Anisotropic Material Behavior Research Articles

Micromechanics-based conversion of CT data into anisotropic elasticity tensors, applied to FE simulations of a mandible

ABSTRACTComputer Tomographic (CT) image data have become a standard basis for structural analyses of bony organs. In this context, regression functions between stiffness components and Hounsfields units (HU) from Computer Tomography, related to X-ray attenuation coefficients, are widely used for the definition of the (actually inhomogeneous and anisotropic) material behavior inside the organ. Herein, we suggest to derive the functional dependence of the fully orthotropic stiffness tensors on the Hounsfield units from the physical information contained in the X-ray attenuation coefficients: (i) Based on voxel average rules for the X-ray attenuation coefficients, we assign to each voxel the volume fraction occupied by water (marrow) and that occupied by solid bone matrix. (ii) By means of a continuum micromechanics representation for bone, which is based on voxel-invariant (species and whole bone-specific) stiffness properties of solid bone matrix and of water, we convert the aforementioned volume fractions into voxel-specific orthotropic stiffness tensor components. The micromechanics model, in combination with the average rule for X-ray attenuation coefficients, predicts a quasi-linear relationship between axial Young's modulus and HU, and highly nonlinear relationships for both circumferential and radial Young's moduli as well as for the shear moduli in all principal material directions. Corresponding whole-organ Finite Element analyses of a partially dentulous human mandible characterized by atrophy of the alveolar ridge show that volumetric strain concentrations/peaks within the organ are decreased when considering material anisotropy, and increased when considering material inhomogeneity.

Finite element model and experimental studies on rubber-cord composite

The rubber-cord composite (CRC) which is created of rubber matrix reinforced with textile cords is used for many applications such as pneumatic membranes, automobile tires, pneumatic air-springs, hydraulic hoses and many others. The CRC is characterized by strongly anisotropic material behaviour and can simultaneously undergo large elastic deformations. In this paper a finite element (FE) model was developed and applied to study the mechanical responses of CRC. This model consists of 8-node hexahedral brick elements describing rubber matrix and 3-D spar elements for modeling of textile cords. The experimental studies in uniaxial and cyclic tension were performed. The material constants of textile cords were fitted to experimentally measured data by approach technique using linear and bilinear elastic models. The simulations of uniaxial tensile tests using proposed FE model were carried out. The numerical results of simulations were compared to experimental ones in order to verify the accurateness of the FE model. The obtained results indicated that the proposed FE model can be applied for the modeling and simulation of mechanical behaviour of CRC.

Large‐scale simulation of arterial walls: mechanical modeling

AbstractBiological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre‐Hadamard condition automatically. When arteries are overstretched, discontinuous damage is observed. For the modeling of this effect we apply a damage model, which basically assumes that the damage occurs mainly in fiber direction. For the numerical simulation we consider an atherosclerotic artery and apply a high internal pressure which is comparable to the pressure applied during a balloon‐angioplasty. The 3D‐discretization results in a large system of equations, therefore, a parallel algorithm using FETI‐DP is applied to solve the boundary value problem. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Least‐squares mixed finite elements with applications to anisotropic elasticity and viscoplasticity

AbstractThe objective of this work is to discuss a least‐squares finite element method with applications to physically nonlinear and anisotropic constitutive equations at small strains. The L2‐norm minimization of the residuals of the given first order system of differential equations leads to a functional, which is a two field formulation in the displacements and the stresses, see e.g. Cai & Starke [1]. These functionals provide the foundation for the formulations of the related least‐squares mixed finite elements. A main focus of the presentation lies on the extension of plane elasticity to anisotropic or nonlinear material behavior. In this context transversely isotropic elasticity and viscoplasticity is considered. Finally a numerical example is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Micromechanics-Based Conversion of CT Data into Anisotropic Elasticity Tensors, Applied to FE Simulations of a Mandible

Computer Tomographic (CT) image data have become a standard basis for structural analyses of bony organs. In this context, regression functions between stiffness components and Hounsfields units (HU) from CT, related to X-ray attenuation coefficients, are widely used for the definition of the (actually inhomogeneous and anisotropic) material behavior inside the organ. Herein, we suggest to derive the functional dependence of the fully orthotropic stiffness tensors on the Hounsfield units from the physical information contained in the X-ray attenuation coefficients: (i) Based on voxel average rules for the X-ray attenuation coefficients, we assign to each voxel the volume fraction occupied by water (marrow) and that occupied by solid bone matrix. (ii) By means of a continuum micromechanics representation for bone, which is based on voxel-invariant (species and whole bone-specific) stiffness properties of solid bone matrix and of water, we convert the aforementioned volume fractions into voxel-specific orthotropic stiffness tensor components. The micromechanics model, in combination with the average rule for X-ray attenuation coefficients, predicts a quasi-linear relationship between axial Young's modulus and HU, and highly nonlinear relationships for both circumferential and radial Young's moduli as well as for the shear moduli in all principal material directions. Corresponding whole-organ Finite Element (FE) analyses of a partially edentulous human mandible characterized by atrophy of the alveolar ridge show that volumetric strain concentrations/peaks within the organ are decreased when considering material anisotropy, and increased when considering material inhomogeneity.

Optimization of ironing processes by means of DOE and FEA

The range of possible applications of ironing processes reaches from the production of cylindrical pressure tanks, shock absorber pipes, battery and beverage cans to the production of flacons for the cosmetics industry. One goal of the finite element analyses of ironing processes is to gain knowledge about process limits regarding maximized forming capacity and optimized surface quality. The ironing process is characterized by parameters like friction, geometry of the ironing ring and anisotropic material behavior. Due to the high complexity of this process, it is complicated to detect these effects on the basis of experimental investigations only. Therefore, one of the aims of this study is to identify relevant process parameters by means of systematic numerical analyses. The design of experiments was utilized to create a model of the ironing process. By means of this model it was possible to reduce the punch load and to minimize the tensile stresses in the first stage of the analyzed ironing process.

A method for evaluating the mechanical performance of thin-walled titanium tubes

A method which was developed to compare the stress–strain properties of three types of thin-walled, commercially pure titanium tubes is presented. The tubes were of types intended for use in large heat-exchanger applications and were to be subjected to significant plastic deformation during subsequent assembly processes. It had been anticipated that small differences in chemical composition and tube-drawing treatment would produce quite different characteristics. It is known that the properties of titanium can exhibit considerable degrees of anisotropy, especially for wrought products; although axial properties of the materials could be evaluated using standard test equipment and procedures, a novel testing system had to be designed to allow the circumferential properties to be assessed. Significant differences between tube types were observed and anisotropic material behaviour was apparent.

The effect of weld metal anisotropy on the creep of a pressurized, circumferentially welded pipe

A finite element implementation of the Hill anisotropic model has been applied to the problem of the determination of the steady-state creep behaviour of an internally-pressurised, two-material, axisymmetric P91 welded pipe. The anisotropic model results have been validated against an available analytical solution using homogeneous steady-state creep properties. In the two-material model, the parent (base) metal is assumed to be isotropic while the weld metal is assigned different creep properties in the welding direction (circumferential) and the direction transverse to welding, consistent with observations on the creep behaviour of P91 weld metal. The anisotropic material behaviour of the pipe was characterised using two constants, p and S, which respectively represent (a) the ratio of transverse to longitudinal creep strain rates in the weld metal and (b) the ratio of weld metal longitudinal to parent metal creep strain rates. The effects of these parameters on the stresses at critical positions were studied parametrically, along with the Norton stress exponent, n, and different end loading conditions.

Constitutive behaviour of anisotropic materials under shock loading

Thermodynamically and mathematically consistent constitutive equations suitable for shock wave propagation in an anisotropic material are presented in this paper. Two fundamental tensors α ij and β ij which represent anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors α ij and β ij , the concept of total generalised “pressure” and pressure corresponding to the thermodynamic (equation of state) response are redefined. The equation of state represents mathematical and physical generalisation of the classical Mie–Grüneisen equation of state for isotropic material and reduces to the Mie–Grüneisen equation of state in the limit of isotropy. Based on the generalised decomposition of the stress tensor, the modified equation of state for anisotropic materials, and the modified Hill criteria, combined with the associated flow rule, a system of constitutive equations suitable for shock wave propagation is formulated. The behaviour of aluminium alloy 7010-T6 under shock loading conditions is considered. A comparison of numerical simulations with existing experimental data shows good agreement of the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels, and suggests that the constitutive equations are performing satisfactorily. The results are presented and discussed, and future studies are outlined.

Non-quadratic anisotropic potentials based on linear transformation of plastic strain rate

In this paper, anisotropic strain rate potentials based on linear transformations of the plastic strain rate tensor were reviewed in general terms. This type of constitutive models is suitable for application in forming simulations, particularly for finite element analysis and design codes based on rigid plasticity. Convex formulations were proposed to describe the anisotropic behavior of materials for a full 3-D plastic strain rate state (5 independent components for incompressible plasticity). The 4th order tensors containing the plastic anisotropy coefficients for orthotropic symmetry were specified. The method recommended for the determination of the coefficients using experimental mechanical data for sheet materials was discussed. The formulations were shown to be suitable for the constitutive modeling of FCC and BCC cubic materials. Moreover, these proposed strain rate potentials, called Srp2004-18p and Srp2006-18p, led to a description of plastic anisotropy, which was similar to that provided by a generalized stress potential proposed recently, Yld2004-18p. This suggests that these strain rate potentials are pseudo-conjugate of Yld2004-18.

A hybrid element formulation for electromechanical problems

AbstractThe difficulty in the modeling of ferroelectric materials is the coverage of the complicated interactions between electrical and mechanical quantities on the macroscale, which are caused by switching processes on the microscale. In the present work we present an electric hybrid element formulation where the stresses and the electric fields are derived by constitutive relations as presented in [1]. Therefore the displacements, the electric potential and the electric displacements are approximated by bilinear ansatz functions. Applying a static condensation procedure we obtain a modified finite element formulation governed by the degrees of freedoms associated to the displacements and the electric potential. The anisotropic material behavior is modeled within a coordinate‐invariant formulation [6] for an assumed transversely isotropic material [4]. In this context a general return algorithm is applied to compute the remanent quantities at the actual timestep. Resulting hysteresis loops for the ferroelectric ceramics are presented. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Elastic-viscoplastic anisotropic modeling of textured metals and validation using the Taylor cylinder impact test

An elastic-viscoplastic model for describing the anisotropic high-strain rate behavior of both low-symmetry and high-symmetry textured materials is proposed. Yielding is described using a recently developed criterion which can capture simultaneously anisotropy and compression–tension asymmetry associated with deformation twinning. The anisotropy coefficients as well as the size of the elastic domain are considered to be functions of the accumulated plastic strain. The specific expressions for the evolution laws are determined using a multi-scale methodology, i.e. experimental measurements of crystallographic texture and uniaxial stress–strain curves, polycrystalline calculations, and macroscopic scale interpolation techniques. An overstress approach is used to incorporate rate effects in the formulation. Applications of the model to the description of the high strain-rate response of low-symmetry (hexagonal-close-packed zirconium) and high-symmetry (body-centered-cubic tantalum) pre-textured metals are presented. The very good agreement between the simulated and experimental post-test geometries of the Taylor impact specimens in terms of major and minor side profiles and impact-interface footprints shows the ability of the model to describe with fidelity the differences in the evolution of anisotropy between zirconium and tantalum.

A generic method for modeling the behavior of anisotropic metallic materials: Application to recrystallized zirconium alloys

A simplified polycrystalline model (the so-called RL model) is proposed to simulate the anisotropic viscoplastic behavior of metallic materials. A generic method is presented that makes it possible to build a simplified anisotropic material texture, based on the principal features of the pole figures. The method is applied to a recrystallized zirconium alloy, used as clad material in the fuel rods of nuclear power plants. An important database consisting in mechanical tests performed on Zircaloy tubes is collected. Only a small number of tests (pure tension, pure shear) are used to identify the material parameters, and the texture parameters. It is shown that six crystallographic orientations (6 “grains”) are sufficient to describe the large anisotropy of such hcp alloy. The identified crystallographic orientations match the experimental pole figures of the material, not used in the identification procedure. Special attention is paid to the predictive ability of the model, i.e., its ability to simulate correctly experimental tests not belonging to the identification database. These predictive results are good, thanks to an identification procedure that enables to consider the contribution of each slip system in each crystallographic orientation.

Transient dynamic fracture analysis using scaled boundary finite element method: a frequency-domain approach

In the scaled boundary finite element method (SBFEM), the analytical nature of the solution in the radial direction allows accurate stress intensity factors (SIFs) to be determined directly from the definition, and hence no special crack-tip treatment, such as refining the crack-tip mesh or using singular elements (needed in the traditional finite element and boundary element methods), is necessary. In addition, anisotropic material behaviour may be handled with ease. These advantages are used in this study, in which a newly-developed Frobenius solution procedure in the frequency domain for solving the governing differential equations of the SBFEM, is applied to model transient dynamic fracture problems. The complex frequency–response functions are first computed using the Frobenius solution procedure. The dynamic stress intensity factors (DSIFs) are then extracted directly from the response functions. This is followed by a fast Fourier transform (FFT) of the transient load and a subsequent inverse FFT to obtain the time history of DSIFs. Benchmark problems with isotropic and anisotropic material behaviour are modelled using the developed frequency-domain approach. Excellent agreement is observed between the results of this study and those in published literature. The effects of the mesh density, the material internal damping coefficient, the maximum frequency and the frequency interval determining the frequency–response functions on the resultant accuracy and the computational cost are also discussed.

Impact of mechanical anisotropy and power-law rheology on single layer folding

The infinitesimal and finite stages of folding in nonlinear viscous material with a layer-parallel anisotropy were investigated using numerical and analytical methods. Anisotropy was found to have a first-order effect on growth rate and wavelength selection, and these effects are already important for anisotropy values (normal viscosity/shear viscosity) < 10. The effect of anisotropy must therefore be considered when deducing viscosity contrasts from wavelength to thickness ratios of natural folds. Growth rates of single layer folds were found to increase and subsequently decrease during progressive deformation. This is due to interference between the single layer folds and chevron folds that form in the matrix as a result of instability caused by the anisotropic material behaviour. The wavelength of the chevron folds in the matrix is determined by the wavelength of the folded single layer, which can explain the high wavelength to thickness ratios that are sometimes found in multilayer sequences. Numerical models including anisotropic material properties allow the behaviour of multilayer sequences to be investigated without the need for resolution on the scale of individual layers. This is particularly important for large-scale models of layered lithosphere.

Electromagnetic and thermal behaviors of multilayer anisotropic composite materials

A methodology based on shell elements, to study the electromagnetic and thermal behavior of multilayered anisotropic conductive composite materials, is presented. The anisotropic behavior and orientation of each layer of the composite material has been taken into account. The model is validated with experimental results. It is then applied to study the induction heating of composite materials, and also to detect the delamination in these materials

Identification and Interpretation of Microplane Material Laws

The present paper addresses the so-called microplane formulation which became recently more and more popular for the description of quasi-brittle materials. The essential feature of this material formulation is a split of the local microplane strains and stresses allowing one to resort to simplified or in certain cases even unidirectional constitutive laws. The main attraction of the microplane concept is that an initial or evolving anisotropic material behavior can be described in a natural and simple way. Motivated from a macroscopic viewpoint, it is advocated to restrict the microplane concept to the pure volumetric-deviatoric split, as a constraint subset of the most often applied volumetric-deviatoric-tangential split. This variant has the particular advantage that typical macroscopic responses are directly reflected on the mesoscale. It will be shown that in certain cases the present version of a microplane formulation is closely related to well-known macroscopic models although being much more general than those macroscopic formulations. This close relation is exploited to derive physically sound microplane constitutive laws. Therefore the characteristic damage mechanisms of materials at two levels of observation, (1) at the macroscale in the sense of classical continuum damage mechanics, and (2) at the mesoscale utilizing the so-called microplane concept, are examined. The comparison of the microplane formulation to a well-known macroscopic one-parameter damage model enables the identification and interpretation of the microplane constitutive laws. The constitutive formulations are embedded in a thermodynamically consistent framework. Finally, the performance of the attained microplane formulation is analyzed in a mixed-mode fracture simulation of concrete.

Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method

The meshless local Petrov–Galerkin method is used to analyze transient heat conduction in 3-D axisymmetric solids with continuously inhomogeneous and anisotropic material properties. A 3-D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem. The cross section is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen subdomains, called local boundary integral equations. These integral formulations are either based on the Laplace transform technique or the time difference approach. The local integral equations are nonsingular and take a very simple form, despite of inhomogeneous and anisotropic material behavior across the analyzed structure. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

Universal microstructural patterns in bone: Micromechanics-based prediction of anisotropic material behavior

Universal microstructural patterns in bone: Micromechanics-based prediction of anisotropic material behavior

Earthquake safety assessment of concrete arch and gravity dams

Based on research studies currently being carried out at Dalian University of Technology, some important aspects for the earthquake safety assessment of concrete dams are reviewed and discussed. First, the rate-dependent behavior of concrete subjected to earthquake loading is examined, emphasizing the properties of concrete under cyclic and biaxial loading conditions. Second, a modified four-parameter Hsieh-Ting-Chen viscoplastic consistency model is developed to simulate the rate-dependent behavior of concrete. The earthquake response of a 278m high arch dam is analyzed, and the results show that the strain-rate effects become noticeable in the inelastic range. Third, a more accurate non-smooth Newton algorithm for the solution of three-dimensional frictional contact problems is developed to study the joint opening effects of arch dams during strong earthquakes. Such effects on two nearly 300m high arch dams have been studied. It was found that the canyon shape has great influence on the magnitude and distribution of the joint opening along the dam axis. Fourth, the scaled boundary finite element method presented by Song and Wolf is employed to study the dam-reservoir-foundation interaction effects of concrete dams. Particular emphases were placed on the variation of foundation stiffness and the anisotropic behavior of the foundation material on the dynamic response of concrete dams. Finally, nonlinear modeling of concrete to study the damage evolution of concrete dams during strong earthquakes is discussed. An elastic-damage mechanics approach for damage prediction of concrete gravity dams is described as an example. These findings are helpful in understanding the dynamic behavior of concrete dams and promoting the improvement of seismic safety assessment methods.