Showing papers on "Viscoplasticity published in 2009"

Applied Mechanics of Solids

Abstract: 1 Overview of Solid Mechanics DEFINING A PROBLEM IN SOLID MECHANICS 2 Governing Equations MATHEMATICAL DESCRIPTION OF SHAPE CHANGES IN SOLIDS MATHEMATICAL DESCRIPTION OF INTERNAL FORCES IN SOLIDS EQUATIONS OF MOTION AND EQUILIBRIUM FOR DEFORMABLE SOLIDS WORK DONE BY STRESSES: PRINCIPLE OF VIRTUAL WORK 3 Constitutive Models: Relations between Stress and Strain GENERAL REQUIREMENTS FOR CONSTITUTIVE EQUATIONS LINEAR ELASTIC MATERIAL BEHAVIORSY HYPOELASTICITY: ELASTIC MATERIALS WITH A NONLINEAR STRESS-STRAIN RELATION UNDER SMALL DEFORMATION GENERALIZED HOOKE'S LAW: ELASTIC MATERIALS SUBJECTED TO SMALL STRETCHES BUT LARGE ROTATIONS HYPERELASTICITY: TIME-INDEPENDENT BEHAVIOR OF RUBBERS AND FOAMS SUBJECTED TO LARGE STRAINS LINEAR VISCOELASTIC MATERIALS: TIME-DEPENDENT BEHAVIOR OF POLYMERS AT SMALL STRAINS SMALL STRAIN, RATE-INDEPENDENT PLASTICITY: METALS LOADED BEYOND YIELD SMALL-STRAIN VISCOPLASTICITY: CREEP AND HIGH STRAIN RATE DEFORMATION OF CRYSTALLINE SOLIDS LARGE STRAIN, RATE-DEPENDENT PLASTICITY LARGE STRAIN VISCOELASTICITY CRITICAL STATE MODELS FOR SOILS CONSTITUTIVE MODELS FOR METAL SINGLE CRYSTALS CONSTITUTIVE MODELS FOR CONTACTING SURFACES AND INTERFACES IN SOLIDS 4 Solutions to Simple Boundary and Initial Value Problems AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC LINEAR ELASTIC PROBLEMS AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC ELASTIC-PLASTIC PROBLEMS SPHERICALLY SYMMETRIC SOLUTION TO QUASI-STATIC LARGE STRAIN ELASTICITY PROBLEMS SIMPLE DYNAMIC SOLUTIONS FOR LINEAR ELASTIC MATERIALS 5 Solutions for Linear Elastic Solids GENERAL PRINCIPLES AIRY FUNCTION SOLUTION TO PLANE STRESS AND STRAIN STATIC LINEAR ELASTIC PROBLEMS COMPLEX VARIABLE SOLUTION TO PLANE STRAIN STATIC LINEAR ELASTIC PROBLEMS SOLUTIONS TO 3D STATIC PROBLEMS IN LINEAR ELASTICITY SOLUTIONS TO GENERALIZED PLANE PROBLEMS FOR ANISOTROPIC LINEAR ELASTIC SOLIDS SOLUTIONS TO DYNAMIC PROBLEMS FOR ISOTROPIC LINEAR ELASTIC SOLIDS ENERGY METHODS FOR SOLVING STATIC LINEAR ELASTICITY PROBLEMS THE RECIPROCAL THEOREM AND APPLICATIONS ENERGETICS OF DISLOCATIONS IN ELASTIC SOLIDS RAYLEIGH-RITZ METHOD FOR ESTIMATING NATURAL FREQUENCY OF AN ELASTIC SOLID 6 Solutions for Plastic Solids SLIP-LINE FIELD THEORY BOUNDING THEOREMS IN PLASTICITY AND THEIR APPLICATIONS 7 Finite Element Analysis: An Introduction A GUIDE TO USING FINITE ELEMENT SOFTWARE A SIMPLE FINITE ELEMENT PROGRAM 8 Finite Element Analysis: Theory and Implementation GENERALIZED FEM FOR STATIC LINEAR ELASTICITY THE FEM FOR DYNAMIC LINEAR ELASTICITY FEM FOR NONLINEAR (HYPOELASTIC) MATERIALS FEM FOR LARGE DEFORMATIONS: HYPERELASTIC MATERIALS THE FEM FOR VISCOPLASTICITY ADVANCED ELEMENT FORMULATIONS: INCOMPATIBLE MODES, REDUCED INTEGRATION, AND HYBRID ELEMENTS LIST OF EXAMPLE FEA PROGRAMS AND INPUT FILES 9 Modeling Material Failure SUMMARY OF MECHANISMS OF FRACTURE AND FATIGUE UNDER STATIC AND CYCLIC LOADING STRESS- AND STRAIN-BASED FRACTURE AND FATIGUE CRITERIA MODELING FAILURE BY CRACK GROWTH: LINEAR ELASTIC FRACTURE MECHANICS ENERGY METHODS IN FRACTURE MECHANICS PLASTIC FRACTURE MECHANICS LINEAR ELASTIC FRACTURE MECHANICS OF INTERFACES 10 Solutions for Rods, Beams, Membranes, Plates, and Shells PRELIMINARIES: DYADIC NOTATION FOR VECTORS AND TENSORS MOTION AND DEFORMATION OF SLENDER RODS SIMPLIFIED VERSIONS OF THE GENERAL THEORY OF DEFORMABLE ROD EXACT SOLUTIONS TO SIMPLE PROBLEMS INVOLVING ELASTIC RODS MOTION AND DEFORMATION OF THIN SHELLS: GENERAL THEORY SIMPLIFIED VERSIONS OF GENERAL SHELL THEORY: FLAT PLATES AND MEMBRANES SOLUTIONS TO SIMPLE PROBLEMS INVOLVING MEMBRANES, PLATES, AND SHELLS Appendix A: Review of Vectors and Matrices A.1. VECTORS A.2. VECTOR FIELDS AND VECTOR CALCULUS A.3. MATRICES Appendix B: Introduction to Tensors and Their Properties B.1. BASIC PROPERTIES OF TENSORS B.2. OPERATIONS ON SECOND-ORDER TENSORS B.3. SPECIAL TENSORS Appendix C: Index Notation for Vector and Tensor Operations C.1. VECTOR AND TENSOR COMPONENTS C.2. CONVENTIONS AND SPECIAL SYMBOLS FOR INDEX NOTATION C.3. RULES OF INDEX NOTATION C.4. VECTOR OPERATIONS EXPRESSED USING INDEX NOTATION C.5. TENSOR OPERATIONS EXPRESSED USING INDEX NOTATION C.6. CALCULUS USING INDEX NOTATION C.7. EXAMPLES OF ALGEBRAIC MANIPULATIONS USING INDEX NOTATION Appendix D: Vectors and Tensor Operations in Polar Coordinates D.1. SPHERICAL-POLAR COORDINATES D.2. CYLINDRICAL-POLAR COORDINATES Appendix E: Miscellaneous Derivations E.1. RELATION BETWEEN THE AREAS OF THE FACES OF A TETRAHEDRON E.2. RELATION BETWEEN AREA ELEMENTS BEFORE AND AFTER DEFORMATION E.3. TIME DERIVATIVES OF INTEGRALS OVER VOLUMES WITHIN A DEFORMING SOLID E.4. TIME DERIVATIVES OF THE CURVATURE VECTOR FOR A DEFORMING ROD References

Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage

Abstract: A unifying thermomechanical framework is presented that reconciles several classes of gradient elastoviscoplasticity and damage models proposed in the literature during the last 40 years . It is based on the introduction of the micromorphic counterpart ϕχ of a selected state or internal variable ϕ in a standard constitutive model. In addition to the classical balance of momentum equation, a balance of micromorphic momentum is derived that involves generalized stress tensors. The corresponding additional boundary conditions are also deduced from the procedure. The power of generalized forces is assumed to contribute to the energy balance equation. The free energy density function is then chosen to depend on a relative generalized strain, typically ϕ- ϕχ , and the microstrain gradient ∇ ϕχ . When applied to the deformation gradient itself, ϕ≡ F , the method yields the micromorphic theory of Eringen and Mindlin together with its extension to finite deformation elastoviscoplasticity by Forest and Sievert. If...

Non-Linear Mechanics of Materials

Abstract: Preface by Jean Lemaitre Chapter 1 Introduction 1.1. Model construction 1.2. Applications to models Chapter 2 General concepts 2.1. Formulation of the constitutive equations 2.2. Principle of virtual power 2.3. Thermodyna~nicso f irreversible processes 2.4. Main class of constitutive equations 2.5. Yield criteria 2.6. Numerical methods for nonlinear equations 2.7. Numerical solution of differential equations 2.8. Finite element Chapter 3 Plasticity and 3D viscoplasticity 3.1. Generality 3.2. Formulation of the constitutive equations 3.3. Flow direction associated to the classical criteria 3.4. Expression of some particular constitutive equations in plasticity 3.5. Flow under prescribed strain rate 3.6. Non-associated plasticity 3.7. Nonlinear hardening 3.8. Some classical extensions 3.9. Hardening and recovery in viscoplasticity 3.10. Multimechanism models 3.1 1. Behaviour of porous materials Chapter 4 Introduction to damage mechanics 4.1. Introduction 4.2. Notions and general concepts 4.3. Damage variables and state laws 4.4. State and dissipative couplings 4.5. Damage deactivation 4.6. Damage evolution laws 4.7. Examples of damage models in brittle materials Chapter 5 Microstructural mechanics 5.1. Characteristic lengths and scales in microstructural mechanics 5.2. Some homogenization techniques 5.3. Application to linear elastic heterogeneous materials 5.4. Some examples. applications and extensions 5.5. Homogenization in thermoelasticity 5.6. Nonlinear homogenization 5.7. Computation of RVE 5.8. Homogenization of coarse grain structures Chapter 6 Finite deformations 6.1. Geometry and kinematics of continuum 6.2. Sthenics and statics of the continuum 6.3. Constitutive laws 6.4. Application: Simple glide 6.5. Finite deformations of generalized continua Chapter 7 Nonlinear structural analysis 7.1. The material object 7.2. Examples of implementations of particular models 7.3. Specificities related to finite elements Chapter 8 Strain localization 8.1. Bifurcation modes in elastoplasticity 8.2. Regularization methods Appendix Notation used A.1. Tensors A.2. Vectors, Matrices A.3. Voigt notation

A new elastoviscoplastic model based on the Herschel–Bulkley viscoplastic model

Abstract: The aim of this paper is to introduce a new three-dimensional elastoviscoplastic model that combines both the Oldroyd viscoelastic model and the Herschel–Bulkley viscoplastic model with a power-law index n > 0 . The present model is derived to satisfy the second law of thermodynamics. Various fluids of practical interest, such as liquid foams, droplet emulsions or blood, present such elastoviscoplastic behavior: at low stress, the material behaves as a viscoelastic solid, whereas at stresses above a yield stress, the material behaves as a fluid. When n = 1 , a recently introduced elastoviscoplastic model proposed by the author is obtained. When 0 n 1 , then the plasticity criteria becomes smooth, the elongational viscosity is always well defined and the shear viscosity shows a shear thinning behavior. This is a major improvement to the previous elastoviscoplastic model. Finally, when n > 1 , the material exhibits the unusual shear thickening behavior.

Modeling the thixotropic behavior of structured fluids

Abstract: A novel approach for modeling the mechanical behavior of thixotropic viscoplastic fluids is presented. Non-monotonic flow curves, stress overshoot during microstructure breakdown flows at constant shear rate, and viscosity bifurcation are some of the common aspects of structured fluids that are predicted by the new model. It involves two evolution equations, one for the stress and the other for the structure parameter. Simple ideas are employed to describe the microstructure, and, as a result, a model with a clear physical basis is obtained. In addition to the flow curve, which by construction is exactly predicted, it is shown that the model is able to predict correctly the behavior observed in the usual rheometric transient flows, among which abrupt changes in shear rate (microstructure buildup or breakdown experiments) and abrupt changes in shear stress (viscosity bifurcation experiments). The model is frame-indifferent and applicable to complex flows.

A thermo-viscoplastic constitutive model to predict elevated-temperature flow behaviour in a titanium-modified austenitic stainless steel

Abstract: The experimental stress–strain data from isothermal hot compression tests over a wide range of temperatures (1073–1473 K), strains (0.1–0.5) and strain rates (0.001–1 s−1) were employed to formulate a suitable constitutive model to predict the elevated-temperature deformation behaviour in a Ti-modified austenitic stainless steel (alloy D9). It was observed that the Johnson–Cook (JC) model in its original form is inadequate to provide good description of flow behaviour of alloy D9 in the above hot working domain. This has been attributed to the inadequacy of the JC model to incorporate the coupled effects of strain and temperature and of strain rate and temperature. A modified constitutive model based on the Zerilli–Armstrong model has been proposed for considering the effects of thermal softening, strain rate hardening and isotropic hardening as well as the coupled effects of temperature and strain and of strain rate and temperature on flow stress. The proposed modified constitutive model could predict the elevated-temperature flow behaviour of alloy D9 over the specified hot working domain of alloy D9 with good correlation and generalization.

Visco-Plastic Behaviour around Advancing Tunnels in Squeezing Rock

Abstract: The visco-plastic behaviour of rocks plays a relevant role in the tunnelling works, especially for deep tunnels subjected to large initial stresses for which squeezing conditions may develop. A rheological model is discussed that accounts for visco-elastic (primary) and visco-plastic (secondary) contributions to rock creep. The effects of tertiary creep are included in the model by way of a gradual mechanical damage governed by the cumulated visco-plastic strains. The parameters of the intact rock are first identified based on laboratory test results presented in the literature. Then, after scaling them to those of the rock mass, the potential applicability of the model is tested through axisymmetric and plane strain finite element analyses of the full face excavation of a deep circular tunnel. The results are discussed with particular reference to the short term redistribution of stresses around the opening and to its convergence. The analyses show the relevant influence of tertiary creep on the tunnel closure. In addition, those based on an axisymmetric scheme turn out to be crucial for the correct long term prediction of the interaction between the rock mass and the supporting structure of the opening.

Micromechanical approach to the strength properties of frictional geomaterials

Abstract: The present paper describes a micromechanics-based approach to the strength properties of composite materials with a Drucker–Prager matrix in the situation of non-associated plasticity. The concept of limit stress states for such materials is first extended to the context of homogenization. It is shown that the macroscopic limit stress states can theoretically be obtained from the solution to a sequence of viscoplastic problems stated on the representative elementary volume. The strategy of resolution implements a non-linear homogenization technique based on the modified secant method. This procedure is applied to the determination of the macroscopic strength properties and plastic flow rule of materials reinforced by rigid inclusions, as well as for porous media. The role of the matrix dilatancy coefficient is in particular discussed in both cases. Finally, finite element solutions are derived for a porous medium and compared to the micromechanical predictions.

A finite-strain model for anisotropic viscoplastic porous media: I – Theory

Abstract: In this work, we propose an approximate homogenization-based constitutive model for estimating the effective response and associated microstructure evolution in viscoplastic (including ideally-plastic) porous media subjected to finite-strain loading conditions. The proposed model is based on the "second-order" nonlinear homogenization method, and is constructed in such a way as to reproduce exactly the behavior of a "composite-sphere assemblage" in the limit of hydrostatic loading and isotropic microstructure. However, the model is designed to hold for completely general three-dimensional loading conditions, leading to deformation-induced anisotropy, whose development in time is handled through evolution laws for the internal variables characterizing the instantaneous "ellipsoidal" state of the microstructure. In Part II of this study, results will be given for the instantaneous response and microstructure evolution in porous media for several representative loading conditions and microstructural configurations.

Development of severe plastic deformation by various asymmetric rolling processes

Abstract: Rolling is the most appropriate process to produce a severe plastic deformation in the sheet or plate type of materials. Compared to symmetric rolling, asymmetric rolling is more efficient in producing plastic deformation since it develops additional shear strain for a given reduction in thickness. In the present investigation, various asymmetric rolling processes were analyzed by the rigid-viscoplastic finite element method. The results of the analyses were compared to each other in terms of plastic deformation: effects of asymmetries due to the differences in size, rotational speed and frictional condition between upper and lower rolls. Rolling pressure distribution, rolling force and rolling torque were also compared.

Drop impact of yield-stress fluids

Abstract: The normal impact of a drop of yield-stress fluid on a flat rigid surface is investigated experimentally. Using different model fluids (polymer microgels, clay suspensions) and impacted surfaces (partially wettable, super-hydrophobic), we find a rich variety of impact regimes from irreversible viscoplastic coating to giant elastic spreading and recoil. A minimal model of inertial spreading, taking into account an elasto-viscoplastic rheology, allows explaining in a single framework the different regimes and scaling laws. In addition, semi-quantitative predictions for the spread factor are obtained when the measured rheological parameters of the fluid (elasticity, yield stress, viscosity) are injected into the model. Our study offers a means to probe the short-time rheology of yield-stress fluids and highlights the role of elasticity on the unsteady hydrodynamics of these complex fluids. Movies are available with the online version of the paper (go to journals.cambridge.org/flm).

Axisymmetric Granular Collapse : A Transient 3D Flow Test of Viscoplasticity

Abstract: A viscoplastic continuum theory has recently been proposed to model dense, cohesionless granular flows [P. Jop, Nature (London) 441, 727 (2006)10.1038/nature04801]. We confront this theory for the first time with a transient, three-dimensional flow situation--the simple collapse of a cylinder of granular matter onto a horizontal plane--by extracting stress and strain rate tensors directly from soft particle simulations. These simulations faithfully reproduce the different flow regimes and capture the observed scaling laws for the final deposit. Remarkably, the theoretical hypothesis that there is a simple stress-strain rate tensorial relationship does seem to hold across the whole flow even close to the rough boundary provided the flow is dense enough. These encouraging results suggest viscoplastic theory is more generally applicable to transient, multidirectional, dense flows and open the way for quantitative predictions in real applications.

Constitutive and Aging Behavior of Sn3.0Ag0.5Cu Solder Alloy

Abstract: We describe double-lap shear experiments on Sn3.0Ag0.5Cu solder alloy, from which fits to Anand's viscoplastic constitutive model, power-law creep model, and to time-hardening primary-secondary creep model are derived. Results of monotonic tests for strain rates ranging from 4.02E-6 to 2.40E-3 s-1, and creep response at stress levels ranging from 19.5 to 45.6 MPa are reported. Both types of tests were conducted at temperatures of 25degC, 75degC , and 125degC. Following an earlier study where Anand model and time hardening creep parameters for Sn3.8Ag0.7Cu and Sn1.0Ag0.5Cu solder alloys were reported, here we report power law model parameters so as to enable a comparison between all three alloys. Primary creep in Sn3.0Ag0.5Cu solder alloy is shown to be significant and are considered in addition to secondary creep and monotonic behavior. Aging influence on behavior is also shown to be significant. On the basis of experimental data, the following four aspects are discussed: 1) difference between testing on bulk versus joint specimen; 2) consistency between the creep and monotonic behaviors; 3) comparison against behaviors of Sn1.0Ag0.5Cu and Sn3.8Ag0.7Cu alloys as well as aganist Sn40Pb, 62Sn36Pb2Ag and 96.5Sn3.5Ag alloys; and 4) comparison of Sn3.0Ag0.5Cu and Sn3.8Ag0.7Cu relative to their aging response.

New Viscoplastic Model for Design Analysis of Tunnels in Squeezing Conditions

Abstract: This paper is intended to describe the SHELVIP (Stress Hardening ELastic VIscous Plastic) model, a new viscoplastic constitutive law which has been developed to incorporate the most important features of behaviour observed in tunnels excavated in severe to very severe squeezing conditions. This model couples the elastoplastic and time-dependent behaviour by using a plastic yield surface, as frequently adopted in tunnel design analysis, and the definition of a state of overstress referred to a viscoplastic yield surface. The model is formulated in all its detailed aspects. The related analytical closed-form solution for representing triaxial creep deformations is developed. Also developed is an incremental numerical solution for describing the triaxial stress–strain behaviour under constant strain rate conditions. The model is shown to fit very satisfactorily the results of creep tests on clay shales and relaxation tests on coal specimens, as recently performed for design analysis of tunnels in squeezing conditions.

Ageing and yield behaviour in model soft colloidal glasses

Abstract: We use multi-arm star polymers as model soft colloids with tuneable interactions and explore their behaviour in the glassy state. In particular, we perform a systematic rheological study with a well-defined protocol and address aspects of ageing and shear melting of star glasses. Ageing proceeds in two distinct steps: a fast step of O(10 3 s) and a slow step of O(10 4 s). We focus on creep and recovery tests, which reveal a rich, albeit complex response. Although the waiting time, the time between pre-shear (rejuvenation) of the glassy sample and measurement, affects the material’s response, it does not play the same role as in other soft glasses. For stresses below the yield value, the creep curve is divided into three regimes with increasing time: viscoplastic, intermediate steady flow (associated with the first ageing step) and long-time evolving elastic solid. This behaviour reflects the interplay between ageing and shear rejuvenation. The yield behaviour, as investigated with the stress-dependent recoverable strain, indicates a highly nonlinear elastic response intermediate between a low-stress Hookean solid and a highstress viscoelastic liquid, and exemplifies the distinct characteristics of this class of hairy colloids. It appears that a phenomenological classification of different colloidal glasses based on yielding performance may be possible.

Determination of the Johnson–Cook Material Parameters Using the SCS Specimen

Abstract: This note addresses the determination of the Johnson-Cook material parameters using the shear compression specimen (SCS). This includes the identification of the thermal softening effect in quasi static and dynamic loading as well as and the strain rate hardening effect in dynamic loading. A hybrid experimental–numerical (finite element) procedure is presented to identify the constitutive parameters, with an application to Ti6Al4V alloy. The present results demonstrate the suitability of the SCS for constitutive testing.

Survey of Four Damage Models for Concrete

Abstract: Four conventional damage plasticity models for concrete, the Karagozian and Case model (K&C), the Riedel-Hiermaier-Thoma model (RHT), the Brannon-Fossum model (BF1), and the Continuous Surface Cap Model (CSCM) are compared. The K&C and RHT models have been used in commercial finite element programs many years, whereas the BF1 and CSCM models are relatively new. All four models are essentially isotropic plasticity models for which 'plasticity' is regarded as any form of inelasticity. All of the models support nonlinear elasticity, but with different formulations. All four models employ three shear strength surfaces. The 'yield surface' bounds an evolving set of elastically obtainable stress states. The 'limit surface' bounds stress states that can be reached by any means (elastic or plastic). To model softening, it is recognized that some stress states might be reached once, but, because of irreversible damage, might not be achievable again. In other words, softening is the process of collapse of the limit surface, ultimately down to a final 'residual surface' for fully failed material. The four models being compared differ in their softening evolution equations, as well as in their equations used to degrade the elastic stiffness. For all four models, the strength surfaces are cast inmore » stress space. For all four models, it is recognized that scale effects are important for softening, but the models differ significantly in their approaches. The K&C documentation, for example, mentions that a particular material parameter affecting the damage evolution rate must be set by the user according to the mesh size to preserve energy to failure. Similarly, the BF1 model presumes that all material parameters are set to values appropriate to the scale of the element, and automated assignment of scale-appropriate values is available only through an enhanced implementation of BF1 (called BFS) that regards scale effects to be coupled to statistical variability of material properties. The RHT model appears to similarly support optional uncertainty and automated settings for scale-dependent material parameters. The K&C, RHT, and CSCM models support rate dependence by allowing the strength to be a function of strain rate, whereas the BF1 model uses Duvaut-Lion viscoplasticity theory to give a smoother prediction of transient effects. During softening, all four models require a certain amount of strain to develop before allowing significant damage accumulation. For the K&C, RHT, and CSCM models, the strain-to-failure is tied to fracture energy release, whereas a similar effect is achieved indirectly in the BF1 model by a time-based criterion that is tied to crack propagation speed.« less