target="_blank" rel="nofollow" href="#fb3_img_img_4e6a66d2-8448-5fe2-a9b4-f34799ea7d6c.png" alt="sigma Subscript i j Baseline equals upper C Subscript i j k l Baseline epsilon Subscript k l Baseline plus beta Subscript i j Baseline left-parenthesis upper T minus upper T 0 right-parenthesis period"/>(2.153)
It should be noted that the stress components are zero everywhere when the strain is defined to be zero everywhere at the reference temperature T0.
The inverse form of the linear stress-strain relations (2.153) is written as
where the compliance tensor Sijkl is such that
where use has been made of (2.15), and where
are anisotropic thermal expansion coefficients.
2.14.1 Isotropic Materials
The situation simplifies when the material is linear thermoelastic and isotropic so that the stress-strain relations are
where E is Young’s modulus, ν is Poisson’s ratio, α is the thermal expansion coefficient 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,
