Sıkıştırılabilir hiperelastik küre ve silindirin radyal hareketleri için bir benzerlik çözümleri sınıfı

Abstract

ÖZET Bu çalışmada iç ve dış basınç etkisindeki sıkıştırılabilir malzemeden yapılmış küre ve sonsuz uzun silindirin radyal hareket lerinin benzerlik çözümleri incelenmiştir. Çalışma yedi bölümden oluşmaktadır. Birinci bölümde konu ile ilgili genel açıklamalar ve kendi çalışmamızın kısa bir özeti sunulmuştur. Bir sürekli ortamın hareketini yöneten temel denklemler ve izotrop elastik malzemeler için gerilme ifadeleri ikinci bölümde ve rilmiştir. Küre ve silindirin hareket denklemlerinin çıkarılması ve denklemlerdeki yapısal benzerlikten faydalanarak denklemlerin birleş tirilmesi üçüncü bölümde sunulmuştur. Dördüncü bölümde de kısmi türevli denklem takımlarının benzerlik çözümlerinin bulunmasında kul lanacağımız bir parametreli Lie dönüşüm grupları hakkında kısa bir önbilgi verilmiştir. Beşinci bölümde ise, üçüncü bölümden çıkarılan hareket denk lemi birinci mertebe kısmi türevli denklem takımına indirgenip, bo yutsuz hale getirilmiştir. Sonra sonsuz küçük grup dönüşümleri uy gulanarak simetri grubunun üreticilerini verecek denklemler elde edilmiştir. Bu denklemlerden keyfi E fonksiyonu için üreticiler bu lunup, benzerlik çözümleri elde edilmiş, kısmi türevli denklemler adi türevli denklemlere indirgenmiştir. Daha sonra da özel seçilmiş bir grup için üreticiler bulunup, benzerlik çözümleri ve bu çözümleri kabul edecek E fonksiyonunun yapısı belirlenmiş, bu hale karşı gelen adi türevli denklemler, gerilme, yer değiştirme ve hız ifadeleri elde edilmiştir. Altıncı bölümde Ko malzemesinin beşinci bölümde bulduğumuz E fonksiyon sınıfı içine girdiği gösterilmiş, bu malzeme için adi tü revli denklemler, gerilme, yer değiştirme ve hız ifadesi elde edilip özel bir çözüm verilmiştir. Yedinci bölümde ise Ko malzemesi için elde ettiğimiz lineer olmayan adi türevli denklem takımının (iki nokta sınır değer proble minin) sayısal çözümünde kullandığımız eş operatör yöntemi hakkında bilgi verilmiştir. En sonunda da sayısal çözümlerimizin sonuçlarını gösteren eğrilerin sunulması ile inceleme bitirilmiştir. iv -

A CLASS OF SIMILARITY SOLUTIONS FOR RADIAL MOTIONS OF COMPRESSIBLE HYPERELASTIC SPHERES AND CYLINDERS SUMMARY In this work we have investigated the similarity solutions of spheres and infinitely long cylinders of an isotropic elastic material, under the applied inner and outer pressures. The inner and outer radii and the inner and outer applied pressures are R, R and P (t), p (t) respectively. 1 2 i 2 We obtained the equations of motion for the both cases, but because of the similarity of these equations, we characterize both the motion of the sphere and the cylinder with a single equation. Then it reduced to the equivalent systems of partial differential equations of the first degree and put in a non-dimensional foriu. Applying the infinitesimal group transformations to these equations which give the generators of the symmetry group are obtained. We obtained the similarity solutions of the above equations for arbitrary functions £ then we determined the similarity solutions for a selected symmetry group and the structure of the T, which admits this group. By this, the system of the partial differential equations are reduced to a system of ordinary differential equations. We have shown that the function E of the Ko material is a special case of ours. For this material, the numerical solution of the afore mentioned ordinary differential equations is done by the adjoint method. The variations of the components of the stress, displacement and the velocity in the radial directions and also with respect to time on the boundaries are investigated. a) Radial Motions of Compressible Hyperelastic Sphere Because of the spherical symmetry we take the motion as r = r(R,t), 6 = e, $ =* (1) where the material and spatial coordinates denoted by (R,Q,$) and (r,8,<(>) respectively. As a result, the Finger and Piola deformation tensors become v -,-ı = [c`lkJl] =, c = I cl) = 1/p2 1/q2 1/q2 (2) where we used the definitions _ 9r __ r P `3R ' q ~R. Invariants of the Finger deformation tensor are taken as I = trc`1 = p2 + 2q2, I 2 II =4 [ (trc`1)2 - trc`2] = 2p2 q2 + q4, III = det c_1 = p2q4 (3) (4) then the strain energy function for the homogeneous isotropic elastic material is given as in Eringen [ 1] E = £(1,11,111). (5) Hence the Cauchy and first P io la-Kir chhof f stress tensors can be written as tH = b c`lk*+ b 6k + b cH, TKk = JX^gK* (6) where the coefficients are. _ 2 3E b 3Î ' -1 AIîî / -^5<`& + T!ht>- `-, --*^s^i (7) components of the stress tensors are tr = 2(İ£ I + 2 -§5_ + a2 -9-E ¦) Z r 13I q 311 q 3III; T q * r 2(1Î + 2`§îîq + 3ÎÎÎ q >» (8) t6e = ^ = 2fi? + ii<P+V+llnq2p]' M^J = 2 rdl, 3S,2 +n2W 3E 2 n2l Rz 6 B2-3I 311 R2 3III Tyy = sın 6T (9) The material form of the equation of the motion is reduced to - VIRr + 2 Rr,R R 06 _ 32r 2rT = p (10) 3t2 When we take into account that E(I,II,III) - E(p,q) we see that 3E _ 0/3E + 3E _2 + 3E _ 4) = TRr = _2 ^r 3p 2(3I 311 q 3III q t `Si 4[-3l 3ÎÎ (p +q ) 3ÎÎI = 4[-£+-^ (p2 +q2) +-^77p2q2]q=2RrT00=2Pqtee (11) making the following abbreviation E =^ z = 3E 32E _ 32E _ P 3p ' q 3q ` 9p2 PP = E 3p3q pq ' = 3p = Şa PR 3R ' qR 3R (12) equation of motion can be put into the form E p` + E qD + 4 (2E - E ) = p -İ-I pp^R pq 4R R p q Ko ^ (13) h) Radial Oscillations of Compressible Hyperelastic Cylinder Because of the axial symmetry the motion can be taken as r = r(R,t), 6=0, z = Z (14) where the material and spatial coordinates denoted by (R,0,Z) and (r,6,z) respectively. The Finger and Piola deformation tensors become (15) invariants of the Finger deformation tensor are I=p2+q2+l, II =pV +P2 +q2, III = p2 q2 (16) components of the stress tensors can be calculated from (6) as r _ `r,3E, 3E N 1,,3E, 3E N, _ 1 `Rr `6 _ or,3E, 3E v 1,,3E, 3E N, _ r J36 fc e-2[(-3î + -3îî)?+ (3Û + 1în)p3qTqT (17) the material form of the equation of motion for vanishing body forces is - vii -t* + î!! - ^ = p *z (18),R K. 3t! As a result of the same procedures used for the case of sphere equation (18) can be reduced to E p_ + E q_ + 4 (I - E ) = p ^ (19) pprR pq R R p q o 9t2 Equations (13) and (19) can be given together as 2 Pt, + Z 1T + ¦£. (nE - £ ) = P - (20) ppFR pq4R R p q' M0 ^2 where we have to put n = 1 for the cylinder and n = 2 for the sphere problems. Here if we make the definition v=f (21) Equation (20) becomes From the equation (3) and (21), equations qt=İ' qR=İ(p`q)' Pt=VR (23) are obtained as the compatibility conditions. If we consider the non-dimensional quantities defined by r = ar', R = aR1, t = - t', v==cvl and 2 = ^E' (24) where c = £ /p the field equations can be put into non-dimensional, 0 0 0 forms i;. -£ = ». ^. `T` <->'-<!') =0, p., -vj,, =0, ^¦».»i- + 1;-,-,4- +^ «¦*,. - si-> - »;. =o <25) Henceforth we shall only use the non-dimensional forms and we drop the primes in the relevant quantities for the sake of notational convenience. c) Infinitesimal Group Transformations With the aim of finding the infinitesimal generators of the Lie group which left the system of equations (25) invariant and which leads us to the similarity solutions, we want to give a brief introductory exposition. - vixi -Assume that we have a system of equations of the first degree in the form of Fa(x, <j>, <j>a) =0; a = 1,2,...,p, q = 1,2,...,m, k = 1,2,...,n (26) J* _ jk, a.,k _ 3(f) * `* (x ), *a-- 5 ox When we transform the independent and dependent variables as xa = xa + eca(xB,/), ^k = 4>k + enk(xB,/) (27) and define the operator as we obtain &£ = ö` + eD ç* + 0(e2), -^ - 6? - eD`Ça + 0(e2) 3x 3x p *a = *a + £( Vk - *gVB) (29) If the system of equations (26) is invariant under the transformation (27), the equation `,-a,k,k. _ _ r a, ra,k, k,k,,_,k,k rj3Nl Fa(x, $, 4>a) = Ffl[ x + e£, <b + en, *a + e(Da<> -cî>gDaÇM)] must be satisfied. As a result of this, we find that equation 3F 3F 3F R - Ca + - Tl* + - (D/ ` CD ?) = 0 (30) 3xa 3<J> 3* a p a a should be satisfied identically for invariance. Ot k This system enables us to find the functions Ç and n. On the other hand, the similarity solutions that we have to find must also remain invariant under the transformations (27), i.e. If Ç is such an invariant of the group we know that it satisfies C°-^ + Tlk^-0 (31) 3x` 3<T as a result of this we arrive at the characteristic system (32) which leads us to the similarity solutions. If we now apply the transformations R = R + £S(R,t,p,q,v), q = q + eQ(R,t,p,q,v) t = t + £T(R,t,p,q,v), v = v + EV(R,t,p,q,v) p = p + £P(R,t,p,q,v) (33) - ix -to the system (25) we obtain from the sys:em (31) the equations Q _ Pias -t =0 xv R v R v (3A). q -£ias -t =o xp R p R p (34). + -J (Q-P +£3 s) =0 (34). `t +l`v -¥ (st +1 V -i (I« +i V +i (is - v) 7°, (34). S - T = 0 v p (34), p + o + İL-İ s -(T + - T ) - V = p ÖR R q K t R V v (34), VPv + tr +£? V ` (st +l sq + V = ° <34>: Sp - TvSpp = 0 (34) j (p-q)[ (P-q)2pq + n£p - Eql (Pv + Tr + £3 Tq) + Pt + Pq ` <VR + ¥ Vq> ` ° (34)< 2 (P - T_ -^ T ) +I[(p-q)Z + nE - E ] (S +T) ppv R Rq Rrrtpq p qv p + s +^s -V =0 t R q p (34) 10 Z P + E Q + E (P `S.-^S + T +^T -V) ppp ppq PPP «. Rq tRq v + 4 T E [ (p-q)E ]+ nE -E ] = 0 R v ppl Vh^ H' qJ p qJ (34) 11 *? < W + ZpqqQ) + V** + ^ V + W QR + V % + Tt + I Tq ` SR ` V Sq ` V ` İ (Tt + ¥ V1 + I f (nZpP ~ V* + (nEpq ` Zqq)Q + ^P ` V x (T + £ T - V. - )] -Z [ <p-q)Z + nE` - Zj tRq v R R pq -(vt+vq)=o (34) 12 x -If we make the definition * = R[Q -E3 S -ÎT] (35) from the first four of the above equations we obtain S = - *, T = - *, P = DD$ p ' v R Q=l` Vp ` /K. v = V (36) where =.(p.q.v.R.t), 1>r = l + qR I,. /=h+/h (3?) By this the derivation of the generators of the symmetry group is reduced to obtaining a potential function $. If we search for a solution of the equations for arbitrary function E(p,q) we find P = 0, Q = 0, V = 0, S = oR, T = at + 3 (38) and from the system (32) we obtain the similarity solutions P=P(Ç), q = q(Ç), v=v(ç), 5=7^1 (39) where a and B are arbitrary constants. If we suppose that S=S(R), T = T(t), P=P(p), Q = Q(q), V=V(v),(40) we obtain the generators of this symmetry group S = (l-a+B)R, T = t+y, p = a +6, Q = aq+6, V = Bv $ =4(l-cd-B)Rp+ (H-f)v] + (B+l)q + 6R (41) and from the system (32) the similarity solutions p+A = (t+Y)0iF(O, q+A = (t+Y)aG(0, v = (rhf)h(O, Ç = /-=f* (42) (t+y)1 °T where a, 8,Y,A and F,G,H are respectively arbitrary constants and functions. The from of the function Z(p,q) admitting the solutions (42) is determined from the equations (34)11_(34)-2 as a result of the following transformation x = P+X, y = q+X (43) For this purpose, by use of the definitions n = xZ + yz - 2yZ, y =- (44) xya the cited equations can be transformed respectively to n =0, (x-y)F + nil -IT =0 (45) xx'^'xyxy from the first and second of these equations and the use of the arbitrary functions f(y) and g(y) we obtain - xi -Tl =xf(y) + g(y) (46) xf'(y) - nf(y) + g' (y) = O (47) respectively. As the result with an arbitrary constant K we obtain y ÎT = [Kyn - yn / £^$- dnl x + g(y) (48) Using above equation of £ and the definitions T(y;y) = - / nn_2y(/ ^^- dv)dn ; *(y;y) = / rf 1_2yg(n)dn y*^1 (49) in (45), we obtain the function E(x,y) as ZC*,y) = y2/ (x/y) + Kxy11 + xy2y`1T(y;y) + y2pK(y;y) (50) where E is an arbitrary function of (x/y) and K is an arbitrary constant. When we use the equations (32) and (45) in (25) we obtain ÇE` İS + Ç[ (2y-l)E' -ÇE` + (2y-l)T + yt] §¦ + l (2+Ç)Z« at, oo at, o - 2yE + 2x - (2y-l)ÇT- xx' - 2yx - yK']i< o + [ (l-cd-B)ÇH' - 6H] ÇG2_2y = 0 (51) where _ dE d2E _ x _ F v, _ o ç y g ' / d? ' d^ Equation (51) is reducible if and only if the equations (2y-l)T + yTf = 0, 2T - 2y< - yic' = 0 (52) are indepent of y. As a result of this the relations T = K y1_2y + c, K = k y`2y + 2K y1_2y + c (53) 1 1 2 1 2 must hold, and E becomes £(x,y) = y2yE (x/y) + Kxy11 + K (x+ny) + K (54) 0 12 Equation (25) reduces to ÇE`(F/G)+Ç[ (2y-l)E'(F/G) - (F/G)E`İG' + (n + F/G)E` - 2yZ (F/G)]G + [ (l-cd-g)Ç2H' - PÇHl GZ Zy = 0 (1-Cd-3)Ç2- aÇG+H =0, Çİ+ G-F=0, (l-at&H - OF + = 0 (55) It is easy to see that the last equation is an identity. - xn -Finally the stress components are found as :r = J. E = L^ [ (t+Y)20`^2^1!' (F/G) + K(t+y)2aG2 r q11 P [(t+Y)aG-A]n ° +K, i t6 =~L^E = 6 npq11`1 1 n[(t+Y)VA][(t+Y)aG-A]n-1 x [ (t+Y)2e-aG2p-1(2yS -Ir)+,*(rW)C*(n~1)aFGn-1-h»K ] (56) 0 G 0 1 Displacement and velocity are also u = r - R = Rtq-1] = R[ (t+Y)aG - A - l] (57) v = u = (t+T)6H = (t+Y)a-1R[ aG-(l-Cf+`3)(F-G)] (58) d) The Ko Material The Ko material whose strain energy function is given as E =T (îîî + 2 IIlîg ` 5) ° (59) is in the class of the 2 function set up in (54) for V`l. For the sphere problem it becomes Z =_L (i + S!_) + pq* - £ Z =ı + lİ, K = 1, q2 2p2 2 ° ' p2 K =0, K = - 5/2 (60) 1 2 and for the cylinder case E =- (1 +-3-) + pq - 2, S =l(l+S_), K = l, K =0, 2q P P K = - 2 (61) 2 The similarity solutions given by (42) are reduced to P = x = (t+rAcs), q = y = (t+y)aG(Ç), v = (t+y)`0lH(O, Ç = 5--*- (62) (t+r)1 2a Also the field equations become F r_,, F ^ j_ _.,_.,xr2r,_31 Ff = £ [n(i - £_) + a(a-l)Ç2GF3] [3-(l-2a)ag2I4U G3 G' =-(F-G) ; (l-2a)Ç2G' - o£G + H = 0, (l-2a)ÇF' - oF + H1 = 0 (63) The stress components are - xiii -r _, 1 = _ p q (t+Y)JUF3Gl t 3 n,,_. N 3+n`3 `n t6e ` X ~ ~^2 ~ 1 ~.,.l+3vrn+2 (64) pq (t+Y) FG and the displacement and the velocity are u = El (t^T)aG - 1] (65) v = u = (t+Y)`°B(C) = (t+Y)a_1R[aG - (l-2a) (F-G)] (66) The last of the equation (63) is satisfied identically. After the elimination of H we arrive at the relations F' = : [n(l - I3-) + a(a-l)£2GF3] [3-(l-2a)2çVlÇ G G' =İ (F-G) The numerical solution of this system of equations are done by the shooting method (adjoint method). The stress components, displacement and velocity fields and their time variations at the inner points and on the boundaries are investigated. - xiv -