Green Lagrange strain tensor

Similarly, the Green-Lagrange strain tensor E j and the infinitesimal strain tensor e-j are related by... 

It is also important to note that the 2nd Piola-Kirchhoff stress tensor is energetically conjugate to the Green-Lagrange strain tensor and the Cauchy stress is energetically conjugate to the infinitesimal strain tensor. In other words, we have... 

Cartesian components of the 2nd Piola-Kirchhoff stress tensor Cartesian components of the Green-Lagrange strain tensor Components of the linear elasticity tensor Increment in the /th displacement component... 

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... 

So far, the shape of the cross-section of the considered beams has not been discussed, while the Green Lagrange strain tensor has been brought up for a continuum only confined with respect to the deformations in the cross-sectional plane by Remark 7.4. Subsequently, a special class of cross-sectional topologies will be examined ... 

It is noted that the above strain can also be expressed by the displacement gradient. For instance, the Green-Lagrange strain in Equation (4.20), after operation in terms of the tensor index, can be rewritten as... 

Finally, the deformation gradient, the Green-Lagrange strain, and rate of deformation tensor can be linked by the following relation ... 

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