On optimal reconstruction of constitutive relations

In this investigation we develop and validate a computational method for reconstructing constitutive relations based on measurement data, applicable to problems arising in nonequilibrium thermodynamics and continuum mechanics. This parameter estimation problem is solved as PDE-constrained optimization using a gradient-based technique in the optimize-then-discretize framework. The principal challenge is that the control variable (i.e., the relation characterizing the constitutive property) is not a function of the independent variables in the problem, but of the state (dependent) variable. The proposed method allows one to reconstruct a smooth constitutive relation defined over a broad range of the dependent variable. It relies on three main ingredients: a computationally friendly expression for the cost functional gradient, Sobolev gradients used in lieu of discontinuous L₂ gradients, and a systematic technique for shifting the identifiability region. The performance of this approach is illustrated by the reconstruction of the temperature dependence of the thermal conductivity in a one-dimensional model problem.     

The generalized quasi-variational principles of non-conservative systems with two kinds of variables

Many investigators have paid much attention to the generalized variational principles in mechanics of deformable bodies. Ref. [1] laid a foundation for the generalized varia-tional principles by studying the principle of complementary energy, from both theoreti-cal and practical points of view. Ref. [2] established the generalized variational princi-ples of elasticity and plasticity and provided the theoretical foundation for the mixed finite element method, which was widely used in the field.…     

Constraints,dissipation functions and cholesteric liquid crystals

Our recently introduced dissipation function theory of molecular liquids and solids is extended to include constrained motions. Lagrangian multipliers are used in the reversible parts of the field equations of motion. Conservation laws of mass, momentum, energy and angular momentum are derived (in both the material and spatial frames) and shown to hold under the same general conditions as in the unconstrained case. Our theory is then applied to liquid crystals. Within our formalism, constitutive relations characterizing linear and nonlinear liquid crystals are given. In the linear regime, for cholesterics, depending on whether the heat flux or the temperature gradient is used in the dissipation function we show that different reciprocal relations follow. Our results for incompressible nematics and cholesterics are compared with the Ericksen-Leslie theory and others.     

Estimation of mechanical properties of nanomaterials using artificial intelligence methods

Computational modeling tools such as molecular dynamics (MD), ab initio, finite element modeling or continuum mechanics models have been extensively applied to study the properties of carbon nanotubes (CNTs) based on given input variables such as temperature, geometry and defects. Artificial intelligence techniques can be used to further complement the application of numerical methods in characterizing the properties of CNTs. In this paper, we have introduced the application of multi-gene genetic programming (MGGP) and support vector regression to formulate the mathematical relationship between the compressive strength of CNTs and input variables such as temperature and diameter. The predictions of compressive strength of CNTs made by these models are compared to those generated using MD simulations. The results indicate that MGGP method can be deployed as a powerful method for predicting the compressive strength of the carbon nanotubes.     

Viscoelastic micropolar continuum and viscoelastic waves

Generalized Kelvin model is applied to isotropic viscoelastic micropolar continuum. Constitutive equations in integral and differential form, generalized Lamé equations, and wave equations for displacement and microrotation are derived for this model. For damped viscoelastic waves, which are realized in the continuum under examination, the wave vectors and vectors of decay are explicitly given as functions of elastic moduli, viscosity coefficients, angular frequency, density, and microinertia coefficient. Analogous relations are derived for further eleven simpler mass models. Results are compared for three models in current use. For two of them (classical elastic and micropolar elastic medium) are the derived results in agreement with the ones usually used, for the third model (viscoelastic continuum) are the usual formulas a limiting case of the relations derived in this paper.     

The equilibrium limit of a constitutive model for two-phase granular mixtures and its numerical approximation

In this paper we analyze the equilibrium limit of the constitutive model for two-phase granular mixtures introduced in Papalexandris (2004) [13], and develop an algorithm for its numerical approximation. At, equilibrium, the constitutive model reduces to a strongly coupled, overdetermined system of quasilinear elliptic partial differential equations with respect to the pressure and the volume fraction of the solid granular phase. First we carry a perturbation analysis based on standard hydrostatic-type scaling arguments which reduces the complexity of the coupling of the equations. The perturbed system is then supplemented by an appropriate compatibility condition which arises from the properties of the gradient operator. Further, based on the Helmholtz decomposition and Ladyzhenskaya’s decomposition theorem, we develop a projection-type, Successive-Over-Relaxation numerical method. This method is general enough and can be applied to a variety of continuum models of complex mixtures and mixtures with micro-structure. We also prove that this method is both stable and consistent hence, under standard assumptions, convergent. The paper concludes with the presentation of representative numerical results.     

Reciprocal relations of transport coefficients in simple materials

The cross effects of viscosity and heat conduction in anisotropic simple materials (solids or liquids) are given in the linear regime, using our dissipation function theory introduced recently. Depending on whether the temperature gradient or the heat flux is used in the dissipation function, we show that two different but unambiguous reciprocal relations between the transport coefficients follow. These are compared and contrasted with the confusing predictions from the Onsager theory, and to the results of rational thermodynamics. The uncertain experimental situation in regard to these reciprocal relations is discussed. Experimental tests are strongly urged.     

A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions

This article presents analytical explicit frequency expressions for investigating the vibrations of single-layer graphene sheets (SLGSs). The interatomic potential is incorporated into a nonlocal continuum plate model through establishing a linkage between the strain energy density induced in the continuum and nonlocal plate constitutive relations. The model which is independent of scattered value of Young's modulus is then applied and explicit frequency formulas for the SLGSs with different edge conditions are derived using static deflection function of the nanoplate under uniformly distributed load. The reliability of the present formulation is verified by the results obtained by the molecular dynamics (MD) simulations and other research workers. The formulas are of a simple short form enabling quick and accurate evaluation of the frequency of the SLGSs and also simple calibration of scale coefficient by the use of MD simulations results.     

Fatigue Life Prediction of a Rubber Mount Based on the Continuum Damage Mechanics

Abstract

Failure analysis and fatigue life prediction are important steps in the design procedure of industrial products to assure the safety and reliability of their components. A new methodology to predict the fatigue life of a rubber mount based on the continuum damage mechanics is proposed in this study. The hyperelastic constitutive model of the natural rubber material in the mount was fitted using the three parameter Mooney-Rivlin model. A damage variable was introduced and the evolution function of cumulative damage in the rubber material was derived. The parameters in the damage function were acquired based on uniaxial tensile tests and fatigue life tests of the natural rubber specimens. Then the finite element analysis (FEA) models of the rubber mount for loads in the X and Y directions were established and the strain contours and the maximum principal strains of the rubber mount under various loads were calculated. The maximum principal strain was used as the fatigue parameter, which was substituted into the natural rubber’s fatigue life damage function to predict the fatigue life of the rubber mount. Finally, the fatigue lives of the rubber mount under various loads were measured on a fatigue test rig to validate the accuracy of the fatigue life prediction method. The test results indicated that the fatigue lives predicted agreed fairly well with the test results and the fatigue prediction method should be applicable to both rubber and other types of components.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

On correct nonlocal generalized theories of elasticity

The paper discusses nonlocal elasticity theories among which are models of media with defect fields, gradient elasticity theories, and hybrid nonlocal elasticity theories. Gradient theories are analyzed, and their correctness properties are examined. Applied theories that satisfy the correctness conditions are developed, and known applied gradient theories are verified for the correctness properties. A new nonlocal generalized theory has been developed for which the operator of balance equations is represented as the product of the equilibrium operator of classical elasticity theory and the Helmholtz operator. It is shown that this theory is one-parameter and is the only representative of hybrid models constructed by a complete system of equations for forces and moments. Unlike classical elasticity that is free from scale parameters characterizing the internal material structure, nonlocal elasticity theories naturally incorporate these parameters. That is why they are suitable for the modeling of scale effects and find application in the solution of numerous applied problems for heterogeneous structures with developed phase interfaces where the degree of influence of scale effects depends on the density of phase boundaries. Nonlocal continuum models are especially attractive for modeling the properties of various micro/nanostructures, elastic properties of composites and structured materials with submicron- and nanosized internal structures in which effective properties are to a great extent defined by the scale effects (short-range interaction effects of cohesion and adhesion). Generalized elasticity theories even for isotropic materials contain many additional physical constants that are difficult or impossible to determine experimentally. Applied models with a small number of additional physical parameters are therefore of great interest. However, the reduction of nonlocal theories aimed at reducing the number of additional parameters is a nontrivial task and may lead to incorrect theories. The goal of this paper is to study the symmetry properties in gradient theories, to analyze the correctness of gradient theories, and to develop applied one-parameter elasticity theories.     

Motion decomposition,frame-indifferent derivatives,and constitutive relations at large displacement gradients from the viewpoint of multilevel modeling

Widespread approaches to generalizing geometrically linear constitutive relations to the case of large displacement gradients have been considered. These approaches are based on the replacement of the material derivatives of stress and strain tensors by frame-indifferent corotational or convective derivatives. The correctness of choosing the indifferent derivatives is analyzed from a more general viewpoint of motion decomposition into rigid and strain-induced motion. It is shown that the use of the Zaremba-Jaumann derivative in constitutive relations corresponds to motion decomposition by the Cauchy-Helmholtz theorem according to which instantaneous rigid rotation of a material particle with small neighborhood is described by the vorticity tensor. The relations derived with the use of the so-called "logarithmic spin" are analyzed. It is noted that the spin tensors entering into these relations are not associated with the material fibers (in particular with the symmetry axes of anisotropic materials) during the entire studied process of deformation. Hence these spins do not describe the rotation of the reference frame (crystallographic one for metals) in which the material property tensor is defined. A new method of motion decomposition is proposed on the basis of a two-level (macro and meso) approach for single and polycrystalline metals. The mesoscopic spin is determined by the rotation rate of the corotational coordinate system associated with the crystallographic direction and crystallographic plane. Mesoscopic constitutive relations are formulated using the proposed spin. The spin of a representative macrovolume is determined by averaging the spins of the crystallites contained in this volume. This spin is used to formulate rate-type elastic constitutive equations. Examples are given to illustrate the stress state determination for loading along closed strain paths and two-segment paths for isotropic and anisotropic (with cubic symmetry, hcp) elastic materials, and an elastoviscoplastic fcc crystallite. The determination is carried out by using the corotational derivatives in the constitutive relations which are obtained by different motion decomposition methods.     

A non-singular continuum theory of point defects using gradient elasticity of bi-Helmholtz type

In this paper, we develop a non-singular continuum theory of point defects based on a second strain gradient elasticity theory, the so-called gradient elasticity of bi-Helmholtz type. Such a generalised continuum theory possesses a weak nonlocal character with two internal material lengths and provides a mechanics of defects without singularities. Gradient elasticity of bi-Helmholtz type gives a natural and physical regularisation of the classical singularities of defects, based on higher order partial differential equations. Point defects embedded in an isotropic solid are considered as eigenstrain problem in gradient elasticity of bi-Helmholtz type. Singularity-free fields of point defects are presented. The displacement field as well as the first, the second and the third gradients of the displacement are derived and it is shown that the classical singularities are regularised in this framework. This model delivers non-singular expressions for the displacement field, the first displacement gradient and the second displacement gradient. Moreover, the plastic distortion (eigendistortion) and the gradient of the plastic distortion of a dilatation centre are also non-singular and are given in terms of a form factor (shape function) of a point defect. Singularity-free expressions for the interaction energy and the interaction force between two dilatation centres and for the interaction energy and the interaction force of a dilatation centre in the stress field of an edge dislocation are given. The results are applied to calculate the finite self-energy of a dilatation centre.     

A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks

We present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling – the heterogeneous multiscale method (HMM). We derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples.     

On homogenization and scaling limit of some gradient perturbations of a massless free field

We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models. This article was processed by the author using the LAT_EX style filepljour1 from Springer-Verlag.