Nonlocal Constitutive Equations Research Articles

On nonlocal residuals and fluctuations

The nonlocal residual is a novel physical quantity introduced in the nonlocal field theory of mechanics. In this paper, the nonlocal residual and some related problems are discussed. Firstly, a representative theorem of nonlocal residual is proved, in which the relation between the nonlocal residual and the spatial distributed fluctuation of the interaction among microstructures in materials is established. The existence of nonlocal residuals of body force, body moment and energy is investigated in detail based on the objectivity of the balance equations. To meet the requirements in physics, an eigen-scale parameter is introduced into the nonlocal kernel. And the properties of nonlocal kernel are then discussed. Finally, the nonlocal hyperelastic constitutive equation is deduced through the representation of the nonlocal residual of energy. Results show that the nonlocality of hyperelastic constitutive equation comes directly from the interaction potential among microstructures within materials.

Non-local constitutive equations for functionally graded materials

In this paper, composites with a graded distribution of heterogeneities are considered. The heterogeneities vary in statistically non-uniform fashion since in a finite layer (or region) properties such as local volume fraction vary gradually. In order to study this class of composites, a procedure of analysis which leads to the effective constitutive non-local operator of the medium is proposed. For two-phase composites, an approximation of Hashin–Shtrikman type for this operator has been obtained in real space and this has been developed explicitly in the case of laminates.

A micromechanics-based nonlocal constitutive equation for elastic composites containing randomly oriented spheroidal heterogeneities

A micromechanics-based nonlocal constitutive equation relating the ensemble averages of stress and strain for a matrix containing a random distribution of randomly oriented spheroidal voids or inclusions is derived. The analysis employs J.R. Willis’ generalization of the Hashin–Shtrikman variational formulation to random linear elastic composite materials and builds on that of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359), who derived completely explicit results for the case of isotropic, nonoverlapping identical spherical inclusions/voids. The model of impenetrable particles employed consists of identical particles with fixed spheroidal shape and random orientation. To facilitate a manageable statistical description, the spheroids are placed within concentric hard “security” spheres. The paper derives three main new results: (i) it is proved within the assumptions just outlined that the effects of inclusion shape and their spatial distribution are separable, for arbitrary inclusion shape (not just spheroids) and arbitrary spatial (statistical) distribution of their security spheres when employing up through two-point statistical information; (ii) closed-form analytical results are obtained from the Verlet–Weis improvement of the Percus–Yevick–Wertheim statistical model of a random distribution of nonoverlapping spherical particles/voids, leading to substantial improvements at higher inclusion/void volume fractions in the nonlocal constitutive equations of Drugan and Willis (1996) and Drugan (2000); (iii) approximate analytical nonlocal constitutive equations are derived for composites consisting of a matrix containing randomly oriented oblate or prolate spheroidal inclusions/voids, using the Verlet–Weis statistical model for the security sphere distribution. Among the specific implications of these new results, it is found that the minimum representative volume element (RVE) size estimate for composites containing spherical inclusions/voids using the Verlet–Weis improvement is significantly larger at higher inclusion/void volume fractions (≈0.3–0.64) than the estimates of Drugan and Willis (1996) and Drugan (2000), who used the Percus–Yevick–Wertheim model. Also, deviations in inclusion/void shape from spherical are shown to cause significant modifications to the nonlocal constitutive equations, as evidenced by nontrivial changes in predicted minimum RVE sizes.

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin–Shtrikman–Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.

Extended gradient plasticity by the subloading surface model with tangential plasticity

The most useful non-local constitutive equation for the analysis of localized plastic deformation would be the gradient plastic constitutive equation proposed by Aifantis. However, it involves the gradient of internal variable and thus the treatment of it at the boundary is difficult. Then, it has been modified mathematically by Vardoulakis transforming it to the relationship between the stress rate and the strain rate with its gradient, and thus the analysis of the boundary value problem has been simplified drastically. In this article the generalized formulation of the gradient plasticity is first given by incorporating the general gradient terms of internal variables. Then, it is extended to the unconventional plasticity by incorporating the subloading surface model. Moreover, it is extended so as to be applicable to the description of plastic instability phenomena by incorporating the tangential plastic strain rate. Besides, the equation for the analysis of shear band thickness in the post-localized deformation is given based on the present constitutive model.

A Nonlocal Model of Materials with Periodic Microstructure Based on Asymptotic Homogenization Method

The asymptotic homogenization method within the framework of the updated Lagrangian formulation is employed to derive a nonlocal constitutive equation for finitely deformed rate-independent materials with a periodic microstructure. Higher-order asymptotic terms naturally introduce strain gradient terms into constitutive equations for macroscopically homogeneous materials. Macroscopic properties, which are the ensemble average of their counterparts over a microscopic unit cell, are discussed. The variational principle of macroscopically homogeneous materials is then established and the complete boundary value problem is formulated.

Micromechanics-based variational estimates for a higher-order nonlocal constitutive equation and optimal choice of effective moduli for elastic composites

A generalization of the Hashin–Shtrikman variational formulation to random composites, due to J.R. Willis, is employed to derive micromechanics-based variational estimates for a higher-order nonlocal constitutive equation relating the ensemble averages of stress and strain, for a class of random linear elastic composite materials. We analyze two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), within a formulation accounting for one- and two-point probabilities, and derive an explicit nonlocal constitutive equation that includes terms up through the fourth gradient of average strain. The analysis is carried out first for an arbitrary comparison medium. Then, a new approach is outlined and applied which employs the nonlocal correction to determine the optimal choice of comparison medium, and hence the optimal effective modulus tensor (as well as the optimal tensor coefficients of the nonlocal terms) for the amount of statistical information employed. The new higher order analysis provides a highly accurate nonlocal constitutive equation, valid down to quite small volume size scales and to rather strong variations of average strain with position. Among several applications illustrated, it permits accurate analytical assessment of the remarkably small predictions derived by Drugan and Willis (1996. Journal of the Mechanics and Physics of Solids 44, 497–524) of the minimum representative volume element (RVE) size needed for accuracy of the standard constant-effective-modulus macroscopic constitutive equation for elastic matrix-inclusion composites that have spherical inclusions/voids. It also affords an analytical assessment of the improved (i.e., reduced) minimum RVE size scale, compared to a standard constant-effective-modulus constitutive equation, to which the leading-order nonlocal constitutive equation derived by Drugan and Willis applies. This improvement is shown to be dramatic in some example cases.

Effective thermoelastic properties of graded doublyperiodic particulate matrix composites in varying externalstress fields

We consider a linear elastic composite medium, which consists of a homogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in a doubly periodicarray and subjected to inhomogeneous boundary conditions. The hypothesis of effective fieldhomogeneity near the inclusions is used. The general integral equation obtained reduces theanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusions insome representative volume element (RVE) . The integral equation is solved by a modifiedversion of the Neumann series; the fast convergence of this method is demonstrated for concreteexamples. The nonlocal macroscopic constitutive equation relating the cell averages of stress andstrain is derived in explicit iterative form of an integral equation. A doubly periodic inclusion fieldin a finite ply subjected to a stress gradient along the functionally graded direction is considered.The stresses averaged over the cell are explicitly represented as functions of the boundaryconditions. Finally, the employed of proposed explicit relations for numerical simulations oftensors describing the local and nonlocal effective elastic properties of finite inclusion pliescontaining a simple cubic lattice of rigid inclusions and voids are considered. The local andnonlocal parts of average strains are estimated for inclusion plies of different thickness. Theboundary layers and scale effects for effective local and nonlocal effective properties as well as foraverage stresses will be revealed.

Triply periodical particulate matrix compositesinvarying external stress fields

We consider a linear elastic composite medium, which consists of ahomogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in aperiodic arrayand subjected to inhomogeneous boundary conditions. The hypothesis of effectivefieldhomogeneity near the inclusions is used. The general integral equation obtained reducestheanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusionsinsome representative volume element (RVE) . The integral equation is solved by theFouriertransform method as well as by the iteration method of the Neumann series ( first-orderapproximation) . The nonlocal macroscopic constitutive equation relating the unit cellaverages ofstress and strain is derived in explicit closed forms either of a differential equation ofasecond-order or of an integral equation. The employed of explicit relations fornumericalestimations of tensors describing the local and nonlocal effective elastic properties aswell asaverage stresses in the composites containing simple cubic lattices of rigid inclusions andvoids areconsidered.

Mechanics of the squeeze flow of planar fibre suspensions

This paper discusses the axisymmetric squeeze flow of concentrated transversely isotropic fibre suspensions in a power-law matrix and relates to the processing of composite materials such as sheet moulding compounds (SMCs) and glass mat thermoplastics (GMTs). A solution to the squeeze flow problem for a transversely isotropic power-law fluid is presented first, followed by a more detailed micromechanical analysis. In the first part of the paper a variational approach is applied to the interpretation of squeeze flow behaviour. This gives a simple expression for the total pressure, which enables the contributions due to extension and shear to be separated. Applying the procedure to GMT data suggests that the dissipation is predominantly extensional, except at very low plate separations. In the second part, a non-local constitutive equation is derived based on a simple drag law for hydrodynamic interactions. This is then used to model the pressure distribution when the effective length of the fibres is comparable to or determined by the dimensions of the squeeze flow plates. The model is shown to describe the observed squeeze flow stresses in both long and short fibre systems and to relate behaviour to the underlying resin flow properties.

A new rate-type gradient-dependent viscoplastic approach for stop-and-go strain band propagation. Numercial vs. physical experiments

It is shown that the non-local rate-type constitutive equation proposed by Faciu & Molinari [1] can predict a large number of experimentally observed features of the kinetics of the Portevin-Le Châtelier (PLC) bands. Numerically obtained serrated yielding curves and staircase creep curves for strain and stress controlled problems, respectively, are compared with experimental curves. A good agreement, between the phenomenological approach and microscopic aspects of the PLC effect is obtained.

Nonlocal models for damage and fracture: Comparison of approaches

The paper analyzes nonlocal constitutive models used in simulations of damage and fracture processes of quasibrittle materials. A number of nonlocal formulations found in the literature are classified according to the type of variable subjected to nonlocal averaging. Analytical and numerical solutions of a simple one-dimensional localization problem are presented. It is shown that some of the formulations inevitably lead to residual stresses even at very late stages of the deformation process and, consequently, they are not capable of modeling complete separation in a widely open macroscopic crack. The mechanisms leading to this specific type of stress locking are explained based on a theoretical analysis of the nonlocal constitutive equations. It is also pointed out that the nonlocal approach distorts the shape of the stress-strain diagram, which has to be taken into account when designing an appropriate local softening law.

Nonlocal stress field of interface dislocations

The basic theory of nonlocal elasticity is stated with emphasis on the difference between the nonlocal theory and classical continuum mechanics. The concept of Nonlocal Interface Residual (NIR) is introduced in nonlocal theory. With the concept of NIR and the nonlocal constitutive equation, we calculate nonlocal stresses due to an edge dislocation on the interface of bi-materials. The nonlocal stress distribution along an interface is quite different from the classical one. Instead of the singularity in the dislocation core, nonlocal stress gives a finite value in the core. A maximum of the stress is also found near the dislocation core.

Local-field effects in nonlinear dielectrics

In a system of identical nonlinear dielectric inclusions immersed in a uniform linear dielectric medium the mean polarization on a macroscopic length scale is related to the Maxwell field by a nonlocal and nonlinear constitutive equation. The method of statistical averaging and cluster expansion is used to derive a formally exact expression for the constitutive equation. The resulting cluster integrals are shown to be absolutely convergent, i.e., independent of the shape of the macroscopic sample in the thermodynamic limit. For a suspension of spherical inclusions a selection of terms leads to a nonlinear Clausius-Mossotti relation. Expressions are derived for the correlation corrections to this mean-field result.

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

On the surface waves in an elastic micropolar and microstretch medium with nonlocal cohesion

A brief review of the main points of Eringen's theory of micromorphic bodies is first given, and balance equations for the linear isotropic micropolar and microstretch body are established. By appeal to the Fourier exponential transformation, nonlocal constitutive equations are derived, and assumptions with regard to the nonlocal moduli are made. The general field equations governing the propagation of a nonlocal surface wave are particularized so as to coincide with the results obtained directly in references [12], [17], [22], and [23], respectively. As an illustrative example, propagation of a microrotation and microstretch wave in a nonlocal medium in the entire Brillouin zone is examined.

Shear Bands in Biaxial Tests on Dry Coarse Sand

A series of plane-strain biaxial compression tests on dry coarse sand is described in which the state of shear-band formation, shear-band orientation, and shear-band growth are investigated. The results show that both the shear strain at shear banding and the shear-band orientation depend on the confining pressure. The experimental results are compared with the theoretical predictions of equilibrium bifurcation theory for a number of local and non-local incremental constitutive equations. Significant deviation between the experimental results and theoretical predictions for local constitutive equations involving the Mohr-Coulomb yield condition is observed at higher confining pressures. Much better agreement is obtained if a yield function possessing a corner is introduced. The analysis of the shear-band growth beyond the onset of localization reveals that simple purely kinematic models cannot describe adequately the observed evolution in the shear-band thickness.

Nonlocal versus local elastoplastic behaviour of heterogeneous materials

Abstract The scale transition methods have been developed for many years in order to obtain the overall behavior of polycrystalline materials from their microscopic behavior and their microstructure. Nevertheless, some basic aspects are absent from such formalisms. The most significant one seems to be the heterogeneization by plastic straining which involves nonlocality of hardening. In this article, a nonlocal theory based upon crystalline plasticity is developed from which a nonlocal constitutive equation at the grain level is derived. With regard to the polycrystal, in order to deduce the behavior of a local equivalent homogeneous medium, an integral equation is proposed and solved for nonlocal inhomogeneous materials by the self-consistent approximation. This scheme is developed in case of a two-phase nonlocal material representing the dislocation cell structure induced during plastic straining. Numerical simulations based on a simplified model show significant effects on the intragranular heterogeneization.

On a three-dimensional Kelvin problem for an elastic nonlocal medium

Making use of the Eringen-Kroener form of the nonlocal constitutive equations and the exponential Fourier transformation, a system of two coupled differential equations of the second order describing the equilibrium of the body is derived. By appeal to the Helmholtz representation, the system is reduced to a single differential equation of the fourth order for the Love function, reminding a Bessel type transform of the biharmonic equation. A solution of this equation is found, and inverse transforms of the stress components using the convolution theorem established. A recourse to the formula of de la Vallee Poussin shows that, in contrast to the classical result, the stress singularity at the point of application of a concentrated force fails to appear, though the stress concentration at that point is extremely high.

One-Dimensional Softening With Localization

The relationship between material softening and structural softening is investigated through the use of a model problem in one dimension. If the size of the softening zone is large the structural softening response is stable under displacement-prescribed loading. For a small size, the softening response is unstable and the loading regime is sensitive to imperfections in stiffness. A nonlocal constitutive equation in which the limit stress is a function of strain and strain gradient is introduced to provide an approach for simulating softening with localization. Implications for the numerical modeling of softening phenomena are given.