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A visco-elasto-plastic multi-surface cyclic model
Un modèle visco-élastoplastique
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Siddiquee, S.A., Islam K.
Civil Engineering Department, BUET, Dhaka, Bangladesh
ABSTRACT: Modeling a visco-elasto-plastic material under cyclic loading has been a big issue due to the complexity of the problem.
Single surface nonlinear kinematic hardening models, two yield bounding surface models, multi-segment backstress models showed
some success in this field. Still, all these models fail in case of modeling cohesionless soil with pressure sensitivity and softening. In
this paper, a visco-elasto-plastic multi-surface model is proposed for cyclic loading. This model is originally proposed by Mroz
(1967) and has been extended to conhesionless soil by Prevost (1985). In this research, the model is extended to include the effect of
rate of loading. The integration of incremental elasto-plastic equation is carried out by steepest descent return mapping algorithm.The
dilatancy of dense sand has been modeled using non-associated flow rule. The non-associated flow has been attained by using Row’s
stress-dilatancy equation relating angle of internal friction to dilatancy angle by material constants. The viscous behavior is introduced
through the incorporation of three component model into the basic constitutive equations. The quality of the simulation is assessed
and limitations are discussed.
RÉSUMÉ : Modélisation d'un matériau visco-élasto-plastique sous chargement cyclique a été un gros problème en raison de la
complexité du problème. Simple surface cinématique non linéaire des modèles de durcissement, deux modèles de rendement de
surface de délimitation, les modèles backstress multi-segments ont montré un certain succès dans ce domaine. Néanmoins, tous ces
modèles ne dans le cas de la modélisation des sols sans cohésion avec sensibilité à la pression et adoucissantes. Dans cet article, un
visco-élasto-plastique multi-surface du modèle est proposé pour le chargement cyclique. Ce modèle est proposé à l'origine par Mroz
(1967) et a été étendu au sol conhesionless par Prévost (1985). Dans cette recherche, le modèle est étendu pour inclure l'effet de la
vitesse de chargement. L'intégration progressive de l'élasto-plastique équation est effectuée par dilatance retour la plus grande pente
cartographie algorithm.The de sable dense a été modélisé en utilisant la règle d'écoulement non associée. Le flux non associé a été
atteint en utilisant stress dilatance Row équation reliant l'angle de frottement interne à l'angle de dilatance par les constantes du
matériau. Le comportement visqueux est introduit à travers la constitution de trois modèle de composants dans les équations de base
constitutifs. La qualité de la simulation est évaluée et limitations sont discutées.
KEYWORDS: multi-surface, cyclic model, elasto-visco-plasticicty, cohesionless, dilatancy.
1 INTRODUCTION
Mathematical modelling of material behavior was of great
interest during the last century due to the progress of digital
computer. As a result, several constitutive models were
proposed by the researchers to simulate the response of
materials under monotonic loading as well as cyclic loading
under the general framework of elasto-visco-plasticity.
Materials are often subjected to transient and cyclic stresses due
to earthquake or other source of dynamic load. A number of
models have been introduced by many researchers to simulate
this cyclic stress-strain behaviour of materials, such as Prager
(1956), Mróz (1967), Hossain
et al.
(2007), Hossain, Siddiquee
and Tatsuoka (2005) etc. These cyclic response calculations of
the material are modeled through kinematic hardening, isotropic
hardening and/or a combination of both. But, most of these
cyclic models are unable to reproduce the memory effect of
materials, which eventually produces a closed or near-closed
hysteretic stress-strain loop. In this paper, multi-surface model
has been put forward with the three component model to
accommodate the viscous property. The elasto-plastic
incremental equations aof multi-surface model are integrated
here by return mapping algorithm. The family of multi-surface
models proposed by Mroz (1967), Prevost (1975), Kohey and
Jamali (1999) and others possess the inherent ability to follow
the Masing’s law. But most of these models were proposed for
pressure independent materials. A pressure dependent one-
dimensional formulation was presented in Tatsuoka
el at
.,
(2003). In this paper, the cyclic behavior in 3D stress space is
simulated by introducing a new framework in which the
dimensionless kinematic hardening rate is varied according to
the instantaneous stress value at that point along the stress path.
When the direction of the loading is reversed, the initial rate of
hardening is restored and the rate of variation of hardening is
scaled according to modified Masing’s law. As a result a closed
hysteretic stress-strain loop is obtained due to cyclic loading.
2 CONSTITUTIVE EQUATIONS
2.1. Small Strain Theory
The kinematic hardening rule evolves with the accumulative
plastic strain. Since the present model has been developed
within the small strain range, the total strain increment can be
divided into its elastic and plastic parts as follows:
(1)
p
ij de
ij d ij d
where,
represents the elastic components of strain defined
by hooks law, represents the incremental plastic strain. As
the material is pressure sensitive, Hooks law can be envisaged
in the form of Bulk’s modulus, K and shear modulus G. In this
paper, both the modulus is variable and depends on the mean
pressure. The plastic strain components are determined by the
flow rule and consistency condition.
e
ij d
p
ij
d
