Type Constitutive Equation Research Articles

Influence of strain rate on hot deformation behaviour and texture evolution of AZ31B

In the present work, the effects of strain rate on the flow behaviour and microstructure evolution of AZ31 Mg alloy were studied by compression testing over a wide range of strain rates (0·01–100 s−1) and temperatures (300–450°C). In addition, the influence of different strain rates on the dynamic recrystallisation (DRX) mechanisms and texture evolution was investigated. The results showed that with increasing strain rate, the twin induced DRX fraction increased at a constant temperature, and the contribution of continuous DRX decreased. On increasing the strain rate, the formation of twins and subsequent twin induced DRX intensified the basal texture in the deformed sample. In addition, the recrystallised volume fraction increased significantly with strain rate. The flow behaviour was fitted to two types of constitutive equations: power law and hyperbolic sine. Average activation energies of about 162 and 135 kJ mol−1 were obtained for the peak and steady state strain respectively.

Review on the Current State of Developing of Advanced Creep Damage Constitutive Equations for High Chromium Alloy <sup></sup>

This paper presents a review of developing of creep damage constitutive equations for high chromium alloy (such as P91 alloy). Firstly, it briefly introduces the background of creep damage for P91 materials. Then, it summarizes the typical creep damage constitutive equations developed and applied for P91 alloy, and the main deficiencies of KRH (Kachanov-Robatnov-Hayhurst) type and Xus type constitutive equations. Finally it suggests the directions for future work. This paper contributes to the knowledge for the developing creep damage constitutive equations for the specific material.

Rate boundary value problems and variational inequalities in rate-independent finite elasto-plasticity

The paper deals with variational inequalities related to the rate quasistatic boundary value problem and associated with a generic stage of the process in our approach to finite elasto-plastic materials. The Eulerian setting of the models can be associated with the various models, firstly postulated with respect to the plastically deformed configurations, by the pushing forward procedure, as well as the Lagragian formalism of the models can be adopted by pulled back to the reference configuration procedure. Only when the constitutive and the evolution functions which describe the behaviour of the elasto-plastic material are compatible with the restrictions imposed by the dissipation postulate, the variational inequalities are characterized by symmetric bilinear forms. In the present formulation of the variational inequalities we used an appropriate procedure that allows us to reformulate pointwise the consistency condition in the form of an inequality. We formulate the variational inequalities to be solved for the unknowns, namely the velocity field and the plastic factor, in the deformed configuration at time t, considered as a reference one. The coefficients of the variational inequalities are dependent on the current values of the stress, the elastic and plastic deformations, as well as the internal state variables. We propose an update algorithm which allows us to find the current state of the process at the time, by integrating the rate type constitutive equations, which characterizes the current state of the elasto-plastic materials through the objective derivative. We prove an existence and uniqueness result concerning the solution, namely the spatial velocity and plastic factor, for the symmetric variational inequality of the first kind and formulate an incrementally objective algorithm in order to update the state of the process. By solving the appropriate variational inequality at a fixed time t, we avoid the application of the Return-Mapping Algorithm proposed by Simo (1998).

Identifying Constitutive Parameters from Heterogeneous Strain Fields using the Virtual Fields Method

The main features of the Virtual Fields Method, which is aimed at retrieving constitutive parameters from full-field measurements, are given in this paper. This method is based on the principle of virtual work which is recalled here. The different aspects of the procedure are then briefly explored, notably the choice of the virtual fields and the different types of constitutive equations that can be considered with this identification technique.

Constitutive Relationship of New Steel 33Mn2V and Its Application in Piercing Process by FEM Simulation

Based on test data from the hot forge experiments on Gleeble 1500, a Kumar type constitutive equation for 33Mn2V steel is established. Applying this constitutive equation in commercial FEM software of MSC/SuperForm 2005, the piercing process of 33Mn2V steel in Mannesmann mill is then simulated. The modeling results visualized the dynamic evolution of equivalent stress, especially inside the workpiece. It is shown that the non-uniform distribution of stress on the internal and external surface of the workpiece is a distinct characteristic of processing tube piercing. The numerical model was verified by comparing the values of calculated force parameters of the piercing process with those measured in laboratory conditions. And it shows that the Kumar-type constitutive relationship meets the practical needs.

Effect of cyclic strain on the mechanical behavior of virgin ultra-high molecular weight polyethylene

Ultra High Molecular Weight Polyethylene (UHMWPE) is a polymeric material employed in critical biomedical applications. Knowledge of its mechanical behavior is essential in order to obtain accurate prediction of stresses and deformations in real components, in particular when cyclic loading is considered. In the present research the effects of alternating and pulsating cyclic strain on the mechanical response of UHMWPE were studied by means of an experimental procedure based on tests carried out in strain control at different mean cyclic strain levels. During the tests the temperature increase due to hysteretic heating was controlled by means of a compressed air cooling apparatus specifically devised. By taking advantage of the possibility to control and stabilize temperature, cyclic steady-state mechanical response was investigated at room temperature and at 37 and 50 °C, comparing the effects of alternating and pulsating loading cycles. A transient thermal analysis using the finite element method (FEM) was also carried out to analyze temperature distribution within the specimen. UHMWPE exhibited cyclic softening as a result of a thermal contribution due to temperature increase and of a mechanical contribution related to the effects of applied load on the microstructure. The material exhibited different peak stress percent reductions for pulsating and alternating loading and during tensile and compressive loading phases. For pulsating tests significant cyclic mean stress relaxation was also observed. Based on the experimental procedure described the cyclic curve was determined as a function of temperature and fitted with a Ramberg–Osgood type constitutive equation, in which material parameters are temperature dependent. In this way the combined effects of temperature rises, such as those that might occur in biological environments or due to frictional heating, and mechanical loads could effectively be taken into account for constitutive modeling purposes of cyclic mechanical behavior of UHMWPE.

Hot Deformation Behaviour of Low Alloy Steel

AbstractHot deformation of a continuously cast low alloyed steel is studied by means of hot compression and tensile tests by using a Gleeble® 1500 machine after austenitization in a wide range of strain rates and between 720–800 °C. The flow data are evaluated to obtain the strain rate sensitivity and the processing maps. A new calculation method is used, yielding on the instability parameter defined as κJ which correlates well with the microstructural changes. The strain rate sensitivity does not predict any instability but all the others instability parameters do, including the new κJ. Flow instability appears at high temperatures and low strain rates where pores are formed at the austenitic grain boundaries, causing a decay of ductility in the tensile test. During hot deformation more ferrite is formed than corresponding heat treatments without deformation. In these conditions, the deformation is concentrated in the softer ferrite phase. The deformation induced ferrite deforms by dynamic recovery forming new grains as revealed by metallography and is correlated with low coefficients of power dissipation. The sinh type constitutive equation represents the flow data well with a stress exponent n = 4.1 and an apparent activation energy Q = 218 kJ/mol.

Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.

An inverse thermal–mechanical analysis of the hot torsion test for calibrating the constitutive parameters

One of the key tasks for mathematical representation of a constitutive model is calibration of its parameters. An inverse computational–experimental solution with the data modelling approach was employed here to estimate the parameters of a hyperbolic sine type constitutive equation. The solution utilizes a multi-layer constitutive model. The computational component of the inverse solution includes a rigid viscoplastic finite element code based on the thermo-mechanical coupling. The hot torsion test data comprises the experimental component of the inverse solution. Determination of the primary constitutive parameters (PCPs) pertinent to a 303 Austenitic stainless steel is presented here as an example. In order to facilitate the calibration of the sub-models, a procedure to provide an initial guess vector for the secondary constitutive parameters (SCPs) is given.

Rheological characterizations and extrudate swell predictions of an LDPE melt by two KBKZ-type constitutive equations

The shear viscoelastic properties of an LDPE (1I2A-1) melt at 160°C were described by using two KBKZ integral type constitutive equations, i.e. Wagner model and the three- parameter rational type model. The present characterizations were compared with the previous calculations by using PSM model. The results show that there are differences between the descriptions on the stress relaxation property in step strain and the first normal stress difference for the three models, and however, all the three models show the almost same predictions on the shear-thinning viscosity of the ldpe melt. Both Wagner model and PSM model are good choice in describing the rheological properties of the melt due to theirs simplicity. The Tanner theory combined with the Wagner model or the three-parameter rational type model was used to predict the swell ratio of the LDPE melt through a long capillary die, which was also compared with the experimental results and the previous calculations on swell. The predictions by Tanner theory combined with the three different constitutive models on the swell ratio of the LDPE melt are similar, which are apparently lower than the experiments and numerical simulations over the whole experimental shear rate range. The results further indicate that Tanner theory cannot predict the swell ratio of the LDPE melt precisely.

Anisotropic and Dissipative Elasto-Plastic Materials with Damaged Structure

The proposed framework for anisotropic damage coupled to finite elasto-plasticity is based on the multiplicative decomposition of the deformation gradient into its elastic (reversible) and plastic components, which involves the existence of a tensorial damage variable, Fd, i.e. F = FeFdFp. Fd appears like a kinematic variable, whose rate in time produces the variation in internal power dissipated during the irreversible process. The material is characterized by an elastic type constitutive equations, which is dependent on Fd, with respect to the stress free and damaged configuration. Fp and Fd are described by the evolution equations written with respect to the stress free and undamaged configuration. The dissipative nature for the irreversible behavior is modeled by the requirement to be satisfied the principle of the free energy imbalance, on the isothermal processes. The constitutive and evolution equations are derived to be compatible with the free energy imbalance.

岩石のピーク強度以前の時間依存性挙動を表す構成方程式とパラメータの取得方法

Authors have proposed constitutive equations for rock based on the theory of non-linear visco-elasticity. The constitutive equations are grouped into two types: compliance or irrecoverable-strain monotonically increases with stress. In earlier studies, it was reported that both types of constitutive equations are applicable at least to the post-failure region.In this study, applicability of the constitutive equations to the pre-failure region was examined. At first, parameters were estimated from the results of unloading-reloading and alternating loading-rate tests under uniaxial compression. The calculated results with the estimated parameters were compared with the experimental results under various loading conditions.If a constitutive equation was properly selected, the equation well simulated the various time-dependent behaviors in the pre-failure region: loading-rate dependency of Young's modulus, creep and generalized relaxation where a set of parameter values was used in not only pre-failure but also post-failure region.

Deformation behaviour of aluminium during machining: Modelling by Eulerian and smoothed-particle hydrodynamics methods

Large-strain deformation behaviour of aluminium that accompanies continuous chip formation during machining (1100 Al) has been studied using experimental and numerical techniques. In experimental studies, local values of plastic strains were determined in the primary and the secondary deformation zones of machined 1100 Al. This was completed through a careful examination of metallographic sections taken from the material ahead of the tool tip, in which orientation changes in the flow lines in the material and shear angles were used to calculate plastic strains. Variations in local flow stresses were estimated from microhardness measurements. The examination of the stresses and strains at each measurement location generated a stress—strain relationship for the 1100 Al material. An important observation from the experimental portion of this research indicated that the material stress—strain response was independent of the feed rates considered in this study. Additionally, the response was observed to obey an exponential relationship with stress saturation occurring at approximately 300 MPa. Parameters associated with the Johnson—Cook constitutive equation were also determined from the experimental work. An Eulerian finite-element method and a relatively new so-called mesh-free method [smoothed-particle hydrodynamics (SPH) method] have been applied to the simulation of machining. The application of these methods permits simulation of the machining process without use of any mesh separation criterion. Appropriate values of the coefficients of friction, for numerical studies, were determined in parametric studies by correlating the experimentally measured chip thicknesses with the numerically predicted values. The effectiveness of the Eulerian and SPH methods in predicting the response of 1100 Al during orthogonal machining has been assessed through a rigorous comparison of the stress—strain distribution within formed chips during steady-state cutting. Both the Eulerian and SPH models showed good overall correlation with the experimentally measured stress—strain distribution when the exponential type material behaviour was assumed in modelling. A maximum stress of 300 MPa at the tool tip was obtained from the numerical simulations using the assumed exponential material behaviour. The location of maximum stress corresponded to the position of maximum strain (8.0). The application of the Johnson—Cook type constitutive equation resulted in predicting a lower maximum equivalent strain (4.5) and higher maximum stresses (325 MPa).

Development of 3-Dimensional Compression Device and its Application to Clarification of Densification Behavior of Urethane-Foam

This paper deals with the results of three dimensional compression tests carried out for high stiffness urethane foams (Penguin-foam, Sunstar Engineering Ltd.), and also deals with the constitutive modelling base on Shima-Oyane’s consolidation condition for the tested foamed urethane. Three kinds of urethane foams, relative densities of which were 0.1, 0.2 and 0.33, were employed in the experiments. Like metallic porous materials, the tested urethane foams show the strong plastic-compressibility. On the other hand, in modelling, unlike metallic porous materials, the identified material constants for different density foams do not take the same (or unified) values but take the different values when Shima-Oyane’s constitutive model is assumed. Furthermore, the experimentally derived stress-relative density curves could not be satisfactorily described by Shima-Oyane’s original constitutive model; the experimental stress-relative density curves show stronger work hardening as compared with the simulated ones especially in the large deformation stage. To avoid those inconvenience, in this paper, a modified Shima-Oyane type constitutive equation was also proposed, and it was shown that the proposed model could well express both the low work hardening area of the stress-relative density curves at the initial deformation stage and the strong work hardening area at the final deformation stage by supposing the stress restriction at initial deformation stage due to the buckling of cell walls of each foam, and the rapid stress increase at the large deformation stage caused by the successive contact and the friction between the bent cellular walls, respectively.

Numerical prediction and experimental validation of viscoelastic annular swell

AbstractBoth swell and sag determine the parison dimensions in a viscoelastic annular extrusion. So, it is highly desirable to be able to predict the swell of an annular parison under the effect of sag during its extrusion. Numerical simulations on the parison extrusion at the conditions corresponding to industrial blow molding applications were carried out by adopting a finite element method and the Kaye–Bernstein–Kearsley–Zapas (K‐BKZ)‐type constitutive equation in the present work. Five dies were used to study the effect of die geometry (mainly die inclination angle) on the parison swells. The results show that the swellings continue to increase with increasing die inclination angle, which is explained according to stress distributions and exit pressure in the die. Reasons for the difference between the numerically predicted and the experimentally measured swells are analyzed. Then, the sensitivity of the predicted swells to the values of material constants appearing in the K‐BKZ model is examined. It is suggested that the material constants should be chosen in accordance with the die geometry to reflect its contribution to the shear and extensional deformation of the polymer melt inside the die. © 2007 Wiley Periodicals, Inc. Adv Polym Techn 25:259–269, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/adv.20077

Optimizing parison thickness for extrusion blow molding by hybrid method

A hybrid method consisting of finite element method (FEM), artificial neural network (ANN), and genetic algorithm (GA) was used to find the optimal parison thickness distribution for a blow molded part with required thickness distribution. Firstly, numerical simulations on the parison inflation were performed using FEM and the K-BKZ integral type constitutive equation. Based on the simulation results, a back propagation (BP) ANN model was then developed to build the relationship between parison thickness distribution and the objective function, which was used to evaluate the wall thickness distribution of part. The predictive ability of the ANN model was verified through FEM simulation results different from those utilized in the training stage. Finally, a GA was developed and used to search for the optimal parison thickness distribution. The results showed that the hybrid method proposed in this work can effectively obtain the optimal parison thickness distribution for a blow molded part with required wall thickness distribution. Compared with the trial and error method, the hybrid method can shorten the part development time and save a lot of material.

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.

Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing

This paper describes a novel numerical algorithm for simulating interfacial dynamics of non-Newtonian fluids. The interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously. The property and evolution of the interfacial layer is governed by a phase-field variable ϕ that obeys a Cahn–Hilliard type of convection-diffusion equation. This circumvents the task of directly tracking the interface, and produces the correct interfacial tension from the free energy stored in the mixing layer. Viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase-field framework. The greatest challenge of this approach is in resolving the sharp gradients at the interface. This is achieved by using an efficient adaptive meshing scheme governed by the phase-field variable. The finite-element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large-scale two-phase systems of complex fluids. In two-dimensional and axisymmetric three-dimensional implementations, the numerical toolkit is applied here to drop deformation in shear and elongational flows, rise of drops and retraction of drops and torii. Some of these solutions serve as validation of the method and illustrate its key features, while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces.

Numerical Analysis of the Rheology of Polymeric Liquid Crystals (3rd Report, Evaluation of a Second Order Tensor Type Constitutive Equation)

We have evaluated numerically a second order tensor type constitutive equation which includes both a molecular shape factor and a quadratic closure approximation, by means of comparing the results with the solutions obtained from the original Doi equation without closure approximations. The second order tensor type constitutive equation can simulate three types of director behaviors such as tumbling, wagging and aligning, by setting the molecular shape factor to be smaller than its inherent value based on the molecular aspect ratio. The shear stress and the first normal stress difference determined from the tensor type constitutive equation qualitatively agree with stresses obtained from the Doi equation, for low shear rate. When shear rate is high, however, even qualitative agreement is not obtained. Therefore, the constitutive equation is applicable to slow flows or situations where a velocity field is so simple that only an orientation field is a consideration.

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.