The thermo-electromagnetic waves in piezoelectric solids

The Maxwell??s equation and stress equilibrium equation of anisotropic media, coupled to the heat conduction equation, are studied here based on the eigen theory of physical presentation, in which a new thermo-piezoelectric constitutive model is induced. The complete sets of uncoupled thermo-electromagnetic wave equations for piezoelectric solids are deduced. The results show that the equations of electromagnetic waves in anisotropic media subjected to heat become fourth order differential ones, and from them, a new phenomenon of thermo-electromagnetic waves with low propagation speed is obtained. In the final part of this paper, we discuss the propagation behavior of the new thermo-electromagnetic waves in the piezoelectric material of class 6?mm.     

Dynamic response of a slab of elastic-viscoplastic material that exhibits induced plastic anisotropy

Various 2-dimensional problems of the dynamic loading of a slab are solved for a material characterization that is elastic-viscoplastic and exhibits anisotropic work-hardening. The governing constitutive equations are based on a unified formulation which requires neither a yield criterion nor loading or unloading conditions. They include multi-dimensional anisotropic effects induced by the plastic deformation history. The theory also considers plastic compressibility which depends on the extent of the anisotropy. A numerical procedure for solving the equations is developed which incorporates the history dependent anisotropic hardening effects. Cases considered are the dynamic penetration of a slab by a rigid cylindrical indenter, and a distributed force rapidly applied over part of the slab surface. Both conditions of fixed and free rear surfaces of the slab are examined. A uniaxial problem is also considered in which different bases for the anisotropic hardening law are examined.     

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.     

On the stress state in rectilinear anisotropic thick plates with blunt cracks

In this paper the analytical solution for the stress fields in the close neighbourhoods of blunt cracks in thick anisotropic plates is provided. As a first step the equations of the three-dimensional theory of elasticity are successfully reduced to two un-coupled equations in the two-dimensional space. Later on, the 3D stress field solution for anisotropic plates with blunt cracks is presented and its degree of accuracy discussed comparing theoretical results and numerical data from 3D FE analyses. Eventually, relevant expressions for the Generalised Stress Intensity Factors for blunt cracks in orthotropic plates under mode I, II and III are provided.     

Applications of tensor functions to the formulation of constitutive equations involving damage and initial anisotropy

In this paper some results of the tensor function theory are applied to the formulation of constitutive equations of isotropic and anisotropic materials in the secondary and tertiary creep stage. The creep process, in its tertiary phase, is characterized by a damage tensor. Because of its microscopic nature, damage has, in general, an anisotropic character even in cases where the material was originally isotropic, i.e. isotropic in its virgin state. Fissure orientation and length cause anisotropic macroscopic behaviour. In the first part of the paper some possible ways of representing constitutive equations involving (initial) anisotropy of the material (e.g. from rolling) and involving anisotropic creep-damage are dealt with. The formulations of such equations are based upon theorems concerning tensor-valued functions. Furthermore, some simplified constitutive equations for more practical use are discussed. The main problem of this part is: to find an irreducible set of tensor generators. Besides the problem of finding such tensor generators it is very important to determine the scalar coefficients in constitutive equations as functions of the invariants and experimental data. The second part of the paper is concerned with the determination of the scalar functions. This can be done by using tensorial interpolation methods as pointed out in detail.     

A constitutive model for the deformation induced anisotropic plastic flow of metals

A mathematical framework is established for the equations governing inelastic deformation under multi-dimensional stress states and for the associated evolution equations of the internal state variables. The formulation is based on a generalization of the Prandtl-Reuss flow law. In the evolution equations for the inelastic state variables that control plastic flow, it is assumed that part of the rate of change is isotropic and the remaining part varies according to the sign and orientation of the current rate of deformation vector. This leads to a minimum of twelve components of the internal state tensor which represents resistance to inelastic deformation. In this manner, both initial and load history induced plastic anisotropy can be modeled. A specific set of equations for anisotropic plastic flow is developed consistent with the inelastic state variables.     

The stress intensity factors for an infinitely long transversely isotropic, thick-walled cylinder which contains a ring-shaped crack

In this study the elastostatic axisymmetric problem for a long thick-walled transversely anisotropic cylinder containing a ring-shaped internal crack is analyzed. The problem is reduced to a singular integral equation which has a simple Cauchy kernel as the dominant part by using Hankel and Fourier transform techniques. These equations are then solved numerically and the stress intensity factors are calculated.The results are given for different transversely anisotropic materials and crack geometries.     

基于解析试函数的各向异性材料厚薄通用板单元

该文采用满足Kirchhoff假设的薄板理论,推导了各向异性材料系列解析试函数,并利用该系列解析试函数构造了一个四边形应力杂交板单元。首先,该文从薄板理论的基本方程出发,推导了各向异性材料薄板中面挠度w应满足的特征微分方程。然后,从该方程出发求得w的系列特征通解,由w特征通解可进一步求得广义位移、广义应变和广义应力的解析试函数。同时,根据广义应力利用平衡条件构造了相应的横向剪力解析试函数。最后,根据已有的广义应力和横向剪力解析试函数构造了一个四边形应力杂交板单元ATF-PH4。数值算例表明:上述方法构造出的单元模型有很好的精度、收敛性,且对网格畸变不敏感,同时能较好地解决板单元的厚薄通用性问题。     

On the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law

The statement that theories of inelasticity at finite strains have arrived at a high level of development is only true in conjunction with isotropic material behaviour. From both points of view (theoretical and computational), the extension to anisotropic material behaviour seems to be a complicated task. The statement is especially true when the multiplicative decomposition of the deformation gradient is considered a basis for the formulation. Of special interest are questions related to the mathematical form of the stored energy function or, equivalently, of the constitutive relation for the material stress tensor as the thermodynamical force. This paper deals with the above issues. The anisotropic formulation is accomplished using the notion of structural tensors. Here we suggest that the privileged directions of the material should be transformed in a specific way under the action of the inelastic part of the deformation gradient. The inelastic behaviour is assumed to be governed by evolution equations of the unified type. Numerically, we deal with the full multiplicative structure of the theory. The numerical treatment is developed in full detail. Expressions concerning the local iteration as well as the tangent operator are derived. Various numerical examples with applications to shells are presented which demonstrate the influence of anisotropy and the applicability of the theory. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of an internal crack in a finite anisotropic plate

In this paper the boundary collocation method is presented for computing the stress intensity factors for an internal crack in a finite anisotropic plate. The stress functions are assumed such that they can represent the stress singularity at the crack tips, satisfying not only the governing equations of the anisotropic plate theory in the domain, but also the stress-free conditions on the crack surfaces. Therefore, only the boundary conditions of the plate need to be considered, and they can be satisfied approximately by the Boundary Collocation Method. Numerical examples demonstrated that the proposed method gives satisfactory results compared with the existing solutions.     

一种新的混凝土各向异性弹塑性损伤本构模型及其数值实施

该文的目的是建立一种新的、相对简单的混凝土各向异性塑性损伤本构模型,以方便的模拟混凝土结构的破坏行为。为了更好地描述混凝土在拉、压荷载作用下的不同损伤机制,建立了拉、压不同的两种损伤演化方程,用于确定各向异性的拉、压损伤变量。另外,根据应变等效假设,假定有效构型和损伤构型的应变相等,该方法不仅大大简化了模型的推导过程,而且可方便的通过解耦算法进行有效应力和损伤及名义应力的计算,也即塑性部分计算可通过现有的隐式算法实现,损伤部分及名义应力的计算则可通过较为简便的显式算法实现,从而可大大提高计算效率。模型结果与试验结果的对比分析表明：该模型能较好地描述混凝土在三维应力状态下的非线性行为;对双边开口四点弯曲梁试件的模拟也表明：该模型能反应混凝土损伤各向异性的特点,计算结果相比ABAQUS软件自带的混凝土损伤塑性本构模型(CDP模型)更符合实际情况,计算效率也更高。     

Stress and vibration analyses of anisotropic shells of revolution

An efficient computational strategy is presented for reducing the cost of the stress and free vibration analyses of laminated anisotropic shells of revolution. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight strain components, the eight stress resultants and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions (Fourier series) in the circumferential co-ordinate, and a three-field mixed finite element model is used for the discretization in the meridional direction. The shell response associated with a range of Fourier harmonics is approximated by a linear combination of a few global approximation vectors, which are generated at a particular value of the Fourier harmonic, within that range. The full equations of the finite element model are solved for only a single Fourier harmonic, and the response corresponding to the other Fourier harmonics is generated using a reduced system of equations with considerably fewer degrees of freedom.     

Boundary integral equations and the boundary element method for fracture mechanics analysis in 2D anisotropic thermoelasticity

This paper develops the Somigliana type boundary integral equations for fracture of anisotropic thermoelastic solids using the Stroh formalism and the theory of analytic functions. In the absence of body forces and internal heat sources, obtained integral equations contain only curvilinear integrals over the solid’s boundary and crack faces. Thus, the volume integration is eliminated and also there is no need to evaluate integrals over the contours in the mapped temperature domain as it was done before. In addition to finite solids, the case of an infinite anisotropic medium with a remote thermal load is also studied. The dual boundary element method for fracture of anisotropic thermoelastic solids is developed based on the obtained boundary integral equations. Presented numerical examples show the validity and efficiency of the obtained equations in the analysis of both finite and infinite solids with cracks.     

On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials

Using the notions of intermediate configuration firstly introduced by Eckart (1948) and of vector directors due to E. and F. Cosserat (1909), we obtain constitutive equations for an anisotropic solid subject to finite strains. This study closely follows Mandel's ideas (1971, 1982) although it differs in presentation and in the way of deriving the final equations. In a second part we establish the form of constitutive equations for some particular anisotropic materials by applying representation theorems. As an application, we show that the introduction of plastically induced rotations suppresses oscillations of Cauchy stress for a material with kinematic hardening submitted to simple shear test: in fact our method consists in defining a new stress rate which is physically motivated. It provides an alternative to the more phenomenological approach of Nagtegaal and de Jong (1980/1982). Furthermore, in contrast to other recent methods, it can be applied to a wide range of anisotropic materials.     

From kinetic theory to hydrodynamics—Uniform anisotropic superfluids

Starting from kinetic theory, and neglecting dissipation, we derive the hydrodynamic equations for anisotropic superfluids. Only the case of a uniform texture in the absence of finite external fields is considered. Other than possessing a normal fluid density which is a diagonal tensor, anisotropic superfluids are characterized by the same hydrodynamics as in the isotropic case. These results constitute a generalization of the kinetic theory of Khalatnikov, which applies to the case of an isotropic quasiparticle spectrum dependent only on density.Supported in part by the National Science Foundation through grant number DMR74-23494.     

A phenomenological theory of transformation superplasticity

A phenomenological theory of transformation superplasticity is presented from the point of view of thermomechanics. The stretching tensor and entropy density are assumed to be decomposed into the elastic part and the part due to the superplastic process. The constitutive equations for the superplastic part are derived by means of a dissipation potential, while the elastic part is shown to be determined from the free energy as usual. The theory developed is applied to the transformation superplastic deformation of a thin-walled tubular specimen of pure iron which is subjected simultaneously to a constant shear stress and temperature cycling. The results clearly show the following characteristic: The increment of the superplastic strain per cycle is proportional to the applied stress. The maximum value of the rate of transformation superplastic strain during a cycle is proportional to the applied stress, as well as to the heating (cooling) rate.     

Elastic structural response of anisotropic sandwich plates with a first-order compressible core impacted by a Friedlander-type shock loading

The foundation of the non-linear theory of asymmetric anisotropic sandwich plates with a first order compressible weak orthotropic core under a Friedlander-type explosive blast is presented. The equations of motion are developed by means of Hamilton’s Principle. Within the theory, the face sheets are asymmetric while adopting the Love-Kirchoff model. In addition, the core layer is assumed to be compressible (extensible) in the transverse direction thereby capturing any wrinkling or global instabilities. The theory is then simplified and applied for the case of sandwich plates with symmetric unidirectional fiber reinforced laminated composite facings with the axes of orthotropy not necessarily coincident with the geometrical axes. The governing solution is developed using the Extended-Galerkin method resulting in two coupled non-linear second order ordinary differential equations which are then solved using the 4th-order Runge–Kutta method for a system of differential equations.     

Construction of anisotropic polyconvex energies and applications to thin shells

In computer simulations where constitutive equations are considered anisotropic polyconvex energies can preferably be used because the existence of minimizers is then automatically guaranteed. In this work we investigate the capability to simulate anisotropy effects of anisotropic thin shells using polyconvex anisotropic energies. The construction of the considered polyconvex transversely isotropic energy is based on specific structural tensors. The iterative enforcement of the zero normal stress condition at the integration points allows the consideration of arbitrary three-dimensional constitutive equations. As a representative example we compare results for isotropic and anisotropic plates.     

复合材料管接头拉扭作用下胶层应力分析

发展了一种复合材料胶接管接头在拉伸和扭转载荷作用下胶层内应力分析模型。该模型利用管接头结构中缠绕法成形的管和套管材料、几何形状和所承受载荷的轴对称性,采用一阶层合板理论分析管壁的应力和变形。通过管接头中管、套管和它们之间胶层的位移协调性确定胶层内的应变和应力,从而建立了管接头结构的平衡方程。本模型计算了缠绕角为54°和26°管接头胶层内剥离应力、轴向剪切应力和切向剪切应力,计算结果与有限元分析结果相吻合。      

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.