Wolff’s Law of Trabecular Architecture at Remodeling Equilibrium

1986 ◽  
Vol 108 (1) ◽  
pp. 83-88 ◽  
Author(s):  
S. C. Cowin
Keyword(s):  
Stress Tensor ◽  
Bone Tissue ◽  
Constitutive Relation ◽  
Cancellous Bone ◽  
Strain Tensor ◽  
Constitutive Relationship ◽  
Trabecular Architecture ◽  
Fabric Tensor ◽  
Principal Axes ◽  
Wolff’S Law

An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff’s law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff’s law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T* H* = H* T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E* H* = H* E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.

scholarly journals MATHEMATICAL MODELING OF CANCELLOUS BONE TISSUE ADAPTATION APPLIED TO THE HUMAN MAXILLODENTAL SYSTEM

2020 ◽  
pp. 35-48
Author(s):  
Alexander Kichenko ◽  
Keyword(s):  
Mathematical Modeling ◽  
Trabecular Bone ◽  
Bone Tissue ◽  
Cancellous Bone ◽  
Strain Tensor ◽  
Kinetic Equations ◽  
Fabric Tensor ◽  
Tissue Remodelling ◽  
Wolff’S Law ◽  
Trabecular Bone Tissue

The bone tissue in different parts of the skeleton conforms to Wolff’s law: it aims to become optimal for the loading which acts on the corresponding bone; the bone is remodelling by means of osteosynthesis and resorption mechanisms. The modern problems of biomechanics demand research on the history of formation of bone structures in the course of time at both physiological and pathological loadings. Ever changing loadings of different nature have influence on development and functioning of the trabecular bone tissue. The mandible is one of the most liable to external and internal changes bones. Very often one has to deal with pathological changes caused by incorrect loading of different regions of bone tissue due to dysfunction of a dentition, a temporomandibular joint and so on. For example, the Popov-Godon’s syndrome which connects with tooth loss is accompanied by pathological remodelling of the surrounding bone tissue. Thus, the mathematical modeling of the cancellous bone tissue behavior in the human maxillodental system is one of the most topical problems of biomechanics and medicine. Trabecular bone tissue is a heterogeneous, porous, anisotropic material. Heterogeneity of spongy structure can be described by methods of quantitative stereology. At the same time, structural features of the trabecular bone can be described by means of the fabric tensor. This is possible to implement if there is both a constitutive relation which connects the stress tensor, the fabric tensor, and the strain tensor, and kinetic equations which describe the evolution of the fabric tensor and bone density. An initial boundary value problem on the trabecular bone tissue remodelling is stated. The effective numerical algorithm allowing to solve the problem is developed. This algorithm is implemented as a complex of problem-oriented programs. Verification of the model and identification of its parameters are carried out. All numerical calculations are performed using the ANSYS software. Trabecular bone tissue evolution is demonstrated on the set of model examples when the stress–strain state is changed. The results demonstrate different character of influence of changes of loading conditions on process of structure formation which follows from Wolff’s law.

Structural Adaptation of Bones

1990 ◽  
Vol 43 (5S) ◽  
pp. S126-S133 ◽  
Author(s):  
S. C. Cowin
Keyword(s):  
Mechanical Properties ◽  
Bone Tissue ◽  
Structural Adaptation ◽  
Connective Tissues ◽  
Environmental Loading ◽  
Principal Axes ◽  
Living Bone ◽  
Wolff’S Law ◽  
Axes Of Symmetry ◽  
Algebraic Formulation

Living bone tissue, like many other connective tissues, is a structural material that adapts its form and microstructure to changing environmental loading conditions. Bone tissue adapts not only its shape, but also its density and the details of its microstructure including its anisotropy. The anisotropy of bone is adapted in both its degree or strength and in the orientation of its principal axes of symmetry. These adaptive features of bone tissue are often referred to as aspects of Wolff’s law, although, strictly speaking, the term “Wolff’s law” applies only to the structural adaptation of spongy or trabecular bone. In this paper the composition, microstructure, mechanical properties and structurally adaptive features of bone are briefly reviewed. An algebraic formulation of Wolff’s law at remodeling equilibrium is described, and the nature of an evolutionary Wolff’s law is sketched.

Computational Modeling of Interaction of Dental Implant with Mandible

2012 ◽  
Vol 245 ◽  
pp. 57-62 ◽  
Author(s):  
Petr Marcián ◽  
Libor Borák ◽  
Ondřej Konečný ◽  
Petr Navrátil ◽  
Zdeněk Florian
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Computational Modeling ◽  
Computational Model ◽  
Bone Tissue ◽  
Dental Implant ◽  
Cancellous Bone ◽  
Strain Analysis ◽  
Trabecular Architecture ◽  
Trabecular Bone Tissue

This paper is focused on computational modeling of an interaction of dental implant with mandible bone. It describes creation of computational model including model of geometry, materials, loads and constraints. There is a comparative stress-strain analysis of the levels of cancellous bone model. Computations are performed with the use of finite element method. Results show differences between the model which includes trabecular architecture of cancellous bone tissue and the model with non-trabecular cancellous bone tissue. For better description of the processes in bone tissue and at the interface between bone tissue and implant, it is necessary to create the computational model on the highest possible level, i.e. with the trabecular bone tissue.

Torsion of a Viscoelastic Cylinder

2000 ◽  
Vol 67 (2) ◽  
pp. 424-427 ◽  
Author(s):  
R. C. Batra ◽  
J. H. Yu
Keyword(s):  
Stress Tensor ◽  
Constitutive Relation ◽  
Linear Relation ◽  
Constitutive Relations ◽  
Strain Tensor ◽  
Time History ◽  
Cauchy Stress ◽  
Kirchhoff Stress Tensor ◽  
History Of ◽  
Energy Dissipated

Finite torsional deformations of an incompressible viscoelastic circular cylinder are studied with its material modeled by two constitutive relations. One of these is a linear relation between the determinate part of the second Piola-Kirchhoff stress tensor and the time history of the Green-St. Venant strain tensor, and the other a linear relation between the deviatoric Cauchy stress tensor and the left Cauchy-Green tensor, its inverse, and the time history of the relative Green-St. Venant strain tensor. It is shown that the response predicted by the latter constitutive relation is in better agreement with the test data, and this constitutive relation is used to compute energy dissipated during torsional oscillations of the cylinder. [S0021-8936(00)00502-X]

An Evolutionary Wolff’s Law for Trabecular Architecture

1992 ◽  
Vol 114 (1) ◽  
pp. 129-136 ◽  
Author(s):  
S. C. Cowin ◽  
A. M. Sadegh ◽  
G. M. Luo
Keyword(s):  
Cancellous Bone ◽  
Continuum Model ◽  
Temporal Evolution ◽  
Feedback Mechanism ◽  
Bone Architecture ◽  
Rate Coefficients ◽  
Stress And Strain ◽  
Trabecular Architecture ◽  
Wolff’S Law ◽  
Anisotropic Elastic

A continuum model is proposed to describe the temporal evolution of both the density changes and the reorientation of the trabecular architecture given the applied stress state in the bone and certain material parameters of the bone. The data upon which the proposed model is to be based consist of experimentally determined remodeling rate coefficients and quantitative stereological and anisotropic elastic constant measurements of cancellous bone. The model shows that the system of differential equations governing the temporal changes in architecture is necessarily nonlinear. This nonlinearity is fundamental in that it stems from the fact that, during remodeling, the relationship between stress and strain is changing as the stress and strain variables themselves are changing. In order to preserve the remodeling property of the model, terms that are of the order strain times the changes in density and/or microstructural properties must be retained. If these terms were dropped, there would be no feedback mechanism for architectural adaptation and no adaptation of the trabecular architecture. There is, therefore, no linearized version of this model of the temporal evolution of trabecular architecture. An application of the model is illustrated by an example problem in which the temporal evolution of homogeneous trabecular architecture is predicted. A limitation of the proposed continuum model is the length scale below which it cannot be applied. The model cannot be applied in regions of cancellous bone where the trabecular bone architecture is relatively inhomogeneous or at a bone-implant interface.

LES Investigation of Boussinesq Constitutive Relation Validity in a Corner Separation Flow

2018 ◽  
Author(s):  
Jean-François Monier ◽  
Nicolas Poujol ◽  
Mathieu Laurent ◽  
Feng Gao ◽  
Jérôme Boudet ◽  
...  
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Constitutive Relation ◽  
Strain Rate Tensor ◽  
Separation Flow ◽  
Compressor Cascade ◽  
Corner Separation ◽  
Reynolds Stress Tensor ◽  
Mean Strain

The present study aims at analysing the Boussinesq constitutive relation validity in a corner separation flow of a compressor cascade. The Boussinesq constitutive relation is commonly used in Reynolds-averaged Navier-Stokes (RANS) simulations for turbomachinery design. It assumes an alignment between the Reynolds stress tensor and the zero-trace mean strain-rate tensor. An indicator that measures the alignment between these tensors is used to test the validity of this assumption in a high fidelity large-eddy simulation. Eddy-viscosities are also computed using the LES database and compared. A large-eddy simulation (LES) of a LMFA-NACA65 compressor cascade, in which a corner separation is present, is considered as reference. With LES, both the Reynolds stress tensor and the mean strain-rate tensor are known, which allows the construction of the indicator and the eddy-viscosities. Two constitutive relations are evaluated. The first one is the Boussinesq constitutive relation, while the second one is the quadratic constitutive relation (QCR), expected to render more anisotropy, thus to present a better alignment between the tensors. The Boussinesq constitutive relation is rarely valid, but the QCR tends to improve the alignment. The improvement is mainly present at the inlet, upstream of the corner separation. At the outlet, the correction is milder. The eddy-viscosity built with the LES results are of the same order of magnitude as those built as the ratio of the turbulent kinetic energy k and the turbulence specific dissipation rate ω. They also show that the main impact of the QCR is to rotate the mean strain-rate tensor in order to realign it with the Reynolds stress tensor, without dilating it.

scholarly journals Constitutive Relation for Large Deformations of Fiber-Reinforced Rubberlike Materials with Different Response in Tension and Compression

2016 ◽  
Vol 44 (1) ◽  
pp. 51-72 ◽  
Author(s):  
Qian Li ◽  
David A. Dillard ◽  
Romesh C. Batra
Keyword(s):  
Constitutive Relation ◽  
Large Deformations ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Strain Tensor ◽  
Fiber Direction ◽  
Fiber Reinforced ◽  
Tension And Compression ◽  
Different Response ◽  
Rubberlike Materials

ABSTRACT Fiber-reinforced rubberlike materials commonly used in tires undergo large deformations and exhibit different responses in tension and compression along the fiber direction. Assuming that the response of a fiber-reinforced rubberlike material can be modeled as transversely isotropic with the fiber direction as the axis of transverse isotropy, we express the stored energy function in terms of the five invariants of the right Cauchy-Green strain tensor and account for different response in tension and compression along the fiber direction. The constitutive relation accounts for both material and geometric nonlinearities and incorporates effects of the fifth strain invariant, I5. It has been shown by Merodio and Ogden that in shear dominated deformations, I5 makes a significant contribution to the stress-strain curve. We have implemented the proposed constitutive relation in the commercial software, LS-DYNA. The numerical solutions of a few boundary value problems studied here agree with their analytical solutions derived by using Ericksen's inverse approach, in which part of the solution is assumed and unknowns in the presumed solution are found by analyzing the pertinent boundary value problem. However, computed results have not been compared with experimental findings. When test data become available, one can modify the form of the strain energy density and replace the proposed constitutive relation by the new one in LS-DYNA.

scholarly journals On continuum damage mechanics

2019 ◽  
Vol 52 (3) ◽  
pp. 125-147
Author(s):  
Kari Juhani Santaoja
Keyword(s):  
Stress Tensor ◽  
Effective Stress ◽  
Elastic Strain ◽  
Damage Mechanics ◽  
Volume Fraction ◽  
Strain Tensor ◽  
Continuum Damage Mechanics ◽  
Continuum Damage ◽  
The Matrix ◽  
Elastic Strain Tensor

A material containing spherical microvoids with a Hookean matrix response was shown to take the appearance usually applied in continuum damage mechanics. However, the commonly used variable damage D was replaced with the void volume fraction f , which has a clear physical meaning, and the elastic strain tensor \Bold {ε}^e with the damage-elastic strain tensor \Bold {ε}^{de}. The postulate of strain equivalence with the effective stress concept was reformulated and applied to a case where the response of the matrix obeys Hooke’s law. In contrast to many other studies, in the derived relation between the effective stress tensor \Bold {\Tilde{σ}} and the stress tensor \Bold {σ}, the tensor \Bold {\Tilde{σ}} is symmetric. A uniaxial bar model was introduce for clarifying the derived results. Other candidates for damage were demonstrated by studying the effect of carbide coarsening on creep rate.

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