Yerel olmayan plastisitede varlık ve teklik teoremleri
Existence and uniqueness theorems in nonlocal plasticity
- Tez No: 21795
- Danışmanlar: PROF. DR. VURAL CİNEMRE
- Tez Türü: Doktora
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Plastisite, Yerel olmayan plastisite, Zımbalama, Plasticity, Nonlocal plasticity, Punching
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışma 5 bölümden oluşmaktadır. Birinci bölümde önce yerel teori ve yerel olmayan teori kısaca tanıtılmış, daha sonra“Yerel olmayan plastisite”ve“Yarım düzlemde hareketli zımba”konusunda literatürde yapılan çalışmalar dan bahsedilmiştir.“Yerel olmayan plastisite teorisinde pekleşmeyen mal zeme için varlık ve teklik teoremleri”adlı ikinci bölüm de, önce plastisite teorisinin temel denklemleri verilmiş, daha sonra teoremlerin ispatı yapılırken kullanılacak olan matematik kavramlar kısaca açıklandıktan sonra yerel olma yan halde pekleşmeyen malzeme için plastisite problemi ta nımlanmıştır. Bundan sonra, önce problemin çözümünün tek olduğu, daha sonra ise çözümün mevcut olduğu gösterilmiş tir. Çalışmanın“îzotropik ve kinematik pekleşme olması durumunda yerel olmayan halde plastisite probleminin zayıf çözümünün varlığı ve tekliği”adlı üçüncü bölümünde, önce kinematik pekleşme ve izotropik pekleşme ayrı ayrı tanım lanmış ve bu halleri ifade eden bünye denklemleri veril miştir. Hem izotropik hem de kinematik pekleşmeyi birara- da ifade eden genel bir bünye denklemi elde edildikten sonra, yerel olmayan halde pekleşen malzeme için plastisi te probleminin tanımı verilmiş ve önce problemin çözümü nün tek olduğu, daha sonra ise problemin çözümünün mevcut olduğu gösterilmiştir.“Yarım düzlemde hareketli sürtünmeli zımba problemi nin yerel olmayan halde çözümü ve çözümün yerel haldeki sonuçlarla karşılaştırılması”adlı dördüncü bölümde, zımba problemi yerel olmayan elastisitede çözülerek sonuçlar yerel problemin sonuçları ile karşılaştırılmıştı r. Çalışmanın“EkA”bölümünde yerel halde yarım düzlemde hareketli zımba probleminin çözümü ayrıntılı şekilde ve rilmiştir.“EkB”adlı bölümde yerel olmayan halde hareketli zım ba probleminin çözümüyle ilgili bilgisayar çıkışları ve rilmiştir.
Özet (Çeviri)
SUMMARY EXISTENCE AND UNIQUENESS THEOREMS IN NONLOCAL PLASTICITY As the first subject of this work,“Existence and uniqueness theorems in the nonlocal theory of perfect elastoplastici ty ”has been handled. The stress boundary value problem can be weakly formulated as follows. A state of stress o..CO is looked for such v J that the following conditions are satisfied : ID In a bounded volume O, for every continuously dif ferentiable vector-valued function v CxD, i = 1,2,3 the following equality should be satisfied : f a £..<v) dx = f g.v. ds + f F.v. dx A tJ VJ di v l ft l v (1) where dCl is the surface of O, g. are the components of the prescribed surface forces, F. are the components of the prescribed volume forces and dv &v 2D a..CO should remain inside the convex flow surface fCoO. = a, that is, for all times 0< t < T fCc<OD < <x (3) where ot is a scalar constant. 3D For all times 0< t < T T f dt TÇ <a(. |x-x' |)a..,,6-,, <x',0 Ct. <x,t> - J JJ. ' ijkl let vj - a (x.t)] dx dx' > > 0 C4) VIwhere T..Cx.tD is a symmetric smooth tensor field «? J satisfying the first two conditions CI 5, C 33. We assume that there exists a c > O such that (T a< |x-x' |)a..,,t.Cx.tD-r, Cx',Odx dx'> ^ ' ' i jkl v j kl c fr. t. dx ; c>0 C53 for all t... <- J The strain tensor e.. is the symmetric part of the gradient of the displacement vector u. e. = - ( u.. + u. ) (6) ij 2 WJ J » «. and it is assumed to be composed of the sum of an elastic strain e.. and plastic strain p : l J t j s.. = e.. + p.. (7) U vj tj e is a non-local linear function of the stress a X. J i j i. <x,t) = f cx(. |x-x' |)a.. u, ^u, <*' »t:> d*' <8) k I k I where a is a smooth not increasing function of distance, a are scalar constants. The third condition C4D l j k I posed in the definition of the stress boundary value problem implies the normality of the plastic strain rate p to convex flow surface p.. = X JL_ C93 *v j da <. J where X is either zero or positive. Whenever both fCoO = O and fCoO > 0, then X > O otherwise X = 0. The uniqueness of the solution can be shown by using the inequality C40 given in the third condition of the stress boundary value problem. We first assume that a = o, t = a where a and a are presumed to be both solutions: VÜ...»..... r..Z... t) _ f dt f f <a(|x-x'l)a.,,ö (x'.tD Co- <x, İ ft h wkl kl lJ - c1 (x,t>3 dx dx' > > 0 ı J 1 2 secondly we write t = c, a = a which gives T f dt ff <a< |x-x# |)a..,,âf, (x'.tD Caf.(x.t) - o - aZ <x,t>3 dx dx' > > O <. J the sum gives T fdt f f <a< |x-x' |)a.., C&' <x',tD-crf, (x'.tDDCo-1.Cx,t> - JJ4 ' ' ı jkl kl kl ij - cZ.<x,t)3 dx dx' > < 0 ı J this means that ff <a( |x-x' |)a..,_ Cff1, (x',tD-^, (x'.t^tcr1.(x.t) - ft ft ' ' <. Jkt kl kl ı j - a2. <x,t>3 dx dx' > < 0 «. J then because of C53 we have a = a. To prove the existence we start from a statically- admissible stress state a CtD which is not necessarily- inside the flow surface. But we assume that it is possible to find y > O such that f(c°<t) + y c-°<t)) < a C10D o for all t in the process. We then follow a method similar to the one used in the book by Hlavâcek et al [30] and we define a penalty functional 1/2 gCc) = f C<(Cf<o') - a 3 + )2 + 1 > - 1 ] dx CUD '?) = f [<Ctf (<y> - aQ3 )z + 1 > where vxiiC fCoO - a ] = o r fCoO-cx if fCe>0>a o o O otherwise This functional is convex. As a result its Gateaux differential satisfies the monotonocity condition: DgC a, cr-iO - DgCT,cr- iO > O The Gateaux differential of gCoO is CI 23 tf<o-)-a 3 Idf/da.3 DgC<r.TD = f 2 Li T. O Cl+<|f<a)-a |+)231/2 ij dx C13Z> We then prove the existence of a statically admissible stress state a CtD and a plastic strain state p CO which satisfy the following properties du. ». J du e _ ı_. t Pij 2 dX. dx where,£. = fa< |x-x'|)a..,,crf, (x' )dx' C14D CJL5Z) and u.Cx.O represents the displacement vector field. 3D p = i [f - a ]+ ^- (1 + (Cf (ex) - a 3 + ) ) r e o der o ı 2 2 def.c.. dfCoO = X CoO ao- C16D where fCoO = a is the prescribed yield condition, and the statical admissibility of <? CU is given weakly by dv. dy. - f °e ? < sr^ + -^r- ^dx = f 9. v. ds + f F v dx CI * i U dx. dx. ^ yv t J i i 7D which is satisfied for all v Cx.O smooth enough. But it is to be noted that a is not required to comply with ixor ?the yield condition. The space S of symmetric second order tensor fields can be written as the direct sum of two subspaces: 1Z> The subspace E of tensor fields which can be written as the symmetric part of the gradient of a vector field 2D The orthogonal complement S of E in S. We define the perpendicular projection operator from S to S by P. Then, because e + p is compatible with a displacement field we have PCe^+p^D =0 CI 83 Pe* = - P [ Xs ^£-2. 3 C19D da O & Now, because both a CtD and a CtD are statically admissible states, their difference a corresponds to zero volume and surface forces which means that it is in S. Let us use the shorthand o A(o0 = f a(|x-x'|)a.,, a Cx'Ddx' C20D Q V J def If PA = B, B has an inverse in S because o CBt,tD = CPAt,tD = CAt,tD > c C t,t D2, t e S o Then if Be = a we have a = -B& -P C X < a +B a ) -=- (. a + B a ) 3 da which with the initial condition a COD =0 gives an initial value problem for a system of first-order differential equations whose unique solution furnishes a and because B is invertible we also obtain a, therefore e a. The following equality and inequality t t _ f C aS,-r 3dt + -Ç DgCo-e,rDdt = O ; f e S C21D Xg Co^CO] < -^- < f la + y a°, a + ya 3dt > C22D *» 2 J O O O 4y o can be easily demonstrated to be valid where the inner product t, ] is defined by [ff,T] = f |<a(|x-x'|)a.,kl<7kl(x',t3T.,(x,t) dx dx' C 23) The inequality makes clear that limCfCcr ) -c* ] = 0, that is, the ultimate stress state lim c CD is not out of the flow surface. As the last step of the demonstration we should prove that the obtained stress state satisfies the compatibility condition which was transformed above into the form t J Za,<y-xl dt > O where t is any statically admissible state with no plastic strain rate CpCTD=OD. Since both a and t are statically admissible, we have a -x e S and therefore J o we can use the above obtained equality C24D for any T e S : o t t _ J laC,oe-x1 dt + i pgia£,a£-x1 dt - o o t -- JDg Ct,o- -t] =0 where the last term is identically zero since pCrD = 0 for the stress state t. Using the monotonocity condition CI 2D we obtain J DgC cr£,x-o£) dt > O C240 o which proves the compatibility for all a. In the thesis, the proof of existence is also given for hardening. In this case we consider a generalised yield function FCo-, y.cO depending on two functions yCtD.cxCD characterising hardening. Therefore the existence problem is formulated weakly for the triple o<0,yCtD.otCtD. The state of stress should satisfy the XIweak condi ti on of statical admissibility and the triple should not leave the flow surface F[o<t:>,?<t:>>act:>] < o casD and because the existence problem to be proven is a stress boundary value problem, the triple should also satisfy a stress compatibility condition which will be called,“ The fundamental variational inequality ”t _ _ _ J* <tb,0-0 1+ i'y,y ~y ) + <«,cn -a )>dt > 0 Ca6D o where c is statically admissible and Fl&, y,cb < O as wel 1. The uniqueness can be proven similarly to the perfect case and it is as easy. Here also a penalty functional formulation has been used for the proof of existence. The penalty functional chosen is g<o-,y»a) = f<<CF(o',j',a)) + ]2 + 1>1/2 - 1) dx C87D First a triple a, y » <* has been determined which satisfy all conditions required for being a solution except FCc,y,a 3 may exceed zero. Secondly it has been shown that, for e*0 the triple approaches the solution of the boundary value problem. Using e^.Cx.tD = fa< |x-x' l)a.., tas, <x',t)dx' CaSD tj r> 'tjktkl and. p£ = - D g CE9D the strain is obtained £.. = e.. + p.. C3CD t J t, J lj By the convexity assumption of yield surface the monotonocity condition has been written in the form DgCo>,y,a;<y-T,y-â,cı-f&-DgCT,y,(3;a>-T,y-6,oı-f& > O C313 where the triple r,6,ft is staticaly admissible. Using xiio £ any statical y admissible a, cr can be decomposed as £ O ~£ a = a + a where a belongs to S. Let us find the nonlocal elastic strain on S ; a = PRer. a can be obtained from a boundary value problem of a system of first -order ordinary differential equations: hf = -PRc^CtD - P - D gCo-°+ CPR5 ^af,y, oû ; a^CCO =0 S ' ?& £ From a, we obtain a and consequently a and we show that for £+0 we find the solution of the problem. In the thesis, it is also given an example for nonlocal elastodynamic boundary value problems. It is the problem of a moving rijid punch on a nonlocal elastic half plane. The kernel of the constitutive equation has been chosen in the form Jx'-x |<a |x'-x|>a where a is the atomic distance. The speed of the punch is constant. The results have been obtained for three cases. In the first case both ends of the punch do not touch the plane, in the second case only one end indents the plane, in the third case both ends of the punch are immersed. There nowhere exists an infinite stress. xiii
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