Constitutive Equation

Classical metal plasticity theory assumes that the hydrostatic pressure has no or negligible effect on the material strain hardening, and that the flow stress is independent of the third deviatoric stress invariant (or Lode angle parameter). However, recent experiments on metals have shown that both the pressure effect and the effect of the third deviatoric stress invariant should be included in the constitutive description of the material. A general form of asymmetric metal plasticity, considering both the pressure sensitivity and the Lode dependence, is postulated. The calibration method for the new metal plasticity is discussed. Experimental results on aluminum 2024-T351 are shown to validate the new material model.

Although the use of Fibre Reinforced Concrete (FRC) for structural applications is continuously increasing, it is still limited with respect to its potentials, mainly due to the lack of International Building Codes for FRC structural elements. Within fib (Féderation Internationale du Béton), the Special Activity Group 5 is preparing a New fib Model Code that aims to update the previous CEB-FIP Model Code 90, published in 1993, that can be considered as the reference document for Eurocode 2. The New Model Code includes several innovations and addresses among other topics, new materials for structural design. In this respect, FRC will be introduced. The Technical Groups fib TG 8.3 ''Fibre reinforced concrete'' and fib TG 8.6 ''Ultra high performance FRC'' are preparing some sections of the New Model Code, including regular and high performance FRC. This paper aims to briefly explain the main concepts behind the structural rules for FRC structural design.

Current seismic design of bridges is based on a displacement performance philosophy using nonlinear static pushover analysis. This type of bridge design necessitates that the geotechnical engineer predict the resistance of the abutment backfill soils, which is inherently nonlinear with respect to the displacement between soil backfill and the bridge structure. This paper employs limit-equilibrium methods using mobilized logarithmic-spiral failure surfaces coupled with a modified hyperbolic soil stress-strain behavior ͑LSH model͒ to estimate abutment nonlinear force-displacement capacity as a function of wall displacement and soil backfill properties. The calculated force-displacement capacity is validated against the results from eight field experiments conducted on various typical structure backfills. Using LSH and experimental data, a simple hyperbolic force-displacement ͑HFD͒ equation is developed that can provide the same results using only the backfill soil stiffness and ultimate soil capacity. HFD is compatible with current CALTRANS practice in regard to the seismic design of bridge abutments. The LSH and HFD models are powerful and effective tools for practicing engineers to produce realistic bridge response for performance-based bridge design.

Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodol-* Corresponding author: Rua Luis Englert, s/n., CEP 90040-040, unstructured and non orthogonal meshes, moving meshes, large variety of interpolation schemes and solvers for the linear discretized system, and the possibility of data processing parallelization. Linear Maxwell, Oldroyd-B, Giesekus, Phan-Thien-Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom-Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was validated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.

A constitutive equation is derived from a Lodge-Yamamoto type of network theory for polymeric fluids. The network junctions are not assumed to move strictly as points of the continuum but allowed a certain "effective slip". The rates of creation and destruction of junctions are assumed to depend on the instantaneous elastic energy of the network, or equivalently, the average extension of the network strand, in a simple manner. Agreement between model predictions and the I.U.P.A.C. data on L.D.P.E. is good.

A methodology to determine the constitutive parameters for the simulation of progressive delamination is proposed. The procedure accounts for the size of a cohesive finite element and the length of the cohesive zone to ensure the correct dissipation of energy. In addition, a closed-form expression for estimating the minimum penalty stiffness necessary for the constitutive equation of a cohesive finite element is presented. It is shown that the resulting constitutive law allows the use of coarser finite element meshes than is usually admissible, which renders the analysis of large-scale progressive delamination problems computationally tractable.

When copper is deformed to high plastic strain (y ~ 34) at high strain rates (~ ~ I04 s -1) a microstructure with grain sizes of ~0.1 am can be produced. It is proposed that this microstructure develops by dynamic recrystallization, which is enabled by the adiabatic temperature rise. By shock-loading the material, and thereby increasing its flow stress, the propensity for dynamic recrystallization can be enhanced. The grain size-flow stress relationship observed after cessation of plastic deformation is consistent with the general formulation proposed by Derby [Acta metall, mater. 39, 955 (1991)]. The temperatures reached by the specimens during dynamic deformation are calculated from a constitutive equation and are found to be, for the shock-loaded material, in the 500-800 K range; these temperatures are consistent with static annealing experiments on shock-loaded specimens, that show the onset of static recrystallization at 523 K. A possible recrystallization mechanism is described and its effect on the mechanical response of copper is discussed.

This paper first presents a complete formulation of a constitutive model that deals with the irreversible behaviour of unsaturated soils under various loading and drying/wetting conditions. A standard form of incremental stress-strain relations is derived. The constitutive model is then cast into the thermodynamical theories and verified using the thermomechanical principles. It is shown that hydraulic hysteresis does not contribute to the plastic dissipation, though it contributes to the plastic work. All plastic work associated with a plastic increment of the degree of saturation is stored and can be recovered in a reversed plastic increment of saturation. The incremental constitutive equations are also reformulated for implementation in finite element codes where displacements and pore pressures are primary unknowns. Qualitative predictions of the constitutive model show that incorporating two suction related yield surfaces and non-associated flow rules into the Barcelona Basic Model opens a full range of possibilities in modelling unsaturated soil behaviour.

A plastic-damage constitutive model for plain concrete is developed in this work. Anisotropic damage with a plasticity yield criterion and a damage criterion are introduced to be able to adequately describe the plastic and damage behavior of concrete. Moreover, in order to account for different effects under tensile and compressive loadings, two damage criteria are used: one for compression and a second for tension such that the total stress is decomposed into tensile and compressive components. Stiffness recovery caused by crack opening/closing is also incorporated. The strain equivalence hypothesis is used in deriving the constitutive equations such that the strains in the effective (undamaged) and damaged configurations are set equal. This leads to a decoupled algorithm for the effective stress computation and the damage evolution. It is also shown that the proposed constitutive relations comply with the laws of thermodynamics. A detailed numerical algorithm is coded using the user subroutine UMAT and then implemented in the advanced finite element program ABAQUS. The numerical simulations are shown for uniaxial and biaxial tension and compression. The results show very good correlation with the experimental data.

This paper describes a combined fracture-plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey-Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.

The response of brick masonry walls to in-plane horizontal cyclic loads analogous to those induced during seismic events is analysed by applying constitutive models which take into account the mechanical behaviour of each component and its interfaces, i.e. decohesion and slipping in the mortar joints and failure in bricks. To this end, a damage model for mortar joints is proposed and then applied in two different approaches to the analysis of brick masonry walls which are described both in the present paper and in the companion paper. The response of the mortar joint model to varying stress and strain is here analysed and applied to the simulation of experimental results from shear tests on mortar-brick assemblages. From the description of the mortar joints and assuming brittle constitutive equations for the brick units, a composite model based on a finite element approach is here developed and applied to the lateral analysis of rectangular shear walls. Even if this model turns out to be computationally burdensome, it may give information on the inelastic mechanisms and related strains by means of a local description of the element motion. Some simulations of the lateral response of experimented walls under cyclic horizontal actions superimposed on vertical loads are carried out and an interpretation of the influence of the wall geometry on the lateral stiffness degradation and on the hysteretic energy dissipation is given. 1997 by John Wiley & Sons, Ltd.

In the present paper, thermal and mechanical behavior of neat polypropylene, 40 wt.% talc-filled polypropylene and 5 wt.% silicate-clay filled polypropylene nanocomposite were tested and evaluated. TGA and DSC results show that filling the talc particle or nano clay into polypropylene can increase the decomposition temperature, but has no effect on glass transition temperature and melt temperature. Tensile results show that 40 wt.% talc particle can increase the modulus, but decrease the yield strength, 5 wt.% nano clay can improve both modulus and yield strength by 90 and 5%, respectively. Fatigue results show that the nanophased polypropylene exhibited the highest fatigue performance, and fatigue strength coefficient, σ f , of nanophased polypropylene was 13.3% higher than that of the neat propyropylene and 50% higher than that of talc-filled polypropylene. At last, a three-parameter non-linear constitutive equation was developed to describe strain rate depend strain softening behavior of three materials.

The material coefficients of Biot's anisotropic poroelasticity are interpreted following micromeehanieal considerations. The adoption of the micro-homogeneity and micro-isotropy assumptions leads to a most practical model for laboratory measurement: the material is defined within 21 drained elastic constants Mijkl, a solid grain bulk modulus Ks, and a Biot modulus M. To facilitate engineering applications, relations among Hookean, engineering and micromechanical constitutive constants are explicitly and extensively listed. These relations are used to convert a set of anisotropic data measured in traditional engineering tests under undrained condition to various continuum mechanical constants.

This article is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters, or buried objects such as cracks. These inverse problems are considered mainly for threedimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e. fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.

The embodied and situated approach to artificial intelligence (AI) has matured and become a viable alternative to traditional computationalist approaches with respect to the practical goal of building artificial agents, which can behave in a robust and flexible manner under changing real-world conditions. Nevertheless, some concerns have recently been raised with regard to the sufficiency of current embodied AI for advancing our scientific understanding of intentional agency. While from an engineering or computer science perspective this limitation might not be relevant, it is of course highly relevant for AI researchers striving to build accurate models of natural cognition. We argue that the biological foundations of enactive cognitive science can provide the conceptual tools that are needed to diagnose more clearly the shortcomings of current embodied AI. In particular, taking an enactive perspective points to the need for AI to take seriously the organismic roots of autonomous agency and sense-making. We identify two necessary systemic requirements, namely constitutive autonomy and adaptivity, which lead us to introduce two design principles of enactive AI. It is argued that the development of such enactive AI poses a significant challenge to current methodologies. However, it also provides a promising way of eventually overcoming the current limitations of embodied AI, especially in terms of providing fuller models of natural embodied cognition. Finally, some practical implications and examples of the two design principles of enactive AI are also discussed.

Current seismic design of bridges is based on a displacement performance philosophy using nonlinear static pushover analysis. This type of bridge design necessitates that the geotechnical engineer predict the resistance of the abutment backfill soils, which is inherently nonlinear with respect to the displacement between soil backfill and the bridge structure. This paper employs limit-equilibrium methods using mobilized logarithmic-spiral failure surfaces coupled with a modified hyperbolic soil stress-strain behavior ͑LSH model͒ to estimate abutment nonlinear force-displacement capacity as a function of wall displacement and soil backfill properties. The calculated force-displacement capacity is validated against the results from eight field experiments conducted on various typical structure backfills. Using LSH and experimental data, a simple hyperbolic force-displacement ͑HFD͒ equation is developed that can provide the same results using only the backfill soil stiffness and ultimate soil capacity. HFD is compatible with current CALTRANS practice in regard to the seismic design of bridge abutments. The LSH and HFD models are powerful and effective tools for practicing engineers to produce realistic bridge response for performance-based bridge design.

A constitutive model to simulate the behavior of ceramic materials under impact loading is proposed in order to achieve a better representation of the damage process due to the material fragmentation. To integrate the proposed constitutive equations, a semi-implicit algorithm (implicit for the stresses and explicit for the damage variable) has been developed, leading to the generalized expressions of the classical return mapping algorithm. The model was implemented in a commercial finite element code and its performance was demonstrated by comparing its predictions with experimental results obtained by other authors.► A constitutive model to simulate the behavior of ceramic materials under impact. ► Yield stress depends on pressure and damage. ► Inelastic flow governed by a non-associated law. ► Developed the corresponding return-mapping algorithm. ► Parameters of the model may be easily determined.

A new computational uid-dynamic (CFD) model with a separate granular temperature (2/3 random particle kinetic energy per unit of mass) equation for each phase or particle size was developed using constitutive equations derived earlier by Huilin, Gidaspow and Manger. In agreement with the experiment and model of Mathiesen, Solberg and Hjertager the new model computes the observed core-annular ow regime. It predicts the trends of the observed radial and axial particle diameter distributions. For elastic particles the computed particle velocity distributions are parabolic. They are close to the laminar type approximate analytical solution for ow in a pipe, where the mean velocity equals the inlet ux divided by the particle density and volume fraction. The computed turbulent intensity is lower for large particles than for small particles, as measured. This is in agreement with an approximate analytical solution for the granular temperature in the developed ow region of a riser for elastic particles. Computations show that for suciently inelastic particles the granular temperature in the center can be lower than near the wall resembling the measured particle uctuating velocity distribution. ?

From MRI velocimetry we measure the local flow characteristics of a Carbopol gel in a Couette geometry under different inner cylinder rotation velocities. Associated with torque data under the same flow conditions we deduce the local, steady-state, simple shear, constitutive equation of the material within a relatively wide range of shear rates [10 −2 ; 100 s −1 ]. Then we show that in this range of shear rates this "local" behaviour is in excellent agreement with the "macroscopic" behaviour deduced from conventional rheometry with cone and plate and Couette geometries. We can conclude that this material effectively behaves as a simple yield stress fluid with a constitutive equation well represented by a Herschel-Bulkley model. This behaviour, likely due to the soft-jammed structure of the fluid, contrasts with that of aggregated systems which exhibit thixotropy and shear-banding at low shear rates.

The conception of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.

The uniaxial monotonic compressive behavior of tuff masonry has been studied by laboratory tests carried out along the directions parallel and orthogonal to the mortar bed joints. Based on the experimental results, different nonlinear stress-strain models have been defined for each loading direction to account for the orthotropy of masonry and to describe strain-softening behavior. Maximum likelihood, mean, and characteristic stress versus strain diagrams have been derived by means of different techniques of statistical analysis to be employed in the structural analysis of both new and existing constructions. Two different analytical models have been obtained via nonlinear regression analysis for defining stress-strain curves for tuff masonry to be used in the current design/assessment procedures. Such constitutive models require only the peak compressive strength and the relevant yielding strain as input data. The allowable strain ductility of masonry can also be explicitly taken into account. Strength, deformability, and ductility parameters have also been identified to run equivalent linear analyses. To this end, simple relationships are provided to define both Young's and shear moduli from compressive strength.

This paper presents a theoretical study of the thermally driven behavior of a shape memory alloy (SMA)/FGM actuator under arbitrary loading and boundary conditions by developing an integrated mathematical model. The model studied is established on the geometric parameters of the three-dimensional laminated composite box beam as an actuator that consists of a functionally graded core integrated with SMA actuator layers with a uniform rectangular cross section. The constitutive equation and linear phase transformation kinetics relations of SMA layers based on Tanaka and Nagaki model are coupled with the governing equation of the actuator to predict the stress history and to model the thermo-mechanical behavior of the smart shape memory alloy/FGM beam. Based on the classical laminated beam theory, the explicit solution to the structural response of the structure, including axial and lateral deflections of the structure, is investigated. As an example, a cantilever box beam subjected to a transverse concentrated load is solved numerically. It is found that the changes in the actuator's responses during the phase transformation due to the strain recovery are significant.

A constitutive model for polycrystalline metals is established within a micromechanical framework. The inelastic deformation is defined by the formation and annihilation of dislocations together with grain refinement due to continuous dynamic recrystallization. The recrystallization studied here occurs due to plastic deformation without the aid of elevated temperatures. The grain refinement also influences the evolution of the dislocation density since the recrystallization introduces a dynamic recovery as well as additional grain and subgrain boundaries, hindering the movement of dislocations through the material microstructure. In addition, motivated by experimental evidence, the rate dependence of the material is allowed to depend on the grain size. Introducing a varying grain size into the evolution of the dislocation density and in the rate dependence of the plastic deformation are believed to be important and novel features of the present model. The proposed constitutive model is implemented in a numerical scheme allowing calibration against experimental results, which is shown using commercial-purity aluminum as example material. The model is also employed in macroscale simulations of grain refinement in this material during extensive inelastic deformation.

Gelatin gel properties have been studied through the evolution of the storage [G' (co)] and the loss [G" (co)] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G'(co)a G"(co)aco ~ and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tanfi = G"/G'= cst= tan(A.n/2) and it may be precisely located using the variations of tan fi versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of A which were generally in the 0.60-0.72 range but could be as low as 0.20-0.30 in some experimental conditions, were compared with published and theoretical values.

In this research, the plastic flow behavior of Al-6%Mg alloy was studied by analyzing the results of hot compression tests in a range of temperature and strain rate. Then, an artificial neural network (ANN) model was trained at which the temperature, strain-rate, and strain parameters were used as the input layer and the flow stress as the output. The comparison of the predicted and experimental results of stress-strain curve proved the prediction capability of the ANN model.

Direct shear experiments on ground surfaces of a granodiorite from Raymond, California, at normal stresses of •6 MPa demonstrate that competing time, displacement, and velocity effects control rock friction. It is proposed that the strength of the population of points of contacts between sliding surfaces determines frictional strength and that the population of contacts changes continuously with displacements. Previous experiments demonstrate that the strength of the contacts increases with the age of the contacts. The present experiments establish that a characteristic displacement, proportional to surface roughness, is required to change the population of contacts. Hence during slip the average age of the points of contact and therefore frictional strength decrease as slip velocity increases. Displacement weakening and consequently the potential for unstable slip occur whenever displacement reduces the average age of the contacts. In addition to this velocity dependency, which arises from displacement dependency and time dependency, the experiments also show a competing but transient increase in friction whenever slip velocity increases. Creep of the sliding surface at stresses below that for steady state slip is also observed. Constitutive relationships are developed that permit quantitative simulation of the friction versus displacement data as a function of surface roughness and for different time and velocity histories. Unstable slip in experiments is controlled by these constitutive effects and by the stiffness of the experimental system. It is argued that analogous properties control earthquake instability.

This paper presents the equations for the implementation of rotational quaternions in the geometrically exact three-dimensional beam theory. A new finite-element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam. The formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form. A strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented. Several numerical examples demonstrate the validity, performance and accuracy of the proposed approach.

Ductile fracture is often considered as the consequences of the accumulation of plastic damage. This paper is concerned with the application of a recently developed damage plasticity theory incorporates the pressure sensitivity and the Lode angle dependence into a nonlinear damage rule and the material deterioration. The ductile damaging process is calculated through the so-called “cylindrical decomposition” method. The constitutive

An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff's law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff's law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium,

A new method of modeling material behavior which accounts for the dynamic metallurgical processes occurring during hot deformation is presented. The approach in this method is to consider the workpiece as a dissipator of power in the total processing system and to evaluate the dissipated power co-contentJ = ∫oσ ε ⋅dσ from the constitutive equation relating the strain rate (ε) to the flow stress (σ). The optimum processing conditions of temperature and strain rate are those corresponding to the maximum or peak inJ. It is shown thatJ is related to the strain-rate sensitivity (m) of the material and reaches a maximum value(J max) whenm = 1. The efficiency of the power dissipation(J/J max) through metallurgical processes is shown to be an index of the dynamic behavior of the material and is useful in obtaining a unique combination of temperature and strain rate for processing and also in delineating the regions of internal fracture. In this method of modeling, noa priori knowledge or evaluation of the atomistic mechanisms is required, and the method is effective even when more than one dissipation process occurs, which is particularly advantageous in the hot processing of commercial alloys having complex microstructures. This method has been applied to modeling of the behavior of Ti-6242 during hot forging. The behavior of α+ β andβ preform microstructures has been exam-ined, and the results show that the optimum condition for hot forging of these preforms is obtained at 927 °C (1200 K) and a strain rate of 1CT•3 s•1. Variations in the efficiency of dissipation with temperature and strain rate are correlated with the dynamic microstructural changes occurring in the material.

This paper presents a new yield criterion for orthotropic sheet metals and its implementation in a theoretical model of the forming limit diagrams. The equivalent stress equation shows that the shape of the yield surface is defined by eight material parameters. The minimisation of an error-function has been used for the numerical identification of these coefficients. The parameters are established in such a way that the constitutive equation associated to the yield surface reproduces the plastic behaviour of the actual material. The uniaxial yield stresses (σ 0 ,  σ 45 , σ 90 ), biaxial yield stress (σ b ), uniaxial anisotropy coefficients (r 0 , r 45 , r 90 ) have been used in identification. The new yield criterion has been implemented in the Marciniak-Kuczynski theory in order to predict the limit strains. The theoretical forming limit curves have been compared with the experimental ones. The friction free tests, the hydraulic bulge test (for the positive minor strains) and the tensile test for plane strain and for uniaxial tensile test (for the negative minor strains) are used. The predicted yield surface and forming limit diagrams for AA5182-0 aluminium alloy sheets are in good agreement with the experimental ones.

Modeling of metal cutting has proved to be particularly complex, especially for tool-chip interface. The present work is mainly aimed to investigate the limiting shear stress at this interface in the case of Titanium alloy (Ti-6Al-4V) dry cutting based on a FE-model. It is first shown that the surface limiting shear stress was linked to the contact pressure and the coefficient of friction (CoF). A relationship between CoF and the limiting shear stress was given, and the effect of the temperature on the limiting shear stress was also considered. After that, an orthogonal cutting model was developed with an improved friction model through the user subroutine VFRIC in Abaqus/ Explicit software. The numerical results obtained were compared with experimental data gathered from literature and a good overall agreement was found. Finally, the effects of cutting speed, CoF and tool-rake angle on chip morphologies were analyzed.

This work deals with the virtual simulation of the sheet metal stamping process. The main objective is to predict when and where the cracks can appear in the workpiece during the forming operation. A local approach based on the strong coupling between anisotropic elastoplasticity with mixed nonlinear work hardening (isotropic and kinematic) and an isotropic ductile damage is proposed. The theoretical and numerical aspects of the constitutive equations are, first, presented. The resolution of the resulting system of equations is carried out via a Vumat user material, using ABAQUS/Explicit finite element code. The results obtained, in the context of Swift's benchmark deep-drawing test show the efficiency and the potential interest of the proposed damage model.

The mechanical behaviour of bonded geomaterials is described by means of an elastoplastic strainhardening model. The internal variables, taking into account the 'history' of the material, depend on the plastic strains experienced and on a conveniently defined scalar measure of damage induced by weathering and/or chemical degradation.

A general framework for the analysis of heterogeneous media that assess a strong coupling between viscoplasticity and anisotropic viscodamage evolution is formulated for impact related problems within the framework of thermodynamic laws and nonlinear continuum mechanics. The proposed formulations include thermo-elasto-viscoplasticity with anisotropic thermo-elasto-viscodamage, a dynamic yield criterion of a von Mises type and a dynamic viscodamage criterion, the associated flow rules, non-linear strain hardening, strainrate hardening, and temperature softening. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. That is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The evolution laws are impeded in a finite deformation framework based on the multiplicative decomposition of the deformation gradient into elastic, viscoplastic, and viscodamage parts. Since the material macroscopic thermomechanical response under high-impact loading is governed by different physical mechanisms on the macroscale level, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (viscoplastic flow) and crack and void interactions. The non-local theory of viscoplasticity and viscodamage that incorporates macroscale interstate variables and their higher-order gradients is used here to describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related viscoplasticity and viscodamage hardening variables. The gradients are introduced here in the hardening internal state variables and are considered independent of their local counterparts. It also incorporates the thermomechanical coupling effects as well as the internal dissipative effects through the rate type covariance constitutive structure with a finite set of internal state variables. The model presented in this paper can be considered as a framework, which enables to derive various non-local and gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions. the macroscale to the mesoscale level. The non-local theories are expanding steadily in order to appropriately overcome this problem, which take into account the influence of the n-the nearest neighbor of the material points or the long-range microstructural interaction.

In this study we have used a convergent and highly accurate mixed finite element technique to model the effect of fluid elasticity on the flow kinematics and the stress distribution in lid driven cavity flow. Our work is motivated by the desire to capture the important physical aspects of the basic flow and thus to better understand the purely elastic instability in recirculating flows which has been reported in the literature elsewhere [A.M. Grillet, E.S.G. Shaqfeh, Observations of viscoelastic instabilities in recirculation flows of Boger fluids, J. Non-Newtonian Fluid Mech. 64 (1996) 141±155; P. Pakdel, G.H. McKinley, Cavity flows of elastic liquids: purely elastic instablities, Phys. Fluids 10 ]. In our numerical investigations we have treated the corner singularities by incorporating a controlled amount of leakage which allows the computation of fully elastic mesh converged solutions. We begin by validating our Newtonian cavity results against previous work to show that the introduction of leakage does not appreciably modify the cavity recirculation flow. Then we examine the polymer stresses to understand how elasticity changes the flow kinematics, slowing the primary recirculation vortex and causing the vortex center to shift opposite of the direction of lid motion. Variations of the cavity aspect ratio are also explored. Focusing on the corners we find that the leakage relieves the corner singularities and moreover, finite leakage helps explain the unusual behavior seen in the radial velocity in experiments. Finally, we have reexamined the previously proposed mechanisms for elastic instability in this flow and put forth a new instability mechanism. Together, these mechanisms may better explain the complex aspect ratio dependence of the onset of elastic instability in lid driven cavity flow. # (B. Khomami) 0377-0257/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 -0 2 5 7 ( 9 9 ) 0 0 0 1 5 -4

Dimensional changes between a pattern die and its corresponding investment cast part occur as a result of complex phenomena such as thermal expansion/contraction and hot deformation (elastic, plastic, and creep) during the processing of the pattern material (wax), mold material (shell), and solidifying alloy. In this study, wax pattern dimensions are determined using computer models that take into account the thermophysical and rheological behavior of the wax. Pattern dimensions are calculated using a three-dimensional finite element model for coupled thermal and mechanical analysis developed within the commercial software ABAQUS TM . Cerita TM 29-51, an industrial wax is considered in this study. The effects of restraint geometrical features by the metal die, and process parameters such as dwell time, die/platen temperature, injection pressure and injection temperature are also considered. The results of numerical simulation are compared with experimental measurements on test patterns. Measurements of the temperature at the injection port, pressure in the die cavity, and the results of numerical simulations provide several insights into the wax injection process. Viscoelastic models exhibit good potential for accurately capturing the details of wax pattern deformation. Conversely, pure elastic models are unsuitable for predicting deformation, as any distortion is completely relaxed once the pattern is removed from the die.

Starting from an analysis of the rheological behavior of the complex modulus predicted by the Cole-Cole formalism, a generalized Cole-Cole ansatz is suggested in order to overcome the related difficulties. The corresponding rheological constitutive equation with fractional derivatives belonging to the generalized Cole-Cole respondance is stated and the characteristic material functions of the linear viscoelasticity theory (like the dynamic modulus and compliance, the relaxation and ratardation functions, the spectra, etc.) are derived. Model predictions of these functions will be compared with experimental results from dynamical measurements and creep data on different polymer systems which show cooperative phenomena (polymeric glasses and gelling systems). One can see that the modified ansatz fits the data very well, in spite of its relative simplicity.

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and twokinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.

The main feature of the interlocking hollow block masonry is the replacement of mortar layers commonly used in bonded masonry with interlocking keys (protrusions and grooves). This study covers the modelling and the analysis of interlocking mortarless ungrouted (hollow) and grouted concrete block system subjected to axial compression loads using FEM. The main features simulated in the developed finite element code are the mechanical characteristics of the interlocking dry joints including the geometric imperfection of the shell beds of the blocks, the interaction between block units, the progressive debonding between the block and grout and material nonlinearity. The applicability of the proposed FE model is investigated by demonstrating the nonlinear structural response and failure mechanism of individual block, ungrouted and grouted interlocking mortarless prisms. The results found show good agreement with the experimental test results.

Formability of sheet metal is usually assessed by the useful concept of forming limit diagrams (FLD) and forming limit curves (FLC) represent a first safety criterion for deep drawing operations. The level of FLC is strongly strain path dependent as observed by experimental and numerical results and therefore nonproportional strain paths need to be incorporated when analyzing formability of sheet metal components. Simulations using finite element method allow accurate predictions of stress and strain distributions in complex stamped parts. However, the prediction of localized necking is a difficult task and the combination of forming limit diagram analysis with finite element simulations often fail to give the right answer, if complex strain paths are not included in these predictions.

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.

AbstractÐA phase mixture model in which a polycrystalline material is regarded as a mixture of a crystalline phase and a grain-boundary phase is presented. The model aims to describe the plastic deformation behaviour of ®ne-grained materials. The mechanical properties of the crystalline phase are modelled using uni®ed viscoplastic constitutive relations, which take dislocation density evolution and diusion creep into account. The total strain rate of a crystallite is calculated by summation of the contributions of dislocation, boundary diusion and lattice diusion mechanisms. The deformation mechanism for the grain-boundary phase is modelled as a diusional¯ow of matter through the grain boundary. Using a simple rule of mixtures, the grain size dependence of the overall plastic deformation behaviour of the material is analysed. Rate eects are also investigated. The results of the calculations are compared with previously published experimental data. #

This paper is a contribution to the study of tunnelling in difficult conditions, with attention paid to large time-dependent deformations, which may develop either during construction, causing instabilities of the tunnel heading and of the face, or during the service life of the tunnel. Under these circumstances the construction costs may rise due to the delays in excavation time, the stabilisation and heavy support measures that need be adopted. Following a review of characterisation and modelling of time-dependent behaviour in rock, the mechanical behaviour of Clay Shales (CS), a structurally complex formation of the Apennines (Italy), is described. Then, the key factors involved in the selection of the constitutive model for CS are identified. Two constitutive models are selected and discussed and their specific material parameters determined. A case study of a large size tunnel is presented where numerical modelling by the finite difference method is carried out. The results of modelling are compared with the monitoring data in terms of radial convergence of the tunnel and extrusion of the tunnel face.