Showing papers on "Viscoplasticity published in 1993"

A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals : application to zirconium alloys

Abstract: We present in this work a visco-plastic self-consistent (VPSC) anisotropic approach for modeling the plastic deformation of polycrystals, together with a thorough discussion of the assumptions involved and the range of application of such approach. We use the VPSC model for predicting texture development during rolling and axisymmetric deformation of Zirconium alloys, and to calculate the yield locus and the Lankford coefficient of rolled Zircaloy sheet. We compare our results with experimental data and find that they are in good agreement with the available experimental evidence. We also compare the VPSC predictions with the ones of a Full Constraints approach and observe that they differ both quantitatively and qualitatively: according with the predictions of the VPSC scheme, deformation is accommodated mostly by the soft systems, the twinning activity is much lower, and fewer systems are active, in average, per grain. These results are a consequence of having accounted for the grain interaction with its surroundings, which is a crucial aspect when modeling plastically anisotropic materials.

Functional and numerical methods in viscoplasticity

Abstract: CHAPTER 1. PRELIMINARIES ON MECHANICS OF CONTINUOUS MEDIA CHAPTER 2. FUNCTIONAL SPACES IN VISCOPLASTICITY CHAPTER 3. QUASISTATIC PROCESSES FOR RATE-TYPE VISCOPLASTIC MATERIALS CHAPTER 4. DYNAMIC PROCESSES FOR RATE-TYPE ELASTIC-VISCOPLASTIC MATERIALS CHAPTER 5. THE FLOW OF THE BINGHAM FLUID WITH FRICTION

Dynamic Models for Structural Plasticity

Abstract: 1 Elastoplastic and Viscoplastic Constitutive Relations.- 2 Principles of Mechanics.- 3 Static Deflection.- 4 Dynamic Rigid-Plastic Response.- 5 Second-Order Effects on Dynamic Response.- 6 More Complex Configurations.- 7 Impact Experiments.

On a class of constitutive equations in viscoplasticity : formulation and computational issues

Abstract: The viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge (Minkowski) function of the convex set. A perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution. It relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation. The operator split methodology and the Newton-Raphson method are used to obtain the numerical solution of the discretized equations of evolution. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations. For several variants of the flow potential function, including some representative stiff functional forms, numerical tests of the integration algorithm based on iso-error maps are provided. Finally, a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach.

Plasticity and Creep Theory Examples and Problems

Abstract: BASIC DEFINITIONS Stress and Strain State Stress tensor Strain tensor Finite Deformations Finite strain tensors in material or spatial coordinates Strain rates tensors Stress tensors in material or spatial descriptions FOUNDATIONS OF PLASTICITY Basic Equations of Perfect Plasticity Uniaxial stress-strain behavior Criteria for yielding in perfect plasticity Stress-strain relations for perfect plasticity Methods of reduction of equations of perfect plasticity Problems Basic Equations of Plastic Hardening Drucker's postulate and the associated flow rule Subsequent yield surfaces for hardening material Theories of plastic hardening Problems Methods of the Theory of Plasticity Analysis of the level of a cross-section Interaction curves on levels of a cross-section or a body Extremum theorems of limit analysis: statically or kinematically admissible solutions Shakedown analysis Integration along characteristics in plane strain problems Problems SOL UTIONS OF ELASTIC-PLASTIC PROBLEMS Elastic-Plastic Torsion and Bending Elastic-plastic torsion of prismatic bars Problems Elastic-plastic bending of prismatic beams and plane frames Problems Elastic-Plastic Analysis of Cylinders, Disks, and Plates Thick-walled tubes, spherical shells and disks Problems Limit analysis of Plates Problems FOUNDATIONS OF CREEP Basic Equations of Uniaxial Creep Models Creep phenomenon Schematizations of creep at constant uniaxial stress Modelling of creep at varying uniaxial stress Linear uniaxial viscoelastic models Modelling of viscoplastic materials Problems Creep Constitutive Equations Under Multiaxial Loading Classical multiaxial creep theories Developed multiaxial creep theories Linear multiaxial viscoelastic equations SOLUTION OF CREEP PROBLEMS Bending, Buckling, and Torsion of Bars Under Creep Conditions Bending and buckling of a prismatic bar made of the linear viscoelastic material Bending of a prismatic be am made of the piece-wise linear elastic/viscoplastic material Bending of a prismatic beam made of the time hardening material Torsion of a circular bar made of the elastic-Bingham material Problems Rotationally Symmetric Creep Problems Creep of a thick-walled tube General formulae for the rotationally-symmetric transient creep problems CREEP RUPTURE Constitutive Equations of Creep Rupture Creep rupture phenomenon Classical creep rupture theories Problems Rotationally Symmetric Creep Rupture Problems Mechanisms of brittle rupture of tubes and disks Design of disks with respect to creep rupture References Author Index Subject Index

Experimental and finite element predictions of residual stresses due to orthogonal metal cutting

Abstract: The development and implementation of a finite element method for the simulation of plane-strain orthogonal metal cutting processes with continuous chip formation are presented. Experimental procedures for orthogonal metal cutting and measurement of distributions of residual stresses using the X-ray diffraction method are also presented. A four-node, eight degree-of-freedom, quadrilateral plane-strain finite element is formulated. The effects of elasticity, viscoplasticity, temperature, friction, strain-rate and large strain are included in this formulation. Some special techniques for the finite element simulation of metal cutting processes, such as element separation and mesh rezoning, are used to enhance the computational accuracy and efficiency. The orthogonal metal cutting experiment is set-up on a shaper, and the distributions of residual stresses of the annealed 1020 carbon steel sample are measured using the X-ray diffraction method. Under nominally the same cutting conditions as the experiment, the cutting processes are also simulated using the finite element method. Comparisons of the experimental and finite element results for the distributions of residual stresses indicate a fairly reasonable level of agreement. The versatility of the present finite element simulation method allows for displaying detailed results and knowledge generated by orthogonal metal cutting processes, such as the distribution of temperature, yield stress, effective stress, plastic strain, plastic strain-rate, hydrostatic stress, deformed configuration, etc. Such knowledge is useful to provide physical insights into the process as well as to better design the process for machining parts with improved performance.

A Dynamical Theory of Structured Solids. I Basic Developments

Abstract: Based on the well-accepted notion of a Bravais lattice of a crystal at the atomic scale and with particular reference to inelastic behaviour of materials, this paper is concerned with the construction of a macroscopic dynamical theory of solids which incorporates the effect of the presence of the atoms and their arrangements. The theory incorporates a wide variety of microstructural processes occurring at various physical scales and has a range approaching the atomic scale. These processes include the effect of the motion of individual dislocations, which are modeled here as continuous distributions at the macroscopic scale. The formulation of the basic theory, apart from the kinematical and kinetical variables employed in classical continuum mechanics, utilizes a triad of independent vector-valued variables - called directors - (or an equivalent tensor-valued variable) which represent the lattice vectors and are determined by additional momentum-like balance laws associated with the rate of change of lattice deformation in the spirit of a Cosserat (or directed) continuum. A suitable composition of the triad of directors and the ordinary deformation gradient is identified as a measure of permanent or plastic deformation, the referential gradient of which plays a significant role in the kinematics of lattice defects. In particular, a uniquely defined skew-symmetric part of the gradient of plastic deformation is identified as a measure of the density of dislocations in the crystal. The additional momentum-like balance laws associated with the rate of lattice deformation include the effect of forces necessary to maintain the motion of dislocations, as well as the inertia effects on the microscopic and submicroscopic scales arising from the director fields. The basic theoretical developments also provide important clarifications pertaining to the structure of the constitutive response functions for both viscoplasticity and (the more usual) rate-independent plasticity.

An analysis of the brittle-ductile transition in dynamic crack growth

Abstract: The brittle-ductile transition in dynamic crack growth is investigated through the numerical analysis of a plane strain edge cracked specimen, subject to impulsive loading at one end The material is characterized as an elastic-viscoplastic solid, with a temperature dependent flow strength Thermal softening due to adiabatic heating and a model for ductile failure by void nucleation, growth and coalescence are incorporated into the constitutive relation The ductile void growth mechanism involves two populations of void nucleating particles; discretely modelled low strength inclusions that give large voids near the crack tip at an early stage and homogeneously distributed small second phase particles that require large strains for void nucleation Cleavage is modelled in terms of a spatially non-uniform, but temperature and strain rate independent, critical value of the maximum principal normal stress The numerical results show a clear transition from cleavage dominated crack growth at low temperatures to purely ductile crack growth at higher temperatures There is an accompanying large increase in the material's resistance to dynamic crack initiation and growth The computed crack growth behaviour is a direct outcome of the material description; no ad hoc dynamic fracture criterion is employed Effects of variations in the material model on the brittle-ductile transition are explored

On the mathematical modelling of material behavior in continuum mechanics

Abstract: The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.

The existence of static yield stresses in suspensions containing noncolloidal particles

Abstract: Many previous investigations have established and/or quantified yield stresses in suspensions of interacting (colloidal or surface active) particles. In this work it is shown for the first time that highly filled suspensions of noncolloidal particles can also exhibit a static yield stress σs. The determination of static yield stresses in these systems was made possible by conducting very long time (up to 6000 h) tensile creep tests. These tests are particularly suitable for high viscosity systems such as the suspensions examined here and also have the advantage that there is minimal influence on the results due to the measurement apparatus. Through these experiments, it is possible to observe an unambigious change of material response above and below the static yield stress σs, with viscoelastic solid‐like behavior exhibited below σs, and viscoelastic liquid‐like behavior above σs. The dependence of this static yield stress on the volume concentration filler is examined in this work.

Thermo-elasto-viscoplasticity of isotropic porous metals

Abstract: A rate and temperature dependent elastic-plastic model for isotropic, moderately porous metallic materials is formulated. This model is intended for material rate-sensitivities in the entire range spanning from highly rate-dependent behavior at high homologous temperatures to nearly rate-insensitive behavior at low homologous temperatures. The predictive capabilities of the constitutive model are verified by comparing results from finite element calculations against results from physical experiments. Specifically, example calculations are presented for: (a) isothermal hot compression of a tapered disk made from an initially porous material. This calculation illustrates the effect of secondary tensile stresses on hot workability of metals, (b) Tension tests, under isothermal conditions at low homologous temperatures, on axisymmetric notched bars made from initially porous materials. This calculation illustrates the effects of nonuniform multiaxial tensile stress states on void growth. Predictions from the computational procedures for both examples agree well with experimental results. The new state variable rate and temperature dependent constitutive model for microporous materials and the associated computational procedures form a basis for the simulation and design of deformation processing operations. This new capability should be useful for the prediction of formation of defects during both cold-working when the material rate sensitivity is low, as well as hot-working when the material is highly rate sensitive. The computational capability should also be useful in simulating the late stages of densification of powder metallurgical products in complex forming operations.

Optimum Design of Forging Die Shapes Using Nonlinear Finite Element Analysis

Abstract: An optimization method is developed for the design of intermediate die shapes needed in the plane strain and axisymmetric forging operations. The approach is based on backward deformation simulation using nonlinear rigid viscoplastic finite element method and shape optimization techniques. The advantage of this optimization approach is that it has the ability to determine the intermediate die shapes from the final product shape by applying constraints on the plastic deformation of the material. This paper presents axisymmetric disk and plane strain case studies to demonstrate the new design procedures for minimizing variations in deformation rates during a multistage forging operation

Dynamic pore collapse in viscoplastic materials

Abstract: Dynamic pore collapse in porous materials is studied by analyzing the finite deformation of an elastic/viscoplastic spherical shell under impulsive pressure loading. Effects of dynamic loading rate, pore size, initial porosity, strain‐rate sensitivity, strain hardening, thermal softening, and mass density of the matrix material on the pore collapse process are examined and results are compared with those from quasistatic analyses of both rate‐independent and rate‐dependent matrix materials. Dynamic (inertia) effects are found to be significant or even dominant in certain shock wave consolidation conditions. An approximate method is proposed to incorporate dynamic effects into quasistatic pore‐collapse relations of viscoplastic matrix materials. Implications of results of current study are discussed in terms of understanding the processes of shock wave consolidation of powders.

Environmental interactions in high-temperature fatigue crack growth of Ti-1100

Abstract: The crack growth behavior of Ti-1100 is investigated for loading frequencies ranging from 30 to 0.0031 Hz at temperature levels extending from 23 °C to 650 °C in both air and vacuum environments. Two types of time-dependent damage mechanisms have been identified: oxidation and creep effects. It is concluded that the effect of oxidation on the crack growth acceleration is rapidly developed and only weakly dependent on total cycle time. Creep effects, on the other hand, are dominant at low frequencies in both air and vacuum and are loading rate dependent. The degree of contribution of each of these two damage modes during the steady state growth region has been phenomenologically determined by examining the frequency dependence on the exponent and coefficient parameters of the Paris-type crack growth equation. It is found that these parameters are largely determined by the extent of the viscoplastic response of the crack tip region below a specific, environment-sensitive transition loading frequency. Furthermore, the physical mechanisms involved in the environment-affected damage are identified with the nature of crack tip plastic work input as a function of loading frequency. The influence of frequency and environment on the anomalous appearance of pronounced stage I/stage II knee regions is also discussed with respect to closure levels and creep transient response.

Extrusion and lubrication flows of viscoplastic fluids with wall slip

Abstract: The simplest model flow which approximates the extrusion (shallow screw channels) and lubrication flow is the steady, laminar flow occurring between two infinitely long parallel plates i.e., the generalized plane Couette flow. Here we develop an analytical model of the generalized plane Couette flow of viscoplastic fluids. The deformation and flow behavior of viscoplastic fluids can be realistically represented with the Herschel-Bulkley constitutive equation, which we have utilized as the basis for the development of our analytical model. Furthermore, as also demonstrated here, the deformation behavior of viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which occurs as a function of the wall shear stress. The wall slip versus the wall shear stress behavior of viscoplastic fluids can be experimentally characterized using viscomelric flows, including steady torsional and capillary flows. Thus determined Navier's wall slip coefficient can then be utilized in ...

Inelastic deformation of metal matrix composites

Abstract: A theoretical model capable of predicting the thermomechanical response of continuously reinforced metal matrix composite laminates subjected to multiaxial loading was developed. A micromechanical model is used in conjunction with nonlinear lamination theory to determine inelastic laminae response. Matrix viscoplasticity, residual stresses, and damage to the fiber/matrix interfacial zone are explicitly included in the model. The representative cell of the micromechanical model is considered to be in a state of generalized plane strain, enabling a quasi two-dimensional analysis to be performed. Constant strain finite elements are formulated with elastic-viscoplastic constitutive equations. Interfacial debonding is incorporated into the model through interface elements based on the interfacial debonding theory originally presented by Needleman, and modified by Tvergaard. Nonlinear interfacial constitutive equations relate interfacial tractions to displacement discontinuities at the interface. Theoretical predictions are compared with the results of an experimental program conducted on silicon carbide/titanium (SiC/Ti) unidirectional, (O4), and angle-ply, (+34)(sub s), tubular specimens. Multiaxial loading included increments of axial tension, compression, torque, and internal pressure. Loadings were chosen in an effort to distinguish inelastic deformation due to damage from matrix plasticity and separate time-dependent effects from time-independent effects. Results show that fiber/matrix debonding is nonuniform throughout the composite and is a major factor in the effective response. Also, significant creep behavior occurs at relatively low applied stress levels at room temperature.

Thermal residual stresses in particle-reinforced viscoplastic metal matrix composites

Abstract: A spherical symmetric thermoviscoplastic model is presented to calculate the thermal residual stresses in particle-reinforced metal matrix composites. It is based on the assumption that the particles behave elastically whereas the matrix exhibits an elastoviscoplastic behaviour. An internal-variable-type constitutive equation is used to predict the behaviour of the matrix, which is taken as Al-1100. The model can predict the change of the stress and displacement fields within the matrix and the particles during cooling of the material as well as the thermal residual stresses induced at room temperature. The effects of cooling rate and volume fraction of particles are emphasized. The stresses on reheating from room temperature and during thermal treatment are calculated. It is shown in particular that holding the composite at moderate temperature (≈ 200 °C) does not lead to substantial relaxation of the stresses. Aging treatment is thus usually carried out under a residual stress field which might influence the precipitation sequence and kinetics.

Modelling the densification of powder composites by power law creep

Abstract: A model describing the densification under isotropic pressure of a powder composite, consisting of viscoplastic particles deforming by power law creep mixed with rigid inclusions of the same size, is presented. A description of the behaviour of each type of contacts is first proposed, from which an expression for the densification rate of the mixture is deduced. The evolution of particle coordination numbers and contact areas during densification is then estimated from an extension of Arzt's model [Acta metall.30, 1883 (1982)] to powder composites. The model can be used up to a density corresponding to the formation of a supportive inclusion network. Finally the validity of the model is discussed in relation with experimental data.

Static and dynamic analysis of space frames using simple Timoshenko type elements

Abstract: In this paper a finite element method for geometrically and materially non-linear analyses of space frames is described. Beams with both solid and thin-walled open cross-sections are considered. The equations of equilibrium are formulated using an updated incremental Lagrangian description. The elements developed can undergo large displacements and rotations, but the incremental rotations are assumed to be small. The material behaviour is described by elastoplastic, temperature-dependent elastoplastic and viscoplastic models with special reference to metals. Computationally, more economical formulations based on the relationship between stress resultants and generalized strain quantities are also presented. In the case of thin-walled beams the torsional behaviour is modelled using a two-parameter warping model, where the angle of twist and the axial variation of warping have independent approximations. This approach yields average warping shear strains directly from the displacement assumptions and no discrepancy between stress and strain fields exists.

A modeling investigation of thermal and strain induced recovery and nonlinear hardening in potential based viscoplasticity

Abstract: Specific forms for both the Gibb's and the complementary dissipation potentials were chosen such that a complete potential based multiaxial, isothermal, viscoplastic model was obtained. This model in general possesses three internal state variables (two scalars associated with dislocation density and one tensor associated with dislocation motion) both thermal and dynamic recovery mechanisms, and nonlinear kinematic hardening. This general model, although possessing associated flow and evolutionary laws, is shown to emulate three distinct classes of theories found in the literature, by modification of the driving threshold function F. A parametric study was performed on a specialized nondimensional multiaxial form containing only a single tensorial internal state variable (i.e., internal stress). The study was conducted with the idea of examining the impact of including a strain-induced recovery mechanism and the compliance operator, derived from the Gibb's potential, on the uniaxial and multiaxial response. One important finding was that inclusion of strain recovery provided the needed flexibility in modeling stress-strain and creep response of metals at low homologous temperatures, without adversely affecting the high temperature response. Furthermore, for nonproportional loading paths, the inclusion of the compliance operator had a significant influence on the multiaxial response, but had no influence on either uniaxial or proportional load histories.

Parameter Evaluation for a Unified Constitutive Model

Abstract: Parameters for the unified constitutive model MATMOD were evaluated for rock salt by using nonlinear least squares to fit the model to isothermal laboratory data. Laboratory data from stress relaxation, constant strain rate, and long term creep tests were used. The latter two test types included staged tests in which the strain rate or stress was changed step wise during the test