Determination of the axisymmetric geometrically nonlinear thermoviscoelastoplastic stress-strain state of shells of revolution

A method is developed for determining the axisymmetric thermoviscoelastoplastic stress-strain state of shells subjected to bending and torsion. The problem is solved in a geometrically nonlinear formulation with allowance for transverse shear. The geometrically nonlinear deformation of an annular plate, the thermoviscoelastoplastic deformation of a cylindrical shell, and the limiting state of a corrugated shell are studied as examples. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 40–48, December, 1999.     

Fatigue strength of metals and composites under repeated high-cycle torsion

The limiting stresses are determined and constant fatigue life diagrams for high-cycle torsion with repeated stress cycle are plotted using the limiting-state models obtained based on the hypothesis of a unified constant-life diagram, which is invariant to the number of cycles to failure. The unified constant-life diagram is given by a transcendental power function whose exponent is an additional parameter characterizing the sensitivity of the material to the asymmetry of the stress cycle. The calculated results and experimental data for carbon and alloyed steels and composite materials are in good agreement __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 19–28, February 2008.     

Plane strain bending of strain-stiffening rubber-like rectangular beams

This paper is concerned with investigation of the effects of strain-stiffening for the classical problem of plane strain bending by an end moment of a rectangular beam composed of an incompressible isotropic nonlinearly elastic material. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress–stretch response, the stresses and resultant moments are obtained explicitly. While such results are well known for classical constitutive models such as the Mooney-Rivlin and neo-Hookean models, our primary focus is on materials that undergo severe strain-stiffening in the stress–stretch response. In particular, we consider in detail two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of bending that beams composed of such materials can sustain is limited by the constraint. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Fatigue strength of metallic and composite materials under repeated high-cycle bending

The problem of fatigue strength analysis of metallic and composite materials subjected to combined static and cyclic bending is resolved. The analysis is based on limit-state models, which allow describing limiting-stress curves of all known shapes, including convex, rectilinear, S-shaped, and concave. The fatigue strength of the following materials is evaluated: carbon and alloyed steels, aluminum alloys, unidirectional organic plastics, laminated plastics, and glassfiber-reinforced plastics. The calculated and experimental data are in satisfactory agreement __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 26–36, May 2006.     

Crack path kinking under generalized stress state

Results of experimental investigations of three-point asymmetric bending applied to beams made of materials with substantially different structural, strength, and strain characteristics are presented. The effect of different types of fracture on the formation of crack kinks under a generalized stress state is revealed. Strength curves of the Coulomb-Mohr type are obtained for Plexiglas and fine-grained concrete, which are characterized by the quasi-brittle and brittle types of fracture, respectively. Dependences of the kink angles of the crack paths on the stress state in the case of three-point bending are described for these materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 205–213, May–June, 2009.     

Stress redistribution during bending fatigue

One of the effects of the application of cyclic stress, in most metals, is a change in stiffness. Some materials become harder while others initially soften as increasing numbers of cyclic stress are imposed. The effect of this changing stiffness on the distribution of stress during bending fatigue is examined. In general, the maximum stress, strain and bending moment are all observed to vary throughout the fatigue life. The manner of variation is dependent on the type of material as well as the loading conditions imposed. Experimental results are compared with calculated values based on cyclic stress-strain curves obtained in axial stress tests.     

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.     

A phenomenological constitutive law for the nonlinear viscoelastic behaviour of epoxy resins in the glassy state

We propose a constitutive model to describe the nonlinear viscoelasticity of epoxy resins in the glassy state. Experimental tests have shown that nonlinear viscoelasticity rules the cyclic behaviour of epoxy resins before the material strength is reached. The proposed model allows the simulation of this cyclic behaviour and, in particular, of the flex which characterises the stress–strain curve upon unloading. The consistent integration algorithm, based on the Central Difference Scheme, of the proposed constitutive law is given. As an example, the model is successfully used to numerically homogenize the cyclic behaviour of hollow sphere-filled epoxy resins (i.e., syntactic foams) by means of unit cell models.     

A state space formalism for anisotropic elasticity. Part I: Rectilinear anisotropy

A state space formalism for anisotropic elasticity including the thermal effect is developed. A salient feature of the formalism is that it bridges the compliance-based and stiffness-based formalisms in a natural way. The displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation, yet it is simple and clear. This is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three-dimensional equations of anisotropic elasticity into a compact state equation and an output equation. The homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of Stroh. Extension, twisting, bending, temperature change and body forces are accounted for through the particular solution. Based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner.     

An experimental study of inhomogeneous cyclic plastic deformation of 1045 steel under multiaxial cyclic loading

An experimental study was conducted on the inhomogeneous cyclic plastic deformation of 1045 steel under multiaxial cyclic loading. Thin-walled tubular specimens were used and small strain gages were bonded on the specimen surface to characterize the local deformation. The controlled loading paths included cyclic tension–compression, cyclic torsion, proportional axial-torsion, 90°-out-of-phase axial-torsion, and fully reversed torsion with a constant axial stress. The maximum stress in each experiment was lower than the lower yield stress of the material. It was found that the cyclic plastic deformation within the gage section of the specimen under multiaxial stress state followed the three-stage process that was observed from uniaxial loading, namely, incubation, propagation, and saturation. The plastic deformation was significantly inhomogeneous during the propagation stage, and the inhomogeneity continued through the saturation stage. The duration of each stage and the saturated strains were dependent on the cyclic stress amplitude and the loading path. Multiaxial stress state reduced the incubation stage. With identical equivalent stress magnitude, the nonproportional loading path resulted in the shortest incubation and propagation stages, and the saturated equivalent plastic strain magnitude was the smallest. Although the deformation over the gage section was inhomogeneous, the plastic deformation in a given local area was found to be practically isotropic.     

Evaluation of cyclic plasticity models in ratcheting simulation of straight pipes under cyclic bending and steady internal pressure

This paper evaluates seven cyclic plasticity models for structural ratcheting response simulations. The models evaluated are bilinear (Prager), multilinear (Besseling), Chaboche, Ohno–Wang, Abdel Karim–Ohno, modified Chaboche (Bari and Hassan) and modified Ohno–Wang (Chen and Jiao). The first three models are already available in the ANSYS finite element package, whereas the last four were implemented into ANSYS for this study. Experimental responses of straight steel pipes under cyclic bending with symmetric end rotation history and steady internal pressure were recorded for the model evaluation study. It is demonstrated that when the model parameters are determined from the material response data, none of the models evaluated perform satisfactorily in simulating the straight pipe diameter change and circumferential strain ratcheting responses. A detailed parameter sensitivity study with the modified Chaboche model was conducted to identify the parameters that influence the ratcheting simulations and to determine the ranges of the parameter values over which a genetic algorithm can search for refinement of these values. The refined parameter values improved the simulations of straight pipe ratcheting responses, but the simulations still are not acceptable. Further, improvement in cyclic plasticity modeling and incorporation of structural features, like residual stresses and anisotropy of materials in the analysis will be essential for advancement of low-cycle fatigue response simulations of structures.     

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.The torsional storage and loss moduli are determined under quasistatic conditions as functions of frequency and axial static pre-strain. The time-strain separability is found to be a resonable approximation in a relatively limited range of static prestrain magnitudes and frequencies considered for the natural gum rubbers investigated. The experimental methodology is discussed in some detail.     

Analytical solution for bending beam subject to lateral force with different modulus

A bending beam, subjected to state of plane stress, was chosen to investigate. The determination of the neutral surface of the structure was made, and the calculating formulas of neutral axis, normal stress, shear stress and displacement were derived. It is concluded that, for the elastic bending beam with different tension-compression modulus in the condition of complex stress, the position of the neutral axis is not related with the shear stress, and the analytical solution can be derived by normal stress used as a criterion,improving the multiple cyclic method which determines the position of neutral point by the principal stress. Meanwhile, a comparison is made between the results of the analytical solution and those calculated from the classic mechanics theory, assuming the tension modulus is equal to the compression modulus, and those from the finite element method (FEM) numerical solution. The comparison shows that the analytical solution considers well the effects caused by the condition of different tension and compression modulus. Finally, a calculation correction of the structure with different modulus is proposed to optimize the structure.     

陶瓷材料动态抗弯性能测试

用改装的Hopkinson压杆试验装置测试了陶瓷材料3点弯曲动态力学性能;定义了量纲一挠度和挠度变化率,给出了几种陶瓷材料在不同挠度变化率下的挠度-最大拉应力曲线,从而给出其抗弯强度。测试结果说明,陶瓷材料的动态抗弯强度具有挠度变化率效应。分析了3点弯曲动态测试的有效性和动态损伤,分析表明,动态损伤因子临界值具有挠度变化率效应。     

Probabilistic character of the microcrack growth in brittle layered materials

Characteristics of microcrack initiation, multiplication and saturation in layered materials are discussed. A probabilistic-analytical method, the ‘characteristic curve method (CCM)’ is developed to correlate the initial defects and the microcrack evolution under static and cyclic loadings. The ‘equivalent applied loading’ and the ‘equivalent crack density’ concepts are introduced to describe different microcrack multiplication features in different layered materials. Microcrack multiplication processes in many layered materials with brittle matrices subjected to static and cyclic loadings can be easily predicted.     

Fluid pressure response in poroelastic materials subjected to cyclic loading

When cyclic loading is applied to poroelastic materials, a transient stage of interstitial fluid pressure occurs, preceding a steady state. In each stage, the fluid pressure exhibits a characteristic mechanical behavior. In this study, an analytical solution for fluid pressure in two-dimensional poroelastic materials, which is assumed to be isotropic, under cyclic axial and bending loading is presented, based on poroelasticity. The obtained analytical solution contains transient and steady-state responses. Both of these depend on three dimensionless parameters: the dimensionless stress coefficient; the dimensionless frequency; and, the axial-bending loading ratio. We focus particularly on the transient behavior of interstitial fluid pressure with changes in the dimensionless frequency and the axial-bending loading ratio. The transient properties, such as half-value period and contribution factor, depend largely on the dimensionless frequency and have peak values when its value is about 10. This suggests that, under these conditions, the transient response can significantly affect the mechanical behavior of poroelastic materials.     

On the effects of characteristic lengths in bending and torsion on Mode III crack in couple stress elasticity

The problem of a stationary semi-infinite crack in an elastic solid with microstructures subject to remote classical K_III field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales. The stress and displacement fields turn out to be strongly influenced by the ratio between the characteristic lengths. Moreover, the symmetric stress field turns out to be finite at the crack tip, whereas the skew-symmetric stress field displays a strong singularity. Ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. The asymptotic fields dominate within this zone, which however has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. In this limiting case the full-field solution recovers the classical K_III field with square-root stress singularity. Outside the zone where the total shear stress is negative, the full-field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude can be adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion. Moreover, the occurrence of a sharp crack profile denotes that the crack becomes stiffer with respect to the classical elastic response, thus revealing that the presence of microstructures may shield the crack tip from fracture.     

混凝土分段曲线损伤模型

在对比分析国内外现有的混凝土受拉损伤模型基础上，结合素混凝土受拉应力应变曲线的试验结果，提出了一个新的混凝土分段曲线损伤模型，并利用应力连续条件确定了损伤模型中各系数的值。运用该模型到力学分析中，推导了矩形截面的混凝土梁在纯弯矩作用下，在损伤较小和损伤较大时的力学方程，算出了梁的最大承载能力，并与材料力学计算的梁承载能力进行了对比。详细分析了损伤区域及损伤应力在横截面上的变化过程，确定了极限状态下含损伤混凝土梁的应力分布和相应的承载能力，画出了损伤状态梁截面的应力分布图。