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Modeling High Temperature Materials Behavior for Structural Analysis

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Abstract

In this chapter we discuss constitutive equations to describe material behavior of high-temperature materials under multi-axial stress state.

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Notes

  1. 1.

    The derivative of the rotation tensor with the fixed axis of rotation is \(\displaystyle \mathrm {d}\pmb Q(\varphi \pmb m)=\pmb m\times \pmb Q(\varphi \pmb m)\mathrm {d}\varphi \), see also Appendix B.7.

  2. 2.

    The rules for derivatives of scalar-valued functions with respect to tensors are given in Appendix B.4.

  3. 3.

    For the sake of brevity tilde is omitted.

  4. 4.

    The orthogonal symmetry group includes also reflections. Since any reflection is the composition of a rotation and the inversion \(-\pmb I\), it is possible to consider the proper orthogonal group first and then apply the inversion.

  5. 5.

    Five different cases of the transverse isotropy can be defined, see, for example, Spencer (1987), Zheng and Boehler (1994).

  6. 6.

    This assumption is deduced from the principle of invariance under superimposed rigid body motions, e.g. Bertram (2012).

  7. 7.

    From this consideration it does not follow that only isotropic materials can be described. Indeed, we assumed that the stress tensor depends only on the velocity gradient, which is not true for anisotropic materials. Extensions will be discussed in Sect. 5.4.3.

  8. 8.

    For example, to identify creep of metals, the creep rate is considered as a function of the stress based on results of uni-axial creep tests.

  9. 9.

    At high temperature the yield point cannot be defined, the \(R_{\mathrm {p} 0.2}\) stress value is used instead in most cases, see Sect. 1.1.1.1.

  10. 10.

    The dependence on the temperature is dropped for the sake of brevity.

  11. 11.

    The change in shape is governed in such a way that the the work per unit time remains unchanged compared to small variations of the stresses within the yield limit. Since the elasticity theory provides a similar relationship between the deformations and the elastic potential, so I call the stress function F also the “plastic potential” or “flow potential”.

  12. 12.

    For the description of elastic material behavior instead of \(\pmb \sigma \) a strain tensor, e.g. the Cauchy-Green strain tensor is introduced. The five transversely isotropic invariants are the arguments of the strain energy density function, see Sect. 5.3.3.

  13. 13.

    The intermediate configuration can only be used as illustrative since it has a number of conceptual shortcomings. Historical essays and critical remarks to the multiplicative decomposition are presented in Naghdi (1990), Bertram (2012), Xiao et al. (2006).

  14. 14.

    See the analysis presented in Sect. 5.3.1.

  15. 15.

    From this consideration it does not follow that only isotropic plasticity can be described. Indeed, we assumed that the stress tensor depends only on \(\pmb \Lambda ^\mathrm {pl}\), which is not true for anisotropic materials.

  16. 16.

    Here we assumed that the tensor \(\pmb V_{\mathrm {el}}\) has distinct principal values.

  17. 17.

    The model was firs published in 1966 in a CEGB report, see Frederick and Armstrong (2007) for historical remarks.

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Naumenko, K., Altenbach, H. (2016). Constitutive Models. In: Modeling High Temperature Materials Behavior for Structural Analysis. Advanced Structured Materials, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-31629-1_5

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