MSU RETURNING MA'IERIALS: Place in book drop to LJBRARJES remove this checkout from “ your record. FINES will ~ » be charged if book is returned after the date stamped below. STUDIES ON VOID FORMATION AND GROWTH FOR INCOMPRESSIBLE NONLINEARLY ELASTIC MATERIALS By MIAO-SZE (OLIVIA) CHOU-WANG 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 1988 ABSTRACT STUDIES ON VOID FORMATION AND GROWTH FOR INCOMPRESSIBLE NONLINEARLY ELASTIC MATERIALS BY Miao-Sze (Olivia) Chou-Wang In this study, a class of bifurcation problems for a solid sphere subjected to uniform tensile dead-loading po at its boundary are examined within the framework of finite elastostatics and elastodynamics. The sphere is composed of a particular class of homogeneous isotropic incompressible nonlinearly elastic materials, namely those of power-law type. First of all, we carry out an investigation of the elastostatic problem. One solution to this problem, for all values of po, corresponds to a homogeneous state in which the sphere remains undeformed while stressed. However, for sufficiently large values of po, there is in addition a second possible configuration involving an internal traction-free spherical cavity. The dependence on constitutive parameters of the critical load at which bifurcation occurs is examined as well as the subsequent void growth. The stress distribution after cavitation occurs is also described. The results are obtained in closed analytic form. Secondly, we study the elastodynamic version of the foregoing problem for the special case of a neo-Hookean material. The sphere is Miao-Sze (Olivia) Chou-Wang set into motion by a suddenly applied uniform radial tensile dead-load p0. One solution to the dynamic problem, for all values of po, corresponds to a trivial homogeneous static state in which the sphere remains undeformed while stressed. However, for sufficiently large values of po, one has in addition another possible radially symmetric motion involving an internal traction-free cavity. The "critical load" at which an internal void may be initiated in the dynamic problem is shown to coincide with that for the static problem. To the Lord my Heavenly Father who endows the heart with wisdom and gives understanding to the mind. iv ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my advisor, Professor Cornelius 0. Morgan for his guidance 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 Dahsin Liu, Thomas J. Pence and David H.Y. Yen. While preparing this dissertation, I held teaching assistantships awarded by the Department of Metallurgy, Mechanics and Materials Science and research assistantships supported jointly by the National Science Foundation under Grant MSM 85-12825 (C.0. Horgan), Department of Metallurgy, Mechanics and Materials Science and the Composite Materials and Structures Center of Michigan State University under a REED Program Grant. The support of these institutions is gratefully acknowledged. A final note of appreciation must go to my husband, Hsien-Ming Chou, for his support and encouragement. LIST OF FIGURES ....................................... Section 1 INTRODUCTION .............................. Section 2 BIFURCATION PROBLEM FOR A SPHERE: FORMULATION AND SOLUTION ------------------ 2'1 Formulation ............................... 2_2 Solutions ................................. 2,3 The Critical Load ......................... Section 3 SOLUTIONS FOR A CLASS OF INCOMPRESSIBLE ELASTIC MATERIALS ......................... 3.1 A class of Incompressible Elastic Materials 3.2 Cavitation Solutions ---------------------- Section 4 ELASTODYNAMIC PROBLEM FOR A NEO-HOOKEAN SOLID SPHERE SUBJECTED TO A SUDDENLY APPLIED DEAD-LOAD ......................... 4_l Formulation ............................... 4.2 Solutions ................................. 4.3 The Basic Differential Equation ----------- 4.4 Oscillations .............................. APPENDIX A ............................................ APPENDIX B ............................................ LIST OF REFERENCES TABLE OF CONTENTS vi OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10. Variation of the stresses 1 LIST OF FIGURES . Behavior of the power law material under uniaxial stress. . Behavior of the power law material under equibiaxial stress. . Behavior of the power law material under pure shear. . Variation of the deformed cavity radius c with applied dead load p0 for a power law material with strain energy density given by (3.1). -- . Variation of the radial stress rRR(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 3/4. . Variation of the radial stress 1RR(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 1. -' . Variation of the radial stress th(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 5/4. . Variation of the stresses 169(r), r¢¢(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 3/4. . Variation of the stresses 199(r), r¢¢(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 1. ~- ee(r), r¢¢(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 5/4. vii OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Page 35 36 37 38 39 40 41 42 43 1. INTRODUCTION Void nucleation and growth in solids have been of concern for a long time because of the fundamental role such phenomena play in fracture and other failure mechanisms. (See e.g. Goods and Brown [1] for a discussion of cavity nucleation in metals). The phenomenon of sudden void formation ("cavitation") has also been observed experimentally in vulcanized rubber by Gent and Lindley [2]. Nonlinear theories of solid mechanics have been used recently to account for such phenomena. The impetus for much of the recent theoretical developments has been supplied by the work of Ball [3]. In [3], Ball has made an extensive study of a class of bifurcation problems for the equations of nonlinear elasticity which model the appearance of a cavity in the interior of an apparently solid homogeneous isotropic elastic body once a critical load has been attained. An alternative interpretation for such problems in terms of the growth of a org-existing micro-void is given in [4]. Further investigations of such bifurcation problems have been carried out in [5-12]. It is worth noting that cavitation can be shown to occur only when finite strain measures are taken into account (see e.g. [4], [9]). The corresponding problems in linearized elasticity or in the infinitesimal strain theory of plasticity do not exhibit such bifurcations. The purpose of the present study is to further investigate this bifurcation approach to void nucleation in two specific contexts. First of all, we carry out an investigation of the problem of static tensile dead-loading of a solid sphere composed of a particular class of homogeneous isotropic incompressible nonlinearly elastic materials, namely those of power-law type. Secondly, we study the elastodynamic version of this problem for the special case of a neo-Hookean material. In Section 2, we formulate the basic boundary-value problem that arises when a solid sphere, composed of an incompressible isotropic elastic material, is subjected to a prescribed uniform radial tensile dead-load po on its boundary. One solution to this problem, for all values p0, corresponds to a trivial homogeneous state in which the sphere remains undeformed while stressed. However, for sufficiently large values of po, one has in addition other possible radially symmetric configurations involving an internal traction-free spherical cavity. Such solutions have been shown by Ball [3] to bifurcate from the homogeneous solution at a critical value of po, say pm” at which the homogeneous solution becomes unstable. The possibility for these bifurcated solutions to exist depends on the constitutive law for the material under consideration. In Section 3, attention is confined to a particular class of homogeneous isotropic incompressible elastic materials, namely those of power-law type. Such nonlinearly elastic materials were first introduced by Ogden [13] and have been employed in a wide variety of problems since then (see e.g. [14], [15]). An extensive discussion of the properties of this class of materials has been provided recently by Zee and Sternberg [16]. Our interest here is in examining the dependence of the critical loads at which cavitation occurs on the constitutive parameter n appearing in the definition of this class of materials (see equation (3.1)). In Section 3, we first examine the behavior of these materials under certain homogeneous deformations namely uniaxial stress, equibiaxial stress and pure shear. These results are shown in Figures 1-3. The explicit relationship between the applied load po and the deformed cavity radius is examined and plotted in Figure 4. It is found that as the hardening parameter n increases, the critical load pa at which bifurcation takes place also increases. For the special case of a neo-Hookean material, for which n - 1, we recover results due to Ball [3]. The stress distribution in the sphere is also described. An interesting feature concerning the principal stresses immediately after cavitation is the presence of a boundary layer near the cavity wall. To see this, we have plotted the stresses in Figures 5-10 for applied dead loads po slightly larger than per. In Section 4, we consider the radially symmetric motion of an isotropic incompressible elastic solid sphere composed of a neo- Hookean material which is set into motion at time t - O by a suddenly applied uniform radial tensile dead-load p0. If the material were compressible, the medium would respond to such a loading by propagating a dilatation wave inward from the boundary. In the incompressible case, the effect of the tensile load is felt immediately throughout the medium and the response takes the form of a nonlinear oscillation. Such oscillation problems were first investigated by Knowles [17], [18] for hollow circular cylinders and have received considerable attention since then (see e.g. [19-22] and the references cited therein). In Section 4, we show that one solution to the dynamic problem described above, for all values of po, corresponds to a trivial homogeneous static state in which the sphere remains undeformed while stressed. However, for sufficiently large values of po, one has in addition another possible radially symmetric motion involving an internal traction-free cavity. A relationship between the applied load po and cavity radius c(t) at time t is obtained in the form of a second-order nonlinear ordinary differential equation (see equation (4.19)). By adapting the techniques of Knowles [17], [18], we show that periodic oscillations can occur if and only if the applied tensile dead-load po is such that po 2 5p/2, (1.1) where p denotes the shear modulus for infinitesimal deformations of the neo-Hookean material. As po‘+ 5p/2+, the deformed cavity radius c(t) 4 0+. It is shown that the value of the "critical load" at which an internal void may be initiated in the dynamic problem coincides with that for the static problem. For values of p0 > 5p/2, following the application of such a load at time t - 0, the cavity would expand until its radius reaches a maximum value given by equation (4.27), then would contract to zero and repeat the cycle. 2. BIFURCATION PROBLEM FOR A SPHERE: FORMULATION AND SOLUTION 2.1 Formulation: We are concerned here with a sphere composed of a homogeneous incompressible isotropic elastic material. Let the undeformed solid sphere be denoted by Do-{(r,0,¢)I0.<_r 0, 0 < r < b; R(O+) z 0, 9 - 0, Q - ¢, on Do’ (2.1) where R(r) is to be determined. The principal stretches associated with the radially symmetric deformation (2.1) are - R(r) Ar - R(r), A6 = A! - , (2.2) where the dot denotes differentiation with respect to the argument. The spherical polar components of the deformation gradient tensor F associated with (2.1) are given by 9 M ' (2.3) F - F - F - F - F - F - O. Incompressibility then requires that the Jacobian determinant J - Det F - l, which upon integration yields R(r) - (r3 + c3)1/3, (2.4) where c 2 O is a constant to be determined. If it is found that c - 0, (2.4) implies that the body remains a solid sphere in the current configuration. On the other hand, if c is found to be greater than zero, then R(0+) - c > O and so there is a cavity of radius c centered at the origin in the current configuration. In this event, the cavity surface is assumed to be traction-free. The strain-energy density per unit undeformed volume for a homogeneous isotropic incompressible elastic material is denoted by W - W( A1, A2, A3), (2.5) where A1 (i - 1,2,3) are the principal stretches. The function W is invariant with respect to interchange of the A1 and is taken to satisfy the normalization condition W(l,1,1) - 0. In the sequel, we proceed formally and assume that W possesses sufficient regularity properties to permit the subsequent analysis. The principal components of the Cauchy stress tensor 1 are given by - A '—- - p, (no sum on i), (2.6) where p is the hydrostatic pressure associated with the incompressibility constraint Adxzxaa- 1. For the radially symmetric deformation with principal stretches given by (2.2), the principal S tress components are r (r) - v"2 w (v‘z. v. v) - pm. RR 1 (2.7) -2 769(r) - rm(r) - V w2(v 9 V, V) ' P(r)o where, following Ball [3], we have introduced the notation R c3 13 v-v(r)-;-(l+-—3)l. (2.8) r Notice that in (2.7) we consider r(r) rather than the more conventional r(R). The subscript notation on W in (2.7) denotes differentiation with respect to the appropriate argument. In (2.7), we have also used W2(v-2, v, v) - W3(v-2, v, v), which follows from the invariance of W with respect to interchange of its three arguments. The dead-load boundary condition now requires that 1mm - PJTIZBTJZ - p0 [v(b>1'2. (2.9) where the constant p0 > O is prescribed. We note that the boundary conditions of vanishing shear tractions are satisfied identically. In addition if c > 0, then the condition for a traction-free cavity surface TRR (0) - 0, (2-10) must also hold. In the absence of body forces, the sphere will be in equilibrium provided that div 1 - O, which will hold provided that ”31 1'1 RR 3r + 2 R “an" '99] - 0, (2.11) holds throughout the sphere. Thus, the problem to be solved is the following: For a prescribed value of the dead-load traction p0 > 0, we seek a pressure field p(r) and a constant c z 0 such that (2.11) and (2.9) are satisfied where r r are given by (2.7) and (2.8). In an’ '99’ addition if c > 0, then (2.10) must also e satisfied. 2.2 Solutions: It may be readily shown that one solution to the foregoing problem, for all values of po, is p(r) - W1 (1, l, l) - p0, c - O. (2.12) This corresponds to the trivial homogeneous state of deformation R(r) - r, (2.13) with corresponding stresses rRRu- Tee - 10° - p0. Next we describe solutions for which c > 0, corresponding to the presence of a traction-free cavity at the origin. For this purpose, we adopt an approach developed by Morgan and Pence [10] and rewrite the differential equation (2.11) in the form —d[v-2 W (v.2 v v) - (r)] + 4 dr 1 ’ ’ p 2" W.1 W (v-2.V.V) - VZW (v-2.V.V)] = 0. r 1 2 on 0 < r < b, (2.14) where we have used (2.7),(2.8). On integration of (2.14), we have p(r) - p(O) - v-2(r) w1(v"‘, v, v) + 2 J(r), o < r < b, (2.15) where J -J {v'51}d—j— , 0 O < r < b. (2.16) On substitution into (2.7) we obtain rRR(r) - -p(0) - 2J(r), O < r < b. (2.17) The traction-free cavity surface condition (2.10), together with (2.17) and J(O) - 0, now yields p(O) — O. (2.18) Finally the boundary condition (2.9) at r - b is satisfied if -2J(b) - p0 [v(b)]-2 . (2.19) The condition (2.19) may be written in a compact fashion on utilizing the change of variables 3 4 v in the integral (2.16). From (2.8) it is seen that this change of variable is one-to-one and invertible if and only if c > 0. Introducing the function W(x) - W(x"’, x, x), (2.20) and adopting the notation W1(x) - 3:1?wa (2.21) (2.19) may be written as c3 2 3 W (v) p - (1+-)’ ————1 dv, c>0. (2.22) 0 3 3 b ea m (V - 1) <1+—3) b Equation (2.22) was first established by Ball [3] for the n- dimensional version of the problem described here (see equation (5.18) of [3]). Thus, for a given dead-load p0, solutions involving a 10 traction-free internal cavity of radius c exist provided that c is a positive root of (2.22). The associated pressure field is given by p(r) - v’2(r)w1(v‘2, v, v) + 2J(r), o < r < b. (2.23) The critical load pc1 at which an internal cavity may be initiated is found by formally letting c 4 0+ in (2.22), and so 131m p - —— dv. (2.24) or (v3 _ 1) This result was first established by Ball [3] in n-dimensions (see equation (5.22) of [3]). In summary then, we have seen that for all values of the applied dead-load traction p0, one obtains the trivial solution (2.12) corresponding to the homogeneous state of deformation (2.13). Moreover, if positive roots c of (2.22) exist, then one obtains the additional solutions involving a traction-free internal cavity described above. Such solutions have been shown by Ball [3] to bifurcate from the trivial solution at the critical value pm at which the trivial solution becomes unstable. 2.3 The critical load: Since the integral in (2.24) is improper; put may or may not be finite, and so cavitation may or may not take place. As regards the lower limit in (2.24), it is shown in Appendix A that dW(l) «1213(1) dv dvz _ 12“, (2.25) 11 where p denotes the shear modulus for infinitesimal deformations of the material. Thus by l'H8pital's rule, the limit of the integrand in (2.24) is finite as v 4 1. An analogous issue was discussed by Horgan and Pence [11] in the context of a composite sphere under tensile dead-loading on its boundary (see equation (17) and the Appendix of [11]). Consequently the question of whether or not p" is finite depends on the behavior of W(v) for large values of stretch v. Sufficient conditions to guarantee that pm be finite were given by Ball [3] for both incompressible and compressible materials. Here we provide an ad _gg treatment of this issue. Suppose, for example, that the strain-energy density per unit undeformed volume for a homogeneous incompressible isotropic elastic material can be written in the polynomial form A W(v) - a0 + alv + azv2 + ------ + anvn , (n > 1), (2.26) so that WICV) - a1 + Zaiv + ------ + naavnqfl (2.27) From (2.24), (2.27) we see that p“ will be finite if v < v for large v. (2.28) Thus if n '< 3, (2.29) the value of pct given by (2.24) will be finite. We now consider some specific examples: Example 1. The neo-Hookean material: The strain-energy density function for this material is given by 12 u 2 z 2 W(Al, A2, A3) - 2 (A1 + A2 4- A3 — 3), AIAZA3 - 1, (2.30) where A1, A2, A3, are the principal stretches, and p > 0 is the shear modulus for infinitesimal deformations. By virtue of (2.2), (2.8) and (2.20) we thus have A p -4 2 W(v) — 2 (v + 2v - 3). (2.31) Therefore W(v) » pvz for large v. (2.32) Thus comparing with (2.26), we get n - 2 and so by (2.29), the critical load pm is finite. In fact Ball [3] has shown that p — 5M2. (2.33) or (See also Section 3 of the present work.) Example 2. The Mooney-Rivlin material: The strain-energy density function for this material is u p 1 2 2 2 2 2 2 2 2 2 2 W(Al, A2, A3) - 2 (,\1 + A2 + A3 - 3) + 2 (A112 + A2A3 + A3A1 - 3), AAA - 1, (2.34) 1 2 3 Where A1, A2 and A3 are the principal stretches, and #1, p2 are positive constants. By virtue of (2.2), (2.8), and (2.20) we thus have A u u W(v) - 21 (v- -i 2 a + 2v2 - 3) + (v‘ + 2v’2 — 3) . (2.35) Therefore n V‘ W(v) -» —;— for large v. (2.36) 13 Thus comparing with (2.26), we see that n - 4, and so by (2.29), the critical load p“ is not finite. Of course it is well known that the Mooney-Rivlin model is not a very accurate constitutive model for large stretches (see, for example, Ogden [15] pp. 492-493 for a discussion of biaxial deformation of a rectangular sheet). Example 3. The Rivlin-Saunders material: Experimental work of Rivlin and Saunders [23] suggests consideration of a strain-energy density function of the form pl 2 2 2 2 2 2 2 2 2 W(A,A,A)-—(A +1 +1 —3)+f(AA +AA +AA -3), 1 2 3 2 1 2 3 1 2 2 3 3 1 AlAzA3 - 1, (2.37) where f is, as yet, an unspecified function, with f(O) - 0 and #1 is a positive constant. By virtue of (2.2), (2.8) and (2.20) we thus have I. A #1 - 2 4 -2 W(v) - -§-(v -+ 2v -— 3) + f(v -+ 2v - 3). (2.38) In what follows, we discuss two special forms of (2.38): l. A ”1 - 2 ”2 a -2 a (i) W(v)--2-(v +2v —3)+—2(v +2v -—3) ,a>0,p2>0. (2.39) Clearly the special case a - 1 corresponds to the Mooney-Rivlin material (2.34) considered in Example 2 above. We see that if 4a > 2, (a > 1/2), 4a A [JV W(v) 4 2 for large v. (2.40) On comparing with (2.26), we have n - 4a, and so by (2.29) to ensure that per is finite, we require that 4a < 3, i.e., a < 3/4, and.so p CI‘ is finite for the material (2.39) if 14 NIH blw < a < (2.41) For a S 1/2, W(v) 4 yivz for large v, (2.42) and so pct is again finite on comparing with (2.26) and (2.29). In summary then, for the material (2.39), per is finite if 3 O 1. For a membrane [26], the corresponding result holds only if a > 2. See also the discussion on pp. 281-282 of the book by Libai and Simmonds [27]. (ii) Another special form of (2.37) has been considered by Gent and Thomas (1958) [24], in which f is taken to be the logarithm function. Thus we have A ll _ p _ W(v) - 2—1(v" + 2v2 — 3) +2—21n(v" + 2v2 — 2), p2 > o, (2.44) so that 3 _3 A pl _5 p2 4V' - 4v W1(v) - §-(-4v -+ 4v) + 2 4 _2 . (2.45) v -+ 2v - 2 Thus W1(v) 4 2piv for large v, (2.46) and so from (2.24) we see that the critical load pc is finite for the 1: material (2.44). 3. SOLUTIONS FOR A CLASS OF INCOMPRESSIBLE ELASTIC MATERIALS 3.1 A class of incompressible elastic materials: We now consider a particular constitutive law, namely that of power-law type, and provide an explicit solution for the bifurcation problem discussed generally in Section 2. Thus consider W(A A A) -#— (.\“+.\211 1’ ’ 3 2n 1 2 2n -1 2 +A3-3),A3-(A1A2),p>0,n>0, (3.1) where A1, A2, A3 are the principal stretches, and the constants p, n are constitutive parameters. Constitutive models of the form (3.1) were first introduced by Ogden [l3] and have been widely investigated since then (see e.g. [14], [15]). The constant p in (3.1) is the shear modulus for infinitesimal deformations and n is the hardening exponent. The special case when n - 1 in (3.1) corresponds to the neo-Hookean material. We recall from Section 2 that the critical load pcr:is given by (2.24), i.e., W1(V) - —-—————-dv, (3.2) P cr 1 (V3 _ 1) where the notation (2.20) is used. Expressed in the notation of (2.20), the strain-energy density (3.1) can be written in polynomial 15 16 form as ,. p W(v) - a; (V4.11 + 2v2 n—3),p>0,n>0. (3.3) To ensure the existence of pm“ we recall from (2.29) that 2n should be less than 3, i.e., n < % . (3.4) It is of interest to observe that a restriction similar to (3.4) also arises in the work of Carroll [28] concerned with the problem of inflation of a hollow sphere composed of the material (3.1). The response of the material described by (3.1) to certain basic pure homogeneous deformations will now be discussed. A recent investigation of these issues was carried out by Zee and Sternberg [l6], and we now summarize those results in [16] which are relevant to our problem here. The pure homogeneous deformations considered are as follows: (i) Uniaxial stress: 2n -n -1/2 111 r22 0, 133(A) p (A A ), A3 A, A1 A2 A , (ii) Eguibiaxial stress: _2 > (3.5) Zn -4n 733-0,722(A)-p(A -A ),A1-A2-A,A3-A , (iii) Pure shear: 2n -2n -1 r -0,111(A)-p(A —A ),A1-A2-A,A3-l. 22 J The normal stresses 131(A), 722(A), as well as 153(A), for each of the pure homogeneous deformations (3.5), are monotonic increasing functions of A for 0 < A < m. The stress-stretch relation (3.5) appropriate to (i) (uniaxial stress) is plotted in Figure 1 for the values of the exponent n given by n - 5/4, 1, 3/4, 1/2, and 1/4. 17 (cf. Figure 3 of [16]). Note that the material hardens as n increases. The graphs of 122(A) and.rll(A), corresponding to the cases (ii) and (iii), are qualitatively similar to Figure 1 and are plotted in Figures 2, 3 respectively. It is of interest to remark on the character of the system of governing partial differential equations, namely the displacement equations of equilibrium [16] -1 Cijkl(§)uk,lj — p.3F31 - O, J - det F - 1, (3.6) where CL“1(F) are the components of the fourth-order tensor defined by 2 W(g) C1jk1(§) - Ck113(§) - 3F 3F (3.7) 1:] RI Necessary and sufficient conditions for ellipticity of the system of equations (3.6), (3.7) have been obtained by Zee and Sternberg in [16]. For the special case of the material (3.1), these conditions are particularly simple. Thus from the results of [16], p.85, ellipticity holds for the material (3.1) t all deformations if :5 IV mlr—I (3.8) In what follows, we assume that (3.8) holds, and so recalling (3.4), we thus have (3.9) NIH IA :3 A NICO 18 3.2 gavitation solutions: Consider a quasi-static loading process in which the solid sphere is subjected to a dead-load po that increases slowly from zero. Cavity formation and growth is described by the relationship po - po(c) given in (2.22). For the material described by (3.1) (recalling the notation (2.2), (2.8)) we have A1 - v-2, A - A3 - v and so the first derivative 2 with respect to A2 is O, i.e., w2(¢1, v, v) - 0, and the first derivative with respect to A1 is given by -.n+2 _ V2142), is n < 2 . (3.10) '2 Vi CV . V. V) ' #(V 2 On using the notation (2.20), we thus obtain W1(v) - 2,.(v2“" — W“), %s n < % , (3.11) and so W (V) v2n-1 _ -4n-1 1 3 - 2p , - S n < — . (3.12) 3 2 2 v - l v — 1 When the relationship (2.22) between the applied pressure po and deformed cavity radius c is specialized to the particular strain- energy function (3.1) (and (3.12) is used), one obtains Zn-l -4n-1 (:3 2,3 v - v p0 - po(c) - Zp (1 + j) dv, b 3 v - l NIH le Sn< (3.13) Before proceeding with an analysis of the relationship (3.13), it is convenient to record here corresponding expressions for the stresses subsequent to cavitation given by (2.7). On using (2.17), (2.18), 19 (2.20), (2.21) we find wlm rRR(r) - —3—— (1V, (3.14) c3143 v -— 1 (1 + —5> r while from (2.7) we obtain -2 - -2 769 - r“ - vW2(v , v, v) - v2W1(v , v, v) + rRR(r). (3.15) On using Wztui, v, v) - 0, (3.10) and (3.12) we obtain Zn-l -4n-1 'V - 1 3 Ga“) "' 2" 3 dv. -2- s n < -2-. (3.16) c3 ya v - 1 (1 + ‘3) r' and Tee ' 'qxp ' Tm“) - #4 Iv"“ - v2"3’3 — TRR(r) - 2p — 1r - 3 1n 3 1/3 3/3 (1 + C 3/ r) - 1 - c3 1/6 1 c3 1/6 c3 1/3 +§1n 1+(1+—3—) —31n 1—(1+-—3) +(1+—3—) _ r r r P 3 - 3 1/6 +%1n l + (1 “PS—QUE + (1 +£§)1/3 - A arctan 2:1“: 3/ r) _ _ r r . J3 J3 _ 3 3 3 1/6 — % 1n 1 + (l +6—3)1/3 + (l +3302” - -—1 arctan 2<1+C 3/ r) +1 _ r r _ ./3 ./3 1 c3 -5/3 1 c -2/3 +§(1+—;) +§(1+3) r r 3 3 1/3 - —1 arctan J30" + c 3Lr3)1 3 , (3.28) J3 2 + (1 + c 3/ r) ’ 23 3 3 (1 +3393” — (1 +33)“ . (3.29) ’99“) - rmu) - rm(r) - u r r The graphs of th(r), 199(r) and 10¢(r) corresponding to (3.24)- (3.29), i.e. for values of n - 3/4, 1, 5/4, are shown in Figures 5-10. An interesting feature concerning these stresses immediately after cavitation is the presence of a boundary layer near the cavity wall. To see this, we have plotted the stresses in Figures 5-10 for applied dead loads p0 slightly larger than pug A.similar boundary-layer phenomenon was observed in [11] for the problem of tensile dead- loading of a composite sphere composed of two neo-Hookean materials. 4. ELASTODYNAMIC PROBLEM FOR A NEO-HOOKEAN SOLID SPHERE SUBJECTED TO A SUDDENLY APPLIED DEAD-LOAD 4.1 Formulation: In this Section, we consider the radially symmetric motion of an isotropic incompressible elastic solid sphere composed of a neo- Hookean material. The undeformed sphere has radius b, and it is set into motion at time t - O by a suddenly applied uniform radial tensile dead-load p0. In this incompressible case, the effect of the tensile load is felt immediately throughout the medium, and the response takes the form of a nonlinear oscillation. Large amplitude oscillations of hollow incompressible elastic cylinders were first considered by Knowles [17,18]. Methods similar to those used in [17] and [18] have been applied to the case of symmetric motions of a hollow thick-walled incompressible elastic sphere in [19], and an unbounded incompressible elastic medium containing a spherical cavity has been treated in [20]. See [21] for a review of some of this work. See also the recent paper [22] for a treatment, using phase-plane arguments, of radial motion of thick spherical shells composed of incompressible materials. The emphasis in [17-20] is on the characteristics of the motion, such as the period and amplitude, and on conditions which will ensure 24 ’ 25 the existence of periodic motions. In this Section we use the techniques developed in [17-20] to investigate the dynamic analog of the bifurcation problem described in Sections 2 and 3. For simplicity of presentation we restrict our attention to the case of a neo-Hookean material. We use similar notation to that introduced in Section 2. Thus a point which at time t has spherical coordinates (R,6,¢) is assumed to have been at the point (r,0,¢) in the undeformed state. The motion is thus described by R - R(r,t) > 0, 0 < r < b; R(0+,t) z 0; 9 - 0, w - ¢, (4.1) where R(r,t) is to be determined. Since the material is assumed to be incompressible, the deformation gradient F obeys det F - l, t 2 0. For the motion (4.1), this implies Rz'DR/)r - r2, which when integrated gives R - R(r,t) - [r3 + c3(c)]3’3, c(t) z o, c 2 o, (4.2) where c(t) is to be determined. The motion is completely determined once c(t) is known. If it is found that c(t) - 0 for t 2 0, (4.2) implies that the body remains a solid sphere in the current configuration. On the other hand if c(t) > O (i.e. R(O+, t) > 0), there is a cavity of radius c(t) centered at the origin in the current configuration. In this event, the cavity surface is assumed to be traction-free. For the neo-Hookean material, the strain energy density per unit undeformed volume is given by g g 2 2 2 _ = W(AI,AZ,>.3) 2 (A1 + 32 + 13 3), 313233 1, (4.3) where A1 (i - 1,2,3) are the principal stretches, and p > 0 is the 26 shear modulus for infinitesimal deformations. For the radially symmetric motion (4.1), the principal stretches are given by Ar -?R(r,t)/>r, A0 - A4, - R(r,t)/r. The principal components of the Cauchy stress tensor 1 are again given by (2.6) which for the material (4.3) and the motion (4.1) can be written as 3 3 4/3 TRR(R,C) =3 [I (R — C) — P(Rot)v R2 (R3 _ C 3)2/3 799(R,t) = TQ¢(R,C) - p - P(R,t), t 2 O, (4.4) where P(R,t) represents the arbitrary hydrostatic pressure. It is assumed that the sphere is in an undeformed state and at rest at time t - 0, so that R(r,0) - r, fi(r,0) - O, and so from (4.2) we deduce that the current cavity radius c(t) must satisfy the initial conditions c(O) - o, é(0) - o, (4.5) where the dot denotes differentiation with respect to time. A dead-load po is suddenly applied and maintained at the surface of the sphere so that the boundary conditions are rRR (A,t) - 0, t < 0, (4.6) b 2 rRR(A.t)-PO(A). tZO. where po is a positive constant and A - R(b,t) - (b3 + c(t)3}1/3 is the deformed outer radius. In addition if c(t) > 0, then the condition for a traction-free cavity surface rRR(c,t) — O, t 2 O, (4.7) must also hold. 27 The equations of motion, in the absence of body force, governing the radially symmetric motion of the sphere reduce to the single equation QT“ +l(2r -‘r 211 R an 99 — 1w) - pli, t Z 0, (4-8) where p is the constant mass density of the material. Thus the fm problem to be solved is the following: For a prescribed value dead-load traction p0 > 0, pg seek a pressure field P(R,t), and a time dependent function c(t) 2 0, such that (4.2), (4.5), (4.8) app (4.6) are satisfied where rRR, Tee, r¢° are givep p1 (4.4). 1p addition if c(t) > 0, then (4.7) mus; also pg satisfied. 4.2 Solutions: It is readily shown that one solution to the foregoing problem, for all values of po, is P(R,t) - p - po , c(t) - O, t a O. (4.9) This corresponds to the trivial homogeneous (static) state of deformation R(r,t) - r, t 2 O, (4.10) with corresponding stresses th*- r - r - p 69 ¢Q 0' Next we describe solutions for which c(t) > 0, corresponding to the presence of a traction-free cavity at the origin. On substitution from (4.4) into (4.8) we obtain 28 3 3 43 3 :34/3 2 Z (R—c)’ _P(Rt)+gg (R c) _ R =.. ‘DR. p 4 ’ R 4 3 3 2m p R R (R. — c ) (4.11) The incompressibility condition (4.2) is now used to compute the acceleration dzR/dt2 in terms of the acceleration d2c(t)/dt2 of particles on the cavity surface, so that we have 2 32—3: - 2cR 3(R3 c.3)(d—‘3)2 + c2 R‘2 192-. (4.12) dt dt Equation (4.12) is now introduced into the right hand side of (4.11) to yield -2 3 _ 3‘u3 3 _ 3‘w3 2 ?_R p (R ‘c) _ P(R,t) +%g (R ‘0) _ a R 3 m R R (R. - c ) -5 3 do 2 2 -2 dzc - 2pcR (R. H)( t) + pc R. -—a (4.13) 2 dt Equation (4.13) is now integrated with respect to R, to yield 3 3 “3 3 3 ya u (R ‘3) — P(R,t) + P(c,t) + 2p [35 5°) - 3 g 3 2/3 ]d§ R c e (e - c) = 2pc(d—c 2 55——L d5 + pc2_ ‘32 ° 95 . (4.14) 2 dt 53 C The integral on the left hand side of (4.14) may be simplified, on integration by parts, to yield 29 3 3 4/3 c ) _ e3 (e (E 5 deg (4.15) 3 3 23 _ c )/ 3 3 4/3 3 _ 3 1/3 - ”(R "‘33 + 2;: 43 2°) d4 — 24 ((53 — c3) 2R 6 2/3 d6. The first integral on the right hand side of (4.15) is also simplified on integrating by parts to yield 3 31J3 § 3 32u3 (g - C) dé _ _ 2H M + 2" 2 R 3 3 5 (€ - C) C C 2” 2/3 (13' (4.16) Thus on combining (4.15), (4.16) and evaluating the integrals on the right hand side of (4.14) directly, we rewrite (4.14) as follows: 3 3 4/3 p 43 'f) -— P(R,t) R 3 3 4/3 3 3 1/3 - — P(c,t) + 2;; (R "°) + (R I; ‘3) 4R do 2 c3 1 3 2 dzc 1 1 + 2pc(—-) — -—+— +pc —-— . (4.17) dt 4R3 R. 4c dtz c R Equation (4.17) is now introduced into the right hand side of the first of (4.4), and then the traction-free cavity surface condition (4.7) is imposed. This leads to P(c,t) - O, t 2 0, and so we obtain 3 3 “a 3 3113 rRR(R,t) - 2p (R ‘C ) + (R E c ) 4R dc 2 c3 1 3 2 dzc 1 1 + 2pc(—- '—- — - + —- + pc - - - . (4.18) dt 4R3 R. 4c dt2 c R Finally the boundary condition (4.6) at R - A - {b3 + c(t)3} is 30 satisfied if b 3 b" b pO 3 3 1/3 - 2p 3 3 4/3 + 3 3 1/3 (b + c ) 4(b + c ) (b + c ) 3 dc 2 c 1 #1 + 2pc(dt 3 3 4/3 - 3 3 1/3 + 4c 4(b + c) (b + c) 2 dzc 1 1 + pc 2 - - 3 3 13 , t Z 0. (4.19) dt c (b + c )’ The relationship (4.19) between the applied load po and cavity radius c(t) is the dynamic counterpart of (2.22), for the neo-Hookean material. In fact, on formally replacing c(t) in (4.19) by the constant c, it is readily verified that one recovers (3.20). 4.3 The basic differential equation: To treat the differential equation (4.19), we adopt the techniques of Knowles [17] and consider the quantity x(t) - C—é—Q > o, (4.20) where b is the original undeformed radius of the solid sphere. In this notation (4.19) becomes a nonlinear second-order ordinary differential equation for the dimensionless cavity radius x(t). From (4.20) we have c(t)-bx(t), fi-bfi, —--b-—. (4.21) On introducing the notation 31 f(x) - —ZA[ 1 + 3 ] , (4.22) pb2 4(1 + x3)"3 (1 + :3)” and using (4.21), we rewrite (4.19) as Po - x2 c13_x l _ 1 pb3(1 + x3)3’3 dtz 3‘ (1 + x3)3’3 dx 2 x3 1 3 + 2x (—) - + — + f(x), t 2 o. (4.23) dt 4(1 + x3)4/3 (1 + x3)1/3 4x Since the motion starts when the sphere is undeformed and at rest (see equation (4.5)), we deduce from (4.5), (4.20), (4.21) that the initial conditions x(0) - o, ggég) - o, (4.24) must also hold. 4.4 Oscillations: With the notation v - dx/dt, dzx/dtz - v dv/dx, it is possible to write the differential equation (4.23) in the form 2 2x1p d—: x3 34-- 1 3 13 v2 + 2x2 f(x) '3 2 :23' (4°25) u+x)’ pba+x)’ Using (4.22), we find that (4.25) may be integrated with respect to x over the interval from zero to x to yield x4 _1__ 1 V2+_4,£ (1+x3)2/3 _ 1 _l x (1 + x3)1/3 pbz 2 4(1 + x3)1/3 4 2pO 3 13 -—[(1+x)l—l],t20. (4.26) pbz 32 It is well known from the theory of vibrations that the motion x(t) is periodic if and only if the 'energy curves' (4.26) are closed curves in the x-v plane with a finite period f dx/v. The energy curve in the x-v plane is symmetric about the x-axis. This curve, given by (4.26), starts at the initial point x - 0, v - 0 at time t - 0. If p0 is sufficiently large to produce an internal cavity, x and v then move into the region x > 0, v > 0 as t increases from zero. If v passes through a maximum and returns to zero as x increases from zero, the curve will be closed. According to (4.26), this will happen for a given po if there is a root x > O of (4.26) when v = 0. Setting v - O in (4.26) we obtain P _g_ 3 u3‘_ -1 3 2n _ 1 [(1+x) l] (1+x) 31/3 " 2(l+x) l - 5 . (4.27) The right hand side of (4.27) is a monotone increasing function of x for x > 0. As x 4 0+ in (4.27), we find, using 1'H6pita1's rule, that 5 -—- 4 5+ . (4.28) For a given p0 > Sp/Z, we denote by x.In the non-zero root of (4.27) (there is nnly nng since the right hand side of (4.27) is monotonic increasing). The quantity x.m is the maximum cavity radius in the oscillation process. If po'< 5p/2, no positive root of (4.27) exists, and hence periodic motions do not occur for this range of applied tensile loads. Thus we have shown that the value of the "critical load" n; which an internal cavity nny bg initiated in the dynamic nroblem coincides with that for the static problem. (Recall equation 33 (2.33)). Thus following application of a pressure p0 > Sp/2, an internal cavity would form and expand until it would reach the value xi, given by the root of (4.27), then would contract to zero and repeat the cycle. It is of interest to note that Knowles and Jakub [20] found that no periodic motions exist for values of pressure above 5p/2 for the problem of an unbounded solid, composed of a neo-Hookean material, containing a spherical cavity which is set into motion by the sudden application of a spatially uniform radial pressure to the cavity wall. In fact, for this problem, the deformed cavity radius tends to infinity as the applied pressure tends to the value Sp/2. A related observation was made by Gent and Lindley [2] and by Ball [3] for the corresponding static problems. 34 n = 1.25 N1 Figure 1. Behavior of the power law material under uniaxial stress . 35 122(k) Gad n'=‘LO Figure 2. Behavior of the power law material under equibiaxial stress. 36 n =1.25 n = 1.0 n = 0.75 n = 0.5 n = 0.25 T 2 3 Figure 3. Behavior of the power law material under pure shear. 37 2.5. n - 0.5 2.0-‘ 1.5.. 1 .0~ Figure 4. Variation of the deformed cavity radius c with applied dead load po for a power law material with strain energy density given by (3.1). 38 1.0002 PC, , 1.002 PC, 0-33 1.02 P CI CI 0.44 0.0 ’ r 1 1 fi 1 ' I ' I ’ 0.0 0.2 0.4 0.8 0.8 1 .0 Figure 5. Variation of the radial stress rfiR(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 3/4. O‘IH 39 1 1.002 P“ 2.0-1 1 1.02 Per '1 ‘ cr Figure 6. Variation of the radial stress 323(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - l. 40 4 0“ 1.002 P“ 1.02 PC, 3.0" 1'1 Per 1.0 , 0'01 i 1 1 l j I ' r ' I, 0.0 0.2 0.4 0.6 0.8 1.0 O‘IN Figure 7. Variation of the radial stress rRR(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 5/4. 41 Figure 8. Variation of the stresses 799(r), 1QQ(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 3/4. 42 1.002 PC, . Pct 2 1 9.002 1 I 1 ' l ‘ l ' l 0.0 0.2 0.4 0.6 0.8 1 .0 Figure 9. Variation of the stresses 199(r), 7Q¢(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - l. 43 5- 4; 1.0202 P c]: 1 ' 1 1 r fi’ I ’ 0.0 0.2 0.4 . Figure 10. Variation of the stresses 166(r), rQQ(r) with undeformed radius r subsequent to cavitation for a power law material (3.1) with n - 5/4. Appendices Appendix A: Verification of (2.25) Equation (2.25) has been established recently by Horgan and Pence [11]. For completeness here, we provide a brief review of their argument. First we recall from (2.20) that 1300 - W(v’3. v. v). (M) and so dW(v) dv - —2v'3w1(v‘3, v, v) + 2w2(v‘3, v, v), (11.2) on using the chain rule and the fact that W2024,v,v) = WSCVq,v,v). Thus dW(1) dv - 2[w2(13 1) 1) _ W1(1, 1’ 1)] - 0, (A°3) which establishes (2.25)1 as desired. To verify (2.25)2, we recall from finite elasticity theory (see e.g. Ogden [15]) that the shear modulus for infinitesimal deformations of an incompressible homogeneous isotropic material with strain-energy density 0(11, 12) is given by 30 30 ”'2571'33—111 -I -3. (3'3) 1 2 1 2 Here I - A2 + A2 + A2, I - AZAZ + AZAZ + AZAZ, are the usual first and 1 1 2 3 2 1 2 2 3 3 1 second invariants. Thus from (A.1) we have 44 45 131m - €1>. where I1(v) - v“ + 2V2, 12(v) - 2v’2 + v". Using the chain-rule, and observing that oil dIz d—v--‘aV—-O, whenv= 1, it is readily verified that (1314(1) of: (1311 3121 031' - -—- + ——-—-- I - I = 3 y 9 dvz 3331 dv2 Iz dv2 3 3 v - 1, and so it follows from (A.4), on using (A 6), that d2W(1) - 12 u. dv2 which establishes (2.25)z as desired. (A.5) (A.6) (A.7) (A.8) (A.9) Appendix B. Verification of (3.18) - (3.21) and (3.22), (3.24), (3.26), (3.23) Here we present the details of the derivation of equations (3.18) - (3.21) and (3.22), (3.24), (3.26), (3.28). We first treat the indefinite integral which is needed to evaluate both (3.13) and (3.16): (Constants of integration will not be written down.) V211—1 _ v-4n-1 I = 3 dv, n < . (B.l) NIH IA le It is convenient to record here the values of Zn - 1 and —4n - 1 corresponding to n - 1/2, 3/4, 1, 5/4, respectively. The integral (B.l) will be decomposed into the two parts involving these exponents. n 1/2 3/4 1 5/4 2n — 1 0 1/2 1 3/2 -4n — 1 -3 —4 —5 —6 (1) Evaluation of I for n - 1/2. When n - 1/2, —3 -3 3 I-JJJ—V—dv-Jv(:-l) dv-Jv33dv--;. (8.2) v --1 46 47 immediately evaluated to yield the desired expressions (3.18), (3.22). (ii) Evaluation of I for n - 3/4. First, we record here the indefinite integrals (2.128) of Ryshik and Gradstein [29], Jdv =_ 1 _b(3£+k—4)J dv ,k#1’ V'Z (k — 1)avkolz:-1 a(k - 1) (8.3) where 23 - a + bva, a # 0, b and i > 0 are constants. When n - 3/4, from (8.1) we see that I=J—°LV—— dv-J ‘3" =11—I. (3.4) (v3 — l) v3(v3 - 1) 2 To evaluate 12, we use (8.3) with k - 4, a - —1, b - l, 2 - 1, and get I.-[ .2 a] :v . v(v —l) 3v v(v '1) The second integral of (8.5) is evaluated as follows: dv v2 dv v2 v2 3 = 3 3 - - (-—; ) dv + 3 dv ‘v(v - l) v (v — l) v (v - 1) 3 --llnv3+-1-ln(v3—1)-—lln-v—-, (8.6) 3 3 3 3 v -— 1 and so, from (8.5), we have 3 1 1 v 12 - 3 + 3 1n 3 . (8.7) 48 In order to evaluate I1 in (8.4) we use a change of variables, i.e., 3/2 r - v , and so ,[v dv 243 dr 1 r - 1 1 v3’2 — 1 I - - - - 1n --- - - ln -————-—-. (8.8) 1 3 2 3 r + 1 3 we v -— 1 r 1- 1 v + 1 Thus on combining (8.7) and (8.8) in (8.4) we obtain an expression for I. The definite integrals in (3.13) and (3.16) are then immediately evaluated to yield the desired expressions (3.19) and (3.24). (iii) Evaluation of I for n = l. When n - 1, -5 -5 6 I- ——V'V dv- 3’ 5"”13dv- v’3(v3+1)dv 3 3 v -1 v -1 -J (v'2 + v‘3) dv = - l - —1. (13.9) V 4 4v 0n using (8.9), the definite integrals in (3.13), (3.16) are immediately evaluated to yield the desired expressions (3.20), (3.26). (iv) Evaluation of I for n - 5/4. When n - 5/4, I -J VJ" 3" -J ‘3" =- 1 - 1‘. (3.10) v3 - 1 v6(v3 - l) 3 49 To evaluate I‘, we use (8.3) with k - 6, a = —l, b = 1, 2 = 1, to get I -—1— + 3 ‘3: . (3.11) V'(V - 1) The integral in (8.11) is evaluated by using (8.3) with k = 3, a = —1, b - 1, 2 - 1, to get 3 ‘33" =i2 + ——33V—. (3.12) V'(V -— 1) 2v v — 1 The last integral in (8.12) can be evaluated using standard integral tables. For example, (2.143) of Ryshik and Gradstein [29] gives dv 1 (1 3 V + ”231/2 1 ,[3v -——-—-——-- — - 1n — - arctan (8.13) v3 _ 1 3 v — 1 J3 2 + v Thus, on using (8.13), (8.12), (8.11), we obtain 2 uz (l + v + v) I = -l- + -l- - l 1n — —l arctan -—1§X—- . (8.14) 4 5v5 2v2 3 v — 1 J3 2 + v In order to evaluate I3, we use a change of variable, i.e., r - v , to get h 13 _ Viv dV' _ :‘dr - :dr + 3rdr . (B 15) v «— l r — 1 r + 1 r — 1 By using (2.145.3) and (2.145.7) of [29], we have 2 I - - 1 1n (1 + JV) + -l arctan 212—:—l 3 6 1 — Jv + v J3 J3 2 + 1 1n (JV - 1) + —l arctan 2 v + l. (8.16) l+Jv+v J3 J3 Thus on combining (8.14) and (8.16), we obtain an expression for I from (8.10). The definite integrals in (3.13) and (3.16) are then readily evaluated to obtain the desired expressions (3.21) and (3.28). List of References References: l. S.H. Goods and L.M. Brown, The nucleation of cavities by plastic deformation. Acta Met,, 22 (1979), 1-15. A.N. Gent and P.8. 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