New exact solution of Euler’s equations (rigid body dynamics) in the case of rotation over the fixed point

A new exact solution of Euler’s equations (rigid body dynamics) is presented here. All the components of angular velocity of rigid body for such a solution differ from both the cases of symmetric rigid rotor (which has two equal moments of inertia: Lagrange’s or Kovalevskaya’s case), and from the Euler’s case when all the applied torques are zero, or from other well-known particular cases. The key features are the next: the center of mass of rigid body is assumed to be located at meridional plane along the main principal axis of inertia of rigid body, besides, the principal moments of inertia are assumed to satisfy to a simple algebraic equality. Also, there is a restriction at choosing of initial conditions. Such a solution is also proved to satisfy to Euler–Poinsot equations, including invariants of motion and additional Euler’s invariant (square of the vector of angular momentum is a constant). So, such a solution is a generalization of Euler’s case.     

Image singularities of Green’s functions for isotropic elastic bimaterials subjected to concentrated forces and dislocations

Green’s functions for isotropic materials in the two-dimensional problem for elastic bimaterials with perfectly bonded interface are reexamined in the present study. Although the Green’s function for an isotropic elastic bimaterial subjected to a line force or a line dislocation has been discussed by many authors, the physical meaning and the structure of the solution are not clear. In this investigation, the Green’s function for an elastic bimaterial is shown to consist of eight Green’s functions for a homogeneous infinite plane. One of the novel features is that Green’s functions for bimaterials can be expressed directly by knowing Green’s functions for the infinite plane. If the applied load is located in material 1, the solution for the half-plane of material 1 is constructed with the help of five Green’s functions corresponding to the infinite plane. However, the solution for the half-plane of material 2 only consists of three Green’s functions for the infinite plane. One of the five Green’s functions of material 1 and all the three Green’s functions of material 2 have their singularities located in the half-plane where the load is applied, and the other four image singularities of material 1 are located outside the half-plane at the same distance from the interface as that of the applied load. The nature and magnitude of the image singularities for both materials are presented explicitly from the principle of superposition, and classified according to different loads. It is known that for the problem of anisotropic bimaterials subjected to concentrated forces and dislocations, the image singularities are simply concentrated forces and dislocations with the stress singularity of order O(1/r). However, higher orders (O(1/r²) and O(1/r³)) of stress singularities are found to exist in this study for isotropic bimaterials. The highest order of the stress singularity is O(1/r³) for the image singularities of material 1, and is O(1/r²) for material 2. Using the present solution, Green’s functions associated with the problems of elastic half-plane with free and rigidly fixed boundaries, for homogeneous isotropic elastic solid, are obtained as special cases.     

轴对称一维六方准晶圆柱的精化理论

将轴对称圆柱的精化分析推广到一维六方准晶中轴对称圆柱的研究当中。利用准调和函数的Bessel算子函数表示以及一维六方准晶中的通解,在不做任何预先假设的情况下,给出了一维六方准晶中轴对称圆柱的精化理论。首先,根据准调和函数的Bessel算子函数表示,利用三个一维待定函数,表示出声子场和相位子场的位移场和应力分量。再根据非齐次边界条件,推导出柱面受径向外载时的精化方程。通过舍弃高阶项,推导出了在径向方向受到柱面载荷的近似解。     

Hill’s class of compressible elastic materials and finite bending problems: Exact solutions in unified form

Hill (1978) proposed a natural extension of Hooke’s law to finite deformations. With all Seth-Hill finite strains, Hill’s natural extension presents a broad class of compressible hyperelastic materials over the whole deformation range. We show that a number of known Hookean type finite hyperelasticity models are included as particular cases in Hill’s class and that Bell’s and Ericksen’s constraints may be derived as natural consequences from Hill’s class subjected to internal constraints. Also we present a unified study of finite bending problems for elastic Hill materials. To date exact results are available for certain particular classes of compressible elastic materials, which do not cover Hill’s class. Here, with a novel idea of circumventing the strong nonlinearity we show that it is possible to derive exact solutions in unified form for the whole class of elastic Hill materials. Reduced results are also given for cases subjected to internal constraints.     

On using mesh-based and mesh-free methods in problems defined by Eringen’s non-local integral model: issues and remedies

With the recent success of nonlocal theories in modeling of engineering problems involving small intrinsic length scales, such as modeling of crack propagation, this paper addresses issues pertaining to cost-ineffectiveness of Eringen’s integral model. The cost effectiveness of the computation may be considered as a twofold issue; one pertaining to the non-local model and another pertaining to the numerical tool. First of all, we shall show that during the solution of problems with Eringen’s non-local integral model, there is no need to consider the integral model for the whole computational domain. In fact, the problems may be solved by just using the integral model close to the boundaries, i.e. a boundary layer effect, or around the points with singularities. In this paper we propose a partitioning strategy to remarkably reduce the computational cost. This may be considered as a gateway for solving some types of two-scale problems, e.g. those with macro/micro and nano scales, in which the small scale effects are localized just at parts of the domain. To demonstrate the efficiency of the numerical tools, we examine the performance of the finite element method (FEM), the element free Galerkin method (EFG) and the finite point method (FPM). This paves the way for using mesh-free methods in the solution of problems with non-local integral models. Examples with smooth and non-smooth solutions are considered for examining the efficiency of the methods. It will be shown that, by considering the boundary layer effect, the FEM and FPM will be efficient enough for being used in problems defined by Eringen’s non-local integral model.

     

Imperfection sensitivity of laminated conical shells

The sensitivity of laminated conical shells to imperfection is considered, via the initial post-buckling analysis, on the basis of three different shell theories: Donnell’s, Sanders’, and Timoshenko’s. Unlike isotropic conical shells or laminated cylindrical shells, in the case of laminated conical shells the thickness and the material properties vary with the shell coordinates, which complicates the problem considerably. The main objective of the study is to investigate the influence of the variation of the stiffness coefficients on the buckling behavior and on the imperfection sensitivity of laminated conical shells. It is felt that by finding the various parameters that influence the shell’s imperfection sensitivity, it is possible to improve the behavior of the whole structure.A special Level-1 computer code ISOLCS (Imperfection Sensitivity of Laminated Conical Shells) had been developed. ISOLCS calculates the classical buckling load and the imperfection sensitivity via Koiter’s theory of laminated conical shells with consideration to the variation of the material properties in the shell’s coordinates. The range of validity of the Level-1 predictions by ISOLCS is verified by the Level-3 code STAGS-A.     

Green’s formula and singularity at a triple contact line. Example of finite-displacement solution

The various equations at the surfaces and triple contact lines of a deformable body are obtained from a variational condition, by applying Green’s formula in the whole space and on the Riemannian surfaces. The surface equations are similar to the Cauchy’s equations for the volume, but involve a special definition of the ‘divergence’ (tensorial product of the covariant derivatives on the surface and the whole space). The normal component of the divergence equation generalizes the Laplace’s equation for a fluid–fluid interface. Assuming that Green’s formula remains valid at the contact line (despite the singularity), two equations are obtained at this line. The first one expresses that the fluid–fluid surface tension is equilibrated by the two surface stresses (and not by the volume stresses of the body) and suggests a finite displacement at this line (contrary to the infinite-displacement solution of classical elasticity, in which the surface properties are not taken into account). The second equation represents a strong modification of Young’s capillary equation. The validity of Green’s formula and the existence of a finite-displacement solution are justified with an explicit example of finite-displacement solution in the simple case of a half-space elastic solid bounded by a plane. The solution satisfies the contact line equations and its elastic energy is finite (whereas it is infinite for the classical elastic solution). The strain tensor components generally have different limits when approaching the contact line under different directions. Although Green’s formula cannot be directly applied, because the stress tensor components do not belong to the Sobolev space H¹(V)$H^1(\text{V})$, it is shown that this formula remains valid. As a consequence, there is no contribution of the volume stresses at the contact line. The validity of Green’s formula plays a central role in the theory.     

基于p型有限元法和围线积分法计算复合型应力强度因子

传统有限元法由于采用低阶插值计算应力强度因子时,需要划分的网格数较多,收敛速度较慢,得到的应力强度因子精度不足。p型有限元法在网格确定时通过增加插值多项式的阶数来提高计算精度,具有网格划分少、收敛速度快、精度高、自适应能力强等特点。本文采用基于p型有限元法的有限元计算软件StressCheck计算得到应力场和位移场,并由围线积分法导出混合型应力强度因子(SIFs)。通过几个经典算例,分析了围线的选择对计算精度的影响,计算了不同裂纹长度、不同裂纹角度和裂纹在应力集中区域不同位置时的应力强度因子。并将数值结果、理论解与文献中其他数值计算方法所得的部分结果进行了对比分析,结果表明自由度数不大于7000时,导出的应力强度因子相对误差最大不超过1.2%,数值解表现出较高的精度及数值稳定性。     

Decoupling coefficients of dilatational wave for Biot’s dynamic equation and its Green’s functions in frequency domain

Green’s functions for Biot’s dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering, rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green’s functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term “decoupling coefficient” for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green’s functions. The correctness of the solution is demonstrated by numerically comparing the current solution with Cheng’s previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green’s functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method (BEM) and other applications.     

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.     

Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media

The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).     

An analysis of nanoindentation in linearly elastic solids

The conventional method to extract elastic properties in the nanoindentation of linearly elastic solids relies primarily on Sneddon’s solution (1948). The underlying assumptions behind Sneddon’s derivation, namely, (1) an infinitely large incompressible specimen; (2) an infinitely sharp indenter tip, are generally violated in nanoindentation. As such, correction factors are commonly introduced to achieve accurate measurements. However, little is known regarding the relationship between the correction factors and how they affect the overall accuracy. This paper first proposes a criterion for the specimen’s geometry to comply with the first assumption, and modifies Sneddon’s elastic relation to account for the finite tip radius effect. The relationship between the finite tip radius and compressibility of the specimen is then examined and a composite correction factor that involves both factors, derived. The correction factor is found to be a function of indentation depth and a critical depth is derived beyond which, the arbitrary finite tip radius effect is insignificant. Techniques to identify the radius of curvature of the indenter and to decouple the elastic constants (E and ν) for linear elastic materials are proposed. Finally, experimental results on nanoindentation of natural latex are reported and discussed in light of the proposed modified relation and techniques.     

三维粘弹性层状半空间埋置集中荷载动力格林函数求解—修正刚度矩阵法

针对三维粘弹性层状半空间埋置集中荷载作用下动力响应问题,在柱面坐标下,结合径向Hankel积分变换,提出了一种新的求解方法—修正刚度矩阵法。方法基于位势函数理论,将三维问题分解为平面内反应(P-SV波型)和平面外反应（SH波型）两个二维问题的叠加;借鉴结构力学中超静定结构的位移法原理,首先固定荷载所在层的上下界面,通过对波动方程的特解和齐次解叠加得到“固端”反力。进而放松两“固端约束”,利用直接刚度法求得各层面位移,荷载作用层内反应另需叠加上该“固定层”内解,并将特解部分积分（直达波）由全空间解析解代替,解决了当接收点和源点作用水平面接近时的积分收敛问题。算例分析表明,对于低频（可退化为静力状态）和高频问题,本文方法均具有很高的计算效率和精度。     

Post-bifurcation equilibria in the plane-strain test of a hyperelastic rectangular block

Of interest here is the bifurcated equilibrium solution of a homogeneous, hyperelastic, rectangular block under finite, plane-strain tension or compression. A general asymptotic analysis of the bifurcated equilibrium path about the principal solution’s lowest critical load is presented using Lagrangian kinematics. The analysis is valid for any compressible hyperelastic material with axes of orthotropy aligned with the block’s axes of symmetry in the reference (stress-free) configuration.The general theory is subsequently applied to blocks of different constitutive laws. Results are presented in the form of bifurcated equilibrium branch’s curvature at the critical load as function of the block’s aspect ratio, since the sign of this curvature determines the branch’s stability. For small aspect ratios there is agreement with existing structural models, while for relatively higher aspect ratios some rather counter-intuitive stability results appear, which strongly depend on the constitutive law.     

Prandtl’s formulation for the Saint–Venant’s torsion of homogeneous piezoelectric beams

The Saint–Venant torsional problem for homogeneous, monoclinic piezoelectric beams is formulated in terms of Prandtl’s stress function and electric displacement potential function. The analytical approach presented in this paper generalizes the known formulation of Prandtl’s solution which refers to homogeneous elastic beams. The Prandtl’s stress function and electric displacement potential function satisfy the so called coupled Dirichlet problem (CDP) in the cross-sectional domain. A direct and a variational formulation are developed. Exact analytical solutions for solid elliptical cross-section and hollow circular cross-section and an approximate solution based on a variational formulation for thin-walled closed cross-section are presented.     

General solution for Eshelby’s problem of 2D arbitrarily shaped piezoelectric inclusions

Eshelby’s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media. For example, it intervenes in the nanostructure design of strained semiconductor devices involving strain-induced quantum dot (QD) and quantum wire (QWR) growth. Using the extended Stroh formalism, the present work gives a general analytical solution for Eshelby’s problem of two-dimensional arbitrarily shaped piezoelectric inclusions. The key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended Stroh formalism applied to Eshelby’s problem. The simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle Eshelby’s piezoelectric problem for a large variety of non-elliptical inclusions. In the present work, explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by Jordan’s curves and Laurent’s polynomials. By considering the piezoelectric material GaAs (110), the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions, and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion.     

Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting

In this article, we investigate a prey– predator model with Allee effect and state-dependent impulsive harvesting. We obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (1.2) by means of the geometry theory of semicontinuous dynamic system and the method of successor function. We also obtain that system (1.2) exhibits the phenomenon of heteroclinic bifurcation about parameter $\alpha $ . The methods used in this article are novel and prove the existence of order-1 periodic solution and heteroclinic bifurcation.     

On the solution of the dynamic Eshelby problem for inclusions of various shapes

In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby’s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation (‘dynamic’ or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby’s inclusion problem (Eshelby, 1957), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions.     

Three-dimensional Green’s function for time-harmonic dynamics in a viscoelastic layer

This paper presents Green’s function for time-harmonic elastodynamic problems for a single layer domain (three-dimensional region bounded by two parallel planes with traction-free boundary conditions). The semi-analytic solution is built in three steps: (a) potential displacement representation; (b) angular Fourier series; (c) radial Hankel transform. Reflection matrices are presented for the plate domain. Kernels are integrally split into a singular closed-form term (the static half-space solution) plus an incremental solution. In order to compute the inverse Hankel transform for displacements and stress components, a modified complex integration path is required. Theoretical considerations allow an adequate delimitation of such a complex path. A specific treatment is proposed for low excitation frequencies where asymmetric Lamb waves play a major role. A series of numerical benchmarks are presented to validate the implementation of the functions (displacements and tractions).