Ibrahim Goda

A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization

Cosserat models of cancellous bone are constructed, relying on micromechanical approaches in order to investigate microstructure-related scale effects on the macroscopic properties of bone. The derivation of the effective mechanical properties of cancellous bone considered as a cellular solid modeled as two-dimensional lattices of articulated beams is presently investigated. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. The asymptotic homogenization technique is involved to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations. Considering lattice microrotations as additional degrees of freedom at both the microscopic and macroscopic scales, an anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. The effective elastic moduli of various periodic cell structures are computed in situations of low and high effective densities to assess the impact of the transverse shear deformation. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure.

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.

Identification of couple-stress moduli of vertebral trabecular bone based on the 3D internal architectures.

The purpose of this paper is to develop a homogeneous, orthotropic couple-stress continuum model as a substitute of the 3D periodic heterogeneous cellular solid model of vertebral trabecular bone. Vertebral trabecular bone is modeled as a porous material with an idealized periodic structure made of 3D open cubic cells, which is effectively orthotropic. The chosen architecture is based on studies of samples taken from the central part of vertebral bodies. The effective properties are obtained based on the response of the representative volume element under prescribed boundary conditions. Mixed boundary conditions comprising both traction and displacement boundary conditions are applied on the structure boundaries. In this contribution, the effective mechanical constants of the effective couple-stress continuum are deduced by an equivalent strain energy method. The characteristic lengths for bending and torsion are identified from the resulting homogenized orthotropic moduli. We conduct this study computationally using a finite element approach. Vertebral trabecular bone is modeled either as a cellular solid or as a two-phase material consisting of bone tissue (stiff phase) forming a trabecular network, and a surrounding soft tissue referring to the bone marrow present in the pores. Both the bone tissue forming the network and the pores are assumed to be homogeneous linear elastic, and isotropic media. The scale effects on the predicted couple stress moduli of these networks are investigated by varying the size of the bone specimens over which the boundary conditions are applied. The analysis using mixed boundary conditions gives results that are independent of unit cell size when computing the first couple stress tensor, while it is dependent on the cell size as to the second couple stress tensor moduli. This study provides overall guidance on how the size of the trabecular specimen influence couple stresses elastic moduli of cellular materials, with focus on bones. The developed approach is quite general and applicable to any heterogeneous cellular and composite materials.

Limit Analysis of Lattices Based on the Asymptotic Homogenization Method and Prediction of Size Effects in Bone Plastic Collapse

Lattice structures possess a huge potential for energy absorbing applications, thus it is important to develop predictive tools for their mechanical response up to collapse. Yielding is generally premonitory of structural collapse for lattice structures, so a comprehensive and quantitative understanding of lattice yielding behavior is indispensable in engineering applications. In the present work, the overall plastic yield and brittle failure behaviors of three-dimensional lattices is investigated by a microstructural modeling approach based on the homogenization of the initially discrete microstructure. The multiaxial yield behavior of the lattice is analyzed to formulate a multiaxial plastic yield criterion. Furthermore, the brittle fracture of the lattice is modeled under triaxial stress states to construct the failure surfaces, defined in the tension–tension quadrant. In plastic yielding, the analyses are performed assuming an elastic perfectly plastic lattice, and a micromechanical model based on an homogenization scheme is applied to a representative unit cell to determine the macroscopic plastic yield surfaces in stress space. This general framework is applied to evaluate the yield and failure properties of trabecular bone, which are of key interest in understanding and predicting the fracture of bones and bone implant systems. The effective strength of trabecular bone is evaluated in the two situations of fully brittle (fracture with no tissue ductility) and fully ductile failure (yield with no tissue fracture) of the trabecular tissue. At high bone volume fraction, the real strut-level ductility is sufficiently high to effectively be fully ductile but at very low bone volume fraction, the real behavior of bone may fail in a brittle mode. An adaptation and extension of the discrete homogenization method towards a micropolar effective medium is introduced in order to construct the plastic yield surfaces for which the material point of the effective continuum supports couple stresses in addition to Cauchy-type stresses. The size effects in the ductile fracture mode are addressed by considering a micropolar behavior, reflecting the influence of additional degrees of freedom and internal bending length effects on the initiation of plasticity. It is observed that when the characteristic size of the microscale structure is comparable to the bending length, a significant difference is shown between the results based on the non-classical theory and those obtained by the classical theory.

Cosserat 3D anisotropic models of trabecular bone from the homogenisation of the trabecular structure

Cellular-solid models are commonly used as simplified – although realistic enough – microscopic models of trabecular bone. A repetitive representative cell is chosen to characterise the main structural features of the trabecular microstructure; a mechanical analysis performed at this cell level is assumed to reflect the ‘effective’ properties of trabecular bone viewed as a continuum material (Rajan 1985; Keaveny 1997). Models developed along this methodology have facilitated a certain basic understanding of the governingmicro-mechanics of bonemechanics, such as differentiation between axial and bending deformations. Simulation of bone remodelling at the trabecular level also requires the knowledge of the relationship between the overall properties of bone and its microstructural parameters. The objective of this work was then to derive the effective mechanical properties of trabecular bone from a micromechanical approach. The novelty advocated lies in the construction of a micropolar constitutive model accounting for the internal length effects due to the cellular microstructure. Trabecular bone is presently modelled as a cellular material with an idealised periodic structure made of open 3D hexagonal cells (Figure 1(a)), which is effectively orthotropic. The cell walls, representing bone tissue, are modelled as linear elastic structural beam elements of uniform cross section; this idealisation of hexagonal cellular solids is supported by microimages of vertebral trabecular bones (Figure 1(b)). The cell walls are treated as Timoshenko thick beams; axial stretching, transverse shearing, flexion and torsion are considered as deformation mechanisms. Periodic 3D anisotropic hexagonal cells are presently considered as the structuralmodel of such typical cellular solids, in linewith Kim and Al-Hassani (2002), in which the authors consider a uniform circular cross section. The effective (homogenised) behaviour of trabecular bone is obtained through a specific discrete homogenisation procedure based on asymptotic expansions (Dos Reis and Ganghoffer 2012). The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus themicropolarmaterial constants. Finite element simulations of the response of bone samples under combined loadings (tension, flexion and torsion) are next performed, relying on the developed homogenised continuum model.

Cosserat Anisotropic Models of Trabecular Bone from the Homogenization of the Trabecular Structure: 2D and 3D Frameworks

Cosserat models of trabecular bone are constructed in 2D and 3D situations, based on micromechanical approaches to investigate microstructure-related scale effects on the macroscopic properties of bone. The effective mechanical properties of cancellous bones considered as cellular solids are obtained thanks to the discrete homogenization technique. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. An anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. Closed form expressions of the equivalent properties are obtained versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations; torsion is additionally considered for a 3D geometry. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the micropolar material constants. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.

Micropolar Models of Architectured Materials from Discrete Homogenization: Case of Textile Monolayers

A general methodology for the determination of the effective behavior of architectured materials endowed with a discrete structure at the microscopic scale is presented. It relies on the discrete homogenization method to derive the effective mechanical properties from the topology of the material and the properties of the micro constituents. This methodology is presently applied to textile monolayers modelled as a network of undulated and interwoven beams representing the yarns in the fabric. The initially discrete fabric architecture is represented at the mesoscopic level by an anisotropic micropolar effective continuum. The effective mechanical properties are validated by FE simulations performed over a representative unit cell.

Micropolar Models of Trabecular Bone

Abstract: This chapter deals with the construction of Cosserat models of trabecular bone in 2D and 3D situations, based on micromechanical approaches to investigating microstructure-related scale effects on the macroscopic properties of bone. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. Closed form expressions of the equivalent properties are obtained versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations; torsion is additionally considered for a 3D geometry. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the micropolar material constants. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. Finite element models over the local architecture of the trabeculae and also over the whole lattice are performed to validate the results from homogenization. The stress distribution in a cracked 2D bone sample is also computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.

Prediction of Size Effects in Bone Brittle and Plastic Collapse

Abstract: The overall plastic yield and brittle failure behaviors of three-dimensional trabecular bone are investigated by a microstructural modeling approach based on the homogenization of the initially discrete architecture. The multiaxial yield behavior is analyzed to formulate a multiaxial plastic yield criterion. Furthermore, the brittle fracture of the lattice is modeled under triaxial stress states to construct the failure surfaces, defined in the tension–tension quadrant. In plastic yielding, the analyses are performed assuming an elastic perfectly plastic lattice, and a micromechanical model based on homogenization scheme is applied to a representative unit cell to determine the macroscopic plastic yield surfaces in stress space. This general framework is applied to evaluate the yield and failure properties of trabecular bone, which are of key interest in understanding and predicting the fracture of bones and bone implant systems. The effective strength of trabecular bone is evaluated in the two situations of fully brittle (fracture with no tissue ductility) and fully ductile failure (yield with no tissue fracture) of the trabecular tissue.

Effective Mechanical Response of Biological Membranes

Abstract: The effective elastic response of biological membranes viewed as repetitive beam networks is evaluated based on the asymptotic homogenization method. A systematic methodology is set up to predict the overall mechanical properties of biological membranes in the nonlinear regime, reflecting the impact of the geometrical and mechanical network micro-parameters on the overall response of the effective substitution continuum. A classification of biomembranes networks is made based on nodal connectivity, so we analyze in this connectivity networks, which are deemed to be representative of most biological encountered networks. The individual network filaments are modeled as undulated beams prone to entropic elasticity, and the persistence length is one important parameter entering the tensile modulus. An effective micropolar substitution continuum of the biological network is evaluated, with a kinematics including nodal displacements and rotations, reflecting the discrete network deformation modes. The ratio of the characteristic bending lengths of the effective micropolar continuum to the unit cell size is a non-dimensional parameter used to quantify the importance of micropolar effects. The peptidoglycan network may present re-entrant hexagonal configuration caused by thermal or pressure fluctuations, for which micropolar effects become important. The predictive nature of the employed homogenization technique allows identifying a strain energy density of a hyperelastic model.