On the effective electroelastic properties of microcracked generally anisotropic solids

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration.     

Single integral representations for nonlinear viscoelastic solids

The finite linear viscoelastic solid and single integral representation with nonlinear dependence on history are investigated in uni-axial stress. Both integral kernels in the stress formulation are determined by single-step constant strain tests, and both kernels in the strain formulation are determined by single-step constant stress tests. Single integral stress and strain formulations are not equivalent. The stress histories required to maintain constant strain-rate for both models are determined from the Volterra integral equations given by the strain formulations once their kernels are determined by constant stress tests. However, known constant strain-rate response does not determine the kernels. Examples are presented to show that variation of the kernel within a given qualitative shape can lead to different shapes of constant strain-rate response, so that both constant stress and constant strain-rate tests may be necessary to deduce the optimum single integral approximation, in preference to multi-step stress tests. It is shown that the apparently simpler finite linear viscoelastic model requires a far lengthier numerical algorithm to solve the Volterra equation, and leads to non-unique and physically unacceptable response, in comparison with the more flexible nonlinear history dependence which yields unique acceptable responses.     

A nonlinear theory for electroelastic shells with relatively large in-plane shear deformation and its implications in nonlinear mode coupling

A set of nonlinear two-dimensional equations for thin electroelastic shells in vibrations with moderately large shear deformation in the tangent plane are obtained from the three-dimensional equations of nonlinear electroelasticity. As an example for application, the equations are used to study nonlinear torsional vibration of a circular cylindrical piezoelectric shell. It is shown that torsion is nonlinearly coupled to axial extension and circumferential extension. The results of this paper emphasize the need for further study of mode coupling induced by nonlinearity.     

A new nonlinear constitutive theory for conducting magnetothermoelastic solids

A new nonlinear theory of constitutive equations for electrically and thermally conducting magnetothermoelastic (MTE) solids is developed. In the theory, the electric current and heat flux vectors are also considered to be independent variables in the argument of each constitutive function. It is shown that the modified Helmholtz free energy (MH FE) density, which is a thermodynamical potential for the specific entropy, the magnetization and the stress tensor, does no longer appear as a function of the temperature, the magnetic field and the strain tensor, but it also depends upon the electric current and heat flux vectors. Furthermore, referring to the mentioned constitutive equations, the Gibbs equation is also generalized. In order to expose the constitutive theory developed here, an appropriate polynomial expression of the MH FE density for the anisotropic materials is proposed, and, exploiting the method of the theory of invariants, its exact expression is also determined. With the use of these two expressions, a set of rather general nonlinear constitutive equations, which governs a lot of magnetoelastothermo-electrical (MET-E) effects, is then obtained explicitly. It is interesting to notice that each of the constitutive equations mentioned above has a pseudo (ir) reversible part in vicinities of the new equilibrium state, namely the thermo-electrical equilibrium (T-EE) state. According to the deductive scheme, the generalized constitutive equations and the Gibbs equation in the present work are finally discussed for special materials, and/or vanishing some of the fields. The resulting expressions are, as they should be, in full mutual agreement with the established theories on the same subject.     

Stress-strain relations in elastoplastic solids with Dugdale-type cracks

Mechanical behavior of a two-dimensional elastoplastic solid with rectilinear cracks is investigated. Plastic strip model is used to reduce plasticity problem to the equivalent linear elasticity formulation. Two realizations of the mixed mode plastic strip model are considered: in-line plastic strips as proposed by Becker and Gross [Int. J. Fract. 37 (1988) 163], and inclined plastic strips of Panasyuk and Savruk [Appl. Mech. Rev. 47 (1994) 151]. The effective mechanical response predictions are based on the procedure presented in Kachanov et al., [Appl. Mech. Rev. 47 (1994) 151]. Stress-strain relations are obtained for parallel and randomly oriented non-interacting cracks. Results are compared with known elastic solutions.     

Universal theorems for total energy of the dynamics of linearly elastic heterogeneous solids

In this paper we consider a sample of a linearly elastic heterogeneous composite in elastodynamic equilibrium and present universal theorems which provide lower bounds for the total elastic strain energy plus the kinetic energy, and the total complementary elastic energy plus the kinetic energy. For a general heterogeneous sample which undergoes harmonic motion at a single frequency, we show that, among all consistent boundary data which produce the same average strain, the uniform-stress boundary data render the total elastic strain energy plus the kinetic energy an absolute minimum. We also show that, among all consistent boundary data which produce the same average momentum in the sample, the uniform velocity boundary data render the total complementary elastic energy plus the kinetic energy an absolute minimum. We do not assume statistical homogeneity or material isotropy in our treatment, although they are not excluded. These universal theorems are the dynamic equivalent of the universal theorems already known for the static case [Nemat-Nasser and Hori, 1993] and [Nemat-Nasser and Hori, 1995]. It is envisaged that the bounds on the total energy presented in this paper will be used to formulate computable bounds on the overall dynamic properties of linearly elastic heterogeneous composites with arbitrary microstructures.     

Two-dimensional equations for electroelastic plates with relatively large in-plane shear deformation and nonlinear mode coupling in resonant piezoelectric devices

Summary A set of two-dimensional equations for electroelastic plates in nonlinear face-shear motion are derived from the three-dimensional equations of nonlinear electroelasticity. The equations can describe the nonlinearity due to moderately large in-plane shear deformation associated with face-shear modes. The equations are used to study nonlinear face-shear vibration of a plate of 6mm crystals.     

Linear stress-strain relations in nonlinear elasticity

Summary Nonlinear strain measures which are compatible with the existence of an elastic potential and lead to a linear stress-strain relation are obtained. An unusual property of these measures is their dependence on material parameters. Standard strain measures commonly used in nonlinear elasticity are shown to be consistent with linear stress-strain relations only for particular cases of Poisson's ratio. Corresponding potentials for these cases are presented.     

Reciprocity relations for nonlinear plasmalike medium in a magnetic field

We consider nonlinear reciprocity of plasmalike medium in a homogeneous magnetic field, which is described by the Vlasov equation in the relaxation approximation. Reciprocal relations for the tensor of nonlinear conductivity are derived. A method of increasing the accuracy of nonlinear functional electronic devices with the aid of obtained reciprocal relations is proposed.

     

Boundary element method for electroelastic interaction in piezoceramics

A boundary element formulation and its numerical realization for the analysis of electroelastic interaction in piezoelectric materials are presented and corroborated numerically. Based on the linear theory of piezoelasticity, the boundary integral equation is developed employing the mechanical displacement and the electric potential as the primary variables. A two-dimensional fundamental solution for transversely isotopic piezoelectric materials is obtained in closed form and numerically implemented using the spline boundary element method. As a numerical example, stress analysis is performed for an infinite piezoceramic medium (PZT-4) containing a cylindrical defect under several mechanical and electric loading conditions. Numerical results are shown to be in good agreement with an available solution.     

Branching of strain histories for nonlinear viscoelastic solids with a strain clock

Summary An important class of constitutive equations for nonlinear viscoelastic response utilizes the concept of a strain clock. The clock takes the form of a material time variable which is defined in terms of the strain history and which increases faster than physical time. Important consequences of the strain clock are that stress relaxation and creep occur faster as strain increases, and the stress may not increase monotonically with time. In this work, we discuss whether this non-monotonic response implies that strain histories may branch into multiple histories.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

Possible nonlinear heat-pulse propagation in solids at Debye temperatures

Molecular dynamics techniques are used to show that heat can be transferred ballistically in threedimensional crystalline materials at temperatures on the order of Debye temperatures. Pis’ma Zh. Tekh. Fiz. 25, 55–59 (March 26, 1999)     

Remote diagnostics of soft solids using nonlinear acoustic methods

A prototype of a new flaw detector for remote noncontact diagnostics of soft solids is proposed. The operating principle of the flaw detector is based on the features of the nonlinear interaction of two ultrasonic waves in the test material with defects. Ultrasonic waves in the test material are excited remotely by ultrasonic beams focused in air. It is demonstrated experimentally that with the help of the proposed flaw detector, subsurface defects with dimensions much smaller than the wavelength of the probing ultrasound can be remotely detected and localized.     

A pseudo-elastic local meshless method for analysis of material nonlinear problems in solids

This paper aims to develop an effective meshless technique for the analysis of elasto-plastic problems. The material nonlinearity will be studied by a new pseudo-elastic local radial point interpolation formulation which is based on the local Petrov–Galerkin form and the radial basis function (RBF) interpolation. Hencky's total deformation theory is used to define the effective Young's modulus and Poisson's ratio, which are treated as spatial field variables, and considered as functions of the final stress state and material properties. These effective material parameters are obtained in an iterative manner using the strain controlled projection method. Several numerical examples are presented to illustrate the effectivity of the newly developed formulation, and the numerical results obtained by the present method closely agree with the results obtained by other methods. It has proven that the present pseudo-elastic local meshless method is effective and easy to apply to the analysis of elasto-plastic materials subjected to proportional loading.     

Temperature relations inside bodies in nonlinear heat-conduction processes

An equation has been obtained which allows the unsteady temperature field inside multidimensional bodies to be calculated from known temperatures along the coordinate axes, in nonlinear heat-condition process.     

Dispersion relations for testing the validity of linear and nonlinear optical spectra

We exploit efficient dispersion relations, which were developed for terahertz spectroscopy, to show their validity for testing linear and nonlinear optical spectra. As an example, we deal with the measured data for complex reflectivity of a KCl crystal and complex nonlinear susceptibility of a polysilane. It is suggested that the spectral data presented in the literature both for the KCl and the polysilane are consistent with the presented spectra analysis method.     

A new nonlinear constitutive theory of electric and heat conductions for magnetoelastothermo-electrical anisotropic solids

In this paper, a new thermodynamic theory of nonlinear constitutive equations for electric and heat conductions is presented. The theory is based on the new physico-mathematical model (P-MM) of conducting magnetothermoelastic (MTE) solids exposed in a previous paper [1], herein after referred to as Paper I. Starting from the results given in Paper I, here we attempt to examine in some details implications of the conduction inequality. In particular, the response functions for the temporal evolution of the electric current and heat flux vectors are determined and the conducting part of the modified Heimholtz free-energy (MHFE) density in the vicinity of the thermo-electrical equilibrium (T-EE) state is specified. Thus the constitutive equations for conduction obtained in the present paper govern “acoupling and bicoupling” effects between the electric and thermal fields, certain relaxation phenomena and a lot of magnetoelastothermoelectric (MET-E) effects. In short, the Maxwell-Cattaneo equation of heat conduction and the Newtonian-Ohm equation of electric conduction are generalized for the considered material. Furthermore, it is shown that one may need at most six material tensors (moduli) to describe the aforementioned thermoelectric effects and to specify the conducting free-energy (CFE) density. Finally, it is shown that some special forms of the constitutive equations obtained in this paper are in full agreement with those in the literature.