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Lani L: :39 211.1%. .. ....x.1~..¥....v.u|flfl.4:.1111}... ...... 3...... 1.11.5.5. i“ 0‘ éfgfiliz\‘kfli y [ngJv’wfilI’sXI‘HJf ou‘v90oov‘cvflV1Id..cVnI¢o.OvaHt§ou1n 1.121 1. t . . r)..0.0n . W r—w—w-mm- .... EHEE‘E LIERAERY Hz f Michigangtmae 1 Univemitx I" w This is to certify that the dissertation entitled Some TupiCS 1n Finite Elasticity presented by Abdol Hossein Jafari has been accepted towards fulfillment of the requirements for ‘ Ph . D . degree in Mechanics Date May 19, 1983 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES m P ‘t- 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. SOME TOPICS IN FINITE ELASTICITY By Abdol Hossein Jafari 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 1983 M" a...) /3 7—— é'erE/E ABSTRACT SOME TOPICS IN FINITE ELASTICITY By Abdol Hossein Jafari This dissertation consists of two parts, both concerned with the investigation of problems in the theory of finite elastostatics. In Part I an analytic approach for obtaining bounds on stress concentration factors in the theory of finite antigplane shear is presented. The problem of an infinite slab with a traction free 'elliptical cavity subjected to a remotely applied finite simple shear deformation is considered. It is assumed that the slab is composed of a homogeneous incompressible elastic material. Explicit estimates are obtained for the stress concentration factor in terms of the dimensions of the cavity, the applied stress and the constitutive parameters. The limiting cases in which the cavity is circular or crack-shaped are also examined. The analysis is based on the application of maximum principles for second-order uniformly elliptic quasilinear partial differential equations. In Part II the finite plane strain deformation of a circular tube of homogeneous compressible elastic material of harmonic type, subjected to simultaneous internal and external pressure, is considered. Explicit closed form solutions for the deformation and stress fields are obtained. The true stress distribution, expressed in terms of undeformed coordinates, is shown to be essentially independent of Abdol Hossein Jafari material properties. The two cases of internal pressure only, and external pressure only, are examined in detail. In the former case there is a finite value of the applied pressure at which the maximum hoop stress in the tube, occurring at the inner surface, becomes unbounded. For the case of external pressure a finite value of the applied pressure exists for which the cavity closes. Furthermore the stability of the equilibrium in the two Special cases described above is investigated by employing a standard perturbation expansion. It is found that an internally pressurized tube is always stable whereas an externally pressurized tube buckles at a certain value'of pressure. In the latter case the smallest buckling load is calculated and the existence of buckling loads corresponding to higher modes established. ACKNOWLEDGEMENTS I should like to express my sincere gratitude to my advisors. Professors Rohan C. Abeyaratne and Cornelius 0. Morgan for their help and support during all stages of this investigation. No graduate student could have hoped for more understanding, patient and generous advisors. Grateful thanks are extended to the other members of the guidance committee, Professors David L. Sikarskie and David H. Y. Yen. Thanks are also due to Mrs. Martha Flores for typing the manuscript at a very short notice. Nhile preparing this dissertation, I held Teaching Assistantships awarded by the Department of Metallurgy, Mechanics and Materials Science and Research Assistantships supported jointly by the Division’ of Engineering Research, Michigan State University, and the National Science Foundation under grants CME 81-0658l and MEA 78-2607l. The support of these institutions is gratefully acknowledged. ii TABLE OF CONTENTS LIST OF FIGURES .................................... PART I: ESTIMATES FOR STRESS CONCENTRATION FACTORS IN FINITE ANTI-PLANE SHEAR INTRODUCTION .............................. STATEMENT OF THE PROBLEM .................. 2.l Displacement Formulation ............. 2.2 Reformation in Terms of Stress Function ............................. COMPARISON THEOREMS ....................... COMPARISON FUNCTION ....................... RESULTS ................................... 5.1 The Load Independent Lower Bound ..... 5.2 The Load Dependent Lower Bound; Results for a Ramberg-Osgood Material. 5.3 Limiting Results for a Thin Ellipse... CONCLUDING REMARKS ........................ 6.l Upper Bound for a Softening Material.. 6.2 Upper Bound for a Hardening Material.. 6.3 Suggestions for Further Work ......... APPENDIX A ......................................... REFERENCES .......................................... PART II: DEFORMATION AND STABILITY OF A PRESSURIZED TUBE FOR HARMONIC MATERIALS INTRODUCTION .............................. THE PRESSURIZED CYLINDRICAL TUBE; HARMONIC MATERIALS ................................. 2.l Statement of Problem ................. 2.2 Harmonic Materials ................... 2.3 Deformation and Stress Fields ........ iii Page IO 13 15 23 23 24 3O 31 31 33 35 36 37 4O 4O 42 Page 3. GEOMETRICALLY PERTURBED PROBLEM .......... 48 3.1 Deformation and Stress Fields ....... 48 3.2 Pressure Boundary Conditions on a Perturbed Surface ................... Sl 3.3 Special Cases ..... .................. 55 4- STABILITY ......................... . ....... 58 4.1 Solution of the Equilibrium Equations 58 4.2 Buckling of an Internally Pressurized Tube ................................ 60 4.3 Buckling of an Externally Pressurized Tube ...... . ......................... 67 5. ILLUSTRATIVE EXAMPLE ..................... 78 6. CONCLUDING REMARKS ....................... 82 APPENDIX A ........................................ 84 REFERENCES ........................................ 86 iv Figure LIST OF FIGURES Cross-section of body, with cavity coordinates and boundary conditions ..... . ......... . ...... Lower bound for the stress concentration factor versus the applied stress at infinity for different ellipses ....................... Lower bound for the stress Concentration factor versus the applied stress at infinity for different ellipses ....................... Lower bound for the stress concentration factor Versus the applied stress at infinity for different ellipses ............... . ....... Lower bound for the Stress Concentration factor for a circular cavity versus the applied Stress at infinity ................... Geometry and coordinate System for the perturbed boundary ............. . ............. External pressure versus B/b for different materials .................................... External pressure versus B/b for different tubes ........................................ Page 26 27 28 29 53 80 Bl PART I ESTIMATES FOR STRESS CONCENTRATION FACTORS IN FINITE ANTI-PLANE SHEAR I. INTRODUCTION Qualitative methods have been used in linear elasticity for a long time (see e.g.Villaggio (1977), Morgan 0982) and references cited therein). The objective of such studies is to find information about the solution of boundary-value problems without actually solving them. The desired results are generally in the form of a priori bounds for fieid quantities_in terms of geometric, constitutive and boundary data. Analogous results in nonlinear elasticity are rare. Estimates of this type are especially important in the finite theory where exact solutionsare seldom available. In addition to their inherent importance. such results are of value as guides in computational analyses. Pointwise stress estimates are particularly important in problems involving stress concentration where localized stresses are of primary concern. In the present study, following on recent results of Abeyaratne and Morgan (1983), we shall consider the application of a priori estimation techniques to a stress concentration problem arising in finite elasticity theory. He confine attention to the simplest possible setting within - the exact theory of finite elasticity:. finite anti-plane shear of an infinitely long cylinder composed of a homogeneous, isotropic, incompressible material. Such deformations have been extensively studied by Knowles (I976, l977) and others. Nhile of less practical interest than their analogs in plane stress or plane strain, these problems are much simpler to analyze analYtically and serve a useful role as pilot problems. We are concerned with the stress concentration arising in the problem of an infinite slab with a traction-free elliptic cavity subject to a state of finite simple shear deformation at infinity. A cross-section of the slab is shown in Figure l. The constitutive law is assumed to belong to a special class of such laws for which nontrivial states of finite anti-plane shear do indeed exist. The analogous problem for a circular cavity was treated recently by Abeyaratne and Morgan (l983). One of the motivations for the present study was to extend their techniques to the elliptical cavity problem, with particular interest in the limiting case modelling a straight crack. When the results of the present investigation are Specialized to the case of a circular cavity. the bounds obtained are sharper than those found by Abeyaratne and Morgan (1983). The boundary-value problem is formulated in Section 2. The maximum shearing stress, of principal interest here, is known to occur on the boundary of the cavity. Our purpose is to provide a means for estimating this quantity. The main results necessary for this task are given in Section 3. In Sections 4 and 5 these results are applied to find explicit bounds on the stress concentration factor for a wide class of materials in terms of the geometry, load and constitutive parameters. The results are illustrated for a particular constitutive law. We conclude with some general remarks in Section 6. III II [alli llli ll 1 III ll I III ‘I .nfl 2. STATEMENT OF THE PROBLEM 2.1 Displacement Formulation Let the three dimensional open region R be the exterior of an infinitely long right elliptical cylinder with semimajor axis A and semiminor axis 8. Suppose that this open region is occupied by the interior of a body in its undeformed configuration- Choose the rectangular cartesian coordinates (x1, x2, x3) with the x3-axis parallel to the generator of the cylinder and the origin at the center. Let D be the cross section of R in the plane x3 = O, and denote by F the boundary of the ellipticalfcavity (Figure 1). Suppose now that the body is subjected at infinity to a simple shear parallel to the (x1, x3) plane. The ensuing deformation maps a point with position vector x in the undeformed configuration to a point with the position vector y: y = 5 + g (x) on R. (2.1) The components of the displacement field are assumed to satisfy (1) u = 0, U3 = kco x2 as xa xdf w , (2.2) where kwi>0) is the amount of applied shear. The deformed surface of the cavity is assumed to be traction-free. (l) The components of all vectors and tensors are taken with respect to the fixed rectangular coordinate system previously chosen. Greek sub, scripts have range (1.2) and summation convention is assumed throughout. A subscript preceded by a comma indicates partial differentiation with respect to the corresponding x-coordinate. i“ u = kwx2 225.. as (x1 + x2) + ZB 2A 7T- Cross-section of body, with cavity, coordinates Figure l. and boundary conditions. Suppose that the body is composed of a homogeneous, isotropic, incompressible elastic material with a strain-energy density function w. Denoting by I], 12, and I3 the fundamental invariants of the left (orrfight) Cauchy-Green deformation tensor we have I1 = I2 = 3 in the undeformed state and I1 3 3, 12 33 for all deformations. Since only locally volume preserving deformations are admissible I3 = l. The elastic potential N depends in general on I1 and 12, w = R(Il, 12). For reasons that will become_apparent later it is convenient to confine attention to the restriction of H(I], 12) to the line I1 = 12 (= I) and define N(I) by w = w(1) = fill, I), 133, w(3) = o, (2.3) where w is assumed to be twice continously differentiable for 133. The reSponse of this material in simple shear is described by i (k) = ka' (3+k2) , - w < k < m, , (2.4) where $(k) is the shear stress associated with an amount of shear k, and prime denotes differentiation with respect to the argument. The (secant) modulus of shear is now given by M(k) = 3%;9.= 2N'(3+k2) (>0). (2.5) In order to satisfy the Baker-Ericksen inequality for the material under consideration we will assume that M(k)>O. (At infinitesimal deformations, we have from (2.5), M(O) = ZN'(3) which we will denote by u: the shear modulus.) Following Knowles (1977) such a material is said to be softening in shear if M'(k) <0 and hardening if M'(k) >0: k 3E'.(k)< $(k) (softening), (2 6) k 2' (k):> $(k) (hardening). . Knowles has shown that for a certain class of materials, the field equations and boundary conditions associated with the problem described above are consistent with the assumption that "a = 0, U3 = u(x], x2) on R, (2.7) corresponding to a state of_anti:plane shear; Two points should be noted. First. forall such deformations I1 = I2 (=3+|Vu|2). Secondly a material governed by an arbitrary strain-energy density function R (1], 12) cannot sustain a nontrivial state of anti-plane shear. The entire class of materials which admit such a deformation has been determined by Knowles (1976) and it is only these materials that we consider here. (An example of such a material is the familiar neo-Hookean material with the elastic potential N= %u(I]-3), u>0-) The governing problem can then be shown, Knowles (1976, 1977), to reduce to the following two dimensional problem for u: div [N'(I) grad u] = O on D, (2.8) with I = 3+|Vu|2 , Vu = grad u, (2.9) u(x], x2) = k0° x2 as xa xd+ m, (2.10) Bu _ -—- - O on P (2.11) an 9 where Bu/an denotes the outward normal derivative of u on P. The corresponding components of Cauchy stress 11.3. are given by = 2w'(I)U,a a (2.12) T0L3 T3o. . 2 TaB = o, r33 = 2w (I)|Vu| . (2.13) Since we have assumed that M(k)>0 it can easily be verified that the quasilinear partial differential equation (2.8) is elliptic at a solution u and at a point (x1, x2) if and only if ¥'(k)>o , k =|Vu| , (2.14) where T (k) is given by (2.4) and the prime denotes differentiation. We shall impose a slightly stronger requirement: we assume that i(k) satisfies b? (k)3k ¥'(k)3 c $(k) for all k 3 o. (2.15) for some positive constants b and c. The right hand side of (2.15) together with (2.5) assures that (2.8) is uniformly elliptic (see Gilbargnand Trudinger (1.977), P. 203) and implies in particular that ;'(k)>0 for all k as well as ;(m)= m» It follows that when (2.5) and (2.15)h01d. T = ¥(k) can be inverted to give k as an odd, monotone strictly increasing function of r: k = k(t) with k(m) e m; It will be seen later that the left hand side of (2.15) is equivalent to a uniform ellipticity assumption foraadifferential equation related to (2.8). Henceforth the ellipticity constants b and c are taken to be the smallest and largest constants respectively for which (2.15) holds. In view of (2.6) we note that a softening material automatically satisfies the left inequality of (2.15) with b = 1, while a hardening material conforms to the right one with c = 1. Consequently, in the following we have Softening: t(k) > k;'(k) c i(k) 0 . .. I (2.16) Hardening: bt(k) 3 kt'(k) >t(k) b>1. The final results derived subsequently will be given in terms of the constitutive functions m(s) and n(s) which we define for all 530 in terms of the response function ¥(k) by m(s) = max ( A T - 1) , (s>0), (2.17) Oftfs kt'(k) n(S) =02; (TZEITTJ - 1). ( S>0). (2.18) and m(0) = n(O) = 0, where k = E(T). From (2.4) we have §'(k) = 2w'(3) =u as k+O and therefore 1im m(s) =1im n(s) = o as 5+ 0. This shows that the functions are continuous at s = 0. By their very definitions m(s) is a non-decreasing and n(s) is a non-increasing function; it then follows that m(s)30 and n(s)50. Thus in view of (2.16) we have Ofm(s) f %—- 1 . (2.19) 03n(s) 3 g). - 1 . (2.20) In the following, the existence of a smooth solution u(x], x2) to the boundary value problem (2.8)-(2.11) will be assumed, where u is twice continuously differentiable on D and once so on P. On linearizing the partial differential equation (2.8) formally by neglecting |Vu|2 in comparison with 3, we recover the analogous problem in classical elasticity. This is a boundarysvalue problem for Laplace's equation which also describes the steady irrotational flow of an inviscid incompressible fluid past an elliptical cylinder. In the flow problem u is identified with the velocity potential and ka,with the free stream speed. The sblution 3 (unique to within a constant) of the linearized problem may be found in standard text books; its explicit form need not concern us here. From (2.12) the corresponding linearized stresses are given by O _ O O T3a " 1111.0, 9 T (231 + i§2)i= ulvfll. (2.21) It is wellknown that gmax occurs on F. For the linearized problem, the stress concentration factor 0 K is defined by ° 0 0 K = Tmax/ too , (2.22) where Tm =“kw. denotes the magnitude of the applied stress at infinity. It can be shown (see e.g. Milne-Thomson (1962),p 171) that O K = l+A/B. (2.23) It should be noted that for a neo-Hookean material, the problem (2.8)-(2.11) specializes exactly (rather than merely 10 O by linearization) to the linear problem. Thus K given by (2.23) is exact for this material. Our main concern here is with the nonlinear problem (2.8)- (2.ll). For this problem, we define a stress concentration factor K by K = Tmax/Too 9 (2024) where 2 2 I T = max (T + T ) , max our 3‘ 32 (2.25) and too = 2k» N'(3+k:) is the magnitude of the applied stress at infinity. Our objective is to develop techniques for obtaining bounds on Tmax, and so on K, which conform to the result (2.23) on linearization. The argument is based on maximum principles and comparison theorems for the second order quasilinear uniformly elliptic equation (2.8). (See Protter and Weinberger (1957). Gilbarg and Trudinger (1977)). Such maximum principles have been used (see e.g. Bers (1958),P-41, Schiffer (l960),P.95) to show that T occurs on the boundary P and so our taSk is to estimate max T on P. 2.2 Reformulation in Terms of Stress Function It is convenient for our purposes to convert the basic problem (2.8)-(2.11) to a problem of Dirichlet type. It follows from (2.8) that there exists a function v, twice continuously differentiable on D and once so on I such that - 1 2 g T301 - 211 (3+ [Vul )u,a €18 v,B on D, (2.26) 11 where 5 a8 €12="€21 = l). The function v is a stress function for the shear is the two-dimensional alternator (eH = 622 = 0, stresses T3a . From (2.26), (2.4) one infers that 2 (Ivul) = |vv| . ‘ (2.27) which upon inversion, yields (w) = Eilwl) . (2.28) We now define a function V by vhz) = 12 (>0).-«m . (2.29) 2N'(3+k (1)) and note that (2.26) may then be written as = _ 2 “la V(|Vv| )edB v (2.30) ’8 It then follows that the stress function v satisfies the differential equation I V’ E div [V(|Vv|2) grad v] = 0 on D. (2.31) It can be verified that equation (2.31) is uniformly elliptic by virtue of the left-hand-side of (2.15). From (2.26). (2.10) and (2.11) v may be shown to satisfy the boundary conditions v(x], x2) = Im x.I as Xa Xa'+ w . (2.32) v = 0 on P. (2.33) 12 It is convenient in the subsequent analysis to restrict attention to 0+, the right half of D where x1>0. The notation L_, L+ is also introduced for the line segments {(xl, x2)| x1 = 0, -w go. (4.1) C sinh 5 sin n - n 5 n < n , x2 where g = go represents the boundary of the elliptic cavity I and 2C is the distance between its foci. In terms of the semimajor and semiminor axes A and B of the ellipse one has 5. .5 2n[(A+B)/(A-B)I. c = (A2 - 32)5. (4.2) The differential operator L appearing in (3.3) can be (1) written as 2 L w = 2v'(|vw|2) { V(lij ) (w + w ) 2V'(|VWI2) ' 55 nn 1 2 2 +3 [”5 "£6 + ZWE "n "En +w11 wnn 1 2 2 - ‘fi'("g + wn ) (hgwg+ hnwn)]} , (4.3) (1) Here and in the sequel subscripts E and nr denote partial differentiation with respect to g and n respectively. 15 16 Where h is the "scale factor" for elliptic coordinates and |le is the magnitude of the gradient of w in these coordinates: h = C (sinh2g+ Sinzn)i , 2 (4.4) + w: )/h . Since V and V' have lvwl2 as their arguments, the form of the operator 1- given by (4.3) is quite complicated. However, we note that by (3.3), we merely require L w 30, and it turns out that a Simpler set of sufficient conditions is obtainable to ensure that this holds. To see this, we first note that from (2.4) and (2.29) one has 2V'(12) = 1 ( t ——7— —:.—-1). §=k(). (4.5) V(T ) ';2' kt'(k) T and so from the definition of m in (2.17) we have 2 2 m(s) = max §I_!%SI_1 . (4.6) Oftfs V(t ) Consequently, one can readily verify that, if for any positive number 5,, w satisfies )+ m(so) [WE wgg+ 2w w w + w2 w' 2 2 ("E + wn)(" E n En n no €€+wnn 1 2 2 - hi 0, (4.13) for g > go where m = m(so) and primes denote differentiation with respect to E. Substitution of (4.9) into the first of (4.8), (3.4) and (3.6) gives f" - f3 0 for g>g° , (4.14) i(g.) = o , 1im (2e'g f(£)/C) <1” . (4.15) g-roo We now construct a function f(§) conforming to (4.10)-(4.15) and ' then show that the second of (4.8) holds. To this end we solve (4.10) with equality subject to (4.15) to find f(5) = Cr0° costh(g), (4.16) where [(sinhgflm coshz;2 F(E) = l (3 0 ) - (4.17) Ill . TTPT (Sinhg) d; cosh; 5o Direct computation shows that (4.16) satisfies (4.14) with inequality. We next rearrange (4.12) and using the fact that f30, m30,cbbtain fH + $1.14”: f"- f+ fig?- f3f" - f>0, (4-18) the last inequality following from (4.14) with inequality. Thus (4.12) is seen to hold. To verify (4.13) we note that 2 ff" + 2mf' - f23f(f" - i)> o, (4.19) where (4.14) has again been used. Finally, we turn to the verification of (4.11). We observe that from (4.10) (with equality) and (4.14) we have costh'3 sinth for E 3 E, , (4.20) which in particular implies that f'(E) 30 for E 3 Es . (4.21) Multiplying (4.11) by (1+m) and making use of (4.20) we find that (4.11) holds provided that Sinth' - costh 30 for E>E° . (4.22) 19 To verify (4.22) let ¢(E) = sinhE f' - coshE f . for E 3 Eo . (4.23) Differentiation yields '(E) = (f" - f) sinhE 30, for E_>E° , (4.24) where the inequality holds by virtue of (4.14). 4(g) is thus a nondecreasing function of E. But Mao) = sinh: Pita): 0. (4.25) by virtue of the first of (4.15) and (4.21). The inequality (4.22) now follows. We now Show that |Vw| attains its maximum at the point 6 =€o.n = 0. From (4.4) and (4.9) we have .2 2 2 . 2 IVWIZ = f cos n + f sin n . (4.26) C2(sinh2g+sin2n) Simple calculations Show that. On using (4.20). 2 2 2 2 2 _2 _ f' cos n+f sin n f'(E) _ 2 VW(€,T]) - < '—E' " m(gs 0) 0 I I C2(sinh2E+sin2n) ' ( CS1" 5‘) I I (4.27) It is readily shown that the right hand side of (4.27) decreases with E so that . 2 . 2 (fl) 5 (653%?) ‘ IMEO. 0H2. (4.28) 20 Thus from (4.27), (4.28), (4.16), (4.17) we have 1 h-ml i o oslvwl s lvwlmax = t” (chsfigz g(Eo;m) . (4.29) where (D £1— m-I- 9(€;m) = I (Sinh%) dc: m = m (so). (4.30) cosh ; Therefore if a positive number so exists such that l - MTT Tco (51:25:20 51%“ f so: m = 111(50): (4031) then the second of (4.8) is satisfied. Assuming for the moment the existence of 50 (>0) we note that w(E,n) given by (4.9), (4.16) and (4.17) satisfies all of the requirements fer an admissible comparison function. We can therefore use it, in conjunction with (3.8), to find a lower bound on the stress concentration factor K: 1 'm Isinhgo) 1 K3 coshao 9mm) . ”-32) where m = m(s.) is defined by (2.17), g(E ;m) is given by (4.30), and So is any positive number conforming to (4.31). We now prove that such a number can always be found. Furthermore s. can be determined in such a way that the right hand Side of (4.32) is maximized. In other words the optimum value for so can be determined. 21 To show the existence of a number so conforming to (4.31), we define a function y(t) by _ l - A» (sinheo) E1T Y(t) ' coshao 915°:t) for t 3 o, . (4.33) Differentiation with respect to t Shows that y'(t) f 0. Now let z(s) = y(m(s)) a s for s 3 O. (4.34) Using the chain rule and recalling that m(s) is a non-decreasing function of S we find z'(s) < 0. Thus 2 is a decreasing function of 5. Moreover 2(0) = y(O) > 0 and z(a0=- «<0. Therefore there exists a unique positive number 5* such that z(S*) = 0, 2(5) <0 for s > 5* and z(s)> 0 for 0< s< 5*. Thus any number S > 5* satisfies (4.31). It can be easily shown that 5* is the optimum value of S. To see this we note that the "best value" of s is a positive number which conforms to (4.31) and maximizes the right hand side of (4.32)which meanSnmximizing y(m(s)) given by (4.33). Since y(m) is a decreasing function of m the optimum value must minimize m(s). There is only one such value of 5 namely 5 = 5*. From (4.34) then we have that 5* is the unique positive root 0f 5* = y(m(5*)). (4.35) and K = It0° 3 5*/T . (4.36) no Tma X The lower bound on the stress concentration factor K given by (4.36) depends in particular on 1”,, the applied stress at infinity. 22 A weaker (and simpler) "load independent" lower bound can be found which is independent of I”. We simply recall that y(m) is a decreasing function of m and m 5 -%7- - 1 (see (2.19)). It then follows that y(m) 3 y(-%— - 1) and so we have s y(-%—-- 1) K 3 -—:F- 3 T (4.37) To summarize, an admissible comparison function has been constructed (see (4.9), (4.16) and (4.17)) and expressionSLfor_lower bounds on the stress concentration factor for a softening material deriVed ((4.37))- 5. RESULTS In this section we first discuss the load independent lower bound given in (4.37). Then we provide an example of how the general load dependent lower bound given by the first of (4.37) can be found explicitly by considering a Special constitutive law. We conclude with a brief discussion of the limiting case of a "thin" ellipse. 5.1 The Load Independent Lower Bound Denoting the load independent lower bound in (4.37) by K* we have 1 K = “E '1): (sinhE.)'c 1 (51) * t W —,-T— . - °° g(EOS‘E‘l) where 1 w ( , h f—c 9(Eo 3 5" 1) = I 51" C d; . 0 o, 0 0 is a material constant and c is.a. softening parameter 0 copoom coppoeucmucoo mmmcum we» com vcaon Logo; .m wczmwa In .1 o..~ wup one e... N? o.._ m. P. N. N. co; to .o._ n o 11, N n o. . e n e O 1.0 II 4..) o.~ o.m m\< n AF. n v cmmeocoa mcwcmacom u Aoo_ n v A~.mv zap o>wo=oeomeoo ego e? oeeemeeo _e_eoeee ” u u o .31 o8 y no; punoa J8M01 27 .mmmawppm ucmcmccwu cot xp_:_$:w um mmmcgm umwpaqe we“ mzmcm> couuoc cowpocucwocoo mmmcum mgu com canon cmzoo .m mcammm o.~ w; e; e; N; o; m. e. e. N. o MI ‘I a . q a q . . . . . 00.... n Fo.P u u N n o. e ... S cm n a m\< n a AN. uv cmumEmgoa mcwcmpwom no AooF nv A~.mv 3MP m>wp=wwpmcou on» c? “cmumcoo Powcmuoe no .21 (48 N.p x no; punog J3M01 28 .mmmawppm pcmcmwmwu cow Auwcww:_ um mmmcum compaao mew msmcm> Locum» cowumcucmucou mmmcum any to» uczon cmzoo .e weaned 1. o o o o o o o o o o N. A om mm at N_ Ne o; w m e- N p. Fo.~ u a (II N n o .11 m\< n u Am. nv cmuosocog mcwcmueom "u Aoo_nv AN.mv zap osmeeoNNmeoo as“ ea eeoeoeoo Peacoeee no m.p o.~ o.¢ o.m x JOJ punog J3MO1 29 .auwcwmcw no mmocum ooppooo on» msmco> xuw>eu copaocwo o coo couuom cowuocpcoocoo «mocum on» cow ocaon ooze; .m ocamwd ON .3 e; e; N; o; N. o. e. N. o IM A J . . _ _ _ o . . q 0 8P 1 L I o.P _.o H U N. n U a m. u o cam LopoEocoo mcwcmowom no Aoop uv Am.mv sop o>wpzpwpmcou ozu cw acoumcoo Powcouoe No 89 am.‘ x JOJ punog J3MO1 30 decreases, Since the material is softening. The decrease in the lower bound at higher loads depends on the softening parameter as is shown in Figure 5. 5.3 Limiting Results for a Thin Ellipse The case of a thin ellipse in which B/A <l) and after some simple calculations strictly analogous to those in the softening case we find * = (sinhg.)'b 1. (6.4) coshgo g(Ea; %__1) 7V 33 where g(Ea; %-- 1) is given by (5.2) with c replaced by b(>l). It can easily be Shown that l+b 914;... ‘3-1) 313,5 [2(1-tanh5all 2 . (5.5) Substitution in (6.4) yields 123 1:9 2 K" 5 (Ma) 1—2—1 i—B-g—g— . (6.6) where A and B are the semimajor and semiminor axes of the ellipse respectively. In the special case of a circle (A = B) we find * K 5 l+b which is the result found by Abeyaratne and Horgan (1983). 6.3 Suggestions for Further Hork We have established lower bounds for softening materials (and upper bounds for hardening ones) (see (4.37), (5.4) and (6.6)). As noted previously, we have been unable to find upper bounds for softening materials (and lower bounds for the hardening case). This issue should be resolved if possible. There are also some places where the present work may possibly be improved. The differential inequalities (4.10)-(4.14) are sufficient conditions for -L w 3 0. While it is not difficult to establish necessary and sufficient conditions they are rather complicated. It would be worthwhile to investigate these and see if sharper results can be established. Another area where improvement may be possible is in connection with the integral on the right hand Side of (5.2). Efforts have been made unsuccessfully to evaluate this integral analytically; furthermore, it does not appear to be evaluated explicitly in the standard integral tables. Finally, it would be 34 of interest to use numerical methods (e.g. finite difference or finite element schemes) to compare with the results obtained here. APPENDIX APPENDIX A An Upper Bound for g(Eo; %-- 1) Defined by (5.2) We wish to find an upper bound for the integral Q . 1-c 915.; 35- 1) = j ‘5‘“ ’ d-C . t.>o. 0'0, with continuous derivatives of all orders. The Cauchy stress tensor g’ associated with a plane deformation is given by ' _ 2 an an 1-33757ng (2.5) = 2u{]jh(R)§+ [H(R)-l] 1}, on o, where we have set h(R) = fliégl- for R> 0, (2.7) and El 715 the second order identity tensor. The Piola stress field 9 associated with the Cauchy stress field T is defined by g = JIF'T = 2 u{h(R)f + J[h(R)-1] f'T}: on Da. (2.8) 42 where PET is the transpose of the inverse of ‘F . In order to ensure a physically reasonable response, one must impose certain restrictions on the constitutive function H(R). Since Cauchy stress and the strain-energy density should vanish in the undeformed state, we must have H(2) = 1, H'(2) = l. (2.9) Furthermore, the strain-energy density function (2.5) should be positive in every state, except the undeformed one. This requirement entails the inequality 2 H(R) > R /4 for all R> 0, R: 2. (2.10) Next, from consideration of the true stress field induced in a plane isotropic deformation and the requirement, on physical grounds, that stress should be monotone increasing with the amount of stretch one deduces that h(R) = H'(R)/R must be monotone increasing, i.e. h'(R) > 0 for 0l for R*< Rcw. (2.12) A more complete discussion can be found in the paper by Knowles and Sternberg (1975). 2.3 Deformation and Stress Fields Returning to the problem under consideration we have from (2.4) and the second of (2.5) that R = 2f + rf', ax r 0 for R* R*, (2.18) where R is given by (2.13), then H'(R) is monotone increasing on the interval of interest and hence may be uniquely inverted. It will be Shown later that (2.18) indeed holds. Here we gggpmg that (2.18) holds and so deduce from (2.17) that 2f(r) + rf'(r) = R0 (constant)for a0). By virtue of the assumed monotonocity 0f h(R) the existence of a unique positive solution to (2.25) is guaranteed provided that its 45 right-hand-Side lies in the open interval (0,1) (see (2.11), (2.12), (2.18)). It will be Shown later that this is always satisfied if the applied pressures are appropriately restricted. The correSponding components of Cauchy stress are found from (2.7), (2.14), (2.15) in conjunction with (2.19), (2.25) and (2.26) to be r - Q TDD = 211 [R;ll'(.Ro) _ 1] - 211 [olr + 02] ’ {23.27) 3 2 2 Q r + Q Tww = 2 ll [gz—E'ga'} '1] = 211 [fir—0'2- , (2.28) 3 ' 2 where _ -2 2 Q] " p1 a (211' p0) " pob (211 " pl) 9 (2-29) 02 = 2)) 4213(1).i - p.). (2.30) - 2 2 Q3 — 211 [b (211 - pi) - a (211 - p.)]. (2.31) The deformed inner radius of the tube a is found from the first of (2.1), (2.20), (2.25) and (2.26)to be a = aR.(b2 - 32) (2)1 'Po) (2 32) 2 2 ° ° 2[b (211 - p1) - 41(211 - 9.)] Examination of (2.27) - (2.31) Shows that the true stress distribution in the cylinder is independent of the constitutive function H(R) and depends on the (harmonic) material at most through its infinitesimal Shear modulus p. 46 He now examine the preceeding results in the two cases of internal and external pressure separately: (i) Internal Pressure Only (p1 > O, p. = O) Considering the hoop stress T we find that - 02(Q]+Q3) <0, WW provided that pi <2t1. Assuming temporarily this to be the case it then follows from (2.28) that T is monotone decreasing with r, 44 so that the maximum hoop stress occurs at the inner surface r-= a and is T “1% (b2 + 32) (2 33) 142 m2 _ a2)_pib2 - ° For this to be positive and bounded, one must have p. q, (1 - azlbz) (< 2 a ). (2.34) 1 It is easy to check that (2.34) implies O<:h(R,)< l, i.e. the right- hand-Side of (2.25) with po = 0 is a number in the open interval (0,1). Thus (2.25) can be solved for a unique value of R. (> 0). Moreover Since h(R) is monotone increasing it follows that Ro> R, and (2.18) is thus verified. (ii) External Pressure Only (pi = 0, p,>>0) 0n consideration of (2.32) with pi = 0, we see that as p, + 2p , 0+0 i.e. the cavity closes at p. = 211 . (For values of p.> 2“ one finds ot< 0!.) He therefore require that p°< 2p . (2.35) This also ensures that 0< h(R.)< 1, i.e. the right-hand-side of (2.25) with pi = 0 lies in the open interval (0,1). The argument 47 in the previous case applies and hence a unique solution R, (> 0) for (2.25) is guaranteed and (2.18)is verified. In the particular case when the applied pressures are small (pi/Zn . p./20 <<‘1) it can be easily verified that upon linearization, (2.27) - (2.31) yield the well known results according to the infinitesimal theory of elasticity (see e.g. Timoshenko and Goodier (l970),p 70) a2b2(p° ‘ P-) p 92 ' pobz T = 2. 2 1 712 I i 2 T ' (2°36) pp b - a r b - a a2b2(p° ' pl) 1 piaz ' pobz (2 37) 't = '-' + . . W b2 - aZ— :72 a - a2 To summarize,in this section the plane axisymmetric deformation of a hollow cylinder subject to simultaneous internal and external pressure has been examined. The stress and deformation fields were determined (see (2.1), (2.20), (2.25), (2.27)-(2.31)) and various features of these fields were examined. In the following sections the stability of the equilibrium solution obtained here will be investigated. This investigation will determine whether the results found here pertaining to an infinite hoop stress (in the internal pressure case) and the closing of the cavity (in the external pressure case) are indeed attainable. 3. GEOMETRICALLY PERTURBED PROBLEM 3.1 Deformation and Stress Fields In order to investigate the stability'LI) of the equilibrium solution found in Section 2, we consider the pressurizing of a right hollow cylinder with "almost" circular boundaries r=a+ 69‘1”) ), r=b+ Ego(¢ )9 (El (<19 05¢: 271: 91.10) = 91(211) . 9.10) = 9o(211) . (3.1) where gi, g, are bounded fuctions on [0,2n]. The resulting ‘deformation is assumed to be a slight perturbation of the purely radial deformation already'discussed.' To this end we introduce o = rf(r) + 6170.9). 4= *9 + \7 (no). (3.2) "3101 where (r,4>) are the polar coordinates of a generic point in the undeformed configuration which is mapped to (0.41) by the deformation (3.2), and e: is a measure of the "imperfection" of the boundaries. Since it is assumed that e: is a small number, in all the developments that follow.. terms which contain powers of 6: higher than one are neglected. (1) As mentioned in the Introduction the study of a geometrically perturbed problem was undertaken in the hope of calculating the buckling pressure for an internally pressurized tube as well as obtaining detailed information regarding the deformation field (from the prebifurcation state to beyond the occurrence of instability). The buckling pressure itself can of course be determined without the introduction of geometrical inhomogenities. ‘ 48 49 The polar components of the deformation gradient tensor '5 associated with (3.2) are given by (see e.g. Malvern (1969), p. 651) = ' ~ = E. ~ - ~ F ‘f +-rf + cur , F r (u fv), rr r0 '9 (3.3) = €(fv)r s F¢¢ = f + $110 where a prime denotes differentiation with respect to r and subscripts r and 4 on E and (fv) denote partial differentiation with respect to r and 4 respectively. For the deformation given by (3.2) we find from (3.3) that T I = tr 155 = I,+eI, .1 = det f = d,+e3, R = (1+2J)5 = R,+cR, (3.4) where 1,,J, and R, are the invariants in the axisymmetric case and are given by the right-hand-sides of (2.3), (2.4), and (2.19) respectively. The "first order invariants" I, J, and R are found to be I = 2 {(f+rf')tlr + .§.[J+(t5)¢] } , (3.5) $1 = for + ;-(f+rf') [fi+(f§7)¢] , (3.5) R= {Ir +1F (mood; . (3.7) Next we expand h(R) = h(R,+cR) in powers of cR about R = R,. To leading order we have ~ h(R)=h(R.) + J). E=Rh'(R.)= gem-(12.) -h(R.)1. (3.8) Substituting for I, J, and h(R) from (3.5), (3.6) and (3.8) into (2.8) we find ° +eo Orr = orr . rr ’ oro =80r¢ (3 9) “4 r g ”or ’ “‘44 . “44> + 8°44 ’ 50 where o:r and 0° are the Piola stress components in the axisymmetric ¢¢ case and are given by the right-hand-sides of (2.14) and Err’ Ero’ 5 , and 5 are found to be ¢F ¢¢ arr = 2n { H"(R,) [or + %. (a + (fV)¢)] - ;.[a + (fV)¢]} . (3.10) 5r¢ = 2n {%-h(Ra) (G¢ - t3) - [h(R,) - 1] (fv)r 1 , (3.11) 3¢r = 2v (h(R.) (f3), - %-[h(R.) - 1] 16¢ - i5) 1 . (3.12) 6¢¢ = 2a {H"(R,) [ar + é-(a + (f;)¢)] - Gr 1 . (3.13) Having found expressions for the stresses, the equilibrium equations, div 9 = Q in polar coordinates, can be written down immediately. (See e.g. Malvern (1969),;L 655). These are H"(R,) { r2 J + r [Ur + (fcqug - U - (f;)¢} rr (3.14) + h(Ra) Iu¢¢ - (f3)¢ - r (f5)r¢] = 0 . on 0,: H"(R,) [(rU + J) + (i?) 1 r ¢ ~ ¢¢ ~ ~ ~ (3.15) - h'(R,) [r2(f;)rr - r(u¢ - fv)r + u¢ - fv] = 0 . on Do It should be noted that the equilibrium equations div 9 =_9 give rise to two groups of terms; one of which does not involve 5 while the other one does. The former consists of one equation which is, of course, exactly the same as that found when considering the equilibrium of the unperturbed cylinder (see (2.16)). The latter consists of (3.14) and (3.15) above. 51 3.2 Pressure Boundary Conditions on a Perturbed Surface we next formulate the boundary conditions for a perturbed boundary subject to pressure. Once this general form is derived, specialization to the particular cases of internal and external pressure will be immediate. To this end let the unit outward normal to a surface S at point M in the undeformed configuration be N. Under deformation the surface S wil be mapped to S and the point M to m. Let p be the unit outword normal to S at m. Denote the area of a surface element surrounding M on S by dA and its image under the given deformation by da. Then (see e.g. Chadwick (1976), p.61) 9 = jE'T ! $9., (3.16) If the deformed surface being considered is subject to a hydrostatic pressure p then the appropriate traction boundary condition is given by I p = - p g on s. (3.17) It is often convenient to transform this condition into one which holds on the (known) undeformed surface 5. To this end, we substitute for I and g in (3.17) from (2.8) and (3.16) respectively. This gives a §= - p Jf’T N on s. (3.13) He now consider the particular case in which the surface S, in the undeformed configuration, is a right cylindrical surface, with the generator perpendicular to the (r,¢) plane. The intersection of S with this plane is given by r = r, + cg(¢), |e|<0) and c are constants and 9(a) is a given 52 bounded smooth function of 4) such tht (11(0). = 0(211). (See Figure 6.) As mentioned previously, 5 determines how much the curve under consideration differs from the circle r = r,. From elementary calculus one has -1 tan 9 ”(3%) . (3.20) where 6 is the angle between the radial line and tangent to the curve at any point. The polar components of the unit normal 3 are "r = - Sine and N¢ = c050. Equation (3.20) in conjunction with (3.19) yields cote = eg'(¢)/r. 4 Mai). ( 3.21) where a prime denotes differentiation with respect to ¢~ 0n recalling the trigonometric identities sine = (1+cot20)'é and, ’ c056 = cote (1+ cot26)'é, expanding their right-hand-sides by the binomial formula, and making use of (3.21) one finds that to first order in e "r = - 1 , N¢ = cg (¢)/r, . (3.22) We turn now to the evaluations of the remainingtquantitiesin (3.18). T By wayiyfillustration, expressions for one element of 5 = JE' and g; each will be derived in detail. The derivation of other elements is accomplished in a similar fashion. From (3.3) and the second of (3.4) it follows that the element All is given by f+e [5 + (f;)¢]/r with r = r, + eg(¢). Now 53 r = r. + 89(9) Figure 6. Geometry and coordinate system for the perturbed boundary. 54 Elf + E- [u + (fV)¢] } = f(r.+69(¢)) r = ro+€g(¢) WEE—9171?) [Giro "’ 89(9).” + fir. + egia))i¢(r. + e914).¢):1. (3.23) After expanding the right hand side in a power series and dropping terms of order higher than unity in 5 one finds w+ étfi+uhpi =RN) r = r'o + €g(¢) +e{rougw)+%woNn+ion%nNn1L (an) In a similar fashion one may write Orr = orrlra + €9(¢).¢) = orr(r..¢) Bo _ + 3,7" ' e9(4). (3.25) 'r=r¢ to leading order in e. Substitution for Orr(r°’¢) in the above from the first of (3.9) and (3.10) gives 3 0rr = 0° (r.)+-6IC3rr(ra.¢) + Orr 9(9)] . r = r? + cg(¢) rr 3r r=r, (3.26) T Calculating the other elements of JE' and 9, making use of (3.18) and (3.22), and after some algebraic manipulation.the order 8 terms in the boundary conditions take the form 55 . 20H"(R.) [err + G (ii)¢] - (2n - p) (a + (fv)¢] f 12v - plra f'g on r = r.. (3.27) and 2u bin.) [roii3)r - 6 + ii] + (Zu - p)(J¢ - f3) 9 LZPH'(Ro) - (R0 - f) (2p -p)] g' on r = r.. (3.28) The boundary condition arising from the term of zero order is of course Cir + pf = 0 as before. 3.3 Special Cases He now specialize the results found in the previous subsection to the two cases of internal and external pressure loading. In both cases the inner and outer boundaries are assumed to have imperfections as described by (3.1). The two cases differ only in the loading. Case (i) Internal Pressure Only (pi> 0, p, = 0). From (3.1), (3.27) and (3.28) with g(o) = gi(¢), p = pi, and r, = a, the boundary conditions on the inner boundapy are found to be ZpH"(R,) [aar + a + (i3)¢1 - (2v p.) [G + (i5)¢1 = (2p - pi)af'gi on r a, (3.29) 2ph(R,) [a(fii)r - G + f3] + (zu - pi) (5¢ - f5) = = [ZuH'(R.) - (Ra-f) (Zn - 91)]9'1 on r a. (3.30) Using the same equations as above with g(o) = g,(¢), p = 0, and r, = b, one finds that the boundary conditions on the traction-free outer boundary are given by 56 H"(R,) [50'r + J + (fG)¢] - u - (inp = bf'g, on r = b, (3.31) h(R,) [b(fv)r - u¢ + fv] + u¢ - fv ' = [H'(Ro)‘Ro + f] 95 on P = D. (3.32) Case (ii) External Pressure Only (pi = 0, p,>0) Again using (3.1), (3.27) and (3.28) with g(o) = gi(¢), p = O, and r1 = a, the boundary conditions on the traction-free inner boundary take the form H"(R,) [aur + u + (fv)¢] - u - (fv)¢ = af'gi on r = a, (3.33) h(R,) [a(fv)r - J + f5] + J - f5 = [H'(R,) - R, + f] 9% ¢ 4 on r = a. (3.34) The above equations with g(o) = g,(¢), p = p. and r0 = b furnish the boundary conditions on the outer boundary as follows zuH"(R.) [b5r + G *. (f3)¢]- (2n - p.) [5 + (fV)¢] = = (2p - p,) f'g, on r = b, (3.35) Zuh(R.) [b(i’v')r - G + ii] + (2n - p.) 16¢ - t5) = ¢ = [ZUH'1RO) ‘ (R0 ’ f) (2“ ’ po)] 93 on r = b. (3.36) The formulation of the equilibrium problem for a pressurized cylinder with "imperfect" boundaries is thus complete. The solution of the equilibrium equations (3.14) and (3.15) subject to the 57 boundary conditions (3.29) - (3.32) or (3.33) - (3.36) will be considered in the following section. A unique solution of this problem 'hnplies stability. 4. STABILITY 4.1 Solution of the Equilibrium Equations Re now turn to the solution of the equilibrium equations (3.14) and (3.15). Assume that the displacement components 5 and f5 can be expanded in trigonometric series of the form 5 = f an(r)cos no , (4.1) n=o f1? - 3"? r bn(r) sin no , (4.2) “:0 where without loss of generality we take b,(r) a 0. Substituting into the equilibrium equations (3.14) and (3.15) we find that, for each n30, H"(R,)(ar" + 1? afl + nbni'- nh(R,) (a;1 + f,— b" + a, an) = 0, (4.3) hit.) in; + $.- bn +32%? 415;, +§an +£2.51] - H"(R,) g2 (al;I + '1,- an + on) = 0, (4.4) for al, by (2.l2), pi>2p(l-k). But for finite hoop stress we must have p{l). Since we require that pi0), An = k" + k'"-2(>o). (4.34) n For instability we must have A = 0, which according to (4.33) yields a quadratic equation for 5. Solving this, we find values of 5 given by s; and s; where k-h - " (4.35) s - . . .]+hni [(l-k)2+4hn]i 3H- Thus if there is a value of internal pressure pi(l, pi0, n3 2. (4.36) Considering first the case where the positive sign is chosen in (4.33) it is obvious that s: is positive by virtue of (4.36). It is easy to see that the negative sign is also inadmissible. Suppose that s; is negative. For this the denominator of (4.35) must be negative. Simple calculations show that this in turn implies that k + hn >2. Now by (4.36) we have k + hn <2k<2 ,'which is a contradiction. Thus we are led to the conclusion that the tube is stable at all values of the internal pressure pi l (by (2.l2)), it is evident that the value of po given by (4.42) is negative. Thus no loss of stability occurs for n = 0. We next consider the case n - l. Substituting from (4.9)- (4.l2) into (4.37)-(4.40) and simplifying yields the following 69 set of four linear equations in three unknowns p [H"h—(H"+h)/4]a [H"h+(H"-h)/2]a-] -2a H"h-(H"+h)/4 -[H~h-(H"-h)/2]a'2 -2a [ZuH"h-(Zu-po)(H"+h)/4]b [H"h+(2u-p.)(H"-h)/2]b'] -2(2u-p,)b'3 _2uH"h-(2u-Po)(H"+h)/4 -[H"h-(Zu-Po)(H"-h)/Z]b'2 -2(2u-I3c.~)b-4 d P ‘ = D . W A1 f q] . -2 D1 (Zu-po)f's] . -Z - J L'[2)JH ’(Ro'f)(2)1'po)]b 5] I. (4.43) We note that the terms containing C1 have dropped out of the equations (4.43). Thus the value of C1 is arbitrary.. As was shown in the case of internal pressure the expressions containing Cl corresponds to a rigid body translation and therefore one can set C1 = 0. (The argument is exactly the same as before and will not be repeated). As in the previous case there are two questions concerning (4.43) which are of importance: Consistency, which is necessary for existence of a sOlution and uniqueness which implies stability. Before investigating these issues, we note that B1 = 0 satisfies the equations. (The calculations leading to this are somewhat lengthy but straightforward.) He can thus remove the second column of the coefficient matrix in (4.43) to obtain a 4 by 2 matrix in which the first row is a multiple of the second and the third row a 70 a multiple of the fourth. The consistency of the equations is established if the same relations hold between the elements of the right hand side column vector. It can be verified that this is indeed the case, hence of the four equations only two are independent and we have the following system h"h-(H"+h)/4 -2/a4 A ZuH"h-(2u-po)(H"+h)/4 -2(2u-po.)/b4 D] -(H'-Ro+f)q]/a2 -[2uH'-(Ro-f)(2u-po)s1Jlbz (4.44) To investigate the uniqueness of solution we look for values of po for which the determinant of the 2 by 2 matrix above vanishes. If any of these values lie in the range (0, 2p) it would be a buckling load. Denoting the determinant by A we find “2“;44-94 [4"[4439;432:5(“k2’(Zn-4°) - 2mm , (4.45) where k = (a/b)2. Setting A = 0 one finds po = 2p and po = 2u(1+k)(2H"-l)/[k(2H"-l)el]. Since we are interested in values of po<2u, we need consider the second expression only. By (2.l2) the numerator is positive. As for the denominator there are two different cases: (i) the denominator is positive,ir|which case po>2u which is inadmissible, (ii) the denominator is nonpositive which results in p°<0 or p° infinite, bothcniwhich are unacceptable. We are thus led to the conclusion that for values of external pressure pO in the range of interest, no buckling will occur in the mode n = l. 71 Finally, we take up the case n 3 2. Upon introduction of (4.9), (4.l0).(4.l3) and (4.14) into (4.37)-(4.40) we obtain. for each n 3 2, a set of four linear algebraic equations for An - Dn which in matrix form can be written as a 72 m Ao¢.ev m-=-aA=._VA.a-;N.- -:-aae+_vaoa-zmv- m-=- ~-=- oA=+pvu «A:+—vu . 23-..:.a-_.~:.-.s-.___£-. i.e. cm.qug-:mv cu =_em-a:..~_-.5- em . A . .5 m-=efic-.vaoa-zmv _-=-nfle\fi..:N.=V-;=Vaoa-3~v+z..zzwg- ~-=afi_-=VA.a-;~. =-:H¢\A..==-;AN+=.3A03-3NV-;..zamH muccAcu—v —I:ucmv\a..zamu:vnccv+z..IHI N-=ea_-=v c-1me\fi..z=-;A~+=vV-z..:H _-=aHe\A..:A~-=v-=e.Aoa-a~v-g..=;~H. camc\fi=A~-=v-..zevAoa-;~v-g..=a~H _-came\A..:A~-:.-g:v-z..zu ccflv\asaNlcvl..==VI£..=H A 73 we again seek those values of po in the range 0l and p°<2u, by (2.12) and (2.35) respectively, T as defined by (4.47) is positive. Recalling that k-hn>0, (4.36), it is evident that T; is positive. 0n the other hand, a simple calculation shows that for thick enough cylinders (k<%~) T; is negative and hence is not of interest. However, for values of k in the range %-o. (4.53) - 1+ 1 ZH'IJRo) (l-k)T: 2H"(R,)-l The inequalities in (4.52) and (4.53) follow from (2.l2) and k< l. Since W+ depends continously on p0 it follows that there i exist numbers p.n 8 (0,2p) such that 4+ (p3) = 0, ‘Y_ (pa) = 0. This establishes the existence (of two sequences) of bUckling pressures p: , p' The former exists for tubes of arbitrary n O 75 thickness ratio k and all modes n3 2, whereas the latter exists for tubes with the thickness ratio k in the range (l/2, l) and sufficiently large modes n. (See paragraph preceeding (4.50)). We show next that the buckling pressures p: corresponding to a given value of n are unique. Clearly it is sufficient for this purpose that W: (p,) be monotone increasing functions on (0, 2p). Differentiation of W+ (p,) with respect to po yields 4u(l-k)Ti H"' I _ n dRo [2H"+(l-k)(2H"-1)T; where dRoldpo may be calculated from (2.25) with pi = 0 to be dR. = _ n(l-k) < o, (4.55) 35: [Zn-k(2u-Po)]7h' (Re) the inequality holds by virtue of (2.ll) and k0. . (4.56) It follows that if (4.56) holds the buckling pressures p; , p; corresponding to a given mode n are unique. He now turn to determining the smallest buckling pressure. We will first show that the buckling pressures p; form a monotone increasing sequence (with respect to n) so that we then have 'p; < p; (n>2). Furthermore we shall also demonstrate that p; 2) as well. Thus p; is the smallest buckling pressure for an externally pressurized tube. 76 To show the monotonocity of p: with respect to n we replace T by T+ in (4.47) and differentiate with respect to po to find dT+ 1 4R .35: = 2". 2(2 )2 [2(2H"-1)H"-pq(2u-p°)H"' 3623) 0. (4.57) ' U'Po “ o The inequality follows from (2.l2), (2.35), (4.55), and (4.56). Treating n as a continuous variable and differentiating (4.49) with respect to n gives dT+/dr1>0. Now + .%E2 = .91;£92_ >0, (4.58) -" dT /dpo which implies that dpgldn >0, i.e. p; form a monotone increasing sequence. To establish p; < p; we first observe that T;>T;>T:(n>2). The last inequality follows since, as already mentioned, T: is monotone increasing with n. Now writing W_ (p,) from (4.51) we have “C(90) = 90 " 21.1 . (4-59) 1} 1 2H" (l-k)T; 2” " + Calculating W_ (p2) we find + ‘1'; (P2) = 2“ I: j: 2"” ' "' 1 1 2H” ] <09 (4'60) 1+.______ 1+ (l-k)T; 2” " (l-k)T; 2" 'l which together with (4.56) verifies our claim. 77 To summarize: a formal solution to the equilibrium equations has been found (See (4.1), (4.2), (4.ll)-(4.l4) and (4.16)). The stability of this solution in the two cases of internal and external pressure was examined. The investigation showed that for an internally pressurized cylinder the equilibrium is always stable in the range of interest, whereas under external pressure the tube becomes unstable. In particular, this implies that under external pressure, the tube buckles before the cavity closes. 0n the other hand,an internally pressurized tube reaches the "bursting pressure" before any instability is encountered. The existence of buckling pressures,ir|the former case, was proved (see remarks following (4.53)) and a sufficient condition for the uniqueness of the buckling loads corresponding to different modes established. Finally, the smallest buckling load for a cylinder subject to external pressure was determined. In the next section we illustrate some of these results using a particular constitutive relation. 5. ILLUSTRATIVE EXAMPLE In this section a special (hypothetical) harmonic material is introduced to illustrate some of the results found preViously. We recall that for harmonic materials the strain energy density function is given by w = 2u[H(R)-J]. In the following we generalize a particular power-law form of H(R) used by Knowles and Sternberg (1977) and suppose that l 2 R H(R) = ,2- R + FT (_%__)m + Pfi’ m30, m¢l, R>0. (5,1) Clearly H(R) is continuous and it is easy to verify that the restrictions (2.9)-(2.l2) are all satisfied. Taking (2.25) with pi = 0 in conjunction with (5.l), we find that the invariant RO is given by l HEW" _ . 2E:£(2Ll'po) R0 ’ 2 .[2u+po_k(2u_p°y] s (502) where k (a/b)2. (Recall that a,b denote the inner and outer undeformed radii of the tube.) Now from the first 6f (2.l), (2.20), (2.25) and (2.26) with pi = 0 one has = (l-kfilRa (5.3) .6. b 2u4k(2u-Pdl Equations (5.2) and (5.3) provide a relation between the applied external pressure po and the deformed outer radius 3. Graphs of po/2p versus B/b for different values of the hardening exponent In 78 79 and the geometric parameter t = b/a are shown in Figures (7) and (8) respectively. Moreover on using (4.34), (4.49), (4.50) and (5.2) (with positive sign chosen in (4.49) and the resulting T+ used in (4.50)) we may calculate the smallest buckling load (pg) of the externally pressurized tube. These buckling pressures are also marked on the graphs. As can be seen from Figure 7, the material hardens with increasing values of m. Moreover the buckling pressure also increasesas the hardening exponent increases. It is also evident from Figure 8 that the buckling pressure increases as the tube becomes thicker which is what one would expect. 80 .5 r m=5 m=l.5 m=.5 4 m=.125 m=.025 I 3 - .2 ~ 1 _ B deformed outer radius b undeformed outer radius a undeformed inner radius m: hardening exponent V: buckling load t = b/a = 3 0 L . 1 .6 .7 .8 .9 Figure 7. External pressure versus B/b for different materials. l.0 01m n‘SDJU‘U) t 8l t 5 II no t t 2 l.l deformed outer radius undeformed outer radius undeformed inner radius hardening exponent (= 1.5) buckling load = b/a Figure 8. External pressure versus B/b for different tubes. :00 l .0 01700 6. CONCLUDING REMARKS The equilibrium ofa pressurized tube of homogeneous, isotropic, compressible material of harmonic type was considered and the stability of the solutions investigated. In summary, the results show that a tube subjected to internal pressure will fail by bursting while an externally pressurized tube fails by buckling. For the latter case the smallest buckling load was calculated and the existence of buckling loads for higher modes proved. Furthermore, a sufficient condition for the uniqueness and monotonocity of these loads with respect to the mode number was established. The results found in this study pertain to a special material (harmonic) and a particular geometry (circular tube). Ideally one would wish to establish results with few restrictions on the consti- tutive law or geometry. This, however, is a formidable undertaking. (Analytical determination of the stress field in a circular tube composed of a general compressible homogeneous isotropic hyperelastic material is a daunting task!) There are more modest goals that one may pursue which can shed some light on these issues. These can be summarized as follows: (i) The effect of geometry; The question to be answered is whether the result found here for an internally pressurized tube (sudden bursting without any instability preceding it) depends on geometry or is a ”property" of the material. Study of tubes with noncircular; bores and/or noncircular outer boundaries would provide, at least, a 82 83 partial answer to this. It would be interesting to see if externally pressurized tubes with noncircular outer boundaries always become unstable by buckling and if so whether all the modes are generated. (ii) The effect of the material: Here there are two questions to be examined. First the effect of hardening. The fact that no buckling instability occurs in a circular tube subject to internal‘ pressure is apparently due to the very rapid hardening of the harmonic material. Examination of other compressible materials (hypothetical or otherwise) could provide some insight into this question.. It would be interesting to see if a "critical" hardening rate can be found such that materials which harden at a lower rate will "permit buckling" while the ones which harden at a higher rate will not. Secondly, it would be interesting to examine whether similar phenomena arise when the material is incompressible. APPENDIX APPENDIX A: Proof of Equation (4.36) We wish to show that k - hn>0 , for all3n 2, (A.l) where k is a real number 02. (A.2) n k"+k'n-2 ' The proof is by induction on n. We first note that for n = 2 one has 2 k-hz = “(“k) >0. (A.3) (1+k) Now assuming that k-hn >0 for any n>2, we have to show that >0. It is sufficient for this to show that hn is a 1 monotone decreasing function of n. To this end we define the k-hn+ function h(x) for all x>l by h(x) = (l-k)2/¢(x). (4.4) where X -X ¢l and therefore the problem reduces to showing that ]_ v (y) =2ny + 2 11% >0. for all y>l. (A.7) We observe that Y(l) = 0 and n- )2 Y'(y) = ——’L’Z >0 for all y>l. (A.8) Thus ¢(x) istmonotone increasing and hence h(x) is monotone decreasing. Therefore, hn is monotone decreasing and the result is thus established. LIST OF REFERENCES REFERENCES Chadwick, P. (1976), Continuum Mechanics, Concise Theory and Emblems _, Allen and Unwin, London. Chu, C. C. (l979), Bifurcation of Elastic-Plastic Circular Cylindrical Shells Under Internal Pressure, J, Appl, Mech., 4_5_, 339-394. Haughton, D. M. and Ogden, R. H. (l979(a)), Bifurcation of Inflated Circular Cylinders of Elastic Material Under Axial Loading-I. J. Mech. Phys. Solids, £1, l79-2l2. Haughton, D. M. and Ogden, R. N. (l979(b)), Bifurcation of Inflated Circular Cylinders of Elastic Material Under Axial Loading-II, ibid., 31, 489-512. Hill, J. M. (T975), Buckling of Long Thick-Walled Circular Cylindrical Shells of Isotropic Incompressible Hyperelastic Material Under Uniform External Pressure, J. Mech. Phys. Solids, 23, 99-ll2. Hill, J. M. (l976), Closed Form Solutions for Small Deformations Superimposed Upon the Simultaneous Inflation and Extension of a Cylindrical Tube, J. Elasticity, g, l25-l36. John, F. (l960), Plane Strain Problems for a Perfectly Elastic Material of Harmonic Type, Comm. Pure Appl. Math..i§3, 239-296. Knowles, J. K. and Sternberg, E. (1975), On the Singularity Induced by Certain Mixed Boundary Conditions in Linearized and Nonlinear Elastostatics, Int. J. Solids Structures, 11, ll73-l20l, Knowles, J. K. and Sternberg, E. (l977), 0n the Failure of Ellipticity of the Equations for Finite Elastostatic Plane Strain, Arch. Rat. Mech. Anal., 33, 321-336. Larsson, M.,Needleman, A.,Tvergaard, V. and StorSkers, B. (1982), Instability and Failure of' Internally Pressurized Ductile . Metal Cylinders, J. Mech. Phys. Solids, 39, l2l-l54. Malvern, L. E. (l969), Introduction to the Mechanics of 3 Continuous Medium, Prentice-Hall, Englewood Cliffs, N.J. , Reddy, B. D. (l982), A Deformation-Theory Analysis of the Bifurcation of Pressurized Thick-Walled Cylinders, 0. Jl. Mech.. Appl. Math.. 35, l83-l96. ' 86 87 Sensenig, C. B. (l964), Instability of Thick Elastic Solids, Comm. Pure Appl. Math., 11, 45l-49l. Timoshenko, S. P. and Goodier, J. N. (l970), Theoryfof Elasticitx. 3rd ed. McGraw-Hill, New York. 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