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Solid mechanics

Strong formulation

The strong form of deformable solids is given by the Cauchy momentum equation:

σ+f=ρu¨\begin{equation} \nabla \cdot \boldsymbol{\sigma} + \boldsymbol{f} = \rho \boldsymbol{\ddot{u}} \end{equation}

where,

  • σ\boldsymbol{\sigma} is the stress tensor
  • f\boldsymbol{f} is the volumetric body force vector
  • ρ\rho is the mass density
  • u\boldsymbol{u} is the displacement field vector
  • u¨\boldsymbol{\ddot{u}} is the acceleration vector

Constitutive equation

Generalized Hooke’s law defines the relation between the stress and strain components:

σ=Cε \boldsymbol{\sigma} = \boldsymbol{C} \boldsymbol{\varepsilon}

where C\boldsymbol{C} is a fourth-order elasticity tensor and ε\boldsymbol{\varepsilon} is the strain tensor.

Strain tensor

  • Small displacement theory: \qquad
ε=12(u+uT) \boldsymbol{\varepsilon} = \dfrac{1}{2} \left( \nabla \boldsymbol{u} + \nabla \boldsymbol{u}^T \right)
  • Large displacement theory: \qquad
ε=12(u+uT+uTu) \boldsymbol{\varepsilon} = \dfrac{1}{2} \left( \nabla \boldsymbol{u} + \nabla \boldsymbol{u}^T + \nabla \boldsymbol{u}^T \nabla \boldsymbol{u} \right)

Voigt notation

In Voigt notation, the stress and strain tensors are represented as vectors. Consequently, the constitutive relation takes the following form:

σv=Hεv \boldsymbol{\sigma}_v = \boldsymbol{H} \boldsymbol{\varepsilon}_v

where H\boldsymbol{H} is the elasticity matrix which is a 6×66 \times 6 symmetric matrix.

Weak formulation

The weak form of solid mechanics is obtained by multiplying the partial differential equation in (1) with the test function of displacement field u\boldsymbol{u}^\prime and integrating over the whole domain Ω\Omega.

Ω  ρu¨u  dΩ+Ω  (σ)u  dΩ+Ω  fu  dΩ=0 \int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; (\nabla \cdot \boldsymbol{\sigma}) \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega = 0

Applying Leibniz rule on the divergence term, we get

Ω  ρu¨u  dΩ+Ω  σ ⁣:u  dΩ+Ω  (σu)  dΩ+Ω  fu  dΩ=0 \int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; -\boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; \nabla \cdot (\boldsymbol{\sigma} \cdot \boldsymbol{u}^\prime) \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega = 0

Applying Divergence theorem on the divergence term, we get

Ω  ρu¨u  dΩ+Ω  σ ⁣:u  dΩ+Ω  u(σn)  dΩ+Ω  fu  dΩ=0 \int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; -\boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; \boldsymbol{u}^\prime \cdot (\boldsymbol{\sigma} \cdot \boldsymbol{n}) \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega = 0

Since the stress tensor is symmetric, the following relation holds

σ ⁣:u=σ ⁣:uT=σ ⁣:12(uT+uT)=σ ⁣:12(u+uT)=σ ⁣:ε\begin{align*} \boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime &= \boldsymbol{\sigma} \colon \nabla {\boldsymbol{u}^\prime}^T \\ &= \boldsymbol{\sigma} \colon \dfrac{1}{2} \left(\nabla {\boldsymbol{u}^\prime}^T + \nabla {\boldsymbol{u}^\prime}^T \right) \\ &= \boldsymbol{\sigma} \colon \dfrac{1}{2} \left(\nabla {\boldsymbol{u}^\prime} + \nabla {\boldsymbol{u}^\prime}^T \right) \\ &= \boldsymbol{\sigma} \colon \boldsymbol{\varepsilon}^\prime \end{align*}

Substituting the above relation σ ⁣:u=σ ⁣:ε\boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime = \boldsymbol{\sigma} \colon \boldsymbol{\varepsilon}^\prime and following the definition of traction vector t=σn\boldsymbol{t} = \boldsymbol{\sigma \cdot n}, we get the final form weak formulation:

Ω  ρu¨u  dΩ+Ω  σ ⁣:ε  dΩ+Ω  fu  dΩ+Γ  tu  dΓ=0 \int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; -\boldsymbol{\sigma} \colon \boldsymbol{\varepsilon}^\prime \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Gamma}\; \boldsymbol{t} \cdot \boldsymbol{u}^\prime \; d\Gamma = 0