires that °R(A.t) = P(t). p(t) > 0. (4.5) 45 46 The constitutive relation for the elastic-plastic material (at infinitesimal deformations) is taken to be the classical Prandtl-Reuss equation with a Von Mises (J2) yield criterion; i.e. . 3 . 35 (0e) Here 5 is the deviatoric stress; Ge is the equivalent stress, oe={3/2 tr(§2)}1/2; A is a loading coefficient. The constitutive function ep(oe) is, of course, given (and gives the value of the corresponding "effective plastic strain"). In the case of uni-axial tension, the relation between stress and strain in loading can be obtained from (4.6) as e =-§ + cp(o) 5 5(0). (4.7) The response function may be inverted to give 0 = 0(6). (4.8) Returning to the problem at hand, from (4.1) and (4.2), the non-zero components of strain-rate are - 2 - 2 ‘ aA - - aA ..-. -2—, e = e = -—--, (409) R3 G 4 R3 where the dot denotes differentiation with respect to t. Equation (4.6) and (4.7), under conditions of loading (A=1), yield 5(0 ) = -——- , o = o - OR’ (4.10) 47 Integrating (4.10) with respect to t and using a = A when t = 0 gives c(oe) = 2u/R. Using (4.8) to invert this gives ‘ 2u 0e a O(-R—- o (4.11) Finally, in the absence of body forces, the equilibrium equations reduce to the single equation :33- 2.39.: D. (4.12) B: 3R R g1 Thus, the problem £9. 92 M ii the following: we wish to find l oR(R,t) and a(t) > A such that the field equations (4.1), (4.11), (4.12) and the boundary condition (4.5) hold. In addition, if a(t) t A it is also required that !’ 0R(O+,t) s 0. (4.13) This requires that when a hole appears at the origin, it must be traction free. First, it is readily shown that, for all value of p > 0, one solution is aR(R,t) = p(t), a(t) = A. (4.14) This corresponds to a homogeneous state of deformation with resulting stresses °R'°e=°o=P(t)o Next we seek a solution for which a(t) > A. Integrating (4.12) with respect to R from R a R to R = A, using the boundary condition (4.5) yields " 3(3!) IP daR = 2 f R dR, 0 < R < A. (4.15) OR R R 48 Employing a change of variable 6 = 2u/R now gives 2(a-A)A2/R3. em¢)=p-§I 3%Hao Pcr the solution tends to another (non-homogeneous) state. It is found that pcr = $3 Sé-Elde. (4.18) Note from (4.17) that by enforcing the boundary condition (4.13) and letting a = A, we may formally recover (4.18). Alternatively, note that (4.18) may also be formally derived from the finite strain formula for Pcre (2.14), by replacing the exponential in (2.14) by the first two terms in its Taylor expansion about a = O. CHAPTER 5 DISCUSSION Here we discuss further the bifurcation predicted by the finite strain formulation of Chapters 2 and 3. For all values of the prescribed radial dead load p one possible configuration is that in which the sphere remains solid (see (2.11)). On the other hand, for a certain range of p one has, in addition, a second possible configuration involving an internal spherical cavity. Equation (2.14) gives the critical value of the load, Pcr: at which a cavity may initiate. It is necessary to examine the stability of these two possible configurations in order to determine whether the homogeneous solution will in fact bifurcate, at p = Pcr: into the one involving a cavity. Figure 5 shows, schematically, a graph of the cavity radius c versus the applied load p. The bold horizontal line coinciding with the positive p-axis corresponds to the homogeneous solution. The curves emanating from (Pcr: 0) correspond to a bifurcated solution involving a cavity. Presumably, if such a curve comes off to the right, the bifurcated solution is locally stable and so the sphere would indeed develop an internal cavity at p = Pcr- Conversely, if bifurcation to the left occurs, the solution is locally unstable and the sphere remains solid. On using a Taylor expansion near Pcr: (2.13) yields _._3 * 2 2|: 3 p = p(C) = pcr + -3- A3 (pa. - -3-) + 0(c) as c + 0. (5.1) 49 50 A where E = o'(O) is the Young's modulus. Thus when Pcr > 2E/3, the slope of the curve at (Pcr. 0) is positive and so bifurcation to the right occurs. On the other hand, when Pcr < 2E/3, the slope is negative. Consequently a void will actually appear at p = Pcr only if Pcr is greater than 2E/3. Of course, the load level at which stable bifurcation is predicted here is unreasonably large. This feature is commonly encountered in bifurcation analyses employing classical flow theories of plasticity. It may be possible to use more elaborate constitutive models or to include the effect of a pre-existing stress concentrator (such as an inclusion) to obtain more realistic values for the critical load. n—J-() 51 up 2 ° \ < a: \ ,. \ t; \ >1 8 ‘1 \ l l l l l l l I I f —————————— —>p APPUED PRESSURE Figure 5. Schematic graph showing variation of cavity radius c versus applied load p. LIST OF REFERENCES REFERENCES J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. §;_Soc. Lond. A306, 557 1982 . “““ C.O. Horgan and R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: Growth of a micro-void. .J_.-o_f_ Elasticity (in press). D. Durban, Large strain solution for pressurized elasto/plastic tubes. g;_gj_Applied Mechanics 46, 228 (1979). J.H. Hutchinson and K.H. Neale, Finite strain Jz-deformation theory. Finite Elasticit . (D.E. Carlson and R.T. Shield, editors), pp. 237-247. Martinus Nijhoff, The Hague (1982). 52