Terms Of Constitutive Equations Research Articles

Effect of Processing Parameters on Superplastic and Corrosion Behavior of Aluminum Alloy Friction Stir Processed

Friction Stir Processing (FSP) is a variant of Friction Stir Welding, and can be used to modify the materials microstructure to functionalize it. Superplastic forming is a technological process used to produce components with very complex shapes. In the last two decades it has been a topic of major development. In Fine Structure Superplasticity (FSSP), the initial grain size exerts a strong influence on the superplastic strain rate and temperatures. Refining grain size (GS) the parameters (temperature and strain rate) of superplastic forming could be optimized. Thermal stability is also an important factor to obtain superplasticity. FSP is used to refine GS, but the optimum processing parameters are still under study over different materials. Corrosion resistance can be affected by FSP too, but the information about it is scarce. In the present study, 7075-T651 aluminium alloy was friction stir processed under different conditions in order to improve superplastic behavior. Tool profile, rotation rate and traverse speed were analyzed. Microstructures with <4 μm grain size were obtained. The maximum superplastic elongations, in a range of 740 to 900%, at 400°C were obtained at 1x10-2s-1 strain rate. The results were discussed in terms of constitutive equations and microstructure evolution. Localized corrosion potentials were obtained. Localized corrosion resistance was affected by friction stir processing.

Failure criteria of fibre reinforced composites in homogenous temperature field

The present paper examines the failure criteria of layered composites with orthotropic properties in the homogenous temperature field. The composite has modeled by two mechanically equivalent families of fibres. The paper formulates constitutive equations in terms of intrinsic ?preferred? directions, which are defined by the orientation of fibers at any point of the composite. A uniformly heated, thermoelastic solid undergoes distortion as well as volume change because it experiences differential expansions in different directions. This effect is more complicated if, in addition of being anisotropic, the material is inhomogeneous, as in the case with laminated materials. In order to illustrate the influence of temperature on the failure of this group of materials constitutive equations are derived and adopted for use in failure criteria, without the influence of temperatures, and with the influence of increased temperature.

Characteristic based split (CBS) meshfree method modeling for viscoelastic flow

A semi-implicit characteristic based split meshfree algorithm is proposed for the numerical solution of incompressible viscoelastic flow in the paper, in which the governing equations are discretized by element free Galerkin method in spatial space and the characteristic based split method is adopted in temporal space. In order to obtain stable and convergent solution at high Weissenberg number, streamline upwind method is chosen to tackle the convective terms in constitutive equations of viscoelastic flow. Meanwhile, mass lumped technique is adopted to accelerate computational efficiency. The method allows the equal order basis to approximate pressure, velocity and extra stress, especially the linear basis that is easy to implement. The planar Poiseuile flow and 4:1 planar contraction flow for an Oldroyd B fluid are investigated. Through these numerical experiments, we find that the method has good stability and the numerical results are in agreement well with those reported by other papers.

The state of the art in the rheology of polymers: Achievements and challenges

The state of the art in the rheology of polymer fluids (polymer solutions and melts) and filled composites is reviewed. This review includes two parts: analysis of the basic principles for the construction of rheological constitutive equations in terms of the continuum mechanics and finding correlations between the rheological characteristics and molecular structure of polymers on the basis of molecular models. Possible approaches to the formulation of constitutive equations are discussed. Special attention is focused on the correct selection of the form of the elastic potential for rubbery deformations induced under the flow of polymer fluids. The use of a power-law potential leads to the best results. To gain unequivocal results and minimize the number of free constants, viscoelastic characteristics of polymer fluids should be described in terms of a continuous relaxation spectrum as a power-law function limited by the maximum relaxation time. To solve the boundary problems by the selected constitutive equation, analysis of the dynamic stability is required, because the combination of viscosity and elasticity controls the limits of flow upon shear and tensile. Deformation can also lead to changes in the phase state of a polymer system. Furthermore, correct formulation of the boundary conditions is necessary because, in many cases, polymer fluids and, in particular, filled materials tend to efficient slip along walls. The existing molecular models adequately describe the characteristics of monodisperse polymers; however, on passing to polydisperse polymers, the additional use of semiempirical approaches is required. The modern level of experimental studies allows test measurements over a wide range of deformation rates, frequencies, and temperatures. However, in this field, the mainstream tendency in experimental studies is concerned with hybrid methods, which combine direct rheological measurements with optical observations of local structure and its evolution in the material. In this case, various physical principles of measurements are applied. In recent years, much interest has been focused on studying polymer compositions containing nanosized fillers, which are able to produce their structures in melt.

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC (finite incremental calculus) pressure stabilization process and the DEVSS (discrete elastic viscous stress splitting) method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU (streamline-upwind) method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP(extended Pom-Pom) constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4:1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of theWeissenberg number, particularly the capability in reproducing the results, which can be used to explain the "die swell" phenomenon observed in the polymer injection molding process.

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.

An analytical–numerical framework for the study of ageing in fibre reinforced polymer composites

In the last few years there has been available a growing amount of information on composites ageing that considers effects such as temperature, oxidation, UV radiation, permanent loading, etc. In this paper, an analytical–numerical framework adequate for the compilation, interpretation and application of experimental data to actual engineering analysis and design is discussed. The formulation proposed includes elastic anisotropic relations; ageing viscoelastic anisotropic constitutive equations in terms of state variables; age-adjusted failure and degradation criteria; all in a setting of large displacements with small strains. The essential differences between hardening and softening ageing processes are described together with the constitutive relations adequate for each case. These equations are written in a form suitable for numerical analysis in a finite elements context, using an incremental-iterative solution procedure that accounts for post-critical effects. Several examples, including elastic, viscoelastic and failure behaviour in a large displacement context are included to check the algorithms.

Explicit exact analysis of infinite periodic structures under general loading

This paper deals with the explicit analysis of infinite periodic structures under arbitrary loadings. In the context or structural stiffness optimization, with its inherent problem of multiple reanalysis, the purpose is to obtain expressions for the stress resultants anywhere in the infinite structure as an explicit function of the stiffnesses of the elements. Following the method of the representative cell the analysis of an infinite structures is reduced to the analysis of single module under transformed loading and boundary conditions by using the discrete Fourier transform. This produces the equilibrium, strain-displacements and constitutive equations in terms of complex-valued displacements, generalized strains and generalized stresses transforms. Next an existing formula is used to write the stress resultants transforms explicitly in terms of the stiffnesses. Finally one computes the stress resultants wherever needed in the real structure by means of the inverse Fourier transform. The exact formula for the stress resultants is usually impractical due to the large number of terms involved in the analytical expressions. What makes the approach practical herein is the very reduced size of the repeating module that is to be analysed, which renders the analytical formula more tractable in many cases. The technique is illustrated with the explicit analysis of an infinite truss with 1D translational symmetry and of an infinite grid of orthogonal beams on elastic supports with 2D translational symmetry.

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule. Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted. Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.

Deformation of Laminated Elastomer Composites

Abstract Flexible elastomer composites subjected to finite deformation are usually associated with significant fiber re-orientation and configuration change which cannot be described by classical composite lamination theory. Utilizing the Lagrangian description and a strain-energy density approach, this work presents a set of 2-D constitutive equations in terms of the overall deformation of laminated composites. This model includes both geometric and material nonlinearities of the laminate. Interlaminar shear deformation is addressed but not studied in this analysis. Theoretical predictions have been compared with the available experimental data about symmetric laminates with various fiber orientations in the finite deformation range. Also, a parametric study has been performed under various ratios of biaxial deformation.

Viscoelastic constitutive modeling of asphalt concrete with growing damage

This paper presents a mechanistic approach to uniaxial viscoelastic constitutive modeling of asphalt concrete that accounts for damage evolution under cyclic loading conditions. An elasticviscoelastic correspondence principle in terms of pseudo variables is applied to separately evaluate viscoelasticity and time-dependent damage growth in asphalt concrete. The time-dependent damage growth in asphalt concrete is modeled by using a damage parameter based on a generalization of microcrack growth law. Internal state variables that describe the hysteretic behavior of asphalt concrete are determined. A constitutive equation in terms of stress and pseudo strain is first established for controlled-strain mode and then transformed to a controlled-stress constitutive equation by simply replacing physical stress and pseudo strain with pseudo stress and physical strain. Tensile uniaxial fatigue tests are performed under the controlled-strain mode to determine model parameters. The constitutive equations in terms of pseudo strain and pseudo stress satisfactorily predict the constitutive behavior of asphalt concrete all the way up to failure under controlled-strain and -stress modes, respectively.

Stress fields in general composite laminates

A direct approach is employed to obtain a general boundary integral formulation for the analysis of composite laminates subjected to uniform axial strain. The integral equations governing the problem are directly deduced from the reciprocity theorem, employing the generalized orthotropic elasticity fundamental solutions expressly inferred. The solution is achieved by the boundary element method, which gives, once the traction-free boundary conditions and the interfacial continuity conditions are enforced, a linear system of algebraic equations. The formulation does not present restrictions with regard to the laminate stacking sequence and it does not require any aprioristic assumption. The interlaminar stress field near the free edge of generally stacked composite laminates subjected to uniaxial extension is investigated through this boundary integral equation formulation. The numerical applications show good agreement with those already available in literature, and they demonstrate the accuracy and the efficiency of the proposed method. approach where a complete three-dimensional analysis is performed was presented by Pipes and Pagano,1 who employed the finite dif- ference technique to obtain the solution of the governing elastic- ity equations. Many solutions obtained by using the finite element method are available.21() These differ from each other in the formu- lation, in the kind of employed elements, and in the discretization schemes. In the literature analytical solutions of approximate the- ories are also present. The techniques employed to achieve these solutions include the perturbation method,11 series solution,12'13 Lekhintskii's complex stress potentials coupled with an eigenfunc- tion expansion14'15 or a polynomial expansion,16'll the extension of Reissner's variational principle,18'19 and the force balance method coupled with the minimum complementary energy principle.20'21 The stress distributions obtained with these approaches show good agreement between them for points away from the free edge. Con- siderable disagreement exists instead for points near the free edge location among the various numerical and analytical solutions. This is to be expected as a result of a priori assumptions or because the boundary conditions of the continuum problem have been trans- formed into conditions on the generalized data. Actually the numer- ical solution techniques are not able to predict the singular trend in the stress field at the free edge, and hence, to achieve satisfactory solutions a lengthy extrapolation procedure is required. On the other hand, the analytical approximate solutions often rest on assumptions about the problem unknowns that enforce the free edge stress field structure. In the present paper, the stress field in multilayered composite laminates under uniform axial extension is analyzed on the basis of the integral equation theory.22'23 The laminate is considered as composed by prismatic elements having different elastic proper- ties. The generic element or ply is treated as homogeneous, and in terms of constitutive equations it is described by a generalized orthotropic law. The integral equations governing the exact elastic- ity solution of the problem are directly obtained by applying the reciprocity theorem with the fundamental solutions of the general- ized orthotropic elasticity.2427 The

Model of Inelastic Response Using Neural Networks.

Inelastic material behavior is usually expressed in terms of constitutive equations. However, it is not easy to estimate parameters included in these equations from the experimental data. This paper describes a model using neural networks for the estimation of material transient responses. To overcome the difficulty for the neural networks to learn the material response curve, two neural networks are independently used to learn the two curves that can be decomposed from the original stress-strain curve based on the internal stress concept. Numerical examples show that the present neural network approach can map the stress-strain curve with good precision.

Continuum Modeling

Common experience with the thermomechanical properties and behavior of matter is usually at the continuum level, i.e., observations concern such phenomena as elastic and inelastic deformation of solids, flow of fluids, and conduction of heat. Continuum descriptions of these phenomena are expressed in terms of partial differential equations representing the principles of balance of mass, momentum, and energy. Since the basic principles apply to all materials, it is apparent that they alone will not suffice for solving specific problems. The peculiarities of individual materials are expressed in terms of constitutive equations, of which Hooke's law of elasticity and Newton's law of viscosity are examples. For the most part, “continuum modeling” refers to the process of devising constitutive equations. This work requires exercise of physical insight at both the macroscopic and microscopic levels, consideration of experimental observations, and application of formal mathematical principles.Much of materials science is devoted to development and application of materials that have been selected for, or designed to have, useful properties different from those of existing materials. Many materials are of interest precisely because they have unusual properties. For example, paints and other coatings are often designed to flow easily when spread, but resist running afterward. Much work is devoted to analyzing manufacturing processes, whether they be processing of foodstuffs, forging metals, drawing films and fibers, or curing polymers. Properties and physical states of materials that are important during manufacture are often very different from those desired under service conditions.

The Einstein equivalence principle, intrinsic spin and the invariance of constitutive equations in continuum mechanics

The invariance of constitutive equations in continuum mechanics is examined from a basic theoretical standpoint. It is demonstrated that constitutive equations which are not form invariant under arbitrary translational accelerations of the reference frame are in violation of the Einstein equivalence principle. Furthermore, by making use of an analysis based on statistical mechanics, it is argued that any frame-dependent terms in constitutive equations must arise from the intrinsic spin tensor and are neglibible provided that the ratio of microscopic to macroscopic time scales is extremely small. The consistency of these results with existing constitutive theories is discussed in detail along with possible avenues of future research.

Progress in Developing Constitutive Equations for Inelastic Design Analysis

A summary is given of the constitutive equations that have been developed for use in design assessments of elevated temperature components of liquid metal fast breeder reactors. The discussion addresses representations of short-term (plastic) and long-term (creep) inelastic material responses. Attention is given to improved representations of the interactions between plastic and creep deformations. Most of the discussion is in terms of constitutive equations that make use of the concept of separating the total strain into elastic, plastic, and creep portions. Additionally, some discussion is given of progress being made toward establishing design equations based on unified measures of inelastic strain that do not distinguish different strain portions.

A note on the slow motion of a bubble in a viscoelastic liquid

The Oldroyd-B fluid has become the starting point for almost all complex flow calculations and analysis involving the behavior of dilute polymer solutions. The reasons for this are clear: based on the kinetic theory of high molecular weight “phantom” chains near equilibrium, the resulting model, in its simplest form, produces a single mode description of the solution constitutive equation in terms of a single symmetric dyad order parameter. The physical connection to the individual chains is therefore unambiguous through the end-to-end configuration tensor. The flow of the fluid resulting from the mathematical solution of the constitutive equation coupled to Cauchy’s equation of motion and continuity has been examined for decades and its strengths and weaknesses in predicting the features of non-linear viscometric and non-viscometric polymer solution flows are well known. Moreover, “simple” extensions (e.g. the FENE-P model) can ameliorate the most serious of these shortcomings. For example, the multi-mode Oldroyd-B model can provide quantitative predictions of the linear viscoelasticity of a wide range of polymer solutions. Remarkably, even with all its shortcomings, the Oldroyd-B fluid has been invaluable in predicting at least qualitatively, features as complex as purely elastic instabilities, turbulent drag modification, and even the effective rheology of particles suspended in polymer solutions. While remaining mindful of the model’s shortcomings, any student of these diverse fields may find the Oldroyd-B fluid an important starting point.

A THEORY OF ANISOTROPIC THERMOELASTICITY AT LOW REFERENCE TEMPERATURE

The linear coupled theory of thermoelasticity, with a modified Fourier heat-conduction law to include the thermal-relaxation effect, has been applied in studying heat pulses in solids at very low reference temperature T 0. The theory is limited to the case in which the temperature change ∣T − T 0∣ is much less than To. This restriction is relaxed in this work by developing a theory in terms of the coldness, which is the reciprocal of the temperature; in any heating process the net change in coldness is always less than the coldness at the reference state. Linear and nonlinear constitutive equations in terms of the coldness variable as well as the balance equations are proposed. They might be useful for investigating heat-pulse propagation in solids at extremely low temperature.

The incremental perturbation method for large displacement analysis of elastic‐plastic structures

AbstractA procedure of applying the perturbation method is presented for the incremental numerical analysis of materially and combined non‐linear problems of discrete and discretized structral systems. Small but finite strain and stress increments are strictly distinguished from the strain rates and stress rates, respectively. It is shown that, by applying the perturbation procedure not only to the non‐linear strain‐displacement relations and equilibrium equations, but also to the constitutive equations in terms of rate quantities, all the governing equations can be satisfied to any desired accuracy at every instantaneous configuration in between the starting and terminal points of an incremental step. The proposed method provides also means of finding, to a desired accuracy, every point on an equilibrium path of a discrete system at which a new element will start yielding or unloading and possible critical points on the path. The significance of the proposed method is expected to be appreciated particularly in numerical investigations of critical behaviours and post‐buckling behaviours.

The use of first order differential equations as constitutive equations in the theory of plasticity

The crucial points which distinguish plastic behaviour of metallic materials from elastic behaviour are the existence of a yield stress and the consequent non-uniqueness of the relation between the stress tensor and the strain tensor. It is shown in this paper that the general pattern of plastic behaviour in quasistatic conditions can be adequately described in terms of constitutive equations which relate the stress, strain, stress-rate and strain-rate tensors.