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Equilibrium and virtual work

Foundation · assumptions, essential equations, full derivation, algorithm, code and verification references.

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FoundationFOUNDATION · infinitesimal quasi-static setting

Connect local force balance, boundary conditions and the weak statement used by displacement finite elements.

01 Model and scope

The continuous model states what must balance at every point and what is prescribed on each boundary. Virtual work rewrites that model in an integral form suitable for approximation. The following derivation assumes small-strain quasi-static mechanics.

Route assumptions and calculation checks

  1. State the domain, loads and essential and natural boundaries.
  2. Choose a stress and strain measure compatible with the kinematics.
  3. Apply the declared constitutive law before discretizing the weak form.

02 Essential equations and symbols

∇ ⁣⋅σ+b=0\nabla\!\cdot\boldsymbol\sigma+\mathbf b=\mathbf0∫Ωε(v):σ dΩ=∫Ωv⋅b dΩ+∫Γtv⋅tˉ dΓ\int_\Omega\boldsymbol\varepsilon(\mathbf v):\boldsymbol\sigma\,d\Omega=\int_\Omega\mathbf v\cdot\mathbf b\,d\Omega+\int_{\Gamma_t}\mathbf v\cdot\bar{\mathbf t}\,d\Gamma

Definitions and conventions

Ω,Γu,Γt\Omega,\Gamma_u,\Gamma_t
Body, displacement boundary and traction boundary.
b,tˉ\mathbf b,\bar{\mathbf t}
Body force per volume and boundary traction.
v\mathbf v
Admissible virtual displacement, zero on the essential boundary.
σ\boldsymbol\sigma
Cauchy stress for this infinitesimal setting.
4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Declare the boundary-value problem

Partition the boundary into the essential and natural parts. Prescribe displacement on the former and traction on the latter. Supports must remove the unintended rigid mechanisms for an ordinary static solve.

∂Ω=Γu∪Γt\partial\Omega=\Gamma_u\cup\Gamma_tu=uˉ(Γu)\mathbf u=\bar{\mathbf u}\quad(\Gamma_u)σn=tˉ(Γt)\boldsymbol\sigma\mathbf n=\bar{\mathbf t}\quad(\Gamma_t)

2 · Test equilibrium

Multiply force balance by any admissible virtual displacement. Integrating the divergence term by parts exposes boundary traction and replaces a stress derivative by a displacement-gradient term.

∫Ωv⋅(∇ ⁣⋅σ+b) dΩ=0\int_\Omega\mathbf v\cdot(\nabla\!\cdot\boldsymbol\sigma+\mathbf b)\,d\Omega=0∫Ω∇v:σ dΩ=∫Ωv⋅b dΩ+∫∂Ωv⋅σn dΓ\int_\Omega\nabla\mathbf v:\boldsymbol\sigma\,d\Omega=\int_\Omega\mathbf v\cdot\mathbf b\,d\Omega+\int_{\partial\Omega}\mathbf v\cdot\boldsymbol\sigma\mathbf n\,d\Gamma

3 · Use symmetry and boundary conditions

With symmetric Cauchy stress only the symmetric part of the virtual gradient contributes. The virtual displacement vanishes on the essential boundary, leaving the prescribed traction on the natural boundary.

∇v:σ=ε(v):σ\nabla\mathbf v:\boldsymbol\sigma=\boldsymbol\varepsilon(\mathbf v):\boldsymbol\sigmaδWint=δWext\delta W_{int}=\delta W_{ext}

4 · Close the mechanical model

A constitutive law closes the equilibrium equations. For homogeneous linear elasticity it maps infinitesimal strain to stress. Changing to finite kinematics requires consistent stress measures, virtual work and integration domains.

σ=C:ε(u)\boldsymbol\sigma=\mathbb C:\boldsymbol\varepsilon(\mathbf u)∫Ωε(v):C:ε(u) dΩ=ℓ(v)\int_\Omega\boldsymbol\varepsilon(\mathbf v):\mathbb C:\boldsymbol\varepsilon(\mathbf u)\,d\Omega=\ell(\mathbf v)

This is a formulation foundation, not verification evidence for an element or material route.

04 Algorithm and implementation

Follow the route sheets below to see how this shared mathematics is implemented.

05 Verification and limitations

FOUNDATION · infinitesimal quasi-static setting

The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.

Capability matrix →Recorded cases and original sources →Run a maintained example →

06 Sources and related sheets

Complete TET4 derivationLinear equilibrium method

Continue the reasoning

Interpolation, mapping and assemblyStrain, stress and rotation measuresUnits, signs and error measures
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