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In the case of thermal strains, sometimes called eigenstrains (terms used interchangeably in this manual) , each phase can have a stress-free strain that comes about by thermal or moisture expansion/shrinkage, or other causes. We denote this strain as eα, where α = 1,6 as usual in the Voigt notation. The stress then becomes σα = Cαβ(εβ − eβ), where εβ is the usual strain defined by the elastic displacements. The elastic energy then becomes
Substituting for the strain using the linear interpolation scheme described earlier, we can then perform the integration over a single pixel, keeping in mind that the thermal strains are constants over the pixel and are not linearly interpolated. The resulting equation in the nodal displacements is
where the first term is identical to the case without thermal strains, the second term is a constant quadratic in the thermal strains, and the third term is linear in nodal displacements, with Trp given by
So even without periodic boundary conditions, there are terms linear and constant in the displacements. Periodic boundary conditions are very similar to the elastic case studied previously, except for an extra term picked up via the Trp term. In a pixel at the face, edge, or corner of the image, the energy becomes
where the components of δrp were given in Table 7, and the u's are the real displacements, brought over from the opposing side or edge or corner of the image in the same way as in the electrical or the no-eigenstrain elastic problems. It is important to note that brp and δrp are linear in the applied strains, and Trp is independent of the applied strains.
In the programs THERMAL3D.F or THERMAL2D.F, the whole system size is allowed to change in order find the overall thermal expansion that minimizes the energy of the system. In this case, the applied strains, Eα, are now called the macrostrains, and become dynamic variables that define the size and shape of the periodic unit cell. Their values are determined in the conjugate gradient relaxation process, on an equal footing with the elastic displacements. In the programs, the length of the u vector changes from ns = nx x ny x nz to nss = ns + 2, where u(ns + 1,1)=Exx, u(ns + 1, 2)=Eyy, u(ns + 1, 3) = Ezz, u(ns + 2,1) = Exz, u(ns + 2, 2) = Eyz, and u(ns + 2, 3)= Exy.
In subroutine DEMBX, the matrix of second derivatives, or the Hessian matrix, is used to update the gradient and conjugate gradient direction in the relaxation process. For the regular nodal displacements, the stiffness matrices make up the Hessian matrix. However, when the six macrostrains are considered to be variables as well, the Hessian matrix goes from being (ns,3) x (ns,3) in size to (ns + 2, 3) x (ns + 2, 3) in size. The extra partial derivatives that have to be evaluated are (1) the mixed second derivatives, or the second derivative of the energy with respect to a nodal displacement and a macrostrain, and (2) the second derivative with respect to a macrostrain squared or two different macrostrains.
Figure 2 illustrates schematically what the full Hessian matrix looks like. The upper left-hand corner is the main section, where the second derivatives of the energy are taken with respect to the nodal displacements. Multiplication of a vector by this piece is taken care of in the large DO loop involving the stiffness matrices in subroutine DEMBX. The upper right and lower left-hand parts of the Hessian involve mixed second derivatives with respect to a nodal displacement and a macrostrain. Examining eq. (24), which is the full elastic energy with periodic boundary conditions and thermal energy terms, one can see that the only term that has dependence on both the nodal displacements and the macrostrains is the term brp urp . The kind of second derivative that is needed is
Figure 2: Schematic view of the structure of the Hessian matrix when thermal strains are used (u stands for elastic displacements, ε represents the macrostrains).
where all the other macrostrains have been set to zero. This is the technique used in the subroutine BGRAD, which computes the terms needed for these parts of the Hessian matrix. The bottom right-hand part of the Hessian matrix involves second derivatives with respect to the macrostrains. Again looking at eq. (24), the only term that has quadratic dependence on the macrostrains is what was called the constant term before, that comes from the periodic boundary conditions. In all the finite element programs with fixed macrostrains, this term is designated C. In the two thermal strain programs, this term is expressed by a matrix called zcon. The second derivative terms are given simply by the various components of this matrix. The subroutine CONST computes the elements of this matrix, which are constant with respect to nodal displacements and macrostrains.
In subroutine DEMBX, the main part of the multiplication by the Hessian matrix is carried out, and then the other three parts of the matrix multiplication are filled in using subroutine BGRAD and the matrix zcon.