E a oij

where a is the linear coefficient of thermal expansion.

69 Inserting the preceding two equation into Hooke's law (Eq. 7.51) yields which is known as Duhamel-Neumann relations.

70 If we invert this equation, we obtain the thermoelastic constitutive equation:

71 Alternatively, if we were to consider the derivation of the Green-elastic hyperelastic equations, (Sect. 7.1.5), we required the constants c\ to c6 in Eq. 7.22 to be zero in order that the stress vanish in the unstrained state. If we accounted for the temperature change © — ©o with respect to the reference state temperature, we would have ck = —fik (© — ©0) for k = 1 to 6 and would have to add like terms to Eq. 7.22, leading to

for linear theory, we suppose that f3ij is independent from the strain and cijrs independent of temperature change with respect to the natural state. Finally, for isotropic cases we obtain

Jkk ^ij which is identical to Eq. 7.82 with ff — 1E''V. Hence

72 In terms of deviatoric stresses and strains we have and in terms of volumetric stress/strain:

p — —Ke + f(© — ©0) and e — + 3a(© — ©0)

0 0

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