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"32113315333233;- 3 I ’9’3‘33’ - 33333‘ :I.‘3:333I3323¥3::!I " :31}, 'IIIIII 332333 ‘ I}; ‘ 1.3;522213II221153f 3322.2 ~u122u 22 2222233 Tfifib‘ls 2 000 lllllllllllllllllllllll\llllHilllllllllll 3 1293 02074 This is to certify that the dissertation entitled Buckling and Post-Buckling Analysis of Neo-Hookean Plates and its Correlation to a Direct Energetic Stability Analysis presented by Sangwoo Kim has been accepted towards fulfillment of the requirements for Ph . D . degree in Mechanics Wlflw M ljm> professor Date August 27, 1999 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE moo mim.po5-m4 BUCKLING AND POST-BUCKLING ANALYSIS OF NEO-HOOKEAN PLATES AND ITS CORRELATION TO A DIRECT ENERGETIC STABILITY ANALYSIS By Sangwoo Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Materials Science and Mechanics 1999 ABSTRACT BUCKLING AND POST-BUCKLING ANALYSIS OF NEO-HOOKEAN PLATES AND ITS CORRELATION TO A DIRECT ENERGETIC STABILITY ANALYSIS By Sangwoo Kim The elastic stability of buckling and post-buckling deformations for incompressible neo-Hookean rectangular plate subjected to a uni-axial thrust is investigated. The buckling deformation is described by the small deformation superposed on finite homogeneous deformations. Throughout the investigation the thickness of the plate is not limited. The resulting nonlinear boundary value problem is analyzed by using the perturbation expansion method in which an associated linear problem is solved at each order. Buckling onset is determined from the first order expansion and can occur in either flexural or barreling mode shapes with any integer number of half wavelength in the direction of thrust. The solutions from the higher order expansion correspond to post- buckling deformations. The higher order problems inherit information from problems of the previous order, both by the expansion procedure and by the application of mathematical solvability conditions. The stability criterion for post-buckling deformations is based on energy competition between the buckled deformation and unbuckled homogeneous deformation in the vicinity of buckling onset. It is formally established that the energetic favoribility correlates with the load following character of the buckled solutions (progressive buckling vs. snap buckling). Based on the expressions obtained by these procedures, it is found that the flexural buckled deformation is energetically favored over the unbuckled homogeneous deformation when mode number is small, otherwise the homogeneous deformation is favored. The barreling buckled deformation is always energetically favored over the homogeneous deformation. This contrasts with previous results of Sawyers and Rivlin (1982), who obtain essentially opposite results for the elastic stability of homogeneous deformation. The approach to evaluate the elastic stability by using the perturbation method considered in this research gives more insights to understand the buckling phenomena and is systematically applicable to higher order analysis. Besides the main topic of stability evaluation, several approximate schemes for the critical buckling load in neo-Hookean three-ply sandwich type plate were developed in view of practical application. The schemes are based mostly on the Rayleigh quotients approach and trial solutions. These schemes can be expanded to general multi-ply composite plates and so reduce the effort to determine the critical buckling load. To my parents for their endless love, support and teaching me the value of challenge and perseverance iv ACKNOWLEDGEMENTS Thank God for giving me the opportunity to write this acknowledgment at last. He always guided and strengthened me through His Words and taught me the values of humility and integrity. During the course of my graduate school experience, I have learned a lot about the knowledge of applied mechanics as well as the life itself from many persons. The first person I should thank was my respectful advisor Prof. Thomas Pence who accepted me, taught me the detailed works of scientific research and advised me with valuable tips of life. Your practical remarks such as “when you stuck, step back and consider a simple example” and “follow the principles” will exist firmly in my memory. Also you showed me the importance of honesty and integrity and I apologize to have you hard times due to my insincerity. I appreciate your continuous encouragement to me when I am suffering. My thanks also goes to my dear committee members - Prof. Dashin Liu, who made me to think about the theoretical and experimental works, Prof. Ronald Averill, who taught me the applied methods of mechanics and the fun of scientific jokes and Prof. Baisheng Yan who granted me the confidence for my works and made me to continue. They deserve my gratitude for their valuable supports and contributions to my work. Very little of this work could have been accomplished without the help of my family for their continuous support and encouragement. I am thankful to my father and mother, Chang-ll Kim and Sung-Si] Baik for their enormous pray and priceless support. I hope I will have a chance to recompense their sufferings. Also I thank to my parents-in- law Hong-Soon Shin and Wol-Hwa Kim for their timeless pray and rigid assurance to my work. Special recognition should be bestowed upon my close family members. I cannot find how to thank to my only blood brother Prof. Junwoo Kim who rescued me from the temptation to give up all my works by numerous emails and phone calls. His existence have given me the strength to overcome every difficulties. And I owe everything to my precious wife Kyoungsook Shin, who always showed the confidence in me and made me comfortable in home. In addition, I deeply thank you for your patience. My first son David made me to consider what is my problems and twin sons Samuel (Gom) and Daniel (Kan) gave me not only sufferings but also the pleasures and reasons to live. The contributions you all gave to my life are indescribable. I also give my thanks to the staffs in Dept. of Materials Science and Mechanics for creating warm and supportive atmosphere. Many friends and elders in Korean community also deserve my thanks for their supports and advice to me and my family during our stay in this town. Finally I want to give my appreciation to the people who maintain the nature centers in Lansing area. In these places, I have relieved, thought, made a decision and recovered my confidence. It is so fortunate for me to have these places in a close distance. Now I am standing top of one small hill in the mountains. Soon I will climb up another hill and may suffer the hard times. However I believe my experiences will help much to overcome these obstacles. Oh God, bless me and my path, and strengthen me just as you have given to me. vi TABLE OF CONTENTS LIST OF TABLES ...................................................... ix LIST OF FIGURES ...................................................... x 1. INTRODUCTION ..................................................... 1 1.1 Overview ....................................................... 1 1.2 Literature Review ................................................ 4 1.2.1 Elastic Stability ............................................ 5 1.2.2 Buckling Instability of Composite Plates ....................... 13 1.3 Thesis Organization .............................................. 14 2. PRELIMINARY WORKS ON NEO-HOOKEAN PLATE ..................... 18 2. 1 Introduction .................................................... 1 8 2.2 Problem Descriptions ............................................. 18 2.3 Bifurcation from Homogeneous Deformation .......................... 22 2.4 Energy Minimization of the Deformed Configuration ................... 26 2.5 Summary ...................................................... 27 3. BIFURCATION ANALYSIS BY PERTURBATION EXPANSION METHODS. . . 28 3. 1 Introduction .................................................... 28 3.2 Perturbation Expansion Methods .................................... 29 3.3 Analysis on Neo-Hookean Plate .................................... 31 3.3.1 Linear Differential Operator ................................. 34 3.3.2 Different Formulations ..................................... 35 3.4 Load Parameters ................................................ 37 3.5 Auxiliary Conditions Associated with Incompressibility Constraint ........ 39 3.6 Relation between Thrust and Load Parameter .......................... 41 3.7 Energy Formulation .............................................. 42 3.8 Summary ...................................................... 43 4. BUCKLING DEFORMATIONS FOR NEO-HOOKEAN PLATE ............... 44 4. 1 Introduction .................................................... 44 4.2 General Solution for the Governing Linear Differential Operator .......... 44 4.3 Nonhomogeneous Ordinary Differential Equation ...................... 49 4.3.1 Nontrivial Solutions to Homogeneous Problem .................. 49 4.3.2 Solutions to Nonhomogeneous Problem ........................ 53 4.3.3 Solvability Condition ....................................... 55 4.4 Buckling Onset (the First Order Solution k=1) ......................... 57 4.4.1 Buckling Initiation by F lexure and by Barreling .................. 58 4.4.2 Load Parameters Associated with Buckling Initiation ............. 60 4.4.3 Asymptotic Expressions for Load Parameters .................... 63 4.5 Post-buckling Deformation (Second Order Solution) .................... 70 4.6 Summary of Full Buckled Deformations .............................. 79 vii 4.7 Reduced Formulations Due to Symmetric Load Parameter ............... 80 4.7.1 Load Parameters .......................................... 80 4.7.2 Energy Equations .......................................... 82 4.7.3 Energy and Load Parameter .................................. 89 4.8 Summary ...................................................... 91 5. STABILITY EVALUATION FOR A NEO-HOOKEAN PLATE ............... 92 5. 1 Introduction .................................................... 92 5.2 Formal Determination of Stability Parameter .......................... 94 5.3 Numerical Determination of Stability Parameter ....................... 97 5.4 Asymptotic Study for Stability Parameter in F lexural Buckling at Low Mode . . ..................................................... 100 5.4.1 Critical Load in Classical Euler Buckling ...................... 101 5.4.2 Buckled Deformations with Asymptotic Equations .............. 104 5.4.3 Stability Parameters with Asymptotic Equations ................ 110 5.5 Discussion .................................................... 1 14 5.6 Summary ..................................................... 1 15 6. APPROXIMATE SCHEMES FOR BUCKLING LOAD OF MULTI-LAYERED COMPOSITE PLATES .............................................. 117 6.1 Introduction ................................................... 1 17 6.2 Buckling Load of Multi-Layered Plates ............................. 118 6.3 Approximate Schemes to Determine Buckling Load ................... 123 6.3.1 Variational scheme ....................................... 125 6.3.2 Trial solutions ........................................... 126 6.4 Discussion .................................................... 129 6.5 Summary ..................................................... 131 7. CONCLUSIONS AND RECOMMENDATIONS ........................... 132 7.1 Conclusions of the Thesis ........................................ 132 7.2 Recommendations for the Future Work .............................. 134 APPENDIX A. STABILITY EVALUATION BY PERTURBATION EXPANSION METHODS ........................................................ 137 A. 1 . Introduction ................................................... 137 A2. The Perturbation Expansion Method ................................ 139 A3. The Analysis Scheme by Sawyers and Rivlin ......................... 141 A4. Example 1: Elastica Problem ...................................... 141 AS. Example 2: Modified Elastica Problem .............................. 151 A6. Example 3: A Higher Order Problem Represented Neo-Hookean Buckling . . 154 A7. Discussions ................................................... 159 APPENDIX B. COEFFICIENTS FOR STABILITY PARAMETER .............. 163 BIBLIOGRAPHY ...................................................... 168 viii LIST OF TABLES Table 6.1 Summary of complete conditions and satisfaction for various schemes 131 Table A.1 Summary of the procedures and their corresponding equations for the perturbation expansion. ..................................... 161 Table A2 Summary of the procedures and their corresponding equations for the Sawyers-Rivlin method ...................................... 162 ix Figure 2.1 Figure 2.2 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 5.1 Figure 5.2 Figure 5.3 LIST OF FIGURES Description of the neo-Hookean rectangular plate under consideration. The thrusts T are applied to the ends of surfaces at X1=ill and the plate has a dimension of 211x212 x213. ............................... 19 The relation between the scaled thrust T5 and load parameters p and A. The thrust is the compressive load so for this study, the load parameter is restricted to 0] .5, asymptotes is quite close to exact expression ...................................... 69 Three-term asymptotic equation of load parameter p0 for barreling deformation when 11 goes to infinity. When n>2.5, asymptotes is quite close to exact expression. Otherwise, two expressions show totally different values. ............................................ 70 Dimensionless stability parameter RS=R3/(ul,M4§23) for the flexural deformation in (5.14) with v=1. At n=1.6283, the curve RS has a discontinuity. Right before this, it is positive and after this, R5 is negative. There are also sign changes in 11:0.6443 and n=1.305. Their details are shown in Figure 5.2 and Figure 5.3. ............................ 98 The detailed curve of dimensionless stability parameter RS for the flexural deformation in (5.14). At 11=nc=0.6443, the sign of RS changes from positive to negative and at 11:] .305 the sign changes from negative to positive. More detail near n=nc is shown in Figure 5.3. ............. 99 The detailed curve of dimensionless stability parameter R5 for the flexural deformation in (5.14). At n=nc=0.6443, the sign of RS changes from positive to negative. Hence when 112’2 ) 213 211 Figure 2.1 Description of the neo-Hookean rectangular plate under consideration. The thrusts Tare applied to the ends of surfaces at X1=i11 and the plate has a dimension 0f 2]] X212 X213. The current configuration is defined by undeformed or reference coordinates so that the deformation tensor is given by x = x(X), (2.1) where X(X,,X2,X3) and x(x,,x2,x3) are undeformed and deformed coordinates, respectively. The deformation gradient tensor and Green’s deformation tensor are given as F = (ax/6X), B = FFT. (2.2) The condition of material incompressibility requires that the volume does not change afier deformation so that 19 det F = 1. (2.3) The Cauchy stress tensor for the incompressible isotropic hyperelastic material is then given by =_ 6W 5W) .. (6W) 2 where p is the hydrostatic pressure due to the incompressibility constraint, 1, and 12 are the first and second invariants of B and W = W(I,,I;) is the strain energy density function of hyperelastic material. The symmetric Cauchy stress tensor 1: described in a current configuration is transformed to the Piola-Kirchoff stress tensor S in a reference configuration which is given by s = F41, (2.5) after applying the incompressibility condition (2.3). The equilibrium equations in a reference frame are then expressed as DwsT=o. (am The plate under consideration is subjected to a compressive load on each surfaces X r = i1 1 . The boundary conditions for a frictionless thrust with an overall stretch ratio of pare S12 = SD = 0, on X] = ill, (2.7) x1 = ipl, on X1 = i1,, (2.8) where the case of compression requires 0 1 . 2.3 Bifurcation from Homogeneous Deformation Motivated by boundary condition (2.11), we restrict attention to states of plane- strain buckling taking place in the (X ,,X2)-plane. The buckling can be described as the bifurcation from the solution of homogeneous deformation (2.14) so that the incremental deformations of buckling are superposed on finite homogeneous deformations. The fully finite deformation is then expressed as X] = PXI +v,(X,,X2), p“X2+ v2(X..X2). (2.18) X3 = X3, x2 and the pressure field is accordingly 22 TS ' w T s i | 4 . I 8 _ I l I l . 2 . comp essroh I . 6 r l , I COITIpI'CSSlOI'I ' o 4 . I l l I I '2 * I 2 r I i i | _4 _ l 0 ' tension ' l l '6 I -2 1 I I : tension -8 : .4 l | I 4 -10 1 - 0 1 2 3 0 1 2 7. 3 p (a) (b) Figure 2.2 The relation between the scaled thrust T s and load parameters p and A. The thrust is the compressive load so for this study, the load parameter is restricted to 0(X..X2>. (3.1) 9(X1rX2) = Zk_,3kp(k)(X1.X2)- In vector notations with v=(v,,v2,q) and u‘k)=(u,("),u2("),p(")), the deviations v away from the homogeneous deformation state are expressed as v = sum + azum + e3u(3) + .... (3.2) Here a is a measure of the amount of deformation away from the homogeneous solution which is defined as 29 e = (v,u“))/(u('),u“)), (3.3) in accordance with the orthogonality condition for the vector functions of each order 11") and u“) (u<1>,um) = o, j¢ 1. (3.4) The brackets ( , ) denote the bilinear inner product on pairs of vector functions in a domain 1'1 so that __ 1 cm» — mliflmmdndxz. (3.5) In order to acknowledge the evolution of the postbuckling path with the thrust, the overall stretch ratio p is also expanded from the stretch ratio on buckling onset p0 as P=%+2bfwr do The stretch ratios p or 1. represent the load parameters as shown in Figure 2.2. and their expansions are related by the expansion of equation (2.17) such that lo = P62, M = 4135391. 12 = -pa3(202-3pa'pi), (3.7) 13 = —2963(p3-3pa‘p1pz+295291), where A=AO+SA1+82A2+83A3+0(84). For the simpler analysis, the expansions of p will be used but for the interpretation of the thrust load, I. will be used afier converting by the relations (3.7). If the odd terms in (3.6) vanish, pk=0, k=l,3,5,..., then the deformation bifurcates symmetrically with respect to the trivial solution path. The symmetric buckling which is common to the perfectly loading plate and the analysis much easier than the case of asymmetric buckling. 30 3.3 Analysis on Neo-Hookean Plate Entering the boundary value problem (2.28) with the expansions (3.1) and (3.6) and collecting together common orders of a, give rise to the following general linearized boundary value problem at each order 8": “(“lfil + 1492) - Pfilpfi‘) = fl")(X,,X2), mum+amrmwa=awana. inn.~ cw) Pfilulii + Pouiii = fSk)(X1./X2), ulk) = 0, “(um +p52ugf5) = 0, on 1“,, “(149+ pazulf‘l) = gi")(X,,i12), on r (3.9) “("53 -Pazui{‘l)—pop<") = g£*)(X,,i12), 2’ here k=1,2,.... The second in F, boundary conditions can be reduced to ugf), = 0 because the first condition gives riff} = O. Primarily the above equation is a nonhomogeneous second order partial differential equation except for order a (k=1). The expressions on the right hand side in (3.8) and (3.9), f“) and g“), depend upon the lower order solutions uU) = [ufl'h up, pm] and pj,j 933: G,u=0 on 1“,,qu = O on F2} and let u e Hh(l‘I) be continuously differentiable. The subscript h as used here is to indicate homogeneous boundary conditions. Notice also that Hh(I'I) is dependent on p0 because the boundary operators G, and 62 depend on p0. In this Section 3.3, we disregard the order superscript k in the equations for simplicity. Let F be the second order differential operator defined in (3.14) and consider (Fu,1—1) = 3717A ln[(Fu)Tfi]dX,dX,. (3.17) Then integration by parts twice upon (3.17) with respect to the variables either X, or X2 yields (F0. ii) = 4—llszl irziiMuL 1 " P62“; 2) - P6'Pll-‘1 + “(142,1 + P62”1,2)§21|rldX2 + irllll(u1,2 + P62u2,1){‘1+{11(u2,2-P62u1,1)- Pupil-lzilrzdxi —J'1‘2[{ M171, 1 - P132522) - P6113} “1 + “(172.1 + P6251,2)u2] ir,dX2 (3.18) -irl[11(l-l1,2 + P132172, ,)u, + {P(l-42,2—P62{‘1. 1) — P0P} “2] lrde‘ +iiniu1{11(1-41,11+ 171,22)- PEIPJ} + “zillU-lz, 11 + l72,22) " P0132} +P(P61{l1,1+ P0172,2)]dX1dX2]- Applying the boundary condition requirements on u inherent in Hh(I‘I) to (3.18) shows that the second of the two terms in the first integral vanishes, as does the entire second integral, and the first of the two terms in the third integral. Hence 34 _ 1 , - _ _ (F0. 11) = miLliMMn’P6‘“2,2)—P61P}“1—P(u2,1 + P62u1,2)u2]|rldX2 _ ' -2' " _ —2" _ " Ir,[p(u"2+p° “21)"1 + {11042.2 P0 ”1.1) Papiuzllrdei (3.19) + iinlu1{P(i-l1,11+51,22)‘P6‘P.1} +u2{P(l—12,11+l72,22)‘P0F,2} +P(P6"-11,1+ Pol-‘2,2)]dX1dX2]- This defines the adjoint operator to the triple consisting of the field operator F and the boundary operators G, and G2. Let the associated adjoint operators be P" and G,°, 02°. Then the integral over F, shows that G ,°=G ,, the integral over F2 shows that GZ°=G2 and the integral over II shows that F‘=F. In other words, the linear differential operator F restricted to Hh(I'I) is self-adjoint. 3.3.2 Different Formulations The nonhomogeneous boundary value problems (3.14) and (3.15) can be formulated in different ways (see Reddy, 1986). The weak (variational) formulation is stated as follows: find u=(u,,u2,p) e H(IT) such that B(w, u) = l(w), for all w e {(w,, w2,r)|w,=0 on F,}, (3.20) where B(w,u) is the bilinear form and l(w) is the linear form given by B(w, u) = HHH[(“1,1W1,1+ “1,2W1,2 + “2.1%, 1 + “2,2W2,2) + P62(u1,2W2,1+ "2,1W1,2 - “2,2W1,1- “1,1W2,1)]dX1dX2 (3-21) -i.in[P(P6'W1,1+ PoW2,2) + '(P61“1,1+ Po“2,2)]dX1dX2- [(W) = —,i,in(w,f, + wzf2 + w3f3)dX,dX2 + in (g,w, + g2w2)|r2dX,. (3.22) The solution space H(I‘I) indicates the boundary conditions that should be satisfied 35 H(IT) = {(u,,u2,p)|G,u = O on F,' qu= g on F2}. (3.23) Note that B(w,u) is symmetric, i.e. B(w,u) = B(u,w). For sufficiently smooth functions, the weak formulation (3.20) is equivalent to the direct formulation (3.14) and (3.15). The energy functional on H(FI) corresponding to symmetric B(w,u) becomes (Reddy, 1986) E(u) ll %B(u, u) - l(u) = gliniuin + “12,2 + ”in + ”£2 + 2P62(u1,2u2, 1 - “1,1“2,2)]dX1dX2 (3.24) ’“n [P(P6'u1,1+ P0112,2)+ (“L/1 + uzfz + “3/3)]dX1dX2 —IF1 (8W1 + g2u2)|r2dX,. The energy (functional) formulation is to find u eH(I'I) which minimizes E(u). If B(u,u) is positive for u eH(FI), then the weak and energy formulations are equivalent. In the problem under study here, it is not clear under what circumstances B(u,u) is positive. The first variation of E(u) gives (3.14) as its Euler equation, when the following boundary conditions are specified: u(u -—p'2u )—-p"p = 0 or u = 0 on F, 1,1 02 2.2 o 1 (3.25) P(“2,1+P6 “1,2) = O 01' “2 = 0 F “(“1,2+P62u2,1) = 81 01' “I = 0 (3 26) on . 2 P(“2,2‘P62u1,1)—P0P = 82 or “2 = 0 In the condition (3.25) and (3.26), the right sides correspond to essential boundary conditions and the left sides to natural boundary conditions. Comparison of (3.25) and (3.26) with (3.9), shows that the boundary condition u, = 0 on F, of the direct formulation contributes the only essential boundary condition. 36 3.4 Load Parameters The load parameters p represent buckling behavior on and after the buckling initiation. The homogeneous problem (3.16) for the case of k=1 will only have nontrivial solutions for certain special values p0 which define buckling initiation modes. At these special values p0, the differential operator trio {F ,G,,Gz} is singular. The same differential operator trio as the homogeneous problem appears in (3.14) for the case of k=2,3,..., and the special values p0 are used here. These cases of k 2 2 will be a problem for solving nonhomogeneous boundary value problem (3.14) and (3.15) for a singular operator trio {F ,G ,,Gz}. For most right hand sides, solutions will not exist. But for certain special right hand side of equation, the solution can exist - eventually this is explained by the Fredholm Alternative Theorem for solvability of the nonhomogeneous equation as developed next for this particular problem. At a fixed value of po, let w=(w,, wz, r) be a nontrivial solution of the homogeneous equation (3.16) and let u=(u,,u2,p) be a solution to the nonhomogeneous equations (3.14) and (3.15) for given 1‘ and g. Then consider the expression (Fu,w)— (u,Fw) = fin“{(Fu)Tw—uT(Fw)}dX,dX. (3.27) Since Fw = 0 and F u = 1' in F1, the left side of the equation (3.27) is equivalent to (f, w). After integration by parts twice and applying the boundary conditions G,u = 0 and G,w = 0 on F, and qu = g and sz = 0 on F2 to the right hand side of equation (3.27), it becomes 1 1 l (‘3 W) = m[,l(W181+ W282)|_2,2dX1~ (3-23) 37 Thus if the nonhomogeneous problem (3.14) and (3.15) is to have solutions, then it is necessary that f, g, and g2 obey the solvability condition (3.28). In particular, since u“) is a nontrivial solution of the homogeneous equation (3.16), any nonhomogeneous solution (u,“",u2“",p(")), k=2,3,... to (3.14) and (3.15) for given fl") and g“) must satisfy llnwumdx— l, gr>+ urgent dX. = o. (3.29) In order for the nonhomogeneous problem of order 82 (k=2) to have a solution “(2), the condition (3.29) must be satisfied with corresponding terms 1(2) and gm in (3.11). Substituting 1‘” and gm into condition (3.29) gives R1+ P1R2 = 0. (3-30) where R, and R2 are constants defined as R. = ”niui”(p“)ul,‘l),.—u£"(p"’ui,'l),,]dx + I in 2P"’)+p9>-p<*>us,'1>1dx — zupaln (ui”u$,‘l - um usnnrzdx. - inpmusuirzdxr (3'32) If R2 is not zero, then p, can be expressed as p, = —R,/R2. (3.33) Solutions “(2) to the linear equations of order two, will exist only if the equation (3.33) is satisfied. Similarly for the nonhomogeneous problem of order 83 (k=3) with 1‘” and g”) in (3.12), the solvability condition (3.29) for the existence of “(3), gives R3+R4P1 +R5P12+R2P2 = 0, (3-34) 38 where R3, R4 and R5 are R. = lintui‘Kp‘Z’ulfl +pu5?1>,,— ut"(p‘2’ui,'l +p<'>u1?1),, + 2p‘”(ui,'lu§,2i - uiiium) (335) + u£}I(p“’ui?l +p<2>u1,'r)-u1}1(pus?) +p<2>ur,'))1dx. R4 = linl-p52(ui”p,‘12’-ui?lp“’) +(u£”p52’-p“’u$?l)]dx (3.36) — 211p53ir1(u1'>u1?1 — u1?1us'>)|,2dX.- lrlp<2>uil>lrzdxh R5 = linlpa3(ui”pfi”-p“’ui,‘l)1dX+3upa4irl(ui”u$,‘l —u£”ui,'l)|r2dX1, (3.37) so that P2 = —{R3 + (P1R4 + PiR5)}/R2, (338) if R2 is not zero. The higher order parameter p,, i=3,4,5,..., can be obtained in a similar way. Note that solving for a specific pN requires full determination of um’s, i=1,..., N. In particular, the conditions (3.33) and (3.38) must be satisfied for the existence of solutions u“) and “(2). The freedom to choose the coefficients p ,, p2,... in (3.6) are used to meet the solvability condition (3.28). Each term p,- in the expansion will give a key to the postbuckling behavior of the system. Budiansky (1974) has discussed the mathematical structure of general post-buckling problems through variational analysis and shows that equation of the form (3.33) and (3.3 8) can be expected in the general case. 3.5 Auxiliary Conditions Associated with Incompressibility Constraint A set of extra conditions from (38);, and (3.9), are derived for simpler calculations of necessary formulations. First double integration on (3.8); gives 39 1351”” ude+ pollnugfwx = ”f/S’de, (3.39) and on (3.9)l gives ”nuif‘ldx = Ilik’lridXz = 0, (3.40) for k=1,2,3...., so that ””2455ng = p51 llnfgwx. (3.41) Using the expressions for f3“) given in (3.10) to (3.13) yields ”n “iiidX1dX2 0, “it “EidXflIXz = PEICDI’ (3.42) ”n “iiidX1dX2 = - P1P62¢1+ p51¢2, ll” ult‘lXmdXz = P62(Pi96' - P2)(D1_ ptp52¢z + 95W.» where «D. = lln(ui,'lu£,'l -u131u53))dxldxs. (3.43) «>2 = ”n[(ui,'lu£?l + uiilu£}I)-(ui,‘1ul?l + ulflulflfldXthz, (3.44) <1). -- M [(434.31 + anus?) + mum) n (3.45) 411131145?) + 111,21 "El + ”Pl “Find/Y1“:- Here the results of each step in (3.42) was used for the calculations of subsequent step. 40 3.6 Relation between Thrust and Load Parameter After bifurcation away from the homogeneous deformation solution (2.14) and (2.15), the associated values of the thrust T is obtained by substituting the expansion (3.6) into the expression (2.16) as follows: T = -4u1213[(po - 953)+(t>1+ 3P6401)€ + (P2 + 3p6‘pz - 695591082 + (P3 + 3P64P3 - 1213559192 + 10966pi)e31+ 0(34) (3.46) 5 TO + ST, + 2:sz + 83T3 + 0(84). The first term T 0 represents the thrust at buckling onset and the other terms in (3.46) will show the change in thrust load after buckling. If the terms except To have a positive value, then the thrust must increase to get a larger buckling deformation after buckling initiates - progressive buckling. For the opposite case, the thrust must decrease when the buckling deformation grows so that there is a possibility of snap buckling in which the buckling mode jumps to another mode. If the bifurcation growth is symmetric with respect to the trivial solution, then p,=p3=...=0 so that the thrust (3.46) becomes T = — 41213P[(Po — p53) + p.(1 + 53.):2] + 0(24) 0 = To + {:sz + 0(84). (3.47) For this symmetric case, if p2 <0, then T is an increasing function of s so that the buckling is progressive. Otherwise, if p2 >0, then T is an decreasing function of a so that the snap buckling is possible. 41 3.7 Energy Formulation The energy equation (2.30) may be expanded accordingly. Substituting the perturbation expansion (3.1) and (3.6) into the energy functional (2.30) and applying the incompressibility (3.8)3, the energy functional for each order becomes AE = 213(3E, + ezEz + 83E3 + 8454) + 0(85). where E ,, E2, E3 and E, are given as follows: E, = ”nMpa'ulfl + poni31)dX1dX2, E. = ”nulp1(ui,‘l — 95214le (pour) + Pa'ui?3)1dXthz + 551.. E3 = ”filliiqulii + P63(P12 " POP2)ui,li} + 91041.21 - pazulil) + (9014,31 + pa'u£?l)ldX1dXz + #512. E4 = ”null psuifl + p6“(- of + 2909192 - 939044535} + {92141.21 + 963(Pf - popz)ul?l} + 9.042) - 9521453) +(pou1‘31 + pa'usfiiidXtdxz + gs... + #313- Here we have introduced the notation: at} = ”nWlpiulfh + ulfzulfb + “if’1u§{)1 + ugf)2u§{)2)dX,dX2. (3.48) (3.49) (3.50) (3.51) (3.52) (3.53) Now direct application of the extra conditions (3.40) to (3.42) into the equations (3.49) to (3.52) gives the following simplifications 42 (3.54) (3.55) 133 = u((-291953)¢1+ 962432) + #312, (356) E4 = Iii P63(3P12P6l " 2P2)+S.(l/C.dX2+L2)} (4.40) + p31C21iisde2 + 17,) + 52(‘ liczdxz + 472)}1. B(Xz) = The function f is specified in the right hand side of (4.11). The complete solution B contains four as yet undetermined constant coefficients E, 1:, 117,, 117, that are available to satisfy the four boundary conditions (4.17). Substituting the solution (4.40) into these four boundary conditions (4.17), the four equations for these constant coefficients are expressed in matrix form as: C4X4J4xl = G4,,“ (4.41) where T J = [L1 ,2 M1 M2] , (4.42) 54 92 F 2 ~ ~ - ' é, (1+ pmro + L3811+ 296“{M1*C2 + 41352} G ___ of _ (1+964){L1C1-L5251}+29641MIC2-M552} (4,3) g2 29641Li51+L2+C1i+962(1+064){M152+M3C2} _2P64 'L151+L2C1} + 962(1 + P64)l—M152 +M2C2}d and the coefficient matrix C is the same as that appearing in the homogeneous problem (4.20). The fianctions with superscripts + or - denote that the functions are evaluated at X2=12 or -12 respectively. Note that the matrix C is singular if and only if p,, satisfies either (4.25) or (4.29). If p,, does not satisfy either of these equations, then (4.41) yields a mnque vector expression for J. On the other hand, if C is singular, then (4.41) gives solutions if and only if G is orthogonal to the null space of CT which generates the solvability condition as we will show in the following section. For this singular case, if the solvability condition is not satisfied, then there is no solution to the problem (4.16) and (4.17). If the solvability condition is satisfied, then there exist infinite solutions which is shown in (4.40). The constant vector J can be obtained by using the psudoinverse matrix of the singular matrix C or by connecting the solvability condition to (4.41). 4.3.3 Solvability Condition r The existence of a solution J = [L 1 L2 M1 M2] for the linear algebraic equation (4.41) when p,, satisfies either (4.25) or (4.29) requires satisfaction of an orthogonality condition with the right hand side vector G and the null space of CT. If p,, obeys (4.25) which corresponds to the symmetric case, then the null space of CT is given by 55 ~ ~ ~ ~ T NSS=Q3(1-p54) s, 4 514 C14 C14 . (4.44) 1+P6 1+Po 2130 2130 If p,, obeys (4.29), which corresponds to the antisymmetric case, then the null space of C T is given by ~ ~ ~ T C C s 5 NS = Q3(l—p‘4) I I I 1 . (4.45) A 0 [1+0641+96429642P64] Together the orthogonality of the null space and the matrix G can be written 67- NSSorA = 0, (4.46) and so yields the following solvability conditions with the relations (4.25) and (4.29). For the case associated with (4.25) and hence a symmetric nontrivial homogeneous solution, [2 i~C CdX=1 4C1~+QS‘~ 447) r2f(1—S 2) 2 —( —Po )[2p—6482 mgi] . (~ ..[2 For the case associated with (4.29) and hence an antisymmetric nontrivial homogeneous solution, S] ~ QC] ~ ]|12 -l £27151 - 552)dX2 = —(1 — Pfi4)[m82 + (1756—4381 (4-48) Thus, if there exist nontrivial solutions 8,, to homogeneous versions of (4.16) and (4.17), then there exist solutions to (4.16) and (4.17) with nonzero f, g, g2 if and only if these functions satisfy either the condition (4.47) or (4.48), as appropriate The solvability conditions (4.47) and (4.48) can be also obtained directly from the ordinary differential equation problem (4.16) and (4.17) via similar procedure developed 56 in Section 3.4 to get (3.28) for the partial differential equation problem. For the problem (4.16) and (4.17), exchanging u to 8, w to 8),, and f to f in the formulation (3.27) gives (BB, Bh> — <13, B8,) = 3711,31. [(138)81. — B(BBMdXZ. (4.49) 2 Here F is the differential operator used in the left hand side of (4.16) so that i: B = f and 133,, = 0. Applying the integration by parts and boundary conditions in (4.17) yields [2 ~ 1 ~ .. (£31.) = —(ngh‘gIBh') - (450) 41,12 —1 Then substituting B, in (4.26) and (4.30) into (4.50) gives the same conditions as (4.47) and (4.48) after rigorous calculations with the relations of (4.25) and (4.29). Hence the condition (3.28) represents solvability at the partial differential equation level, the condition (4.50) represents solvability at the ordinary differential equation level and conditions (4.47) and (4.48) represent solvability at the linear algebra level. The three conditions (3.28), (4.50) and either (4.47) or (4.48) for solvability are perfectly matched to each other. 4.4 Buckling Onset (the First Order Solution k=1) The first order solution u“)=(u,(”, uz“), pl”) of homogeneous boundary value problem (3.16) represents the status of buckling initiation and was investigated by Sawyers and Rivlin (1974). Since all the right hand side terms 1“) and g“) vanish, the terms 1,, J,, K, in (4.7) and G, H, in (4.8) as well as the right hand sides of the equations (4.11) _and (4.13) vanish. Therefore the boundary value problem for the case of k=1 becomes 57 Bf;1.””—912.n(1 + P6039?" + 914;..pr ’ = 0. (4-51) (1 IL” 81.1.2“ 024541312 0. F - 4.52 31.L"”-Q?;.(1+2064)B,‘.‘,2'—0, °“ 2 < > In view of the relations (4.12), the other coefficients in (4.3) become 2 A112 = 5" 8112', C112= up—§(Bf.1"'-Qin8112)'- (4.53) i;n aim The first order solution of (4.51) and (4.52) can be derived from the solution of homogeneous problem in (4.26) for the symmetric case, and the solutions of (4.30) for the antisymmetric case by substituting 85],) into B), and adding subscript i to Q and 7]. 4.4.1 Buckling Initiation by Flexure and by Barreling Two different kinds of solution in view of symmetry represents two shapes of deformation: flexural and barreling deformation. For the flexural deformation, the lateral deflection uz“) is symmetric with respect to X2, so that B};‘,,)(X2) is an even function. It follows essentially from (4.25) - (4.27) as 811209) = M1cosh(n.-..Xz)-s.,.cosh(n.,.paZX2)1, (454) under the solvability condition of (1 + p64)2tanh(nmpaz)—4pa‘5tanhn,-.. = 0- (455) Here 1],”, which is given from (4.1) and (4.23) as 11,," = lez. The constant M denotes the amount of buckling from homogeneous deformation so that M will be determined according to the normalization convention. The value of M will be determined later in this 58 section. For the barreling case, the lateral deflection is antisymmetric so that B§,',,)(X2) is odd function and expressed from (4.29) - (4.31) as 31,1206) = Mlsinh(Q.-;nX2) - S.;nsinh(91;nt>62X2)l, (456) under the solvability condition of (1+ pa“)2tanhn.-.n-4paétanh(n,~..paz) = 0. (4.57) Nontrivial solutions will only exist, at fixed mode number n and initial geometries I, and 12 (which will be shown in 0,," and 11,"), for particular values of p,,. The conditions (4.55) and (4.57) show the relations between p,, and 11,," which represent the load parameter curve. The aspect ratio 5 for both types of deformation is _(1+ 1» ) sinhzn... 2p0 Sinh“ + 962M”, + vsinh(1— p62)n,m’ (4-58) i;n where v=1 (v=-1) for flexural (barreling) deformation. The value of the constant coefficient M in either (4.54) or (4.56) is determined from normalization for which we define as 4171,,” [(11,113 (Lf:’)2+(8,‘1'_’)jdx,dx, = 1- (4.59) This is in contrast to the normalization used by Sawyers and Rivlin (1982) who instead require that u§‘)(l,, 0) = :1 for flexure and u5')(l,, 12) = i1 for barreling deformation. Their results are simple and procedures are relatively easy by introducing above special rules of normalization. But in this work, we will follow the definition of normalization, Hum" = 1, as shown in (4.59). Applying the solutions (4.2) for k=1 to (4.59) with the relations (4.53) and integrating over F, give 59 ii ( p6 B"")2+('B— "2+3 pg 1”"— 92 B“)' 2 (1X — 1 460 412 F2 (T3171 1;" [2 Dim i;n 1;") 2 _ ° ( ' ) Then substituting the solution expression for B,_.,,“) in the separate cases of flexural deformation (4.54) and barreling deformation (4.56) and the aspect ratio 5 in (4.58) into (4.60) and applying the solvability conditions (4.55) and (4.57) give the following equation, M2 2 4 nh(2n.,.) sinh(2952n.-.n)) _ + +0 2 2 2 _ —4 2 _ 4—I§[{V(l 0' Po) (1 Po) T—Smn “711,115 90(1 Po ) (V 296271.35. nh2( 052111..) +s2 v l—cr2 +1+cr2 4.61 l ( ) ( )Si 295211,. ( ) sinhn,.,,(1- P62) SiflhT], n(1 + p02) —2svl—0'22 ' +l+o'22 =1. i( W 111511-952) ( p0) n1,"(l+p5) where o=lz/l, and v=1 for flexural deformation, and v=-1 for barreling deformation. When the geometry 1,, 12 and mode number i, n of given plate are supplied, the value of M for flexural or barreling deformation is determined by (4.61). 4.4.2 Load Parameters Associated with Buckling Initiation The load parameter values p,, at which the plate may initiate buckling, is dependent upon the initial geometry through mode number, n”. Their relations for the case of flexure and barreling deformations are given in (4.55) and (4.57) and are shown in Figure 4.1. Sawyers and Rivlin (1974, 1982) first reached these first order solutions in terms of A=p0’2 and n by using the variational approach. It can be shown for each fixed value of i and for each n (which then specifies a value for n,,.,,>0) that there exists exactly one solution of which satisfies (4.55). Also for each fixed value of i and for each n, there 60 Po 0.9 b Flexure mode 0.8 . 0.7 - 0.5 . p,,, p—————————_— 0.5 .. 0.4 . Barreling mode 0.3 . 0.2 - 0.1 Figure 4.1 The load parameters for flexure and barreling modes. When 11 goes to infinity, both modes converge to pm=0.543689 as shown in (4.65). exists exactly one solution p}? which satisfies (4.57). Therefore there exist only two possible solutions for each fixed Q”. The indices i=1 or 2 and n=1,2,3,... determine special values for each solution po = p57(i;n), p63(i;n), i = 1,2, n = 1,2,3, (4.62) If p,, is not equal to one of the two special values, then no solution, other than the trivial solution u")=0, exists for the homogeneous problem of k=1. That is buckling can not initiate at loads other than those given by (4.62). On the other hand, if p,, is equal to one of the special values given by (4.62), then the solution of the first order problem consists of 61 the single function from each of the infinite series in (4.2) that corresponds to the particular i and n which satisfy (4.62). Note that the two curves in Figure 4.1 are each monotonic and approach the common asymptote p,,, as n—>oo in which they share the same load parameter. When n-—>oo, both tanhnw and tanhpaznw have the same value since of is finite and hence the conditions (4.55) and (4.57) lead to (1 + 9.54? - 49.3" = 0. (4.63) The solution except for trivial case of pw=1 in (4.63) is the real root of 95.6 - 312.54 - 13;} - 1 = 0. (4.64) and becomes by using Cardano’s solution for cubic equation (see Qiu, et al., 1993) poo = [1 +3JR+ iQ3+R2+3~/R_ ,lQ3+R2]-“2 = 0.543689... (4 65) Q = —(4/3), R = 2. The deformed shape of the plate at n—wo involves an infinite number of wrinkles. In view of the loading mechanism, when the thrust T is increasing, the load parameter p,, is decreasing starting from p0=1 according to (2.16) so that the buckling initiation modes occur sequentially such as Tf0 and n—mo. This gives four separate cases corresponding to: (i) flexural deformations at low mode number ((4.55) as 1190); (ii) barreling deformation at low mode number ((4.57) as 71-90); (iii) flexural deformation corresponds to wrinkling ((4.55) as n—-)oo); (iv) barreling deformation corresponds to wrinkling ((4.57) as '11—)00). Later in Section 5.4.2, the low mode number flexure expansion (i) is used in an asymptotic stability analysis. The other cases (ii)-(iv) are given here for completeness. (i) p,, for flexural deformation when 11 goes to zero 63 When 11 goes to zero for the flexural deformation in (4.55), the parameter p,, goes to one as shown in Figure 4.1, so that we assume the series polynomial expansion of p,, as p,, = 1+k,n+k,n2+k3n3+... (4.71) The hyperbolic tangent can be expanded in a series form when n has small value as 1n3+3n5—... (4.72) tanhn=n—§ 1 Introducing (4.71) and (4.72) to the condition (4.55) gives the polynomial equation of n in which each coefficient function vanishes simultaneously. The lowest order becomes 113 and its coefficient shows k, = 0. (4.73) Then the coefficient firnctions after substituting (4.73) become l3§k,(1+ 3k,)n5 + 161.3(1 + 215,),6 3 + 8(— $9., — 1:31., — 112k, + 16k; + 13—6k4 + 32k,k,)n7 (4.74) 4 19 1 +16(fik3 — -3—k,k3 — 211531., + 2k,k, + ik, + 2k,k, n8 + 0(119) = 0. Because of small 11, each term in (4.74) vanishes simultaneously so that the first term gives k, = __. (4.75) excluding the trivial case, k,=0. Subsequently the other coefficients show that _1 19 k5 = 0, k6 = "Tg'g—O, k7 = O. (4.76) 64 Therefore the asymptotic equation for p,, when 1] goes to zero gives 1 1 19 which is shown in Figure 4.2 up to the fourth order accompanied with the exact values which is computed by using (4.55). As we can see in (4.77), p,, is an even function of n. The asymptotic equation (4.77) were also obtained by Sawyers and Rivlin (1982) in terms oflask = p57- =1+§n2+gn4+0(n6). Po 0.8 . Exact expression (4.55) 0.6 . Asymptotic expression 0-4 - up to the fourth order ' (4.77) 0.2 . 0 A 1 A A J 0 l n 2 Figure 4.2 Asymptotic equations of load parameter p,, for the fiexural deformation when n has small value by using (4.77) up to the fourth order. For n<1, asymptotic expression is quite close to exact expression. 65 (ii) p,, for barreling deformation when 11 goes to zero For the case of barreling deformation shown in (4.57), the parameter p,, goes to zero when 11 goes to zero. However npo'2 in hyperbolic tangent does not go to zero. A consistent analysis of possible forms for the expansion of p,,(n) near n=0 shows that the correct expansion form is Po = km “2 + km + km” + 15,112 + km” + 0013). (478) Then the expansion 11ng = k72—2k2kf3n “2 + (3k3-k,‘4 — 2k3k,‘3)n + (4.79) so that lim0(np52) = kfz. (4.80) n-> Substituting (4.78) into the condition (4.57) and the Taylor series expansion of tanh(np0'2) with respect to the value of (4.80) give an algebraic equation of n in which each coefficient function vanishes. Then the first term becomes k73(1 - 4k',’-tanhk,‘2)n‘3 = 0 (4.81) which shows the numerical value of k,=0.500169... The following terms show that k,, k;, and k4 equal zero and the fifth term shows k,(1—6k?) k5 = 24(- tanth;2 + 315301511153) (4.82) Using the value of k,, the equation (4.82) shows k5=-0.05234..., numerically. Hence the asymptotic equation of p,, for barreling deformation when 11 goes to zero becomes p0 ~ 050016911“2 - 0.05234115/2 + 0(03) (4.83) 66 which is shown in Figure 4.3 with exact equation. 0.6 Po Exact expression (4.57) 0.5 - 0.4 . Asymptotic expression (4.83) 0.3 1- 0.2 - 0.] I Figure 4.3 Two-term asymptotic equation of load parameter p,, for barreling deformation when n has small value by using (4.83). Again when n<1, the asymptote is quite close to exact expression. (iii) p,, for flexural deformation when 1] goes to infinity When 1] goes to infinity, the parameter (3,, goes to a finite value pw so that p,, can be written as Po = 9.11 +800}. (4.84) where 6 is remainder term that vanishes as 1] goes to infinity. The hyperbolic tangents are expanded by using infinite series of exponential equation as 67 = __ -_— _ —2 —4 _ —6 tanhn 1+e-20 1 2e rI+2e '1 28 0+... (4.85) tanh(n 952) = 1 — 262"“? + 2.540962 — 2e'6np52 + where npa2 = np;.2(1—25+382—483+...). (4.86) The possible form of 6(1)) can be obtained from considering (4.85) with the conditions (4.55) or (4.57) as 5 = k,e’7-‘1 + k,e‘4" + (4.87) Substituting (4.84) and (4.85) into (4.55) for flexural deformation and equating each coefficient function of each order of e to zero give algebraic equations which contain the unknowns k,. The first term which is independent on 11 shows (1 + p54)2-4p;1’ = 0 (4.88) and the real solution to (4.88) give a limit value of pw=0.5437... excluding the case of unloading, p0=1, by using Cardano’s rule as shown in (4.65). The next leading terms show that 2 k1= p. , 1-3p$+pw 489 k =(9-21p.2.+51>;1)Ic%--2p.2.(1+6k.) (' ) 2 204917-91) ’ with the numerical values of p,,, the parameter p,, when 11 goes to infinity becomes p0 ~ p,,,(l + 1.47395e72'1 + 2.98066e‘4" + ...), (4.90) which is shown in Figure 4.4 with exact solution. 68 Po» 0.9 - 0.8 - Exact expression (4.50) / 0.7 - 0.6 . f Asymptotic expression (4.90) 0.5 0 l 2 3 4 5 6 7 8 9 1110 Figure 4.4 Three-term asymptotic equations of load parameter p,, for flexural deformation when n goes to infinity. When n>1.5, asymptotes is quite close to exact expression. (iv) p,, for barreling deformation when r] goes to infinity For barreling deformation, the similar approaches can be used in the case of flexural deformation when 11 goes to infinity. The expansions (4.84) and (4.87) are also possible candidates for barreling. Then the results shows that k _ (1 + PW ‘ _ 4(1- 393. + pt)’ (4.91) _ (13-4295+1098)ki+(1+P$)(1+P$-8k1) 2 _ 4(1-3p?.+p:‘.) ’ which determines the asymptotic equation numerically as p,, ~ pw(1—1.47391e-2'1 + 34,94543—40 + ...). (4.92) 69 The asymptotic equation (4.92) with exact equation are shown in Figure 4.5. and it 0.6 2 . Po Asymptotic expression (4.92) 0.55 . 1 1 Exact expression (4.57) 05 4 A n . . . . 1 1 0 1 2 3 4 5 6 7 8 9 10 1] Figure 4.5 Three-term asymptotic equation of load parameter p,, for barreling deformation when 11 goes to infinity. When n>2.5, asymptotes is quite close to exact expression. Otherwise, two expressions show totally different values. is shown that the asymptotic equations are well matched when 1] has larger values in both Figure 4.4 and Figure 4.5. 4.5 Post-buckling Deformation (Second Order Solution) The higher order solutions “(2), u‘”,... are the expansion in e as given by (3.1) to account for the postbuckled deformation. This is determined by the nonhomogeneous nature of the equations which effect f and g in (3.8) and (3.9). To determine the second order solution bifurcated from one mode of the first order solution, we now choose either 70 fiexure or barreling and also fix i=1 or 2 and mode number n=1, 2, 3... in the first order buckling solution u“) in (4.68) to (4.70). It will be convenient to rename i to j and n to m as we wish to use j and m in what follows as the expansion indices for the chosen first order mode. Thus for fixed j=1 or 2 and m=1,2,3,... as determined from the bifurcation mode under consideration, we seek the coefficient functions BM”), A ”(2), Cm”), i=1,2, n=1,2,3,..., in the second order case. For the zeroth terms, the general equations (4.14) and (4.15) with the right hand sides of the second order equation (3.11) become II p8 ' I I I I I 11382) “—90%” = -u,—,—,—;(B,..B11.2 ) 5 90362) = OMB/($13}; ) , " 5 (4.93) . Po . . on F, 11832) —poC62) = #155519» 3,93 . where Brim = B};1ni"_'sz;InBJ(;iri° (4°94) To within the rigid body motion given by a constant in 80(2), the solution for (4.93) is I I I p8 I I 812’= 1258112810. C19 = {(311.2319) Jrq‘im 311.2] (4.95) For the other coefficient functions, we will utilize the generalized solution for nonhomogeneous equation, (4.43) with the solvability conditions (4.47) and (4.48). Substituting {(2), gm in (3.1 1) into the equations (4.7) and (4.8) shows the following results for the right hand sides of the equation (4.9) and (4.10): (A) If the mode of u“) is j=1 and m, then only the expressions for i=1, n=m and for i=2, =2m-1 of “(2) are non-vanishing: 71 Iizrii = —PIPBZQI;mCii}n Jig): = plCiirir'a K132. == -p1(31!.’.'+p5201..A1!.2.). (496) G132. = 2119195391....3111. H132. = 91(C112.+21195301;mr41!3.), and .... NIH [\N‘“b 112.3..-1 01,m(C1!.1.'31§2.-C1!,2.31!2.'). 0an 915461354112.-C1§2.All.2.'), (4.97) Kiym-l = iQI;m(Aiirir’Bilrir _AilrirBiirir')a l l GSQm-l = ioleiirirCiirir’ Hiiim—r = iQI'JnAi‘IiIBiirir (B) If the mode of u“) is j=2 and m, then only the expressions for i=2, n=m and for i=2, n=2m of "(2) are non-vanishing: [£2121 = _plp0202mC3irin J39: = plciirir'a K113. = -p.(31!.2.'+p5202-,.A11.2.), (4.98) 0132. = 20910530253131, H132. = 91(C122.+ZMP5302..A1!,2.), and 1 . . 119.. = —,92..(C1!.2. 8121-6111810. 1. 1 I I J19. = 792.4610. A112. $121419. 1, 1 (4.99) K59»: = —§QZ;M(A$irir'B£lrir-A$irirBii,rir')9 1 1 059m = —§QZ;InB£itiIC£irin H53»: = —§QZ.MA$LZIB£BV For the modes which are not mentioned above, the right hand sides of the equation (4.9) and (4.10) vanishes so that these modes have the same solutions as those for the homogeneous case or u“). However, according to the orthogonality rule (3.4), the solution 72 of these modes must be eliminated. For the cases of n=m in (4.96) and (4.98), the differential operator are the same as those of homogeneous equation so that in most cases, the solution does not exist. But for the special arrangement of the nonhomogeneous terms which satisfy the solvability condition, the solutions exist. Substituting (4.96) and (4.98) into the right hand side of (4.11) and (4.13) with the relations (4.53) for Am,“ and C111,," gives ]=-4plp5'(B11."'-Q-2-3.1”)” (4.100) 1’" 1’" and g“, = ‘2P196'(B}-,in)"—sz;m 9643193) 2'. = 49195'(B,1.,.""-Q?BB-'-1'2) Jv’" 1’" (4.101) Substituting from (4.100) and (4.101) into the solvability conditions (4.50) with Bh=B“)J-_.m and integration by parts to the left hand side gives 2p1p5'[{(B,<,0" + 0,2,..prB<1.2)B<1.> 1|: (4.102) —2 1: {(31.202 + 9,2..(B1.‘,.2)214X2] = 0. Considering the first order Fz-boundary condition (4.52),, the condition (4.102) becomes p,=0 since the integral is positive definite. Hence the solution for the case of n=m exists only if p,=0. If the first order load parameter p,=0, the right hand sides, (4.96) and (4.98) vanish so the modes n=m have the same solutions as the homogeneous solutions. Due to the orthogonality between the first order and higher order solutions, the solutions of modes n=m must be eliminated. Therefore if the mode of j=1, m is chosen for u“), then only the mode associated with i=2 and n=2m-1 in the second order solution expansion “(2) 73 is governed by an equation that has nonhomogeneous terms. That is, only Bfim, has nonhomogeneous solutions so that 81%,} = 0 for all n and B5?) = 0 unless n=2m-1. On the other hand, if j=2, m of u“) is chosen, then only the mode i=2 and n=2m, that is, 8533", has nonhomogeneous solutions. Thus for fixed mode variables m=1,2,3,... and j=1,2, there is exactly one nonzero B3,? which corresponds to only BS} governed by a nonhomogeneous equation. The index i for this nonzero second order B3,) is always i=2 and n is given by n=2m-1 if j=1 and n=2m if j=2. Note that the mode number n of the second order is different from the mode number m of the first order so that the differential operators of the second order are different from those of the first order. Hence there exists a unique nontrivial solution to the second order equation (4.16). After combining the two cases in which the nontrivial solutions are possible to exist, the right hand sides become 1 l 1 Q—TWYIMSan—la 1 g—lfi—MKlmsnflm—l’ 11?) = 31198 1 J1?) = 51198 , -mY2m5n2m9 -mx2;m8n;2m’ (4.103) 1 0’] 'msn‘Zm— 19 K?) = _ 2( ’ ’ in 2‘30 —O'2:m6n;2m2 and '_1_'Bl Biirirsn'Zm-lr LBlunBiirir'an'Zm-l’ , 1 9....2"'~ ,1 or... ' G1,.) = 51198 1 H§,,2= 511123 1 (4.104) 'KTz—BimBiirirfinfim’ _EB'ZynBilyir'éan’ where the upper terms are for j=1 and the lower terms are for j=2 and A1,," and C”, have been converted to Bj_.,,,. The notations used above are 74 (1., = B};1Iri"Bj(;lni‘-Bj(;lni'Bj(;iri'9 71;»: = B" BU)... iLMBJ(#i'i" Jv’" 1;"! 14’" = " BU)’ — 8,.,,,B<.1>" Kj;m j;m jLM 1,"! 9 (4.105) and SM is the Kronecker delta (6M = 1 ifp = q and SW, = 0 ifp i q). For the equations (4.103) and (4.104), the mode numbers i and n in the left hand side of (4.9) and (4.10) are changed accordingly. And from the definitions of mode number in (4.1), Q," can be written in terms of Q”, as 02:2711-1 = 291m: Q2;2m = 292;».- (4-106) so that the formulations for the second order can be considered as the generalized formulations developed in Section 4.2 with 29m in place of Qw- Applying the relations (4.106) and the right hand side terms in (4.103) and (4.104) into the boundary value problem (4.11) and (4. 13) for both cases of j=1 and 2, have the following format B12)” — 4912,,(1 + p54)B(2)" + 169,1, (5543(2) ' 1 .. ' ~ (4.107) = mPO(Y/‘;m — 2Kj;m + iaJ‘M—ZQJZWGJW) 51(2)’ 011 F2, II 1 ' I ~ 8(2) + 4szynp6430) = mp0(§aj;m + BjJflBJ(.lni) E gin, B(z)m-4QJ'2;,,,(1 + 2p64)B(2)' (4.108) 1 . . r " = wpo[Yj;m + iajmr " 2(1 + P04)sz;majzm " zflj;mBj(;iri :15 81(22): where 3522m_ 2 19 for . = 1’ B<2> = ’ ' 55 = ( I, (4.109) 359m, _1’ for J = 2 75 According to the relations (4.12), the other coefficients become 2 Am 90 (3121233100, m) 2:2. 2 m 4.110 C121 —- 1153 B121"'—4Q.2.,,,B(21'wp0 —4Q2.55 +2 ( ) _ 40}, l ‘T(°‘Bm" m a)... 175)], where A2 , c2m_ , -___ A12) = 53”“ Co) = 5'3 ‘ J 1’ (4.111) A13... C13... = 2- Applying the first order solutions (4.54) and (4.56) with zero right hand side into (4.94) and (4.105), then the right hand sides of (4.107) and (4.108) becomes 1‘” = -5500 2,..(p5 _15') 5,.,,,, (4.112) and 812’ = l912591-2-...(7p5“+ 03,15,231!» .III 2 ’ (4.113) g1” = -255,a,2,,(2p,4 —1)(B}}.2')2 Substituting 3,1,1”; for flexure in (4.54) or for barreling in (4.56) reduces the equations (4.112) and (4.113) to im= ,3,,-va2s..9.-..p5(1-pr)21v(1-pa2)sinh{n,..<1+55%} +(1+p5z)sinh{Q--,m(1—952)X2}]. (4.114) gi2) = 4911421300130'1'1)Qj3yn[51nh(29j;n-X2)+5};m9625mh(291m902X2) —Sj;m{ (1 + 962)Sinh{ 0);..(l 7‘ 962))(21 + V(1 - 962)Sinh{Q,-,m(1 - (262)21’2} } 1. (4.115) 76 812’ = —90sz5(21>54 -1)Qj‘,m[cosh(2§2).mX2)+ 3,2;m964005h(201;m1362X2) (4.116) - V(1 + .9295“) - 28);..9521 cosh { 9,...(l + p52)X2} - vcosh { Q,...(1 - p52)X2} 1]. where v=1 for flexural mode and v=-1 for barreling mode. The general solution to the differential equation (4.107) with the boundary conditions (4.108) has the expression B121 = N, sinh(2p52(2,,,,X,) + N, sinh(20,-.,,,X,) . . (4.117) +1\1_w.smh{(p52 + 1)Q,,..X2} 211’5smh{(p52 ~1)Q),..X2}- Here the first two terms are from the homogeneous part and the rest are from nonhomogeneous part with constants N,’s. Introducing (4.117) to the differential equation (4.112) and matching the coefficients give N3 and M, as _ 3 (954-1)(p52+1) N3 — mesjimflji’"p°(3p52+1)(p52+3)’ 3 (p54— 1)(p52- 1) N4 = -O)VWSj-;mflj;mp0(3p62_1)(p62—3), (4.118) and to the boundary conditions (4.113) give the values of other coefficients as _ wngmpa ' _ 32A (011129.. p0 = 1"" N2 16A [4954D1905h(211)‘ (1 + p54)Dzsinh(2n)], (4.119) [p5zDzsinh(295zn) — (1 + PB4)D1°05h(290211)], where = 4p5fisinh(2952n)cosh(20) — (1 + p54)zcosh(295zn)sinh(20). (4.120) and 01 = 211(952) + vd1(—95)—(7p5“ +1)[sinh(20)+51%mp5zsinh(2952n)]. (4.121) 77 D2 = d2(962)+vd2(-962) _4(2964 -1)[(cosh2n -— v) + 512;»: 904(90511206211. _ v)], (4.122) 1— 2 2 ' , 6W) = "31""(1 + ”if (2215,??? L“ + 7501521120 + 5..)11, (4.123) 512(E.) = 4.431149“ “112134“ 752) (1 +3§)(3+e=,) +8§(1—2§2)]cosh(1+§)n. (4.124) The denominator A represents the combination of the conditions (4.55) for fiexure and (4.57) for barreling such as A = 45153213132101: p54)2tanhp5zn -4p56tanhn} .. .. (4.125) + C1C2{(1+ p54)2tanhn — 4955tanhp52011. The solution (4.117) can be also obtained by modifying a generalized solution (4.43) with replacing 0,," by 20m. The particular solution (4.40) contributes to the last two terms including N3 and N4 in (4.117). Using the variational analysis, Sawyers and Rivlin (1982) also found the same type of second order solution (4.117) which only differs in the notations. Finally, the corresponding second order nonhomogeneous solutions u‘2’=(u,(2), u212),p(21) become from (4.2) with (4.95) for the zeroth term and (4.110) and (4.1 1 I) for the higher order as up) = .A12)sin(ZQ,-,MX1)2 1 u)” '2'PoB(l)B(1)'+B<21cos(20,;mX1). 1;"! 1;"! (4.126) 4 2‘” = 202191218) + 12)-18.818? “220828.220, Jim where Bfig has only homogeneous solution and 853,} has nonhomogeneous solutions 359m , or 8533", for j=1 or 2, respectively. The nature of coefficients B12) depends on the 78 choice of the first order solution - flexure or barreling mode and j =1 or j=2 and mode number m. 4.6 Summary of Full Buckled Deformations The fully finite buckled deformations v,(X,,X,), v,(X,,X,) with pressure p(X,,X,) after perturbation expansion are defined in (3.1) and solutions of the first and second orders are determined in the previous two sections. Now we will summarize the results as a reminder of complex procedure. The buckled deformation with respect to X, axis is given from (3.1), as v2(X1.X2) = eu1'>(X.,X2)+ e2u1221X..X2)+ 0(22). (4.127) More specific result is from (4.1), (4.69) and (4.126), as V2(X1,X2) = 1 , . 2m— 831,2,(X,)srn( 211 7tX,) + 52{.p2_°B(g,),(X,)B(§,2,'(X,) + 81?;(X2)cos(2m, lit/1’1) } + 0023), I (4.128) < cB,l,2,(X,)cos(lflrtX,) 1 + e2{9,-°Bs!.2.(X.)B12.2.'(X.) + 812.1(X2)cos(2,—’"-nX.) } + 00:2). I where the upper solution is for j=1 and the lower solution is for j=2. The coefficients 8,9,} are given in (4.54) for flexural deformation and (4.56) for barreling deformation and the coefficients 8,13,} are given in (4.117). The subscripts j=1,2 and m=1,2,3,... are indices of the chosen onset bifurcation modes. Then deformation v,(X,,X,) along the X, axis and pressure p(X,,X,) follow v, solution (4.128) in an appropriate order of e from (4.2), (4.53) 79 and (4.110). In conclusion, there exist four different types of solution which depend on the deformation types such as flexure or barreling and chosen value of j=1 or 2. In each type, there exist infinite modes of solution. 4.7 Reduced Formulations Due to Symmetric Load Parameter The first order solutions u“) obtained in (4.68) - (4.70) may affect the formulations on the load parameter and the energy equations so that their equations will be reduced to simpler forms. These simplified forms will make less efforts to investigate the stability near buckling initiation. 4.7.1 Load Parameters With the results on the first order solution in hand, the postbuckling formulations given in solvability conditions can be much reduced. In particular, we now show that p, as given by (3.33) must vanish. Applying integration by parts once with respect to X, to the first two terms of R, in (3.31) yields 8. = ir2P(')(uI”“i,li"ui')ul,'i)|r|dX2 (4.129) + 11,3121'2018431 — u131u13))dx.dx.. Then applying the F,-boundary conditions in (3.9), to the first term, the numerator R, in (3.33) becomes 12. = sllnpuxumum -u131u13))dX.dX.. (4.130) Substituting from either option for the first order solution (4.68) into (4.130) and performing the associated integration gives 80 R, = 0. (4.131) Rearranging R, in (3.32) yields R. = 41111552491201 — u12>p<0>dx + 11np(pa2u131— agndxr. (4.134) I 2 Substituting the first order solutions (4.68) - (4.70) into (4.134) and integration with respect to X, now yields p8 I II II I B. = -41211,72-B1"(Q%p5431"+31" )|,2— lrzuBs') )2+Q..(B1'> )2)dX.. (4.135) The F, boundary term in (4.135) vanishes by virtue of the homogeneous form of the F,- boundary condition (4.13),. Then R. = 4011%11,1181”")2+£2.2(B12>')2)dX.. (4.136) 81 It is significant to note that R, > 0, (4.137) for nonconstant 8,1”. The results on R, in (4.131) and R, in (4.137) in conjunction with (3.33) show that p, = 0. (4.138) This result simplifies the expression of p, in (3.38) to p, = 772. (4.139) The numerator R3 can be reduced via integration by parts with respect to X, on the first two terms in (3.35) such as R3 = irz[u(|>(p(2)u§_')+p‘11u§?3)- u§1)(p(2)u(]) +p(l)u(?))]|rldX, 41,, 029013142) — 4131u12))+p<2>(41}14531— :43) u13))1dx (4.140) + 112500131421 - u131u121)dx n + 11” 15944314412)—u131u12))+p<2>)|,2992+ 1, (u111u12>-43141201,de5 (4.143) 9. = 12012142413342” 4314420122. 1 (4.144) - ..(u121u12>—u111u122+ 4431412221.Xm- Substituting F, - boundary condition in (3.9),, all the first integrals in (4.142) to (4.144) vanish so that 9. = -l,lu131u1'>|,zdx., (D2 = i,l(u1,‘lu12’-22131u12’)|,2dX1, (4.145) 9. = -l, —u131u12>+ 43149)), 420. Similarly, after integration by part with respect to the variables for 12,01, the definition E,J,’s in (3.53) are reduced to E..- = 1201:1410 + us: 1221”)|,,dX2 + 1,011.4» + u155u22)|,de. (4.146) —lln 104:1. + 2415.291) + 04:21. + u1€2.)uy>ldx, where i=1,2 and j=1 ,2,3,.... For i=1, the coefficients of u,” and 11,11) in the third integral are substituted by the governing equations in (3.8),, for k=1 and due to the F,-boundary condition in (3.9),, the first integral vanishes. Hence E..- = 1, 043114» + u131u1/1)|,24X1--,l-,iln(p5‘p‘,'lu1”+ P0P(,liuy))dx- (4.147) 83 Integration by parts with respect to the variables of p“) in the second integral in (4.147) and application of the boundary condition (3.9), yield ._ . . P . =.,- = ((5213154) + 431119795949) 4X. r2 (4.148) 1 . . + T1 iin(p5'u1,’1 + p5u2’2)p‘”dx- The parentheses inside of the second integral in (4.148) is the left hand sides of incompressibility conditions in (3.8), so that substituting the right hand sides of the condition to (4.148) becomes for each j 3.. = 1,1041%!”u1,')usl>—%’p<'>u1”) 4X5 (4.149) a. = l, (24,042+u131u122—39pmu122) 4X. ' ,1 ’2 (4.150) + 5,1 inp‘”(u1,'1u1.'l — u131u13))dx. 3.. = ((53,50).. u131u12>-%p<1>u12>) 426 (4.151) +fillnp152dx (“52) -,l,11n1u12>(p<3141}1—p<3)u131)+ 2422029124131 -p<,') 4111))dx. Then, from the abbreviated equations (4.145) and (4.149) to (4.152), the terms in the expansion of energy difference (3.55) to (3.57) have reduced formulation. Substituting (4.145) and (4.149) into (3.55) yields dX,. (4.153) r2 B. = ,1, H013) - p52u1fl)-P,fp1"}u1”+(u111u1‘2- 5524311410] The coefficient of u,(” in the first term in (4.153) vanish due to the F,-boundary condition (3.9),, and after integration by part with respect to X, to the second term and applying the F,-boundary condition (3.9),, E, becomes E. = 51,013) +pa2u1}))u1'>|,2dX.. (4.154) Substituting again the F,—boundary condition in (3.9), to the coefficient of u,‘” in (4.154) yields E, = 0. (4.155) Similarly E3 in (3.56) with (4.138), (4.143) and (4.150) becomes 85 E3 = ”(P62¢2+Elz) Po = 1_-21__(1) 2) 1 —21 2) 1,1101.) 5.221)) ,p }u1+(u1,)+p.u1,1)u1] 4X. (4.155) r2 + ,1 11594511840 — 431439444X. Applying the F, -boundary conditions in (3.9), to the first integral, E3 becomes 3: p u u —u u , ,. . E 31”“ (0(1 3 1 ])dXdX (4157) The integrand of E3 in (4.157) is the same as that of R, in (4.130) so that the procedure leading to (4.131) also gives E3 = 0. (4.158) Based on the results (3.54), (4.155) and (4.158), the first nonzero term in the energy difference is, at minimum, the fourth order term 13., so that AB = 213E434+ 0(85). (4.159) With (4. 138), the formulation of E4 in (3.57) becomes 1... .— E4 = P(- 29133sz1 '1' 962(1’3 '1' 5:22 '1' :43)- (4-160) Here (I), requires use of u“), 3,, requires use of u“) and “(2), and CD3 and 3,, requires use of u“), “(2), and um. We now establish, however, that E, can in fact be determined without first having obtained “(3), Using (4.145), and (4.151) we may write 86 5,25, +2, = 1r. [-pa2(u12)u12>—u1}1u12>+ 440:4”) dX, + u111u12>+ u1})u12>—%9pmu9>] r2 (4.161) +fillnp1pz+ u121u12> + u12)u12>+ 1,95%?) 4X. I.2 + 1 15940182112) + ut2)u1}1)- 0431542) + 2:121u13)))dx 1 (4.165) + , llnp<2>| dX (4 168) 2 Ian ’ r2 1 2 1‘, ’ O ’ r2 1. . Again substituting the F, boundary condition in (3.9), for k=2 into the second integral yields 11,506,251}, - u12>u131 4.,an- (4.159) Hence R6 in (4.165) becomes 88 R. = 11,015,250, —u12>u131)|,de. + 11115940131421 + u12)u131)—(u111u1.2) + u121u13))1dx (4.170) + i1 1np<2>1u131u91 — 4943092 51151212991213 91,8481) + 1420;015:131 —p<31 u13))1dx. 4.7.3 Energy and Load Parameter In equation (3.47), the relation between the thrust T and the load parameter p, is revealed for the case of symmetric bifurcation. Now we will investigate the relation between the energy difference AE and the postbuckling behavior. The first non-zero term of AE appears in E4 and the postbuckling behavior depends on the sign of p,. If p, is negative, then the progressive buckling occurs in view of the relation between T and p,. Right afier the buckling initiates, there exist an extension of trivial solution path and the buckled paths. The actual deformation will follow the energy minimizer between these solution paths — stable deformation. Integration by parts to the last integral for R6 in (4.170) with respect to the variables of p“) gives _ % Jr2p(1)(u(2)u§1) — u§2)u(1))|r‘dX2 -%irlp‘”(u12’u1,'1 - u,2)u(1])|r2dX, (4 17,) «1 11159013142) +u131u121—u131u12)—u13)u121)dX- The first integral in (4.171) vanish after applying the boundary condition on F, in (3.9),. Then R6 in (4.170) becomes 89 B. = ,1 Inp<1>uu10u121 + u12)u12))— 0:131:42) + u121u13).)1dx , (4.172) + 11np<2>(u1})u131- u1})u13))dx. Comparing R6 in (4.172) with the result on R, in (4.141) and using the relation for R3 in (4.139), establish the following relation. 1 1 ,R, = —-R,p,. (4.173) R6= 4 Note also that R, in (4.134) and R7 in (4.166) are related by R, = R,. (4.174) NIH In conclusion, the first non-zero term in the energy difference E4 in (4.164) with (4.173) and (4.174) becomes 1 B, = Z8,5,. (4.175) The relation (4.175) shows that E, is simply related to p,. Since in (4.136), R, is always positive for the nontrivial solutions so that the sign of E4 depends on the sign of p,. If p,<0, then E4<0 so that AE<0. In other words, the solutions corresponding to buckling have lower energy than the trivial solution at the same load - the buckled path is stable. Therefore when the progressive buckling occurs, the buckled path is stable in the vicinity of the buckling onset. The other case is also clear. When the snap buckling is possible, then the trivial solution path is stable in the vicinity of buckling onset. In view of the above statement, if the values of p, are known, then the postbuckled behavior and the stability of each path can be obtained. 90 The parameter p, is given by (4.139) involves R, and R3 so that the energy (4.175) becomes 154 = --R3. (4.176) The parameter R3 shown in (4.141) involves the first and second order solutions u“) and “(2) determined in Section 4.6. 4.8 Summary The solutions of the first and second order in the expanded linear type boundary value problems are obtained by using the separation of variables and infinite series method. Physically, the higher order solutions are explained as the bifurcation from one of infinite modes of the first order solution or buckling initiation. With the first order solution, it is shown that the first load parameter p, vanish so that the buckling behaviors become symmetric with respect to the trivial or homogeneous deformation. According to the symmetric nature, the formulations on the higher order load parameters and energy formulations are much reduced. By substituting the first order solutions and pl=0 into the energy formulations, it is revealed that the first nonzero terms come from the fourth order energy equation for which only the first and second order solutions are necessary to solve. 91 CHAPTER 5 STABILITY EVALUATION FOR A NEO-HOOKEAN PLATE 5.1 Introduction The energy difference (3.48) between the buckled and homogeneous deformations determines the energy minimal, or stable, path after bifurcation occurs. The equations (3.54), (4.155) and (4.158) show that the first nonzero term in the energy difference is the fourth order, E4, so that the energy difference (3.48) becomes AE = 213E424 + 0(85). (5.1) Also as we have seen in (4.175), E, is linearly related to the second order load parameter p, as E4=R,p,/4. Specifically, the sign of E4 is the same as the sign of p, since R,>0 in (4.137) so that if p,<0, then E40, then the configurations on the bifurcated path have less energy than those on the homogeneous solution path. Hence R3>O gives that the nonhomogeneous deformation is more stable. Conversely if R3<0, then the homogeneous deformation is more stable. In this Chapter, the stability of the bifurcated nonhomogeneous deformations and homogeneous deformations near the buckling initiation will be investigated by using the reduced formulations and the buckling solutions. Our comparison is to Sawyers and Rivlin (1982) who analyzed the stability of those type of homogeneous deformations in a neo-Hookean rectangular plate by comparing the energy of homogeneous deformation with that of the bifurcated deformation. This was done in the vicinity of the bifurcation points. Their analysis, however, is apparently not a direct energy comparison between the homogeneous and nonhomogeneous bifurcated deformations, either flexure or barreling, at the same level of loading condition as analyzed in this work. On the basis of their analysis, the following conclusions were obtained: (1) From Figure 3 of Sawyers and Rivlin (1982), the homogeneous deformation is more stable than flexural buckled deformation when n<0.32 and the flexural buckled deformation is more stable than homogeneous deformation when n>0.32. This is based on their equation (8.4). (2) From Figure 4 of Sawyers and Rivlin (1982), the homogeneous deformation is more stable than barreling buckled deformation at all values of n. This is based on their equation (8.7). Note from (4.55) and Figure 4.1 that the value n=0.32 on the flexure branch corresponds to p=0.966. Also n—mo corresponds to poo-90.5437. Thus the analysis of Sawyers and Rivlin predicts the 93 followings: For increasing compression ratio, and hence p decreasing from p=1, and near bifurcation onset, (1) the homogeneous deformation is more stable than flexural buckled deformation for p:l—>O.966, (2) the flexural buckled deformation is more stable than homogeneous deformation for p:O.966—->O.5437, and (3) the homogeneous deformation is more stable than barreling buckled deformation for p:0.5437—>O. 5.2 Formal Determination of Stability Parameter We first examine p,, which is related with stability parameter R3. The denominator R, of p, in (4.139) is always positive for the nontrivial solution in view of (4.137). Hence the formulation (4.139) shows Sign(p,)=Sign(-R3). Direct substitution of the first order solutions (4.68) - (4.70) and the second order solutions (4.126) into the simplified equation of R3 in (4.141) give upon collecting values with X, and X,: + Q ,E '9: ‘2': 8 + N h; E ‘72 C5 r—‘fi Cl)! ~N Ql LL 1 \ + Q 55 65 OR 13 A '-.J L/ (5.3) +02) Aume _ _ +A(”B(‘>' _ _ Casi C3Ci +-C62’ Aurgm _ +A(1)B(')' _ 2 5.2 cf Here the following trigonometric notations have been used, C, = cosQXl, S, = sinflXl, ' sinZQXl, _ (5.4) C3 = cos2QX1, S3 94 and upper (lower) terms are for j=1 (j=2). Note that the subscripts j and m of the coefficients A“), 3"", Cl") and (2 which denote the chosen mode of the first order solution, are ignored for simpler expressions. Integration (5.3) with respect to X 1 gives R3 = Q]l J12 [-coC(')(A(2>’B(”+A“)B(2)'+2A“)'B(2)+2A‘2)B“)') 42 (5.5) —o)C‘2)(A(”B“)' _Am'Bm) + C(1)A(11362)' + C52>(A<‘>B“))']dX,. where m=1 or -1 for j=1 or 2. According to the relations A“) and CI") to 8"" shown in (4.53) and (4.110), the equation (5 .5) becomes the function of only B“) as 2 p4 R3 = 1491i]! [,2'3'5'2(2(0961QA + QB)dX2’ (5-6) where _25' { 3(2)"B(1) + 4(8“)'B(2))'} + a(B(2)"—4QZB(2))', -a{a" - 40201 + 2([3"B(‘) — B'B(”')} + 213'(a'B(') + 2018“") (5.7) + 3(3(1)3(1)')1{2p'3(1)'+ szflgamury}, QA Q3 Here B and a are defined in (4.94) and (4.105). Note from the new notations that Q, is function of B“) and 3(2) and that Q, is function of only 8‘”. Note also that the second order solution 8‘” is more complex than B“). One approach to evaluating (5.6) is directly substituting B“) in (4.54) for flexure or (4.56) for barreling and 3‘” in (4.117) into (5.6) and integration over -l,0 and corresponding values of load parameter p0 according to the relation (4.55) or (4.57) into s in (4.58) and A in (4.120), and then substituting obtained values into the stability parameter R3 in (5.13). For the purpose of numerical setting, we introduce the dimensionless stability parameter Rs = R3/ (ullM4Q3). Then Rs = 2{nn2,01+2,‘.1.(1'11.k1+ ”é; kl) 41111121111}. (5.14) The Figure 5.1 and Figure 5.4 show the relations between RS and n for flexure and barreling modes, respectively. For the flexural deformation, Figure 5.1 and its detailed Figure 5.2 and Figure 5.3 show that Rs is positive when 01.6283. According to 97 the relation (5.2), the signs of R and R5 are different from the sign of AE. Therefore when n1.6283, the unbuckled state is energetically favored and hence stable. Note that when 11 goes to zero, the undeformed geometry looks like a rod subjected to thrust at its ends which is similar to the conventional elastica problem. At n=1.6283 which corresponds to 7&0 = po'2 = 3, there exists discontinuity. This comes from the fact that N4 becomes infinity at this value according to (4.118). For the case of barreling, Figure 5.4 shows that RS is RS . . I 400 . l I I 200 . l I I I I I 0 I I I . . I . I . . - Posmve I Negative I Posrtive : Negative I I I '200 '- I I I a | | I I l l -400 - I I I -1 I | l 1 . 4 l k 4 . J . n r . o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 TI 2 Figure 5.1 Dimensionless stability parameter RS=R3/(ul,M“Q3) for the flexural deformation in (5.14) with v=1. At n=1.6283, the curve Rs has a discontinuity. Right before this, it is positive and afier this, RS is negative. There are also sign changes in 11:0.6443 and n=1.305. Their details are shown in Figure 5.2 and Figure 5.3. 98 _n d 1 0.8 - 0.6 - 0.4 . 0.2 ~ ' -0.2 , -0.4 . -0.6 . 1.4 1.6 1.8 2 11 Figure 5.2 The detailed curve of dimensionless stability parameter R5 for the flexural deformation in (5.14). At n=rIc=0.6443, the sign of Rs changes from positive to negative and at n=l .305 the sign changes from negative to positive. More detail near n=m, is shown in Figure 5.3. positive in a whole range of n so that the buckled state is always stable. Hence all buckled barreling deformations has larger energy than homogeneous deformation. Compared with results by Sawyers and Rivlin (1982), they conclude in Figure 3 and 4 of their paper that the homogeneous state at which bifurcation occurs is stable when nj vanish in the process. The undetermined coefficients am, a“, a,0,.. ., ay- are calculated by substituting the proposed solution (5.25) into (5.23) and (5.24) with (4.77) and equating the coefficients of various products of C; and 11 to zero. In fact, we immediately find that aI-0=0, for i=1,2,3,... by considering the 0(11‘") term in (5.23). From (5.24), after substituting (5.25), the 0(1) term gives 1 all = —§. (5.26) Substituting (5.26) into the expanded equations of (5.23), the 0(1) terms give 1 022 = —§. (5.27) The same result is separately obtained by considering the 0(r1) term in (5.24),. The other coefficients can be obtained similarly. Namely the 0(Q2112) term in (5.23) gives a33 = -1_;—44 and the 0(112) term in (5.24)I gives a,l = %. Then the 0(112) term in (5.23) after substituting known coefficients give a,, = ——1-. The same result for 03, is obtained 18 simultaneously considering the 0(113) term in (5.24),. Finally the 001‘) term in (5.24)] gives a3l = 415-. In summary, the first order solution B(11,§) becomes 2 4 6 B(n.€)= C{1—-2-n2+(3 (9— -§§-4)n‘+(4-% —%—I-I--I—%—4)n6+0(n8)}. (5.28) 105 The constant coefficient C will be obtained from the normalization process explained in Section 4.4.1. The reduced normalization equation (4.60) can be rewritten by using new variable (5.22) as 2 1 6263 d 2 d 1 6213 _ m -1I—n'2—(21g) + Bz+pg{E[?a?—BII Idg — 1, (5.29) where o=l,/l 1- Substituting the solution (5.28) into (5.29) with p0 in (4.77) gives C = [21211 + 0012)}. (5.30) The 0(112) correction to (5.30) will put terms of order 112", k=2,3,4,... in (5.28), but we have previously shown that aI0=0, for all i. Hence we conclude that C = J21, so that = C2 2 (C2 C 4 (4? C“ C6) 6 3 B(‘LQ filz{I-7Tl + 3‘? 11 + E-Tg-m 11 +001) - (5-31) The complete solution (5.28) with (5.30) may be compared with the direct small 11 expansion of analytic solution (4.54). Here one finds that = 22(16Q4Il6 263996 8 B(TlaC) M{§Tl + 4—5—‘3- '1 + ‘9—43—75-‘1- 11 +001) , (5-32) and the coefficient M in (4.61) for small 11 expanded flexural deformation is found as _ 3 _2 4 244 2 4 Note that M is required to be positive by virtue of (4.61). Therefore this expansion is valid only for 11 « 1.37. Thus both solutions (5.31) and (5.32) with (5.33) are the same in the view of small 11. The same expression for solution (5.31) can be also found in (7.7) of 106 Sawyers and Rivlin (1982) except for the constant coefficient C = J21, which is caused by adopting the different normalization condition. For the second order solution B(l,§) = BI2’(X,), the boundary value problem (4.107) and (4.108) is rewritten here in terms of B(11,§) and B(n, Q) as: 1 4141? “+0604”? 4- _ 30) (064-1) d( c123 dBd 4'44?— TEEHM B ‘ 5690—17—12 IRE-2742" (534) And onq = i1, 1d21‘3 _4_ _ 1(1) _4 dB TEE-MPG B — 212—90090 +1)Bd—§ (5.35) 31:11 4(1+2pa“ 0)d3_ _9 (2pa4-1)d_ 1134K3 11 dc lzp" Then substituting small 11 expansion of the first order solution (5.31) into the right hand sides of (5.34) and (5.35) gives 1d473 (1+064 )dzB fid—Cr 4—nz°_d__§2+16p543 = —2ml,§{8114+(%+ +-§2)11 6}+0(118), (5.36) andonq = :1, d2? _ lid—2+ 40643 = -2wlz(4n2+gn‘+§06) + 0(113), n C 3 15 (5 37) 1d31'3 (1+2p54 )dB I 3 5 5 7) 9 ' 1-1—33E— T3: = -2(Dl, 21’] +411 +3311 +00] ). After considering the solution (4.117), the expression for 73(11, Q) in (5.36) and (5.37) must be an even function in T] and an odd function in Q. This motivates E(Tl, C) = —2wl,§{b”112 +(b21+ b,,§2)114 +(b31'I’ b32C2 “I b33C‘)n6 (5.38) +(b41 ‘I' b42§2 + b43C4 ‘I' bug‘s)“8 + 00110) l- 107 Substitution of (5.38) into (5.36) and (5.37) and equating the coefficients of left and right hand sides in terms of the product Q and 11, give the undetermined coefficients b,,—’s. The details to obtain bIj’s are as follows: 0(112) term in (5.37)I and 0(11) term in (5.3 7), give b” and b,,, 0(Q112) term in (5.36) gives b,,, 0(Q211‘I) term in (5.36) gives b411, 0(114) term in (5.37), and 0(113) term in (5.37), give b,I and b,,, 0(Q114) term in (5.36) gives b,,,, and 0(116) term in (5.37), and 0(115) term in (5.37), give b3, and b,,,. In summary, the asymptotic second order solution 73(11, Q) for flexural deformation when 11 is small, becomes 3014) = -21912C l112- Z-ICZ 114+ g-ZCHIC‘ 116 4 6 2 72 9 6 (5.39) 2 110741-33? + 5%“ + 4:5?)118 + 0(11 10) I The coefficient b,,, is undetermined yet but it is not necessary for future calculation. As we expected, since the B matches the second order solution of Sawyers and Rivlin (1982), the asymptotic second order solution (5.39) is the same as the series equation (7.14) of theirs except for the sign which is due to the difference of definition of (0 and the normalized coefficient. The asymptotic solution (5.39) can be compared with the small 11 expansion of analytic solution (4.117) for verification of its accuracy. Now we find from (4.118) and (4.119) that o 9 69 6537 = _ ___— 2 __ 4 6 N3 ”C2112I32 160TI +22400'I +0“ I}: (5.40) 3 1 3 1 2107 4 6 - mCZQI—§+§11 “274-611 +0(11)}, 108 and N _mczoI 21 107 103 I- _ __ __ 2114 6 112 64+640'I+40'I +0“ )I _ 29 3 169 2_447116 N2“°°C1?I671+a—o" 2'37)" “10‘ I} (5.41) where C = J21, Here small 11 expansions of coefficient M in (5.33) and the notation A in (4.120) as have been used. Then the second order solution B(n, Q) in (4.117) becomes 7301, o = 22.411 + §0.594, the homogeneous deformation has less energy. These stability conclusions are similar to those obtained by the numerical procedures in Section 5.3, except that the asymptotic analysis predicts a critical 11=0.594, while the numerical analysis gave a critical 11=O.6443. Note that the curve generated by 112 (5.53) can not be directly compared with the numerical curve of Figure 5.3 because of the presence of the normalizing M in (5.13). Performing a similar normalization on (5.53) using asymptotic coefficient M in (5.3 3) makes dimensionless stability parameter RS as R301) _ 16 9I 7 = —_ _ _ __ 2 4 15,01) 01.444423 8111 2 511 +0011}. (555) where 11 = Q], was used. The transition value here is 11=l.l95 but this value is not important since the expansion for M in (5.33) is valid only for 11 « 1.37. The comparison of RS in numerical results (5.14) and asymptotic results (5.55) is shown in Figure 5.6. 0.15 . . . . . 4 Rs . Asymptotic result (5.48) 0.1 . \ I Numerical result (5.14) 0.05 t \ . o I l | -005 . I l | I I | -0.1 . . I 4 _ =1.19 -0.15 _ 11—0.6443 '1 I -0.2 A A A . A A 0 0.2 0.4 0.6 0.8 1 1.2 Figure 5.6 Comparison of dimensionless stability parameters Rs of numerical result (5.14) and asymptotic result (5.55) for flexural deformation. The asymptotic result is valid only for very small 11. 113 When 11 is near zero, the signs and slopes are close so that the results are consistent with the numerical procedures. 5.5 Discussion The sign of p, is opposite to the sign of R3 according to (4.173) and the sign of E4 is the same as that of p, according to (4.175). E, is the leading term in the energy difference AE. Positive (negative) values of p, denotes that the load must decrease (increase) after the buckling onset value p0 in order to follow the bifurcated branch of buckling solutions. In other words, by converting p, to 71, with 7L, = —2 p,ij3 , the load must decrease if K,<0. For flexural deformation the numerical results show if 11<0.6443 then Rs>0 which means the bifirrcated path involves less energy than the trivial solution so that the homogeneous deformation near the bifurcation initiation is unstable. Otherwise when 11>0.6443, the homogeneous deformation near the bifurcation initiation is stable. The numerical results also suggest additional stability transitions near 11=1.305 and 11=1.6283 (Figure 5.1). However, these results are highly sensitive to the numerical evaluation procedure and so are rather suspect since they involve R5—>oo. Further it is not obvious how to treat these by asymptotic or perturbation procedures. These Rs—mo transitions that are only detected numerically will be dismissed from further discussion. In contrast, the stability for small 11 on the flexural branch is consistent with the asymptotic analysis near 11=0 (although the value of 11 for stability transition found by the asymptotic procedure is different). The numerical results also show for barreling deformation that Rs>0 for all 11 so that the homogeneous deformation is unstable compared to the barreling deformation. 114 These trends are in fact opposite to those found by Sawyers and Rivlin (1974) in the view of stability evaluation. They also find that 11=0.32 gives the transition in stability on the flexural branch. 5.6 Summary The stability of post-buckled deformation near buckling onset was evaluated by using energy minimization scheme. Extensive use of symbolic algebra procedures enabled certain simplifications, but the problem still remained very complex. Accordingly, a combination of asymptotic and numerical procedures were employed to attempt to determine stability transitions. The physical buckling behaviors are as follows. As the thrust load increases from the original zero value, a family of infinitesimal flexural deformation competes with the homogeneous deformation until the thrust reaches the value associated with p0=0.5437 (TS=5.6786 in (2.16)). This thrust is known as the wrinkling load. Then as the thrust exceeds the wrinkling load, the infinitesimal flexural deformation family ceases to exist and is replaced by an infinitesimal barreling deformation, which again competes with the homogeneous deformation family. The energy analysis shows that the infinitesimal flexural deformation family is energetically favored over the homogeneous deformation family at small loads (implying small mode number), but that the homogeneous deformation family is energetically favored at large loads (again dismissing Rs—wo transitions). Thus there is a transition load value, and a corresponding transition mode value 11=0.6443 (See Equation (5.14)) for this exchange in stability. In contrast, the infinitesimal barreling deformation family is always found to be energetically favored over the homogeneous deformation family. 115 Asymptotic analysis, valid only for small 11, was employed to attempt to verify the behaviors of stability pattern for the flexural deformation. The results agree with those of the numerical approach. A leading order value for transition of stability gives 11=1.195 (See Equation (5.55)). This precise value (11=1.195) is not of importance since the asymptotic analysis is only valid for 11 near zero. Clearly there remain significant questions with respect to this work, especially with regard to precise numerical transition value. It must also be admitted, since the stability interpretation is dependent on the (+/-) sign of very complicated expressions (Equation (5.13)), that additional efforts are necessitated for confirming those results. This, however, should not obscure the fundamental basis provided by this work. Notably the consistent perturbation analysis provides strict order expansions both of the energy competition between homogeneous and bifurcated solutions (Equation (4.159)) and of the relation to the bifurcated path near buckling onset (Equations (4.2), (4.69), (4.126), (4.127)). Here the essential and consistent coupling between these is provided by (4.141) and (4.176). 116 CHAPTER 6 APPROXIMATE SCHEMES FOR BUCKLIN G LOAD OF MULTI-LAYERED COMPOSITE PLATES 6.1 Introduction The previous Chapters were concerned with the buckling and post-buckling behavior for a noncomposite single ply plate and the stability of the various competing solutions with respect to each other. However in this Chapter, we will investigate somewhat practical topic - the critical buckling load on a composite plate. In a structure, the critical buckling load plays an important role since it gives the lowest load to resist against the compressed load. We had the critical buckling load on a single ply plate by solving rather simple equations in (4.55) and (4.57) and showed the result in Figure 4.1 as the relation between the load parameter p0 and the mode ntunber 11. The curves of buckling onset which give the load at the buckling onset for specified geometry and mode number, are monotonic with mode number (increasing for the flexural deformation and decreasing for barreling deformation). Hence the critical (lowest) buckling load is always mode-1 flexural deformation. But for a multi-layered plate such as the three-dimensional geometry of Figure 6.1, this behavior may be seriously altered. Pence and Song (1991) and Qiu et. al. (1994) showed that in symmetric three-ply plate composed of two different types of neo-Hookean material, there exist another family of buckling paths and they are not always monotonic. This means the mode of the lowest critical buckling load is not always mode-1 flexural deformation. As the number of layers in a composite plate increases, the direct algebraic analysis of the bifurcation conditions becomes increasingly complicated since it involves 117 I A( / 13 <— 211 >/ Figure 6.1 Geometry of the symmetric three-ply composite layer. The buckled configurations involve deformations in the (X,,X,)-plane. seeking roots to a determinant equation for which the matrix dimension grows with the numbers of plies. Thus it is useful to seek alternate methods for determining buckling onset load in the manner of simpler approaches and closer to the exact values. The purposes of this final chapter is to present some developed observations on these issues. 6.2 Buckling Load of Multi-Layered Plates The prediction for buckling onset load may change if the material is composed of multi-layered composite plate. In this Chapter, we will consider the general N-ply plate stacked along the X, direction. The undeformed configuration of whole plate occupies the region 21,x21,><213 and all plies are neo-Hookean materials. The shear modulus of each ply is either III or 11" and alternate by ply. Perfect bonding is assumed across the ply interfaces. Again our attention is restricted to plane strain deformation where buckling takes place in the (X 1,X,)-plane as shown in (2.18). Then the mathematical formulation of composite plate problem is similar to those of single ply plate studied in Chapter 2. The differences 118 are (i) the shear modulus p of single ply plate alternates between Ill and u", and (ii) the assumption of perfect bonding yields the following interface conditions on the traction and displacement. XXX?) = X.(Xi ) + on interfaces i = 1,2,3. (6.1) 521(le = 521(X2 ) Here the conditions for i=3 are automatically satisfied in a plane strain setting. Let A” be the sum of original areas normal to X 1 direction of plies whose shear modulus is p’ (i=I,II) so that AI +AII = 41,13. Then the total thrust on X, = ill for homogeneous deformation can be modified from (2.16) to T = -(P - 9’3)(H1AI+ WA”)- (62) The buckling onset can be analyzed by the incremental deformation superposed onto homogeneous deformation such as (2.18). The complete boundary value problem of a composite plate then consists of that of noncomposite plate (2.28) in which It changes to p" and the interface continuity conditions [Pj(V1,2 + 13-2121) + qu,1IX2+ = IHJIVLz + 13.2121)+ 9V2, 11x, [11102.2 ' P72V1, 1) -‘I(V1,1 'I‘ NIX; = [111.(v,,, — p-2v1, 1) —9(V1,1 + 9)]X5 (6 3) IVIIX3 = [VIIXi IV2IX; = [VZIXE on interface. Here 11! is the shear modulus of top ply and pi is of bottom ply on that interface. Since the buckling onset occurs at the first order (k=1) in perturbation expansion of deformation, we will consider only the homogeneous boundary value problem modified from (3.16) with appropriate interface conditions (6.1) such that, 119 Flu = 0 in IT, GIu = 0 on I], (6.4) 0 on F,, and [61'2“]3 = [szulXi on interfaces (6.5) IG3“IX2+ = [63“],i where superscript j=I,lI, denotes the differential operator of ply j in a composite plate. For simplicity in the expression of the first order equation, the superscript I” will be suppressed here and after. Here 0, is the constant matrix 4 = 1:, 1 31 The difference in potential energy E in (2.30) for noncomposite plate can be used for composite plate as the sum of the energy of each ply. For buckling onset, the energy equation have the value up to the second order so that 0') E2 = 213IIIIE5'12902(“1,2“2,1 ‘ “1,1“2,2) + ”1,1 + “22.2 + ui, + “i, 1 }dXIdX, (6+7) where the integration on X, is sum of the integrations of each ply. Following the similar analysis to noncomposite case, this composite plate problem is reduced to one homogeneous ordinary differential equation, 4 boundary conditions and 4(N-1) interface conditions. Introducing the proper general solutions which is similar to (4.18) with discrete constants L1“), L5"), MI"), My"), m=1,2,...,N of differential equation (6.4), to the boundary and interface conditions (6.4),, and (6.5) form a homogeneous 120 4Nx4N matrix equation. For the buckling onset load (nontrivial solution), the determinant of this matrix must vanish. In particular case of symmetric plate, the dimension of matrix can be reduced to two 2Nx2N - one for symmetric mode and the other for antisymmetric mode as explained in Chapter 4. For the simple example of symmetric plate, the three-ply composite plate (N=3) was considered by Pence and Song (1991, 1993). Here the central ply (material 11) has the thickness 2R (<21,) and the shear modulus Ir" so that the shear moduli of outer plies (material I) are 11'. This problem then simplifies to 12 homogeneous linear equations for the 12 constants L1“), L5”), MI"), My") , m=1,2,3. The vanishing of the determinant of 12x12 coefficient matrix gives the loads for buckling onset. Due to the aspect of symmetry, this problem can be considered by two 6x6 matrix equation - symmetric (flexure) deformation and antisymmetric (barreling) deformation along the X, direction with four dimensionless parameters as 2.0 = p52, 11 = (21,, B = pII/IrI, or = R/l,. (6.8) When [3:] or a=0 or CF], this problem reduces to the noncomposite case which is analyzed previously by Sawyers and Rivlin (1974,1982). The numerical computation for three-ply plate under various parameter sets shows in Figure 6.2 that (i) the buckling onset load for composite plate does not guarantee its monotonicity, i.e., the critical load is either mode-1 flexural deformation or wrinkling deformation in which the mode number is infinity (Pence and Song, 1991 ), (ii) there exists additional solutions for each original solution of flexure and barreling - we categorize these into a new family and the original family of solutions, respectively, and (iii) the wrinkling load of original family converges to that of noncomposite plate which is 121 A 15 1 fl I 0 \I ' \ (I3,a)= \ new barreling (2.0,0.5) ' I \\//i” 11 2 new flexure _ Aw 3 , flexure l .1 l l 1 l o 2 4 6 a 10 I I I = 71'0 194 _I (Baa) (05,05) 17 — 15 _ new barreling ~ 13 4 _ 1, I. new flexure l a u 9 — _ 11 It 7 _ 2 5 a barreling 4 kw 3 1— _. 1 flexure _ 1 l l o 2 4 6 a 10 Figure 6.2 The buckling onset prediction curve for a symmetric tri-layer with different values of B=uII/I.rI. The shear moduli of shaded plies are twice as large as those in the unshaded plies. In both cases, the volume fraction of central ply, a, is 1/2. Nonmonotonic behaviors, new curves at large 3. values, and asymptotes of all curves are shown (Qiu, et a1, 1993) 122 constant for the material parameters and the wrinkling load of the new family converges to a higher value than that of original family depending on the stiffness ratio 13 (Qiu, et al., 1993). An asymmetric two-ply plate (N=2) is discussed in Qiu, et al. (1993). Due to the lack of symmetric property in shape, the buckling deformations become a mixed mode of flexure and barreling. Solving the resulting 8x8 determinant shows that there are three solutions. The lower two curves are original family and the higher one is new family. The wrinkling load of original family have the same values of noncomposite plate and the wrinkling load of new family depends on the shear modulus ratio. 6.3 Approximate Schemes to Determine Buckling Load To construct buckling onset prediction curves of buckling load vs. mode parameter at fixed values of appropriate volume fraction and stiffiress ratio, will be a key to determine the critical buckling load and eventually require complicated numerical procedures. For the general N-ply sandwich plate, this problem is reduced to solving a 4Nx4N determinant equation. An exact analytic solution satisfies the complete conditions:_ the nonlinear constitutive equation (CE) in (2.13) for noncomposite plate, equilibrium equation (BE) in (2.6), boundary conditions of a free surface condition (FSC) in (2.9) and conditions of interface displacement continuity (IDC) in (6.1), and traction continuity (ITC) in (6.1),. The simultaneous satisfaction of all these conditions gives much difficulty when the plate consists of large number of ply stacking. This difficulty is stems from the fact that standard numerical procedures to find the roots of the necessary deterrrrinant are subject to various numerical errors and numerical instabilities. The possible approximation schemes may involve procedures (specifically trial fimctions) that do not 123 satisfy certain conditions mentioned above. Satisfaction of all these conditions gives an exact solution and thus an exact prediction of the buckling load. Therefore the goal of the research described in this Chapter is to construct useful approximation schemes which by sacrificing some of the conditions, give a simpler mathematical formulation. The effect on accuracy of these sacrifices will then be examined. The simplest approximate scheme is based on equivalent modulus where the composite structure is treated as a homogeneous media with volume averaged stiffness modulus. For example as described in Figure 6.3, the three-ply plate which the central ply 1 IA11 1th 11“ I211 212 -> =“I(l’°‘)+“"a 212 I I‘ v <—fll—> <——211———> Figure 6.3 In the equivalent modulus scheme, the composite plate is treated as a single layer of volume averaged stiffness, ucq. has It" and top and bottom plies have 11' considered previously, is treated as a single ply which has the equivalent stiffness modulus ucq = IrI(1-01) + uIIa where a is the volume fraction explained in (6.8). Note that this scheme does not satisfy constitutive equation pointvvise, but does so in a volume averaged sense. However the buckling onset prediction curves of single ply plate does not show the dependency on system parameters so that any combinations of plies have the same buckling onset prediction curves as the noncomposite 124 plate. Therefore the critical buckling load for equivalent modulus scheme is always mode- 1 flexure. 6.3.1 Variational scheme Most approximation methods used in structual analysis are based on variational mechanics in which the approximate solutions satisfy the weak (variational) form or minimizes the energy functional (Reddy, 1986). The buckling load in a structual problem or the natural frequency in a vibration problem can be determined approximately by so called Rayleigh quotient obtained from the variational method. The critical buckling load for the composite plate considered here can be constructed from the boundary value problem of composite plate (6.4) and (6.5). The inner product (u,Ffu) = 0 (6.9) followed by the integration by parts once and applying boundary conditions (6.4),, and continuity conditions (6.5) gives the critical buckling load as (I) IIin(ui,1+ “3,2 + “i,2 + “inldxrdxz (6.10) x0 = on _ .I.In “(1)011, 1u2,2 " “1.2“2, 1)dX1dX2 where kozpo‘z is the load parameter and the integration on X, is sum of the integrations of each ply. This quotient can be also obtained by energy formulation based on the fact that deformed configurations with less strain energy than the unbuckled homogeneous configuration only become available once the buckling load is attained. The difference in energy from the homogeneous state to buckled state of a composite plate in (6.7) must be 125 negative when the buckling takes place. At the buckling initiation, E, becomes zero. Equating E,=0 gives the same quotient for the buckling load A, = p52 as (6.10). If we have the exact solutions a, and u, of buckling onset, then the quotient (6.10) will give the exact buckling onset load. However the procedure to determine the exact buckling solutions u, and u, of the general ply composite plate is not that easy. So the approximate (trial) solutions which will satisfy part of required complete conditions must be considered. These will give a closer prediction to the exact buckling load. 6.3.2 Trial solutions Recall that the exact solutions satisfy the requirements of complete conditions: CE, EE, FSC, ITC, IDC. A simple approximation for the composite plate is to use exact solutions of noncomposite plate (4.68) with (4.26) for flexure and (4.56) for barreling. One approach is that the composite material can be considered as the combination of corresponding single plies (combined single ply solution). For example as shown in Figure 6.4, the geometry of mode-m flexural buckling of composite plate is similar to mode-m flexure in each ply, while mode-m barreling of composite material is similar to mode-m barreling in the central layer with mode-m flexure in the outer layers. Note that in this approach, the length of X, as well as 11 in each single ply are scaled to those of single ply and the X, coordinates in each ply are transformed to the origin. This approximation does not satisfy the interface displacement (6.5),, although the displacements are close. For a symmetric three-ply plate with (B,a)=(0.l,0.5), the buckling onset curves in Figure 6.5, are generated by the quotient with this combined single ply solutions. The exact curves and the curves by equivalent modulus scheme are also shown. For flexure, the variational scheme with combined single ply solutions gives better results than the 126 =1 _/_ _/— flexure f f I131:)lrure -> . I _/ f. 321.... :12'R f f =1 flexure 2R , f fl:)l(ure ter f f _ —/— /— garielin 2’! x I" , N k x XI rfrl=l n=1 barreling Figure 6.4 In the direct energy scheme, the overall deformation is approximated by the single layer deformations as shown. The deformation of the central layer distinguishes overall flexure from overall barreling. equivalent scheme. This is because the single deformed shapes are well matched to the three-ply plate for flexure. However they are not well matched for the barreling case. The other approach is that the single ply solutions can be applied directly to all the plies (direct single ply solution) since the final displacement of deformed shape of composite plate is similar to that of single ply plate. For example of three-ply case, the mode-m flexural buckling of composite plate may use the solutions of single ply mode-m 127 equivalent modulus I combined single ply solution \ exact o l l l o 2 4 6 8 10 TI A0 5 1 1 1 1 4 2 / combined single ply solution _ equivalent modulus 3 l— A ‘/ exact 2 A 4 l P _1 o l l l l o 2 4 6 a 10 TI Figure 6.5 The onset prediction curves as given by the equivalence scheme, exact scheme and the variational scheme with combined single ply solution for flexure (top) and barreling (bottom) for tri-layer with (B,a)=(0.1,0.5). 128 flexure. This approximate solution satisfies the boundary and interface conditions, but not the constitutive equation. For a three-ply plate with fixed pairs of (B,a)=(0.5,0.5) shown in Figure 6.6, the buckling onset curves are close to the exact solutions for both flexural and barreling modes. They form an upper bound as expected from the fact that the Rayleigh quotient gives an upper bound. 6.4 Discussion For an analysis for determining buckling onset load of a composite plate, we examined some approximation schemes in which the approximate trial solutions satisfy some parts of the complete conditions. Since general multi-ply plate is combination of even or odd stacking, the approximate schemes developed previously can be applied to a different ply stacking (like two-ply plate) in order to determine whether these schemes can predict the buckling onset curves well enough. Also we can consider other trial solutions based on combined single ply solution in the variational scheme so as to satisfy the interface boundary conditions (IDC) in (6.1) 1- One alternative is obtained by multiplying a suitable mollifier function of X, to a single layer solution (4.26) for flexure or (4.56) for barreling. The example of mollifier function is a simple polynomial forms with undetermined coefficients and play a role as a single ply solution in each separated ply in a composite plate according to its shape. The disadvantage of using mollifier function is the sacrifice of satisfaction on field conditions (CE) and (EE) but the satisfaction of (IDC) may improve the accuracy of whole approximation. Satisfaction with complete conditions of various schemes studied here and suggested scheme is summarized in TABLE 6-1. 129 lo 4 1- .1 equivalent modulus \ / 3 ~ _. direct single ply solution 2 exact ‘ 1 1 0 l l 1 J 1 1 l o 1 2 3 4 5 TI 5 1 1 1 r 1 o lo 4 e equivalent modulus 3 .— \ direct single ply solution 2 r exact . 1 1— —1 0 l J_ L l l l 1 l 1 5 1'I 0 1 2 3 4 Figure 6.6 Comparison of the flexure (top) and barreling (bottom) onset prediction curves for symmetric tri-layer with (B,or)=(0.5,0.5) as generated by the exact scheme, the equivalent modulus scheme, and the variational scheme with direct single ply solutions. The upper bound property of variational scheme is evident. 130 Table 6-1 Summary of complete conditions and satisfaction for various schemes CEl EEI FSCI ITC I IDCI Exact solution yes yes yes yes yes Equivalent modulus scheme no2 yes yes yes yes Variational scheme yes yes yes yes no3 (combined single ply solution) Variational scheme no2 yes yes yes yes (direct single ply solution) Variational scheme - suggested no5 no5 possible4 possible4 yes (mollified single ply solution) Condition (6.1).. IBI'ITIS. 2. satisfied only in a volume averaged sense, not pointwise. 1. CE: Constitutive Equation (2.13); EE: Equilibrium Equation (2.6); FSC: Free Surface Condition (2.9); ITC: Interface Traction Condition (6.1),; IDC: Interface Displacement 3. however expect the result to be close since the mode shapes should be well approxi- mated by the single layer theory. 4. these interface conditions can be ensured for mollifier functions of sufficiently many 5. greater sacrifice of CE and EE will yield more improvement in FSC and ITC. 6.5 Summary Three approximate schemes to determine the critical buckling load for three-ply neo-Hookean plate were investigated. Each scheme satisfies with parts of the perfect conditions and gives reliable values of critical load. The schemes developed here can estimate the critical buckling load of general multi-ply plate in a simpler manner. 13] CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 7.1 Conclusions of the Thesis The elastic stability analysis for the post-buckled and homogeneous deformations of single-ply neo-Hookean plate and the approximate schemes for buckling load of multi- ply neo-Hookean plate have been investigated. The elastic stability near buckling onset gives the prediction for physically existed deformation and can be evaluated by comparing the energy of all possible deformations. The buckling behavior are understood by the bifurcation theory in mathematical terminology. Under these basis, two dimensional nonlinear boundary value problem for single ply neo-Hookean rectangular plate was generated in the context of finite elasticity. The perturbation expansion method was then applied to analyze the nonlinear problem into the set of linear equations by the order of a. Each set of linear equations gives rise to a nonhomogeneous boundary value problem (except for the first order) and the solutions are related to the solutions and parameters of previous order. The first order equation is homogeneous and gives the thrust on buckling onset which is the critical buckling load. Also the load parameter that is barometer for thrust, are determined by Fredholm Alternative Theorem as a solvability conditions. The process for obtaining the solutions and load parameter for each order was developed in a systematic way and the more accurate results can be obtained from investigation of higher order equations. The deformations and their behavior at buckling initiation was at first obtained from the first order problem and shows the same results as other works. 132 For the stability evaluation of post-buckled deformation, the solutions and load parameters of second and third orders were obtained to analyze the energy difference which has the fourth order as the first appearing term (see Equation (4.159)). Since formulations is quite complex, a numerical analysis was applied at final evaluation of stability parameter (Rs in (5.14)). The numerical analysis shows that for flexural deformation, the buckled deformation has less energy than the homogeneous (unbuckled) deformation when 11<11c=0.6443 and 1.305<11<1.6283 and the opposite is true when 11c<11<1.305 and 11>1.6283. For barreling deformation, the buckled deformation has always less energy than the homogeneous deformation (see Section 5.3). The complexity of formulation and non-explicity of load parameter lead to an asymptotic analysis for post-buckled deformation. The analysis on extreme shape gives the same critical load as Euler buckling. Also for extreme case of flexural deformation, the buckled deformation has less energy than the homogeneous defamation when 11<1.l95 and the opposite is true when 11>1.195 (see Equation (5.55) and Figure 5.6). This asymptotic analysis is valid when 11 is extremely small. Hence the analytic results of energy comparison for the flexural deformation are true for limiting case of small 11. In this study, the perturbation expansion approach was used for evaluating the stability of homogeneous deformation. As a comparative work, Sawyers and Rivlin (1974) applied the variational methods to determine the stability of homogeneous deformation near the critical state. Their results are that the homogeneous deformation of flexural deformation is stable (has less energy) when 11<0.32 and unstable elsewhere and the homogeneous deformation of barreling deformation is always stable. Hence the results of their works and this study have the opposite pattern. Also the transitional points 11 133 obtained by Sawyers and Rivlin are different from those found here. The perturbation method used in this study is more direct and reasonable compared to the variational methods but the procedure is equally complex. One possible source of the difference in results stems from the different predictions that may be expected in general, as discussed in Appendix A in the context of some simpler problems. A second possible source of difference may be due to the difference in norrnalizations as explained in (4.59). As an extended work from the stability of homogeneous deformation, the analytical way of the determination of critical load for composite plate was also studied (see Section 6.2). The critical load which can be found from the first order equation, plays a major role in the design of load-bearing structures. However for multiple stack composite plates, the determination of critical load requires the solution of a nonlinear eigenvalue problem for a 4Nx4N matrix. As stack grows, the equation will be so complex that we need simpler albeit approximate methods. In a beginning stage, three-ply symmetric plate was analyzed for the critical load in three different schemes. Compared with exact critical load in previous study by Qiu, et a1. (1993), the results are quite close (see Figure 6.5 and Figure 6.6). These schemes can be expanded to more general ply plates. 7.2 Recommendations for the Future Work Direct energy comparison for the stability of homogeneous deformation was performed and the results showed the differences compared with those of previous works. Hence as a verification of results, other limiting cases of infinite 11 for flexure, near zero 11 for barreling and infinite 11 for barreling should be investigated. 134 Though the computations on the values of higher order will give more accurate behaviors of post-buckling for hyperelastic plate, this will also be confined in local behavior because of the limitation of perturbation method. However the local stability criterion such as the stability near critical buckling load will be ensured. 135 APPENDICES 136 APPENDIX A STABILITY EVALUATION BY PERTURBATION EXPANSION METHODS A.1. Introduction The analysis for the stability of buckled deformations in the vicinity of buckling initiation for a neo-Hookean plate requires complex mathematical calculations, so that the clarity of evaluations for each step is confused. In this Appendix, more simplified examples than considered problem in the context will be investigated in order to obtain the thorough acknowledgments for the concept of stability. The well-known elastica problem will be considered at first as the simplest model. A modified elastica problem will be considered next to investigate the relation between the post-buckling behavior which is dependent upon a second order term in load parameter and the stability. Finally a higher order problem, which in certain ways resembles the neo-Hookean plate problem, will be examined. For the methods on stability evaluation near the buckling initiation, the perturbation analysis (PA) which has been used in the context and the analysis method adopted by Sawyers and Rivlin (SR) in (1982), will be used for the comparison. In particular, it is shown that these methods can give different stability predictions. These examples will consider an energy equation E(th) where 71 and u are load parameter and buckled deformation in the buckling problem or eigenvalue and solution in the mathematical bifurcation problem, respectively. Then the first variation of the energy equation gives a governing equation in a domain 9 and boundary conditions on a boundary F, 137 111).. = 0 in :2, 6(1):. = 0 on r, (A.l) where F and G are differential operators. The statement in (A1) constitutes a nonlinear boundary value problem. With respect to (Al), it is assumed that there is an obvious trivial solution um, for all values of 7.. Thus u = u,,,vOt) which is the family of trivial solutions. We now seek additional solutions (competitors) that bifurcate from this trivial solution. These additional solutions would also depend on 3., say u = u,,/(1.), so that a continuous parametric dependence on 7t also defines a family or branch of those additional solutions. Unlike the trivial solutions, the family u,,/(71.) may exist for only a restricted range of 7.. Now the solution family u,,-10.) is said to bifurcate from the trivial family u,,-,0.) at the value 3.0 if quO) = ”611(10)- The stability evaluation is well explainable under the concept of energy minimization. If, at a given load parameter A, the energy of the one equilibrium solution is less than that of another competing equilibrium solution, then the original solution is energetically preferable to that of the competitor (it is more stable). The energy difference between the trivial solution and the buckled solution at certain load level 1.. 415(1) = £14.51). 111-150....0). 1) (42) will be considered in the following analysis. According to the energy rrrinirnization scheme, if ADO, then the state corresponding to the trivial solution is stable. 138 A.2. The Perturbation Expansion Method One of the well-established approaches to solve the nonlinear boundary value problem is by using the perturbation method. This approach utilizes an expanded solution which is perturbed from the trivial solution u0=u,,,-, with a small parameter s such as u = u0+sul +82u,+83u3+ (A.3) The load parameter A is also expanded accordingly, 7» = lo+sll +szl,+s3k3+ (A.4) Here 3.0 is the critical load for bifurcation initiation so that the bifurcated solution u =ubI-f in (A.3) is branched from the bifurcation initiation u, = u,,-,0.) at k=1, Substituting (A.3) and (A.4) into (Al) and (A2), and collecting along the same order of a, give the set of linearized boundary value problem for each order of a. By solving each set of equations, we can construct a complete expansion on u and 1.. Budiansky (1974) also used the similar procedures to this study for post-buckling analysis. For the purpose of stability evaluation, the energy difference AE compares the energy between the bifurcated and trivial path at a load level it as shown in (A2). The general solution (A.3) in this approach is an expansion in the vicinity of buckling initiation so that the results will be limited to the local analysis. Again by using the expansions (A.3) and (A.4), the energy difference (A.2) becomes along the order of e as AE = 22E, + 23E, + 24E, + (A.5) Here E1 is vanished automatically if we substitute the equilibrium solution. For i-th set of the order 8, the solution u,- can be solved by the conventional way used in the analysis of linear differential equation and the eigenvalue X,,, can be obtained through the Fredholm 139 Alternate Theorem (FAT). The solutions and eigenvalues obtained are substituted into the energy equation then EH, can be determined. Since a is small, the first nonzero term on energy equation (A.5) becomes the leading term on energy difference. If AE becomes positive then E(ube) > E(uI,,-,,) so that the trivial solution is energetically preferable at the same load level 71.. For the opposite case, the bifurcated solution is preferable. The schematic diagram of the procedure is shown in Figure A.1. During the process, it is sometimes hard to find all the solutions uI-. Instead of direct application, we introduce a certain orthogonal condition to make some terms in energy equation vanish. This condition can be obtained by using the integration by parts to the linearized equation. Detailed calculation will be explained later for a specific examples. Governing Equation ‘__. Energy Expression Boundary Condition 5 u = 110+ 8111+ azu,+ e3u3 +... ‘ 2 3 . l=lo+skl+8 k,+8 1.3+... BVP on First Order (a) —> Mu. —> E,(u,,>.,) BVP on Second Order (82) ‘V 7m u, —> E3(u,,[u,], 7(0, 7+1) / BVP on Third Order (83) ‘> 12, [113] —> E4(u1,u2.[u3l, A'09 kl, A'2) Figure A.1 The procedure for a perturbation method. Here the variables with a bracket are not necessary if we introduce a certain orthogonal conditions. 140 A.3. The Analysis Scheme by Sawyers and Rivlin The approach to evaluate the stability of fundamental state which is the buckling initiation for the buckling problem, makes reference to the work of Koiter (1981). The energy of the admissible displacement u from the fundamental state characterize the stability of the fundamental state as the second variation to the energy is positive definite. Later Sawyers and Rivlin (1982) applied this approach to evaluate the stability of buckling initiation for the neo-Hookean rectangular plate. If the second variation P,(u) for the potential energy difference P(u) which is eventually the same as AE in (A2) under the dead loading condition, is deterrrrined as positive definite then the fundamental state is regarded as stable. The neutral equilibrium solution u which becomes buckling deformations, can be obtained by a zero minimum of the second variation P,(u). Here P,(u) represents a linear version in the whole energy analysis so that P,(u) vanishes with the neutral equilibrium solution. Hence for the further condition on stability, the bifurcated path u is decomposed into a linear version of the neutral equilibrium it and an additional term v. Substituting this new solution into the energy equation P(u) leads to a new energy equation P(v). Again by solving the equilibrium solution for v and substituting the solution into P(v). one can evaluate whether P(v) as well as P(u) is positive definite. This approach is also based on energy minimization scheme but the objects for competition is difi‘erent from the perturbation expansion scheme. A.4. Example 1: Elastica Problem An elastica problem in which a long slender beam is compressed axially, has been a model for the buckling analysis (Thomson and Hunt, 1969). The vertical deformation or 141 the buckled shape u is described mathematically as the bifurcation from the unbuckled deformation at a critical load parameter 110. The potential energy for a slender beam with a The primes denotes the differentiation with respect to x. The governing equation and boundary condition of (A6) through the first variation lead to u" + ksinu = 0, u'(O) = u'(1) = 0, (A7) where l. is an eigenvalue which represents the load parameter applied to the beam axially and the boundary conditions are characterized as a natural condition. Clearly one solution to the problem (A.7), valid for all possible A, is u(7t) = 0. This is therefore the trivial solution to this problem, um, =0. The linearized version for the nonlinear boundary value problem (A7) is stated as u"+?tu = 0, u'(0) = u'(l) = 0. (A8) For a variable v, the inner product (u" + ku, v) = 0 gives the adjoint problem to the linearized problem (A.8) through the integration by parts. The adjoint problem with respect to v has the same differential operator as that in (A8) so that the linear differential operator in (A8) is self-adjoint. The bracket used in inner product is defined as 14g) = III/mane. (A9) The solution to the linearized problem (A.8), u,,-,I becomes u,,-,I = Acosmrtx, A = (M102, m = 1,2,3, (A.10) 142 where A is an arbitrary constant. The linear solution reveals that there is an infinite sequence of bifurcation load (one for each m). A.4.l Perturbation Analysis (PA) We now seek additional solutions u= u,,,f, bifurcated from uo = u,,II, =0 in (A.3) such as u = 8u1+82u,+s3u3+... (A.11) Here the perturbation parameter a is defined as e= with an orthogonality condition (u,, uI) = 0, i1: 1. (A.12) The bracket denotes the inner product defined in (A9). Substituting the expanded solution (All) and load parameter (A.4) into the boundary value problem (A.7) leads to the set of governing equations and boundary conditions which form the separate boundary value problems that originate from the various order of 8. Each set of governing equation consists of a linear differential operator with i-th order deformation u, in the left hand side. The right hand side of i—th equation involves all of the previous solutions including u,,, and X,,]. The boundary conditions of each equation becomes uI'(O) = uI'(1) = 0, i = 1,2,... (A.13) For the coefficient of s, the first order equation becomes u," + 1.0a, = O. (A.14) The solutions of (AM) with the normalized coefficient are u1 = J2cosm11x, 1.0 = (mn)2, m = 1,2,3, (A.15) 143 The first order equation (A. 14) and the solutions (A. 15) are in fact the same as those found in linearized problem (A8) and (A.10) except for the subscript. Hence the first order equation represents the linearized problem. There is an infinite sequence of bifurcation loads according to the eigenvalue parameters 2.0 in (A.15) but, from now on, we will consider the lowest value, that is the case of m=1. The second order equation is shown as u,” + kou, = 4.114,. (A.16) Since the differential operator in the left hand side of (A.16) is same as that in (A.14), the solutions u, will exist only if the right hand side of (A.16) is orthogonal to the solutions of (A.14), namely (—}.Iu1, 111) = 0 according to the Fredholm Alternate Theorem (FAT). The existence condition and orthogonality (A.12) give rI = 0. (A17) Introducing (A. 17) into (A. 16), the solutions with the orthogonality (A. 12) become u, = 0. (A18) In a similar way, the third order BVP is expressed as u3"+}\.0u3 = éloU?—xlu2‘}\.2ul. (A.19) Applying the FAT again to the right hand side of (A.19) and the first order solutions in (A.14) gives l.2 = -)l‘0' (A.20) With the eigenvalues (A.17) and (A20) into (A.19), the solution for the third order becomes 144 u, = ~36‘71cos3m11x. ~ (A.21) For the fourth order problem, the similar approaches give the boundary value problem as u,," + 2.0114 = 52.011111, — k,(u3 — £11?) — 74,21, — 13m, (A22) and the results are A, = O and u,, = 0. (A23) The higher order solutions can be obtained in a similar way. Then the buckled solution and load parameter up to the fourth order are summarized as follows. u = u,,If = (J2cos1tx)e — (gai- cos3rtx) e3 + 0(85), (A.24) )4 = 2.0 + £71,082 + 0(34), 10 = 112. (A25) The equation (A.25) represents the post-buckled solution path in A-e curve. To evaluate the stability of the equilibrium solution near the buckling initiation, the energy minimization between the bifurcated and trivial solutions at the load level it, is used. The energy difference (A.2) becomes II 1 AB = (II-2.152 + A(cosu—1)]dx. (A.26) If AE >0, then the trivial solution is stable. Substitution the bifurcated solution 14be in (A.24) and eigenvalues (A.25) into (A.26) give the even-ordered series of energy difference 145 AE = 82E2 + 84E4 + (A.27) The odd-order terms vanish automatically. The second order term, E2 in (A27) is _ 1 J1 :2 2 With the solutions u] in (A.15), E2 vanishes. The fourth order term, E4 is E =f[u'u'-kuu—lkuf+iku4]dx (A29) 4 0 1 3 0 l 3 2 2 24 O l ' ' Substituting all the solutions and eigenvalues in (A.24) and (A25) into (A.29) gives In fact, it is not necessary to know u3 explicitly to evaluate (A.29). To see this, multiplying the first order problem (A.14) by uk, k=1,2,3,... and integrating over the domain 0O in (A.21) so that the load is increased when the bifurcation grows. Hence, considering the local behavior near the first bifurcation, m=1, there exists only one solution before bifurcation initiation, which is trivial and stable. After bifurcation initiation, there exist three local solutions of which one solution is trivial and unstable, and two other solutions follow the bifurcated path and are stable. A.4.2 The Sawyers and Rivlin (SR) Scheme Now consider the elastica problem with the approach used in the paper by Sawyers and Rivlin (1982). Starting from the potential energy equation for the admissible displacement u defined in (A.6). The energy difference which is in (A26) is now rewritten as P(u) = E(u,l)—E(u,,,v,k) = £{éu'2+k(cosu— 1)}dx, (A.34) where u = um], the bifurcated branch of solutions that we seek to construct. With the power series expansion of cos u, equation (A.34) becomes P(u) = i1 lu’2+k(— lu2+-l—u“- ) dx (A 35) o 2 2 24 ' ' The second variational term in the expansion (A35) is P2 = %£(u'2—lu2)dx. (A36) 147 The necessary condition for stability of trivial solution is non-negative P2 2 0. To find the stationary P2, set 8P2=0 8P2 = £(u’5u’—Au5u)dx = O. (A.37) After integration by parts, equation (A.3 7) yields u" + Au = 0, u'(0) = u’( l) = 0. (A.38) The solutions of (A38) are u = Acosmnx, X = (mn)2, m = 1,2,3, , (A.39) which retrieves the bifurcation initiation values A0 and u, previously given in (A.15) and also the linearized solution u,,-n in (A.10). Again we will stick to the lowest bifurcation value m=1. Here u is only a linearized version of the solution branch u,,,f near the point of bifurcation initiation. By substituting (A.39) into (A.36), the result is P2 = O. (A.40) Hence the state for which P2 has a stationery value, is regarded as a state of neutral equilibrium. Now we decompose u = u,,,f into the linearized solution (A39) and the remainder term v as u = a£4+v it = cosnx, (A.41) where it = A cos nx comes from the linearized u in (A39) and a is the coefficient given by a = (fl, u)/ (fl, 1?). The solution components 2": and v have the orthogonality condition 148 (a, v) = 0. (A.42) Substituting the bifurcation solution u,,,f as given in (A.41) into the energy difference (A.35) gives P(u) = £{éazfiz' - 1545+ a(a'v' — 7m) + 5.0/2 — W) + ilzuaa + v)4 + }dx. (A.43) It is to be noted that, unlike the PA method, the SR scheme always use 7L = 1.0 associated with bifurcation initiation. Now consider the multiplication v to the equation (A3 8) with changed variable 2? and integration over the domain O0, then from (A.61) and (A.64), A2O. Since E4 is the dominant term in energy difference, the trivial solution has smaller energy than the bifurcated solution, that is, energetically stable. If a<0, there exist three solutions before bifurcation initiation and A2>O and E40, P(u) becomes negative so that the trivial solution is unstable. This result is opposite to the PA method. A.6. Example 3: A Higher Order Problem Represented Neo-Hookean Buckling The buckling and post-buckling problem for a neo-Hookean plate considered in the context by using the perturbation analysis, have a slightly different form from the previous two examples. Namely, the neo-Hookean plate involves u2 at 0 (sec (4.128)) and A1=0 (p,=0, (4.138))so that the expansion for the deformation u and eigenvalue A become u = 8u1+82uZ+83U3+ ..., A.69 A = A0+32A2+84A4+.... ( ) In order to understand this type of expansion, it is useful to examine a simpler model than the neo-Hookean plate buckling problem. Such a simple model may be provided by considering the following energy expression 2 1 '2 1 2 1 ' 2 '2 E(u, A) Mia 7M +5u(u +u )]dx, (A70) for a function u obeying u(0) = u( 1) = O. The governing Euler equation for the energy equation (A.70) are given by the first variation as u"+Au+u'u" = 0. (A71) 154 The trivial solution for the problem (A.71) is obviously u,,,v=0. The linearized problem to (A71) is stated as u"+Au = O, u(O) = u(1) = 0. (A.72) The boundary condition in (A.72) is different from previous two examples. However this does not affect the self-adjointness for the linear operator. The linear solution is obtained by solving (A.72) directly u = Asimrx, A = n2. (A.73) Here we considered the lowest value of bifurcation (m=1). A.6.] PA Scheme The bifurcated solution u=u,,,-f is an expansion from the trivial solution u,,,v=0. Substituting the expansions in (A.69) into the problem (A.71), the resulting boundary value problems for each order become 0(8): u,"+A0ul = 0, (A74) C(82): u2"+A0u2 = —A.u, -u,'u,", (A.75) 0(83): u3"+Aou3 = -A1u§-A2u, —u,'u2"—u2'u1", (A.76) and so on. The boundary conditions are u,.(0) = u,(1) = 0 i= 1,2,... (A.77) Since the trivial solution for this problem is u,,-50, the energy difference between the bifurcated and trivial solutions is the same as (A.70). By using the expansions (A.69), this energy difference AE can be expanded accordingly. 155 AE = E282 + E383 + E484 + 0(85), (A.78) where the components are 1 E2 = vi:§(u,'2—A0u12)dx, (A.79) _ f I I 1 1 I l ' E3 — O[u, u, —A0ulu2+§ —x,u,2+§ul .4430.l )3 ]dx, (A.80) E4 = £%{2(u1'u3' — A0u1u3) + (142')2 - 7‘0“? ‘ 211“]“2 — A2u%}dx 1 1 (A.81) + 2 Ji){§(2“1u1'u2 + “12“2') + “1 'uz'}dx, and so on. Now we consider the first order equation. By solving directly the solution becomes for the lowest value of bifurcation (m=1) as u, = Jisinnx, A0 = 1:2. (A.82) In a similar way in (A.32), consider the following integration kw," + A0u1)ukdx = O = 1, 2, 3, (A.83) Integration by part once and applying the boundary condition in (A.77) lead to the equation (A.32). This equation can apply to E2 in (A.79) and the first two terms in (A.80) and (A.81) so that it is not necessary to solve 143 in (A.81). The solution in (A.82) is substituted into the second order solution and the results become x u2 = —Tosin21tx, A, = o. (A.84) 156 Then applying the inner product with the right hand side of (A.76) and ul and the FAT gives the next order eigenvalue as 1 With the previous solutions (A.82), (AM) and (A.85), the third order solution for the equation (A.76) becomes A u3 = isin3nx. (A.86) 4f2 However the third order solution (A.86) is not necessary to evaluate E4 since there are no u3 terms in (A.81) after considering the equation (A.83)]. Substituting all the solutions and eigenvalues obtained previously into the energy equation, then E2 in (A.79) and E3 in (A.80) vanish and E4 in (A.81) gives E4 = —A3, (A.87) so that the energy difference becomes AE = 115138” 0(85). (A88) In this case, A2 is always negative and then AE is always positive so that the trivial solution near the bifurcation initiation is always stable. 1. This parallels the results in Section 4.7.2 where the u”) terms are eliminated in the energy expression (4.164) by using the boundary conditions (3.9)2 for k=1. 157 A.6.2 SR Scheme The energy equation for the neo-Hookean plate is expressed in (A.70). The second variation in energy P2 to (A.70) is given by P = f (l(u')2 -1Au2)dx (A 89) 2 o 2 2 ' ' After first variation to P2, the linearized governing equation becomes u" + Au = O, u(O) = u(1) = 0. (A.90) The solution to the boundary value problem (A90) is u = Asinnx A = n2. (A.91) Again we stick to the first mode m=1. Substituting the solution (A.91) into (A.89) gives P, = 0. (A92) Then the decomposition of u=u,,,~finto the linearized solution u=u,,,, (A91) and remainder term as u = ail + v, (A.93) where a is a small coefficient and the following orthogonality holds (it, v) = 0. (A.94) After applying the new solution (A.93) to the energy equation (A70) and set v = 022-4, the energy equation after ignoring the order higher than 4 becomes P(u)=a4£[{%fi'(&§+£¢"+§) +504 +(“'))+2((u )2 Au )H . (A95) 158 The first variation for the equation (A.95) gives the governing equation about {I as 17" + A2} = —z‘4'£4" 27(0) = 5(1) = O. (A.96) The solution for (A.96) with orthogonality condition (A.94) becomes - A2 . u = Knsrn(2nx). (A.97) Substitution the solution (A.97) and linearized solution (A91) to the energy equation (A.95) gives 9.2.44 4 P = - 48 a . (A98) The equation (A.98) reveals that P is negative so that the trivial solution is regarded as unstable. A.7. Discussions In this Appendix, we examined the stability in the vicinity of buckling initiation for more simplified problems than the buckling problem of neo-Hookean plate by using the perturbation method and the method used in the paper by Sawyers and Rivlin (1982). The whole procedure and their corresponding equations are summarized in Table 1. and 2. The perturbation approach compares the energy between the bifurcation and the trivial solutions on the same load level of A and on the first mode (m=1). The sign of an energy difference determines the lower energy level of two competing solutions which shows more energetically preferable solution. Also the results tell the relation between the post-buckling behavior (A2) and the energy difference (E4). From the results on A2 and E4, (A20) and (A30) for the elastica problem, (A61) and (A.64) for the modified elastica 159 problem and (A85) and (A.87) for the simplified neo-Hookean buckling problem, it can be shown that E, = "1,. (A99) Therefore the signs of A2 and E4 are different so that, if A2 is negative then the trivial solution is energetically preferable. Otherwise the bifurcated solution is preferable. In the SR method, the eigenvalue is not expanded so that the energy difference used in this method just compares the energy on the bifurcation initiation. This will give a confusion that on the eigenvalue (load parameter) at the bifurcation initiation. It also can generate opposite conclusions as to the stability of the bifurcated branch of solutions. 160 Table A.1 Summary of the procedures and their corresponding equations for the perturbation expansion. Modified Simplified Neo-Hookean Step Elastica Elastica neo-Hookean plate (context) 1. Nonlinear (A6), (A26) (A55),(A.62) (A.7l) (2.28), (2.30) BVP, Energy equation 2. Applying the (A. 1 1),(A.4) (A. 1 1),(A.4) (A. l 1),(A.4) (3.1) expansion 3-1. lst order (A.14) (A57) (A.74) (3.8), (3.9), BVP (3.10) 4-1. A0 (A.15) (A.61) (A.82) (4.62) 5-1. u, (A.15) (A.60) (A.82) (4.68)-(4.70) 6-1. E2 (A.28), E2=0 (A.64), E2=0 (A.88), E2=O (4.155),E2=O 3-2. 2nd order (A.16) (A58) (A.75) (3.8), (3.9), BVP (3.11) 4-2. A, (A.17), AI=O (A.61), A1=0 (A.84), AI=O (4.138), p,=0 5-2. uz (A.18), u2=0 (A.60), u2=0 (A.84) (4.126) 6-2. E3 E3=O E3=O (A.88),E3=0 (4.158),E3=O 3-3. 3rd order (A.19) (A59) (A.76) (3.8),(3.9), BVP (3.12) 4-3. A2 (A20), (A61), (A.85), A2=1/4 A0 A2=-3/2 0er A2=-1/3 A02 5-3. u3 (A.21) (A.60) (A.86) N/A 6-3. E4 (A.30), (A.64), (A.87), E4=-1/16 A0 < E4=3/8 0er E4=l/12 A02> O O 161 Table A2 Summary of the procedures and their corresponding equations for the Sawyers-Rivlin method. Modified Simplified Neo-Hookean Step Elastica Elastica neo-Hookean plate (1984) 1. Energy (A.35) (A65) (A70) (2.12) equation, P(u) 2. Second vari- (A.36) (A.36) (A.89) (3.1) ation, P2 3. Solving for (A.37) - (A.39) (A.37) - (A.90) - (A.91) (3.8) neutral CQUlilb- (A.39), (A.66) (314) (3.16) rium ’ 4. Define new (A.4l) (A.41) (A.93) (4.1) solution 5. Energy with (A.48) (A.67) (A.95) (4.7), (4.10) new solution 6. Solving for (A.49) - (A52) (A.49) - (A52) (A.97) (4.13) new solution (5.5) 7. Evaluate (A53), (A.68), (A.98), (6.1) energy “,9 P=1/64 AA > o P=-3/32 (1AA =-1/48 2.2.44 Fig.3, Fig.4 new solution (4th order) 162 APPENDIX B COEFFICIENTS FOR STABILITY PARAMETER The coefficient functions Y[i,j] used in equations (5.11) and (5.12) for stability parameter R3 are defined as follows: 1 _ _ _ _ Y“, 1] = mlymao)+N1)’1,2(7~o)-N2.Y1,30~0)+N3y1,4(}~o)-VN4)’1,5(7¥0)] Ylll, 2] = 3—2ngb’1,1(‘xo)—N|yl,2(—xo)-Nzy1,3(‘xo)—VN4)’1,4(-}t-o) + N3J’I, 5910)] ”193] = figiyraao)+N1Y1,7()~0)—VN4}’1,8(7~0)], i Y[l, 4] = -§;T§D’1,6(—7\o)—N1y1,7(-7~0) +N3Y1,8(-7~0)], O 1’11. 51 = gi—at—zwo — A3)(1 + 317,)- 1-)-’3)’1,9(7~o)‘r mum—m]. v _ ._ _ Y[I, 6] = —m[3210(1- 1110+ 16M“ — Ad)(1+ 2523-193‘ N3y1,1o(}~o) + VN4y1,1o(-7¥o)la (1+Ao) — — - ”1,71 = "Wb’mlafi-ZWNWI,12(10)‘24N1+N2V1,13(7~0)}], l—A _ _ _ Y[l, 8] = “%§18L)U1,11(4~o)—Zi—N3J’1,12(-}~o)+ 24N1+N2y1,13(—}~o)}], V(1+ Ao) Y[1,9] = ———41—8—{3VM—st(l 40)}, Y[l 10] = M9137 +Ns(1+A )} , 41-8 3 2 0 9 111,11] = 0, N132 1,12 = — l-A2 . Yl ] 2x0( 0) 163 1'12. 01 = —1—2§73(1—7~8)2{—N3(1—ko)+vF/4(1 + m} +(1+A%) W{16(1+s4x3)+52(3x3+ 58A3 + 3)}, 1'12, 11 = —%}:gly2,1(7~o) + N1y2,2(?~o) + 1852,30.) + 7852,4001]. 112. 21 = —1—6V-,7,-1yz,.(—xo)-N1y2,z(—xo) + My. 3(—x0)—vK/4y2, 46-14)]. 1'12. 31 = 5362,41.» N1y2,.(xo)— vN4y2,7(xo)1. Y12. 41 = fi—gm, .(-xo)—N1y2,.(—xo) + NW2, 76710)]. 1'12, 51 = gg—igtso — W3 + 6418 — 313) — 6131mm.) - VN4y2,s(-}~o)}], Y[2, 6] = .3:—:B[s(1 - x3)(3 + 32x3 — 313 + 32:21.3) + 6{]V3y2,8(A0) — mum—1.0)} ], Y[2, 7] = 64—17L3-[y2’9(AO)+ 1927\7.(1— A8) + Nzyz,10(A0)+ VIV4y2, ”(A0)], 1’12, 81 = gawk—ka—wzml — x8) + N212, Mao-My. ”Hon. 1’12. 91 = —-4—‘,:—8 Uz, no.) + vN4y2,13(7»o) + My), I.01..)1. 1’12, 101 = 7&8 Dz, 12(-7~o)-N3y2, ”(280) + N216, ”(4491. Y12. 111 = 71-0. 112,12] = —‘2—41.0(1 —21.g). 164 Here the 27 notations yum) used in coefficient functions Y[i, j] are defined as; yi,i(€) = -S§(1-€2){4+52(1-2§+5€2)}, yi,2(C) = 85C(1+§)(10+§+C2), yi,3(C) = 24S(I+€)(1+3§), yi,4(€) = 8{SZC(1+5€+2§2+4€3)-3(3+€)}, yi,s(C) = -8(1+C){(3-C)+52€2(7-5€)}, yi,o(C) = 52(1-€)(1+C)3~ yi,7(€) = -8(1+ €)(1-€)(6+C), yi,s(€) = 85(1+§)(1+24), yi,9(€) = -(1-€)(3 +23C-3CZ+€3), yi,io(€) = 5(1-€)(5-31C-13€2-€3), yi,u(€) = -52C(1-C)(1+€)2, yi,12(C) = -S(3 + 134-762-519, yi,13(§) = 852C(1+€+C2), y2,i(€) = S(1+C){(7+5€+17CZ+3€3)+SZCZ(15-19C+41C2-5C3)}. y2,2(€) = 128€(1-€)(1+§)(2+3€-€2). y2,3(€) = -125(1-€)(1 + C)(1+ 3C), y2,4(€) = -6(1-§){2(3+€)-52€(1 +3€)(1+C2)}. y2,5(C) = 2S2§(1-§)(1-3C-5€2+ 3C3), y2.o(€) = -6(1 + 92(2 + 3C - C2), yz,7(§) = -3S(1 +€)(1-3€)(1+€2), y2,s(§) = (1-€)(1+6€-16€2-6€3-§4), 165 y2,9(C) = s2(3 +34C-19Q2—4Q3+173Q4—62§5+_3§6), y2,io(€) = -9652€(1-C2)(1+€2). y2,n(§) = 125(1-C2)2(1+€). y2,iz(€) = 28C(1-€)(1+2§-C2), y2,13(C) = -3(1 +€)(3 -§), y2,14(C) = 35(1 +C)(1-C)(1-3§). 166 BIBLIOGRAPHY 167 BIBLIOGRAPHY Antman, Stuart S. (1995). Nonlinear Problems of Elasticity, Springer-Verlag, New York. Bajenitchev, A. (1996). A numerical procedure for a non-linear elastic problem for incompressible material based on a perturbation method, Comput. Methods Appl. Engrg., 131, 31-39. Ball, J. M. (1982). Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A, 306, 557-611. Ball, J. M. and Schaeffer, D. G. (1983). Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions, Math. Proc. Camb. Phil. Soc., 94, 315-339. Bauer, L and Reiss, E. L. (1965). Nonlinear buckling of rectangular plates, J. Soc. Indust. Appl. Math. (SIAM, 13, 603-626. Beatty, M. F. (1965). Some static and dynamic implications of the general theory of elastic stability. Arch. Rational Mech. Anal, 19, 167-188. Beatty, M. F. (1971). Estimation of ultimate safe loads in elastic stability theory, J. Elast, 1, 95-120. Beatty, M. F. (1987). Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues - with examples, Appl. Mech. Rev, 40, 1699-1734. Beatty, M. F. and Dadras, P. (1976). Some experiments on the elastic stability of some highly elastic bodies. Int. J. Engng. Sci, 14, 233-238. Beatty, M.F. and Pan, F. (1998). Stability of an internally constrained, hyperelastic slab, Int. J. Non-Linear Mechanics, 33, 867-906. Biot, M. A. (1963). Exact theory of buckling of a thick slab, Appl. Sci. Res, 12, 183-198. Budiansky, B. (1974). Theory of buckling and post-buckling behavior of elastic structures, Advances in Applied Mechanics, Vol. 14, Academic Press, New York, 1-65. Burgess, I. W. and Levinson, M. (1972). The instability of slightly compressible rectangular rubberlike solids under biaxial loadings, Int. J. Solids Structures, 8, 133- 148. Dym, Clive L. (1974). Stability Theory and Its Applications to Structual Mechanics, Noordhoff. 168 Elgindi, M. B. M., Yen, D. H. Y. and Wang, C. Y. (1992). Deformation of a thin-walled cylindrical tube submerged in a liquid, J. Fluids Struct., 6, 353-370. Erbay, H. A. (1997). On the asymptotic membrane theory of thin hyperelastic plates, Int. J. Engng Sci, 35, 151-170. Haughton, D. M. (1992). Boundary layer solutions for incompressible elastic cylinders, Int. J. Engng Sci, 30, 1027-1039. He, L. W. and Cheng, C. J. (1992). Nonlinear buckled states of rectangular sandwich plates and their stabilities, Int. J. Non-Linear Mechanics, 27, 423-435. Hill, R. (1957). On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids, 5, 229-241. Holden, J. T. (1964). Estimation of critical loads in elastic stability theory, Arch. Rat. Mech. Anal, 17, 171-183. Horgan, C. O., Lang, K. W. and Nemat-Nasser, S. (1978). Harmonic waves in layered composites: New bounds on eigenfrequencies, J. Appl. Mech., 45, 829-833. Horgan. C. O. and Pence, T. J. (1989). Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere. J. Elast, 21, 61-82. Horgan. C. O. and Polignone, D. A. (1995). Cavitation in nonlinearly elastic solids: A review, Appl. Mech. Rev., 48, 471-485. James, R. D. and Spector, S. J. (1991). The formation of filamentary voids in solids. J Mech. Phys. Solids, 39, 783-813. Keller, Joseph B. and Antman, S. Eds. (1967). Bifurcation Theory and Nonlinear Eigenvalue Problems, Courant Institute of Mathematical Sciences, New York. Kirchgassner, K. (1975). Bifurcations in nonlinear hydrodynamic stability, SIAM Rev., 17, 652-683. Koiter, W. T. (1981 ). Elastic stability, buckling, and post-buckling behavior, Proceedings of the I UT AM Symposium on Finite Elasticity, Carlson DE. and Shield R.T. Eds., Martinus Nijhoff Publishers, 13-24. Lang, K. W. and Nemat-Nasser, S. (1977). Vibration and buckling of composite beams. J. Struct. Mech., 5(4), 395-419. Lazopoulos, K. A. (1996). Second order displacement fields of a thick elastic plate under thrust - the incompressible case, Int. J. Solids. Structures, 33, 433-449. 169 Levinson, M. (1968). Stability of a compressed neo-Hookean rectangular parallelepiped, .1. Mech. Phys. Solids, 16, 403-415. MacSithigh, G. P. and Chen, Y. C. (1992). Bifurcation and stability of an incompressible elastic body under homogeneous dead loads with symmetry Part 1: general isotropic materials and Part II: Mooney-Rivlin material, Quart. J. Mech. app]. Math, 45, 277- 313. Matkowsky, B. J. and Putnick, L. J. (1974). Multiple buckled states of rectangular plates, Int. J. Non-Linear Mechanics, 9, 89-103. Meijers, P. (1987). Stability of a rubber half-space, J. Elast., 17, 143-158. Nemat-Nasser, S. and Minagawa, S. (1975). Harmonic waves in layered composites: comparison among several schemes, J. Appl. Mech., 699-704. Nowinski, J. L. (1969). On the elastic stability of thick columns, Acta Mech., 7, 279-286. Ogden, R. W. (1992). Nonlinear elasticity: Incremental equations and bifurcation phenomena, Nonlinear Equations in the Applied Sciences, 437-468. Ogden, R. W. (1995). Nonlinear effects associated with waves in pre-stressed elastic solids near bifurcation points, Nonlinear Waves in Solids, ASME Pub., 109-113. Parker, R. G. and Mote, C. D. Jr. (1996). Exact boundary condition perturbation solutions in eigenvalue problems, J. Appl. Mech., 63, 128-135. Pearson, C. E. (1955). General theory of elastic stability, Quart. Appl. Math, 14, 133- 144. Pence, T. J. and Song, J. (1991). Buckling instabilities in a thick elastic three-ply composite plate under thrust, Int. J. Solids Structures, 14, 1809-1828. Polignone, D. A. and Horgan, C. O. (1993). Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres, Int. J. Solids Structures, 30, 3381-3461. Qiu, Y., Kim, S. and Pence, T. J. (1994). Plane strain buckling and wrinkling of neo- Hookean laminates, Int. J. Solids Structures, 31, 1149-1178. Reddy, J. N. (1986). Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York. Rivlin, R. S. (1974). Stability of pure homogeneous deformations of an elastic cube under dead loading, Q. Appl. Math, 32, 265-271. 170 Sagan, H. (1961). Boundary and Eigenvalue Problems in Mathematical Physics, Dover, New York. Sawyers, K. N. (1976). Stability of an elastic cube under dead loading: two equal forces, Int. J. Non-Linear Mechanics, 11, 11-23. Sawyers, K. N. and Rivlin, R. S. (1974). Bifurcation conditions for a thick elastic plate under thrust, Int. J. Solids Structures, 10, 483-501. Sawyers, K. N. and Rivlin, R. S. (1982). Stability of a thick elastic plate under thrust, Int. J. Solids Structures, 12, 101-124. Sawyers, K. N. (1996). On the thickness limitation for Euler buckling, Nonlinear eflects in fluids and solids, Carroll, M.M and Hayes, M., Eds, Plenum Press, 297-320. Song, J. and Pence, T. J. (1992). On the design of three-ply nonlinearly elastic composite plates with optimal resistance to buckling, Struct. Optimization, 5, 45-54. Tagg, Randall (1994). The Coutte-Taylor problem, Nonlinear Science Today, 4, 1-25. Thompson, J. M. T. and Hunt, G. W. (1969). Comparative perturbation studies of the elastica, Int. J. Mech. Sci, 11, 999-1014. Timoshenko, S. P. and Gere, J. M. (1963). Theory of Elastic Stability, McGraw-Hill. Weinberger, H. F. (1965). A first course in partial diflerential equations, with complex variables and transfer methods, Dover Publications. Wu, C. H. and Widera, O. E. (1969). Stability of a thick rubber solid subject to pressure loads, Int. J. Solids Structures, 5, 1107-1117. 171 UV. MICHIGAN STATE NI lllllllllllllllllllllllllWIN 31293020 RRRIES llllllll 1'7 LIB 7410