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Published in:
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2022 | OriginalPaper | Chapter

13. Computational Setting

Author : Paul Steinmann

Published in: Spatial and Material Forces in Nonlinear Continuum Mechanics

Publisher: Springer International Publishing

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Abstract

This chapter sketches the consequences for computational mechanics by outlining the material force method based on finite element discretization of the material virtual work principle and highlights its applicability to geometrically nonlinear fracture mechanics by some computational examples.

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previous chapter Consequences of Thermodynamical Balances
Footnotes
1
Moreover, for example, the Piola-Kirchhoff and Mandel stress read concretely as
$$ \boldsymbol{S}=\mu [\boldsymbol{I}-\boldsymbol{B} ]+\lambda \ln J\boldsymbol{B}\quad \text{ and } \quad \boldsymbol{M}=\mu [\boldsymbol{C}-\boldsymbol{I} ]+\lambda \ln J\boldsymbol{I}, $$
whereas the Cauchy stress and the Piola-type Eshelby stress expand as
$$ J\boldsymbol{\sigma }=\mu [\boldsymbol{b}-\boldsymbol{i}]+\lambda \ln J\boldsymbol{i}\quad \text{ and } \quad J\boldsymbol{p}=\psi _\mathrm{m}\boldsymbol{F}^t-\mu \boldsymbol{F}^t\cdot [\boldsymbol{b}-\boldsymbol{i}]-\lambda \ln J\boldsymbol{F}^t. $$
.
 
Literature
1.
Ackermann D, Barth FJ, Steinmann P (1999) Theoretical and computational aspects of geometrically nonlinear problems in fracture mechanics. In: Proceedings (CD-ROM) of the European conference on computational mechanics ECCM’99 (ECCOMAS), August 31 to September 3, Munich, Germany 1999
2.
Steinmann P (2000) Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int J Solids Struct 37:7371–7391
3.
Steinmann P, Ackermann D, Barth FJ (2001) Application of material forces to hyperelastostatic fracture mechanics. II. Computational setting. Int J Solids Struct 38:5509–5526
Metadata
Title
Computational Setting
Author
Paul Steinmann
Copyright Year
2022
Book
Spatial and Material Forces in Nonlinear Continuum Mechanics

Print ISBN: 978-3-030-89069-8

Electronic ISBN: 978-3-030-89070-4

Copyright Year: 2022

DOI
https://doi.org/10.1007/978-3-030-89070-4_13

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