Abstract
Constitutive equations have to be formulated in continuum mechanics that characterizes the material response of a solid body.
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Notes
- 1.
By expressing the invariants in terms of principal stretches
$$ I_C = \lambda _1^2 + \lambda _2^2 + \lambda _3^2 \,\,,\,\, II_C = \lambda _1^2\,\lambda _2^2 + \lambda _2^2\,\lambda _3^2 + \lambda _3^2\,\lambda _1^2 \,\,,\,\, III_C = \lambda _1^2 \, \lambda _2^2 \, \lambda _3^2 $$the strain energy function assumes the form
$$ W({\varvec{C}}) \equiv W \,(\lambda _1^2,\,\lambda _2^2,\,\lambda _3^2). $$This approach is not considered within this book.
- 2.
- 3.
The latter conditions can be physically interpreted in the way that stresses have to approach \(-\infty \) for a volume going to zero and \(+\infty \) for a volume approaching \(\infty \).
- 4.
Nonlinear elastic material behaviour is often described in engineering literature by a linear relation between the second Piola–Kirchhoff stress tensor and the Green–Lagrange strain tensor (St. Venant material)
This constitutive equation corresponds to Hooke’s laws of the infinitesimal theory of elasticity with the Lamé constants \(\varLambda \) and \(\mu \) (these constants can be converted to the modulus of elasticity \(E = \frac{(3\varLambda +2\mu )\,\mu }{\varLambda +\mu }\) and Poisson’s ratio \(\nu =\frac{\varLambda }{2\,(\varLambda +\mu )}\) ) . Generally it can be shown that this constitutive equation is restricted to deformations with large displacements and finite rotations but small strains. St. Venant’s law depicts major deficiencies in the compressible range: in the limit case of the compression of a body to volume “0” the stress \({\varvec{\sigma }}\) approaches zero instead of \(\lim _{J_F \rightarrow 0} {\varvec{\sigma }}\rightarrow -\infty \). Thus the St. Venant material equation is not applicable for general simulations of solids within the finite deformation range. However it can be successfully used for large deflection analysis of thin structural members like beams or shells.
- 5.
- 6.
Here the flow conditions depend upon the stress \({\varvec{\sigma }}\) and not solely on the deviatoric stresses \({\varvec{s}}\) as in classical \(J_2\)-plasticity. The reason for this is that inelastic processes of general non metallic materials or of metals in which damage has to be considered are pressure sensitive and hence the flow condition has to depend upon the full stress tensor.
- 7.
The equation system can be solved either for the stresses \({\varvec{\sigma }}\) and the back stress \({\varvec{q}}\) or the plastic strains \({\varvec{\varepsilon }}^p\) and the hardening variable \({\varvec{\alpha }}\) at step \(n+1\). In the following the solution for the strain variables is preferred. Algorithms that yield the stress variables can be found in e.g. Simo (1998), Simo and Hughes (1998), Wriggers (2008) and de Souza Neto et al. (2008).
- 8.
The vectorization operator \(\mathrm {vec(\mathbf{{x}})}\) converts tensors into column vectors while accounting for the symmetry and skipping the zero entries. It relates to the Voigt notation used in classical finite element formulations.
- 9.
It is well known that the equations describing the von Mises plasticity model lead to a single nonlinear equation for the plastic multiplier, see Simo (1998), Simo and Hughes (1998) and de Souza Neto et al. (2008). Hence the more general approach followed in this section seem to be an overkill. However when comparing the efficiency of both approaches it turns out that the more general method is as efficient—when using AceGen —as the special elimination procedure leading to only one equation. Thus we follow in this book the approach discussed in Sect. 5.2.3 that can easily be adopted to more general plasticity formulations.
- 10.
It is well known that by using the wrong rates for the description of the elastic response physically meaningless stress states can occur at large deformations, see e.g. Atluri (1984). However it should be noted that special stress rates can be defined that circumvent these problems, see Xiao et al. (1997).
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Korelc, J., Wriggers, P. (2016). Materials. In: Automation of Finite Element Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-39005-5_5
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