The Non-Linear Field Theories of Mechanics

Truesdell, C.; Noll, Walter

doi:10.1007/978-3-662-10388-3

Cited by 866 publications

(652 citation statements)

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“…Their unimodular parts are defined as C = J −2/3 C and b = J −2/3 b. For an elaborate treatment see, e.g., Truesdell and Noll [36], Holzapfel [15] and Gurtin et al [13]. Next, we exploit the representation theorem of invariants according to Spencer [34] and Holzapfel [15], and define the three irreducible isotropic invariants…”

Section: Motion and Deformation In An Anisotropic Continuummentioning

confidence: 99%

“…This formulation typically uses the unimodular deformation gradient. However, contributions by Sansour (Eur J Mech A Solids 27: [28][29][30][31][32][33][34][35][36][37][38][39] 2007) and Helfenstein et al (Int J Solids Struct 47:2056-2061 conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples.…”

Section: Introductionmentioning

confidence: 99%

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On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

2018

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Quasi-incompressible behavior is a desired feature in several constitutive models within the finite elasticity of solids, such as rubber-like materials and some fiber-reinforced soft biological tissues. The Q1P0 finite element formulation, derived from the three-field Hu-Washizu variational principle, has hitherto been exploited along with the augmented Lagrangian method to enforce incompressibility. This formulation typically uses the unimodular deformation gradient. However, contributions by Sansour (Eur J Mech A Solids 27: [28][29][30][31][32][33][34][35][36][37][38][39] 2007) and Helfenstein et al. (Int J Solids Struct 47:2056-2061 conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples. The results corroborate the new concept, show its numerical efficiency and reveal a direct physical interpretation of the fiber stretches.

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Section: Motion and Deformation In An Anisotropic Continuummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

2018

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“…For , , T , q ,p , , , , p , and represented generically by f , we have (79) Here, g = grad , G s = grad C s (a third-order tensor), and d = grad . This type of constitutive relation represents materials devoid of memory (Truesdell and Noll, 1992), able to describe thermoelasticity and viscoelasticity of the Voigt type (Bowen, 1968). While this list of dependent variables seems daunting, it is justified by our effort to consider both solid and fluid constituents (hence the need for C s , and for the solid, and for the fluids), and the fact that the entropy inequality of Eq.…”

Section: State Variablesmentioning

confidence: 99%

On the theory of reactive mixtures for modeling biological growth

Ateshian

2007

Biomech Model Mechanobiol

183

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Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes. A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents is lacking in the biomechanics literature. This study provides the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents. The constituents are intrinsically incompressible and may carry an electrical charge. The interface jump condition on the mass flux of individual constituents is shown to define a surface growth equation, which predicts deposition or removal of material points from the solid matrix, complementing the description of volume growth described by the conservation of mass. A formu-lation is proposed for the reference configuration of a body whose material point set varies with time. State variables are defined which can account for solid matrix volume growth and remodeling. Constitutive constraints are provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochem cal potential of the fluids. Simplifications appropriate for biological tissues are also proposed, which help reduce the governing equations into a more practical format. It is shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework.

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“…For this reason, we will refer to relevant literature [19,20,21,22,23]. Thus, for the stochastic strain energy function we obtain…”

Section: A Stochastic Variation Of Ogden's Materials Modelmentioning

confidence: 99%

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

Penner¹,

Caylak²,

Mahnken³

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Abstract. In many engineering applications the material behavior, e.g. hyper-elasticity and plasticity, is described by appropriate mathematical models. However, these become uncertain due to different types of uncertainty such as variation in the manufacturing process, measurement errors and missing or incomplete information on material properties. This contribution presents a framework for nonlinear elastic stochastic material model at large deformations. As a key idea uncertainty of the material is described by parameters, which are modeled as stochastic variables.To this end, 150 specimens for three different rubber materials are experimentally investigated in tensile tests. Based on experimental results 1000 artificial data are generated by aid of an ARMA process [1]. The artificial data are used for parameter identification of an Ogden material model for rubber materials . Furthermore, statistical analysis of material parameters including their correlations is studied. The number of material parameters define the dimension of the stochastic space. Usually, the stochastic material parameters are considered as stochastically independent. However, in our work we consider the dependency including the correlation obtained from experimental data.The hyperelastic stochastic material parameters are expanded with the multivariate PCE. In this context, the stresses depend on stochastic variables. To determine the corresponding PC coefficients for non-independent stochastic material parameters we use a Cholesky decomposition. As a numerical example we consider the static problem for uniaxial tension of the rectangular plate. This structure is investigated in order to represent the experimental setup conditions.

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The Non-Linear Field Theories of Mechanics

Abstract: Library of Congress Cataloging-in-Publication Data Truesdell, C. (Clifford),1919-The non-linear field theories of mechanics I C. Truesdell, W. Noll-3rd ed./ edited by StuartS. Antman. p. em. Includes bibliographical references and index.

Cited by 866 publications

References 0 publications

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

On the theory of reactive mixtures for modeling biological growth

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

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