İnce hiperelastik plakların asimptotik teorisi

Abstract

ÖZET Bu çalışmada, nonlineer elastisite teorisinin referans konu mundaki üç boyutlu denklemlerinin asimptotik analizi yardımıyla geo metrik nonlineer ligi olduğu gibi fiziksel nonlineer ligi de içeren bir ince plak teorisi geliştirilmiştir. Çalışmanın ilk bölümünde asimptotik açılım tekniğinin plak teo rilerinin çıkarılmasında kullanılmasının şimdiye kadarki örnekleri üze rinde durulmuştur, önceleri lineer plak teorilerini daha sonra geomet rik nonlineerliği içeren plak teorilerini elde etmede kullanılan yön temin gelişimi kısaca özetlenmiştir. ikinci bölümde ilk olarak nonlineer elastisite teorisinin alan denklemleri maddesel koordinatlarda sunulmuştur. İnce plak tarafından işgal edilen bölge karşılaştırılabilir boyutlara sahip yeni bir bölgeye dönüştürülmüş ve daha sonra yerdeğiştirme vektörü ve gerilme tansörü kalınlık parametresinin yardımıyla von Karman teorisinde olduğu gibi ölçeklenmiştir. Üçüncü bölümde yerdeğiştirme vektörü ve gerilme tansörünü ka lınlık parametresi cinsinden asimptotik kuvvet serisi olarak açtıktan sonra bu açılımın denge denklemleri ve sınır şartlarında kullanılmasıy la, bu parametrenin kuvvetlerinin katsayılarını sıfıra eşitleyerek, denklem hiyerarşileri elde edilmiştir. Benzer bir yaklaşım ile, izo- trop elastik cisimlere ait bünye denklemi hiyerarşisi de çıkarılmıştır. Bu şekilde elde edilmiş denklem hiyerarşilerinin ilk iki mertebesi ay rıntılı olarak incelenmiş ve sıfırıncı mertebe yaklaşımın von Kirman teorisine karşı geldiği gösterilmiştir. Ayrıca fiziksel nonlineerliğin etkisinin birinci mertebede ortaya çıktığına işaret edilmiş ve sıkışa bilir hal için bulunmuş sonuçlardan sıkışmaz elastik cisimler için olanlara nasıl geçileceği açıklanmıştır. Son bölümde, önceki bölümde geliştirilen teorinin bir uygulaması olarak üst yüzünden uniform yüke maruz sonsuz şerit problemi incelenmiş tir. Hem ankastre hem de basit mesnetli haller için çözümler analitik olarak elde edilmiştir. Ko, Mooney-Rivlin ve Neo-Hookyen cisim halinde bulunan sayısal sonuçlar sadece geometrik nonlineerlikten hareket ede rek bulunanlar ile karşılaştırılmış ve fiziksel nonlineerliğin önem kazandığı bölge için çökmelerin mertebesi açısından yaklaşık bir sınır verilmiştir. - iv -

AN ASYMPTOTIC THEORY OF THIN HYPERELASTIC PLATES SUMMARY There exist several works in the literature dealing with asymptotic derivation of the nonlinear equilibrium theory of plates. In these works, the plate theories of various orders are obtained by asymptotic expansions of field variables in terms of a small thickness parameter. However, it should be noted that approximations assumed in those papers are mostly valid only for linear stress-strain relations. The aim of present work is to develop a theory of' thin plates which is physically as well as geometrically nonlinear, using an asymptotic analysis of three dimensional equations of nonlinear elasticity in the reference configuration. To this end the nonlinear field equations, the asymptotic expansion technique and the results will be briefly summarized. As is known, the strain-displacement relation» in the curvi linear material coordinates Xr (K=l,2,3) are given by 2EKL = /;L + UL;K + V/jL where U is the displacement vector, E is the Lagrangian strain tensor, semicolon denotes the partial covariant differentiation and the summation convention is employed. For a body occupying a region R in the reference configuration the equilibrium equations are in the form o» ¦ W^ ¦ P/ ~ o where T is the second Pio la-Kir chhoff stress tensor, p is the initial density and FK is body force per unit mass. Similarly? at the boundary 3R of R, or on a part of it, the surface tractions /F- satisfy i-P- + W}1!)/ - tL where N is the unit exterior normal to 3R. On the other hand, the constitutive equations of isotropic nonlinear elasticity are _KL,32. T 32. TT 32 vrKL,32 T 32 * JSL ^ = «5Ç + ^ TffÇ + XIE 3ÎÎÇ)G ` <3ÎÇ + h 3ÎÎÇ)E. 32 EK^ML 3IIIE - v -where G is the metric tensor, Z is the strain energy function and 1^, II` and IIL, are the basic invariants of E. Eovever for an incompressible nonlinear elastic material~when the Cayley-Hamilton theorem is used these equations take,the fom where I_ and II are the basic invariants of C_, Green deformation tensor which is defined by C_T = GL^ + 2E_. The plate occupies the region bounded by the two faces, X = h and X3 = - h, where 2h is the uniform thickness of the plate, and a cylindrical surface having generators normal to the middle plane such that X3 = 0 represents the middle plane of the plate. If the diameter of the plate is denoted by 2L then the thickness parameter which specifies the order of deflection of the plate is defined by E = h/L. Moreover X0 (a=l,2) which are the curvilinear coordinates in the middle plane are chosen such that X°£[-L,l] and X3 shows the coordinate in the normal direction of the middle plane. The dimensionless coordinates £ are introduced by xa = Lça, x3 = eLç3, Ki,e,eei-i,i] This mapping transforms the domain occupied by a thin plate to a domain of. comparable dimensions. Later it is assumed that the displacement components are scaled as in the von Karman theory. Namely, the dimensionless displacements and stress components in Ç coordinates are described in the form Ua(X) - e2Lua(), U3(X) = eLu3() T06») = T e2aap(Ç), Ta3(X) = T e3o-a3(Ç), ~ o - ` o - T33(X) = T eV3(Ç) ~ o ~ where Tft is an appropriately chosen factor of stress dimension. Moreover, if it is assumed that the surface tractions t^ and t are prescribed on the upper and lower faces, respectively and the stress vector on the lateral surface is denoted by J then the nondimensional surface and body force densities are defined by t? (X6) = T e3g` Jth, t» _<XB) = i e»gj _(ÇB) ta(X) = T e2Ta(Ç), t3(X) = T eV(Ç), XQC, l£c p LFa(X)/T = e2fa(0, p LF3(X)/T = e3f 3(Ç) - viwhere C and c are the boundaries of the middle plane of the plate in X and Ç coordinates, respectively. Using the above equations the equilibrium equations in the dimensionless coordinates become,aB 2 cry B 2û3Bx, 3B, 2 sa B (er* + e*o VV + e2cT up ) + (a H + e2a u »Y »3 *a »a + E2c33uB,) + f3 = 0 » 3 » 3 (aa3 + a^u3 v + oa3u3 3) ¦ (a33 + a3V v + a3 V J 3 + f 3 = 0 where derivatives are taken with respect to Ç coordinates. Similarly the boundary conditions are obtained in the `form 3B. 2 3a 8. 2 33 B - 8 a p + e2a up + e2a3V =+ gj_ on u and w o33 + a3V + a33u3 3 =+ gf _ (a*3 + e2aaYue + e2aa3uB )n = tB »Y »3 a on a /.^t*3 j. ~OY 3 j. ~a3 3 N _,3 (a + 0 'u + a u )n = t »Y »3 a where w and u are the upper and lower plane faces of the plate in the transformed domain, respectively, a is the lateral surface and n is the exterior unit normal to a. In this case the strain-displacement relations take the form E ` = i e2(u ` + u` +u u`) + i £*u u* a aB ' t v a;B 8;ct 3, a 3,B' * Y;a ;B E = i e(u +u +u u ) + i e3u. uY a' a,3 3,0 3, a 3,3 Y»a »3 E =u +iu u +i e2u ua 33 3,3 3,3 3,3 a, 3 »3 V ' '. ¦ ~ * ` ~ ` The following asymptotic expansions for u(Ç) and o(Ç) in the dimensionless coordinates are defined: «o 00 u(Ç) = Z e2nnu(Ç), a(Ç) - S e2nn0(O n=0 ~ ~ n=0 If these expansions are introduced into the governing equations in £ coordinates successive systems of the equations are obtained by equating coefficients of the like powers of E in both sides of the equations. The equilibrium equations corresponding to the lowest two order members in the hierarchy of equations so obtained are given by - vi 1 -»aa3 + °a30 + fB = 0 ;a,3 °aa3 + V3 + (V0* °u3 + °ffa3 °u3 ) + (°c3Y V ;a,3,y.3 »a,y + °a33 °u3 ) + f3 - 0 »3,3 for the zeroth order approximation and >aa& + »a*B + (»a0* °uB + °aa3 °u3 ) + (°a3a °uB ;y »3 ;a ; ;a »3 0`33 0.8 a + `ct33 uuK ) =0 »3 »3 iffa3 + l8, + (0ffOY iu3 + iCT«Y ou3 + oaa3 xu3 ;a,3,Y »Y »3 + iaa* °u3 ) + (°a3Y V + VY °u3 + V3 V ts to,Y »Y »3 + Ia33 °u3 ) =0 »3 i3 for the first order approximation. Similarly the boundary conditions for the lowest two_order are obtained in the form » V3 + °cr3a °u3 + V3 V =+ g3 »a,3 s+,~ °aag n = X3 a (°CTa3 + «ff0* °u3 + °0a3 °u3 )n = T3 on a) and oj on a »Y,3 a and V3 + °a3a °u3 + °a33 °u3 = 0 »a,3 1«33. 0 3d i. 3 i~3<* o`3 i_33. u *`-» i 9 ? a_wv* u 3 ? 0_33 13 ? 1 - 33 0 3 r/ O + O U + CT u + G U + (J u = 0,a,a,3 »3 (i0aB + offaY o B + oCTa3 0 B } = 0 5Y »3 a CoT + »a0* 1u3 + ij* °u3 + °aa3 V + to** °u3 )n = 0,Y »Y »3 »3 a If the asymptotic expansion where E ` and *E R depend on the displacement components as follows <x8 Ja8 °E. = i (°u fl + °uQ + °u V 0) a3 a;8 8;a 3, a,8 XE _ = i (Ju 0 + xu0 + °u aB V ` + xu 0..3 o`Y 8;a 3, a,8 a, a,8 Y.a »8 3 ? u U 1 / u _ + u u `) - viaa -is employed in the constitutive relations, then, after some calculations, the stress components °aa& and 1a<*3 take the following form °aaP = 2A (A V A*3 + °Eae) i o Y »a0* = 2A (A >EY Aag + 1E0tB) + A V3 A0*5 + [* A (°EY °E* loy o u 3 Y Ğ - °e/ »E6 ) + A °EY °E6jAaB - A V °EaB 6 Y * Y $ 3 Y + A °Ea °EYB 2 Y otS where A is the metric tensor of the middle plane and A (n=0, 1,2,3,4) are the material constants evaluated as some derivatives of the stress potential E* with respect to invariants at the natural state. The values of these coefficients are given below for some well-known materials. For the Ko solid the strain energy function is defined as where y is constant. Choosing the scale factor T = p it is found that o 1 4 4 A=T,A=1, A=-8, A=T, A = - -s- 0-Jl 2 3 J k 9 The strain energy function for the Murnaghan solid is given by Z = i (X + 2y)l- 2yIIE + £l+ mlgllg** nlllg where X, y and Ü, m, n are Lame and Murnaghan' s constants respectively. In this case the above coefficients are obtained as a _ ^ AiAnA 2(m + n) Ao - XTTjT ' Aa = *? A2 = y ' A3 = X+2y A - [8y2(3Jl + m) - 4X(X + y)(m + n) - X3n/y]/(X + 2y)* where TQ = y. It is obvious that A2 = A3 = A^ = 0 for the classical linear stress- strain relations. As is seen from the above constitutive equations for the lowest two order the effect of physical nonlinearity comes into effect at the first order approximation. If the condition that the components of stress tensor with a negative supercript vanish in the hierarchy of constitutive equations is imposed then this leads to forms V(Ç) = Vç1,?2), °ua(Ç) =.v°(çl,ç*) - e Va ~ ~ » for the displacement field. After some calculations the first order displacement components are also found in the form V(Ç) = »wC?1,?2) + Va1,?2,?3) - ix -where 1U3 and XU are known functions of only the zeroth order terms. These equations prove that the lowest order displacement field is a Kirchhoff-Love field. But it is not possible to forward the same claim for the first order approximation. The theories corresponding to the lowest two order members in the hierarchy of field equations so obtained is. studied in detail and it is shown that the zeroth order theory corresponds to. the well- known von Ka'rman theory. On the other hand, in first approximation it has been observed that the total stress cannot be decomposed into bending and membrane stress components due to the coupling between bending and stretching. Moreover the stress components in the transverse direction have been determined and the Cauchy stress tensor which represents the actual state of stress in the deformed plate is also evaluated. A similar expansion is given for an incompressible nonlinear elastic material and the incompressibility condition is used to eliminate the arbitrary pressure function. Later, it is shown that the results obtained for the compressible elastic solid are also valid for the incompressible solid under the following transformation A ?*? 1, A -*. A, A -* - 4(A -A), A.*? 4A, A -*- - 4(A +A) Q 10 2 0 13 0`+ 01 where A and A, are the material constants for the incompressible solid. °As is known, for the Mooney- Rivlin material the Z function is given as Z = C (I_ - 3) + C (II_ - 3) 1 C 2 C where C and C are the material constants. In this case A and A take the following values ° * A = 2(1 + 6), A = - 26 o i where 6 = C /C and T is chosen as C. If 5 = 0 Cor C2 = 0) is substituted2 *in the0 above expressions then the results are valid for a material which is called Neo-Hookean. Finally, as an example the problem of infinitely long strip under uniform lateral load is studied for various edge conditions. Using the plane strain assumption the solutions are calculated analytically for both simply supported strip and clamped strip problems. Later these problems are numerically investigated for the various thickness parameters and for the various materials. In the special cases of Ko, Mooney-Rivlin and Neo-Hookean solids, the maximum deflections are plotted versus the lateral load. The results are compared with the solutions based upon the geometric nonlinearity and at very large deflections it is found that the material nonlinearity plays an important part. Moreover, for this range of deflections the variations of the axial force, the bending moment and the Cauchy stress along the length of the strip are plotted. From these figures it is seen that the solutions are dominated by membrane field. - -x -