Thermomechanics of solids: Nonequilibrium and irreversibility

Reconciliation of thermal and mechanical interdependence is made possible by casting away the concept of heat. Defined are the dissipation and available energy density as two mutually exclusive quantities; thermal and mechanical changes become two aspects of the same process that coexist and are in continual operation. This is accomplished by interlacing the rotation, deformation and change in element size into one single operation; irreversibility is embedded inherently into the theory. No a priori assumption needs to be made on the constitutive relations which are, in fact, derived for each individual element and time increment. Only the initial slope of the reference material and load history need to be specified. Instead, the surface and volume energy density are assumed to be exchangeable without letting the change of volume with surface area to vanish in the limit, a simplification of classical physics and continuum mechanics that results in the decoupling of thermal and mechanical effects. Complete nonlinearity and finiteness of deformation are retained such that boundary problems can be solved directly by specifying the tractions and/or displacements. Nonequilibrium/irreversible solutions are shown to possess definite limits and to be bounded by the equilibrium/irreversible states whose solutions are proved to be unique. The existence of the isoenergy density function provides an elegant means of resolving the multidimensionality of the problem; the translation of unidimensional data to multidimensional states.     

Differential geometrical method in elastic composite with imperfect interfaces

I.IntroductionWhethertheinterfacesofcompositematerialsareperfectornotwillaffectitsmacromechanicaloreffectivepropertiesimportantly.Butsofar,almostallofthestudiesontheeffectivepropertiesofcompositematerialsarebasedontheassumptionthattheinterfacesareperfectl"2].Infact,thisisnotappropriateforallinterfaces[31.Thusthestudiesonmechanicalpropertyofcompositematerialswithimperfaceintert'acehavebeenconsideredrecentlyinsomeliteratures.Hashin16]hasextendedtheelasticextremumprinciplesofminimumpotentialandm…     

Magnetoelectric effects in multiferroic laminated plates with imperfect interfaces

Two-dimensional(2D)equations for multiferroic(MF)laminated plates with imperfect interfaces are established in this paper.The interface between two adjacent sublayers,which are not perfectly bonded together,is modeled as a general spring-type layer.The mechanical displacements,and the electric and magnetic potentials of the two adjacent layers are assumed to be discontinuous at the interface.As an example,the influences of imperfect interfaces on the magnetoelectric(ME)coupling effects in an MF sandwich plate are investigated with the established 2D governing equations.Numerical results show that the imperfect interfaces have a significant impact on the ME coupling effects in MF laminated structures.     

Stiffness and strength of hierarchical polycrystalline materials with imperfect interfaces

In this study we investigate the effect of imperfect (not perfectly bonded) interfaces on the stiffness and strength of hierarchical polycrystalline materials. As a case study we consider a honeycomb cellular polycrystal used for drilling and cutting tools. The conclusions of the analysis are, however, general and applicable to any material with structural hierarchy. Regarding the stiffness, generalized expressions for the Voigt and Reuss estimates of the bounds to the effective elastic modulus of heterogeneous materials are derived. The generalizations regard two aspects that are not included in the standard Reuss and Voigt estimates. The first novelty consists in considering finite thickness interfaces between the constituents undergoing damage up to final debonding. The second generalization consists of interfaces not perpendicular or parallel to the loading direction, i.e., when isostress or isostrain conditions are not satisfied. In this case, approximate expressions for the effective elastic modulus are obtained by performing a computational homogenization approach. In the second part of the paper, the homogenized response of a representative volume element (RVE) of the honeycomb cellular polycrystalline material with one or two levels of hierarchy is numerically investigated. This is put forward by using the cohesive zone model (CZM) for finite thickness interfaces recently proposed by the authors and implemented in the finite element program FEAP. From tensile tests we find that the interface nonlinearity significantly contributes to the deformability of the material. Increasing the number of hierarchical levels, the deformability increases. The RVE is tested in two different directions and, due to different orientations of the interfaces and Mixed Mode deformation, anisotropy in stiffness and strength is observed. Stiffness anisotropy is amplified by increasing the number of hierarchical levels. Finally, the interaction between interfaces at different hierarchical levels is numerically characterized. A condition for scale separation, which corresponds to the independence of the material tensile strength from the properties of the interfaces in the second level, is established. When this condition is fulfilled, the material microstructure at the second level can be efficiently replaced by an effective homogeneous continuum with a homogenized stress–strain response. From the engineering point of view, the proposed criterion of scale separation suggests how to design the optimal microstructure of a hierarchical level to maximize the material tensile strength. An interpretation of this phenomenon according to the concept of flaw tolerance is finally presented.     

Two circular inclusions with inhomogeneously imperfect interfaces in plane elasticity

This paper is concerned with the problem of two circular inclusions with circumferentially inhomogeneously imperfect interfaces embedded in an infinite matrix in plane elastostatics. Infinite series form solutions to this problem are derived by applying complex variable techniques. The numerical results demonstrate that the interface imperfection, interface inhomogeneity, and interaction among neighboring inclusions (fibers) will exert a significant influence on the stresses along the interfaces and average stresses within the inclusions.     

On Saint Venant - Kirchhoff imperfect interfaces

Using matched asymptotic expansions with fractional exponents, we obtain original transmission conditions describing the limit behavior for soft, hard and rigid thin interphases obeying the Saint Venant-Kirchhoff material model. The novel transmission conditions, generalizing the classical linear imperfect interface model, are discussed and compared with existing models proposed in the literature for thin films undergoing finite strain. As an example of implementation of the proposed interface laws, the uniaxial tension and compression responses of butt joints with soft and hard interphases are given in closed form.     

Fracture analysis of magnetoelectroelastic bimaterials with imperfect interfaces by symplectic expansion

A Hamiltonian-based analytical method is used to study the mode Ⅲ interface cracks in magnetoelectroelastic bimaterials with an imperfect interface. By introducing an unknown vector, the governing equations are reformulated in sets of first-order ordinary differential equations. Using separation of variables, eigensolutions in the symplectic space are obtained. An exact solution of the unknown vector is obtained and expressed in terms of symplectic eigensolutions. Singularities of mechanical, electric, and magnetic fields are evaluated with the generalized intensity factors. Comparisons are made to verify accuracy and stability of the proposed method. Numerical examples including mixed boundary conditions are given.     

New phenomena concerning a screw dislocation interacting with two imperfect interfaces

Dislocation mobility and stability in nanocrystals and electronic materials are influenced by the material composition and interface conditions. Its mobility and stability then affect the mechanical behaviors of the composites. In this paper, we first address, in detail, the problem of a screw dislocation located in an annular coating layer which is imperfectly bonded to the inner circular inhomogeneity and to the outer unbounded matrix. Both the inhomogeneity-coating interface and coating-matrix interface are modeled by a linear spring with vanishing thickness to account for the possible damage occurring on the interface. An analytic solution in series form is derived by means of complex variable method, with all the unknown constants being determined explicitly. The solution is then applied to the study of the dislocation mobility and stability due to its interaction with the two imperfect interfaces. The most interesting finding is that when the middle coating layer is more compliant than both the inner inhomogeneity and the outer unbounded matrix and when the interface rigidity parameters for the two imperfect interfaces are greater than certain values, one stable and two unstable equilibrium positions can exist for the dislocation. Furthermore, under certain conditions an equilibrium position, which can be either stable or unstable (i.e., a saddle point), can exist, which has never been observed in previous studies. Results for a screw dislocation interacting with two parallel straight imperfect interfaces are also presented as the limiting case where the radius of the inner inhomogeneity approaches infinity while the thickness of the coating layer is fixed.     

On material interfaces in thermomechanical solids

This contribution derives thermodynamically consistent balance equations for material interfaces in thermomechanical solids. Thereby, both displacement and temperature jumps are admissible. To this end, the interface is equipped with its own thermodynamic life, i.e. we assume separate interface free energy, entropy and the like. The thermodynamical admissibility then follows from proper definitions of the interface displacement and the interface temperature, whereby in particular, the latter follows in an unexpected format. The formulation is exemplified for the example of thermoelasticity.     

A continuum description of partially coherent interfaces

A continuum model of a two-phase crystal-crystal system is constructed in which the structure of the interface between the phases is determined by energy minimization, rather than by being specified a priori. The interfacial structure is parameterized by a variable? corresponding to the jump in the surface deformation gradient (or strain) at the interface, so that coherence is defined locally by the condition? = 0. The energy of the system is taken to be the sum of the bulk and interfacial energies, where the interfacial energy densityf ^xs depends on?. In order to explore how the equilibrium interfacial structure depends on the functionf ^xs (?), a model system consisting of an elastic film on a rigid substrate is studied, and the interfacial energy density is taken to be nonconvex with a sharp minimum associated with coherence. In this case, it can be shown that the energy of the system is driven to its infimum by separating the interface into coherent and incoherent regions, which may be viewed as a continuum analog to a partially coherent interface. Further, this solution only appears above a certain critical thickness of the film, in agreement with misfit dislocation models of partially coherent interfaces.     

Influences of imperfect interfaces on effective properties of multiferroic composites with coated inclusion

In order to predict the effective properties of multiferroic composite materials, the effective material constants of multiferroic composites with the coated inclusion and imperfect interface are investigated. Based on the generalized self-consistent theory, the closed-form solutions of the effective material constants are derived. For the composites with piezomagnetic inclusion, piezoelectric coating and polymer matrix, numerical calculations are performed to present the influences of the imperfect interface cooperating with the coating on the effective material constants. From the results, it can be observed that the effective constants can be enhanced by the coating but reduced by the imperfect interface. Moreover, the coating has the shielding effects on the imperfect interface for the composite structures with its higher filling ratio.     

Propagation of weak waves in elastic-plastic and elastic-viscoplastic solids with interfaces

A numerical technique based on the method of singular surfaces has been developed for the computation of wave propagation in solids exhibiting rate-independent elastic-plastic or rate-dependent elastic-viscoplastic behavior. The von Mises yield condition and associated flow rule is taken to represent the rate-independent behavior, while the Perzyna dynamic overstress model is taken to represent the rate-dependent behavior. For 1100-0 Al, a good empirical fit with published experimental data was found to be: $J_2{}^{1/2}?\kappa \left(W^p\right)=\left({\tau }_0/{\gamma }_0\right)\left(\stackrel{0}{W^p}/J_2{}^{1/2}\right)$ where:J₂ is the second invariant of the stress deviator;_k(W^p) is the static hardening curve;W^p is the plastic work and the parameter (τ₀/γ₀) = 0 (rate-independent model) or (80)^?1 to (70)^?1 MPa · s. In the numerical technique, the “connection equations” which provide relations between discontinuities in space and time derivatives lend themselves naturally to finite difference representations. A five-point space-time grid (center point coincident with the instantaneous location of the singular surface) is sufficient for the differenced form of the connection equations and suggests a natural marching scheme for the calculation of all necessary variables at each time step. Supplementing these equations which hold in the interior of the specimen are interface equations which assure continuity in stress and velocity across boundaries which separate materials with dissimilar properties. Application of the technique is made to wave propagation in pure shear for the purpose of comparing numerical predictions with relevant experimental data. The measurements of Duffyet al.[10] which are obtained from the torsional Kolsky apparatus (one dimensional torsional shear wave propagation in a thin-walled tube) were compared with predictions obtained numerically. By using the experimental input pulse history and the constitutive equation reported above, excellent agreement between the predicted and observed histories of reflected and transmitted pulses was obtained when the viscoplastic model was used. Poorer agreement was observed when the rate-independent model (τ₀/γ₀=0) was used. It is concluded that the Perzyna model gives good results for the behavior of 1100-0 Al at high rates of strain.     

Dynamic effective property of fibrous piezoelectric composites with spring- or membrane-type imperfect interfaces

The present paper studies the dynamic effective property of piezoelectric composites embedded with cylindrical piezoelectric fibers under anti-plane harmonic electro-elastic waves. By using the dynamic generalized self-consistent method (DGSM) of electro-elastic coupling wave, the problem of randomly distributed cylindrical fibers in a piezoelectric medium can be analyzed in terms of a representative volume element with a coated fiber embedded in an equivalent effective medium. The interfaces between the fibers and the matrix are assumed to be imperfect which are here modeled as spring- or membrane-type interfaces. Through wave function expansion method and an iterative method, the effective piezoelectrically stiffened shear modulus and the effective wave number are obtained. Examples are conducted to verify the present solutions and to illustrate the dependence of the effective piezoelectrically stiffened shear modulus on the wave number (frequency) as well as the interface properties. The special size effect related to interfacial imperfection is also discussed.     

Electroelastic interaction between piezoelectric screw dislocation and circularly layered inclusion with imperfect interfaces

The interaction between a piezoelectric screw dislocation and an interphase layer in piezoelectric solids is theoretically investigated.Here,the dislocation located at arbitrary points inside either the matrix or the inclusion and the interfaces of the interphase layer are imperfect.By the complex variable method,the explicit solutions to the complex potentials are given,and the electroelastic fields can be derived from them.The image force acting on the dislocation can be obtained by the generalized PeachKoehler formula.The motion of the piezoelectric screw dislocation and its equilibrium positions are discussed for variable parameters.The important results show that,if the inner interface of the interphase layer is imperfect and the magnitude of degree of the interface imperfection reaches the certain value,two equilibrium positions of the piezoelectric screw dislocation in the matrix near the interface are found for the certain material combination which has never been observed in the previous studies(without considering the interface imperfection).     

弱粘结的斜交铺设层合圆柱壳的柱形弯曲

导出了两端简文的具有弱粘结界面的任意斜交铺设层合圆柱壳柱形弯曲问题的一个精确弹性理论解。分析中采用线性弹簧模型来表征界面的弱粘结特性。引进新的物理量改写了基本方程,导出了对应的状态空间列式,并利用变量替换技术将该状态方程转换成常系数状态方程,从而方便求解。最后给出了数值算例,并讨论了弱界面的影响。     

Analysis of plastic deformation in nanoscale metallic multilayers with coherent and incoherent interfaces

Nanoscale metallic multilayered (NMM) composites possess ultra high strength (order of GPa) and high ductility, and exhibit high fatigue resistance. Their mechanical behavior is governed mainly by interface properties (coherent and/or incoherent interfaces), dislocation mechanisms in small volume, and dislocation-interface interaction. In this work, we investigate these effects within a dislocation dynamics (DD) framework and analyze the mechanical behavior of two systems: (1) a bi-material system (CuNi) with coherent interface and (2) a newly developed tri-material system (CuNiNb) composed of both coherent and incoherent interfaces. For the bi-material case we analyze the influence of networks of interfacial dislocations whose nature and distribution are commensurate with the level of relaxation and loading of the structure. Misfit and pre-deposited interfacial dislocation arrays, as well as combinations of both, are studied and the dependence of strength on layer thickness is reported, along with observed dislocation mechanisms. It is shown that interfacial defect configurations significantly alter the strength and mechanical behavior of the material. Furthermore, it is shown that the implementation of penetrable interfaces in DD captures the strength dependence at layer thicknesses on the order of 3-7 nm. For the tri-material case we analyze the effects of coherent and incoherent interfaces in large-scale simulations. The results show that these materials have strong strength-size dependence but are limited by the strength of the incoherent (CuNb) interface which is weak in shear. The weak interface acts as a dislocation sink. This in turn induces an internal shear stress field that activates cross-slip in the adjacent CuNi interlace and thus causing softening. Moreover, it is shown that the yield stress of the CuNiNb system is controlled by the volume fraction of the Nb. Because Nb is the most compliant of the three materials, an increase in volume fraction of Nb decreases the overall yield strength of the material.