A Simple Derivation of Representations for Non‐Polynomial Constitutive Equations in Some Cases of Anisotropy

Boehler, J. P.

doi:10.1002/zamm.19790590403

Cited by 187 publications

(110 citation statements)

References 22 publications

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“…From this follows a formulation that is free from dependence on the choice of any special coordinate system, and does not restrict the reinforcement to any special geometrical arrangement. This approach has close connections with the theory of anisotropic tensor representations based on the use of structural tensors that was initiated by Boehler [7] and developed and extended by Zheng [8]. The fibre vector formulation has been applied to many kinds of material behaviour.…”

Section: Introductionmentioning

confidence: 99%

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Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Spencer¹,

Soldatos²

2007

International Journal of Non-Linear Mechanics

129

253

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To cite this version:A.J. M. Spencer, K.P. Soldatos. Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. International Journal of Non-Linear Mechanics, Elsevier, 2007, 42 (2) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. In memory of Ronald RivlinAbstract.In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length . To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.

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Section: Introductionmentioning

confidence: 99%

“…To take a simple example for illustration, let 7) and N is an odd integer. Thus d is a measure of the scale of the inhomogeneity of the material.…”

mentioning

confidence: 99%

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Spencer¹,

Soldatos²

2007

International Journal of Non-Linear Mechanics

129

253

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“…with the material symmetry group G and the special orthogonal group SO(3), the isotropicization theorem of anisotropic tensor functions, [2,3], is applied. However, as structural tensors we consider second-order, symmetric and positive definite metric tensors G = HH T , which can be interpreted as the push-forward of the cartesian metric of a fictitious reference configuration to the real reference configuration.…”

Section: Crystallographic Motivated Structural Tensorsmentioning

confidence: 99%

Polyconvex Models for Arbitrary Anisotropic Materials

Ebbing

Schröder

Neff

2008

Proc Appl Math and Mech

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In this work we provide polyconvex models for arbitrary anisotropic materials. The fundamental idea is the introduction of second-order, symmetric and positive definite structural tensors which are motivated by some basic crystallographic geometric relations. The deduced invariant functions automatically satisfy the polyconvexity condition and ensure the requirement of a stress free reference configuration. Restrictions coming along with the polyconvexity condition and the usage of second-order structural tensors will be pointed out. Furthermore the results of three representative fittings show the adaptability of the proposed models for the description of real anisotropic material behavior.

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“…In this contribution, we restrict ourselves to the modelling of hyper-elasticity, which requires the definition of a free energy function 0 , depending on the deformation gradient F. To satisfy the invariance under superposed rigid body motions, the dependency of 0 on the deformation gradient is commonly realized by a dependency on the right Cauchy-Green tensor C. Since the free energy of an anisotropic material depends, in general, on the orientation of the material, an exclusive dependence of 0 on C would lead to an anisotropic free energy function, i.e. for transverse isotropy the works of Lokhin and Sedov [27] and Boehler [28], we extend the tensor function isotropically by means of structural tensors. This leads to the following isotropic representation of the free energy function for transverse isotropy…”

Section: Transversely Isotropic Hyper-elasticitymentioning

confidence: 99%

Time‐dependent fibre reorientation of transversely isotropic continua—Finite element formulation and consistent linearization

Himpel

Menzel

Kuhl

et al. 2007

Numerical Meth Engineering

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SUMMARYTransverse isotropy is realized by one characteristic direction-for instance, the fibre direction in fibrereinforced materials. Commonly, the characteristic direction is assumed to be constant, but in some cases-for instance, in the constitutive description of biological tissues, liquid crystals, grain orientations within polycrystalline materials or piezoelectric materials, as well as in optimization processes-it proves reasonable to consider reorienting fibre directions. Various fields can be assumed to be the driving forces for the reorientation process, for instance, mechanical, electric or magnetic fields. In this work, we restrict ourselves to reorientation processes in hyper-elastic materials driven by principal stretches.The main contribution of this paper is the algorithmic implementation of the reorientation process into a finite element framework. Therefore, an implicit exponential update of the characteristic direction is applied by using the Rodriguez formula to express the exponential term. The non-linear equations on the local and on the global level are solved by means of the Newton-Raphson scheme. Accordingly, the local update of the characteristic direction and the global update of the deformation field are consistently linearized, yielding the corresponding tangent moduli. Through implementation into a finite element code and some representative numerical simulations, the fundamental characteristics of the model are illustrated.

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A Simple Derivation of Representations for Non‐Polynomial Constitutive Equations in Some Cases of Anisotropy

Cited by 187 publications

References 22 publications

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Polyconvex Models for Arbitrary Anisotropic Materials

Time‐dependent fibre reorientation of transversely isotropic continua—Finite element formulation and consistent linearization

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