Mixed-mode crack growth in plates under three-point bending

A combined theoretical and experimental study of the problem of crack growth in a plate subjected to unsymmetrical three-point bending was undertaken. The opening-modeK _I and sliding-modeK _II stress-intensity factors describing the local stress field around the crack tip were determined by a finite-element computer program. The crack growth was analyzed by the maximum circumferential stress and the minimum strain-energy density criteria. The critical loads for crack growth and the crack trajectories were determined both by theory and experiment. The experimental results corroborated the theoretical predictions.     

Crack kinking in sandwich structures under three-point bending

Sandwich beams under three-point bending containing cracks in the core material very close to the upper skin interface are investigated. The cracks considered parallel or with an imperceptible inclination to the longitudinal beam axis and at different distances from the upper skin interface, are analyzed with static non-linear elastic two and three dimensional finite element analyses. From the proposed analyses stress intensity factors are calculated using linear elastic fracture mechanics. It is demonstrated that the crack propagation on the compression side of the core is mainly subjected in shear. The strain energy density criterion is used in order to determine the angle of kinking of the crack into the core.     

混凝土三点弯曲梁软化特性的数值计算方法

本文推导了基于虚拟裂纹模型的混凝土梁荷载位移全曲线的计算公式,研究了不同网格剖分、不同的拉伸软化曲线模型和不同的预制裂纹长度等对荷载位移全曲线的影响。计算结果表明,其结果不受网格划分的影响,通过与实验结果对比,证明了其有效性。     

Optimal design of box-section sandwich beams in three-point bending

Minimum mass designs are obtained for box-section sandwich beams of various cross-sections in three-point bending. The overall compliance of the hollow, tubular beams are decomposed additively into a global contribution due to macroscopic bending (Timoshenko beam theory) and a local contribution associated with transverse deflection of the walls of the hollow beam adjacent to the central loading patch. The structural response is analysed for beams of square sections with various internal topologies: a solid section, a foam-filled tube with monolithic walls, a hollow tube with walls made from sandwich plates, and a hollow tube with walls reinforced by internal stiffeners. Finite element analysis is used to validate analytical models for the overall stiffness of the tubes in three-point bending. Minimum mass designs are obtained as a function of the overall stiffness, and the relative merits of the competing topologies are discussed.     

Nonlinear behavior of composite sandwich beams in three-point bending

The load-deflection behavior of a composite sandwich beam in three-point bending was investigated. The beam was made of unidirectional carbon/epoxy facings and a polyvinyl chloride closed-cell foam core. The load-deflection curves were plotted up to the point of failure initiation. They consist of an initial linear part followed by a nonlinear portion. A nonlinear mechanics of materials analysis that accounts for the combined effect of the nonlinear behavior of the facings and core materials (material nonlinearity) and the large deflections of the beam (geometric nonlinearity) was developed. The theoretical predictions were in good agreement with the experimental results. It was found that the effect of material nonlinearity on the deflection of the beam is more pronounced for shear-dominated core failures in the case of short span lengths. It is due to the nonlinear shear stress-strain behavior of the core. For long span lengths, the observed nonlinearity is small and is attributed to the combined effect of the facings nonlinear stress-strain behavior and the large deflections of the beam.     

Large deflections of a beam subject to three-point bending

In this paper, a solution for the equilibrium configuration of an elastic beam subject to three-point bending is given in terms of Jacobi elliptical functions. General equations are derived, and the domain of the solution is established. Several examples that illustrate a use of the solution are discussed. The obtained numerical results are compared with the results of other authors. An approximation formula by which the beam load is given as a polynomial function of beam deflection is also derived. The range of applicability of the approximation is illustrated by numerical examples.     

Elastic buckling of I-beams under linear moment gradient

This paper investigates the elastic lateral–torsional buckling of I-beams under linear moment gradient that very precisely incorporates the effects of moment gradient and various end restraints. The elastic critical buckling moments are obtained independently by using: (1) the Bubnov–Galerkin method and (2) the finite element method. The present formula of the moment gradient correction factor cannot satisfactorily predict the buckling capacities of doubly symmetric and monosymmetric I-beams with various end restraints. We propose alternative equations for evaluating the moment gradient correction factor, considering end restraint conditions.     

Peridynamic modelling of impact damage in three-point bending beam with offset notch

The nonlocal peridynamic theory has been proven to be a promising method for the material failure and damage analyses in solid mechanics.Based upon the integrodifferential equations,peridynamics enables predicting the complex fracture phenomena such as spontaneous crack nucleation and crack branching,curving,and arrest.In this paper,the bond-based peridynamic approach is used to study the impact damage in a beam with an offset notch,which is widely used to investigate the mixed I-II crack propagation in brittle materials.The predictions from the peridynamic analysis agree well with available experimental observations.The numerical results show that the dynamic fracture behaviors of the beam under the impact load,such as crack initiation,curving,and branching,rely on the location of the offset notch and the impact speed of the drop hammer.     

粘接多胞管三点弯曲实验与数值研究

基于实验和数值模拟方法,论文对一种易制备粘接多胞薄壁结构的弯曲性能进行了研究,分析了粘接多胞管在横向三点弯曲载荷作用下的变形和能量吸收性能.三点弯曲准静态实验表明由于粘接的作用,通常情况下粘接多胞管的能量吸收性能高于其基本构成单胞管能量吸收的总和,某些情况下粘接可以最高带来近70%的性能提升.基于有限元软件LS-DYNA,对三点弯曲实验进行了数值模拟仿真,计算得到的结构变形模式和力-位移曲线与实验结果吻合良好.此外,采用数值方法,对三种不同接触条件下的结构响应进行了对比分析,结果表明若未出现明显的粘接脱开,则多胞管的吸能特性与完整的多胞结构相当,否则其能量吸收性能会被严重削弱.     

Simulation of three-point bending at large deflections by an elastoplastic contact analysis

Three-point bending is simulated by an elaborate numerical procedure based on an elastoplastic, large deflection, contact analysis. A minimization formulation is used, which is equivalent to the incremental form posed as partial differential equations with inequalities. A sequential quadratic programming approach based on the finite-element technique is adopted as a method of solution. To examine the validity of the simulation method, experiments are carried out for specimens that have various widths.Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

Determination of dynamic fracture-initiation toughness using three-point bending tests in a modified Hopkinson pressure bar

We present a procedure for measuring the dynamic fracture-initiation toughness of materials. The method is based on three-point bending tests at high loading rates, performed in an experimental device which is a modification of the classical split Hopkinson pressure bar. Coupled with the loading device, a high-speed photography system was used to measure the crack mouth opening displacement (CMOD) directly on the specimen. The stress intensity factor was calculated by three different simplified methods and the time to fracture was obtained from an appropriate specimen instrumentation. To evaluate the results derived from the simplified methods, a two-dimensional full-numerical analysis of the dynamic bending fracture test was made. The model includes the specimen, the input bar, the impacting projectile and the supporting device and takes into account the possible loss of contact during the experiment between the input bar and the specimen and between the specimen and its supports. From the tests and numerical results, it can be concluded that the CMOD procedure, together with the knowledge of the time to fracture determined using crack gages, seems to be the best method for measuring dynamic fracture-initiation toughness.     

Calculations of fracture process zones on meso-scale in notched concrete beams subjected to three-point bending

The paper describes investigations on fracture process zones (FPZ) at meso-scale in notched concrete beams subjected to quasi-static three-point bending. The simulations were carried out with the FEM using isotropic damage constitutive model enhanced by a characteristic length of micro-structure by means of a non-local theory. Concrete was modelled as a random heterogeneous three-phase material. The effect of the beam size, aggregate distribution, aggregate density, aggregate shape, aggregate size and characteristic length on the width and shape of FPZ and load-displacement curve was numerically investigated. The numerical results were compared with own test results using Digital Image Correlation method (Skar?yński et al., 2009a), the tests by Le Bellěgo et al., (2003) and the size effect law by Ba?ant (2004).     

混凝土楔入劈拉和三点弯曲断裂过程的细观数值模拟

基于离散元思想和Voronoi单元划分技术,利用混凝土细观刚体弹簧元模型,开展了混凝土楔入劈拉试件和三点弯曲切口梁的断裂过程数值仿真分析。从裂缝开展过程、试件破坏形态、荷载-张口位移曲线(P-CMOD)和断裂能等方面,将数值分析结果与已有的试验结果进行了对比。结果表明,细观刚体弹簧元法较准确地模拟了混凝土楔入劈拉试件和三点弯曲切口梁断裂过程。最后,分析了缝高比和骨料体积含量对混凝土断裂过程的影响规律,发现断裂能随骨料体积含量呈单调递增趋势。     

Elastic-plastic secondary indeterminate problem for thin-walled pipe through the inner-wall loading by three-point bending

The inner-wall loading by three-point bending about thin-walled pipe is an elastic-plastic secondary indeterminate problem in the symmetrical three-roller setting round process. In this study, the shifting of the tangent point between the pipe and lower roller is ignored. The bilinear hardening material model is adopted, and the static equilibrium condition, physical relationship of elastic-plastic deformation, and deformation compatibility condition are taken into account. Based on the geometrical discrete idea, a semi-circular pipe is meshed equably into N micro-pipe-wall elements with same geometric parameters along the circumferential direction. Deformation characteristics of each element are calculated, and then the deformation history response of the whole pipe is resolved by the load increment method. The finite element model of static bending in three-roller setting round process is established by using the software package ABAQUS. The theoretical and simulated results show that the cross section of pipe has two positive bending regions and two reverse bending regions; the maximum bending curvature appears in the bottom section of pipe, the minimum bending curvature appears in the section corresponding to the tangent point of the pipe and lower roller. The quantitative relationships between the upper roller load, maximum(minimum) bending curvature and reduction are given. Finally, the reliability of theoretical calculation is proved by numerical simulation.     

Localization in NiTi tubes under bending

Nearly equiatomic NiTi can exhibit pseudoelastic behavior due to reversible solid-to-solid stress induced phase transformation at room level temperatures. In tension, the transformation leads to localized deformation of several percent that tends to spread at nearly constant stress. The deformation is recovered upon unloading while again localized deformation is exhibited. Under compression, while still pseudoelastic, the transformation strains are smaller, the stress is higher, the response is monotonic, and the deformation is essentially homogeneous. This study examines how this texture-driven, complex material asymmetry affects a simple structure: the bending of a tube. To this end, NiTi tubes are bent in a custom four-point bending facility under rotation control and isothermal conditions. The phase transformations lead to a closed moment-rotation hysteresis comprised of loading and unloading moment plateaus. During loading, localized nucleation of martensite results in a high curvature for the transformed sections of the tube and low curvature for the untransformed. Martensite, which corresponds to the higher curvature regime, spreads gradually while the moment remains nearly constant. The nucleation of martensite is in the form of bands inclined to the axis of the tube that organize themselves into diamond shaped deformation patterns on the tensioned side of the structure. The patterns are similar to those observed in bending of steel tubes with Lüders bands, however, for NiTi they develop only on the tensioned side due to the material asymmetry. A lower moment plateau is traced upon unloading with similar localized bending and the erasure of the diamond deformation patterns. This complex behavior was found to repeat for a number of temperatures in the pseudoelastic regime of NiTi with the moment-rotation hysteresis moving to higher or lower moment levels depending on the temperature.     

混凝土三点弯曲梁裂缝断裂全过程数值模拟研究

考虑裂缝黏聚力的作用,基于Paris位移公式推导出混凝土三点弯曲梁裂缝扩展过程中断裂过程区上的裂缝张开位移的解析表达式.采用起裂韧度作为裂缝起裂及扩展的判断标准,提出了荷载作用下混凝土裂缝起裂、扩展及失稳破坏全过程的数值模拟方法,并分别与国内外断裂试验实测值及有限元计算值进行了比较.结果表明,本文提出的数值模拟方法形式简单且精度较好.     

Nonlinear bending response and buckling of ring-stiffened cylindrical shells under pure bending

In this paper, the nonlinear bending response of finite length cylindrical shells with stiffening rings is investigated by using a modified Brazier approach. The basic assumptions for the present study are that the deformation of a shell subjected to pure bending can be simplified into a two-stage process. One is that the shell ovalizes but its axis remains straight; the other is that the bending of the shell is regarded as a beam with nonuniform ovalization. The nonlinear bending response is derived by applying the minimum potential energy principle and the corresponding critical moment, associated with local buckling, is determined by employing the Seide–Weingarten approximation. Numerical results are shown and compared with those obtained from other methods, which demonstrates that the assumptions used in the present study are reasonable.