A Fully Intrinsic Approach to the Inversion of Fourth-Order Material Tensors

Continuum finite element (FE) models of bones and bone-implant configurations are usually based on clinical CT scans. In virtually all of these models, material properties assigned to the bone elements are chosen as isotropic. It has been shown, however, that cancellous bone can be highly anisotropic and that its elastic behavior is best described as orthotropic. Material models have been proposed to derive the orthotropic elastic constants from measurements of density and a fabric tensor. The use of such relationships in FE models derived from CT scans, however, is hampered by the fact that the measurement of such a fabric tensor is not possible from clinical CT images since the resolution of such images is not good enough to resolve the trabecular micro-architecture. In this study, we explore an alternative approach that is based on the paradigm that bone adapts its micro-architecture to the loading conditions, hence that fabric and stress tensors should be aligned and correlated. With this approach, the eigenvectors and eigenvalues of the element continuum-level stress tensor are used as an estimate of the element fabric tensor, from which the orthotropic material properties then are derived. Using an iterative procedure, element orthotropic material properties and fabric tensors are updated until a converged situation is reached. The goals of this study were to investigate the feasibility and accuracy of such an iterative approach to derive orthotropic material properties for a human proximal femur and to investigate whether models derived in this way can provide more accurate results than isotropic models. Results were compared to those obtained from models of the same femurs for which the fabric was measured from micro-CT scans. It was found that the iterative approach could well estimate the orientation of the fabric principal directions. When comparing the stress/damage values in the models with material properties based on estimated and measured fabric tensors, the differences were not significant, suggesting that the material properties based on the estimated fabric tensor well reflected those based on the measured fabric tensor. Errors were less than those obtained when using isotropic models. It is concluded that this novel approach can provide a reasonable estimate of anisotropic material properties of cancellous bone. We expect that this approach can lead to more accurate results in particular for models used to study implants, which are usually anchored in highly anisotropic cancellous bone regions.

Fabric and compliance tensors of a cube of cancellous bone with a complicated three-dimensional trabecular structure were obtained for trabecular surface remodeling by using a digital image-based model combined with a large-scale finite element method. Using mean intercept length and a homogenization method, the fabric and compliance tensors were determined for the trabecular structure obtained in the computer remodeling simulation. The tensorial quantities obtained indicated that anisotropic structural changes occur in cancellous bone adapting to the compressive loading condition. There were good correlations between the fabric tensor, bone volume fraction, and compliance tensor in the remodeling process. The result demonstrates that changes in the structural and mechanical properties of cancellous bone are essentially anisotropic and should be expressed by tensorial quantities.

ANN-aided evaluation of dual-phase microstructural fabric tensors for continuum plasticity representation

Relationship between void- and contact normal-based fabric tensors for 2D idealized granular materials

The major objective of this work is to relate continuum damage mechanics introduced through the concept of fabric tensors to composite materials within the framework of classical elasticity theory. A model of directional data-damage mechanics for composite materials is formulated using fabric tensors. In addition, a general hypothesis for damage mechanics is postulated. It is seen that the two available hypotheses of elastic strain equivalence and elastic energy equivalence may be obtained as special cases of the postulated general hypothesis. This general hypothesis is then used to derive the sought relationship between the damage tensor for composite materials and the fabric tensors. Two approaches to link the fabric tensors damage effect to the behavior of composite materials are adopted. The first approach is the continuum approach, which introduces damage with fabric tensors to the composite media; where the latter is treated as a homogenized material. Properties of the constituents are homogenized before the damage with fabric tensors is introduced. The second approach is the micro-mechanical approach, where damage with fabric tensors is introduced to the constituents rather than to the homogenized material. Within the framework of classical elasticity theory, both approaches should lead to equivalent results. Thus, a comparison between the two approaches is carried out to verify their equivalency. Damage evolution for both approaches is derived in a mathematically consistent manner that is based on sound thermodynamic principles. Numerical examples and application to the theory developed herein are presented. Micro-crack distributions in different constituents of the composite material are thoroughly investigated.

Fabric tensor has proved to be an effective tool statistically characterizing directional data in a smooth and frame‐indifferent form. Directional data arising from microscopic physics and mechanics can be summed up as tensor‐valued orientation distribution functions (ODFs). Two characterizations of the tensor‐valued ODFs are proposed, using the asymmetric and symmetric fabric tensors respectively. The later proves to be nonconvergent and less accurate but still an available solution for where fabric tensors are required in full symmetry. Analytic solutions of the two types of fabric tensors characterizing centrosymmetric and anticentrosymmetric tensor‐valued ODFs are presented in terms of orthogonal irreducible decompositions in both two‐ and three‐dimensional (2D and 3D) spaces. Accuracy analysis is performed on normally distributed random ODFs to evaluate the approximation quality of the two characterizations, where fabric tensors of higher orders are employed. It is shown that the fitness is dominated by the dispersion degree of the original ODFs rather than the orders of fabric tensors. One application of tensor‐valued ODF and fabric tensor in continuum damage mechanics is presented.

To quantitatively describe the influence of the contact characteristics of granular materials on their mechanical response, the true-triaxial tests with different particle shapes are simulated by the discrete element method (DEM), and the connection between the evolutions of particle contact fabric and the anisotropic mechanical responses is studied. The contact normal vector of the particle in 3D space is described by two independent angles, by which the contact fabric tensor is defined. The amplitude parameters in three orthogonal directions are defined by the invariants of the plane fabric tensor, which are scalars and describe the degree of anisotropy of the contact fabric in each plane. The expression of orthotropic fabric tensor is derived from the amplitude parameters, with the change of geometric space of contact normal vector, which is reduced to the different tensor of transverse isotropic naturally. The fabric tensor can be directly applied to the constitutive equation to describe the effects of the particle contact on the mechanical response. For verifying the rationality of contact characteristics described by fabric tensor, four particle shapes are clumped by PFC3D. The mechanical properties of specimens with different particles are simulated under the true-triaxial loading path, and the data of contact normal vector is extracted in real time. The simulation results showed that the particle shapes have a significant effect on the 3D stress-strain relationship and strength, which showed apparent anisotropy, and the invariants of fabric tensor can be used to describe the evolution of particle contact in the loading process.

The major objective of this work is to apply continuum damage mechanics introduced through the concept of fabric tensors to composite materials within the framework of elasticity theory. A model of directional data damage mechanics for composite materials is formulated using fabric tensors. The physical meaning of damage is enhanced and understood better through the introduction of fabric tensors into the analysis of damage of composite materials. The `continuum approach' is used here to link the fabric tensors' damage effect to the behavior of composite materials. In this approach, damage with fabric tensors is introduced to the composite medium, where the latter is treated as a homogenized material. Also, the overall properties of the composite system as a whole are used without using the constituent properties. A generalized formulation of damage evolution is derived in a mathematically consistent manner that is based on sound thermodynamic principles. A numerical example is presented to show the applicability. In addition, damage evolution for the one-dimensional tension case is also illustrated.

Inversions related to the stress-strain-fabric relationship

We present and validate a general three-dimensional continuum model for predicting the coupled fabric and stress transient response in 3D dense granular flows for the quasi-static regime. The model is inspired by isotropic and kinematic hardening theory, which is widely applied to plastic loading cycles in metals, which constitutes a connection between two different flowing materials through the same plastic modeling framework. The first part of the model consists of a differential evolution equation for the fabric tensor, which incorporates a new parameter called contact persistence to model the capacity of the fabric network to keep its contacts according to the relative direction of the shear-rate. The second part of the model is an expression for the shear stress comprised of a backstress, proportional to the fabric tensor, and a term proportional to the shear-rate direction. This shear stress decomposition was obtained from DEM data extracted within a 3D Couette cell during unsteady processes wherein the shear-rate direction rotates instantaneously with respect to the axis perpendicular to the walls of the cell. The results of the model are compared with DEM simulations for different changes in shear orientation, achieving a good agreement for the evolution of the fabric and deviatoric stress tensors. The model is shown to be compatible with the second law of thermodynamics, revealing that the origin of the backstress flow resistance in granular media is distinct from the cause of backstresses in metals; rather than arising from stored defect energy, it arises from the dependence of dilatancy on the alignment of the fabric and flow-rate.

This study provides experimental evidence of the correlation between different fabric tensors in 2D granular assemblies subjected to shearing. Two testing assemblies with different elliptical rods were used and the shearing tests were conducted in a tailor-made biaxial cell. Both the rods and cell were produced using the 3D printing technique. Throughout the shearing, three fabric tensors, i.e., the particle orientation-based, the contact normal-based and the void-based fabric tensors, were studied. In particular, to obtain the void-based fabric tensor, the best-fit ellipse method was used, which can better describe the irregular void geometric information compared with the conventional methods described in the literature. In both samples, a strong linear relationship between the anisotropy intensity factors of the three fabric tensors was found. However, such a correlation was not identified when the void-based fabric tensor was quantified using the conventional methods. The discrepancy demonstrates the importance of properly quantifying the fabric tensor, and by using the best-fit ellipse method, the correlation between the fabric tensors can be better observed, which is crucial to understand the constitutive relationship in granules.

Form-function relationship between trabecular bone architecture and biomechanical function in the horse humerus

Mechanical properties of cancellous and cortical bone have been investigated. The fabric tensors used in the relevant literature have been discussed. Nonlinear elastic and elastic-perfectly plastic constitutive relationships have been proposed within the framework of small deformations. To this end the theory of representation of tensor functions has been used. It has been shown that the fabric tensor plays the role of a parametric tensor. Orthotropic linear elasticity has been carefully examined from the point of view of interrelations of classical material constants with the proposed material parameters and eigenvalues of the fabric tensor. Hoffman's strength criterion has been extended by incorporating the fabric tensor. Anisotropic properties of human cancellous and cortical bones have been investigated by using the relations derived.

Fabric tensor based boundary element analysis of porous solids

In an effort to develop a constitutive equation associated with more deformation and failure features of granular materials, the constitutive formulation is studied taking into account the fabric tensors as internal variables and their evolution modes. In order to render the constitutive equation satisfy the principle of material objectivity, the representation theorem of isotropic functions is effectively used. The theorem is reviewed with a particular reference to the rate type constitutive equation which gives rise to the rate of deformation tensor as a function of the stress, stress rate and fabric tensors. After discussing the relationship between essential independent variables in constitutive equations and the state of granular materials from a viewpoint of the evolution mode of the fabric tensors, particular problems are studied with the result that: (1) Granular materials can show less summetry (more anisotropic) than orthotropic when the general stress states and paths are concerned; (2) In two dimensions, the introduction of the (anisotropic) fabric tensor is essential in order to describe the non-coaxial and dilatant behavior under the rotation of principal stress axes.