Cyclic plasticity models and application in fatigue analysis

An analytical procedure for prediction of the cyclic plasticity effects on both the structural fatigue life to crack initiation and the rate of crack growth is presented. The crack initiation criterion is based on the Coffin-Manson formulae extended for multiaxial stress state and for inclusion of the mean stress effect. This criterion is also applied for the accumulated damage ahead of the existing crack tip which is assumed to be related to the crack growth rate. Three cyclic plasticity models, based on the concept of combination of several yield surfaces, are employed for computing the crack growth rate of a cracked plane stress panel under several cyclic loading conditions.     

The comparison between three different elements within a center crack plate considering elastic-plastic action

Stress near a crack tip in plasticity was analyzed using three different finite element modelings; a constant strain triangle, eight-noded quadrilaterial, and a crack tip singularity element all considering viscoplasticity. The specimen under consideration was a center cracked plate made from IN-100, a nickel-base superalloy containing a half-crack length equal to 0.1367 in. (3.472 mm). An elastic solution was formulated in conjunction with two different loadings to generate plasticity. Fine mesh and coarse mesh solutions for the higher order elements were generated and compared considering equal number of degrees of freedom in two specific regions referred to as the near field and the far field regions.

The authors determined that the elements whose elastic solutions conformed best to linear elastic fracture mechanics predictions were the constant strain triangle and the eight-noded quadrilateral in a fine mesh. The crack tip element did not perform as well as expected. For the plastic analysis, the constant strain triangle exhibited the largest plastic region. The eight-noded isoparametric element came within 15% of the stress levels generated from the constant strain triangle. The stress singularity that is characteristic of the crack tip element forced that element to behave unnaturally stiff immediately adjacent to the crack tip.

Because it is not as stiff as either the crack tip element or the eight-noded element, the constant strain triangle offered the most promising solutions as verified through experimentation. It was therefore determined that the constant strain triangle offered the best solution to elastic-plastic finite element problems for the center cracked plate.     

A singular ES-FEM for plastic fracture mechanics

The stress and strain fields around the crack tip for power hardening material, which are singular as r approaches zero, are crucial to fracture and fatigue of structures. To simulate effectively the strain and stress around the crack tip, we develop a seven-node singular element which has a displacement field containing the HRR term and the second order term. The novel singular element is formulated based on the edge-based smoothed finite element method (ES-FEM). With one layer of these singular elements around the crack tip, the ES-FEM works very well for simulating plasticity around the crack tip based on the small strain formulation. Two examples are presented with detailed comparison with other methods. It is found that the results of the presented singular ES-FEM are closer to reference solution, which demonstrates the applicability and the effectiveness of our method for the plastic field around the crack tip.     

Finite element mesh for a complete solution of a problem with a singularity

It is well known that the stress field at the tip of a crack in an elastic body exhibits a singularity. In this paper, a complete elastic solution for a center cracked plate loaded under uniform tension is obtained by a finite element plane-stress analysis using constant strain elements. No a priori assumptions about the form of the stress singularity are required. The numerical results are compared with the exact analytic solution. It is shown that a particular mesh arrangement in which the size of the elements decreases in a geometric series as the crack tip is approached yields stress and strain fields which are accurate over the entire plate, even at distances very close to the crack tip. The effects of changes in mesh arrangements on the accuracy of the solution are considered. The computations are carried out on the CRAY-1 computer and the advantages of vectorization are discussed for this problem.     

Harmonic analysis of the vibrations of a cantilevered beam with a closing crack

In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure.     

Effect of loading types and reinforcement ratio on an effective moment of inertia and deflection of a reinforced concrete beam

In the design of reinforced concrete structures, a designer must satisfy not only the strength requirements but also the serviceability requirements, and therefore the control of the deformation becomes more important. To ensure serviceability criterion, it is necessary to accurately predict the cracking and deflection of reinforced concrete structures under service loads. For accurate determination of the member deflections, cracked members in the reinforced concrete structures need to be identified and their effective flexural and shear rigidities determined. The effect of concrete cracking on the stiffness of a flexural member is largely dependent on both the magnitude and shape of the moment diagram, which is related to the type of applied loading. In the present study, the effects of the loading types and the reinforcement ratio on the flexural stiffness of beams has been investigated by using the computer program developed for the analysis of reinforced concrete frames with members in cracked state. In the program, the variation of the flexural stiffness of a cracked member has been obtained by using ACI, CEB and probability-based effective stiffness model. Shear deformation effect is also taken into account in the analysis and the variation of shear stiffness in the cracked regions of members has been considered by employing reduced shear stiffness model available in the literature. Comparisons of the different models for the effective moment of inertia have been made with the reinforced concrete test beams. The effect of shear deformation on the total deflection of reinforced concrete beams has also been investigated, and the contribution of shear deformation to the total deflection of beam have been theoretically obtained in the case of various loading case by using the developed computer program. The applicability of the proposed analytical procedure to the beams under different loading conditions has been tested by a comparison of the analytical and experimental results, and the analytical results have been found in good agreement with the test results.     

鼠笼局部断裂的刚度特性及其转子动力学特性分析

本文针对带有局部断裂的鼠笼刚度特性以及其支承的转子系统动力学特性开展研究.首先基于力学分析推导了鼠笼局部断裂刚度理论计算公式.其次,基于带有局部断裂的鼠笼有限元模型进行其刚度计算,并采用最小二乘法对有限元刚度计算结果进行分析,提出了鼠笼局部断裂后的拟合经验刚度公式,通过对比分析得出局部断裂对鼠笼刚度特性的影响.最后,建立带有鼠笼局部断裂的两支点弹性支承转子系统的动力学模型,并仿真获得了带有鼠笼局部断裂转子系统的固有特性和振动响应的变化规律.结果表明:鼠笼局部断裂导致其刚度下降以及支承刚度不对称,引起转子系统临界转速下降,且水平、垂直振动有差异,轴心轨迹呈椭圆形.     

Application of genetic algorithm in crack detection of beam-like structures using a new cracked Euler–Bernoulli beam element

In this paper, a crack identification approach is presented for detecting crack depth and location in beam-like structures. For this purpose, a new beam element with a single transverse edge crack, in arbitrary position of beam element with any depth, is developed. The crack is not physically modeled within the element, but its effect on the local flexibility of the element is considered by the modification of the element stiffness as a function of crack's depth and position. The development is based on a simplified model, where each crack is substituted by a corresponding linear rotational spring, connecting two adjacent elastic parts. The localized spring may be represented based on linear fracture mechanics theory. The components of the stiffness matrix for the cracked element are derived using the conjugate beam concept and Betti's theorem, and finally represented in closed-form expressions. The proposed beam element is efficiently employed for solving forward problem (i.e., to gain accurate natural frequencies of beam-like structures knowing the cracks’ characteristics). To validate the proposed element, results obtained by new element are compared with two-dimensional (2D) finite element results as well as available experimental measurements. Moreover, by knowing the natural frequencies, an inverse problem is established in which the cracks location and depth are identified. In the inverse approach, an optimization problem based on the new beam element and genetic algorithms (GAs) is solved to search the solution. The proposed approach is verified through various examples on cracked beams with different damage scenarios. It is shown that the present algorithm is able to identify various crack configurations in a cracked beam.     

Nonlinear finite element analysis of reinforced concrete plates and shells under monotonic loading

Plane stress constitutive models are proposed for the nonlinear finite element analysis of reinforced concrete structures under monotonic loading. An elastic strain hardening plastic stress-strain relationship with a nonassociated flow rule is used to model concrete in the compression dominating region and an elastic brittle fracture behavior is assumed for concrete in the tension dominating area. After cracking takes place, the smeared cracked approach together with the rotating crack concept is employed. The steel is modeled by an idealized bilinear curve identical in tension and compressions. Via a layered approach, these material models are further extended to model the flexural behavior of reinforced concrete plates and shells. These material models have been tested against experimental data and good agreement has been obtained.     

Dynamic stress intensity factors for ultrasonic fatigue test specimens

Ultrasonic fatigue test methods are increasingly being used to study crack growth and threshold behaviour of metallic materials. In such studies a particular problem is the accurate determination of the dynamic stress intensity factor range, ΔK. A direct time integration finite element analysis of a center cracked specimen tested at its lowest natural frequency is presented. The effects of opening and closing of the crack tip when the test specimen is subjected to a fully reversed load cycle, i.e. zero mean load, is investigated. Numerical results correlate well with experimental data. It is shown that old published data are in error with up to at least 20% due to insufficient evaluation of dynamic stress intensity factors.     

Time-dependent modelling of RC structures using the cracked membrane model and solidification theory

A non-linear finite element model is presented for the time-dependent analysis of reinforced concrete structures under service loads. For the analysis of members in plane stress, the model is based on the cracked membrane model using a rotating crack approach combined with solidification theory for modelling creep. The numerical results are compared with a variety of long-term laboratory measurements, including development of deflections and cracking with time in a reinforced concrete beam, time-dependent change in support reactions of a continuous beam subject to support settlement and creep buckling of columns. The numerical results are in good agreement with the test data.     

Detection of cracks in simply-supported beams by continuous wavelet transform of reconstructed modal data

This paper proposes a new approach for damage detection in beam-like structures with small cracks, whose crack ratio [r = H_c/H] is less than 5%, without baseline modal parameters. The approach is based on the difference of the continuous wavelet transforms (CWTs) of two sets of mode shape data which correspond to the left half and the right half of the modal data of a cracked simply-supported beam. The mode shape data of a cracked beam are apparently smooth curves, but actually exhibit local peaks or discontinuities in the region of damage because they include additional response due to the cracks. The modal responses of the damaged simply-supported beams used are computed using the finite element method. The results demonstrate the efficiency of the proposed method for crack detection, and they provide a better crack indicator than the result of the CWT of the original mode shape data. The effects of crack location and sampling interval are examined. The simulated and experimental results show that the proposed method has great potential in crack detection of beam-like structures as it does not require the modal parameter of an uncracked beam as a baseline for crack detection. It can be recommended for real applications.     

Stability and convergence proofs for a discontinuous-Galerkin-based extended finite element method for fracture mechanics

We prove the optimal convergence of a discontinuous-Galerkin-based extended finite element method for two-dimensional linear elastostatic problems over cracked domains. The method, which we proposed earlier [1], has two distinctive traits: a) it enriches the finite element space with the modes I and II singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region, and b) it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain. The treatment for this discontinuity, generally a non-polynomial function, is facilitated by a specially designed discontinuous Galerkin method based on the Bassi–Rebay numerical flux. The stability of the method is contingent upon an inf–sup condition, which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes. We have also shown the optimal convergence of the displacement and stress fields, and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions.     

On the use of XFEM within the Arlequin framework for the simulation of crack propagation

The Arlequin method is a generic numerical method that allows, by local superposition and coupling of models, to address multimodel and multiscale mechanical problems. In particular, this method has already been used to super-impose cracked patches on sound structures, reducing this way the global simulation resources a classical finite element approach would have required.In this paper, one of the key features of the extended-finite element method, namely the heaviside enrichment function, is used within the Arlequin framework to further reduce the costs of crack propagation simulations. The main goal of the paper is to describe the proposed methodology and to assess its performance through numerical experiments.     

Non-conservative stability of a cracked thick rotating blade

A finite element model is applied to the non-conservative stability problems of a cracked thick rotating blade. This finite element model can satisfy all the geometric and natural boundary conditions of a rotating blade. The blade is considered to be subjected to follower moments and aerodynamic forces. The effects of crack locations and crack sizes are studied. It is found that the rotation speed and crack can change the stability characteristics of a non-conservative system.     

Elastoplastic analysis of tubular joints of offshore platforms

The subject of this paper is the analysis and design of complex tubular joints, eventually including internal and external gussets and stiffener rings, corresponding to fixed and mobile offshore structures.Two different joints are analysed. In the first case an X joint is studied for elastic and elastoplastic behaviour, loading up to collapse in order to determine ultimate strength and safety factor. Finite elements which can reproduce the elastic and plastic singularities of the stress and the strain fields in the crack tip, are then used for the analysis of a T joint. Both direction and rate are considered in the crack propagation, and an elastoplastic analysis is carried out, to determine the crack opening displacement (COD).Finally, the consideration of fatigue effects in tubular joints is discussed, and techniques for evaluating fatigue life are outlined.     

A generalised technique for fracture analysis of cracked plates under combined tensile,bending and shear loads

The objective of this paper is to propose a generalized technique called numerically integrated modified virtual crack closure integral (NI-MVCCI) technique for fracture analysis of cracked plates under combined tensile, bending and shear loads. NI-MVCCI technique is used for post-processing the results of finite element analysis (FEA) for computation of strain energy release rate (SERR) components and the corresponding stress intensity factor (SIF) for cracked plates. NI-MVCCI technique has been demonstrated for 4-noded, 8-noded (regular and quarter-point) and 9-noded (regular and quarter-point) isoparametric plate finite elements. These elements are based on Mindlin’s plate theory that considers shear deformation. For all the elements, reduced integration/selective reduced integration techniques have been employed in the studies. In addition, for 9-noded element assumed shear interpolation functions have been used to overcome the shear locking problem. Numerical studies on fracture analysis of plates subjected to tension–moment and tension–shear loads have been conducted employing these elements. It is observed that among these elements, the 9-noded Lagrangian plate element with assumed shear interpolation functions exhibits better performance for fracture analysis of cracked plates.     

Post buckling behavior of transverse cracked columns

Post buckling behavior of a column with a transverse surface crack on the one side is studied, considering the local flexibility because of the crack. This flexibility, also called compliance, is known to be related to the stress intensity factor. This relation is generalized and expressed in the form of a complete local stiffness matrix of the cracked section of the beam. The Paris equation for the deflection of cracked members is extended for this purpose to give the generalized influence coefficients, being considered as incremental deflections because of the presence of the crack.Eigenvalue solutions for the buckling load are developed which do not differ for appreciably slender columns from known solutions based on some only of the local flexibility coefficients reported in the literature.Moreover, two distinct buckling modes have been identified to closing cracks, because of the different behavior of the cracked region in compression and tension or positive and negative bending modes.The post buckling behavior has been studied, for both buckling modes solving numerically the nonlinear equation for the elastica with the local flexibilities because of the crack.As expected, this behavior of the column is, in general, stable with positive slopes. However, due to the character of the closing cracks, jumping phenomena are governing the transition from the zero equilibrium to the post buckling equilibrium paths.The postbuckling behavior is finally tabulated for a simply supported column as a function of the crack depth.