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In terms of computer code, the 2-D programs, both finite element and finite difference, were created directly from the 3-D programs. This was done essentially by leaving out the third (z) dimension everywhere. For the finite difference programs, only two conductance vectors, gx and gy, are needed to store the conductance information. For the finite element programs, the stiffness matrices dk are only dimensioned for four nodes, not eight nodes, and the elastic programs have only two degrees of freedom per node, not three as in 3-D. The elastic modulus tensor is 6 x 6 in 3-D, but only 2 x 2 in 2-D. For the thermal strain programs, in 3-D the last two entries of the displacement vector contain the six macrostrains, three in each entry. In 2-D, this is also the case, but as there are only three macrostrains in all, the second component of the last entry is not used in the program. The rest of the changes between 2-D and 3-D are obvious, and are in the programs.
Analytically relating 2-D to 3-D elasticity can be a problem. Engineers usually always think in 3-D, and to go to 2-D requires them to think in terms of plane strain (no z strains) or plane stress (no z stresses). However, it is possible to set up the equations of elasticity in 2-D, independent of but analogous to 3-D. This is the approach used in the finite element programs. The result looks like a plane strain approach. If one wishes to use the programs as a plane strain or stress limit from three dimensions, then simply substitute the correct moduli for the 2-D moduli in the programs.
The following formulas are used in the beginning of the finite element elastic programs, and are worth repeating here, for isotropic phases. The subscripts indicate the dimensionality of the modulus. K is the shear modulus and G is the bulk modulus. For isotropic elasticity, the full elastic moduli tensor can be expressed in terms of two independent constants. These are usually taken to be either K and G, or E and ν, the Young's modulus and Poisson's ratio, respectively. The 2-D and 3-D relations interrelate the two pairs, as only two of them are independent for isotropic elasticity. In 3-D,
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