Multi-mode parametric excitation of a simply supported plate under time-varying and non-uniform edge loading

In this study, multi-mode parametric excitation of a simply supported plate under time-varying and non-uniform edge loading is modeled and the solution is found. Equations for multi-mode parametric excitation of a simply supported plate are derived using stress distributions within the plate as well as on the edges, considering both the effects of non-uniform edge loading and the non-linearity caused by the large deflection. The multi-mode equations are coupled by first-order linear terms, even in the case of simply supported boundary conditions, due to the non-uniform edge loading. The perturbation solutions of two-mode parametric excitation are examined by the method of multiple scales. For the edge loading, which consists of a uniform term as well as a non-uniform one, equations could be coupled or de-coupled by parametric excitation terms, and the numbers and values of the resonance frequencies of the parametric excitations could also differ, depending on whether the non-uniform term of the edge loading is time-varying or not. In addition to the resonant frequencies of the case when only the uniform term of the edge loading is time-varying, there are additional combination resonances at the vicinity of the sum of two natural frequencies of each mode when the non-uniform term of the non-uniform edge loading is time-varying.     

宽翼缘梁非线性温差自应力级数解

为了客观反映宽翼缘梁在非线性温差作用下的自应力分布特点,克服按梁理论计算温度自应力时的不足,本文从弹性力学的分析方法入手,按平面应力状态分析翼缘板和腹板的温度应力。通过翼缘板与腹板连接处的变形协调条件及平衡条件建立补充方程,求解艾瑞应力函数中的积分常数,推导了翼缘板与腹板的纵、横向正应力、剪应力及位移分量解析式。导出的翼缘板纵向正应力公式可自动考虑宽翼缘梁沿横截面宽度自应力不均匀分布的特点。对一宽翼缘T梁的计算表明,当翼缘板相对于腹板发生温差变化时,沿其宽度的纵向温度自应力分布很不均匀,在翼缘板根部自应力较大而在悬臂端则显著减小。按通常基于梁理论的温度自应力计算方法,无法反映这种应力分布规律。     

Instability analysis of non-homogeneous materials under biaxial loading

In this paper, we formulate the instability problem of non-homogeneous materials under biaxial loading, in the sense that the mechanical parameters, and more specifically the strain hardening or softening and the strain-rate sensitivity, are spatially dependent functions. We first study the time behavior of the process and present the evolution of strain non-uniformities before the critical time, to show that it depends on the interplay between the spatial distribution of mechanical parameters and the initial non-uniformities. We next study the instability modes of a material exhibiting inhomogeneities along one direction by an effective instability analysis [Dudzinski, D., Molinari, A., 1991. Perturbation analysis of thermoviscoplastic instabilities in biaxial loading. International Journal of Solids and Structures 27 (5), 601–628] adapted to the non-homogeneous case. This method is based on a suitable non-uniform reference solution and allows to select the localization modes activated at different deformation levels for different “deformation paths” of the material zones.     

Numerical analysis of an electrode embedded between dissimilar electrostrictive materials

The asymptotic problem of an electrode that is embedded between dissimilar electrostrictive materials and subjected to electric loading is numerically analyzed by using the finite element method. Electrostatic analysis for the asymptotic problem is conducted under the small-scale saturation condition on the basis of the mathematical equivalence between anti-plane shearing and electrostatics. The distribution of the electric displacement fields is obtained. It is shown that the shapes of the saturation zones are affected by the ratios of the permittivities and the saturated electric displacements between the dissimilar electrostrictive materials. Stress fields that are generated for matching incompatible strains, which are induced by non-uniform electric displacement fields, are numerically calculated for various combinations of the material properties. Also, stress intensity factors for arbitrary small cracks that are initiated from the edge of an electrode are evaluated. Behaviors of cracking that may take place at the edge of an electrode are discussed.     

Analysis of dynamic stress intensity factors of three-point bend specimen containing crack

A new formula is obtained to calculate dynamic stress intensity factors of the three-point bending specimen containing a single edge crack in this study. Firstly, the weight function for three-point bending specimen containing a single edge crack is derived from a general weight function form and two reference stress intensity factors, the coefficients of the weight function are given. Secondly, the history and distribution of dynamic stresses in uncracked three-point bending specimen are derived based on the vibration theory. Finally, the dynamic stress intensity factors equations for three-pointing specimen with a single edge crack subjected to impact loadings are obtained by the weight function method. The obtained formula is verified by the comparison with the numerical results of the finite element method (FEM). Good agreements have been achieved. The law of dynamic stress intensity factors of the three-point bending specimen under impact loadings varing with crack depths and loading rates is studied.     

Analysis of dynamic stress intensity factors of three-point bend specimen containing crack

A new formula is obtained to calculate dynamic stress intensity factors of the three-point bending specimen containing a single edge crack in this study. Firstly, the weight function for three-point bending specimen containing a single edge crack is derived from a general weight function form and two reference stress intensity factors, the coefficients of the weight function are given. Secondly, the history and distribution of dynamic stresses in uncracked three-point bending specimen are derived based on the vibration theory. Finally, the dynamic stress intensity factors equations for three-pointing specimen with a single edge crack subjected to impact loadings are obtained by the weight function method. The obtained formula is verified by the comparison with the numerical results of the finite element method (FEM). Good agreements have been achieved. The law of dynamic stress intensity factors of the three-point bending specimen under impact loadings varing with crack depths and loading rates is studied.     

单箱三室箱梁剪力滞效应的试验研究

为开展单箱三室箱梁剪力滞效应的试验研究,制作了有机玻璃简支箱梁模型,对试验模型进行了分级加载。对该试验箱梁进行集中加载,分别作用于跨中截面四腹板上方、两对称边腹板上方和两对称中腹板上方。采用DH3816应变采集仪测得跨中及四分之一跨截面各关键点应变值,用百分表测得箱梁各关键截面挠度值。测量得到的截面应力分布规律验证了箱梁截面剪力滞效应的存在。对该有机玻璃简支箱梁,利用有限元方法和模型试验方法,研究了3种集中力工况下截面的剪力滞分布规律。结果表明,集中力作用下单箱三室箱梁各翼板间存在明显的剪力滞效应,且荷载的横向作用位置对箱梁截面剪力滞效应影响较大。     

Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements

Three-dimensional enriched finite elements are used to compute mixed-mode stress intensity factors (SIFs) for three-dimensional cracks in elastic functionally graded materials (FGMs) that are subject to general mixed-mode loading and constraint conditions. The method, which advantageously does not require special mesh configuration/modifications and post-processing of finite element results, is an enhancement of previous developments applied so far on isotropic homogeneous and isotropic interface cracks. The spatial variation of FGM material properties is taken into account at the level of element integration points. To validate the developed method, two- and three-dimensional mixed-mode fracture problems are selected from the literature for comparison. Two-dimensional cases include: inclined central crack in a large FGM medium under uniform tensile strain and stress loadings, a slanted crack in a finite-size FGM plate under exponentially varying tensile stress loading and an edge crack in a finite-size plate under shear traction load. The three-dimensional example models a deflected surface crack in a finite-size FGM plate under uniform tensile stress loading. Comparisons between current results and those from analytical and other numerical methods yield good agreement. Thus, it is concluded that the developed three-dimensional enriched finite elements are capable of accurately computing mixed-mode fracture parameters for cracks in FGMs.     

A Dynamic-induced Direct-shear Model for Dynamic Triggering of Frictional Slip on Simulated Granular Gouges

This study presents a dynamic-induced direct-shear model to investigate the dynamic triggering of frictional slip on simulated granular gouges. An incident P-wave is generated as a shear load and a normal stress is constantly applied on the gouge layer. The shear stress accumulates in the incident stage and the frictional slip occurs in the slip stage without the effect of the reflected wave. The experimental results show a non-uniform shear stress distribution along the gouge layer, which may be induced by a shear load induced torque and by normal stress vibration along the layer. The shear stress at the trailing edge strongly affects the frictional slip along the P-wave loading direction, while the rebound stress at the leading edge propagates along the opposite direction. The frictional slip is triggered when the maximum shear stress at the trailing edge reaches a critical value. The normal stress influences the maximum shear stress at the trailing edge, the maximum slip displacement and the slip velocity. The advantages and the limitations of this model are discussed at the end.     

A solution for an isotropic sector under anti-plane shear loadings

In this study, the problem of an isotropic sector subjected to anti-plane shear loadings is investigated. The loadings were applied to the arc of the sector, and the radial edges of the sector were under traction-free or fixed conditions. Depending on these conditions, three problems, namely, free–free, fixed–free, and fixed–fixed edges were studied. A procedure using the finite Mellin transform combined with the Laplace transform was proposed for solving these problems. Explicit closed form solutions for the displacement and stress fields throughout the sector were obtained. The stress intensity factor (SIF) for each problem was analyzed using the obtained stress fields. It was determined that the SIF disappeared under the special condition of a fixed–fixed edge. Other special cases having anti-symmetric conditions were deduced from the derived solutions, and the results of these verified those cited in the literature as well as those obtained using finite element analysis (FEA).     

A general failure criterion for spot welds under combined loading conditions

The circumferential failure mode of spot welds is investigated under combined loading conditions. Failure mechanisms of spot welds under different loading conditions are first examined by the experimental observations and a plane stress finite element analysis. An approximate limit load analysis for spot welds is then conducted to understand the failure loads of spot welds under combinations of resultant forces and resultant moments with consideration of the global equilibrium conditions only. The approximate limit load solution for circumferential failure is expressed in terms of sheet thickness, nugget diameter and combinations of loads. Failure contours are generated for spot welds under opening and shear loading conditions. The results indicate that failure contours become smaller when the ratio of the sheet thickness to the nugget diameter increases. Based on the approximate limit load solution, a general quadratic failure criterion for spot welds under combined three resultant forces and three resultant moments is proposed with correction factors determined by fitting to the experimental results of spot welds under combined loading conditions. The failure criterion can be used to characterize the failure loads of spot welds with consideration of the effects of sheet thickness, nugget diameter and combinations of loads. Experimental spot weld failure loads under combined opening and shear loading conditions and those under combined shear and twisting loading conditions are shown to be characterized well by the proposed failure criterion. Finally, a simplified general failure criterion for spot welds under three resultant forces and three resultant moments is proposed by neglecting the coupling terms of the resultant forces and moments for convenient use of the failure criterion for engineering applications.     

压电圆柱形夹杂对邻近裂纹三维应力场的影响

研究了压电复合材料薄板中压电圆柱形夹杂与邻近宏观钝裂纹间的相互作用。重点分析了外加电场，裂尖与压电圆柱形夹杂间韧带长度对裂尖三维应力场的影响。计算结果表明：在不同的外加电场作用下，压电体不仅能改变裂尖张开应力的大小，还能改变其分布。所得结果对进一步探讨线弹性介质中裂纹的启裂控制有参考价值。     

Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets

Wrinkles are commonly observed in stretched thin sheets and membranes. This paper presents a numerical study on stretch-induced wrinkling of hyperelastic thin sheets based on nonlinear finite element analyses. The model problem is set up for uniaxial stretching of a rectangular sheet with two clamped ends and two free edges. A two-dimensional stress analysis is performed first under the plane-stress condition to determine stretch-induced stress distribution patterns in the elastic sheets, assuming no wrinkles. As a prerequisite for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio of the sheet and the applied tensile strain in the longitudinal direction. A phase diagram is constructed with four different distribution patterns of the stretch-induced compressive stresses, spanning a wide range of aspect ratio and tensile strain. Next, an eigenvalue analysis is performed to find the potential buckling modes of the elastic sheet under the prescribed boundary conditions. Finally, a nonlinear post-buckling analysis is performed to show evolution of stretch-induced wrinkles. In addition to the aspect ratio and tensile strain, it is found that the critical condition for wrinkling and the post-buckling behavior both depend sensitively on the sheet thickness. In general, wrinkles form only when both the magnitude and the distribution area of the compressive stresses are sufficiently large. The wrinkle wavelength decreases with increasing strain, in good agreement with the prediction by a scaling analysis. However, as the tensile strain increases, the wrinkle amplitude first increases and then decreases, eventually flattened beyond a moderately large critical strain, in contrast to the scaling analysis.     

Analytic solutions to crack tip plastic zone under various loading conditions

Based on stress field equations and Hill yield criterion, the crack tip plastic zone is determined for orthotropic materials and isotropic materials under small-scale yielding condition. An analytical solution to calculating the crack tip plastic zone in plane stress states is presented. The shape and size of the plastic zone are analyzed under different loading conditions. The obtained results show that the crack tip plastic zones present “butterfly-like” shapes, and the elastic–plastic boundary is smooth. The size of the plastic zone for orthotropic composites is less at the crack tip for various loading conditions, compared with the case of isotropic materials. Crack inclination angle and loading conditions affect greatly the size and shape of crack tip plastic zone. The mode I crack has a crucial effect on the plastic zone for mixed mode case in plane stress state. The plastic zone for pure mode I crack and pure mode II crack have a symmetrical distribution to the initial crack plane.     

Influence of flaw inclination angle and loading condition on crack initiation and propagation

With reference to the experimental observation of crack initiation and propagation from pre-existing flaws in rock specimens under compression, the influences of pre-existing flaw inclination angle on the cracking processes were analyzed by means of finite element method (FEM) and non-linear dynamics method. FEM analysis on the stress field distribution induced by the presence of a pre-existing flaw provided better understanding for the influence of flaw inclination angle on the initiation position and initiation angle of the potential cracks. Numerical analysis based on the non-linear dynamics method was performed to simulate the cracking processes. The resultant crack types, crack initiation sequences and the overall crack pattern were different under different loading conditions. Under a relatively low loading rate or a small magnitude of maximum loading pressure, tensile cracks would tend to initiate prior to shear cracks. In contrast, under a relatively high loading rate and a large magnitude of maximum loading pressure, shear cracks would tend to initiate prior to tensile cracks instead.     

A General Formulation of Stress Distribution in Cylinders Subjected to Non-Uniform External Pressure

This work presents a two-dimensional stress analysis for elastic solid cylinders subjected to combined loading. The loading is generally formed with a number of concentrated and partially distributed forces all applied radially on the outer surface. The distributed forces cause pressures with non-uniform intensity along the circumferential direction. The cylinder is assumed to be long so that a state of plane-strain is valid. To obtain the stress distribution for the problem of partially distributed forces a new approach is followed first introduced in this paper. It is based on the expressions formed after using the theory of simple radial stress distribution when point-forces are applied on the cylinder and leads to the solution after direct integration. The total stresses due to both concentrated and distributed forces are obtained using the method of superposition. Apart from its simplified formulation, this general solution is always preferable since it proved to have a great advantage. As a result of not containing Fourier series, it eliminates some problems of convergence of the series at the boundaries that appear due to the Gibbs phenomena when the boundary conditions are a discontinuous function. Numerical results are presented for some interesting cases of loading conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Equilibrium States of Inextensible Membranes under Gravity

We discuss the possible shapes and stress distributions of an inextensible membrane acted on by a gravitational force. It is shown that there are smooth shapes which cannot be an equilibrium solution under gravity for any stress distribution. In addition there are infinitely many shapes which are equilibrium solutions under gravity and for which the stress distribution is unique. Finally, there are infinitely many solution surfaces – each consisting of straight line generators parallel to the direction of gravity. These solutions are possible under infinitely many different stress distributions. In several cases we will solve the equations explicitly to find actual shapes and stress distributions. Mathematics Subject Classifications (2000) 74.     

Modelling of crack growth under cyclic contact loading

The paper describes a general computational model for modelling of subsurface fatigue crack growth under cyclic contact loading of mechanical elements. The model assumes that the initial fatigue crack develops along the slip line in a single crystal grain at the point of the maximum equivalent stress. The position and magnitude of the maximum equivalent stress are determined with the Finite Element Analysis of the equivalent contact model, which is based on the Hertzian contact conditions with the addition of frictional forces. The Virtual Crack Extension method is then used for simulation of the fatigue crack propagation from the initial to the critical crack length, when the surface material layer breaks away and a pit appears on the surface. The pit shapes and relationships between the stress intensity factor and the crack length are determined for various combinations of contacting surface curvatures and contact loadings. The computational results show that the model reliably simulates the subsurface fatigue crack growth under contact loading and can be used for computational predictions of surface pitting for various contacting mechanical elements.     

Numerical study of rate-dependent strength behavior under ramp and shock wave loading

The main objective of the current study was to gain a detailed understanding on the rate-dependent strength behavior under ramp and shock wave loading. A forward, numerical-simulation-based cause and effect analysis was used to address the research objective. The apparent strength associated with shock and ramp wave loadings with different risetimes and shapes was investigated. It was shown that intrinsic material strength could vary with pressure, temperature, and deformation history, but the apparent strength, which was larger than the intrinsic strength, was a result of the interaction between the rate sensitivity of the strength and the rate of the external loading. The degree of interaction led to different levels of mechanical and thermal dissipations and their partition, which was manifested by different temperature, stress, and deformation histories.