Green’s strain tensor

Although the deformation is described by U, other measures of strain can be useful. One example is provided by Green s strain tensor G, defined as... 

Green s strain tensor vanishes in an undeformed system G = 0. For small deformations, it converges to the strain tensor e, defined in section 2.2.2. 

For small deformations, the terms of the form duu/dii are small. The product in the second term thus becomes small and can be neglected. For small deformations, Green s strain tensor thus converges to e. 

Fij thus contains only diagonal entries. Green s strain tensor G can be calculated... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

The general and detailed constitutive relations of E.H. Lee s elastic-plastic theory at finite strain have been derived by Lubarda and Lee [5]. In this work, let the specid constitutive relations which are employed in the general purpose finite element program be listed as follows. First, the Helmholtz free energy density, E, as a function of the invariants of the elastic Cauchy-Green tensor, c/y, may be expressed as... 

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