Type Constitutive Equation Research Articles

Assessment of Constitutive Equations Used in Machining

Machining of metals is characterised by plastic deformation occurring at high strains, strain-rates and temperatures. In the few predictive machining theories reported in literature, two types of constitutive equations, the power-law relation and Johnson-Cook equation, have mainly been used for describing the material's plastic behaviour. This paper aims at assessing the above constitutive equations in terms of their ability to describe the material behaviour for a wide range of strain-rates and temperatures encountered in machining. The focus is on the plain carbon steel and copper work materials. Using the above constitutive equations, flow stress results were determined for conditions encountered in machining and then compared with experimental results obtained from literature which are compatible with the rate equations applicable to the microscopic deformation mechanisms under considered conditions. It was concluded that considerable improvements are required for the aforementioned constitutive equations for accurate prediction of the work material behaviour under machining conditions and that further development is needed for more reliable and accurate constitutive equations.

Particle Fluid Separation in Shear Flow of Dense Suspensions: Experimental Measurements on Squeezed Clay Pastes

AbstractParticle fluid separation is studied in the case of slow squeezing flow of dense clay suspensions. The fluid pressure gradient generated by the test induces heterogeneity in the sample. Experimental water content measurements at different time points through the test allow the quantification of this separation phenomenon. The problem equations are written in the case of purely extensional flow. Based on Terzaghi principle, Darcy’s law and a Cam Clay type constitutive equation, the influence of the permeability function on the predicted void ratio evolution is studied. It is then shown that a certain water amount is strongly linked to the grains and cannot be extracted from the sample using simple compression. This critical water amount is then taken in account in the permeability function in order to predict the compression load through the test.

알루미늄 합금박판 비등온 성형공정 스프링백 해석용 유한요소 프로그램 개발 (2부 : 이론 및 해석)

The implicit, finite element analysis program for analyzing the springback in the warm forming process of aluminum alloy sheets was developed. For the description of planar anisotropy in warm forming temperatures, Barlat`s yield function is employed, and the power law type constitutive equation is used in terms of working temperatures for the depiction of work hardening in high temperatures. Also, Jetture`s 4-node shell elements are introduced for reflecting the mechanical behavior of aluminum alloy sheet and the non-steady heat balance equations are solved for considering heat gain and loss during the forming process. For the springback evaluation, Newton-Raphson iteration method is introduced for overcoming the geometric nonlinearlity problem. In order to verify the validity of the FEM program developed, the stretching bending and springback processes are simulated. Though springback analysis results are slightly bigger than experimental ones, they have the same trend of the decreasing springback as the forming temperature increases.

Extended irreversible thermodynamics and the Jeffreys type constitutive equations

A postulate of extended irreversible thermodynamics is considered, according to which the entropy density is a function of the internal energy, the specific volume, and their material time derivatives. On the basis of this postulate, entropy balance equations and phenomenological equations are obtained, which directly lead to the Jeffreys type constitutive equations.

Predictions and Applicability of the BKZ Type Models in Extensional Flows of Polymer Melts

Predictions on shear, planar, uniaxial and biaxial extensional flows are obtained from the BKZ type constitutive equations with the assumption of time-strain separability. Using a single relaxation mode, the effects of adjustable parameters on the stress growth coefficients are investigated in detail. A parameter describing the normal stress ratio governs the biaxial extensional flow behavior. It also controls the planar extensional flow behavior in the direction of constant width (perpendicular to the extension direction). Applicability of two representative models with this parameter is investigated for multiple relaxation times by comparing with a set of data for polyisobutylene (literature data) and polystyrene melts. We propose a method to obtain the model parameters from shear, uniaxial and biaxial damping functions. With the introduction of the parameter describing the normal stress ratio, both the PSMLT model (Papanastasiou-Screven-Macosko as modified by Luo-Tanner) and the WDFO model (Wagner-Demarmels as modified by Fegel-Ottinger) give good agreements with all the experimental data. It is suggested that the PSMLT model is preferable for engineering purposes, because the model gives non-divergent stresses in almost of all cases investigated.

Generalized slit flow of an ellis fluid

AbstractIsothermal, steady‐state and fully developed flows of Ellis fluids in planar and annular slits are discussed. The flow equations derived for Ellis fluids describe also the flows of Newtonian and power law fluids as specific cases. The most important flows resulting from thp general theory, i.e., the pressure flows in flat and annular slits for stationary channel walls and at transverse and longitudinal movements of a wall, are analyzed in detail. Numerical verification of the results leads to the conclusion that the planar and annular flows of Ellis and power law fluids are qualitatively similar The quantitative differences resulting from the slit curvature and the type of constitutive equation are relatively large only for flows of strongly non‐New‐tonian liquids.

The application of an integral type constitutive equation to numerical flow analyses of viscoelastic fluid in unsteady flow

AbstractThe applicability of the Wagner model to numerical flow analyses of the injection molding process is investigated under the following approximations: the strain and stress histories of the molten polymer before the injection is negligible, and the flow field in a mold cavity is treated as Hele‐Shaw flow. A comparison between the results for simple step‐strain‐rate flow calculated with the Wagner model and that calculated with the Leonov model suggests that the Wagner model is superior to the Leonov model for unsteady flow because of its stability and accuracy. Therefore, numerical flow analysis software of a viscoelastic fluid in the injection molding process is developed using the Wagner model. For the analysis, the velocity profile of a Newtonian fluid is used instead of that obtained through iterative calculation. The validity of the developed program is confirmed through a comparison of the results of the computation for two simple flow velocity histories with the analytical results from the Wagner model. Furthermore, the computation time of the developed software is only 1.4 times greater than that of the previous numerical flow analysis of a viscous fluid.

岩石のコンプライアンス可変型構成方程式とそのパラメータの取得法

Authors had proposed two types of constitutive equations which were developed based on the experimental data of some Japanese rocks. Since proposal of the equations, a lot of questions were asked us by researchers and engineers who took part in construction of tunnels, underground storages and so on. Through the discussion with questioners, it was found that there were many common points in questions, especially on how to obtain the parameters in the equations, which arose mainly because explanations and assumptions concerning the constitutive equations were changed many times and it was hard for readers to understand the whole systems starting at experimental work, through determination of parameters, and ended at numerical simulation and evaluation of the calculated results. In this manuscript, we tried to explain the present constitutive equations in the form of FAQ (Frequently Asked Question), which was chosen for convenience of busy readers. A variable-compliance-type constitutive equation is mainly stated, however, most questions and answers are also applicable to other types of equations with slight modifications.

Computational implementation of stress integration in FE analysis of elasto-plastic large deformation problems

In this paper, we compare different numerical implementation algorithms for the rate type constitutive equation and present an integration scheme based on the physical meaning of the stress. Numerical implementation of various schemes is investigated in conjunction with the return mapping algorithm and the conditions to maintain plastic consistency. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. An alternative numerical treatment for rate of deformation tensor D ij is presented and is shown to maintain incremental objectivity. Numerical examples included a single element under rigid body rotation, a necking bifurcation of a bar in tension and a punch indentation process. It is shown that the use of Truesdell stress rate with specific numerical integration procedure gives more accurate results than other procedures presented.

Generalization of Integration Methods for Complex Inelastic Constitutive Equations with State Variables

The prediction of the inelastic behavior of the structure is an essential part of reliability assessment procedure, because most of the failures are induced by the inelastic deformation, such as creep and plastic deformation. During decades, there has been much progress in understanding of the inelastic behavior of the materials and a lot of inelastic constitutive equations have been developed. These equations consist of the definition of inelastic strain and the evolution of the state variables introduced to quantify the irreversible processes occurred in the material. With respect to the definition of the inelastic strain, the inelastic constitutive models can be categorized into elastoplastic model, unified viscoplastic model and separated viscoplastic model and the different integration methods have been applied to each category. In the present investigation, the generalized integration method applicable for various types of constitutive equations is developed and implemented into ABAQUS by means of UMAT subroutine. The solution of the non-linear system of algebraic equations arising from time discretization with the generalized midpoint rule is determined using line-search technique in combination with Newton method. The strategy to control the time increment for the improvement of the accuracy of the numerical integration is proposed. Several numerical examples are considered to demonstrate the efficiency and applicability of the present method. The prediction of the inelastic behavior of the structure is an essential part of reliability assessment procedure, because most of the failures are induced by the inelastic deformation, such as creep and plastic deformation. During decades, there has been much progress in understanding of the inelastic behavior of the materials and a lot of inelastic constitutive equations have been developed. These equations consist of the definition of inelastic strain and the evolution of the state variables introduced to quantify the irreversible processes occurred in the material. With respect to the definition of the inelastic strain, the inelastic constitutive models can be categorized into elastoplastic model, unified viscoplastic model and separated viscoplastic model and the different integration methods have been applied to each category. In the present investigation, the generalized integration method applicable for various types of constitutive equations is developed and implemented into ABAQUS by means of UMAT subroutine. The solution of the non-linear system of algebraic equations arising from time discretization with the generalized midpoint rule is determined using line-search technique in combination with Newton method. The strategy to control the time increment for the improvement of the accuracy of the numerical integration is proposed. Several numerical examples are considered to demonstrate the efficiency and applicability of the present method.

Modified Robertson projection operator method and irreversible thermodynamics

A modified Robertson projection operator is used to obtain the evolution equation for the projected distribution function as well as various macroscopic evolution equations. These evolution equations, although not as yet proved to be consistent with the thermodynamic laws, form a basis for a formal theory of macroscopic processes. On introducing some approximations to the projected propagator we construct an approximate, thermodynamically consistent evolution equation (kinetic equation) and develop a thermodynamically consistent theory of irreversible processes with the approximate evolution equation so obtained. The mathematical structure of the theory of irreversible processes thus formulated can be brought into harmony with the theory based on the Boltzmann equation and the generalized Boltzmann equation, which was previously developed and reported in the literature. The Maxwell–Cattaneo–Vernotte type constitutive equations are derived on making a number of approximations on the exact evolution equations for the stress and the heat flux.

NUMERICAL PROCEDURE FOR THE COMPUTATION OF FLUID FLOW WITH ARBITRARY STRESS-STRAIN RELATIONSHIPS

A finite-volume method is presented that allows for general stress-strain constitutive equations to be incorporated into a standard momentum-pressure-correction procedure. The method is sequential and segregated in nature, the various equations for mass and momentum conservation and for the evolution of the stress tensor are solved following a predefined order, and one of its features is the use of nonstaggered, and generally nonorthogonal, computational meshes. Two types of constitutive equations are used to test the method: the standard explicit and algebraic Newtonian model, and one of the simplest implicit differential equations, the upper-convected Maxwell model. In spite of its apparent simplicity, this latter model is known to pose the most severe numerical difficulties. However, the results in this article show the method to be effective in solving the equations for the flow of Newtonian and viscoelastic fluids through abrupt planar contractions with an area reduction of 4 to 1, one typical benchmark problem. The results are compared with available data and with solutions from a standard and validated code, and good agreement and consistency is found. A new formulation to evaluate stresses at cell faces is presented and shown to lead to improved results.

Existence results for steady flow of weakly compressible viscoelastic fluids with a differential constitutive law

In this paper we study flows of viscoelastic weakly compressible fluids having a differential constitutive equation. In both cases, Jeffreys and Maxwell type constitutive equations, we establish a result of existence and uniqueness of solutions. We also show that, when the compressibility goes to zero, then the weakly compressible steady solution goes to the incompressible one.

Recent developments in the incremental flow formulation for the numerical simulation of metal forming processes

This paper describes the use of the incremental flow formulation for the numerical analysis of forming processes. This formulation is based on the approximation of the flow type of constitutive equations by means of an algorithmic equation based on geometry changes over a timestep. The resulting equilibrium equations are functions of nodal positions. Hence geometrical constraints such as contact and incompressibility can be enforced in an exact manner. Two types of incremental equations leading to first and second order time approximations are presented. The more accurate case is based on the incremental logarithmic stretches. Several examples will be used to demonstrate the validity of the method.

High weissenberg number simulation of an annular extrudate swell using the differential type constitutive equation

AbstractAnnular extrudate swell simulations at high Weissenberg numbers were made using a differential type constitutive equation. The streamline‐upwinding method with a sub‐element for extra stress components, which is called SU4 × 4, is one of the best mixed finite element methods for computation of viscoelastic flows. Planar and capillary extrudate swell calculations at high Weissenberg numbers (We > 1000) were accomplished by SU4 × 4. However, annular extrudate swell simulations at high We by SU4 × 4 were not successful. The calculated We was less than about 4. A new calculation technique using a Newton‐Raphson discretization of the equation of motion was developed. This technique is called a “new under‐relaxation method.” The calculated We of annular extrudate swell simulation by the new under‐relaxation method with SU4 × 4 was about 6∼250 times larger than those by SU4 × 4. Reasonable calculation results were obtained in an annular flow and a capillary extrudate swell by this method, and the reliability and the utility of the new under‐relaxation method are shown. It is now possible to consider the swell shapes of annular extrudate under industrially useful conditions. The calculated swelling ratios were also compared with experimental ones.

Finite element analysis of the superplastic forming of thick sheet using the incremental flow formulation

The paper discusses the finite element analysis of the superplastic forming of thick sheet components. The incremental formulation proposed is based on a geometrical approximation of the flow type of constitutive equations that describe the behaviour of the alloy during forming. The spatial discretization is achieved using eight-noded finite elements. An algorithm capable of predicting the correct forming pressure is also presented in a form consistent with the incremental flow formulation. Some experimental validation of these techniques will be shown together with a number of more realistic applications which will illustrate the generality of these techniques and their ability to simulate the forming of complex components. Most of the material in this section is standard but has been included for the purpose of completeness and to introduce the reader to the notation used in the paper. © 1997 John Wiley & Sons, Ltd.

Effects of Changes in Geometry on The Linear Elastic Consolidation Deformation

Linear elastic finite consolidation deformation analysis has been made to investigate the effects of changes in geometry on a plane strain embankment problem in a subsiding ground below the water table. A theoretical formulation is presented for the consolidation of a linear elastic soil foundation subject to finite deformations. With the use of the Green-Naghdi effective stress rate in the linear rate type constitutive equation, a finite element solution is obtained based on the Updated Lagrangian scheme. Since the traction boundary is continually modified as the deformation proceeds, the traction force is treated by dividing into two components: the force due to static water pressure and the force due to embankment load. The deformation characteristics, including one dimensional consolidation, are discussed with comparisons of infinitesimal deformation theory. Geometry change affects the ground rigidity, giving stiffened time-settlement response during consolidation and reduced lateral deformations when compared with the infinitesimal deformation theory. The load concentration effects, which come from the changes in traction boundary geometry, can reduce the extent of stiffened time-settlement response even to a lower magnitude than in infinitesimal case. The lateral deformations at the embankment toe are always lower than that the infinitesimal case. Unlike an elasto-plastic material, the deformation behavior for an elastic material is independent of the loading history and dependent only on the magnitude of the load non-linearly.

Rheological Considerations on the Grain-Boundary Sliding of Polycrystalline Ceramics

The grain-boundary sliding and interlocking of polycrystalline ceramics are investigated by means of the model arrays, i.e., two-dimensional arrays of square or hexagonal elastic grains embedded in a contiguous grain-boundary melt. A nonlinear Maxwell type constitutive equation is derived in an universal manner by considering two fundamental processes associated with elastic deformation of grains and squeezing-in/out of a viscous fluid between adjacent grains. The equation applied to creep deformation and stress relaxation yields several important predictions for the nonlinear viscoelastic behavior of polycrystalline ceramics with grain-boundary phase.

Generalized network type constitutive equations

The network model is extended by including a Coulombic force. The rates of creation and loss of junctions are assumed to be functions of the second invariant of the rate-of-deformation tensor. The resulting constitutive equation is used to predict the viscosity and the normal stress difference in viscometric flow, as well as the elongational viscosity. The predicted viscometric functions agree with experimental data.

Viscoelastic entrance flow of pipe with K‐BKZ type model ‐ Numerical Analysis

Viscoelastic flow at a region of a circular pipe entrance was examined numerically. The numerical calculation was achieved by using I.C.E. (implicit continuous‐fluid Eulerian) technique with the K‐BKZ type constitutive equation. The transient flow characteristics with the sudden stand‐up were obtained. The resultant flow field and its associated flow parameters of the K‐BKZ integral constitutive equation were compared with the Newtonian case and the Oldroyd‐B model. It is revealed that flow is largely affected by a choice of the Deborah number. When the Deborah number is high enough for a fixed Reynolds number, there would be a circulation zone at the entrance region of the pipe by the effects of the viscoelasticity, which cannot be observed with the Newtonian flow.