Properties for Design of Composite Structures. Neil McCartney
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and The inverse form is
When the stress field is hydrostatic so that σij=−pδij, and because σkk≡σ11+σ22+σ33, it follows from (2.157) that
The inverse form (2.158) may be written as
where λ and μ are Lamé’s constants, μ being the shear modulus, which can be calculated from Young’s modulus and Poisson’s ratio as follows:
2.15 Introducing Contracted Notation
The general formulation for describing the elastic constants of anisotropic materials involves fourth-order tensors that are difficult to apply in many practical situations where analytical methods can be used. A simplified contracted notation is usually used for such analyses where the fourth-order tensors of elastic constants are replaced by a second-order matrix formulation that is now described. The matrix formulation makes use of the fact that the stress and strain tensors are symmetric. These symmetry properties enabled the derivation of the relationships (2.149)–(2.151).
The components of the stress and strain components are now assembled in column vectors of length six so that
It should be noted that a factor of two has been applied only to the shear terms of the relation involving the strains so that the quantities 2εij for i≠j correspond to the widely used engineering shear strain values. General linear elastic stress-strain relations, including thermal expansion terms, have the contracted matrix form
where CIJ are symmetric elastic constants, which are components of the second-order matrix C, and where UI are thermoelastic constants associated with the tensor βij, which are components of the vector U, the uppercase indices I and J ranging from 1 to 6. For orthotropic materials the stress-strain relations have the simpler matrix form
The stress-strain relations (2.163) may be written, using a repeated summation convention for uppercase indices over the range 1, 2, …, 6, as
The inverse of the matrix CIJ is denoted by the symmetric matrix SIJ such that
where δIK is the Kronecker delta symbol having the value 1 when I=J and the value 0 otherwise. On multiplying (2.165) on the left by SLI and on using (2.166), it can be shown that
The