Isotropic Material Behavior Research Articles

Finite element analysis of a bone-implant system with the proximal femur nail

Static analysis with finite element of a realistic femur nail bone-implant system in a typical proximal femoral fracture under physiological load bearing situations provides results for stress, displacement and strain. The question to be answered is, if simulation with the finite element analysis is able to explain biomechanically clinical observed patterns of failure. Surface-Reconstruction with CT database of a proximal femur and reconstruction with CT based density data was done. Next steps were to unite the bone structure with the Proximal Femoral Nail and to model two relevant fractures (31-A2.2 and A2.3 according AO). After modelling of geometry, isotropic material behaviour and load application numeric calculation of the femur-nail system with FE-software was performed. FE simulation mainly shows an axial dislocation of the femoral head screw with nearly no dislocation of the antirotation screw. This so-called z-effect therefore means: (1) Tilting of the proximal main fragment around the sagittal axis between the screws and (2) relative movement of both screws in the frontal plane. Relative movement of the two screws against each other could be the reason for implant failure, the so called cut out. Furthermore simulation shows different gliding of the screws explaining the so called z-telescoping. The analyzed stress patterns have to be relativized, because isotropic material behaviour of cancellous bone was assumed. Further examinations for this issue are necessary.

An Engineering Model for Three-Dimensional Elastic-Plastic Rolling Contact Analyses

Fatigue life of heavily loaded rolling bearings is strongly dependent on elastic-plastic material properties. For bearing steels these elastic-plastic properties can be accurately obtained by performing monotonic or half-compressive tests. A three-dimensional strain deformation analysis based on the incremental theory of plasticity and the use of Prandtl-Reuss relations in conjunction with the von Mises yield criterion was developed in order to evaluate the permanent deformation in dry contacts loaded above the elastic limit in case of normal loading. The Ramberg-Osgood stress-strain relation for two martensitically hardened variants of SAE 52100 bearing steel considered the nonlinear kinematic and/or isotropic material behavior. Parameters describing the influence of retained austenite are modeled by using a nonlinear isotropic law. Pressure distribution and contact surface displacements during incremental loading are evaluated by using a conjugate gradient method and the internal stress field is derived by using the superposition principle. Further, a fast analysis of smooth surfaces in elastic-plastic static and rolling contact is developed based on analytical relations for the internal stress field. Cyclic evaluation of plastic strains and residual stresses is carried out until shakedown. In order to verify the theoretical model, rolling contact tests under high normal load were performed. Residual stresses and residual profiles measurements show excellent agreement between numerical and measured cyclic values.

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.

Fabrication and probabilistic fracture strength prediction of high-aspect-ratio single crystal silicon carbide microspecimens with stress concentration

Single crystal silicon carbide micro-sized tensile specimens were fabricated with deep reactive ion etching (DRIE) in order to investigate the effect of stress concentration on the room-temperature fracture strength. The fracture strength was defined as the level of stress at the highest stressed location in the structure at the instant of specimen rupture. Specimens with an elliptical hole, a circular hole, and without a hole (and hence with no stress concentration) were made. The average fracture strength of specimens with a higher stress concentration was larger than the average fracture strength of specimens with a lower stress concentration. Average strength of elliptical-hole, circular-hole, and without-hole specimens was 1.53, 1.26, and 0.66 GPa, respectively. Significant scatter in strength was observed with the Weibull modulus ranging between 2 and 6. No fractographic examination was performed but it was assumed that the strength controlling flaws originated from etching grooves along the specimen side-walls. The increase of observed fracture strength with increasing stress concentration was compared to predictions made with the Weibull stress-integral formulation by using the NASA CARES/Life code. In the analysis isotropic material and fracture behavior was assumed — hence it was not a completely rigorous analysis. However, even with these assumptions good correlation was achieved for the circular-hole specimen data when using the specimen data without stress concentration as a baseline. Strength was over predicted for the elliptical-hole specimen data. Significant specimen-to-specimen dimensional variation existed in the elliptical-hole specimens due to variations in the nickel mask used in the etching. To simulate the additional effect of the dimensional variability on the probabilistic strength response for the single crystal specimens the ANSYS Probabilistic Design System (PDS) was used with CARES/Life. The ANSYS-PDS & CARES/Life simulation correlated better to the elliptical-hole specimen data than CARES/Life predictions based on average specimen dimensions.

A New Approach for Single Crystal Materials Analysis: Theory and Application to Initial Yielding

A new approach used for single crystal (SC) materials analysis is described. Its principle is based on the extension of predictive models for isotropic material behavior to anisotropic materials such as SC nickel base superalloys. A viscoplastic model describes the material in the macroscopic level while a factor, based on the crystallographic approach, accounts for the global state of micro slip on the crystal. The combination of both elements defines the so-called “combined approach” (CA). This paper presents the development of the theory and its applications to the determination of initial yielding and tension–compression asymmetry.

Modeling the Nanoindentation of Elastoplastic Materials With Nonlinear Adaptive Springs (NASs)

In this paper, the /spl beta/-material concept helps to elaborate and explore a new model for the indentation cycle of elastoplastic materials. The proposed approach takes into account the nonlinear behavior of homogeneous and isotropic materials. It uses the idea of a nonlinear adaptive spring (NAS) with changing properties according to the depth of penetration to accurately reproduce the material behavior in loading and unloading stages. The properties of the adopted NAS are included in its own stiffness function /spl kappa/ appearing in the form of an infinite sum of which the convergence and some properties are discussed in detail. This new model, which allows the indentation cycle to be reproduced whatever the penetration depth, permits at the same time a direct calculation of the involved energy terms. It also provides the possibility to perform separate analysis of the plastic energy, which allows distinguishing between different types of the material behavior and a better understanding of its nature. A validation is accomplished by applying the method to three different materials.

Effect of the actual distribution of applied stresses on global buckling of isotropic and transversely isotropic thin-walled members: Numerical examples

Thin-walled members made of fiber-reinforced polymer (FRP) composites are increasingly being used for structural applications, due to the advantageous material properties such as low weight, high tensile strength and insensitivity to corrosion. When manufactured by “traditional” pultrusion, these members display a transversely isotropic constitutive behavior, with plane of isotropy perpendicular to the fiber direction. Previous research showed that, considering a thin-walled member subjected to two arbitrary distributions of applied stresses on the extreme faces, the state of stress and the global buckling load (moment) of the member may be influenced by the actual distribution of the applied pressures, to an extent depending on the values of the elastic constants. The reduction in buckling load (moment) due to this effect was shown to be possibly not negligible for transversely isotropic material behavior, and particularly for low ratios of transverse to longitudinal elastic constants. In order to evaluate how significant this weakening effect can be for design purposes, this work aims at quantifying it as a function of various influential parameters. The influence of symmetry properties of the cross-section on results is also studied. In particular, the I-section and the semicircular sections are taken as examples of a doubly symmetric and a singly-symmetric open cross-section, respectively.

Modelling of anisotropic growth in biological tissues

In this contribution, we develop a theoretical and computational framework for anisotropic growth phenomena. As a key idea of the proposed phenomenological approach, a fibre or rather structural tensor is introduced, which allows the description of transversely isotropic material behaviour. Based on this additional argument, anisotropic growth is modelled via appropriate evolution equations for the fibre while volumetric remodelling is realised by an evolution of the referential density. Both the strength of the fibre as well as the density follow Wolff-type laws. We however elaborate on two different approaches for the evolution of the fibre direction, namely an alignment with respect to strain or with respect to stress. One of the main benefits of the developed framework is therefore the opportunity to address the evolutions of the fibre strength and the fibre direction separately. It is then straightforward to set up appropriate integration algorithms such that the developed framework fits nicely into common, finite element schemes. Finally, several numerical examples underline the applicability of the proposed formulation.

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

AbstractThis study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.

On the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law

AbstractThe statement that theories of inelasticity at finite strains have arrived at a high level of development is only true in conjunction with isotropic material behaviour. From both points of view (theoretical and computational), the extension to anisotropic material behaviour seems to be a complicated task. The statement is especially true when the multiplicative decomposition of the deformation gradient is considered a basis for the formulation. Of special interest are questions related to the mathematical form of the stored energy function or, equivalently, of the constitutive relation for the material stress tensor as the thermodynamical force. This paper deals with the above issues. The anisotropic formulation is accomplished using the notion of structural tensors. Here we suggest that the privileged directions of the material should be transformed in a specific way under the action of the inelastic part of the deformation gradient. The inelastic behaviour is assumed to be governed by evolution equations of the unified type.Numerically, we deal with the full multiplicative structure of the theory. The numerical treatment is developed in full detail. Expressions concerning the local iteration as well as the tangent operator are derived. Various numerical examples with applications to shells are presented which demonstrate the influence of anisotropy and the applicability of the theory. Copyright © 2003 John Wiley & Sons, Ltd.

ESTIMATION OF 3D FIELD QUANTITIES AND ENERGY FLUX ASSOCIATED WITH ELASTIC WAVES IN A BAR

A method to estimate dispersion relations and warping associated with elastic wave propagation in a bar is presented. The method is based on Hamilton's principle. It is shown how the theoretical model together with strain measurements can be used to evaluate three dimensional (3D) field quantities like displacements and stresses at an arbitrary position in the bar, as well as energy flux through an arbitrary cross-section of the bar. It is also shown how redundant measurements can be used to increase the accuracy. The method is general and can be applied to any mode of wave propagation, isotropic or anisotropic linearly elastic material, and any cross-sectional geometry. Here, it is applied to longitudinal waves in a split Hopkinson pressure bar with linear elastic isotropic material behaviour and square cross-section. In particular, axial displacement, axial stress and energy flux are evaluated at a free end of the bar in order to test the method. The method is also used to estimate the Poisson ratio of the bar material, by measuring axial and transverse strains at the same axial position.

Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry.

An Analytical Model to Predict Fracture of Off-Axis Unidirectional Composites

An analytical model based on isotropic homogeneous material behaviour is proposed to predict fracture in unidirectional composites under general loading. The model calculates the internal stress distribution corresponding to the applied load. In conjunction with the respective strength values, the model is capable of assessing the dominant stress component for failure initiation. For uniaxial tensile loading a comparison of calculated analytical stress distributions reasonably agrees with results from FE-analysis. A comparison of the analytical predictions with fractographic results for different off-axis angles provides good agreement.

Study of Rheological Properties of Liquid and Solidlike Materials Using Burgers' Model

It was shown that the use of Burgers' model and introduction of the effective viscosity function make it possible to account for general trends in rheologic behavior of different isotropic liquid and solidlike materials: stress relaxation and strain delay, influence of loading or shear rate on a strain curve, curve hysteresis on loading and unloading, and the presence of a stress maximum and its shift at different shear rates for nonlinear-viscous systems. Generally, the rheological model consists of the binomial rheological equation and the modified Maxwell and Kelvin–Voigt differential equations. The model was tested for the case of tangential stresses when studying the polyacrylamide–aluminium acetate solutions and for the normal stresses, when studying the strain of the rock samples saturated with oil.

Vector neural network hysteresis model

A neural network model of scalar hysteresis phenomena has been developed for modeling the behavior of isotropic magnetic materials. The function approximation ability of artificial neural networks has been applied. The virgin curve and a set of the first-order reversal branches can be stored preliminarily in a system of three neural networks. Different properties of magnetic materials can be simulated by a knowledge-based algorithm. Finally, hysteresis characteristics of different materials predicted by the introduced model are compared with the results of the classical Preisach simulation. Theoretical achievement and results of vector generalization of the method are also introduced.

Large inelastic strain analysis by multilayer shell elements

The objective of this contribution is to model large inelastic strains of ductile metals, to couple this material model with a multilayer shell kinematics and finally to achieve a finite element formulation applicable in general form to shell analysis. Elasto-plastic constitutive law is formulated by using the multiplicative decomposition of the deformation gradient and Neo-Hookean model for elastic strains assuming an overall isotropic material behavior. These 3D-material model is then enforced directly into a multilayer shell kinematics which provides a very accurate consideration of local effects, particularly stresses across the thickness. Finite element formulation is accomplished by means of the enhanced strain concept. Thus the well known deficiencies due to incompressible deformations and the inclusion of transverse strains are avoided. Several examples are given to demonstrate the performance of the algorithms developed concerning various aspects.

Tests in von Mises and isotropic Hoffman elastoplasticity

Plane strain constitutive behaviour of von Mises and isotropic Hoffman materials is examined using single element tests. Two kinds of tests are conducted – (a) prescribed displacement tests; and (b) tests with a mixture of displacements and boundary tractions prescribed. While (a) are used to understand the manner of stress traversal on the yield surface in principal stress space, (b) are employed to study the load displacement response and the possibility of ensuing localization. Associated plasticity is assumed throughout. The tests are conducted using perfect and strain softening plasticity. It is found that for the von Mises criterion limited exact solutions can be evolved even under softening (or hardening) conditions. For isotropic Hoffman materials the nature of the stress traversal, load deflection response and the satisfaction of the localization conditions are strongly influenced by the ratio and difference of uniaxial yield strengths, in tension and compression, as well as by the softening parameters.

Isotropic vector Preisach particle

A Preisach-type pseudoparticle is proposed, to describe macroscopic vector magnetic behaviour of isotropic materials. The magnitude of magnetization is computed by means of the classical Preisach model with the projection of field strength on its direction as input. The orientation of the magnetization vector is governed by the equilibrium between its vector product with the field strength and a friction-like opposing torque proportional to the square of its magnitude. The behaviour of the model is studied in various conditions and the results compared to experiments reported in literature.

Simulation of Sheet Metal Forming on the Basis of the Deformation Theory and Optimization Techniques

AbstractThe basic approach for the simulation of the sheet metal forming in this paper is a reduction of the whole process to a mapping between the undeformed and the deformed configuration. One major aim consists in a fast code that can be used in the design stage of a workpiece. The approach has to permit first statements to the producibility of the workpiece. Since the configurations are given, the central question raises is the placement of a material points in the deformed configuration. In this approximation the necessary consideration of plastic material behaviour is possible only on the basis of the deformation theory of plasticity. The present paper is restricted, for simplicity, to the membrane stress state in the shell and to isotropic material behaviour.

Research on features of behaviour of isotropic and anisotropic materials under impact

Summary In paper, the influence of anisotropy degree and orientation of strength properties on destruction of materials under impact is investigated numerically and experimentally. The numerical modeling was carried out in three-dimensional statement by a method of finite elements in frameworks of the continual approach of the mechanics of deformable solid. The destruction is described by tensor polynomial criterion of the fourth degree, which takes into account the influence of hydrostatic pressure.