Eine elastisch‐plastische Übergangsbedingung für die Tropfenschlagerosion von Gesteinen und Betonen

An elastic‐plastic transition criterion for the erosion of rocks and concrete materials by impinging liquid drops Depending on loading regime and material type, mineralic materials behave either elastic or elastic‐plastic if eroded by impinging water drops. A simple transition number that combines fracture toughness and compressive strength (respectively Vickers hardness), can be used to distinguish between both modes. Conventionally ‘hard’ materials, namely granite and feldspars, own low transition numbers (high hardness values) and respond elastic. Conventionally ‘soft’ materials, namely limestone, cement mortar and schist, are characterised by high transition numbers (low hardness values) and show an elastic‐plastic response.     

Direct analysis of elastic–plastic structures with ‘overlay’ materials during cyclic loading

Several computer codes have incorporated the ‘overlay’ material models: the volume element, which is characteristic of the material, is composed of sub-elements with different kinematic hardening, perfectly plastic or even viscoplastic flow rules and different elastic properties, these sub-elements exhibiting all, however, the same total strain.¹,² In this paper it is demonstrated how the simple mathematical framework we first proposed for elastic-plastic structures with kinematic hardening material,³ and we extended to some elastic viscoplastic ones,⁴ can easily be applied to these particular ‘overlay’ materials. One of the interesting advantages of this approach is a straightforward analysis of structural response under cyclic loadings by applying the linear elastic analysis.     

Buckling of Carbon/Glass Composite Tubes Under Uniform External Hydrostatic Pressure

Abstract: This paper describes an experimental and an analytical investigation into the collapse of 44 circular cylindrical composite tubes under external hydrostatic pressure. The results for 22 of these tubes were from a previous investigation and the results for a further 22 models are reported for the first time in this paper. The investigations concentrated on fibre‐reinforced plastic tube specimens made from a mixture of three carbon and two E‐glass fibre layers. The lay‐up was 0°/90°/0°/90°/0; the carbon fibres were laid lengthwise (0°) and the E‐glass fibres circumferentially (90°). The theoretical investigations were carried out using a simple solution for isotropic materials, namely a well‐known formula by ‘von Mises’. The previous investigation also used a numerical solution based on ANSYS, but this was found to be rather disappointing. The experimental investigations showed that the composite specimens behaved similarly to isotropic materials previously tested, in that the short vessels collapsed through axisymmetric deformation while the longer tubes collapsed through non‐symmetric bifurcation buckling. Furthermore, it was discovered that the specimens failed at changes of the composite lay‐up due to the manufacturing process of these specimens. These changes seem to be the weak points of the specimens. For the theoretical investigations, two different types of material properties were used to analyse the composite. These were calculated properties derived from the properties of the single layers given by the manufacturer and also the experimentally obtained properties. Two different approaches were chosen for the investigation of the theoretical buckling pressures, of the previously analysed models, namely a program called ‘MisesNP’, based on a well‐known formula by von Mises for single‐layer isotropic materials, and two finite element analyses using the famous computer package called ‘ANSYS’. These latter analyses simulated the composite with a single‐layer orthotrophic element (Shell93) and also with a multi‐layer element (Shell99). The results from Shell93 and Shell99 agreed with each other but, in general, their predictions were higher than the analytical solution by von Mises. The von Mises solution agreed better than the finite element solutions for the longer vessels, which collapsed by elastic instability, particularly when the experimentally obtained material properties were used. Thus, it was concluded that the results obtained from the finite element analyses predicted ‘questionable’ buckling pressures. The report provides design charts by all approaches and material types, which allow the possibility of obtaining a ‘plastic knockdown factor’ for these vessels. The theoretical buckling pressures obtained using the computer programs MisesNP or ANSYS can then be divided by the plastic knockdown factor obtained from the design charts, to give the predicted buckling pressures. It is not known whether or not this method can be used for the design of very large vessels.     

Nonlinear mode III crack stress fields for materials obeying a modified Ramberg‐Osgood law

In this paper, an analytical study is carried out on the work‐hardening, elastic‐plastic stress distributions in a cracked body under antiplane shear deformation. A modified Ramberg‐Osgood law is introduced to describe the material behaviour, and stress and strain fields are derived in closed form. Compared with the conventional Ramberg‐Osgood formulation, the new law includes the effect of a new parameter, κ, which allows the transition from the ideally elastic behaviour (low stress regime) to the power law behaviour (large stress regime) to be controlled, thus providing 1 more degree of freedom to better fit the actual behaviour of engineering materials. A discussion is carried out on the features of stresses and strains close to and far away from the crack tip.     

State‐of‐the‐art review on extended stress intensity factor concepts

The stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K₁, K₂ and K₃ are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K_1ρ, K_2ρ and K_3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K_1p, K_2p and K_3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

Fracture analysis of dissimilar Al‐Al friction stir welded joints under tensile/shear loading

In this research, fracture of dissimilar friction stir welded (FSWed) joint made of Al 7075‐T6 and Al 6061‐T6 aluminum alloys is investigated in the cracked semi‐circular bend (CSCB) specimen under mixed mode I/II loading. Due to the elastic‐plastic behavior of the welded material and the existence of significant plastic deformations around the crack tip at the propagation instance, fracture prediction of the FSWed specimens needs some failure criteria in the context of the elastic‐plastic fracture mechanics which are very complicated and time‐consuming. For this purpose, the Equivalent Material Concept (EMC) is used herein by which the tensile behavior of the welded material is equated with that of a virtual brittle material. By combining EMC with the 2 brittle fracture criteria, namely the maximum tangential stress (MTS) and mean stress (MS) criteria, the load‐carrying capacity (LCC) of the FSWed CSCB specimens is predicted. Comparison of the experimental results and theoretical predictions from the 2 criteria showed that both criteria could accurately predict the LCC of the cracked specimens. Moreover, as the contribution of mode II loading increases, the size of the plastic region around the crack tip at failure increases, leading to increasing the LCC.     

Generalized B‐bar: unlocking multiple compliance modes of finite elements with stiff‐fiber anisotropy

Element locking is often seen in homogenized models of elastic fiber‐reinforced materials, and splitting the material compliance into two separate terms isolates troublesome strain modes. Once isolated, the locking modes can be addressed with tailored integration schemes or the opportune introduction of field variables. The canonical application of this approach is seen in the dilatational‐deviatoric split used to treat so‐called ‘volumetric locking’. In the present work, we invoke the spectral decomposition of the material compliance to provide a generalized split. Doing so naturally parses the response into six independent strain modes, with varying propensity for locking. This split can be used to generalize fundamental techniques, such as selective reduced integration and the B‐bar method. This broadened approach works to remedy locking suffered by lower order finite elements used to discretize troublesome materials. Applying these generalized methods to achieve the dilational‐deviatoric split is trivial. However, the compliance spectrum's ability to naturally isolate stiff material response modes makes it a uniquely valuable tool for use on homogenized anisotropic materials. Applying the split, defined by only the first compliance mode, has given rise to the generalized methods, which have proven effective in unlocking finite element models of anisotropic materials. In the present work, the generalization is broadened to treat more than one constrained mode. While treating six modes is equivalent to simple reduced integration techniques, up to five compliance modes are now separated for advantageous treatment. However, some attention must be paid to the stability of the resulting finite element stiffness matrices. We focus here on the treatment of two principal compliance modes. These ‘two‐mode’ applications of the generalized B‐bar method are shown to be a more robust default treatment of linear hexahedral elements than is provided by classical selective reduced integration. This is achieved with a negligible computational overhead. A framework for assessing element stability is delineated, and commonly arising instabilities are analyzed. Copyright © 2016 John Wiley & Sons, Ltd.     

Influence of Wooden Board Strength Class on the Performance of Cross‐laminated Timber Plates Investigated by Means of Full‐field Deformation Measurements

Although cross‐laminated timber (CLT) plates are increasingly used in high‐performance building structures, a tailored composition of them or, at least, a performance‐based classification scheme is not available. Especially, the influence of the quality of the ‘raw’ material (wooden boards) on the load carrying capacity of CLT elements is hardly investigated yet. For this reason, within this work, bending tests on 24 CLT plates consisting of wooden boards from three different strength classes have been carried out. The global mechanical response as well as the formation of failure mechanisms were investigated, including a full‐field deformation measurement system, which allowed for a qualitatively as well as quantitatively identification of board failure modes. Interestingly, no influence of the board strength class on the elastic limit load of the CLT plates was observed, but the situation was different for the load displacement history beyond the elastic regime, where basically, two different global failure mechanisms could be distinguished. The obtained knowledge about the ‘post‐elastic’ behaviour of CLT plates may serve as a basis for the optimisation of CLT products and the development or improvement of design concepts, respectively. Moreover, the obtained large ‘post‐elastic’ capacity reserve of CLT consisting of high quality boards could lead to a better utilisation of the raw material.     

Structural integrity of pipelines: T-stress by line-spring

The elastic T‐stress is an important constraint parameter for characterizing elastic–plastic crack‐tip fields and in fracture assessment procedures. However, many of the methods reported in the literature for estimating T‐stress are not easily suited for surface‐cracked pipes because these are three‐dimensional in nature. Here, the line‐spring method is demonstrated to be an efficient and accurate tool for the constraint estimation in surface‐cracked pipes. Detailed three‐dimensional analyses are performed to verify the line‐spring results. Using the line‐spring method, the effects of different crack geometries and diameter‐to‐thickness ratio on stress‐intensity factor (SIF) and T‐stress in circumferentially surface‐cracked pipes are examined. Further, a compendium of normalised SIF and T‐stress values for surface‐cracked pipes in remote tension and bending, calculated from a total of 1000 analyses, is tabulated. Finally, the application of an ‘elastic–plastic’ T‐stress under large‐scale plasticity is explored.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

A new approach for ctod evaluation in slow crack growth situations

In ductile materials slow crack growth often follows crack initiation, and the R-curve analysis characterizes both the crack initiation and crack growth behaviour of a material. The existing standards determine the crack tip opening displacement (CTOD) from the total clip gauge output by separating it into elastic and plastic components. The present work outlines the need for further separation of the plastic part of CMOD into the displacements due to slow crack growth and true plastic yielding. While the CTOD values for the elastic and true plastic yielding parts may be inferred according to the present methods, a new and simplified approach is proposed to infer crack growth CTOD from the corresponding displacement. The crack growth displacements may be obtained by using unloading compliance variation with crack growth. The present work is conducted on two materials, namely a steel and a 7004 grade aluminium alloy. The crack growth displacement in aluminium is found to be three times that in steel for a given crack growth. Geometrical as well as analytical comparison between the CTOD values obtained by the standard method and the proposed method is made. The comparison reveals that the existing method overestimates the CTOD value as reported by others. However, as the crack growth continues, the difference in CTOD values is found to diminish.     

Non‐linear analysis of structures using high performance hybrid elements

In this work, we describe the formulation and implementation for stress‐based hybrid elements for conducting non‐linear analysis of elastic structures. The motivation behind developing these elements is that they should be as simple to use as standard displacement‐based isoparametric brick elements, but at the same time, be relatively immune to the shortcoming that these elements suffer from, namely, ‘locking’ problems which occur when they are used to model plate/shell geometries, almost incompressible materials or when the elements are distorted, and so on. The formulation is based on a two‐field mixed variational principle. Numerical examples are presented to demonstrate the excellent performance of the proposed elements on a variety of challenging problems involving very large deformations, buckling, mesh distortions, almost incompressible materials, etc. Copyright © 2006 John Wiley & Sons, Ltd.     

A rate‐consistent and intrinsically path‐dependent incremental method for plastic buckling problems

This paper presents an incremental predictor–corrector method which is able to handle the continuous spreading of elastic unloading and, therefore, is particularly well suited to solve plastic bucking problems. The method, which deals explicitly with rate variables and equations, is (i) rate consistent, because it leads to the ‘true’ tangent matrix, and (ii) intrinsically path‐dependent, because it enables an adequate identification and characterization of the onset of elastic unloading within each incremental step. Copyright © 2000 John Wiley & Sons, Ltd.     

Equivalent localization element for crack band approach to mesh‐sensitivity in microplane model

The paper presents in detail a novel method for finite element analysis of materials undergoing strain‐softening damage based on the crack band concept. The method allows applying complex material models, such as the microplane model for concrete or rock, in finite element calculations with variable finite element sizes not smaller than the localized crack band width (corresponding to the material characteristic length). The method uses special localization elements in which a zone of characteristic size, undergoing strain softening, is coupled with layers (called ‘springs’) which undergo elastic unloading and are normal to the principal stress directions. Because of the coupling of strain‐softening zone with elastic layers, the computations of the microplane model need to be iterated in each finite element in each load step, which increases the computer time. Insensitivity of the proposed method to mesh size is demonstrated by numerical examples. Simulation of various experimental results is shown to give good agreement. Copyright © 2004 John Wiley & Sons, Ltd.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.     

Abschätzung von Bruchzähigkeitskennwerten aus der Bruch‐ oder Kerbschlagarbeit

Estimation of Fracture Toughness from the Charpy Fracture Energy By means of a simplistic mechanical model a mathematical relation between the total fracture energy of an edge‐cracked beam under bending and the fracture toughness was derived. Inserting the parameters of the standard Charpy specimens and accounting for the finite root radius and the shallowness of the notch, an equation to calculate fracture toughness from the Charpy fracture energy was obtained. Unlike the commonly used empirical correlation formulas, the presented equation is applicable to any elastic‐plastic material. From its theoretical basis and the underlying assumptions it is expected to be conservative in any case. Although the derivation only holds for the upper‐shelf regime, it also is applicable to the brittle‐to‐ductile transition regime as a lower bound. In this range, the degree of conservatism can be reduced by accounting for the well‐known shift of the transition temperature. Comparison with experimental data confirm these features of the presented formula.     

Experimental and numerical rolling contact fatigue study on the 32CrMoV13 steel

The aim of this work is to study pure rolling contact fatigue in 32CrMoV13 quenching and tempering steel. The study involves both experimental and numerical work. The influence of the roughness and the residual stresses on the mechanisms and zones of cracking were studied. The results show a rapid reduction in roughness during the first minute of test but even so there will be specimen deterioration. The residual stress profile after rolling contact tests have high compression values in the surface and at a depth of approximately 240 μm, which is related with the Hertzian maximal shear stress. The numerical simulation of the Hertzian loading was used both to determine the elastic shakedown of the material and to apply a high‐cycle multiaxial fatigue criterion. The three‐dimensional finite element analysis used in the numerical calculation includes elastic‐linear kinematic hardening plastic material and allows the introduction of an initial residual stress state. Taking into account the elastoplastic load induced by the Hertz pressure, low‐cycle fatigue tests were used to characterize the mechanical properties of the material. In order to validate the numerical simulation, the results of the calculation after elastic shakedown were compared with the values measured by X‐ray diffraction after rolling contact tests. The results showed a reasonable agreement between calculated and measured stresses. The Dang Van high‐cycle multiaxial fatigue criterion showed a good relationship with the experimental findings.