On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

Chen, Mingxiang; Qi, Peng; Huang, Junjie

doi:10.1002/nme.4707

Cited by 5 publications

(21 citation statements)

References 30 publications

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“…Because of material isotropy, those response functions can be represented in terms of three isotropic invariants of their tensor arguments. Chen et al have shown that a convenient set of invariants is given by Haigh–Westergaard coordinates.…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

“…Haigh–Westergaard coordinates of ϵ are defined by

ξ_{bold-italicϵ} = \frac{I_{1, bold-italicϵ}}{\sqrt{3}}, ρ_{bold-italicϵ} = \sqrt{2 J_{2, bold-italicϵ}}, θ_{bold-italicϵ} = \frac{1}{3} \arccos ((), \frac{3 \sqrt{3}}{2} \frac{J_{3, bold-italicϵ}}{J_{2, bold-italicϵ}^{3 / 2}}),

where { I 1, ϵ , J 2, ϵ , J 3, ϵ } are the three usual isotropic invariants of ϵ :

I_{1, bold-italicϵ} = tr bold-italicϵ, J_{2, bold-italicϵ} = \frac{1}{2} tr {bold-italice}^{2}, J_{3, bold-italicϵ} = \frac{1}{3} tr {bold-italice}^{3} .

In particular, ξ ϵ is the trace of ϵ divided by

\sqrt{3}

, ρ ϵ is the length of the deviatoric part of ϵ , and θ ϵ is the Lode angle. The trihedron constituted by the derivatives of the Haigh–Westergaard coordinates h ϵ ={ ξ ϵ , ρ ϵ , θ ϵ } with respect to the tensor ϵ (for instance, ), is then introduced:

{bold-italicf}_{1, bold-italicϵ} = \frac{\partial ξ_{bold-italicϵ}}{\partial bold-italicϵ}, {bold-italicf}_{2, bold-italicϵ} = \frac{\partial ρ_{bold-italicϵ}}{\partial bold-italicϵ}, {bold-italicf}_{3, bold-italicϵ} = ρ_{bold-italicϵ} \frac{\partial θ_{bold-italicϵ}}{\partial bold-italicϵ} .

In Appendix A, classical form...…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

“…Analogous formulas hold for the invariants of the stress σ and of the increment of plastic strain Δ ϵ p . For later use, transformation rules of trihedra associated to generic coaxial tensors ϵ and η are recalled (Appendix A):

\begin{matrix} f_{1, η} & = f_{1, ϵ}, \\ f_{2, η} & = \cos (θ_{η} - θ_{ϵ}) f_{2, ϵ} + \sin (θ_{η} - θ_{ϵ}) f_{3, ϵ}, \\ f_{3, η} & = - \sin (θ_{η} - θ_{ϵ}) f_{2, ϵ} + \cos (θ_{η} - θ_{ϵ}) f_{3, ϵ} . \end{matrix}

…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

“…It is defined by

{double-struckC}^{ep} = \frac{{normald}^{2} scriptE}{normald {((), {bold-italicϵ}^{e,tr})}^{2}} = \frac{normald bold-italicσ}{normald {bold-italicϵ}^{e,tr}},

where all quantities are evaluated at time step t n + 1 . Rosati and Valoroso and Chen and co‐workers have shown that such tangent operator can be decomposed into the sum of two linear mappings:

{double-struckC}^{ep} = {double-struckC}^{ep, 1} + {double-struckC}^{ep, 2},

respectively, arising from the variation of the Haigh–Westergaard coordinates of ϵ e,tr keeping fixed its principal directions and from the variation of the principal directions of ϵ e,tr keeping fixed its Haigh–Westergaard coordinates. In the following, simple representation formulas, requiring no matrix inversion, are presented for both contributions.…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

“…Rosati and co‐workers have shown that such computation can be performed in a coordinate‐free setting and reduces to the inversion of a matrix of dimension three . Exploiting the framework of Haigh–Westergaard coordinates, this derivative is here performed explicitly and yields expressions , , and , similar to the ones reported in .…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

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State update algorithm for isotropic elastoplasticity by incremental energy minimization

Nodargi

Bisegna

2015

Numerical Meth Engineering

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Summary An original state update algorithm for the numerical integration of rate independent small strain elastoplastic constitutive models, treating in a unified manner a wide class of yield functions depending on all three stress invariants, is proposed. The algorithm is based on an incremental energy minimization approach, in the framework of generalized standard materials with convex free‐energy and dissipation potential. Under the assumption of isotropic material behavior, implying coaxiality of trial stress, increment of plastic strain, and updated stress, the problem is reduced from dimension six to three. Then, exploiting the cylindrical tensor basis associated with Haigh–Westergaard coordinates, the problem is recast in terms of two nested scalar equations. The proposed algorithm (i) exhibits global convergence even for yield functions with difficult features, such as not being defined on the whole stress space, or implying high‐curvature points of the yield domain, and (ii) requires no matrix inversion. After the tensor reconstruction of the unknowns, a simple expression for the algorithmic consistent material tangent is derived. The algorithm is validated by comparison with benchmark semi‐analytic solutions. Numerical results on single material points and finite element simulations are reported for assessing its accuracy, robustness, and efficiency. A Matlab implementation is provided as supplementary material. Copyright © 2015 John Wiley & Sons, Ltd.

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Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

“…Haigh–Westergaard coordinates of ϵ are defined by

ξ_{bold-italicϵ} = \frac{I_{1, bold-italicϵ}}{\sqrt{3}}, ρ_{bold-italicϵ} = \sqrt{2 J_{2, bold-italicϵ}}, θ_{bold-italicϵ} = \frac{1}{3} \arccos ((), \frac{3 \sqrt{3}}{2} \frac{J_{3, bold-italicϵ}}{J_{2, bold-italicϵ}^{3 / 2}}),

where { I 1, ϵ , J 2, ϵ , J 3, ϵ } are the three usual isotropic invariants of ϵ :

I_{1, bold-italicϵ} = tr bold-italicϵ, J_{2, bold-italicϵ} = \frac{1}{2} tr {bold-italice}^{2}, J_{3, bold-italicϵ} = \frac{1}{3} tr {bold-italice}^{3} .

In particular, ξ ϵ is the trace of ϵ divided by

\sqrt{3}

{bold-italicf}_{1, bold-italicϵ} = \frac{\partial ξ_{bold-italicϵ}}{\partial bold-italicϵ}, {bold-italicf}_{2, bold-italicϵ} = \frac{\partial ρ_{bold-italicϵ}}{\partial bold-italicϵ}, {bold-italicf}_{3, bold-italicϵ} = ρ_{bold-italicϵ} \frac{\partial θ_{bold-italicϵ}}{\partial bold-italicϵ} .

In Appendix A, classical form...…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

\begin{matrix} f_{1, η} & = f_{1, ϵ}, \\ f_{2, η} & = \cos (θ_{η} - θ_{ϵ}) f_{2, ϵ} + \sin (θ_{η} - θ_{ϵ}) f_{3, ϵ}, \\ f_{3, η} & = - \sin (θ_{η} - θ_{ϵ}) f_{2, ϵ} + \cos (θ_{η} - θ_{ϵ}) f_{3, ϵ} . \end{matrix}

…”

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

“…It is defined by

{double-struckC}^{ep} = \frac{{normald}^{2} scriptE}{normald {((), {bold-italicϵ}^{e,tr})}^{2}} = \frac{normald bold-italicσ}{normald {bold-italicϵ}^{e,tr}},

where all quantities are evaluated at time step t n + 1 . Rosati and Valoroso and Chen and co‐workers have shown that such tangent operator can be decomposed into the sum of two linear mappings:

{double-struckC}^{ep} = {double-struckC}^{ep, 1} + {double-struckC}^{ep, 2},

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

Section: Formulation In Haigh–westergaard Coordinatesmentioning

confidence: 99%

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State update algorithm for isotropic elastoplasticity by incremental energy minimization

Nodargi

Bisegna

2015

Numerical Meth Engineering

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An improved return‐map stress update algorithm for finite deformation analysis of general isotropic elastoplastic geomaterials

Huang

Chen

2013

Num Anal Meth Geomechanics

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SUMMARYThis paper develops a novel return mapping algorithm for the numerical integration of general isotropic finite strain elastoplastic constitutive models for geomaterials. The constitutive formulation is founded on multiplicative decomposition of the deformation gradient. The logarithmic strain measure as well as the exponential approximation of the plastic flow rule is utilized to restore the standard infinitesimal format return mapping algorithm. Central to the algorithm is the exploitation of a set of three mutually orthogonal unit base tensors for the representation of constitutive relations and the corresponding integration of the rate form of the constitutive equations. The base tensors constitute a local cylindrical coordinate system in the principal space, which allows to formulate the return mapping algorithm in the three-dimensional space and reduce the dimension of the problem to be analyzed from six down to three. With the proposed approach, direct determination of the principal axes and the transformation procedure between the general space and the principal space, as required in traditional spectral decomposition, are avoided. Furthermore, the matrices that are involved in the inversion evaluation take simple forms, leading to extremely easy inverse computation. As a result, the consistent tangent operator can be streamlined into a form simpler and more compact than those by conventional integration methods. Following the formulation of the integration procedure, a numerical experiment is performed to assess the accuracy and efficiency of the proposed algorithm.

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A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function

Meng

Chen

2022

Acta Mech. Solida Sin.

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On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

Cited by 5 publications

References 30 publications

State update algorithm for isotropic elastoplasticity by incremental energy minimization

State update algorithm for isotropic elastoplasticity by incremental energy minimization

An improved return‐map stress update algorithm for finite deformation analysis of general isotropic elastoplastic geomaterials

A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function

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