Constitutive Functions of Elastic Materials in Finite Growth and Deformation

Chen, Yi-Chao; Hoger, Anne

doi:10.1007/978-94-010-0728-3_12

Cited by 34 publications

(44 citation statements)

References 8 publications

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“…The result (3.5) was obtained in [21] for a growing compressible elastic material. Together, (2.5), (3.3), and (3.5) produce the growth continuity equation…”

Section: $Cncpeg Qh /Cuumentioning

confidence: 75%

“…Consequently, the compounding effect of the growth of new material was not accounted for during the growth process. The theory of [17] was further developed by Chen and Hoger [21]. Recently, Hoger [22] further extended the theory and presented an implementation in which the growth law is defined on the current configuration of the growing material; this implementation was used to model the growth of aortic tissue by Van Dyke and Hoger [23].…”

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confidence: 99%

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A Theory of Volumetric Growth for Compressible Elastic Biological Materials

Klisch

Dyke²,

Hoger

2001

Mathematics and Mechanics of Solids

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A general theory of volumetric growth for compressible elastic materials is presented. The authors derive a complete set of governing equations in the present configuration for an elastic material undergo ing a continuous growth process. In particular, they obtain two constitutive restrictions from a work-energy principle. First, the authors show that a growing elastic material behaves as a Green-elastic material. Sec ond, they obtain an expression that relates the stress power due to growth to the rate of energy change due to growth. Then, the governing equations for a small increment of growth are derived from the more general theory. The equations for the incremental growth boundary-value problem provide an intuitive description of the quantities that describe growth and are used to implement the theory. The main features of the theory are illustrated with specific examples employing two strain energy functions that have been used to model biological materials.�� : Growth, elasticity, biomechanics $!

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“…The result (3.5) was obtained in [21] for a growing compressible elastic material. Together, (2.5), (3.3), and (3.5) produce the growth continuity equation…”

Section: $Cncpeg Qh /Cuumentioning

confidence: 75%

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confidence: 99%

A Theory of Volumetric Growth for Compressible Elastic Biological Materials

Klisch

Dyke²,

Hoger

2001

Mathematics and Mechanics of Solids

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“…This framework differs in a number of important details from that adopted by previous investigators (Rodriguez et al, 1994;Chen and Hoger, 2000;Klisch et al, 2003;Humphrey and Rajagopal, 2002;Guillou and Ogden, 2006). Some of these earlier descriptions have been explicitly limited to volume growth, to ensure a one-to-one mapping between the reference and current configurations, thereby allowing a decomposition of the motion into a contribution from growth and a contribution from loading.…”

Section: Discussionmentioning

confidence: 99%

“…Since the early work of Hsu (1968), Cowin and Hegedus (1976) and Skalak et al (1982), considerable advances have been made in the field of growth and remodeling of biological tissues. Some of the recent developments address various challenges in the identification of natural reference configurations for the tissue, whether modeled as a single solid constituent (Rodriguez et al, 1994;Chen and Hoger, 2000;Guillou and Ogden, 2006), a mixture of solid constituents Rajagopal, 2002, 2003), or a mixture of solid and fluid constituents (Klisch et al, 2003).…”

Section: Reference Configurationmentioning

confidence: 99%

On the theory of reactive mixtures for modeling biological growth

Ateshian

2007

Biomech Model Mechanobiol

185

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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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“…More recent theories have instead represented volumetric growth by a tensor quantity (Skalak et al 1996(Skalak et al , 1997 and were first used to study the growth of vascular tissues (Rodriguez et al 1994;Taber and Eggers 1996;Taber 1998). In addition to our work on the development of cartilage growth mixture models, in recent years there has been much interest in the development of continuum growth models for single constituents (Chen and Hoger 2000;Epstein and Maugin 2000;DiCarlo and Quiligotti 2002;Lubarda and Hoger 2002;Kuhl and Steinmann 2003;Huang 2004;Volokh 2004;Lappa 2005;Menzel 2005), mixtures (Quiligotti 2002;Ganghoffer and Haussy 2005) and mixtures that employ a stress balance hypothesis (Humphrey and Rajagopal 2002;Preziosi and Farina 2002;Breward et al 2003;Garikipati et al 2004).…”

Section: Introductionmentioning

confidence: 99%

A nonlinear finite element model of cartilage growth

Davol

Bingham

Sah

et al. 2007

Biomech Model Mechanobiol

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The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MAT-LAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

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Constitutive Functions of Elastic Materials in Finite Growth and Deformation

Cited by 34 publications

References 8 publications

A Theory of Volumetric Growth for Compressible Elastic Biological Materials

A Theory of Volumetric Growth for Compressible Elastic Biological Materials

On the theory of reactive mixtures for modeling biological growth

A nonlinear finite element model of cartilage growth

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