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A.2.2 Fourth-Order Tensors
The dyadic product of four vectors is a fourth-order tensor , defined as The Cartesian components of a fourth-order tensor are given by such that Therefore, a fourth-order tensor transforms a vector into a third-order tensor, The double dot product of a fourth-order tensor with a second-order tensor is a second-order tensor defined as from which it can be shown that or equivalently, .
A fourth-order tensor can exhibit three levels of symmetry, which can be represented using Cartesian components as Whereas a general fourth-order tensor may have 81 distinct components, a tensor with one minor symmetry has 54 distinct components; a tensor with both minor symmetries has 36 distinct components; and a tensor with minor and major symmetries has 21distinct components. We may represent the major symmetry of as , whose Cartesian representation is provided above. It follows from this definition that