Anisotropic Constitutive Law Research Articles

Combining the Finite Element Method with Structural Connectome-based Analysis for Modeling Neurotrauma: Connectome Neurotrauma Mechanics

This article presents the integration of brain injury biomechanics and graph theoretical analysis of neuronal connections, or connectomics, to form a neurocomputational model that captures spatiotemporal characteristics of trauma. We relate localized mechanical brain damage predicted from biofidelic finite element simulations of the human head subjected to impact with degradation in the structural connectome for a single individual. The finite element model incorporates various length scales into the full head simulations by including anisotropic constitutive laws informed by diffusion tensor imaging. Coupling between the finite element analysis and network-based tools is established through experimentally-based cellular injury thresholds for white matter regions. Once edges are degraded, graph theoretical measures are computed on the “damaged” network. For a frontal impact, the simulations predict that the temporal and occipital regions undergo the most axonal strain and strain rate at short times (less than 24 hrs), which leads to cellular death initiation, which results in damage that shows dependence on angle of impact and underlying microstructure of brain tissue. The monotonic cellular death relationships predict a spatiotemporal change of structural damage. Interestingly, at 96 hrs post-impact, computations predict no network nodes were completely disconnected from the network, despite significant damage to network edges. At early times () network measures of global and local efficiency were degraded little; however, as time increased to 96 hrs the network properties were significantly reduced. In the future, this computational framework could help inform functional networks from physics-based structural brain biomechanics to obtain not only a biomechanics-based understanding of injury, but also neurophysiological insight.

Lekhnitskii’s formalism of one-dimensional quasicrystals and its application

By generalizing the complex potential approach developed by Lekhnitskii, plane problems of one-dimensional quasicrystals are solved first by using an octet formalism for which there are four pairs of complex roots. The approach uses a representation of stresses and proceeds by integration of the expressions for deformations and application of the anisotropic constitutive law and the compatibility of displacements. To illustrate its utility, the generalized Lekhnitskii’s formalism is used to analyse the coupled phonon and phason fields in an infinite quasicrystal medium containing an elliptic rigid inclusion.

The Axisymmetric Problem of a Partially Insulated Mixed-Mode Crack Embedded in a Functionally Graded Pyro Magneto-Electro-Elastic Infinite Medium Subjected to Thermal Loading

This article describes our investigation of the influence of an axisymmetric partially insulated mixed-mode crack on the coupled response of a functionally graded magneto-electro-elastic material (FGMEEM) subjected to thermal loading. The crack is embedded at the center of an infinite medium, and the material is graded in the direction orthogonal to the crack plane and is modeled as a nonhomogeneous medium with anisotropic constitutive laws. The heat conduction equation is first solved using the Hankel transform to yield the temperature field in the medium. Using the same integral transform, the magneto-electro-elasticity equations are converted analytically into a system of four singular integral equations that are solved numerically to yield the crack-tip mode I and II stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor. The main objective of this research is to study the influence of material nonhomogeneity on the fields’ intensity factors for the purpose of gaining better understanding on the behavior of graded pyro magneto-electro-elastic materials.

An embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper focuses on the study of the influence of a mixed-mode crack on the coupled response of a functionally graded magnetoelectroelastic material (FGMEEM). The crack is embedded at the center of a 2D infinite medium subjected to magnetoelectromechanical loads. The material is graded in the direction orthogonal to the crack plane and is modeled as a nonhomogeneous medium with anisotropic constitutive laws. Using Fourier transform, the resulting plane magnetoelectroelasticity equations are converted analytically into singular integral equations which are solved numerically to yield the crack-tip mode I and II stress intensity factors, the electric displacement intensity factors and the magnetic induction intensity factors. The main objective of this paper is to study the influence of material nonhomogeneity on the fields’ intensity factors for the purpose of gaining better understanding on the behavior of graded magnetoelectroelastic materials.

Simulations and experiments of open hole tension tests – Assessment of intra-ply plasticity, damage, and localization

Laminated composites with various layups of unidirectional carbon fiber reinforced epoxy are investigated by numerical modeling and experimental testing. Open hole tension configurations are studied with special considerations of the non-linear behavior of the plies. Computational predictions are based on an anisotropic constitutive material law which includes plasticity and damage, the latter comprise hardening as well as softening. Experimental evaluations include a 3D image correlation photogrammetry system for capturing the strain fields of the sample surfaces.The open hole tension configuration represents a structural problem, in which the stress distribution is highly non-uniform. The measured and the predicted strain fields as well as the localization zones show very good agreement which indicates that the governing intra-ply mechanisms are captured well by the constitutive law.

A two-dimensional homogenized model for a pile of thin elastic sheets with frictional contact

The present paper deals with the computational simulation of a pile of thin sheets. The sheets are not laminated or glued, but they interact by frictional contact. In general, it is not possible to perform a full-scale finite element contact computation for piles containing thousands of sheets; the problem size becomes too large, and numerical solution methods suffer from severe convergence problems due to the large number of strongly coupled contact conditions. In this paper, a macroscopic material model is presented for the two-dimensional case. The pile of sheets is homogenized by introducing an effective anisotropic constitutive law, which is motivated by formulations of the theory of elasto-plasticity. This macroscopic material law models the behavior of a pile of sheets, allowing for no tensile stresses in the direction normal to the sheets and obeying Coulomb’s law of friction in the tangential contact plane. Applying this macroscopic material model, an equivalent homogeneous body can be treated using much coarser discretizations. Computational results for the problems are provided, and a comparison with simplified contact computations is done.

A simple framework for full-network hyperelasticity and anisotropic damage

A formulation of a constitutive behaviour law is proposed for hyperelastic materials, such that damage induced anisotropy can be accounted for continuously. The full-network approach with directional damage is adopted as a starting point. The full-network law with elementary strain energy density based on the inverse Langevin is chosen as a reference law which is cast into the proposed framework. This continuum formalism is then rewritten using spherical harmonics to capture damage directionality. The proposed formalism allows for an efficient (and systematic) expansion of complex non-linear anisotropic constitutive laws. A low order truncated expression of the resulting behaviour is shown to reproduce accurately the stress–strain curves of the exact behaviour laws.

Predicting sintering deformation of ceramic film constrained by rigid substrate using anisotropic constitutive law

Sintering of ceramic films on a solid substrate is an important technology for fabricating a range of products, including solid oxide fuel cells, micro-electronic PZT films and protective coatings. There is clear evidence that the constrained sintering process is anisotropic in nature. This paper presents a study of the constrained sintering deformation using an anisotropic constitutive law. The state of the material is described using the sintering strains rather than the relative density. In the limiting case of free sintering, the constitutive law reduces to a conventional isotropic constitutive law. The anisotropic constitutive law is used to calculate sintering deformation of a constrained film bonded to a rigid substrate and the compressive stress required in a sinter-forging experiment to achieve zero lateral shrinkage. The results are compared with experimental data in the literature. It is shown that the anisotropic constitutive law can capture the behaviour of the materials observed in the sintering experiments.

Implémentation éléments finis du modèle hyperélastique anisotrope HGO

Anisotropic hyperelastic constitutive laws are often used to determine strain and stress in biological soft tissues such as ligaments, tendons or arterial walls. In this paper, the implementation of the HGO model in the finite element code FER is presented. Three numerical examples are studied: homogeneous uniaxial tension test where analytical solutions are available; uniaxial tension test highlighting the anisotropic behavior (contracting and swelling of the section in two perpendicular directions); contact and impact between hand and soft biological tissues in the framework of application using a virtual mannequin generator.

Application of Numerical Modelling to Dowel-Welded Wood Joints

Numerical models using a 3-D finite element analysis method and the behaviour of dowel-welded wood joints are presented. Simulation results for step butt wood joints with two welded wood dowels under shear are analyzed and a good agreement with the experimental results is shown. Anisotropic elasto-plastic constitutive law with hardening associated with material densification, without distinction between radial and tangential properties, was used for the compressive behaviour of wood. The good coherence of the results obtained demonstrates clearly the capability of the model developed to simulate accurately the non-linear behaviour of dowel-welded wood joints to failure.

Ideal Flow in Plasticity

Ideal plastic flows constitute a class of solutions in the classical theory of plasticity based on, especially for bulk forming cases, Tresca’s yield criterion without hardening and its associated flow rule. They are defined by the condition that all material elements follow the minimum plastic work path, a condition which is believed to be advantageous for forming processes. Thus, the ideal flow theory has been proposed as the basis of procedures for the direct preliminary design of forming processes, which mainly involve plastic deformation. The aim of the present review is to provide a summary of both the theory of ideal flows and its applications. The theory includes steady and nonsteady flows, which are divided into three sections, respectively: plane-strain flows, axisymmetric flows, and three-dimensional flows. The role of the method of characteristics, including the computational aspect, is emphasized. The theory of ideal membrane flows is also included but separately because of its advanced applications based on finite element numerical codes. For membrane flows, restrictions on the constitutive behavior of materials are significantly relaxed so that more sophisticated anisotropic constitutive laws with hardening are accounted for. In applications, the ideal plastic flow theory provides not only process design guidelines for current forming processes under realistic tool constraints, but also proposes new ultimate optimum process information for futuristic processes.

Flow-induced anisotropy in polar ice and related ice-sheet flow modelling

As fibers or other crystalline materials exhibiting hexagonal symmetry, the crystal of ice can be orientated by using only one single vector, i.e. its c-axis. Such a characteristic allows to apply specific methods to deal with the properties of the polycrystalline aggregate. Among others, the fabric (texture) of the ice polycrystal can be described by an ODF, i.e. a scalar function of two angles that gives the distribution of the orientation of all the constituents (grains). This paper presents a strain-induced anisotropic flow law for polycrystalline ice and the associated equations describing the evolution of its fabric. This constitutive law is formulated at the polycrystal scale and tabulated using a micro–macro model. The fabric is defined by the second- and fourth-order orientation tensors for the c-axes, assuming the so-called “invariant-based optimal fitting closure approximation”. Both the anisotropic constitutive law and the fabric evolution equations have been implemented in a finite element code in order to solve large scale ice flow problem. As an application, the flow of an idealized ice sheet over a bumpy bed is studied.

Anisotropic constitutive laws for sintering bodies

Material microstructure and properties become anisotropic during the sintering of cofired ceramics, thin films as well as hot pressed powder compounds. The well-known isotropic constitutive laws are no longer able to describe the sintering process in these cases. Experimental evidence of material anisotropy is given in this paper for alumina thin films and hot forged samples. A new transversely isotropic formulation is developed and the constrained sintering of ceramic films and sinter-forging of powder compacts analyzed using this formulation. Finally, comments regarding the expected values of the parameters and experimental techniques to measure them are presented.

Analysis of geometrically nonlinear anisotropic membranes: theory and verification

A new anisotropic constitutive model is introduced to simulate the geometrically nonlinear dynamic behavior of general anisotropic membranes experiencing large deformations. This model relates the second Piola–Kirchhoff stress with the Green–Lagrange strain in a 3-D convected curvilinear coordinate frame. The construction of the anisotropic constitutive law from the predefined principal material coordinate system to the final convected curvilinear coordinate system using coordinate transformations is presented. The proposed theory is implemented into a finite element code and several numerical examples are given for validation. An application problem is described to demonstrate the importance of the new model in real applications.

Dislocation theory as a 3‐dimensional translation gauge theory

AbstractWe consider the static elastoplastic theory of dislocations in an elastoplastic material. We use a Yang‐Mills type Lagrangian (the teleparallel equivalent of Hilbert‐Einstein Lagrangian) and some Lagrangians with anisotropic constitutive laws. The translational part of the generalized affine connection is utilized to describe the theory of elastoplasticity in the framework of a translation gauge theory. We obtain a system of Yang‐Mills field equations which express the balance of force and moment.

A large displacement formulation for anisotropic constitutive laws

This work has been motivated by configurations designed in our laboratory for which wood samples undergo large displacements that result in more accurate measurements. From a theoretical point of view, such configurations require a formulation able to take into account the geometrical non-linearities in the case of a highly anisotropic material. This paper proposes a solution to this specific problem in the case of large displacements, where the rotations may be large but the pure strain field is assumed to remain small. We propose to show how it is possible to extend existing rate-form small displacement constitutive laws in order to use them in a large displacement formulation. The strategy adopted in order to account for the anisotropy of the material and, at the same time, to respect the principle of tensorial objectivity, is to express the constitutive law in a co-ordinate system, which rotates during the deformation. Therefore we use tensors whose eigenvectors have a Lagrangian orientation while the eigenvalues have a sense in the Eulerian configuration. In order to exhibit the possibilities of the resulting formulation, two different configurations are tested: • The curvature due to shrinkage of a board section submitted to a linear moisture content profile, • The deflection of a cantilever beam. Both examples are compared to an analytical calculation obtained using the material resistance results. The numerical calculation uses a viscoelastic orthotropic constitutive law, which is integrated using an implicit time scheme and eight node isoparametric finite elements.

The architecture of an anisotropic elastic‐plastic sea ice mechanics constitutive law

The architecture for a new large‐scale anisotropic constitutive law intended for use in a sea ice dynamics model is presented here. This architecture accounts directly for refrozen lead systems in the pack ice strength (with an anisotropic failure surface) and in the ice thickness distribution (with an oriented thickness distribution). The lower limit (5 km) of the model resolution is controlled by the fracture spacing of old, thicker ice and the maximum lead width. The upper limit of the model resolution (100 km) is controlled by curvature in the lead directions and variations in the lead width. These, in turn, are controlled by the variations in internal ice stress due to driving forces (winds and currents), which set the time resolution. This architecture features abrupt changes in the failure surface and the associated flow rule that cannot be averaged over a time step. In addition, the principal stress normal to a new lead must be zero as it opens. This model has subscale simulations that allow for the inclusion of phenomena such as ridging, rafting, buckling, and fracture on the behavior of the ice. With this new ice constitutive law, it is possible to test directly the ice failure strength, plastic flow rule, and ice thickness distribution. The data most useful for this testing come from ice stress and position buoys together with synthetic aperture radar deformation data. Some data comparisons have already been made.

Elastoplastic analysis of inclusion reinforced structures

An analytical model for assessing the global elastoplastic behaviour of inclusion-reinforced materials is presented in this contribution. It is based upon a description of the reinforced material as a two-phase composite system, namely a matrix material and the reinforcements which are assumed to behave as tensile-compressive load carrying elements. An anisotropic elastoplastic constitutive law exhibiting work-hardening is then derived in an explicit form. It involves a number of hardening parameters equal to the number of reinforcing directions. Such a model, which is readily implementable in a finite element computer code, is applied to the numerical simulation of the settlement of a shallow strip footing resting upon a soil reinforced in two symmetric directions (“micropiling technique”). The load-settlement curve predicted from using the work-hardening model is finally compared with that deduced from a previously-adopted elastic perfectly plastic schematization of the reinforced soil.

An investigation into the tensile behaviour of a schistose rock

An experimental investigation was conducted into the behaviour of a quartzitic gneiss in a direct tensile test. It was found that the stress-strain relation is quasi-linear up to failure, which occurs in a brittle way. The values of the stiffness modulus and of the apparent tensile strength are dependent on the inclination of the schistosity with respect to the loading direction. It is shown that the behaviour prior to failure can be modelled by a linear elastic anisotropic constitutive law. To describe the variation of strength with the schistosity inclination, a theory is formulated that allows the observed data to be reproduced and the inclination of the failure plane to be predicted. Predictions show an overall agreement with observed data. However, for schistosity inclination close to the loading direction, the theory overpredicts the experimental results. It is suggested that a different mechanism may be the cause of failure in such cases. Finally, for use in the solution of boundary value problems, the extension of the strength criterion to full 3-D loading conditions is formally given.