Lagrange strain tensor

Frequently it is useful to express the coordinates of the Lagrange strain tensor by means of the displacement gradients (3.10) ... 

It is possible to gain a rather direct insight into the meaning of the diagonal coordinates Vn,V22 and F33 of the Lagrange strain tensor by a suitable choice of the material line element under consideration. Let us take first N = 1, A 2 = 0, N2 = 0. Then the relative change in length of the material line element becomes... 

In a similar way relations containing F31 and F12 may be derived. The general result is that the off-diagonal elements of the Lagrange strain tensor are related to the... 

A spatially periodic simulation box represents the contents of the atomistic inclusion. Again, the shape of this box is expressed by a scaling matrix h = [abc] (cf. Fig. 1). The scaled coordinates of atom s are used as degrees of freedom via x = hs . The Lagrange strain tensor of the atomistic box, can be calculated from Eq. (1) by replacing and with h and ho, respectively. The energy in the atomistic box can be expressed as (h, s ) and can be obtained via any energy model for atomistic simulation, e.g., a force-field approach. 

The material morphology is specified by a set of nodal points in the continuum description. The inclusion boundary is defined by a mesh of vertices xf b for boundary). The exterior of the inclusion contains the vertices xt (c for continuum). Inside the atomistic system, the (affine) transformations obtained by altering the scaling matrix from ho to h can be expressed by the overall displacement gradient tensor matrix M(h) = hho, The Lagrange strain tensor [40] of the atomistic system is then... 

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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