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The following article provides a detailed summary of the weak form equations in elastic mechanics. However, since it is inconvenient for quick reference, the detailed explanations are omitted here, and only the flow is presented.
Elastic Mechanic
Displacement and Strain
\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)Stress and Strain (with Elasticity Matrix)
The elasticity matrix (D) represents the material properties and describes the relationship between stress (σ) and strain (ε) at a material point.
\sigma = \mathbf{D} : \varepsilonBalance Equation
\nabla \cdot \sigma + \mathbf{f} = 0Unified Form
\nabla \cdot \left( \mathbf{D} : \nabla_s \mathbf{u} \right) + \mathbf{f} = 0Stiffness Matrix
The stiffness matrix (K) represents the behavior of the entire structure and relates nodal forces (F) to nodal displacements (u).
\boldsymbol{F}=\boldsymbol{K} \boldsymbol{u} \boldsymbol{K} = \int_v \boldsymbol{B}^T \boldsymbol{D} \boldsymbol{B} dV \\B: Strain-displacement matrix (derived from shape function gradients).
C: Elasticity matrix.
