Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties

Ganapathysubramanian, Shankar; Zabaras, Nicholas

doi:10.1016/j.cma.2004.04.004

Cited by 68 publications

(39 citation statements)

References 18 publications

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“…One should also note that although each of these subspaces belongs to exactly one element of LatE 4 3 / ∼, their tensors might belong to several classes simultaneously. Let K (s) be a class of elastic tensors with K (s) ⊂ E 4 3 as a representative subspace.…”

Section: Elasticity Class Identificationmentioning

confidence: 99%

“…It seems reasonable to associate any T ∈ E 4 3 with K ∈ LatE 4 3 / ∼, if K contains a subspace, to which T belongs. In this regard, a union of all congruent subspaces contained in the element K ∈ LatE 4 3 / ∼ is considered to mean the class of 4-rank tensors, K = ∪K, or, in the related context, the elasticity class. Given some representative subspace, K ∈ K, the following definition can be written:…”

Section: Elasticity Class Identificationmentioning

confidence: 99%

“…First, define the congruence relation, «∼», between two linear subspaces, T, S ⊂ E 4 3 , in such a way that T ∼ S holds if and only if there exists a tensor O ∈ O such that T = O * S. As it is readily seen, this is an equivalence relation, so that it can be used to form the quotient set, LatE 4 3 / ∼, from the set, LatE 4 3 , of all subspaces of E 4 3 . The elements of LatE 4 3 / ∼ are the classes of congruent subspaces.…”

Section: Elasticity Class Identificationmentioning

confidence: 99%

“…Control of the fragmentation mechanisms and rotational modes of inelastic deformation offers considerable scope for producing metals and alloys with a grain structure exhibiting the best properties for the preset operating conditions. Different issues related to the problems of designing functionally graded materials are discussed, e.g., in [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. General methodology of designing such materials based on multi-level modeling [15][16][17][18][19] is described in detail in [8].…”

Section: Introductionmentioning

confidence: 99%

“…An alternative way to represent ODF in a reduced form is considered in [1,4,5,[11][12][13][14]23]. To control the texture-dependent parameters, a finite element representation of the ODF conservation equation [3,24] within a fundamental region of the Rodrigues space [24][25][26] is used.…”

Section: Introductionmentioning

confidence: 99%

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On Elastic Symmetry Identification for Polycrystalline Materials

Трусов¹,

Ostapovich²

2017

Symmetry

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Abstract:The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.

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Section: Elasticity Class Identificationmentioning

confidence: 99%

Section: Elasticity Class Identificationmentioning

confidence: 99%

Section: Elasticity Class Identificationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Elastic Symmetry Identification for Polycrystalline Materials

Трусов¹,

Ostapovich²

2017

Symmetry

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A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

Acharjee

Zabaras

2006

Int. J. Numer. Meth. Engng

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SUMMARYA methodology is introduced for rapid reduced-order solution of stochastic partial differential equations. On the random domain, a generalized polynomial chaos expansion (GPCE) is used to generate a reduced subspace. GPCE involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the Askey series. A proper orthogonal decomposition (POD) approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain. POD methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas. They have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom. This concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations (PDEs) in natural convection processes involving randomness in the porosity of the medium and the Rayleigh number. The results indicate that owing to the multiplicative nature of the concurrent model reduction, extremely large computational gains are realized without significant loss of accuracy.

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Hyper‐reduction of mechanical models involving internal variables

Ryckelynck¹

2008

Numerical Meth Engineering

168

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SUMMARYWe propose to improve the efficiency of the computation of the reduced-state variables related to a given reduced basis. This basis is supposed to be built by using the snapshot proper orthogonal decomposition (POD) model reduction method. In the framework of non-linear mechanical problems involving internal variables, the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model. This drawback is due to the fact that, using a Galerkin formulation, the size of the reduced basis has no effect on the complexity of the constitutive equations. In this paper we show how a reduced-basis approximation and a Petrov-Galerkin formulation enable to reduce the computational effort related to the internal variables. The key concept is a reduced integration domain where the integration of the constitutive equations is performed. The local computations being not made over the entire domain, we extrapolate the computed internal variable over the full domain using POD vectors related to the internal variables. This paper shows the improvement of the computational saving obtained by the hyper-reduction of the elasto-plastic model of a simple structure.

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Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties

Cited by 68 publications

References 18 publications

On Elastic Symmetry Identification for Polycrystalline Materials

On Elastic Symmetry Identification for Polycrystalline Materials

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

Hyper‐reduction of mechanical models involving internal variables

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