Asymptotic analysis of the behavior of an elastic bar under aperiodic intensive loading

It is established that when aperiodic loads of large intensity act on an elastic bar, the higher modes of stability loss have the highest rates of growth of deflections. A method is indicated for determining the numbers of these modes, when the effect of the longitudinal vibrations on the transverse vibrations is taken into account and when it is not taken into account. A comparison of the results obtained with results of other authors [1–7] is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 160–170, May–June, 1972.     

基于三维Hopkinson杆的混凝土动态力学性能研究

混凝土、岩石类材料在复杂应力状态下的动态力学性能研究一直备受关注,但鉴于动态实验的复杂性,对真三轴应力状态下材料的动态加载一直未曾实现。本文中研制了一套真三轴静载作用下混凝土、岩石类材料的“三维Hopkinson杆”动态力学实验系统,为冲击载荷作用下材料动态各向异性特性的研究提供了一种有效的实验测试技术。该系统采用液压伺服控制对立方体试件施加三向独立的0~100 MPa真三轴静载,再利用分离式Hopkinson压杆对试件施加冲击动载,具体研究了C30混凝土材料在不同真三轴静载条件下的动态压缩性能,得到了不同条件下X、Y、Z方向上的动态应力应变关系。     

Linear stability analysis for a thermoviscoplastic material under cyclic axial loading

Some mechanical properties exhibit a very strong dependence upon temperature; these evolutions can be properly analyzed by the steady state response in cyclic loading. To relate experimental conditions to thermomechanical characteristics, the existence and the stability of steady state solutions are studied for cylinders submitted to cyclic compression. The material, considered as rigid viscoplastic, is modeled by a non-Newtonian temperature dependent viscous law. Closed form solutions are obtained in the framework of a large deformation theory by neglecting thermal expansion and inertia effects. Steady state regime is analyzed. The stress versus strain rate response and the temperature distribution are established as functions of the geometry of the cylinder, the loading characteristics and the material parameters. The stability of steady state solutions is analyzed with use of a linear perturbation scheme.Received: 4 July 2002, Accepted: 5 August 2004, Published online: 24 February 2005PACS: 46.15.Ff, 83.60.St Correspondence to: F. Dinzart     

Waves in a viscoelastic bar surrounded by soils under smooth loading

The results of numerical solution of the nonstationary wave problem of longitudinal wave propagation in the bar-soil system are given. The barmaterial and the soil are assumed to be linearly viscoelastic (standard linear body). A nonlinear interaction condition is satisfied on the bar-soil contact boundary. The laws of propagation of longitudinal waves and variations in the longitudinal stresses in the bar are determined depending on the mechanical characteristics of the soil and the bar and on the interaction parameters.     

The Saint-Venant problem of plane bar under an axial force

I.PrefaceAsweknow,theSaint-Venanttheoryisawell-knownanduniversallyacceptedtheory.Butasoneofthegeneraltheoriesofelasticmechanics,itisnotprovedperfectely,althoughitistestedbyexperiments.Infact,onlyforthesimplestproblemofelasticity,itisnoteasytofindanalytics…     

Stress response of a rubber block with frictional contact under axial loading

Rubber blocks (or springs) are structural components widely used in many applications. Design characteristics of a rubber block under axial loading are an apparent modulus (or compression modulus) and normal and shear stresses on contact surfaces. These are affected by a contact condition of the rubber block in contact with two rigid plates and the shape of the block. The problem of a rubber block bonded to two rigid plates has been solved using various approaches. In contrast, for a rubber block whose one surface is bonded to a rigid plate and the other surface in contact with a frictional surface, there is little work in spite of practical applications. For this contact condition, approximate solutions for rectangular blocks in plane strain and for axi-symmetric discs are derived under the assumption of Coulomb frictional contact. The problem is treated as an extension of the problem of an incompressible rubber block bonded to two rigid plates with one of the plates having a frictional interaction with the rubber block. In the linear range of deformation, finite element analysis and experimental results for rubber blocks with shape factors ranging from 1 to 6 are compared for the validation of analytic results. It is found that friction coefficients play important roles in the design characteristics of the rubber block.     

Free vibrations of a viscoelastic noncircular cylindrical shell under homogeneous axial loading

Free low-frequency vibrations of a noncircular cylindrical shell subject to an axial static load are investigated. The case where the vibrations are localized near a certain generatrix, called the weakest, is considered. The complex WKB-method is used to determine the natural modes of vibrations that damp with time and drop down far from the weakest generatrix. An example on the vibration of a viscoelastic shell is considered with the relaxation rate function given and in the presence of a static axial force. State Polytechnic Academy of Belarus, Minsk. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 68–74, November, 1999.     

Sudden tensile loading of a rubberlike bar

In uniaxial tension, the stress–strain curve for rubber changes curvature from concave to convex as the strain increases. For sudden tensile loading of a bar, a one-dimensional model that reflects this behavior leads to an under-determined problem reminiscent of that arising in materials capable of undergoing phase transitions. In the latter setting, adding the kinetic relation underlying the phase change to the conventional statement of the problem removes the indeterminacy; the same is true when such a relation is used in a formal way in the problem for rubber. This presents a physical question: What is the evolutionary process at the microscale whose kinetics are needed in the dynamics of rubber?     

Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution

This work investigates the nonlinear longitudinal forced vibrations of a bar with hysteretic dynamics typical of rocks and man-made geomaterials. The material properties are described by a constitutive relation that includes energy dissipation effects consistently with hysteretic dynamics. The harmonic balance method and a perturbation technique that uses the material's modulus defect as perturbation parameter are employed to solve the equation of motion. The spatial dependence of the modulus defect, which is disregarded in earlier solutions of this problem, is duly accounted for. According to this model, the resonance frequency shift is proportional to the amplitude of the excitation, while the inverse of the quality factor increases as the square root of the latter. Further, the spatial variation of the modulus defect is shown to affect the modulation of the fundamental component along the bar. The nonlinear spectral components are of odd order, with amplitude proportional to the maximum value of the modulus defect and to the square of the excitation's amplitude, and decreases non-monotonically with increasing harmonic order. The motivation for this work is twofold. Analytical models may improve our understanding of the dynamics of hysteretic materials in general, and of the mutual interaction of material defects in particular. Secondly, this work establishes a bench-mark result on a mathematically simple hysteretic system against which alternative mathematical approaches, possibly concerning material with complex constitutive relations, could be tested.     

Fracture behavior of a cohesive soil under crushing stress loading

In this study the concepts of engineering fracture mechanics are used to investigate the behavior of cohesive soil crumbling. Prior to the analysis, the physical structure was investigated using the electron scanning method. Cohesive soil-crushing tests were conducted to evaluate the fracture parameters. The results of both phenomenal and experimental approaches indicated that the concept of engineering fracture mechanics can be applied to evaluate soil fracture leading to crumbling.     

Pressing a rigid plastic surface by a wedge-shaped stamp under the Coulomb-Mohr yield condition

The problem of treating a surface with a wedge-shaped stamp is considered using the model of an ideal rigid plastic body. The strain fields in the vicinity of singularities of the displacement velocity field (on the discontinuity lines of the displacement velocities and at the center of the fan of characteristics) are investigated taking into account irreversible compressibility.     

Exact solutions for buckling of non-uniform columns under axial concentrated and distributed loading

In this paper, the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading is studied. The expression for describing the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a non-uniform column with arbitrary distribution of flexural stiffness or axial forces is reduced to a second-order differential equation without the first-order derivative by means of functional transformations. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 12 cases, several of which are important in engineering practice. The exact solutions that represent a class of exact functional solutions for the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading are obtained. In order to illustrate the proposed method, a numerical example is given in the last part of this paper.     

A multi-dimensional composite model for plastic continua under polyaxial loading condition

By replacing a medium with reinforcing components oriented and distributed uniformly in a multi-dimensional space, a constitutive model is constructed. The components are extended/compressed compatibly with the strain and the resultant of load exerted on them to balance the stress. Their load-elongation relation can be determined from a conventional material test. Each component undergoes different elongation history depending on its own orientation during deformation, so that the model can simulate elasto-plastic behavior of materials under polyaxial loading conditions. The incremental constitutive matrix has been derived for application in numerical analysis and a yield criterion is also introduced. A few subsequent yield surfaces have been predicted and compared with experiments.     

Shearing properties of sand under a repeated loading representing the ground pressure distribution of a tracked vehicle

This paper deals with experiments concerning the shearing properties of sand under repeated loading simulating the ground pressure distribution of a tracked vehicle. Shearing tests were carried out in sand by using a direct shear apparatus. It was confirmed that the shear strength of sand under repeated loading decreases with an increase in the number of loadings. This influences the soil thrust exerted by a highspeed tracked vehicle.     

Experimental investigation of the yield condition in relation to dynamic loading of compressible soils

The yield function has been investigated in relation to the dynamic loading of various compressible soils (clays, loams, and loesses). The form of the yield function depends importantly on the composition and moisture content of the soils and on the loading conditions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 2, pp. 131–133, March–April, 1970.     

Inelastic behaviour of polycrystalline metals under complex loading condition

In this paper, a simple mechanical model of polycrystal is suggested on the basis of the crystal plasticity. This model incorporates interactions among grains in polycrystal and interactions among slip systems in a crystal grain component. Especially, an equation for a slip strain of slip system is based on a theory of thermo-activation motion of dislocation. In this way, the model can take into account rate (creep) properties and temperature effects as well as an induced plastic anisotropy depending on deformation path, under multi-axial condition.Some computations are done by using the model for typical sets of strain paths and compared with corresponding experimental results of aluminum alloy at 200°C. The computational results reproduce accurately typical features observed in the experiments.     

Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading

The stability of a viscoelastic plate strip, subjected to an axial load with the Kelvin–Voigt fractional order constitutive relationship is studied. Based on the classical plate theory, the structural formulation of the plate is obtained by using the Newton’s second law and the aerodynamic force due to the fluid flow is evaluated by piston theory. The Galerkin method is employed to discretize the equation of motion into a set of ordinary differential equations. To determine the stability margin of plate the obtained set of ordinary differential equations are solved using the Laplace transform method. The effects of variation of the governing parameters such as axial force, retardation time, fractional order and boundary conditions on the stability margin of fractional viscoelastic panel are investigated and finally some conclusions are outlined.     

The propagation of elastic stress waves in a conical shell under axial impulsive loading

The propagation of elastic stress waves in a conical shell subjected to axial impulsive loading is studied in this paper by means of the finite element calculation and model experiments. It is shown that there are two axisymmetrical elastic stress waves propagating with different velocities, i.e., the longitudinal wave and the bending wave. The attenuation of these waves while propagating along the shell surface is discussed. It is found in experiments that the bending wave is also generated when a longitudinal wave reflects from the fixed end of the shell, and both reflected waves will separate during the propagation due to their different velocities. Southwest Institute of Structural Mechanics