Tensor Versus Matrix Formulations
The constitutive relations defined in Equation 6-2 relate the change in resistivity matrix (a rank 2 tensor) to the stress or strain matrix (also a rank 2 tensor). This means that the piezoresistivity tensor and the elastoresistance tensor are both rank 4 tensors. A formal tensor formulation of Equation 6-1 therefore takes the form:
(6-8)
Here the Einstein summation convention is used. Note that this equation implies that Δρij=ΠijklSkl and Δρij=Mijklεkl. Tensors Π and M have a number of symmetry properties. Tensors S, ε, and Δρ are symmetric and can be represented as six component vectors. There are two conventions for ordering the components in these vectors, and the Piezoresistivity interfaces all use the Voigt notation prevalent in the literature.
The Structural Mechanics branch interfaces use a distinct standard notation by default but does also support the Voigt notation.
Cubic materials, such as silicon, possess symmetries that mean the two notations are, in any case, equivalent when defining material properties. Within Voigt notation, the vectors that correspond to S, ε, and Δρ are:
This notation is known as the reduced subscript notation, since the number of subscripts required to represent the stress, strain, or change in resistivity is reduced from two to one. The relationship between the stress or strain and the resistivity change becomes:
Where π and m are now matrices related to the components of the corresponding Π and M tensors in the following manner:
Components of the Π and M tensors that do not appear in these matrices are always equal to one of those that does appear as a result of symmetry constraints.
For the most general triclinic crystal groups, the π and m matrices are not necessarily symmetric, as is the case for the elasticity matrix in structural mechanics.
This is because the rank four tensors Π and M lack one of the symmetry properties of the elasticity tensor (Πijkl ≠ Πklij and Mijkl≠ Mklij see Ref. 1 and Ref. 2 for details). Finally the relationship between π and m is given by:
(6-9)
where D is the elasticity matrix (s=D⋅e), which is defined in Voigt notation as: