By Mase, G. Thomas; Mase, George E.; Smelser, Ronald E.
Continuum TheoryContinuum MechanicsStarting OverNotationEssential MathematicsScalars, Vectors and Cartesian TensorsTensor Algebra in Symbolic Notation - Summation ConventionIndicial NotationMatrices and DeterminantsTransformations of Cartesian TensorsPrincipal Values and primary DirectionsTensor Fields, Tensor CalculusIntegral Theorems of Gauss and StokesStress PrinciplesBody and floor Forces, Mass DensityCauchy Read more...
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The determinant of the matrix A is designated by either det A, or by |Aij |, and for a 3 × 3 matrix A, det A = |Aij | = A11 A21 A31 A12 A22 A32 A13 A23 A33 . 38) A minor of det A is another determinant |Mij | formed by deleting the ith row and jth column of |Aij |. 39) where superscript (c) denotes cofactor of matrix A. Evaluation of a determinant may be carried out by a standard method called expansion by cofactors. In this method, any row (or column) of the determinant is chosen, and each element in that row (or column) is multiplied by its cofactor.
These equations provide the focal points around which studies in elasticity, plasticity, viscoelasticity and fluid mechanics proceed. Mathematically, the fundamental equations of continuum mechanics mentioned above may be developed in two separate but essentially equivalent formulations. One, the 1 2 Continuum Mechanics for Engineers integral, or global form, derives from a consideration of the basic principles being applied to a finite volume of the material. The other, a differential, or field approach, leads to equations resulting from the basic principles being applied to a very small (infinitesimal) element of volume.
In some instances, higher order dimension spaces play integral roles in continuum topics. Because a scalar has only a single component, it will have the same value in every system of axes, but the components of vectors and tensors will have different component values, in general, for each set of axes. In order to represent vectors and tensors in component form we introduce in our physical space a right-handed system of rectangular Cartesian axes Ox1 x2 x3 , and identify with ^1 , e ^2 , e ^3 , shown in Fig.