Anisotropic Constitutive Law Research Articles

Modeling of textile composite using analytical network-averaging and gradient damage approach

In this contribution, we present a gradient damage model for anisotropic textile reinforcements including fiber inextensibility and fiber sliding. In contrast to previous works, the gradient damage formulation stems not from a numerical regularization basis but from the thermodynamics of internal variables. It results in a nonlocal term as the internal energy of fiber bending with measurable nonlocal parameter. Furthermore, to guarantee a priori that rotations and reflections determined by orthogonal tensors among the symmetry group do not affect the response function of the anisotropic constitutive law, a novel mesoscopic kinematic measure for the representative volume element of the fabric is defined on the basis of the analytical network-averaging concept. Such kinematic measure is of crucial importance for material modeling of damage-elastoplasticity in anisotropic textile reinforcements, and allows for analytical descriptions of inter- and intra-ply sliding of fibers. A mixed finite element formulation is then presented for textile reinforcements taking into account fiber inextensibility. The predictive capability of the computational model is demonstrated by comparing with multiple experimental datasets of dry textile fabrics.

Numerical homogenization of fiber reinforced layer in large elastic deformation using a decoupled iterative method

We propose a procedure to approximate the large elastic deformations of a fiber reinforced layer by a two-scale decoupled homogenization numerical procedure. The nonlinear micro and macroscopic scales are strongly coupled in most homogenization methods which is very costly. Our method consists in decoupling the micro and macro scales by considering separate boundary value problems and an intermediate anisotropic constitutive law optimized over a training set. We propose an iterative procedure based on this method which allows to improve the quality of the approximation to get closer to the coupled homogenization and keeping a reasonable computational cost. We perform representative numerical studies for a layer with heterogeneous hyperelastic material in order to demonstrate the capability and reliability of the proposed method and test several intermediate constitutive laws.

On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

Data-driven constitutive modeling is an emerging field in computational solid mechanics with the prospect of significantly relieving the computational costs of hierarchical computational methods. Additionally, this data-driven paradigm could enable a seamless connection of experimental data probing material responses with numerical simulations at the structural level. Traditionally, these surrogates have just been trained using datasets which map strain inputs to stress outputs for elastic and inelastic materials directly. Recently, artificial neural networks (ANNs) have instead been trained to additionally incorporate the underlying physical laws in the construction of these models. However, ANNs do not offer convergence guarantees from an engineering point of view and are majorly reliant on user-specified parameters. In contrast to ANNs, Gaussian process regression (GPR) is based on nonparametric modeling principles as well as on fundamental statistical knowledge and hence allows for strict convergence guarantees. Motivated by the recent work by Frankel et al. (2021) which is based on rewriting the stress output as a linear combination of an irreducible integrity basis, in this work we present a physics-informed and data-driven constitutive modeling approach for isotropic and anisotropic hyperelastic materials at finite strain. The trained surrogates are able to respect physical principles such as material frame indifference, material symmetry, thermodynamic consistency, stress-free undeformed configuration, and the local balance of angular momentum. Our approach is based on probabilistic machine learning and uniquely can be used in the big data context while maintaining the benefits of GPR. As sampling in the mixed invariant space poses a unique challenge, we additionally present the first sampling approach that directly generates space-filling points in the invariant space corresponding to a bounded domain of the deformation gradient tensor. The sampling technique is based on simulated annealing and provides more efficient and reliable physics-informed constitutive models. Overall, the presented approach is tested on synthetic data from isotropic and anisotropic constitutive laws and shows surprising accuracy even far beyond the limits of the training domain, indicating that the resulting surrogates can efficiently generalize as they incorporate knowledge about the underlying physics.

Explicit Analysis of Sheet Metal Forming Processes Using Solid-Shell Elements

To simulate sheet metal forming processes precisely, an in-house dynamic explicit code was developed to apply a new solid-shell element to sheet metal forming analyses, with a corotational coordinate system utilized to simplify the nonlinearity and to integrate the element with anisotropic constitutive laws. The enhancing parameter of the solid-shell element, implemented to circumvent the volumetric and thickness locking phenomena, was condensed into an explicit form. To avoid the rank deficiency, a modified physical stabilization involving the B-bar method and reconstruction of transverse shear components was adopted. For computational efficiency of the solid-shell element in numerical applications, an adaptive mesh subdivision scheme was developed, with element geometry and contact condition taken as subdivision criteria. To accurately capture the anisotropic behavior of sheet metals, material models with three different anisotropic yield functions were incorporated. Several numerical examples were carried out to validate the accuracy of the proposed element and the efficiency of the adaptive mesh subdivision.

Experimentally-validated orthotropic elastic model for Wire-and-Arc Additively Manufactured stainless steel

Abstract Wire-and-Arc Additively Manufactured (WAAM) alloys are characterized by specific mechanical properties which can largely differ from the conventionally-manufactured alloys. In detail, the printing process results in a peculiar microstructure, characterized by preferential crystallographic orientation with reference to the printing direction, that leads to an anisotropic mechanical behavior of the printed part. Previous experimental tests on WAAM-produced stainless steel plates showed in particular a strong anisotropic elastic behavior. Based on the above, the present work formulates a specific anisotropic elastic model for a WAAM-processed austenitic stainless steel, considering an orthogonally anisotropic (or orthotropic) constitutive law and a procedure to calibrate the elastic parameters based on the experimental results. In detail, the procedure is applied to calibrate the numerical values of the elastic parameters of a specific WAAM 304L austenitic stainless steel. For this aim, specific investigations on both the mechanical and microstructural features were carried out. Experimental tensile tests were performed on specimens with different orientations with reference to the printing direction. In detail, Young’s modulus and Poisson’s ratios were evaluated for samples oriented along three different orientations with regard to the printing deposition layers: longitudinally (L), transversally (T) and diagonally (D) to them. Digital Image Correlation (DIC) optical measuring system was used to acquire the full strain fields during the test. Microstructural analysis was also carried out to study the inherent microstructure, characterized by a distinctive grain growth direction, and to assess the preferred crystallographic orientations of specimens extracted along the three considered directions. The experimental results are used to calibrate the orthotropic elastic model. From the calibrated model additional material properties in terms of Young’s and shear modulus for any printing direction are derived. The resulting values exhibit very large variations with the printing angle, with ratios between minimum to maximum values around 2 for the Young’s modulus and 3.5 for the shear modulus. This marked orthotropic behavior could open unexplored design possibilities based on deformability issues. Additionally, the calibrated orthotropic model can also be used for future experimental explorations of the mechanical properties of WAAM alloys and for stiffness-based structural design optimizations.

Coupled simulation of flow-induced viscous and elastic anisotropy of short-fiber reinforced composites

The present work discusses the impact of the back coupling of the fiber orientation distribution on the base flow and on the fiber orientation itself during mold filling simulations. Flows through a channel and over a backward-facing step are investigated. Different closure approximations are considered for modeling the flow-induced evolution of anisotropy. The results corresponding to the decoupled approach, in which the effect of fibers on local fluid properties is neglected, build the basis of comparison. The modeling is limited to a laminar, incompressible, and isothermal flow of a fiber suspension consisting of rigid short fibers suspended in an isotropic Newtonian matrix fluid. A linear, anisotropic constitutive law is used in combination with a uniform fiber volume fraction of 10% and an aspect ratio of 10. To evaluate the impact of back coupling and of different closure methods in view of the manufactured solid composite the resulting anisotropic elastic properties are investigated based on the Mori–Tanaka method combined with an orientation average scheme. Relative to the range [0, 1] the pointwise difference in fiber orientation between the decoupled and the coupled approach is found to be pm 5{%} in the channel and pm 30{%} in the backward-facing step, respectively. The viscosity and the elasticity tensor show both significant flow-induced anisotropies as well as a strong dependence on closure and coupling.

Experimental 3D characterization and parametrization of an anisotropic constitutive law for a synthetic nonwoven

The usage of synthetic nonwoven materials is gaining importance in filtration. However, the complex material behavior of these synthetic, fiber-based materials is challenging for the manufacturing departments. The materials are exhibiting a highly anisotropic elastic-plastic material behavior. A material testing program was conducted to characterize the unknown characteristic material behavior of synthetic nonwoven filter media in all three spatial directions for the load cases of tension, compression, and shear for the first time. For conversion processes like cutting, embossing, calendering, and pleating, the out-of-plane properties are essential for the accurate description of the material behavior. Using an ARCAN device, it is possible to do the necessary tests to determine the out-of-plane properties. The test results were used to calibrate a suited anisotropic material model for fibrous and porous materials. Parallels and differences in the characteristic material behavior between cellulose and synthetic nonwoven filter media are discussed.

A stabilised displacement–volumetric strain formulation for nearly incompressible and anisotropic materials

The simulation of structural problems involving the deformations of volumetric bodies is of paramount importance in many areas of engineering. Although the use of tetrahedral elements is extremely appealing, tetrahedral discretisations are generally known as very stiff and are hence often avoided in typical simulation workflows.The development of mixed displacement–pressure approaches has allowed to effectively overcome this problem leading to a class of locking-free elements which can effectively compete with hexahedral discretisations while retaining obvious advantages in the mesh generation step. Despite such advantages the adoption of the technology within commercial codes is not yet pervasive.This can be attributed to two different reasons: the difficulty in making use of standard constitutive libraries and the implied continuity of the pressure, which makes the application of the method questionable in the context of multi-material problems. Current paper proposes the adoption of the volumetric strain instead of the pressure as a nodal value. Such choice leads to the definition of a modified strain making the use of standard strain-driven constitutive laws straightforward. At the same time, the continuity of the volumetric strain across multimaterial interfaces can be understood as a sort of kinematic constraint (stresses can still remain discontinuous across material interfaces). The new element also opens the door to the use of anisotropic constitutive laws, which are typically problematic in the context of mixed elements.

Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: An experimental and numerical approach

Our aim is to estimate regional mechanical properties of the annulus fibrosus (AF) using a multi-relaxation tensile test and to examine the relevance of using the transverse dilatations in the identification procedure. We collected twenty traction specimens from both outer (n = 10) and inner (n = 10) sites of the anterior quadrant of the annulus fibrosus of one pig spine. A 1-h multi-relaxation tensile test in the circumferential direction allowed us to measure the force in the direction of traction and the dilatations in all three directions. We performed a specific-sample finite element inverse analysis to identify variations, along the radial position, of material and structural parameters of a hyperelastic compressible and anisotropic constitutive law. Our experimental results reveal that the outer sites are subjected to a significantly greater stress than the inner sites and that both sites exhibit an auxetic behavior. Our numerical results suggest that the inhomogeneous behavior arises from significant variations of the fiber angle taken into account within the hyperelastic constitutive law. In addition, we found that the use of the measured transverse dilatations in the identification procedure had a strong impact on the identified mechanical parameters. This pilot study suggests that, in quasi-static conditions, the annulus fibrosus may be modeled by a hyperelastic compressible and anisotropic law with a fiber angle gradient from inner to outer periphery.

A note on the volumetric-deviatoric split on the anisotropic constitutive model for fiber-reinforced materials

In the literature, it has been suggested that for a class of anisotropic constitutive laws for fiberreinforced materials, the volumetric-deviatoric split should only be performed on the isotropic (matrix) term, but not on the anisotropic (fiber) term. In this research note, we follow up on the theoretical and numerical analyses adopted in these early publications with an intuitive example that allows us to directly analyze the effect of this split. We demonstrate that performing such split on the anisotropic term leads to non-physical volume growth of the material sample. Therefore, we consolidate the observation that the volumetric-deviatoric split should not be applied to the anisotropic (fiber) term of the total strain energy.

Breaking the state of the heart: meshless model for cardiac mechanics.

Cardiac modeling has recently emerged as a promising tool to study pathophysiology mechanisms and to predict treatment outcomes for personalized clinical decision support. Nevertheless, achieving convergence under large deformation and defining a robust meshing for realistic heart geometries remain challenging, especially when maintaining the computational cost reasonable. Smoothed particle hydrodynamics (SPH) appears to be a promising alternative to the finite element method (FEM) since it removes the burden of mesh generation. A point cloud is used where each point (particle) contains all the physical properties that are updated throughout the simulation. SPH was evaluated for solid mechanics applications in the last decade but its capacity to address the challenge of simulating the mechanics of the heart has never been evaluated. In this paper, a total Lagrangian formulation of a corrected SPH was used to solve three solid mechanics problems designed to test important features that a cardiac mechanics solver should have. SPH results, in terms of ventricle displacements and strains, were compared to results obtained with 11 different FEM-based solvers, by using synthetic cardiac data from a benchmark study. In particular, passive dilation and active contraction were simulated in an ellipsoidal left ventricle with the exponential anisotropic constitutive law of Guccione following the direction of fibers. The proposed meshless method is able to reproduce the results of three benchmark problems for cardiac mechanics. Hyperelastic material with fiber orientation and high Poisson ratio allows wall thickening/thinning when large deformation is present.

Multiscale modelling of asymmetric rolling with an anisotropic constitutive law

A parametric study is presented, which employs a new anisotropic constitutive law in order to study the influence of anisotropic plasticity on the deformation field of the Asymmetric Rolling (ASR) process. A version of the facet method is presented, where an analytical yield function is restricted to the subspace of the stress and strain rate space relevant for 2D Finite Element Analysis (FEA), but can still accurately reproduce the plastic anisotropy of an underlying Crystal Plasticity (CP) model. The influence of anisotropy on the deformation field and corresponding texture evolution is examined in terms of the changes in texture component volume fractions and formation of texture gradients. It is found that a material with the anisotropy of a sharp cold-rolled aluminium alloy is more beneficial than that of a recrystallised hot-rolled aluminium alloy, and this influence of anisotropy suggests that Asymmetric Rolling (ASR) may be best carried out in the latest stages of cold rolling.

A Computational Tool for the Microstructure Optimization of a Polymeric Heart Valve Prosthesis.

Styrene-based block copolymers are promising materials for the development of a polymeric heart valve prosthesis (PHV), and the mechanical properties of these polymers can be tuned via the manufacturing process, orienting the cylindrical domains to achieve material anisotropy. The aim of this work is the development of a computational tool for the optimization of the material microstructure in a new PHV intended for aortic valve replacement to enhance the mechanical performance of the device. An iterative procedure was implemented to orient the cylinders along the maximum principal stress direction of the leaflet. A numerical model of the leaflet was developed, and the polymer mechanical behavior was described by a hyperelastic anisotropic constitutive law. A custom routine was implemented to align the cylinders with the maximum principal stress direction in the leaflet for each iteration. The study was focused on valve closure, since during this phase the fibrous structure of the leaflets must bear the greatest load. The optimal microstructure obtained by our procedure is characterized by mainly circumferential orientation of the cylinders within the valve leaflet. An increase in the radial strain and a decrease in the circumferential strain due to the microstructure optimization were observed. Also, a decrease in the maximum value of the strain energy density was found in the case of optimized orientation; since the strain energy density is a widely used criterion to predict elastomer's lifetime, this result suggests a possible increase of the device durability if the polymer microstructure is optimized. The present method represents a valuable tool for the design of a new anisotropic PHV, allowing the investigation of different designs, materials, and loading conditions.

へき開面における原子間分離に基づく異方性損傷構成則

へき開面における原子間分離に基づく異方性損傷構成則

Modelling of tear propagation and arrest in fibre-reinforced soft tissue subject to internal pressure

The prediction of soft-tissue failure may yield a better understanding of the pathogenesis of arterial dissection and help to advance diagnostic and therapeutic strategies for the treatment of this and other diseases and injuries involving the tearing of soft tissue, such as aortic dissection. In this paper, we present computational models of tear propagation in fibre-reinforced soft tissue undergoing finite deformation, modelled by a hyperelastic anisotropic constitutive law. We adopt the appropriate energy argument for anisotropic finite strain materials to determine whether a tear can propagate when subject to internal pressure loading. The energy release rate is evaluated with an efficient numerical scheme that makes use of adaptive tear lengths. As an illustration, we present the calculation of the energy release rate for a two-dimensional strip of tissue with a pre-existing tear of length \(a\) under internal pressure \(p\) and show the effect of fibre orientation. This calculation allows us to locate the potential bifurcation to tear propagation in the \((a,p)\) plane. The numerical predictions are verified by analytical solutions for simpler cases. We have identified a scenario of tear arrest, which is observed clinically, when the surrounding connective tissues are accounted for. Finally, the limitations of the models and further directions for applications are discussed.

Fluid-structure interaction simulations of cerebral arteries modeled by isotropic and anisotropic constitutive laws

This paper considers numerical simulations of fluid-structure interaction (FSI) problems in hemodynamics for idealized geometries of healthy cerebral arteries modeled by both nonlinear isotropic and anisotropic material constitutive laws. In particular, it focuses on an anisotropic model initially proposed for cerebral arteries to characterize the activation of collagen fibers at finite strains. In the current work, this constitutive model is implemented for the first time in the context of an FSI formulation. In this framework, we investigate the influence of the material model on the numerical results and, in the case of the anisotropic laws, the importance of the collagen fibers on the overall mechanical behavior of the tissue. With this aim, we compare our numerical results by analyzing fluid dynamic indicators, vessel wall displacement, Von Mises stress, and deformations of the collagen fibers. Specifically, for an anisotropic model with collagen fiber recruitment at finite strains, we highlight the progressive activation and deactivation processes of the fibrous component of the tissue throughout the wall thickness during the cardiac cycle. The inclusion of collagen recruitment is found to have a substantial impact on the intramural stress, which will in turn impact the biological response of the intramural cells. Hence, the methodology presented here will be particularly useful for studies of mechanobiological processes in the healthy and diseased vascular wall.

Multiscale approach to predict crack initiation in unidirectional off-axis laminates

Crack initiation in unidirectional off-axis laminates made of carbon fiber and epoxy resin is predicted based on multiscale modeling. This multiscale modeling consists of two finite-element analyses (FEA) on different scales. One is macroscopic FEA, based on the assumption of homogeneous materials, and the other is microscopic periodic unit-cell (PUC) analysis using a micromechanical model. The macroscopic FEA is performed by applying uniaxial tension to off-axis laminates, in which we employ an anisotropic elasto-plastic constitutive law to obtain accurate deformation fields in laminates. In the microscopic PUC analysis, the strain history at a point in laminates obtained from the macroscopic FEA is applied as external forces, and crack initiation is predicted using two failure criteria for the matrix resin. The first is the dilatational energy density criterion under elastic deformation, and the second is a ductile damage growth law under plastic deformation. The simulated predictions are compared with the experiments results.

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.

Influence of random uncertainties of anisotropic fibrous model parameters on arterial pressure estimation

This paper deals with a stochastic approach based on the principle of the maximum entropy to investigate the effect of the parameter random uncertainties on the arterial pressure. Motivated by a hyperelastic, anisotropic, and incompressible constitutive law with fiber families, the uncertain parameters describing the mechanical behavior are considered. Based on the available information, the probability density functions are attributed to every random variable to describe the dispersion of the model parameters. Numerous realizations are carried out, and the corresponding arterial pressure results are compared with the human non-invasive clinical data recorded over a mean cardiac cycle. Furthermore, the Monte Carlo simulations are performed, the convergence of the probabilistic model is proven. The different realizations are useful to define a reliable confidence region, in which the probability to have a realization is equal to 95%. It is shown through the obtained results that the error in the estimation of the arterial pressure can reach 35% when the estimation of the model parameters is subjected to an uncertainty ratio of 5%. Finally, a sensitivity analysis is performed to identify the constitutive law relevant parameters for better understanding and characterization of the arterial wall mechanical behaviors.

Finite element modeling of mitral leaflet tissue using a layered shell approximation.

The current study presents a finite element model of mitral leaflet tissue, which incorporates the anisotropic material response and approximates the layered structure. First, continuum mechanics and the theory of layered composites are used to develop an analytical representation of membrane stress in the leaflet material. This is done with an existing anisotropic constitutive law from literature. Then, the concept is implemented in a finite element (FE) model by overlapping and merging two layers of transversely isotropic membrane elements in LS-DYNA, which homogenizes the response. The FE model is then used to simulate various biaxial extension tests and out-of-plane pressure loading. Both the analytical and FE model show good agreement with experimental biaxial extension data, and show good mutual agreement. This confirms that the layered composite approximation presented in the current study is able to capture the exponential stiffening seen in both the circumferential and radial directions of mitral leaflets.