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Concept# Yield surface

Summary

A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity.
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space ( \sigma_1, \sigma_2 , \sigma_3), a two- or three-dimensional space spanned by stress invariants ( I_1, J_2, J_3) or a version of the three-dimensional Haigh–Westergaard stre

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In quasi-brittle materials such as concrete, failure is preceded by highly localized deformation patterns. Localized failure is caused by strain softening, which cannot be described by classical local models, because they lack a length scale. Mathematically, the use of such models leads to ill-posed boundary value problems. If properly formulated, nonlocal models can provide a sound description of strain softening and model the complete failure process of quasi-brittle materials. A bifurcation analysis is performed for a number of nonlocal plasticity models of the gradient and integral type, and the load-displacement response up to complete failure is computed. The nonlocal models are constructed mainly as enhancements of the standard smallstrain, rate-independent, isotropic local plasticity model. Most nonlocal models respond in a physically appropriate manner at initial bifurcation, but for some models, the plastic zone widens at later stages of the loading process, and stresses remain at a spuriously high level. For the integral format of two nonlocal models selected for further development, the VermeerBrinkgreve model and the ductile damage model, the bifurcation analysis is extended to hardening and the two-dimensional, plane strain setting. For both models, the enhancement is based on a modified hardening law, which depends in a specific way on the local hardening variable and its nonlocal counterpart. The analysis determines the range of material parameters for which the solution is regular and provides an analytical expression for the initial distribution of the rate of plastic strain. The use of the Vermeer-Brinkgreve model is restricted to softening, while the ductile damage model can be used for hardening and softening. An efficient and robust iterative numerical algorithm for the stress return of integral-type nonlocal models that use the local and nonlocal hardening variable in the yield function is developed and implemented. Due to weighted averaging, the stress return is a coupled nonlinear complementarity problem. It is solved by a Jacobi-like iterative technique, derived in a rigorous manner for a special case and generalized based on a physical interpretation of the resulting equations. In view of the numerical modeling of concrete, the nonlocal enhancement is applied to plasticity models with pressure-dependent yield functions and non-associated flow rules. Simulations of laboratory tests demonstrate that the nonlocal model regularizes localization due to softening and non-associated plastic flow. Neither the load-displacement response nor the propagation of the process zone depend in a spurious manner on the computational grid. For eccentric compression of a prismatic column under plane strain conditions, the trend for the size effect on structural strength and post-peak response is correctly predicted.

Although the majority of soils are unsaturated, their mechanical behaviour is still not well defined. This dissertation presents first an experimental study of an unsaturated silt and then proposes an elasto-plastic constitutive law to model the behaviour of unsaturated soils. The first part of this work deals with the characterisation of the mechanical behaviour of unsaturated soils by means of laboratory experiments on a remoulded silt. Suction is imposed by controlling the excess air pressure in the sample. Several techniques for the measurement of volume changes of samples are developed and compared. The saturated behaviour is studied by means of imposed suction and controlled tests under triaxial and oedometric (uniaxial strain) conditions. Other test performed in Richards cells are used for defining the purely so-called "hydric" behaviour of the soil. The analysis of test results were done in the effective stress plane (σ'=σ-uw,) as well as in the net stress plane (σ*=σ-ua). The hydric tests confirmed the hysteretic nature of the wetting-drying cycles. The hydric and mechanical response changes when the suction raises above the air entry value. Isotropic tests show that the preconsolidation pressure increases with the suction in the log(p')-e plane. However, this increase is rather limited in the net stress diagrams. Deviatoric tests show an increase of stiffness and peak strength with suction at constant mean effective or net pressure. Suction appears to have an effect similar to a mechanical overconsolidation when representing the result in the effective stress plane. Softening appears post peak for the cases of strong suction or weak net mean pressure. The evolution of the critical state is interpreted in terms of effective and net stresses. In the p'-q plane, the critical state line is shown to be independent of the suction (in the studied range of stress and suction); this is not true in the p*-q plane. The yield surfaces F derived from the test results grow with suction and both p'-q and p*-q planes, and the behaviour corresponds to non-associated plasticity. An existing elasto-plastic model originally developed for saturated soils (HISS-δ1) was improved and extended to integrate suction effects. The new proposed model, called δ1-unsat, is described by means of two independent stress state variables: the effective stress σ' and the suction s. Two yield surfaces, coupled through the hardening internal variable, are used in the model; the first one is linked to the mechanical behaviour, while the second one is used for hydric loading. This new model requires five new material parameters: three of them are used for the purely hydric behaviour, one describes the increase of peak strength with the suction and the last one describes the growth of yield surface with suction. The δ1-unsat model is shown to be able to reproduce the principal features of the unsaturated soils: (a) existence on a drying path of a saturated domain with non-zero suction where plastic (irrecoverable) strains appear, (b) elastic behaviour on drying path when the suction is larger then the air-entry value, (c) wetting collapse, (d) influence of mechanical stress state on the hydric behaviour, (e) increase of preconsolidation pressure with suction, (f) increase of peak strength with suction. The quantitative reliability of the δ1-unsat model is good quantitatively. In particular the prediction of the behaviour in the domain of low strains (pre-peak) is well described. The introduction of a damage concept (DSC - disturbed state concept) in the δ1-unsat model allowed the modelling of the post-peak behaviour as well. In conclusion, the mechanical behaviour of an unsaturated silt was characterised. A constitutive elasto-plastic model integrating suction and damage effects was developed and was proved to represent the principal constitutive features of unsaturated soils.

Claudio Bacciarini, Raphaël Charvet, Etienne Combaz, Willy Dufour, Andreas Mortensen

It has been proposed in the literature, based on theoretical considerations and on finite-element calculations, that all three stress tensor invariants govern the yield surface of cellular materials. Recent experiments on 75 gm pore size aluminium replicated foams (Combaz E, Bacciarini C, Charvet R, Dufour W, Mortensen A. Multiaxial yield behaviour of AI replicated foam, submitted for publication) showed such a dependence of the yield surface in axisymmetric tests. This study explores the yield behaviour of 400 gm pore size aluminium replicated foams: experiments confirm the influence of the third invariant on the yield surface shape, together with the observations that (i) the yield surface shape does not depend on relative density and (ii) measured flow vectors conform with normality. A simple parabolic model fitting data in the previous study also captures well the present data under all tested stress states (biaxial, axisymmetric and Pi-planes in stress space). Biaxial and axisymmetric tests are also performed on 400 pm pore size polyurethane (PU) replicated foams with a similar mesostructure. Results show yield to occur at a value lower than predicted by micromechanical models for both matrix materials (aluminium and PU). This suggests that the "knock-down" factor usually observed between predicted and observed stress values probably cannot be explained by a lowered yield stress in the material making the foam. The data also suggest an influence of the matrix nature on the yield surface geometry. (C) 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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