Showing papers on "Constitutive equation published in 2001"

An approach to micro-macro modeling of heterogeneous materials

Abstract: A micro-macro strategy suitable for modeling the mechanical response of heterogeneous materials at large deformations and non-linear history dependent material behaviour is presented. When using this micro-macro approach within the context of finite element implementation there is no need to specify the homogenized constitutive behaviour at the macroscopic integration points. Instead, this behaviour is determined through the detailed modeling of the microstructure. The performance of the method is illustrated by the simulation of pure bending of porous aluminum. The influence of the spatial distribution of heterogeneities on the overall macroscopic behaviour is discussed by comparing the results of micro-macro modeling for regular and random structures.

Fluid mechanics and rheology of dense suspensions

Abstract: ▪ Abstract We review the fluid mechanics and rheology of dense suspensions, emphasizing investigations of microstructure and total stress. “Dense” or “highly concentrated” suspensions are those in which the average particle separation distance is less than the particle radius. For these suspensions, multiple-body interactions as well as two-body lubrication play a significant role and the rheology is non-Newtonian. We include investigations of multimodal suspensions, but not those of suspensions with dominant nonhydrodynamic interactions. We consider results from both physical experiments and computer simulations and explore scaling theories and the development of constitutive equations.

Nonlinearity in piezoelectric ceramics

Abstract: The paper presents an overview of experimental evidence and present understanding of nonlinear dielectric, elastic and piezoelectric relationships in piezoelectric ceramics. This topic has gained an increasing recognition in recent years due to the use of such materials under extreme operating conditions, for example in electromechanical actuators and high power acoustic transducers. Linear behaviour is generally confined to relatively low levels of applied electric field and stress, under which the dielectric, elastic and piezoelectric relationships are described well by the standard piezoelectric constitutive equations. Nonlinear relationships are observed above certain ‘threshold’ values of electric field strength and mechanical stress, giving rise to field and stress-dependent dielectric (e), elastic (s) and piezoelectric (d) coefficients. Eventually, strong hysteresis and saturation become evident above the coercive field/stress due to ferroelectric/ferroelastic domain switching. The thermodynamic method provides one approach to describing nonlinear behaviour in the ‘intermediate’ field region, prior to large scale domain switching, by extending the piezoelectric constitutive equations to include nonlinear terms. However, this method seems to fail in its prediction of the amplitude and phase of high frequency harmonic components in the field-induced polarisation and strain waveforms, which arise directly from the nonlinear dielectric and piezoelectric relationships. A better fit to experimental data is given by the empirical Rayleigh relations, which were first developed to describe nonlinear behaviour in soft magnetic materials. This approach also provides an indication of the origins of nonlinearity in piezoelectric ceramics, in terms of ferroelectric domain wall translation (at intermediate field/stress levels) and domain switching (at high field/stress levels). The analogy with magnetic behaviour is also reflected in the use of Preisach-type models, which have been successfully employed to describe the hysteretic path-dependent strain-field relationships in piezoelectric actuators. The relative merits and limitations of the different modelling methods are compared and possible areas of application are identified.

Computational Mechanics of the Heart

Abstract: Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties of the passive tissue are described here by a “pole–zero” constitutive law, whose parameters are derived in part from a model of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating the observed regional distributions of fibre and sheet orientations. Pressure–volume relations and 3D principal strains predicted by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments, is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges for future modellers.

A computational model of viscoplasticity and ductile damage for impact and penetration

Abstract: A coupled constitutive model of viscoplasticity and ductile damage for penetration and impact related problems has been formulated and implemented in the explicit finite element code LS-DYNA. The model, which is based on the constitutive model and fracture strain model of Johnson and Cook, and on continuum damage mechanics as proposed by Lemaitre, includes linear thermoelasticity, the von Mises yield criterion, the associated flow rule, non-linear isotropic strain hardening, strain-rate hardening, temperature softening due to adiabatic heating, isotropic ductile damage and failure. For each of the physical phenomena included in the model, one or several material constants are required. However, all material constants can be identified from relatively simple uniaxial tensile tests without the use of numerical simulations. In this paper the constitutive model is described in detail. Then material tests for Weldox 460 E steel and the calibration procedure are presented and discussed. The calibrated model is finally verified and validated through numerical simulations of material and plate perforation tests investigated experimentally.

A model of crystal plasticity based on the theory of continuously distributed dislocations

Abstract: This work represents an attempt at developing a continuum theory of the elastic–plastic response of single crystals with structural dimensions of ∼100 μm or less, based on ideas rooted in the theory of continuously distributed dislocations. The constitutive inputs of the theory relate explicitly to dislocation velocity, dislocation generation and crystal elasticity. Constitutive nonlocality is a natural consequence of the physical considerations of the model. The theory reduces to the nonlinear elastic theory of continuously distributed dislocations in the case of a nonevolving dislocation distribution in the material and the nonlinear theory of elasticity in the absence of dislocations. A geometrically linear version of the theory is also developed. The work presented in this paper is intended to be of use in the prediction of time-dependent mechanical response of bodies containing a single, a few, or a distribution of dislocations. A few examples are solved to illustrate the recovery of conventional results and physically expected ones within the theory. Based on the theory of exterior differential equations, a nonsingular solution for stress/strain fields of a screw dislocation in an infinite, isotropic, linear elastic solid is derived. A solution for an infinite, neo-Hookean nonlinear elastic continuum is also derived. Both solutions match with existing results outside the core region. Bounded solutions are predicted within the core in both cases. The edge dislocation in the isotropic, linear theory is also discussed in the context of this work. Assuming a constant dislocation velocity for simplifying the analysis, an evolutionary solution resulting in a slip-step on the boundary of a stress-free crystal produced due to the passage and exit of an edge dislocation is also described.

Plastic potentials for anisotropic porous solids

Abstract: The aim of this paper is to incorporate plastic anisotropy into constitutive equations of porous ductile metals. It is shown that plastic anisotropy of the matrix surrounding the voids in a ductile material could have an influence on both effective stress–strain relation and damage evolution. Two theoretical frameworks are envisageable to study the influence of plastic flow anisotropy: continuum thermodynamics and micromechanics. By going through the Rousselier thermodynamical formulation, one can account for the overall plastic anisotropy, in a very simple manner. However, since this model is based on a weak coupling between plasticity and damage dissipative processes, it does not predict any influence of plastic anisotropy on cavity growth, unless a more suitable choice of the thermodynamical force associated with the damage parameter is made. Micromechanically-based models are then proposed. They consist of extending the famous Gurson model for spherical and cylindrical voids to the case of an orthotropic material. We derive an upper bound of the yield surface of a hollow sphere, or a hollow cylinder, made of a perfectly plastic matrix obeying the Hill criterion. The main findings are related to the so-called ‘scalar effect’ and ‘directional effect’. First, the effect of plastic flow anisotropy on the spherical term of the plastic potential is quantified. This allows a classification of sheet materials with regard to the anisotropy factor h ; this is the scalar effect. A second feature of the model is the plasticity-induced damage anisotropy. This results in directionality of fracture properties (‘directional effect’). The latter is mainly due to the principal Hill coefficients whilst the scalar effect is enhanced by ‘shear’ Hill coefficients. Results are compared to some micromechanical calculations using the finite element method.

Applying Anand Model to Represent the Viscoplastic Deformation Behavior of Solder Alloys

Abstract: A unified viscoplastic constitutive law, the Anand model, was applied to represent the inelastic deformation behavior for solders used in electronic packaging. The material parameters of the constitutive relations for 62Sn36Pb2Ag, 60Sn40Pb, 96.5Sn3.5Ag, and 97.5Pb2.5Sn solders were determined from separated constitutive relations and experimental results. The achieved unified Anand model for solders were tested for constant strain rate testing, steady-state plastic flow and stress/strain responses under cyclic loading. It is concluded that the Anand model can be applied for representing the inelastic deformation behavior of solders at high homologous temperature and can be recommended for finite element simulation of the stress/strain responses of solder joints in service. @DOI: 10.1115/1.1371781#

The Linearized Theory of Elasticity

Abstract: Preface List of Figures List of Tables 1. Review of Mechanics of Materials 2. Mathematical Preliminaries 3. Kinematics 4. Forces and Stress 5. Constitutive Equations 6. Linearized Elasticity Problems 7. Two-Dimensional Problems 8. Torsion of Noncircular Cylinders 9. Three-Dimensional Problems 10. Variational Methods 11. Complex Variable Methods Appendix: General Curvilinear Coordinates References Index

Differential constitutive equations for polymer melts: The extended Pom–Pom model

Abstract: The Pom‐Pom model, recently introduced by McLeish and Larson @J. Rheol. 42, 81‐110~1998!#, is a breakthrough in the field of viscoelastic constitutive equations. With this model, a correct nonlinear behavior in both elongation and shear is accomplished. The original differential equations, improved with local branch-point displacement, are modified to overcome three drawbacks: solutions in steady state elongation show discontinuities, the equation for orientation is unbounded for high strain rates, the model does not have a second normal stress difference in shear. The modified extended Pom‐Pom model does not show the three problems and is easy for implementation in finite element packages, because it is written as a single equation. Quantitative agreement is shown with experimental data in uniaxial, planar, equibiaxial elongation as well as shear, reversed flow and step-strain for two commercial low density polyethylene ~LDPE! melts and one high density polyethylene ~HDPE! melt. Such a good agreement over a full range of well defined rheometric experiments, i.e., shear, including reversed flow for one LDPE melt, and different elongational flows, is exceptional. © 2001 The Society of Rheology. @DOI: 10.1122/1.1380426#

Multi-axial electrical switching of a ferroelectric: theory versus experiment

Abstract: Samples of the polycrystalline ferroelectric ceramic PZT-5H were poled by applying an electric field at room temperature. Subsequently, an electric field was applied to the samples at a range of angles to the poling direction. The measured non-linear responses in electric displacement are used to construct “yield surfaces” in electric field space corresponding to the onset of ferroelectric switching. The results are compared with predictions from three models: (i) a previous self-consistent polycrystal calculation with rate-independent, non-hardening crystal plasticity; (ii) a simplified crystal plasticity model with viscoplastic (rate-dependent) behaviour and a sufficient number of transformation systems to reproduce the polycrystalline behaviour; (iii) a phenomenological model based on rate-independent flow theory, using kinematic hardening and a quadratic yield surface in electric field and stress space. The experiments suggest that the self-consistent crystal plasticity formulation is most able to reproduce the multi-axial electrical response and yield surface of the polycrystal. The phenomenological model is able to reproduce the uniaxial response accurately, but gives relatively poor performance for multi-axial loading paths, in its present form. A tolerable compromise in multi-axial modelling is the simplified crystal plasticity approach. This is able to reproduce multi-axial constitutive behaviour with reasonable accuracy, whilst offering computational simplicity and speed similar to that of the phenomenological model.

Micromechanical analysis of quasi-brittle materials using fracture-based interface elements

Abstract: A microstructural model for the mechanical behaviour of quasi-brittle materials is developed and verified for concrete and bone specimens. The model is based on interface elements equipped with a constitutive law representing non-linear fracture, while continuum elements remain linear elastic. The interface constitutive model is implemented with a sub-stepping scheme. Non-linear geometric effects due to large displacements are included in the model by means of an incremental Lagrangian formulation, although strains in the continuum and relative displacements in the interfaces are assumed to remain small. An arc-length procedure is used to ensure convergence during the highly non-linear behaviour in the post-peak regime. Concrete and bone specimens are idealized as two-phase particle composites and are discretized into finite elements, including interface elements along the main potential crack paths. The numerical results in tension and compression are described and compared with experimental observations. The need of considering non-linear geometric effects in this type of calculations is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Shear strength of rock joints based on quantified surface description

Abstract: One of the primary objectives of this work is to better understand the frictional behavior of joints under shear loads, including the creation of damage zones. Discontinuities have an important influence on the deformational behavior of rock systems. The choice of a general criterion to determine the shear strength of rough rock joints is a general problem that has been investigated for many years. Numerous shear models have been proposed in the last decades to relate shear-strength to measurable joint parameters, but their limitations have to be recognized. The problem is how to measure and then to express the roughness with a number (e.g. JRC) or a mathematical expression in order to introduce the morphology of the joint into a shear strength criterion. In the frame of this work it has been pointed out that the geometry of roughness influences the size and distribution of contact areas during shearing. In order to locate and estimate the contact area during the shearing, it was argued that only the zones of the surface faced to the shear direction, and steeper than a threshold inclination are involved in the shearing. An empirical relation between the potential contact area and the minimal apparent dip inclination of the surface is proposed. The close agreement between this empirical description of the potential contact area, and experimental points permits to predict the real contact area involved in the phenomena. A new constitutive law, relating stress and displacements, is proposed to model the shear resistance of joints under constant normal load conditions. It is based on the empirical surface description, and on the results from more than fifty constant-normal-load direct-shear tests performed on both replicas of tensile joints, and induced tensile fractures for seven rock types. It is shown that this constitutive model is able to describe experimental shear tests realized in laboratory. Moreover, the parameters required in the model can be easily obtained through standard laboratory tests. The proposed model was also used to estimate the JRC value. The expression obtained to evaluate the joint roughness coefficient is capable of predicting the JRC. It was successfully compared with JRC values obtained by back analysis of shear tests. In the current research no attention was paid to investigate the influence of the scale on the shearing. The results have been validated only in the range of the samples tested in laboratory. Further studies are needed to explore the applicability of the proposed model in field conditions.

Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements

Abstract: This paper presents experimental studies on the mechanical behavior of fiber fabrics using a biaxial tensile device based on two deformable parallelograms. The cross-shaped specimens are well adapted to fabrics because of their lack of shear stiffness. Tension versus deformation curves, for different strain ratios, are obtained in the case of composite woven reinforcements used in aeronautic applications. It is shown that the tensile behavior of the fabric is strongly nonlinear due to the weaving undulations and the yarn contraction, and that the phenomenon is clearly biaxial. A constitutive model is described and identified from the experimental data. The essential role played by the yarn crushing will be pointed out.

Incompatibility and a simple gradient theory of plasticity

Abstract: In the continuum theory, at finite strains the crystal lattice is assumed to distort only elastically during plastic flow, while generally the elastic distortion itself is not compatible with a single-valued displacement field. Lattice incompatibility is characterized by a certain skew-symmetry property of the gradient of the elastic deformation field, and this measure can play a natural role in nonlocal theories of plasticity. A simple constitutive proposal is discussed where incompatibility only enters the instantaneous hardening relations. As a result, the incremental boundary value problem for rate-independent and rate-dependent behaviors has a classical structure and rather straightforward modifications of standard finite element programs can be utilized. Two examples are presented in this paper: one for size-scale effects in the torsion of thin wires in the setting of an isotropic J2 flow theory and the other for hardening of microstructures containing small particles embedded in a single crystal matrix.

Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity

Abstract: Simple shear of a constrained strip is analyzed using discrete dislocation plasticity and strain gradient crystal plasticity theory. Both single slip and symmetric double slip are considered. The loading is such that for a local continuum description of plastic flow the deformation state is one of homogeneous shear. In the discrete dislocation formulation the dislocations are all of edge character and are modeled as line singularities in an elastic material. Dislocation nucleation, the lattice resistance to dislocation motion and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. A complementary solution that enforces the boundary conditions is obtained via the finite element method. The discrete dislocation solutions give rise to boundary layers in the deformation field and in the dislocation distributions. The back-extrapolated flow strength for symmetric double slip increases with decreasing strip thickness, so that a size effect is observed. The strain gradient plasticity theory used here is also found to predict a boundary layer and a size effect. Nonlocal material parameters can be chosen to fit some, but not all, of the features of the discrete dislocation results. Additional physical insight into the slip distribution across the strip is provided by simple models for an array of mode II cracks.

The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release

Abstract: The molecular stress function theory for polymer melts is extended to include a new, dissipative convective constraint release process. First the Helmholtz free energy of tube segments with strain-dependent tube diameter is established neglecting constraint release, and it is demonstrated that the molecular stress is a function of the average logarithmic stretch under these conditions. Then convective constraint release is introduced as a dissipative process in the energy balance of tube deformation, which leads to a strain-dependent evolution equation for the molecular stress function. Constraint release is considered to be the consequence of different convection mechanisms for tube orientation and tube cross section. Our new, dissipative constraint release model emphasizes that tube kinematics are fundamentally different for rotational and nonrotational flows, and therefore distinguishes explicitly between simple shear and pure shear (planar extension). For the startup of simple shear and extensional flows, the predictions of our set of constitutive equations consisting of a history integral for the stress tensor and a differential evolution equation for the molecular stress function with only two nonlinear material parameters are in excellent agreement with experimental data of a polydisperse high-density polyethylene (HDPE) and a polydisperse low-density polyethylene (LDPE) melt. Also, stress relaxation after step-shear strain is described for both the HDPE and the LDPE melt.