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• W3147477804 abstract "The purpose of this paper is to introduce and to discuss several main variation principles in nonlinear theory of elasticity namely the classic potential energy principle, complementary energy principle, and other two complementary energy principles (Levinson principle and Fraeijs de Veubeke principle) which are widely discussed in recent literatures. At the same time, the generalized variational principles are given also for all these principles. In this paper, systematic derivation and rigorous proof are given to these variat)onal principles on the unified bases of principle of virtual work, and the intrinsic relations between these principles are also indicated. It is shown that, these principles have unified bases, and their differences are solely due to the adoption of different variables anrT Legendre tarnsformation. Thus, various variational principles constitute an organized totality in unified frame. For those variational principles not discussed in this paper, the same frame can also be used, a diagram is given to illustrate the interrelationships between these principles. I. In t roduct ion (1) In the non-linear elasticity theory, is it like the linear theory that the stress field is the complementary energy principle of the only independent variation? This is a problem of wide interest in the recent years. The center of interest of this problem lies in the question whether the relationship between Piola stress tensor and the deformed gradient can be reversible? i.e. whether the corresponding Legendre transformation [ 1,2 ] can be tenable? Up to now the complementary energy principle of the non-linear elastic variation principle which have been set forth now-a-days, may be generalized into three classifications: First of all, let us mention Ressner principle [ 3 -7 ] which is based on Kirchhoff's stress tensor. But it is usually considered that it is not the pure complementary principle. Owing to the fact, besides the stress tensor, the displacement vector is an independent variation. After this, there is the principle that solely based on Piola stress tensor x and sometimes it is declared as Levinson principle [ 8-17 ] of the true complementary energy principle. But this principle is not always tenable. Because even in the simple condition of the isotropy, it is only when x* . x (2) on (1) This article was completed in the author's stay in Ruhr University, Federal Republic of Germany. He wishes to acknowledge his gratitude to Alexander yon Humboldt Foundation, Ruhr University, especially to Prof. Th. Lehmann for his hearty hospitality. (2) The asterisk signifies the conjugation of the tensor. every point of the body is with dissimilar principal value. ( It is hardy predicable that this condition is satisfactory ), the function relationship of Piola stress tensor and deformed gradient can be reversible. Although the condition in choosing the proper branch is given, the multivalue under the action of the significance of the converse integration of the function concerned ( regarding the various isotropies, there are at least four branches ) also brings in adjunct difficulties in the practical application, that which is always established is Fraijs de Veubeke principles [ 12-17 ] which is based on the polar decomposition. But here besides Piola stree tensor, the rotational tensor also appears a s the independent variation. Nobody has achieved the purity of the complementary energy principle which has been awaited for years. Evidently whether the question should be thus set forth or not as well as the complementary energy principle and other problems concrrned are waiting for to be further studied. The purpose of this article is to prove that the non-linear elastic principle and the various variation principles ( pote:ntial energy principle and their extensions ) can be derived from the unique principle of virtual work, so as to form an organic whole body within a unified frame. This article also presents the intrinsic relationship of the principles concerned and their relational diagram. II.Mathematical Notation This work makes use of the two-point tensor field method. L e t be the reference configuration ( undeformed ) of the body .~ and let. -t be the present configuration ( already deformed ). They are fitted with two arbitrary curvilinear coordinate scries{ X ~ } and {x } respectively G~, G A, G.4B, G aB gl , gi , gil, gii From .~ to -~ deformation embodiment of the body is: x = x ( X ) x ' = x ( X A) ( 2 . 1 ) It shows the space location x which the mass point X of body occupies after the deformation. All the vectors, which act on the mass point X , form the tridimentional Euclidean space. It might as well note down as . All its arbitrary elements may be shown as G a , or the linear combination G • , e.g. V=V,~GA--=-V 4GA. Similarly there is ~,., e.g. v-----v~g;-----v;g ~. Furthermore, we have to consider the four tensor product spaces: ff/~@5~ , @ . , , ~(~)~. and ~| . Their arbitrary elements may be shown as ,GAQGs,. GAQg.',. g;(~)GA~ and g;~)gi ( They may be substituted for the contravariant basis vector too ). l~rom now on we leave out the tensor product notation and adopt the linear combination of the Gibbs dyad notation, being noted down as, e.g. R =R-4BGAGn, S--~-S~GAg, T-~TI'~g;GA, U=U~Jg;91 By means of dot product operation, the element of the tensor product ( 9-dimensional ) space becomes the linear transformation derived from the 3-dimentional, e.g. W = R . V----(RA~GAGB) ' (VoGL')=R~VBGA-----WAGA w----TV = ( T i l g i G A ) (VsGA)=T~AV~g;=w'g, The elements of 5~@-~or @ , e.g. S and T, are called two-point tensor. The order of the dyad is elementary. indicates the conjugation of the tensor: The unit element ( unit tensor ) of 2 R*=RAAGaG~, T * = T ' A G A g , o J~| is: I = G~BG'4G~ = G ~G . . . . I = o ; j g g = g ; g i . . . . and the unit element ( it is called the transfer tensor ) is: ( ) I =o;~GAg~----oAIGAgl . . . . >< I = g ;Ag~GA=o '~g ;GA . . . . The geometric significance of the transfer tensor is to let the vector or 5~ invariably translating to 5~ or .Amongthem g = G A g~----g~ Gx g~.A: , therefore they may be uniforrnly noted down as g . In handling g ~, , it is likewise. With the help of the transfer tensor, i t is possible to reduce the two-point tensor to the one-point tensor. Or just the other way around, ff let the dot product to be instead of the tensor product. Among the basis vector dyads, we may obtain the trace of the tensor, e.g. : t rR~RABG.~ 9 G a =R.~A t r S S~.~G~ 9 g' =g~S':~ The trace of the tensor is the scalar. The double-point product of the two-tensor is also the sca]~: R : S . t r f R * 9 S) The absolute calculuses of the tensor fields of R and U are: d R = ( R V ) 9 dX ax Among them the absolute differential quotients ( gradients ) RV =RB~AGaGc GA Uu = U J i g i g g t ( ); A and ( ); .;indicate the convariant differential quotients atS~ and -~ respectively. R 9 V=RB~AGaGc 9 G~=RB~AGa R XV=R~L~GaGcXG A They are called the divergence and the curl of R respectively. Regarding the absolute differential quotient, the unit tensor and the rotational tensor are like the constant. From now on, in the whole discussion we consider that the various tensors, which appear in this article have been through the transfer tensor to reduce to the one-point tensor of previously, now let us indicate I. with I. This method is called Lagtange descriptive method. III. Strain and Stress In deformation ( 2.1 ), the displacement vector of the body mass point X is: u ( X ) = x ( X ) X the deformed gradient is F = x V = l + u u its polar decomposition is (3.1)" @default.
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• W3147477804 date "2005-01-01" @default.
• W3147477804 modified "2023-09-23" @default.
• W3147477804 title "Unified Theory of Variation Principles in Non-linear Theory of Elasticity" @default.
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