u Subscript r Baseline plus StartFraction partial-differential u Subscript theta Baseline Over partial-differential theta EndFraction right-parenthesis comma 2nd Row epsilon Subscript phi phi Baseline equals StartFraction 1 Over r sine theta EndFraction StartFraction partial-differential u Subscript phi Baseline Over partial-differential phi EndFraction plus StartFraction u Subscript r Baseline Over r EndFraction plus StartFraction u Subscript theta Baseline cotangent theta Over r EndFraction comma epsilon Subscript r theta Baseline equals one-half left-parenthesis StartFraction 1 Over r EndFraction StartFraction partial-differential u Subscript r Baseline Over partial-differential theta EndFraction plus StartFraction partial-differential u Subscript theta Baseline Over partial-differential r EndFraction minus StartFraction u Subscript theta Baseline Over r EndFraction right-parenthesis comma 3rd Row epsilon Subscript theta phi Baseline equals StartFraction 1 Over 2 r EndFraction left-parenthesis StartFraction 1 Over sine theta EndFraction StartFraction partial-differential u Subscript theta Baseline Over partial-differential phi EndFraction plus StartFraction partial-differential u Subscript phi Baseline Over partial-differential theta EndFraction minus u Subscript phi Baseline cotangent theta right-parenthesis comma epsilon Subscript r phi Baseline equals one-half left-parenthesis StartFraction 1 Over r sine theta EndFraction StartFraction partial-differential u Subscript r Baseline Over partial-differential phi EndFraction plus StartFraction partial-differential u Subscript phi Baseline Over partial-differential r EndFraction minus StartFraction u Subscript phi Baseline Over r EndFraction right-parenthesis period EndLayout"/>(2.142)
2.14 Constitutive Equations for Anisotropic Linear Thermoelastic Solids
As we are concerned in this book with various types of composite material, it is necessary to define a set of constitutive relations that will form the basis for the development of theoretical methods for predicting the behaviour of anisotropic materials. Consider a general homogeneous infinitesimal strain εkl (applied to a unit cube of the composite material) defined in terms of the displacement vector uk and the position vector xk by (2.107), namely,
With the assumption that the strain field is uniform in the composite, the displacement field is linear and may be written in the form
where it is assumed that the displacement vector is zero when x1=x2=x3=0.
The local equation of state (2.111) is not of a form that can easily be related to experimental measurements as one of the state variables is assumed to be the specific entropy η. It is much more convenient, and much more practically useful, if the state variable η is replaced by the absolute temperature T. A local equation of state for the specific Helmoltz energy is assumed to have the following form (equivalent to (2.111) as implied by (2.67)–(2.69))
where εij is the infinitesimal strain tensor introduced in Section 2.12 (see (2.107)). For infinitesimal deformations, and because ψ≡υ−Tη from (2.64), the following differential form of (2.145) may be derived from (2.112)
where the specific entropy η and the stress tensor σij are now defined by
For infinitesimal deformations, a linear thermoelastic response can be assumed so that the Helmholtz energy per unit volume ρ0ψ^ has the form
where Cijkl are the elastic constants having the dimensions of stress or modulus, βij are thermoelastic coefficients and where T0 is a reference temperature. As the strain tensor is symmetric, it follows that βij=βji and that
On substituting (2.148) into (2.147), it follows that
