Unified elasto-plastic associated and non-associated constitutive model and its engineering applications

doi:10.1016/S0045-7949(98)00306-X

Computers & Structures

Volume 71, Issue 6, June 1999, Pages 627-636

https://doi.org/10.1016/S0045-7949(98)00306-X Get rights and content

Abstract

A unified elasto-plastic associated and non-associated constitutive model, which is based on the unified theory, is proposed in this paper. This unified constitutive model can be easily used and has been implemented in UEPP (unified elasto-plastic finite element program). This new elasto-plastic constitutive model can be widely used in engineering.

Section snippets

Unified strength theory

Strength theory is usually called the yield criterion for metallic materials, the failure criterion for granular materials (concrete, soil and rock) or the material model for non-linear finite element analysis. It is important and fundamental for the strength calculation and design of engineering structures. Generally speaking, for any engineering materials, two basic strength properties must be studied, i.e. initial and subsequent yield properties. Initial yield solves the problem of when the

Unified elasto-plastic constitutive model

Associated and non-associated constitutive model based on the unified strength theory can be worked out. For isotropic materials, for the convenience of numerical computation, the unified strength theory is usually expressed in terms of stress (deviatoric) invariants, that is, the first invariant of stress tensor I₁, the second and third invariants of stress deviatoric tensor J₂ and J₃ and stress angle θ. The stress angle θ is defined so as to reflect the σ₂ effect $θ= 13 cos^{−1} 332 J_{3} J^{3}_{2} (0≤θ≤π/3)$ It

Determination of the unified associated elasto-plastic stiffness matrix

According to Ref. [8], here {a}=∂F/∂{σ} (or {a}=∂F′/∂{σ}) is defined as the flow vector of the unified strength theory (Eq. (5)), thus ${a}= ∂F ∂{σ} = ∂F ∂I_{1} ∂I_{1} ∂{σ} + ∂F ∂ J_{2} ∂ J_{2} ∂{σ} + ∂F ∂θ ∂θ ∂{σ} =C_{1} {a_{1}}+C_{2} {a_{2}}+C_{3} {a_{3}}$ where ${a_{1}}= ∂I_{1} ∂{σ}, {a_{2}}= ∂ J_{2} ∂{σ}, {a_{3}}= ∂J_{3} ∂{σ}$ $C_{1} = ∂F ∂I_{1} = 13 (1−α)$ $C_{2} = ∂F ∂ J_{2} + ctg 3θ J_{2} ∂F ∂θ = 1+ α 223 cos θ+ α(1−b) 1+b sin θ+ ctg 3θ α(1−b) 1+b cos θ− 1+ α 223 sin θ$ $C_{3} = − 32 J^{3}_{2} sin 3θ ∂F ∂θ = 3 2J_{2} sin 3θ 1+ α 223 sin θ− α(1−b) 1+b cos θ$ Similarly, C′₁, C′₂ and C′₃ connected with F′ of the unified strength theory can be expressed as $C′_{1} = ∂F′ ∂I_{1} = 13$

Singular points on the yield surface of the unified elasto-plastic constitutive model

The singular point is the point where the flow vector {a} is not determined uniquely. There are two kinds of singularities, as shown in Fig. 1. One is point at θ=θ_b where flow vector {a} can be determined either by F or by F′. Because of the existence of difference between the two flow vectors, the singular phenomenon occurs. The other kind of singularity is at point θ=0° and θ=60° where flow vectors {a} can be determined by F and F′, respectively. The singular phenomenon exists when substitute

The implementation of non-associated elasto-plasticity

Because plastic volume dilation calculated by associated plasticity exaggerates the realistic situation for some engineering materials such as soil and concrete, this makes the introduction of non-associated flow rule necessary, as shown in Fig. 3. A common treatment of non-associated plasticity can be found in Refs. [10], [11] by letting plastic potential have the same form as the yield (loading) function but using different frictional angles, i.e. ψ for plastic potential and ϕ for yield

Tests, applications and verification

In order to test the validity of the above discussion and make the application possible, a computer code, unified elasto-plastic finite element program (UEPP) has been worked out. Two examples are presented here.

Conclusions

This paper discusses some problems in the implementation of a new unified elasto-plastic constitutive model including associated and non-associated flow rules, solutions of singularities and the choice of material parameters. Explicit formulae are given, which makes the elasto-plastic finite element application easier. Two examples are used to compare the test, computation and theoretical results. Agreements have be achieved among these results, which, in turn, verify the discussion proposed in

Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC) (No. 59779028) to Xi’an Jiaotong University, China. The generous financial support is gratefully acknowledged.

References (12)

Mao-hong Yu
Twin shear stress yield criterion
Int J Mech Sci
(1983)
H.S. Yu
A close-form solution of stiffness matrix for Tresca and Mohr–Coulomb plasticity models
Comput Struct
(1994)
Mao-hong Yu et al.
Strength theory for rock and concrete: history, now and development
Prog Nat Sci
(1998)
Mao-hong Yu et al.
Twin shear stress theory and its generalization
Sci Sin, Ser A
(1985)
Mao-hong Yu et al.
Generalized twin shear stress yield criterion and its generalization
Chin Sci Bull
(1992)
Mao-hong Yu et al.
A new model and theory on yield and failure of materials under the complex stress state

There are more references available in the full text version of this article.

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Unified elasto-plastic associated and non-associated constitutive model and its engineering applications

Abstract

Section snippets

Unified strength theory

Unified elasto-plastic constitutive model

Determination of the unified associated elasto-plastic stiffness matrix

Singular points on the yield surface of the unified elasto-plastic constitutive model

The implementation of non-associated elasto-plasticity

Tests, applications and verification

Conclusions

Acknowledgements

Int J Mech Sci

Comput Struct

Strength theory for rock and concrete: history, now and development

Prog Nat Sci

Twin shear stress theory and its generalization

Sci Sin, Ser A

Generalized twin shear stress yield criterion and its generalization

Chin Sci Bull

A new model and theory on yield and failure of materials under the complex stress state