均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

Using the logarithmic strain measure for solving torsion problems

The problems of free and constrained torsion of a rod of solid circular cross-section are solved numerically using a tensor linear constitutive relation written in terms of the energy compatible Cauchy stress and Hencky logarithmic strain tensors. The only function used to determine the properties of the isotropic incompressible material of the rod is a power-law function that approximates the shear diagram and corresponds to an elastoplastic material with power-law hardening. The solution obtained shows that, despite the tensorial linearity of the state law, the use of the logarithmic strain measure allows one to describe qualitatively the effect of significant elongation of the rod in free torsion (the Poynting effect) as well as the arising normal longitudinal, radial, and circumferential stresses, whose values are commensurable, at large deformations, with the maximum tangential stresses in the cross-section. Computational dependences of the torsional moment on the angle of twist in free and constrained torsion are obtained. These dependences are found to be significantly different from each other; the limitmoment and the correspondingmaximum angle of twist for free torsion are found to be considerably lower than those for constrained torsion. It follows that the shear strength, which is traditionally calculated from the maximum torsional moment, becomes indeterminate. For constrained torsion, the dependence of the longitudinal compressive force on the angle of twist is obtained.     

Torsional waves of finite amplitude in elastic rod

We propose mathematical models generalizing the Coulomb and Vlasov equations of torsional vibrations of rods by taking the geometric nonlinearity into account. In the general case, the nonlinearity is taken into account both in the system of displacements (because the displacement vector in the case of rod torsion can be finite even for small strains) and in the relations between displacements and strains. We analyze nonlinear torsional stationary waves and find the effect of splitting of soliton-like unipolar waves in countercollisions. We also show that, in several cases, the existence of nonlinearities can also induce dispersion and that nonlinear stationary waves can also exist in the absence of dispersion in the linear medium.     

异质双材料粘接梁的界面端部应力及端部龟裂破坏的实验应力分析

本文采用了高灵敏度的云纹干涉法对异质双材料粘接梁在弯曲载荷作用下的位移进行了测量，用局部杂交法对界面端部区域的应变和应力进行了计算。通过对该区域内的实验应力分析发现：拉应力σｘ是影响结构强度的关键因素。本文还对在基体材料表面近角点区域可能出现的龟裂破坏的原因进行了分析。     

A study of elasticity and plasticity equations under arbitrary displacements and strains

Equations of geometrically nonlinear theory of elasticity with finite displacements and strains are analyzed. The equations are composed using three versions of physical relations and applied to solve the problem of tension-compression of a straight bar. It is shown that the use of the classical relations between the components of the stress tensor and the Cauchy-Green strain tensor in the problem of compression of the bar results in the appearance of “spurious” static loss of stability such that the bar axis remains straight if the stresses are referred to unit areas before the deformation (conditional stresses). However, in the problem of tension, the classical relations do not permit one to describe the phenomenon of static instability (neck formation as the plastic instability occurs). These drawbacks disappear if one uses the third version of the physical equations, composed as relations between the true stresses referred to unit areas of the deformed faces on which they act and the true elongations and shears. The relations of the third version are most correct; they permit one to pass to self-consistent equations of elasticity and plasticity under small strains and finite displacements, and they should be recommended for practical use. As an example, such relations are composed for the flow theory.     

A crack on the interface between a linear elastic medium and a stress-state dependent physically nonlinear medium

A crack on the interface between a linear elastic medium and a stress-state dependent physically nonlinear medium is studied. A numerical method to solve such problems is proposed. Some asymptotic distributions of stresses, strains, and displacements near the crack tip are obtained under the assumption that the forces and displacements are continuous on the interface.     

Free-edge stresses and progressive cracking in surface coatings of circular torsion bars

This paper considers the explicit solutions of free-edge stresses near circumferential cracks in surface coatings of circular torsion bars and their application in determining the progressive cracking density in the coating layers. The problem was formulated within the framework of linear elastic fracture mechanics (LEFM). The free-edge stresses near crack tip and the shear stresses in the cross-section of the torsion bar were approached in explicit forms based on the variational principle of complementary strain energy. Criterion for progressive cracking in the coating layer was established in sense of strain energy conservation, and the crack density is thereby estimated. Effects of external torque, aspect ratio, and elastic properties on the density of progressive cracking were examined numerically. The present study shows that, in the sense of inducing a given crack density, compliant coating layer with lower modulus has much higher critical torque than that of a stiffer one with the same geometries and substrate material, i.e., compliant coating layer has greater cracking tolerance. Meanwhile, the study also indicates that thicker surface coating layer is more pliant to cracking than the thinner ones. The present model can be used for analyzing the damage mechanism and cracking tolerance of surface coatings of torsion shafts and for data reduction of torsional fracture test of brittle surface coatings, etc.     

Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications

This two-part contribution presents a beam theory (BT) with a non-uniform warping (NUW) including the effects of torsion, and shear forces and valid for any homogeneous cross-section made of isotropic elastic material. In part I, the governing equations of the NUW-BT has been established and simplified-NUW-BT versions has been deduced, wherein the number of degrees of freedom is reduced. In this part II, these theories are used to analyze, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not), the elastic behavior of cantilever beams subjected to torsion or shear-bending. For bi-symmetrical-CS, torsion and shear-bending are analyzed separately: analytical and numerical results are given for the distributions along the beam axis of the cross-sectional displacements and stresses, for the NUW-BT and its simplified versions. Numerical results are also given for the three-dimensional stress distributions close to the embedded section: the stress predictions of the NUW-BT are compared to those obtained by three-dimensional finite elements computations. It can be drawn from all these results indications that can help to decide when the simplified theories may be applied, and hence when the warping parameters may be reduced. As specified in NUW-BT, torsion and bending are coupled for non-symmetrical-CS, even if the bending moments refer to the centroid while the torsional moment refers to the shear center. To illustrate this coupling effect, the particular example of the channel-CS presented in Kim and Kim [Kim, N.-I., Kim, M.-Y., 2005. Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-Walled Structures 43, 701–734.] is analyzed and the results are compared.     

Elastoplastic state of flexible conical shells with a circular hole under axial tension

The elastoplastic state of thin conical shells with a circular hole is analyzed assuming finite deflections. The distributions of stresses, strains, and displacements along the hole boundary and in the zone of their concentration are studied. The stress–strain state of shells around the hole under axial tension is analyzed taking into account two nonlinear factors. The numerical results are presented as plots and tables     

Numerical and experimental study of elastoplastic tension-torsion processes in axisymmetric bodies under large deformations

We consider a numerical solution technique for generalized axisymmetric problems with torsion for elastoplastic bodies of revolution of arbitrary shape under large strains, as well as simple or complex loading, and the conditions of inhomogeneous stress-strain state. The processes of elastoplastic deformation, strain localization, and fracture of solid axisymmetric steel samples of variable thickness are studied experimentally and numerically for the cases of proportional and nonproportional kinematic torsional and/or tensile loading until failure. The mutual influence of torsion and tension on the deformation and failure under large strains is estimated.     

On the existence of stress concentrations in loaded bodies made of polycrystalline materials

We study the character of stress distributions near the corner point of the interface between the two joined crystals. The interface forms a dihedral angle. The joined crystals have a cubic symmetry and consist of the same material. They have a single common principal direction of elasticity, which is parallel to the edge of the dihedral angle. The other principal directions do not coincide and are oriented arbitrarily.In the framework of elasticity, we consider problems of out-of-plane and plane strain of the twocrystal. We show that, in the case of longitudinal shear in the direction of the common principal axis of elasticity, there is no stress concentration near the corner point of the interface between the two joined crystals.For the case of plane strain in which all displacements and strains occur only in the planes perpendicular to the common principal direction, we use separation of variables to construct the characteristic equation that determines the stress concentration degree and find the roots of this equation, which determine the order of singularities of the stresses.     

Variation in the length of perfectly elastic rods under torsion

On the basis of elastic constitutive relations that reflect geometrically nonlinear second-order effects, we refine the theory of torsion of rectilinear rods of an arbitrary transverse cross-section. In particular, we obtain a universal formula, independent of the material properties, that determines the longitudinal strain arising as the rod undergoes free torsion. According to this formula, the length of a rod made of an isotropic perfectly elastic material can, in contrast to the traditional concepts, either increase or decrease as the rod undergoes torsion. Moreover, the variation in the length depends only on the geometry of the transverse cross-section.     

Antiplane strain in a nonlinearly elastic incompressible body

The stress-strain state of an incompressible cylindrical elastic body with antiplane strain under the action of potential forces and surface loading constant along the body is considered in a nonlinear formulation in actual variables. The stresses are expressed via the pressure and independent strains, the pressure is expressed via the force and elastic potentials, and nonlinear boundary-value problems are posed for strains (and displacements). Various methods for solving these problems are developed. For the nonlinear equations obtained, some analytical solutions containing free parameters are given, which can be used as a basis for solving particular problems. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 93–101, November–December, 2006.     

A solution of torsional problem by energy method in case of anisotropic cross-section

We proposed an original method to investigate the problem of torsion of anisotropic cross-section. We implemented an energy method to calculate the stress function represented by infinite series of trigonometric functions adapted to rectangular cross-section. After validation, we implemented a parametric sensitivity study to investigate the influence of the cross-section aspect ratio and the anisotropy level on the stress function, the strain energy density and the torsion stiffness. The process showed a fast convergence with a very good accuracy. The model showed a potential interest for the experimental identification of anisotropic material properties.     

On Compression of a Heavy Compressible Layer of an Elastoplastic or Elastoviscoplastic Medium

The problem of deformation of a horizontal plane layer of a compressible material is solved in the framework of the theory of small strains. The upper boundary of the layer is under the action of shear and compressing loads, and the no-slip condition is satisfied on the lower boundary of the layer. The loads increase in absolute value with time, then become constant, and then decrease to zero.Various plasticity conditions are consideredwith regard to the material compressibility, namely, the Coulomb–Mohr plasticity condition, the von Mises–Schleicher plasticity condition, and the same conditions with the viscous properties of the material taken into account. To solve the system of partial differential equations for the components of irreversible strains, a finite-difference scheme is developed for a spatial domain increasing with time. The laws of motion of elastoplastic boundaries are presented, the stresses, strains, rates of strain, and displacements are calculated, and the residual stresses and strains are found.     

Solution of generalized coordinate for warping for naturally curved and twisted beams

A theoretical method for static analysis of naturally curved and twisted beams under complicated loads was presented, with special attention devoted to the solving process of governing equations which take into account the effects of torsion-related warping as well as transverse shear deformations. These governing equations, in special cases, can be readily solved and yield the solutions to the problem. The solutions can be used for the analysis of the beams, including the calculation of various internal forces, stresses, strains and displacements. The present theory will be used to investigate the stresses and displacements of a plane curved beam subjected to the action of horizontal and vertical distributed loads. The numerical results obtained by the present theory are found to be in very good agreement with the results of the FEM results. Besides, the present theory is not limited to the beams with a double symmetric cross-section, it can also be extended to those with arbitrary cross-sectional shape.     

横观各向同性圆柱体端部问题的三维弹性理论解

从位移的通解出发，用分离变量法得到横观各向同性圆柱体的位移和应力的特征函数展开式，并把位移势函数的解用付里叶积分的形式表示。利用留数运算，该积分解可以转换成类似于特征函数的展开式。通过混合端部边界问题，得到与特征函数解成双正交关系的另一组函数。利用这种双正交关系，可以处理不同的端部边界问题。