New integral formulation and self-consistent modeling of elastic–viscoplastic heterogeneous materials

Predicting the overall behavior of heterogeneous materials, from their local properties at the scale of heterogeneities, represents a critical step in the design and modeling of new materials. Within this framework, an internal variables approach for scale transition problem in elastic–viscoplastic case is introduced. The proposed micromechanical model is based on establishing a new system of field equations from which two Navier’s equations are obtained. Combining these equations leads to a single integral equation which contains, on the one hand, modified Green operators associated with elastic and viscoplastic reference homogeneous media, and secondly, elastic and viscoplastic fluctuations. This new integral equation is thus adapted to self-consistent scale transition methods. By using the self-consistent approximation we obtain the concentration law and the overall elastic–viscoplastic behavior of the material. The model is first applied to the case of two-phase materials with isotropic, linear and compressible viscoelastic properties. Results for elastic–viscoplastic two-phase materials are also presented and compared with exact results and variational methods.     

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.     

Basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials

The basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials was studied in this paper using the generalized Almansi’s theorem. To make the analysis tractable, it was assumed that the shear modulus varies exponentially along the horizontal axis parallel to the crack. The problem was formulated through a Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the gradient of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

Homogenization of the viscoelastic heterogeneous materials with multi-coated reinforcements: an internal variables formulation

In this work, a new homogenization method to estimate the effective behavior of viscoelastic heterogeneous materials with multi-coated reinforcements is presented. Unlike classical methods that are based on the Laplace transform, the present internal variables formulation operates directly in the time domain. Using the Green’s function techniques, the micromechanical approach is based on establishing a new integral equation adapted to scale transition methods. Using this integral equation, we apply a generalized self-consistent scheme to determine the local stress concentration equations and the effective behavior of multi-coated inclusion-reinforced materials. To assess the reliability of our model, some applications to the isotropic viscoelastic heterogeneous materials with homothetic spherical inclusions are given. The model is applied to the case of two-phase and three-phase materials, and the results are compared to exact solutions. Results for three-phase materials are presented regarding the influence of soft and stiff viscoelastic interphase on the effective behavior of heterogeneous materials.     

A Variational Approach to the Macroscopic Electrodynamics of Anisotropic Hard Superconductors

We consider Bean’s critical state model for anisotropic superconductors. A variational problem solved by the quasi-static evolution of the internal magnetic field is obtained as the Γ-limit of functionals arising from Maxwell’s equations combined with a power law for the dissipation. Moreover, the quasi-static approximation of the internal electric field is recovered, using a first order necessary condition. If the sample is a long cylinder subjected to an axial uniform external field, the macroscopic electrodynamics is explicitly determined.     

Development of a turbulent boundary layer after a step from smooth to rough surface

The flow developing downstream of a step change from smooth to rough surface condition is studied in the light of Townsend’s wall similarity hypothesis. Previous studies seem to support the hypothesis for channel and pipe flows, but there are considerable controversies about its application to boundary layers and in particular to surface roughness formed by spanwise bars. It has been suggested that this controversy arises from insufficient separation of scales between the boundary layer thickness and the roughness length scale. An experimental investigation has therefore been undertaken where the flow evolves from a fully developed smooth wall boundary layer at high Reynolds numbers over a step in surface roughness (Re _θ = 13,400 at the step). The flow is mapped through the development of the internal layer until the flow is fully developed over the rough wall. The internal layer is found to grow as δ ∼ X ^0.73, and after about 15 boundary layer thicknesses at the step, the internal layer has reached the outer edge of the incoming layer. At the last rough wall measurement station, the Reynolds number has grown to Re _θ ≈ 32,600 and the ratio of boundary layer to roughness length scales is δ/k ≈ 140. The outer layer differences between the smooth and the rough wall data were found to be sufficiently small to conclude that for this setup the Townsend’s wall similarity hypothesis appears to hold.     

Closed-form solutions for Euler–Bernoulli arbitrary discontinuous beams

Euler–Bernoulli arbitrary discontinuous beams acted upon by static loads are addressed. Based on appropriate Green’s functions here derived in a closed form, the response variables are obtained: (a) for stepped beams with internal springs, as closed-form functions of the beam discontinuity parameters, without enforcing neither internal nor boundary conditions; (b) for stepped beams with internal springs and along-axis supports, as closed-form functions of the unknown reactions of the along-axis supports only, to be computed by enforcing pertinent conditions. A remarkable reduction in computational effort is achieved, in this manner, compared to competing methods in the literature.     

非均质材料动力分析的广义多尺度有限元法

自然界和工程中的大部分材料都具有多尺度特征,当考察尺度小到一定程度后,都将表现出非均质性.针对非均质材料的动力问题,提出了一种广义多尺度有限元方法,其基本思想是利用静态凝聚法以及罚函数法构造能够反映单元内部材料非均质特性的多尺度位移基函数.与传统扩展多尺度有限元法中的基函数构造方式不同,广义多尺度有限元法的基函数无需通过在子网格域上多次求解椭圆问题得到,而可直接通过矩阵运算获得.其主要步骤如下:利用数值基函数将一个非均质单胞等效为一个宏观单元,进而形成整个结构的等效刚度矩阵,并得到宏观网格的节点位移,最后再次利用数值基函数得到微观尺度上的位移结果.该广义多尺度有限元法是扩展多尺度有限元法的一种新的拓展,可模拟具有更加复杂几何的非均质单胞的力学行为.通过数值算例,模拟了非均质材料的静力问题、广义特征值问题以及瞬态响应问题,计算结果表明:在边界条件一样的情况下,广义多尺度有限元法的计算结果与传统有限元的计算结果保持高度一致.与传统有限元相比,该方法在保证计算精度的同时极大地提高了计算效率.研究结果表明,广义多尺度有限元法能够很好地模拟非均质单胞的力学行为,具有良好的工程应用潜力.      

An unsteady point vortex method for coupled fluid–solid problems

A method is proposed for the study of the two-dimensional coupled motion of a general sharp-edged solid body and a surrounding inviscid flow. The formation of vorticity at the body’s edges is accounted for by the shedding at each corner of point vortices whose intensity is adjusted at each time step to satisfy the regularity condition on the flow at the generating corner. The irreversible nature of vortex shedding is included in the model by requiring the vortices’ intensity to vary monotonically in time. A conservation of linear momentum argument is provided for the equation of motion of these point vortices (Brown–Michael equation). The forces and torques applied on the solid body are computed as explicit functions of the solid body velocity and the vortices’ position and intensity, thereby providing an explicit formulation of the vortex–solid coupled problem as a set of non-linear ordinary differential equations. The example of a falling card in a fluid initially at rest is then studied using this method. The stability of broadside-on fall is analysed and the shedding of vorticity from both plate edges is shown to destabilize this position, consistent with experimental studies and numerical simulations of this problem. The reduced-order representation of the fluid motion in terms of point vortices is used to understand the physical origin of this destabilization.      

Stress state of an orthotropic material with an elliptic crack under linearly varying pressure

The static equilibrium of an elastic orthotropic medium with an elliptic crack subject, on its surface, to linearly varying pressure is studied. The stress state of the elastic medium is represented as a superposition of the principal and perturbed states. Use is made of Willis’ approach based on the triple Fourier transform in spatial variables, the Fourier-transformed Green’s function for an anisotropic material, and Cauchy’s residue theorem. The contour integrals are evaluated using Gaussian quadratures. The results for particular cases are compared with those obtained by other authors. The influence of orthotropy on the stress intensity factors is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 73–81, July 2006.     

A Discontinuous Elastic Interface Transfer Model of Thin Film Nanoindentation

A new model of thin film indentation that accounted for an apparent discontinuity in elastic strain transfer at the film/substrate interface was developed. Finite element analysis suggested that numerical values of strain were not directly continuous across the interface; the values in the film were higher when a soft film was deposited on a hard substrate. The new model was constructed based on this discontinuity; whereby, separate weighting factors were applied to account for the influence of the substrate in strain developed in the film and vice-versa. By comparing the model to experimental data from thirteen different amorphous thin film materials on a silicon substrate, constants in each weighting factor were found to have physical significance in being numerically similar to the bulk scale Poisson’s ratios of the materials involved. When employing these material properties in the new model it was found to provide an improved match to the experimental data over the existing Doerner and Nix and Gao models. Finally, the model was found to be capable of assessing the Young’s modulus of thin films that do not exhibit a flat region as long as the bulk Poisson’s ratio is known.     

A thermodynamic theory of damage in elastic inorganic and organic solids

This contribution presents the foundations of a thermodynamic theory of damage in elastic solids, developed in collaboration with the late J. Kestin and with E. Honein and T. Honein. The theory is rooted in the so-called conservative or conventional thermodynamics of irreversible processes, where the concept of a local thermodynamic state plays a prominent role. An elastic body prone to damage is regarded as a thermodynamic system characterized by a set of extensive variables that can be defined in both equilibrium and nonequilibrium states and assigned approximately the same values in both the physical space and the abstract state space (i.e., the Gibbsian phase space of constrained equilibria). The extensive variables introduced include internal parameters which describe the damaged state of the body and whose conjugate intensive variables, or affinities, constitute a generalization of Eshelby’s concept of a “force on an elastic singularity”. The local state approximation is applied by assigning to the entropy and temperature in physical space local values which can be calculated in the Gibbsian phase space by the well-established methods of equilibrium thermodynamics. This leads to an explicit expression for the entropy production. The rate equations for the damage are then postulated in such a way as to conform to the second part of the second law of thermodynamics. The resulting theory captures many features of real inorganic material behavior in which no mass loss is sustained. By contrast, damage of organic materials, such as compact bone subject to osteoporosis, is accompanied by bone mass loss. This feature can be accommodated in the theory proposed by a suitable adjustment of the expression of the Gibbs free energy.     

An efficient method for nonlinear aeroelasticy of slender wings

This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of some other works and show a good agreement between results.     

Basic solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in?piezoelectric materials

The solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in piezoelectric materials were given by using the generalized Almansi’s theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack,the shape of the rectangular crack and the distance between two rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.     

Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.     

SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES

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Theory of elastoplastic processes and study of structure-mechanical properties of materials

The prospects of application of A. A. Il’yushin’s theory of elastoplastic processes when studying structure-mechanical properties of materials are discussed. It is shown that the idea to classify material strain processes by their complexity, which was proposed in the theory of elastoplastic processes, allows one to form task-oriented data banks (DB) of structure-mechanical properties of materials in a new way. These DB can also be used to estimate the physical reliability of solutions of boundary value problems in the process of obtaining products with prescribed functional properties by using some information about the structure state of the material.     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: II Computational Validation

In Part I Moyne and Murad [Transport in Porous Media 62, (2006), 333–380] a two-scale model of coupled electro-chemo-mechanical phenomena in swelling porous media was derived by a formal asymptotic homogenization analysis. The microscopic portrait of the model consists of a two-phase system composed of an electrolyte solution and colloidal clay particles. The movement of the liquid at the microscale is ruled by the modified Stokes problem; the advection, diffusion and electro-migration of monovalent ions Na⁺ and Cl⁻ are governed by the Nernst–Planck equations and the local electric potential distribution is dictated by the Poisson problem. The microscopic governing equations in the fluid domain are coupled with the elasticity problem for the clay particles through boundary conditions on the solid–fluid interface. The up-scaling procedure led to a macroscopic model based on Onsager’s reciprocity relations coupled with a modified form of Terzaghi’s effective stress principle including an additional swelling stress component. A notable consequence of the two-scale framework are the new closure problems derived for the macroscopic electro-chemo-mechanical parameters. Such local representation bridge the gap between the macroscopic Thermodynamics of Irreversible Processes and microscopic Electro-Hydrodynamics by establishing a direct correlation between the magnitude of the effective properties and the electrical double layer potential, whose local distribution is governed by a microscale Poisson–Boltzmann equation. The purpose of this paper is to validate computationally the two-scale model and to introduce new concepts inherent to the problem considering a particular form of microstructure wherein the clay fabric is composed of parallel particles of face-to-face contact. By discretizing the local Poisson–Boltzmann equation and solving numerically the closure problems, the constitutive behavior of the diffusion coefficients of cations and anions, chemico-osmotic and electro-osmotic conductivities in Darcy’s law, Onsager’s parameters, swelling pressure, electro-chemical compressibility, surface tension, primary/secondary electroviscous effects and the reflection coefficient are computed for a range particle distances and sat concentrations.     

Feedback stabilization of a class of nonlinear second-order systems

The goal of this paper is to study stabilization techniques for a system described by nonlinear second-order differential equations. The problem is to determine the feedback control as a function of the state variables. It is shown that the following controllers can asymptotically stabilize the system: linear position feedback, linear velocity feedback and a group of nonlinear feedbacks. The stability of the corresponding closed-loop system is proved by imposing a suitable Lyapunov function and then using LaSalle’s invariance principle. The results of numerical computations are included to verify theoretical analysis and mathematical formulation. Some application examples from robotics, mechanics and electronics are presented.