Abstract
Within this chapter, the case of the nonlinear elasticity, meaning strain-dependent modulus of elasticity, will be considered. The problem will be illustrated with the example of bar elements. First, the stiffness matrix or alternatively the principal finite element equation will be derived under consideration of the strain dependency. For the solving of the nonlinear system of equations three approaches will be derived, namely the direct iteration, the complete Newton–Raphson iteration and the modified Newton–Raphson iteration, and will be demonstrated with the help of multiple examples. Within the framework of the complete Newton–Raphson iteration the derivation of the tangent stiffness matrix will be discussed in detail.
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Notes
- 1.
At plastic material behavior remaining strains occur. This case will be covered in Chap. 11.
- 2.
For this, see the treatment of bar elements with variable cross-sectional areas \( A= A(x)\) in Chap. 3.
- 3.
At this point it was assumed that for \(\varepsilon = 0\) the stress turns 0. Therefore, for example no residual stress exists.
- 4.
One considers that the associative law applies for matrix multiplications.
- 5.
In the context of the finite element method Newton’s iteration is often referred to as the Newton–Raphson iteration [4].
- 6.
Alternative names in literature are HESSIAN, JACOBIAN or tangent matrix [1].
- 7.
One considers that the calculation of the inverse has to be carried out numerically in commercial programs.
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Öchsner, A., Merkel, M. (2013). Nonlinear Elasticity. In: One-Dimensional Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31797-2_10
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DOI: https://doi.org/10.1007/978-3-642-31797-2_10
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