By Lekhnit︠s︡kiĭ S. G.

Anisotropic.Plates.Lekhnitskii.1984.0677206704(600dpi)(T)(ISBN 0677206704)(546s)

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We shall sometimes refer to the invariant functions /i(T), fiCT), and /3(T) as the primary invariants to distinguish them from other invariant functional forms. The trace of a tensor is simply the sum of its diagonal components. We use the operator "tr" to designate the trace. Thus, tr(T) = T^ is the first invariant of the tensor T. The second and third invariants can also be expressed in terms of the trace operator. Let us introduce the notation of a tensor raised to a power asT^ = XT and T^ = TTT, where the components are given by the formula for products of tensors, Eqn.

90) are equivalent. The convenience of one form over another depends upon the application. Integral Theorems The divergence theorem. There is an mtegration theorem worth mentioning here because it comes up repeatedly in solid mechanics. We call it the divergence theorem because it involves the divergence of a vector field. Consider again a region S of arbitrary size and shape, with boundary Q described by its normal vectors n. The divergence theorem can be stated as follows divydV = ndA (91) This remarkable theorem, also known as Green's theorem or Gauss's theorem, relates an integral over the volume of a region to an integral over the boundary of that same region.

If the eigenvalues and eigenvectors are known, we can express the original tensor in terms of those objects in the following manner 3 T = 2]/^,n,(g)n, (62) /=1 Note that we need to suspend the summation convention because of the number of times that the index / appears in the expression. This form of expression of the tensor T is called the spectral decomposition of the tensor. How do we know that the tensor T is equivalent to its spectral decomposition? As we indicated earlier, the operation of a second-order tensor is completely defined by its operation on three independent vectors.