Motivated by applications in orthopaedic and maxillo-facial surgery, the mechanical behaviour of cortical bone tissue in cyclic overloads at physiological strain rates is investigated The emphasis is on the development of appropriate constitutive laws that faithfully reproduce the loading, unloading, and reloading sequence observed during experimental in vitro uniaxial testing To this end, the models include three distinct modes of evolution, namely a linear elastic mode due to bone cohesion, a damage mode where microcracks are generated and a plastic mode corresponding to sliding at the microcracks The proposed models use the internal state variable approach common in continuum damage mechanics and allow a straightforward interpretation of the constitutive behaviour of cortical bone They are derived within the generalized standard materials formalism and are thus thermodynamically consistent The mathematical formulation of the models is based on the definition of two internal state variables: a damage variable that represents the microcrack density reducing the tissue stiffness, and a plastic strain variable representing the deformation associated with these microcracks Firstly, two one-dimensional models describing the uniaxial quasistatic behaviour of cortical bone are developed The first one includes a single scalar damage variable, whereas the second one is based on tensile and compressive damage variables, which improves the simulation results Both models are then extended into rate-dependent alternatives by relating the rate of damage accumulation to some high power of the damage threshold stress All four models consider different tensile and compressive damage threshold stresses as it is the case for cortical bone Secondly, the material constants characterizing the one-dimensional models are identified on experimental grounds To this end, a series of in vitro uniaxial overloading tests were carried out on bovine cortical bone Reliable measurements were obtained in tension using dumbbell specimens, avoiding thus undesirable boundary effects Thirdly, a three-dimensional rate-independent constitutive law inspired by the one-dimensional models is formulated and implemented in a finite element code It includes porous fabric-based orthotropic elasticity and rate-independent plasticity with damage The onset of damage is characterized by an orthotropic stress-based damage criterion described by porosity and fabric, which takes into account distinct tensile and compressive damage threshold stresses Finally, the potential of the new three-dimensional elastic plastic damage constitutive law for cortical bone is demonstrated by means of a finite element analysis of the compression of a vertebra

Elastic plastic damage laws for cortical bone

Image-based finite element analysis (FEA) to predict and understand the biomechanical response has become an essential methodology in musculoskeletal research. An important part of such simulation models is the constitutive material model of which recent advances are summarized in this review. The review shows that existing models from other fields were introduced, such as cohesion zone (cortical bone) or phase-field models (trabecular bone). Some progress has been made in describing cortical bone involving physical mechanisms such as microcracks. Problems with validations at different length scales remain a problem. The improvement of recent constitutive models is partially obscured by uncertainties that affect overall predictions, such as image quality and calibration or boundary conditions. Nevertheless, in vivo CT-based FEA simulations based on a sophisticated constitutive behavior are a very valuable tool for clinical-related osteoporosis research.

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A Review on Recent Advances in the Constitutive Modeling of Bone Tissue

Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

Homogenization methods in bone mechanics

Summary Properties of the apparent stiffness tensor of bone are investigated. The computer reconstruction composed of computer microtomograph imaging, nite element reconstruction and numerical tests results in apparent stiffness tensor of bone. Next the stiffness tensor is subjected to spectral and harmonic decompositions. Kelvin moduli and invariants of orthogonal projectors provided by spectral analysis as well as ve invariant parts resulting from harmonic decomposition are obtained. These scalar and tensorial invariants enable the analysis of properties of the stiffness tensor such as material symmetry. Finally, the closest isotropic stiffness tensor is derived and the possible orthotropic approximations are proposed and their accuracy is studied.

/pdf/approximations-of-stiffness-tensor-of-bone-determining-and-4c4om3atp3.pdf

M. Ma

Papers

Approximations of stiffness tensor of bone - determining and accuracy