CrisfieldTrussElement

General description

The Crisfield Truss Element is a truss element in 3D space.(Currently it is only applied in 2D, but we can change it also for 3D). For this element, both geometrial nonlinearity and material nonlinearity are considered here.

Element formulation

Element position vectors:

• current configuraton: $\mathbf{x^{e}}=(\mathbf{x^{e1}},\mathbf{x^{e2}})^T$
• reference configuraton: $\mathbf{X^{e}}=(\mathbf{X^{e1}},\mathbf{X^{e2}})^T$

Element displacement vector:

• $\mathbf{u^{e}} = (\mathbf{u^{e1}},\mathbf{u^{e2}})^T = \mathbf{x^{e}}-\mathbf{X^{e}}$

Element lengths:

• current configuraton: ${l^{2}}=\mathbf{x^{e}}\cdot\mathbf{A}\mathbf{x^{e}}$
• reference configuraton: ${L^{2}}=\mathbf{X^{e}}\cdot\mathbf{A}\mathbf{X^{e}}$
• with $\mathbf{A}=\begin{pmatrix} \mathbf{I} & -\mathbf{I} \\ -\mathbf{I} & \mathbf{I} \end{pmatrix}$, $\mathbf{I}$ is a second-order unity tensor.

Green-Lagrange strain:

• ${E}_{11}(\mathbf{u^{e}}) = \tfrac{1}{L}(\mathbf{X^{e}} +\tfrac{1}{2}\mathbf{u^{e}})\cdot\mathbf{A}\mathbf{u^{e}}$

2nd-Piola-Kirchhoff stress:

• St. Venant model: ${S}_{11}(\mathbf{u^{e}}) = \mathrm{E} {E}_{11}(\mathbf{u^{e}})$
• Scalar damage model: ${S}_{11}(\mathbf{u^{e}}) = \big(1-d(\kappa)\big)\mathrm{E} {E}_{11}(\mathbf{u^{e}})$

Residual force vector:

• $\mathbf{r} = \mathbf{r_{e}}-\mathbf{r_{i}}(\mathbf{u^{e}})$
• external force vector: $\mathbf{r_{e}}$
• internal force vector: $\mathbf{r_{i}}(\mathbf{u^{e}})= \tfrac{A}{L}\big(\mathbf{X^{e1}}+\mathbf{u^{e1}}-\mathbf{X^{e2}}-\mathbf{u^{e2}} , -(\mathbf{X^{e1}}+\mathbf{u^{e1}}-\mathbf{X^{e2}}-\mathbf{u^{e2}})\big)^T$

Tangent stiffness matrix:

• $\mathbf{{K}^{e}_{T}} = \mathbf{{K}^{e}_{m}}(\mathbf{u^{e}})+\mathbf{{K}^{e}_{g}}(\mathbf{u^{e}})$
• material stiffness matrix: $\mathbf{{K}^{e}_{m}}(\mathbf{u^{e}})= \tfrac{\mathrm{E}A}{L^3} \big[\mathbf{A}\cdot(\mathbf{X^{e}}+\mathbf{u^{e}})\big]\big[(\mathbf{X^{e}}+\mathbf{u^{e}})\cdot\mathbf{A}\big]$
• geomatrical stiffness matrix: $\mathbf{{K}^{e}_{g}}(\mathbf{u^{e}})= \tfrac{A}{L}{S}_{11}(\mathbf{u^{e}})\mathbf{A}$

How to use this element

The CrisfieldTrussElement is part of the KratosStructuralApplication and can be generated in the .elem input file using the name CrisfieldTrussElement3D2N.