The transfer of heat in solids is a thermal conduction process and is governed by the equation:
βtβTβ=βββ
q+Qββ
where,
- q is the heat flux density (W/m2)
- Q is the volumetric heat source (W/m3)
- T is the Temperature field (K)
Constitutive equation
Heat flux density due to conduction is given by Fickβs law of diffusion:
q=βΞΊβTββ
where ΞΊ is the thermal conductivity of the material (W/m2K).
The governing equation becomes:
ββtβTβ+ββ
(ΞΊβT)+Q=0ββ
The weak form of solid mechanics is obtained by multiplying the partial differential equation in (3) with the test function
of Temperature field Tβ² and integrating over the whole domain Ξ©.
β«Ξ©βββtβTβTβ²dΞ©+β«Ξ©βββ
(ΞΊβT)Tβ²dΞ©+β«Ξ©βQTβ²dΞ©=0
Applying Leibniz rule on the divergence term, we get
β«Ξ©βββtβTβTβ²dΞ©+β«Ξ©ββΞΊβTβ
βTβ²dΞ©+β«Ξ©βββ
(ΞΊβTTβ²)dΞ©+β«Ξ©βQTβ²dΞ©=0
Applying Divergence theorem on the divergence term, we get
β«Ξ©βββtβTβTβ²dΞ©+β«Ξ©ββΞΊβTβ
βTβ²dΞ©+β«Ξβ(ΞΊβTβ
n)Tβ²dΞ+β«Ξ©βQTβ²dΞ©=0
Substituting (2) in the boundary term, we get the final form weak formulation:
β«Ξ©βββtβTβTβ²dΞ©+β«Ξ©ββΞΊβTβ
βTβ²dΞ©+β«Ξβ(qβ
n)Tβ²dΞ+β«Ξ©βQTβ²dΞ©=0