A new exact integration method for the Drucker–Prager elastoplastic model with linear isotropic hardening

This paper presents the exact stress solution of the non-associative Drucker–Prager elastoplastic model governed by linear isotropic hardening rule. The stress integration is performed under constant strain-rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory. Based on the time-continuous stress solution, a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex. Explicit expression for the algorithmically consistent tangent tensor is also derived. In addition, a fully analytical strain solution is also derived for the stress-driven case using constant stress-rate assumption. In order to get a deeper understanding of the features of these solutions, two numerical test examples are also presented.     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

Elastomers in large-amplitude oscillatory uniaxial extension

In this work, we subject elastomers to a fixed pre-stretch in uniaxial extension, ε _p , upon which a large-amplitude, ε ₀, oscillatory uniaxial extensional (LAOE) deformation is superposed. We find that if both ε _p and ε ₀ are large enough, the stress responds with a rich set of higher harmonics, both even and odd. We further find the Lissajous-Bowditch plots of our measured stress responses versus uniaxial strain to be without twofold symmetry and, specifically, to be shaped like convex bananas. Our new continuum model for this behavior combines a new nonlinear spring, in parallel with a Newtonian dashpot, and we call this the Voigt model with strain-hardening. We consider this three-parameter (Young’s modulus, viscosity, and strain-hardening coefficient) model to be the simplest relevant one for the observed convex bananas. We fit the parameters to both our LAOE measurements and then to our uniaxial elongation measurements at constant extension rate. We develop analytical expressions for the Fourier components of the stress response, parts both in-phase and out-of-phase with the extensional strain, for the zeroth, first, second, and third harmonics. We find that the part of the second harmonic that is out-of-phase with the strain must be negative for proper banana convexity.     

Proposal of FEM implemented with particle discretization for analysis of failure phenomena

A new method for numerical simulation of failure behavior, namely, FEM-β, is proposed. For a continuum model of a deformable body, FEM-β solves a boundary value problem by applying particle discretization to a displacement field; the domain is decomposed into a set of Voronoi blocks and the non-overlapping characteristic functions for the Voronoi blocks are used to discretize the displacement function. By computing average strain and average strain energy, FEM-β obtains a numerical solution of the variational problem that is transformed from the boundary value problem. In a rigorous form, FEM-β is formulated for a variational problem of displacement and stress with different particle discretization, i.e., the non-overlapping characteristic function of the Voronoi blocks and the conjugate Delaunay tessellations, respectively, are used to discretize the displacement and stress functions. While a displacement field is discretized with non-smooth functions, it is shown that a solution of FEM-β has the same accuracy as that of ordinary FEM with triangular elements. The key point of FEM-β is the ease of expressing failure as separation of two adjacent Voronoi blocks owing to the particle discretization that uses non-overlapping characteristic functions. This paper explains these features of FEM-β with results of numerical simulation of several example problems.     

Inversion of a full-range stress-strain relation for stainless steel alloys

This paper presents an approximate inversion of the stress-strain relation for stainless steel alloys. Using currently available stress-strain relations based on a modified Ramberg-Osgood equation, a new expression for the stress σ as an explicit function of the total strain ε is obtained. The new expression is valid over the full-range of the stress well beyond the 0.2% proof stress σ_0.2, defined as the stress level corresponding to the plastic strain value of 0.2%. The validity of the inverted expression is tested over a wide range of material parameters. The tests show that the new expression results in stress-strain curves which are both qualitatively and quantitatively consistent with the fully iterated numerical solution of the full-range stress-strain relation.     

Viscoelastic computations for reverse roll coating with dynamic wetting lines and the Phan-Thien-Tanner models

The computational modelling of reverse roll coating with dynamic wetting line has been analysed for various non-Newtonian viscoelastic materials appealing to the Phan-Thien-Tanner (PTT) network class of models suitable for typical polymer solutions, with properties of shear thinning and strain hardening/softening. The numerical technique utilizes a hybrid finite element-sub-cell finite volume algorithm with a dynamic free-surface location, drawing upon a fractional-staged predictor-corrector semi-implicit time-stepping procedure of an incremental pressure-correction form. The numerical solution is investigated following a systematic study which allows for parametric variation in elasticity (We-variation), extensional hardening-softening (ε), and solvent fraction (β). Under incompressible flow conditions, linear PTT (LPTT) and exponential PTT (EPTT) models were used to solve the paint strip coatings, under reverse roll-coating configuration. This involves two-dimensional planar reverse roll-coating domains, considering a range of Weissenberg numbers (We) up to critical levels, addressing velocity fields and vortex development, pressure and lift profiles, shear rate, and stress fields. Various differences are observed when comparing solutions for these constitutive models. Concerning the effects of elasticity, increase in We stimulates vortex structures, which are visible at both the downstream meniscus and upstream narrowest nip region, whilst decreasing the peak pressure and lift values at the nip constriction. At low values (ε > 0.5, β = 0.1) of extensional viscosity, the LPTT flow fields were much easier to extract, attaining critical We levels up to unity, in contrast to critical We levels of 0.4 for EPTT solutions. This finding is reversed at higher extensional viscosity levels (ε < 0.5). This trend reveals qualitative agreement with theoretical studies. Noting flow behaviour under EPTT solution, increasing the peak level of strain hardening/softening is found to stimulate vortex activity around the nip region, with a corresponding increase in peak pressure and lift values.     

Pressure and velocity of air during drying and storage of cereal grains

A common method of drying cereal grains is to ventilate a large static mass of grain with an even flow of air at near ambient temperature. After the grain has been dried it is often stored in the same container and kept cool by aeration with a lower velocity of air than is used in drying. To analyse the airflow through this mass of grain a nonlinear momentum equation for flow through porous media is used where the resistance to flow is a + b ¦ν¦. This equation, together with the assumption that the air is incompressible, defines the problem which is solved numerically, using the finite element method, and the results compared with experimental values. The small parameter ε = bν _r /a, where ν _r is the velocity scale, is used in a perturbation analysis to examine the nonlinear effects of the resistance on the airflow. When ε = 0 the equations reduce to those for potential flow, while for small values of ε there are first-order corrections to the pressure p ₁ and the stream function χ ₁. The nonlinear problem is simplified by changing to curvilinear coordinates (s, t) where s is constant on the potential flow isobars while t is constant on the streamlines. General conclusions are derived for p ₁ and χ ₁, for example that they depend on the curvature of the potential flow solution with a large curvature of the isobars leading to larger values of p ₁ and similarly for the streamlines. The potential flow solution p ₀ and the first order solution p ₀ + εp ₁ are close to the solution of the full nonlinear problem when ε is small. To illustrate this for a typical grain storage problem, the solution p ₀ is shown to be very close to the finite element solution (with a difference of less than 1%) when ε < 0.03 while for the first order solution p ₀ + εp ₁ the difference is less than 1% when ε < 0.1.     

A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction

A new low-Reynolds-number k–ε turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k–ε turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89–108]. In extending the range of application to higher values of drag reduction, three main improvements were incorporated: a modified eddy viscosity closure, the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate, and a new closure for the cross-correlations between the fluctuating components of the polymer conformation and rate of strain tensors (NLT_ij). The NLT_ij appears in the Reynolds-averaged evolution equation for the conformation tensor (RACE), which is required to calculate the average polymer stress, and in the viscoelastic stress work in the transport equation of k. It is shown that the predictions of mean velocity, turbulent kinetic energy, its rate of dissipation by the Newtonian solvent, conformation tensor and polymer and Reynolds shear stresses are improved compared to those obtained from the earlier model.     

Local Energy Dissipation Rate in an Agitated Vessel—a Comparison of Evaluation Methods

There are two main groups of local turbulent energy dissipation rate (ε) evaluation methods, namely (i) velocity gradient methods and (ii) fitting the energy spectrum function. We calibrated our measurements and then applied these methods to evaluate ε from our measurements. The experiments were carried out in the region below the impeller, wherewe assumed the existence of local isotropic turbulence, which is the main assumption for all ε evaluation methods. However, the results differed from each other. Predictions obtained by the methods dealt with here are compared on the basis of our experimental program, which also brings in data obtained on relatively large vessels 0.3 and 0.4 m in inner diameter and two liquid viscosities under fully developed turbulence (Re > 50 000).     

Strain-hardening extrusion—II. Analysis of slip-line fields based on experimental flow patterns

A method of constructing slip-line fields based on experimental extrusion flow patterns was described in a previous paper. Such fields provide a basis for constructing representative solutions of strain-hardening flow problems. In this paper, after proving some theoretical results on the structure of such solutions, a method of analyzing slip-line fields is described and some typical results presented. The analysis comprises computation of the velocity solution and construction from it of a set of streamlines for comparison with the original flow pattern; computation of the total strain by integration along the streamlines, its graduation and adjustment to meet the boundary conditions; estimation of the stress field (shear stress k and mean pressure p) and check for consistency; and computation of the forces on the boundary of the plastic region and their resultant.     

The analysis of cyclically loaded creeping structures for short cycle times

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±iε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.     

Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids

An exact asymptotic analysis is presented of the stress and deformation fields near the tip of a quasistatically advancing plane strain tensile crack in an elastic-ideally plastic solid. In contrast to previous approximate analyses, no assumptions which reduce the yield condition, a priori, to the form of constant in-plane principal shear stress near the crack tip are made, and the analysis is valid for general Poisson ratio ν. Specific results are given for ν = 0.3 and 0.5, the latter duplicating solutions in previous work by L.I. Slepyan, Y.-C. Gao and the present authors. The crack tip field is shown to divide into five angular sectors of four different types ; in the order in which these sweep across a point in the vicinity of the advancing crack, they are : two plastic sectors which can be described asymptotically (i.e., as r → 0, where r is distance from the crack tip) in slip-line terminology as ‘constant stress’ and ‘centered fan’ sectors, respectively ; a plastic sector of non-constant stress which cannot be described asymptotically in terms of slip lines; an elastic unloading sector; and a trailing plastic sector of the same type as that directly preceding the elastic sector. Further, these four different sector types constitute the full set of asymptotically possible solutions at the crack tip. As is known from prior work, the plastic strain accumulated by a material point passing through such a moving ‘centered fan’ sector is O(ln r) as r → 0 ; it is proved in the present work that the plastic strain accumulated by a material point passing through the ‘constant stress’ sector ahead of a growing crack must be less singular than In r as r → 0. As suggested also in earlier studies, the rate of increase of opening gap δ at a point currently at a distance r behind, but very near, the crack tip is given for crack advance under contained yielding by

$\text{δ}\text{?}\text{=}\text{α}\text{J}\text{?}\text{σ}{}_0\text{+β(}\text{σ}{}_0\text{E}\text{)}\text{a}\text{?}\text{ln(}\text{R}\text{r}\text{)}$

where a is crack length, σ0 is tensile yield strength, E is Young's modulus, J is the value of the J-integral taken in surrounding elastic material, and the parameters α and R are undetermined by the asymptotic analysis. The exact solution for ν = 0.3 gives β = 5.462, which agrees very closely with estimates obtained from finite element solutions. An approximate analysis based on use of slip line representations in all plastic sectors is outlined in the Appendix.     

Complex variable formulation for non-slipping plane strain contact of two elastic solids in the presence of interface mismatch eigenstrain

In this paper, the problems of non-slipping contact, non-slipping adhesive contact, and non-slipping adhesive contact with a stretched substrate are sequentially studied under the plane strain theory. The main results are obtained as follows:(i) The explicit solutions for a kind of singular integrals frequently encountered in contact mechanics (and fracture mechanics) are derived, which enables a comprehensive analysis of non-slipping contacts. (ii) The non-slipping contact problems are formulated in terms of the Kolosov–Muskhelishvili complex potential formulae and their exact solutions are obtained in closed or explicit forms. The relative tangential displacement within a non-slipping contact is found in a compact form. (iii) The spatial derivative of this relative displacement will be referred to in this study as the interface mismatch eigenstrain. Taking into account the interface mismatch eigenstrain, a new non-slipping adhesive contact model is proposed and its solution is obtained. It is shown that the pull-off force and the half-width of the non-slipping adhesive contact are smaller than the corresponding solutions of the JKR model (Johnson et al., 1971). The maximum difference can reach 9% for pull-off force and 17% for pull-off width, respectively. In contrast, the new model may be more accurate in modeling the non-slipping adhesion. (iv) The non-slipping adhesions with a stretch strain (S-strain) imposed to one of contact counterparts are re-examined and the analytical solutions are obtained. The accurate analysis shows that under small values of the S-strain both the natural adhesive contact half-width and the pull-off force may be augmented, but for the larger S-strain values they are always reduced. It is also found that Dundurs’ parameter β may exert a considerable effect on the solution of the pull-off problem under the S-strain.These solutions may be used to study contacts at macro-, micro-, and nano-scales.     

Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations

Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.     

Large strain deformation of polycrystalline metals at low homologous temperatures

Large strain compression data (true strains to about ?3.0) are presented for polycrystalline α U and α Fe at room temperature. The results, together with other published data at low homologous temperatures (≈0.2 T_m), where T_m is the absolute melting temperature, suggest that a steady-state flow stress σ_s is approached after extensive strain-hardening, α U exhibits a very high strain-hardening rate, with σ_s ≈ 2900 MPa (420 ksi) indicating that cold-working is a very potent method of strengthening this metal. All the data evaluated can be fit by the stress-strain relation σ = σ_s? exp (?(Nε)^p)(σ_s? σ_y), where σ_y is the yield stess, p is a constant equal to a for the metals analyzed, N is a constant associated with the strain-hardening characteristics of a material, σ is true stress, and ε is true strain.     

Crack paths in three-dimensional elastic solids. ii: three-term expansion of the stress intensity factors—applications and perspectives

This work continues the calculation of the stress intensity factors, as a function of position s along the front of an arbitrary (kinked and curved) infinitesimal extension of some arbitrary crack on some three-dimensional body. More precisely, ε denoting a small parameter which the crack extension length is proportional to, what is studied here is the third term, proportional to εfn2 = ε and noted K ⁽¹⁾ (s) ε, of the expansion of these stress intensity factors at the point s of the crack front in powers of ε. The novelties with respect to previous works due to Gao and Rice on the one hand and Nazarov on the other hand, are that both the original crack and its extension need not necessarily be planar, and that a kink (discontinuity of the tangent plane to the crack) can occur all along the original crack front. Two expressions of K ⁽¹⁾ (s) are obtained; the difference is that the first one is more synthetic whereas the second one makes the influence of the kink angle (which can vary along the original crack front) more explicit. Application of some criterion then allows to obtain the apriori unknown geometric parameters of the small crack extension (length, kink angle, curvature parameters). The small scale segmentation of the crack front which is observed experimentally in the presence of mode III is disregarded here because a large scale point of view is adopted; this phenomenon will be discussed in a separate paper. It is shown how these results can be used to numerically predict crack paths over arbitrary distances in three dimensions. Simple applications to problems of configurational stability and bifurcation of the crack front are finally presented.     

The inhomogeneous strain distributions within finite cylinders under the axial point load tests and the effects on the valence band of Si1−xGex alloy

An exact solution for inhomogeneous strain and stress distributions within a finite cubic isotropic cylinder of Si_1?xGe_x alloy under the axial Point Load Strength Test (PLST) is analytically derived. Lekhnitskii’s stress function is used to uncouple the equations of equilibrium, and a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. The solution for isotropic cylinders under the axial PLST is covered as a special case. Numerical results show that the strain and stress distributions in the central region within half height and radius are relatively homogeneous, but strain and stress concentrations are usually developed near the point loads. The largest tensile strain and stress are always induced along the line joining the point loads, which gives theoretical explanation why most of the cylindrical specimens are split apart along the line joining the point loads under the axial PLST. In addition, by using envelope-function method, the effect of strain on the valence-band structure of Si_1?xGe_x alloy is analyzed. It is found that strain changes the band quantum gap and the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1?xGe_x alloy.     

Inhomogeneous elastostatic problem solutions constructed from stress-associated homogeneous solutions

In this paper, we present a theorem that provides solutions for anisotropic and inhomogeneous elastostatic problems by using the known solution of an associated anisotropic and homogeneous problem if the associated problem has a stress state with a zero eigenvalue everywhere in the domain of the problem. The fundamental property on which this stress-associated solution (SAS) theorem is built is the coaxiality of the eigenvector associated with the zero stress eigenvalue in the homogeneous problem and the gradient of the scalar function ? characterizing the inhomogeneous character of the inhomogeneous problem. It is shown that most of the solutions of anisotropic elastic problems presented in the literature have this property and, therefore, it is possible to use the SAS theorem to construct new exact solutions for inhomogeneous problems, as well as to find—using the SAS theorem—solutions for the shape intrinsic and angularly inhomogeneous problems.