The strong form of deformable solids is given by the Cauchy momentum equation
∇⋅σ+f=ρu¨,
where
- σ is the stress tensor
- f is the volumetric body force vector
- ρ is the mass density
- u is the displacement field vector
- u¨ is the acceleration vector
Constitutive equation
Generalized Hooke’s law defines the relation between the stress and strain components
σ=Cε,
where C is a fourth-order elasticity tensor and ε is the strain tensor.
Strain tensor
- Small displacement theory:
ε=21(∇u+∇uT)
- Large displacement theory:
ε=21(∇u+∇uT+∇uT∇u)
Voigt notation
In Voigt notation, the stress and strain tensors are represented as vectors.
Consequently, the constitutive relation takes the form
σv=Hεv,
where H is the elasticity matrix which is a symmetric 6×6 matrix.
The weak form of solid mechanics is obtained by multiplying the partial differential equation in (1) with the test function of displacement field u′ and integrating over the whole domain Ω.
∫Ω−ρu¨⋅u′dΩ+∫Ω(∇⋅σ)⋅u′dΩ+∫Ωf⋅u′dΩ=0
Applying Leibniz rule on the divergence term, we get
∫Ω−ρu¨⋅u′dΩ+∫Ω−σ:∇u′dΩ+∫Ω∇⋅(σ⋅u′)dΩ+∫Ωf⋅u′dΩ=0.
Applying Divergence theorem on the divergence term, we get
∫Ω−ρu¨⋅u′dΩ+∫Ω−σ:∇u′dΩ+∫Ωu′⋅(σ⋅n)dΩ+∫Ωf⋅u′dΩ=0.
Since the stress tensor is symmetric, the following relation holds.
σ:∇u′=σ:∇u′T=σ:21(∇u′T+∇u′T)=σ:21(∇u′+∇u′T)=σ:ε′
Substituting in the above relation σ:∇u′=σ:ε′ and following the definition of traction vector t=σ⋅n, we get the final form of our weak formulation:
∫Ω−ρu¨⋅u′dΩ+∫Ω−σ:ε′dΩ+∫Ωf⋅u′dΩ+∫Γt⋅u′dΓ=0