Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.     

Saint-Venant's Problem for Porous Linear Elastic Materials

We use a semi-inverse method to study deformations of a straight, prismatic, homogeneous body made of a porous, linear elastic, and isotropic material and loaded only at its end faces by self equilibrated forces. As in the classical theory, the problem is reduced to solving plane elliptical problems. It is shown that the Clebsch/Saint-Venant and Voigt hypotheses are not valid for this problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

Second-order effects and Saint Venant’s principle in the torsion problem of a nonlinear elastic rod

The torsion problem of a circular nonlinear elastic rod loaded by end moments is considered. The solution constructed by the method of successive approximations taking into account second-order effects is compared with the solution obtained by a semi-inverse method. It is shown that the dead-loading assumption breaks the symmetry of the Cauchy stress tensor in a certain region. A refined formulation of Saint Venant’s principle is proposed for the problem of determining integral strain characteristics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 129–136, November–December, 2006.     

Bifurcation phenomena in the rigid inclusion power law matrix composite sphere

Bifurcation of interface separation related to cavity nucleation is analyzed for a radially loaded composite sphere consisting of a rigid inclusion separated from a power law matrix by a uniform, non-linear cohesive zone. Equations for the spherically symmetric and non-symmetric problems are obtained from a hyperelastic finite strain theory by a limiting process that preserves non-linear matrix and interface response at infinitesimal strain. A complete solution to the symmetric problem is presented including bifurcation load, stresses, and evolution of elasto-plastic boundary and interface separation. An analysis of non-symmetric bifurcation, under symmetric conditions of geometry and loading, yields the bifurcation load and first non-symmetric mode shape associated with rigid inclusion displacement. An energy analysis is carried out for both symmetric and non-symmetric problems in order to assess stability of spherically symmetric states to spherically symmetric and non-symmetric “rigid body mode” perturbations.Results are provided for an interface force law that captures interface failure in normal mode and linear response in shear mode. For the symmetric problem, (i) there are threshold parameter values above which bifurcation will generally not occur, (ii) threshold values below which there do not exist equilibria in the post bifurcation regime, (iii) bifurcation occurs after attainment of the maximum interface strength. For the non-symmetric problem, (i) bifurcation always occurs, although it can be delayed by interfacial shear, (ii) for the smooth interface, non-symmetric bifurcation occurs after attainment of the maximum interface strength and always precedes symmetric bifurcation.     

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.     

Some anomalies of the curved bar problem in classical elasticity

The classical formulation of the homogeneous problem of a curved bar loaded only by and end force involves the assumption of an appropriate stress function with four arbitrary constants and the determination of these constants from the boundary conditions. Since there are five boundary conditions, four on the curved edge and one at the end, the solution is only possible because the coefficient matrix of the resulting algebraic equations is singular. This in turn means that certain inhomogeneous problems in which the curved edges are loaded by sinusoidally varying tractions cannot be solved using apparently appropriate stress functions.Additional stress functions which resolve this difficulty are introduced and an example problem is solved, which exhibits qualitatively different behavior from that in more general cases of loading. The problem is then reconsidered as a limiting case of sinusoidal loading of arbitrary wavelength. It is shown that the solution of the latter problem appears to become unbounded as the special case is approached, but that when the end conditions have been correctly satisfied by superposing an appropriate multiple of the end-loaded solution, the limit can be approached regularly and the correct special solution is recovered. The limiting process reveals a general procedure for determining the additional stress functions required for the special case.Similar relationships between homogeneous and inhomogeneous solutions for other common geometries are discussed.     

相空间中有二阶线性单面约束的非完整系统的Lie对称性与守恒量

研究相空间中有二阶线性单面约束的非完整系统的Lie对称性与守恒量。首先根据微分方程在无限小变换下的不变性建立Lie对称性所满足的确定方程和限制方程，给出结构方程和守恒量；其次讨论系统的Lie对称性逆问题。最后举一实例说明结果的应用。     

Saint-Venant Decay Rates for an Isotropic Inhomogeneous Linearly Elastic Solid in Anti-Plane Shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

Lie symmetries and conserved quantities of second-order nonholonomic mechanical system

The Lie symmetries and the conserved quantities of the second-order nonholonomic mechanical system are studied. Firstly, by using the invariance of the differential equation of motion under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equation and the conservative quantities of the Lie symmetries are obtained. Secondly , the inverse problems of the Lie symmetries are studied . Finally , an example is given to illustrate the application of the result.     

On some accurate finite-difference methods for laminar flame calculations

The ozone-decomposition flame has been studied by means of fourth- and second-order accurate schemes. The fourth-order methods include a method of lines, a time-linearization algorithm, and a majorant operatorsplitting technique. The second-order schemes include two time-linearization methods which use different temporal approximations. It is shown that the fourth-order techniques yield comparable results to those obtained with very accurate finite element and adaptive grid finite-difference algorithms. The results of the second-order methods are in good agreement with second-order explicit predictor-corrector methods but predict a lower flame speed than that obtained by means of fourth-order techniques. It is also shown that the temporal approximations are not as important as the spatial approximations in flame propagation problems characterized by the presence of several small time scales.     

Piezoelectric Analogy of Generalized Torsion in Anisotropic Elasticity

We consider a piezoelectric body bounded by a cylindrical surface in which all cross sections are of the same geometry. Suppose on the lateral surface the body is loaded in such a way that the stress and electric displacement do not vary along the axial direction. In addition the end of the cylinder is also subjected to forces reducing to bending moments, twisting moment, axial force and electric charge. We follow Lekhnitskii's formalism to characterize the deformation of the considered problem. It is found that, when the solid possesses a material symmetry plane normal to the axial direction, the simplified field equations together with suitably chosen boundary conditions are entirely analogous to those of a generalized torsion problem in anisotropic elasticity. In particular, we show that by setting a linkage between two sets of 21 material constants, any problem in one field can be resolved as another problem in the other area. This revised version was published online in August 2006 with corrections to the Cover Date.     

Implicit BEM formulations for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept

This paper presents boundary element method (BEM) formulations for usual and sensitivity problems in (small strain) elasto-plasticity using the concept of the local consistent tangent operator (CTO). “Usual” problems here refer to analysis of nonlinear problems in structural and solid continua, for which Simo and Taylor first proposed the use of the CTO within the context of the finite element method (FEM). A new implicit BEM scheme for such problems, using the CTO, is presented first. A formulation for sensitivity analysis follows. It is shown that the sensitivity of the strain increment, associated with an infinitesimal variation of some design parameter, solves a linear problem which is governed by the (converged value of the) same global CTO as the one that appears in the usual problem. Numerical results for both usual and sensitivity problems are shown for a one-dimensional example. They demonstrate the effectiveness of the present approach. In particular, accurate sensitivities with respect to material parameters (e.g., exponent of the power-type hardening law) are obtained even with few integration cells and for large load increments.     

CrMnFeCoNi纳米晶温度相关的拉伸行为研究

摘要：高熵合金是一种由多种主元元素组成的新型合金。实验研究表明等原子比CrMnFeCoNi高熵合金在低温下具有比室温更高的拉伸强度和断裂韧性。本文针对这一现象,利用分子动力学模拟对平均晶粒尺寸为6 nm的CrMnFeCoNi纳米晶在300、200和77 K下分别进行拉伸模拟。模拟研究揭示了纳米尺度CrMnFeCoNi高熵合金力学行为的温度效应和强韧机理。微结构演化分析表明：低温下,塑性变形阶段,滑移系开动的较少,位错滑移所受的阻力越大,屈服强度和抗拉强度越大;模型破坏时,孔洞缺陷形核较慢,更多孔洞缺陷演化成断口,更多的断口分摊拉伸应变,使得高熵合金纳米晶的低温韧性更好。     

Buckling delamination of a rectangular plate containing a rectangular crack and made from elastic and viscoelastic composite materials

This paper studies a three-dimensional buckling delamination problem for a rectangular plate made from elastic and viscoelastic composite material. It is assumed that the plate contains a rectangular band-crack (Case 1) and a rectangular edge-crack (Case 2) and that the edge-surfaces of these cracks have an initial infinitesimal imperfection. The evolution of this initial imperfection with an external compressive loading, acting along the crack (for an elastic composite) or with duration of time (for a viscoelastic composite under fixed external loading) is investigated within the framework of three-dimensional geometrically non-linear field equations of the theory of the viscoelasticity for anisotropic bodies. To determine the values of the critical force or critical time as well as the buckling delamination mode, the initial imperfection criterion is used. The corresponding boundary-value problems are solved by employing boundary form perturbation techniques, Laplace transform and FEM (Finite Element Method). The influence of the materials and/or the geometrical parameters of the plate on the critical values are discussed. In particular, it is established that for the considered change range of the problem parameters, the buckling form depends only on the initial infinitesimal imperfection mode of the crack edges.     

A nonlocal species concentration theory for diffusion and phase changes in electrode particles of lithium ion batteries

A nonlocal species concentration theory for diffusion and phase changes is introduced from a nonlocal free energy density. It can be applied, say, to electrode materials of lithium ion batteries. This theory incorporates two second-order partial differential equations involving second-order spatial derivatives of species concentration and an additional variable called nonlocal species concentration. Nonlocal species concentration theory can be interpreted as an extension of the Cahn–Hilliard theory. In principle, nonlocal effects beyond an infinitesimal neighborhood are taken into account. In this theory, the nonlocal free energy density is split into the penalty energy density and the variance energy density. The thickness of the interface between two phases in phase segregated states of a material is controlled by a normalized penalty energy coefficient and a characteristic interface length scale. We implemented the theory in COMSOL Multiphysics\(^{\circledR }\) for a spherically symmetric boundary value problem of lithium insertion into a \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\) cathode material particle of a lithium ion battery. The two above-mentioned material parameters controlling the interface are determined for \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\), and the interface evolution is studied. Comparison to the Cahn–Hilliard theory shows that nonlocal species concentration theory is superior when simulating problems where the dimensions of the microstructure such as phase boundaries are of the same order of magnitude as the problem size. This is typically the case in nanosized particles of phase-separating electrode materials. For example, the nonlocality of nonlocal species concentration theory turns out to make the interface of the local concentration field thinner than in Cahn–Hilliard theory.     

Deformation measurements by digital image correlation: Implementation of a second-order displacement gradient

This paper outlines the procedure for refining the digital image correlation (DIC) method by implementing a second-order approximation of the displacement gradients. The second-order approximation allows the DIC method to directly measure both the first- and second-order displacement gradients resulting from nonlinear deformation. Thirteen unknown parameters, consisting of the components of displacement, the first- and second-order displacement gradients and the gray-scale value offset, are determined through optimization of a correlation coefficient. The previous DIC method assumes that the local deformation in a subset of pixels is represented by a first-order Taylor series approximation for the displacement gradient terms, so actual deformations consisting of higher order displacement gradients tend to distort the infinitesimal strain measurements. By refining the method to measure both the first- and second-order displacement gradients, more accurate strain measurements can be achieved in large-deformation situations where second-order deformations are also present. In most cases, the new refinements allow the DIC method to maintain an accuracy of ±0.0002 for the first-order displacement gradients and to reach ±0.0002 per pixel for the second-order displacement gradients.