# KinematicLinear

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## General description

KinematicLinear is an element to solve the momentum balance in a kinematic linear formulation, it uses Cauchy stresses and linear strains assuming infinite deformations. The element formulation is done using a B-Operator to compute the strains. As the element formulation is purely linear the problem has converge within a single step if a linear material law (like Isotropic3D) is used. This element has been implemented to be used within students training and for the testing of new implementations as its formulation represents the most basic and simplest formulation to solve structural problems.

## Element Formulation

$\mathbf{r}^{int}=\int_\Omega \mathbf{B}\cdot\boldsymbol{\sigma}d\Omega=\sum_{q=1}^{NQ} \mathbf{B}\left(\boldsymbol{\xi}^q\right)\cdot\boldsymbol{\sigma}\left(\boldsymbol{\xi}^q\right)\gamma^q\mbox{det}J$

$\mathbf{r}^{ext}= \int_\Omega \mathbf{N}\cdot\rho\mathbf{g}d\Omega=\sum_{q=1}^{NQ} \mathbf{N}\left(\boldsymbol{\xi}^q\right)\cdot\rho\mathbf{g}\gamma^q\mbox{det}J$

• Linearization (LeftHandSideMatrix)

$\mathbf{K}=\int_\Omega\mathbf{B}\cdot\mathbf{C}\cdot\mathbf{B}^Td\Omega=\sum_{q=1}^{NQ} \mathbf{B}\left(\boldsymbol{\xi}^q\right)\cdot\mathbf{C}\cdot\mathbf{B}^T\left(\boldsymbol{\xi}^q\right)\gamma^q\mbox{det}J$

• B-Operator

$\mathbf{B}\left(\boldsymbol{\xi}^q\right)_i=\left[\begin{array}{ccc}N^i_{,1}&0&0\\0&N^i_{,2}&0\\0&0&N^i_{,3}\\N^i_{,2}&N^i_{,1}&0\\0&N^i_{,3}&N^i_{,2}\\N^i_{,3}&0&N^i_{,1}\end{array}\right]$ , where $N^i_{,j}=\frac{\partial N^i}{\partial \xi_k}\frac{\partial \xi_k}{\partial X_j}$

• strain vector

$\boldsymbol{\epsilon}\left(\boldsymbol{\xi}^q\right)=\mathbf{B}\left(\boldsymbol{\xi}^q\right)\boldsymbol{u}^e$

## How to use this element

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

• KinematicLinear3D4N for a 4 node tetrahedron with linear shape functions,
• KinematicLinear3D10N for a 10 node tetrahedron with quadratic shape functions,
• KinematicLinear3D8N for a 8 node hexahedron with linear shape functions,
• KinematicLinear3D20Nfor a 20 node hexahedron with quadratic shape functions,
• KinematicLinear3D27Nfor a 27 node hexahedron with quadratic shape functions.

## Test Example

The test example consists of a 1*1*1m cube with a linear material ($\rho=7620\frac{kg}{m^3}, E=210\frac{N}{mm^2}, \nu= 0.3$) subjected to gravity loads ($\mathbf{g}=[0\;0\;-9.81]^T\frac{m}{s^2}$). The correct solution is a displacement of − 0.1322mm at the top of the cube. An KinematicLinear3D27N has been used.