Application of the multiscale FEM to the modeling of cancellous bone

Ilić, Sandra; Hackl, Klaus; Gilbert, Robert P.

doi:10.1007/s10237-009-0161-6

Cited by 63 publications

(59 citation statements)

References 61 publications

(52 reference statements)

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“…Computing the stiffness tensor through FEM simulations is still under active research [72,34,76]. One of the most important problems faced by researchers is that the results can have a large variation for the same sample by applying different boundary conditions, homogenization schemes and methods to generate nodes for the FEM simulations.…”

Section: Solid Mechanics Approachmentioning

confidence: 99%

Techniques for Computing Fabric Tensors: A Review

Moreno

Borga

Smedby

2014

Mathematics and Visualization

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The aim of this chapter is to review different approaches that have been proposed to compute fabric tensors with emphasis on trabecular bone research. Fabric tensors aim at modeling through tensors both anisotropy and orientation of a material with respect to another one. Fabric tensors are widely used in fields such as trabecular bone research, mechanics of materials and geology. These tensors can be seen as semi-global measurements since they are computed in relatively large neighborhoods, which are assumed quasi-homogeneous. Many methods have been proposed to compute fabric tensors. We propose to classify fabric tensors into two categories: mechanics-based and morphology-based. The former computes fabric tensors from mechanical simulations, while the latter computes them by analyzing the morphology of the materials. In addition to pointing out advantages and drawbacks for each method, current trends and challenges in this field are also summarized.

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Section: Solid Mechanics Approachmentioning

confidence: 99%

Techniques for Computing Fabric Tensors: A Review

Moreno

Borga

Smedby

2014

Mathematics and Visualization

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“…Already our former works [1,2] elaborate the topic of the application of the multiscale FEM to the modeling of cancellous bone. This method belongs to the group of the homogenization methods where the simulation of the macroscopic body requires investigation of a so called representative volume element (RVE).…”

Section: Multiscale Conceptmentioning

confidence: 99%

“…In [1,2] we presented a model for a RVE (RVE I), where the solid phase consists of thin walls modeled by using shell elements (Fig. 1a) and the fluid phase is considered as a barotropic fluid simulated using 8-node cubic elements.…”

Section: Multiscale Conceptmentioning

confidence: 99%

Effective parameters of cancellous bone

Ilić

Hackl

Gilbert

2008

Proc Appl Math and Mech

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Cancellous bone is a two-component structure consisting of the bone frame and interstitial blood marrow. In the scope of this presentation, the multiscale finite element method is used for its modeling. This method results from a combination of homogenization theory and the theory of finite elements and is based on the calculation of effective material parameters by investigating representative volume elements (RVEs). For the particular kind of material considered here, a cubic two-phase RVE is assumed where the dry skeleton is modeled in different ways. Apart from the variations of the geometry, the influence of the usage of different types of finite elements is studied in this context. Note that the presence of a liquid phase requires dynamic investigation including the viscous phenomena. To this end, acoustic excitation and an analysis in the complex domain are chosen. The method permits calculation of the effective material parameters such as Young's modulus, bulk modulus and Poisson's ratio and furthermore the simulation of the behaviour of the complete bone or of its parts. Multiscale conceptAlready our former works [1,2] elaborate the topic of the application of the multiscale FEM to the modeling of cancellous bone. This method belongs to the group of the homogenization methods where the simulation of the macroscopic body requires investigation of a so called representative volume element (RVE). The connection of the scales is based on the equality of the macrowork with the volume average of the microworkwhich is known as Hill-Mandel macrohomogeneity condition. Here the overbar symbol denotes the quantities related to the macroscopic body and averaging is carried over the RVE Ω of the volume V . The notation typical for the theory of small deformations is applied: σ denotes stress and strain tensor. Moreover relation (1) is used for the derivation of possible boundary conditions for the RVE. Among them the periodic boundary conditions [3] are the most convenient for modeling of cancellous bone. In such a case the microdeformation u depends on macrostrain tensor¯ and microfluctuationsũ (2. a) which additionally have to be periodic on the periodic boundary of the RVE (2. b,c)Hereũ + andũ − are microfluctuations on the boundary parts with oppositely oriented normal vectors N + and N − . The calculated microfluctuations permit also the calculation of the microstrains (3. a,b) and furthermore of the microstresses σ whose volume average represents sought counterpart on the macroscaleσ (3. c)The numerical interpretation of the definition of the elasticity tensor (3 .d) is the final output from the microscale. Note also that the activation of viscous phenomena in the interstitial marrow requires a dynamic interrogation of the RVE and that harmonic behaviour in time is assumed for all quantities. In such a case, the load p(x, t) and the initiated displacements u(x, t) read p(x, t) = p(x)e iωt , u(x, t) = u(x)e iωtwhere p(x) and u(x) may belong to the complex domain, i is imaginary unity and ω is excitation freque...

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“…In the aspect of numerical simulation, various analytical and computational models have been proposed to predict mechanical properties of bone at these different structural scales. Sandra Ilic(2010) considers the application of multiscale finite element method to the modeling of cancellous bone as an alternative for Biot's model [4]. Hamed,Elham(2010) predict analytically the effective elastic constants of cortical bone by modeling its elastic response at these different scales [3].…”

Section: Introductionmentioning

confidence: 99%

Multiscale Modeling for Mechanical Properties of Cancellous Bone Based on the Schwarz Surface

et al. 2017

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Abstract. Microstructure and the mechanical properties of cancellous bone were modeled by multiscale finite element method in this paper. Microstructure of the cancellous bone determines its mechanical properties and a rational geometry modeling of the cancellous bone is fundamental to predict the material properties. In this paper, we generate a new parameterized microstructural unit cell model based on triple periodic minimal surfaces (TPMS) which is established by the Schwarz surface with various volume fractions. This model truly simulates the microstructure of the cancellous bone and satisfies some biological and mechanical requirements. By using the new unit cell model, the second-order two-scale (SOTS) method is applied to predict the mechanical properties of cancellous bone. The predictive the mechanical properties of numerical results are in good agreement with the experimental data reported in literature, which demonstrate that the established unit cell model is applicable and the second-order two-scale method is valid to predict the mechanical properties of cancellous bone.

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Application of the multiscale FEM to the modeling of cancellous bone

Cited by 63 publications

References 61 publications

Techniques for Computing Fabric Tensors: A Review

Techniques for Computing Fabric Tensors: A Review

Effective parameters of cancellous bone

Multiscale Modeling for Mechanical Properties of Cancellous Bone Based on the Schwarz Surface

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