Cosserat point element (CPE) for finite deformation of orthotropic elastic materials

Jabareen, Mahmood; Sharipova, Leah L.; Rubin, M. B.

doi:10.1007/s00466-011-0654-x

Cited by 8 publications

(9 citation statements)

References 36 publications

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“…Using the notation in , the element reference directors D i and the present directors d i are related to the nodal directors by the expressions

{boldD}_{i} = \sum_{j = 0}^{7} A_{ij} {\overset{boldD}{false}}_{j}, {boldd}_{i} = \sum_{j = 0}^{7} A_{ij} {\overset{boldd}{false}}_{j}, (i = 0, \dots, 7),

where A i j is a constant matrix , which is recorded in Appendix A for the sake of completeness. It should be emphasized that the element shape functions and the standard nodal shape functions

{\overset{N}{false}}^{i} (i = 0, \dots, 7)

, which possess the Kronecker property, are related by

{\overset{N}{false}}^{i} = \sum_{j = 0}^{7} A_{ji} N^{j}

.…”

Section: Basic Equations Of a 3d Brick Cosserat Point Elementmentioning

confidence: 99%

“…Also, the strain energy function that controls the inhomogeneous deformations has to be restricted for satisfaction of the patch test, which ensures that the CPE can exactly reproduce all three dimensionally homogeneous deformations of an arbitrary reference element geometry. Specifically, it can be shown that when the strain energy function Ψ has the following quadratic form,

m normalΨ = \frac{V}{2} \sum_{i = 1}^{12} \sum_{j = 1}^{12} B_{ij} b_{i} b_{j}, B_{ij} = B_{ji} = B_{ij} ((), H_{k}, D_{kl}, {\overset{K}{true}}_{klmn}^{*}, {boldM}_{k}),

the CPE automatically satisfies the patch test . The constant coefficients B i j = B j i in control the accuracy of the CPE for bending and torsion of thin structures.…”

Section: Basic Equations Of a 3d Brick Cosserat Point Elementmentioning

confidence: 99%

“…The stability for shell elements with enhanced strain formulation was examined in by comparing different nonlinear constitutive models, and it was found that the stability of enhanced strain shell elements in large deformations radically depends on the mathematical form of the constitutive law. In addition to the hourglass instability, it was demonstrated in that some of these improved EAS element formulations in commercial codes can produce inelastic behavior, even though they attempt to model a hyperelastic material with a strain energy function.…”

Section: Introductionmentioning

confidence: 99%

“…By modifying the strain energy function that controls the inhomogeneous deformations, Jabareen and Rubin removed the previously mentioned insensitivity, and the developed CPE therein proved its accuracy and robustness in modeling isotropic elastic materials. Jabareen et al developed a CPE for modeling orthotropic materials that undergo large deformations. Specifically, the strain energy function for homogeneous deformations depends on the physically based nonlinear invariants for orthotropic solids .…”

Section: Introductionmentioning

confidence: 99%

“…As a result, the latter formulation was limited for accurately modeling structures with a specific orientation of orthotropy and undistorted elements. Mtanes and Jabareen avoid these limitations by adding coupling terms in the strain energy function of the inhomogeneous deformations, which were not included in . However, the modification in was limited to the plane strain elasticity because of its complexity.…”

Section: Introductionmentioning

confidence: 99%

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A 3D Cosserat point element (CPE) for nonlinear orthotropic solids: Generalization for an initially distorted mesh and an arbitrary orientation of material orthotropy

Jabareen

Mtanes

2015

Numerical Meth Engineering

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Summary The objective of the present paper is to develop a Cosserat point element (CPE) formulation for the numerical solution of three‐dimensional problems of general hyperelastic orthotropic materials under finite deformations with initially distorted element shapes. Generally speaking, the accuracy of the CPE depends on the constitutive coefficients of the strain energy function that controls the inhomogeneous deformations. In the present study, a new methodology for the determination of the constitutive coefficients is presented, which allows the CPE to model any elastic material including isotropic, orthotropic, and anisotropic materials and with initially distorted element geometry. A number of example problems are considered, which verify and compare the performance of the developed CPE with other 3D brick elements that exist in the commercial finite element package ABAQUS. These examples demonstrate that the developed CPE is accurate, robust, free of hourglass instabilities, and can be used for modeling both 3D and thin structures that undergo large deformations. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Using the notation in , the element reference directors D i and the present directors d i are related to the nodal directors by the expressions

{boldD}_{i} = \sum_{j = 0}^{7} A_{ij} {\overset{boldD}{false}}_{j}, {boldd}_{i} = \sum_{j = 0}^{7} A_{ij} {\overset{boldd}{false}}_{j}, (i = 0, \dots, 7),

where A i j is a constant matrix , which is recorded in Appendix A for the sake of completeness. It should be emphasized that the element shape functions and the standard nodal shape functions

{\overset{N}{false}}^{i} (i = 0, \dots, 7)

, which possess the Kronecker property, are related by

{\overset{N}{false}}^{i} = \sum_{j = 0}^{7} A_{ji} N^{j}

.…”

Section: Basic Equations Of a 3d Brick Cosserat Point Elementmentioning

confidence: 99%

m normalΨ = \frac{V}{2} \sum_{i = 1}^{12} \sum_{j = 1}^{12} B_{ij} b_{i} b_{j}, B_{ij} = B_{ji} = B_{ij} ((), H_{k}, D_{kl}, {\overset{K}{true}}_{klmn}^{*}, {boldM}_{k}),

the CPE automatically satisfies the patch test . The constant coefficients B i j = B j i in control the accuracy of the CPE for bending and torsion of thin structures.…”

Section: Basic Equations Of a 3d Brick Cosserat Point Elementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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