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Effects of Soil Deposition on the Initial Stress State in Model Tests: Experimental Results and FE Simulation Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Improved Integration of High-Cycle Accumulated Strain Using Hierarchical and EAS Finite Elements

Authors : A. Niemunis, I. Melikayeva

Published in: Holistic Simulation of Geotechnical Installation Processes

Publisher: Springer International Publishing

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Abstract

The spatial variability of high-cycle accumulated strain is larger than the variability obtained from monotonic loading. Gradients of strain cause that the fit of a given strain field \(\epsilon _{ij}^{\text {acc}}\) is difficult for the conventional elements (if no refinement is used). Usage of identical mesh for monotonic and cumulative deformations leads to numerical self-stresses in elements. More flexible elements are therefore examined. They can reduce the self-stresses considerably.

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previous chapter Neohypoplasticity—Estimation of Small Strain Stiffness
next chapter Simulation of Soils Under Rapid Cyclic Loading Conditions
Footnotes
1
Obtained from \( u^{\text {acc}}_{1,122} + u^{\text {acc}}_{2,211} - (u^{\text {acc}}_{1,212} + u^{\text {acc}}_{2,212} ) = 0\).
 
2
The brackets in the index part denote symmetrization \( u_{(i,j)} = \frac{1}{2} \left( u_{ i,j } + u_{ j,i } \right) \).
 
3
Similarly one could also weaken the constitutive relationship and instead of the usual strong connection \(\sigma _{ij} = E_{ijkl} (u_{k,l} {- \epsilon ^\text {acc}_{kl} }) \) we could introduce an independent unknown field of stress \(\sigma _{ij}\) which should be compatible with the constitutive stress \(\sigma _{ij}^\epsilon = E_{ijkl} (\epsilon _{kl} - \epsilon _{kl}^\text {acc})\). This \(\sigma _{ij}^\epsilon \) would be treated as a slave of the master field \(\epsilon _{kl}\). The weak for of the connection \(\sigma _{ij} - \sigma _{ij}^\epsilon \) is \( \lambda _{ij}^{\sigma } (\sigma _{ij} - \sigma _{ij}^\epsilon ) \) with the Lagrangian multipliers \(\lambda _{ij}^{\sigma }\).
Similarly, we may also proceed with the essential BCs in the form \(u_i - \breve{u_i} = 0\) defined on \(S_u\). They will be added to the functional with the help of Lagrange multipliers \(\lambda ^u_{i}\) in the form \( - \int _{S_u} \lambda ^u_{i} (u_i - \breve{u_i}) \mathrm{d}S \)
This term should be added to the extended potential. The result is called the Hu-Washizu functional.
 
4
Note that \(\lambda ^{\epsilon }_{ij} = \sigma _{ij}\) must be symmetric and hence \(\lambda ^\epsilon _{ij} \delta u_{(i,j)} = \lambda ^\epsilon _{ij} \delta u_{ i,j }\) i.e. symmetrization may be omitted in (11), (12).
 
5
The enforcement of full orthogonality of \( \tilde{\epsilon }_{ij} \) with \(\sigma _{ij} = E_{ijkl} \epsilon ^{el}_{kl}\) would imply \( \tilde{\epsilon }_{ij} = 0\).
 
6
\(\int _\square \ldots \mathrm{d}\varvec{\xi }\) represents \(\mathop {\int }\limits _{-1}^1 \int \limits _{-1}^1 \ldots \mathrm{d}\xi \mathrm{d}\eta \).
 
7
hence we may omit symmetrization brackets in \(u_{(i,j)}\).
 
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Metadata
Title
Improved Integration of High-Cycle Accumulated Strain Using Hierarchical and EAS Finite Elements
Authors
A. Niemunis
I. Melikayeva
Copyright Year
2015
Book
Holistic Simulation of Geotechnical Installation Processes

Print ISBN: 978-3-319-18169-1

Electronic ISBN: 978-3-319-18170-7

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-18170-7_10