Type Constitutive Equation Research Articles

Modelling the plane strain compression test to obtain constitutive equations of aluminium alloys

Hot plane strain compression tests on 1050, 1198, 3003 and 3004 aluminium alloys have been conducted. Based on these experiments and on a set of internal type constitutive equations for hot working, the values of the parameters in the constitutive functions are determined. The constitutive equations proposed here, with the constitutive functions and material parameters associated, accurately reproduce the basic tests. The procedure used to fit the material parameters is improved, in comparison with classical slip line analysis, by using a finite element modelling of the plane strain compression test. It is demonstrated that accurate plane strain or three-dimensional large strain finite element analysis can be used to correct the friction and lateral spread effects. Furthermore, it is demonstrated from comparison with the experimental observations that microstructural parameters can be accurately determined from numerical modelling. The constitutive equations and finite element procedure proposed here can be useful for obtaining an improved analysis of hot rolling of aluminium alloys.

Behavior of Laminar Boundary Layer with Shear Thickening-Thinning Characteristics of Non-Newtonian Flows

Non-Newtonian characteristics of boundary layer flow passing a flat external surface with zero incident angle were studied using Phan-Thien Tanner (PTT) type constitutive equation with the network theory. A consideration is given to the shear thickening and the shear thickening-thinning characteristics of fluid in the flow problem. The Von Karman integral equation is solved numerically in order to obtain the boundary layer thickness and associated friction coefficient. Two important parameters in the network theory, the segment number and chain slippage parameter, are discussed to investigate their effects on the boundary layer characteristics.The calculated results indicate that there are large jumps of boundary layer thickness at the leading edge of the plate for small slip parameters when the effect of normal stress is not included in the calculation. On the contrary, the jumps are eliminated, particularly for smaller Reynolds number, with the inclusion of the effect of normal stress.

Rate type equations for the diffusion in polymers: Thermodynamic constraints

AbstractConditions imposed by the second law of thermodynamics on viscoelastic rate type constitutive equations for the diffusive mass flux are considered. The analysis of three different rate type models proposed in the literature points out that presently physically unrealistic predictions are possible in desorption processes. The thermodynamic analysis of such models, based on the entropy inequality and on the stability requirement of the equilibrium states, leads to precise relationships among relaxation times, diffusion coefficients, and entropy equations of state. In particular, the analysis shows that relaxation times and diffusion coefficients cannot be simply constant numbers. When the thermodynamic constraints imposed on the constitutive equations are introduced, the models do not show physically unrealistic behaviors any more; Fickian diffusion close to the pure penetrant or pure polymer regions is also recovered. Finally, it is shown that the stability requirement for the equilibrium states may introduce very rigid requirements for the model feasibility, well beyond what appears explicitly from the kinetic equations alone.

Viscosity of passive human neutrophils undergoing small deformations

At issue is the type of constitutive equation that can be used to describe all possible types of deformation of the neutrophil. Here a neutrophil undergoing small deformations is studied by aspirating it into a glass pipet with a diameter that is only slightly smaller than the diameter of the spherically shaped cell. After being held in the pipet for at least seven seconds, the cell is rapidly expelled and allowed to recover its undeformed, spherical shape. The recovery takes approximately 15 s. An analysis of the recovery process that treats the cell as a simple Newtonian liquid drop with a constant cortical (surface) tension gives a value of 3.3 x 10(-5) cm/s for the ratio of the cortical tension to cytoplasmic viscosity. This value is about twice as large as a previously published value obtained with the same model from studies of large deformations of neutrophils. This discrepancy indicates that the cytoplasmic viscosity decreases as the amount of deformation decreases. An extrapolated value for the cytoplasmic viscosity at zero deformation is approximately 600 poise when a value for the cortical tension of 0.024 dyn/cm is assumed. Clearly the neutrophil does not behave like a simple Newtonian liquid drop in that small deformations are inherently different from large deformations. More complex models consisting either of two or more fluids or multiple shells must be developed. The complex structure inside the neutrophil is shown in scanning electron micrographs of osmotically burst cells and cells whose membrane has been dissolved away.

Comparison of computational efficiency of flow simulations with multimode constitutive equations: integral and differential models

A decoupled finite-element method (INT/FEM) is presented for calculation of two-dimensional viscoelastic flows with integral constitutive models. The momentum and continuity equations are solved by Galerkin's method with the viscoelastic stress treated as a fixed body force. The viscoelastic stress is computed by using the stream function to track fluid particles upstream, integrating a system of ordinary differential equations that govern the displacement-gradient tensor, and evaluating the integral constitutive equation by numerical quadrature. The quasi-linear upper-convected Maxwell and Oldroyd-B models, as well as the nonlinear model of Papanastasiou, Scriven and Macosko (PSM), are used in the simulations. The efficiency of the integral method is compared to that of the recently developed finite-element method (EVSS/FEM) for differential constitutive models. Convergence and accuracy of the INT/FEM are shown by calculations for flow between eccentric cylinders. The upper limit in De attainable by using the INT/FEM is comparable to values for the EVSS/FEM only for constitutive models with a shear-thinning ratio of the first normal stress difference to the shear stress and a large solvent contribution to the solution viscosity. The INT/FEM becomes the more efficient technique for simulation with this type of constitutive equation when three or more relaxation modes are included in the memory function.

Plastic deformation and damage accumulation below the worn surfaces

Delamination of material layers adjacent to the worn surfaces is a commonly observed form of wear in unlubricated or poorly lubricated surfaces. In ductile materials, the delamination process usually involves large plastic deformation and subsurface damage. In this study, metallographic techniques have been used to determine the extent of plastic deformation and strain localization events during the sliding wear of annealed OFHC copper samples. Tests were performed using a block-on-ring type wear machine under constant load and constant velocity conditions. Subsurface displacement and microhardness gradients were measured as a function of sliding distance. It was observed that both the magnitude of plastic strain (and stress) gradients and the depth of highly deformed layers vincreased with the sliding distance. The flow stress and strains at the subsurface regions are shown to obey a Voce type constitutive equation. Wear proceeded mainly by a mechanism of delamination via subsurface crack growth. It is proposed that the competition between the plastic strain which enhances void growth and the hydrostatic pressure which suppresses it is responsible for the generation of a damage gradient so that the delamination takes place at a certain depth where the damage accumulation rate is maximum. A model based on the Rice and Tracey analysis of ductile void growth is developed and used to determine the location of subsurface crack propagation.

Finite strain elastic closed form solutions for some axisymmetric problems

In this paper, rate type constitutive equations using the Jaumann and 2nd Piola-Kirchhoff stress rate are used to develop axisymmetric finite strain, elastic, closed form solutions for a variety of loading conditions. We examined several cases comprizing compression-extension loading conditions, simple shear and cavity expansion conditions. Results from small strain analyses are used to indicate strain ranges for which such analyses will provide satisfactory solutions. For all the cases examined, except simple shear, the 2nd Piola-Kirchhoff stress rate does not appear to be a suitable stress rate to describe a material which follows a rate-type constitutive equation for strains greater than about 40%. The Jaumann stress rate solution shows oscillatory shear stress for axisymmetric simple shear similar to that found earlier by many other authors for rectangular (or cuboidal) condition. Negative excess pore water pressure (suction) at the cavity wall during the expansion of a cylindrical cavity was also observed in Jaumann stress rate solution.

On the Solid-Fluid Transition Zone in Welding Analysis

Equilibrium and energy equations have been developed in describing the solid-fluid transition zone during the melting and solidification of the binary alloys. Due to the existence of the transition region exhibiting both solid and fluid phases at the same material point from continuum point of view, mixture theory was utilized to analyze the region. Unlike the Stefan problem, the latent heat due to the phase change appears as a source term in the heat equation. The molten fluid is treated as a thermoviscous and incompressible fluid, whereas the solid is thermoviscoplastic described by the Bodner-Partom/Walker type of constitutive equations. Thermal mechanical behaviors of the solid and the fluid phases are determined separately because of insignificant mechanical interactions between them. Volume fractions of the phases are obtained according to the equilibrium phase diagram. The simulation process of the transition zone and the welding process was carried out by FEM. The molten fluid motion, the sizes, and the contours of the transition zone were presented.

Delamination wear in ductile materials containing second phase particles

Plastic deformation and damage accumulation below the contact surfaces play an important role in sliding wear of ductile materials. In this study, metallographic techniques have been developed and used to determine the magnitude of the shear strains and the microhardness gradients in near surface regions in an aluminum-7% silicon alloy. Under dry sliding wear conditions, both the magnitude of plastic strains and the depth of heavily deformed zones increased with sliding distance and applied load. The flow stress and the plastic strains in the deformed zones are shown to obey a work hardening law which can be expressed in the form of a Voce type constitutive equation. Scanning electron microscopy (SEM) studies on the longitudinal sections below the worn surfaces indicated that thin flake-shaped debris particles were generated by a process of subsurface delamination occurred via cracks which originated from silicon particles within the deformed zones (but not at the contact surface) and propagated parallel to the surface. A model based on the hypothesis that delamination cracks are formed by the coalescence of voids at a critical depth below the worn surfaces has been proposed. It is shown that the critical depth for maximum rate of damage accumulation is determined by a competition beetween the plastic strain which enhances void growth and the hydrostatic pressure which suppresses it.

高ひずみ速度下での広いひずみ域に適用できる修正マルバーン型構成式中の材料定数Ｋの決定法について

高ひずみ速度下での広いひずみ域に適用できる修正マルバーン型構成式中の材料定数Ｋの決定法について

Modeling Welding by Surface Heating

A mathematical model has been developed in describing the temperature distribution, the flow of the molten fluid and the stress field in the solid during welding. In modeling the properties of the material during welding, the solid phase is assumed to behave as a thermoviscoplastic solid obeying Bodner-Partom/Walker type constitutive equation, whereas the fluid phase as a thermoviscous incompressible fluid. Three regions exist: pure solid, pure fluid, and the transition (solid-fluid mixture). In the formulation of the boundary value problem, the energy equation is coupled to the equation of motion through the terms of mechanical work and the latent heat of the phases, whereas the equations of motion of the solid and the fluid are decoupled. Appropriate thermal and traction boundary conditions are detailed in the text. Phase transformation activities during cooling are monitored by CCT diagram and Avrami equation. An arbitrary Lagrangian and Eulerian method is used to accommodate the kinematic description of both the solid and the fluid phases. A representative plane perpendicular to the moving heat source is analyzed. Results of sample calculations are presented to show the temperature and the stress evolution in time. Residual stress and microstructure patterns are presented.

Stress-strain relations of irradiated stainless steels below 673 K

Abstract Most specimens, irrespective of thermo-mechanical treatment, exhibited proof stress levels of above 800 MPa and uniform elongations below 1% after irradiation in the High Flux Isotope Reactor (HFIR). Only the solution annealed specimens irradiated at a low temperature of 328 K showed uniform elongations larger than 5% and proof stresses smaller than 800 MPa. Irradiation in the High Flux Reactor (HFR) caused more hardening than did irradiation in the HFIR. Ductility loss and change in work hardening characteristics by HFR irradiation were evaluated from reduction of area values. Residual ductility was revealed to be larger than 0.5 in natural strain, and the irradiation was estimated to have a small effect on work hardening characteristics and on fracture stress. The ductility of the irradiated alloys was found to be about 58% of that for the unirradiated alloys, as has been previously reported for irradiation in the HFIR. It was also demonstrated that true stress-strain relations, except for the fracture conditions, could be represented by Swift's type constitutive equation.

An exact solutions of a viscoelastic ekman layer with suction

Because of the higher order derivatives the second-order fluid model requires an additional boundary condition in some flow fields. In this paper we discuss the problem of the missing boundary condition by solving the Ekman layer with suction for two types of constitutive equations. In the case of a viscoelastic fluid model of integral type (see eqn. 25) we obtain a “well-posed problem” with an exact unique solution. In the case of a second-order fluid model (see eqn. 16) we encounter the problem of the missing boundary condition. For weakly non-Newtonian second-order fluids with eqn. (25) as the original (generating) equation this difficulty can be avoided by iteration, obtaining an asymptotic solution. This procedure can be applied only if the “second-order fluid approximation” holds true in the whole flow field.

The constitutive equation resemblance of elastic — Plastic multiphase solid

Based on composite materials, the equivalent elastic-plastic constitutive equations of multiphase solid are researched. According to the suggested definition of constitutive equivalence it is demonstrated that the multiphase solid, composed of several kinds of homogeneous elastic-plastic media that conform to the generalized normality rule, has the same type of constitutive equations as its constituents have that also conform to the generalized normality rule.

Comparison of experimental data with the numerical simulation of planar entry flow: Role of the constitutive equation

AbstractThe goal of this research was to determine whether there is any interaction between the type of constitutive equation used and the degree of mesh refinement, as well as how the type of constitutive equation might affect the convergence and quality of the solution, for a planar 4:1 contraction in the finite eiement method. Five constitutive equations were used in this work: the Phan‐Thien–Tanner (PTT), Johnson–Segalman (JS), White–Metzner (WM), Leonov‐like and upper convected Maxwell (UCM) models. A penalty Galerkin finite element technique was used to solve the system of non‐linear differential equations. The constitutive equations were fitted to the steady shear viscosity and normal stress data for a polystyrene melt. In general it was found that the convergence limit based on the Deborah number De and the Weissenberg number We varied from model to model and from mesh to mesh. From a practical point of view it was observed that the wall shear stress in the downstream region should also be indicated at the point where convergence is lost, since this parameter reflects the throughput conditions. Because of the dependence of convergence on the combination of mesh size and constitutive equation, predictions of the computations were compared with birefringence data obtained for the same polystyrene melt flowing through a 4:1 planar contraction. Refinement in the mesh led to better agreement between the predictions using the PTT model and flow birefringence, but the oscillations became worse in the corner region as the mesh was further refined, eventually leading to the loss of convergence of the numerical algorithm. In comparing results using different models at the same wall shear stress conditions and on the same mesh, it was found that the PTT model gave less overshoot of the stresses at the re‐entrant corner. Away from the corner there were very small differences between the quality of the solutions obtained using different models. All the models predicted solutions with oscillations. However, the values of the solutions oscillated around the experimental birefringence data, even when the numerical algorithm would not converge. Whereas the stresses are predicted to oscillate, the streamlines and velocity field remained smooth. Predictions for the existence of vortices as well as for the entrance pressure loss (ΔPent) varied from model to model. The UCM and WM models predicted negative values for ΔPent.

Fabric Tensors in Constitutive Equations for Granular Materials

ABSTRACT In an effort to develop a constitutive equation associated with more deformation and failure features of granular materials, the constitutive formulation is studied taking into account the fabric tensors as internal variables and their evolution modes. In order to render the constitutive equation satisfy the principle of material objectivity, the representation theorem of isotropic functions is effectively used. The theorem is reviewed with a particular reference to the rate type constitutive equation which gives rise to the rate of deformation tensor as a function of the stress, stress rate and fabric tensors. After discussing the relationship between essential independent variables in constitutive equations and the state of granular materials from a viewpoint of the evolution mode of the fabric tensors, particular problems are studied with the result that: (1) Granular materials can show less summetry (more anisotropic) than orthotropic when the general stress states and paths are concerned; (2) In two dimensions, the introduction of the (anisotropic) fabric tensor is essential in order to describe the non-coaxial and dilatant behavior under the rotation of principal stress axes.

An internal variable constitutive model for hot working of metals

A set of internal variable type constitutive equations which model large elastic-viscoplastic deformations of metals at high temperatures is formulated. For a first-order representation of large deformation constitutive behavior we use a scalar as an internal variable to represent the isotropic resistance to plastic flow offered by the internal state of the material. Hot compression tests on an iron-2% silicon alloy and a commercially pure aluminum have been conducted. Based on these experiments, specific viscoplastic constitutive functions are proposed, and the values for the materials parameters appearing in these functions are determined. The constitutive equations with these functions and material parameters accurately reproduce the basic tests and also accurately predict the response of the metal to some critical experiments which were not used to determine the material parameters in the constitutive functions. Although the use of a single scalr internal variable limits the constitutive equations to describing the deformation behavior of initially isotropic metals up to deformation levels where significant internal damage and polycrystal texturing has not developed, the constitutive equations presented here should be useful for obtaining an improved analysis of various hot deformation processing operations of metals.

Rigid-Plastic Finite Element Method Using Rate-Type Constitutive Equation

The rigid plastic finite element method is frequently used to predict the deformation behaviour of the materials during the forming processes. The constitutive equation employed is usually restricted to flow type one so that the normality of the strain rate to the yield surface is entailed. Then the deformation can not follow the abrupt change in stress history. This often yields the over stabilization of the deformation. In this paper, to overcome this situation, a rate type constitutive equation with a one-to-one relation between stress rate and strain rate is employed, and the formulation of the rigid plastic finite element method is presented. The method is applied to the prediction of the flow localization of the strip under tension. The feature of the present results is clarified by comparing the results with those due to J2 flow and corner theory.

Rigid-plastic finite element method using rate type constitutive equation.

The rigid plastic finite element method is frequently used to predict the deformation behaviour of the materials during the forming processes. The constitutive equation employed is usually restricted to flow type one so that the normality of the strain rate to the yield surface is entailed. Then the deformation can not follow the abrupt change in stress history. This often yields the over stabilization of the deformation. In this paper, to overcome this situation, a rate type constitutive equation with a one-to-one relation between stress rate and strain rate is employed, and the formulation of the rigid plastic finite element method is presented. The method is applied to the prediction of the flow localization of the strip under tension. The feature of the present results is clarified by comparing the results with those due to J2 flow and corner theory.

Onset and growth of wrinkles in thin square plates subjected to diagonal tension

A corner type constitutive equation is developed using the general quadratic anisotropic yield function. The wrinkling point of square plates subjected to diagonal tension was obtained by the bifurcation and Mindlin type plate theories, in conjunction with the finite element approximation. The growth of wrinkles was traced by a three-dimensional analysis in terms of an isoparametric shell element. The numerical investigation was performed for the thin plates characterized by Hill's anisotropic yield function. The effects of such material characteristics as yield stress, hardening rate, orthotropy and normal anisotropy on the wrinkling limit and wrinkle height have been clarified.