An integral expression that is domain independent in curvilinear coordinates and compatible with zero divergence of Eshelby's (Phil. Trans. Roy. Soc. (London) 244 (1951) 87.) energy momentum tensor was obtained from the principle of virtual work. By applying Eshelby's definition of the force on a material defect a general expression of the crack extension force for a curved crack in three dimensions, here called the F-integral, was derived from the domain independent integral expression. The F-integral is given explicitly for a number of curved cracks and found to be in agreement with previously known solutions, when available. The influence of crack surface and crack front curvature upon the various forms of the F-integral is discussed. The F-integral presented in this work is a generalisation of the J-integral (Rice, J. Appl. Mech. 35 (1968) 379.) to curved cracks in orthogonal curvilinear coordinates.

A method for calculation of fracture toughness requirements for ordinary structural steel is described. The method is based on typical loading cases and takes distributed load and fracture toughness into account while the size of an assumed crack is kept deterministic. Transferability has been studied by means of full scale testing of structural elements and laboratory specimens extracted from these. Particularly for older and inhomogeneous steels transferability is obtained under similar constraint and for full thickness laboratory specimens. Examples of toughness as a function of required safety are given.

The expression for the J-integral at a point on a three-dimensional crack front, obtained from a surface independent integral, is in general a sum of a contour integral and an area integral. In this work a general expression of an area integral for a crack with a curved front is derived in curvilinear coordinates. In certain situations the area integral vanishes and previously known cases are a straight crack front in plane stress or plane strain. The general conditions for a vanishing area integral are studied. It is shown that the area integral is non-zero for cracks with a curved front in the direction of crack extension. Some examples of curved cracks are given, for which the area integral vanishes and that are of interest in practice.

Steel profiles with slightly tapered cross section sides are not uncommon in older structures. The steels are often inhomogeneous and the fracture toughness varies across the thickness. In this project, the geometry of the standard three point bend specimen (ASTM E-1820) is modified to allow full thickness testing of tapered samples. The part of the project reported here deals with the J-integral of the modified specimen. The J-integral is calculated analytically and numerically with the finite element method for a linear and a non-linear material model and different specimen cross section proportions. A simple, single test fracture toughness evaluation procedure is proposed.

Certain types of structural elements, like riveted or rolled beams and girders, etc., of older steels, often have tapered flanges and the steels themselves are as a rule mor or less inhomogeneous. The fracture toughness at in-plane loading of related structural details, has been observed to vary significantly in the thickness direction and generally increases outwardly from the mid-section part. The most common fracture toughness test specimen used for testing of samples taken from oldes steel structures, in e.g. assessment work, is the three point bend specimen. This specimen, like all standard specimen types, is plane-parallel and if machined from a tapered part it will not necessarily yield a fair estimate of the effective fracture toughness for a through crack in situations where fracture toughness varies across the thickness. In view of this, a modified three point bend specimen with partly tapered sides has been designed so as to accomodate samples with this feature. The project includes calibration , testing and evalutation of the modified specimen. The firt part of the project, which is reported here, is mainly analytical and is aimed at obtaining a first and rough estimate of the order of influence the modified geometry upon crack tip parameters.

Steel profiles, such as angle sections in beams and girders, with slightly tapered cross section sides are not uncommon in older structures. Further, the fracture toughness of older and inhomogeneous steels varies in general across the thickness of a sample. The thickness of a machined, standard parallel-sided specimen of a tapered sample is in practice seldom more than 60-70 % of the full sample thickness and such a specimen captures in general inferior core material only. In view of this conundrum, a modified three point bend specimen with partly tapered sides has been designed so as to accommodate tapered samples. The project includes calibration, testing and evaluation of the modified specimen. The part of the project, which is reported here, is part analytical and part numerical and aimed at calculation of the stress intensity factor for various crack lengths, taper and specimen cross section proportions.

In a recent work Z.-H. Jin and C. T. Sun (2004) presented a derivation of the J-integral from the potential energy of a system, which is thought to circumvent previous cumbersome or flawed derivations. Two items in this work call for discussion; one is related to the effect of a singularity upon different types of contour integrals and the other to the strain energy difference term.

The complex solution method of Okubo for the deflection of a thin circular aelotropic plate with simply supported edge and uniform lateral load was extended to an elliptic plate by Ohasi. In his work however several inconsistencies appear, of which at least one disqualifies a central part. From a revisit to the works of Okubo and Ohasi a new solution for the deflection of a thin elliptic aelotropic plate with simply supported edge and uniform lateral load emerged. The solution is a generalisation of Okubo's solution and is valid for any angle between material and geometric principal axes. Previously known solutions, including those for circular plates, are reproduced as special cases of the new solution and results of numerical calculations in new situations appear reasonable.

The stress intensity factor and the J-integral have been derived analytically and numerically for a modified three-point bend specimen with partly tapered sides, for various crack lengths, taper and specimen cross-section proportions, in order to allow full-thickness testing of tapered samples, common in older steel structures, to obtain a fair effective fracture toughness value for a through thickness crack in inhomogeneous materials. The stress intensity factor is obtained with the approximate analytical method of Kienzler and Herrmann, based on the concept of material forces. The J-integral is calculated numerically with a 3D finite element model for a linear elastic material and an elastic ideal-plastic material. A simple single specimen fracture toughness evaluation procedure is proposed. It is found that the effect of taper in the range encountered in practice is small, of the order of a few percent.

A general method to determine the crack extension force F related to a finite segment of a curved crack is presented. A path independent integral expression that holds in curvilinear coordinates is derived from the principle of virtual work. An appropriate virtual displacement field allows variation of the position of a crack tip. F is related to the path independent integral expression through variation of a total energy expression. To illustrate the applicability of the method F is derived for the conical crack in axisymmetric loading. For a plane crack with a straight front F is identical to the J-integral, Rice (1968).

Vanishing divergence of Eshelby's energy momentum tensor allows formulation of path or domain independent integral expresssions of the crack extension force. In this work, a decomposition scheme of this tensor is presented, which results in zero divergence decomposed parts that allow formulation of expressions yielding the Mode I, II and III crack tip parameters J and K, with particular emphasis on Mode III, at present. By using the Mode III decomposed part of Eshelby's tensor and the virtual crack extension method, a path and a domain independent integral, both new, for the crack extension force of a plane cirular crack in axi-symmetric Mode III loading, are derived as examples of applicatioin.

A domain independent integral expression that is derived from the principle of virtual work and holds in curvilinear coordinates is used to derive the energy release rate for a penny-shaped crack in a linear piezoelectric solid. The virtual mechanical and electric displacement fields are chosen to allow variation of the crack tip position. Results for the energy release rate for a finite crack segment and point-wise show the effect of crack front curvature in a piezoelectric material.

method for estimating the residual fatigue life of service damaged and corroded structural members are developed. Scratch and hit marks, misplaced or empty holes that are exposed to corrosion are considered. The method is based on Palmgren-Miner's damage accumulation rule and field registered stress collectives. Standard methods are used to take effects of stress concentration and corrosion into account. If stress concentration and corrosion combine as is usually assumed, then corroded scratch marks may affect residual life

Brottmekanisk provning, dragprovning och slagprovning av konstruktionsstål från undre ramstång i huvudfackverk, från långbalk och från tvärbalk i den nedmonterade järnvägsbron över Åby älv, uppförd 1951.

A compilation of data from very different sources shows that correlation between notch toughness and fracture toughness cannot be universal. It is shown that common empirical relationships grossly underestimate fracture toughness of inhomogeneous base material and, worse, that toughness requirements on weld metal in terms of notch toughness do not necessarily guarantee the desired safety against fracture

A domain independent integral is obtained from the principle of virtual work. A suitable choice of the virtual displacement field allows variation of the position of a crack tip. For materials possessing a strain energy function Eshelby's definition of the force on a point defect is used to obtain the crack extensioin force. The method is general and allows treatment of a crack whose surfaces and front are curved by unsing curvilinear coordinates. To illustrate the applicability of the method three examples of the point-wise crack extension force are given, with different combinations of crack surface and crack front curvature. A general expression of the crack extension force for curved cracks is suggested. In Cartesian coordinates the proposed expression reduces to the J-integral of Rice.

The effect of corrosion and stress concentration on the fatigue life of older structural steel is studied for a typical loading history and some types of damage encountered in practice. The structural steel specimen represents a railway beam that is severely corroded during the 80 years of service. Standard uniaxial tension testing at ambient temperature and notch toughness testing at -20 deg C are carried out. It is found that the residual life is, in general, reduced both by mechanical damage and corrosion. The combined effect may be reduce the residual life significantly.

A point-wise value of the J-integral for a radially expanding plane axisymmetric crack, which is path independent an yields the energy release rate, has been derived from Eshelby's energy momentum tensor taken in cylindrical coordinates. In particular the area integral in the expression for J by Carpenter et al. has not been found to be zero, as recently advanced by Jonsson and Nilsson, but in agreement with results of others, e.g. Bergkvist and Lan Huong

The blend rule for the effective fracture toughness of a layered material was originally derived from the special case of a through crack in a globally elastic material and later extended to accomodate non-linear behaviour. It is now derived from a general case by considering material elements of finite size and of different toughness along and around the tip of a crack. Experimental results obtained with an inhomogeneous ordinary structural steel which support the blend rule are presented. It is shown that the effective fracture toughness governs the load-bearing capacity of a cracked full-scale structure. Some further results found in the literature for the heat-affected zone material of a high-strength microalloyed quenched and tempered structural steel and computational results for a structural steel typical of a nuclear pressure vessel are shown to support the blend rule.

A previous analytical solution of the deflection of a thin circular aeolotropic plate, with simply supported edge and uniform lateral load, has been used to derive approximate series expressions for the plate support reaction, which are directly applicable in practice. The support reaction, which has been calculated for some typical anisotropic materials of varying degree of anisotropy, varies significantly along the plate perimeter and strongly anisotropic materials require in general a higher order series solution. Certain solution constants of previous deflection approximations were not found to harmonize and are therefore recalculated.

Older, ordinary structural steel sections are often inhomogeneous in their microstructure, having a low-toughness "core" material and a higher toughness surface material. The overall behaviour of such structural components has been found to be more ductile than predicted by "weakest-link" arguments. This behaviour is analysed by extending the theory of the globally elastic fracture of layered sandwich materials to accommodate non-linear behaviour and mixed toughness. The extended theory has been applied to the behaviour of full scale beams typical for bridges.

Because of the nonlocal interparticle forces inherent in peridynamics, surface, boundary, and end effects appear in 3D, 2D and 1D body problems, respectively. In certain situations, the effect is seen as a disturbance, and various efforts, mostly centering on 2D and 1D problems, have been made to reduce it. A simple method has been derived to remove the end effects in a 1D body by homogenizing the body. When a certain body type, common in practice, is homogenized, its linear elastic behavior, independent of the interparticle force range and with a finite number of material points, in the limit infinite, is identical to that of a corresponding classical continuum mechanics body.

In peridynamics, boundary effects generally appear due to nonlocality of interparticle forces; in particular, end effects are found in 1D bars. In a previous work by Eriksson and Stenström (J Peridyn Nonlocal Model 2(2):205–228, 2020), a simple method to remove end effects in certain types of 1D bars, or to homogenize such bars, was presented for bars with constant micromodulus. In this work, which is a continuation of Eriksson and Stenström (J Peridyn Nonlocal Model 2(2):205–228, 2020), the homogenizing procedure is applied to bars with a linear, or “triangular,” micromodulus. For the examples studied, common in practice, the linear elastic behavior of a homogenized bar, is identical to that of a corresponding classical continuum mechanics bar, independently of the interparticle force range and total number of material points of the bar.

The deformation and failure in bending of precracked broad flange beams of type HEB 400 and similar was studied. Six m long beam elemments were slowly loaded to fracture or plastic deformation at the testing temperature -30 deg C. The J contour integral for the full scale beam element was calculated with the finite element method for two typical crack lengths. J as a function length was approximated with the R6-method using analytical estimates of the fully plastic bending moment for thin-walled beam cross-sections. Critical values of J and CTOD were obtained with standard compact and SENB specimens machined from the beam flanges. Although scatter is considerable the fracture toughness ranges of the laboratory specimens and of the full scale tests are very much in coincidence. By means of a residual strength diagram the borderline between long and short cracks is illustrated. Here long cracks are those which caused fracture and short cracks just plastic deformation but no crack growth.

In order to model the wood chipping process, the primary process parameters have been identified and their first order interaction studied. The model is analytical and incorporates, in particular, the influence of sliding friction between the wood chipping tool and the log. To estimate the accuracy of the analytical model, a Finite Element (FE) analysis of the problem considered was also performed. The analytical model and the FE analysis are both restricted to small deformations and linear elastic orthotropic material behaviour. The most severe limitation with both the analytical and the FE model is the assumption of linearly elastic material. On the other hand it is felt that existing models of anisotropic plasticity in metals are lacking too much of physical relevance, if applied to wood. The analytical model predicts the normal and shear strain distribution in the crack-plane prior to crack initiation. The analytical distributions are in reasonable agreement with the corresponding distribution of the FE analysis. Based on experimental findings, it is suggested that the stress field over the entire crack-plane, in conjunction with the stress field close to the tip of the chipping tool, are critical for chip creation, rather than just the latter.

A path independent integral expression for the crack extension force of a two-dimensional circular arc crack is presented. The integral expression, which consists of a contour and an area integral, is derived from the principle of virtual work. It is implemented into a FEM post-processing program and the crack extension force is calculated for a circular are crack in a linear elastic material. Comparison with exact solutions by Cotterell and Rice for the effective elastic stress intensity factor shows acceptable accuracy for the numerical procedure used

Three-point bend and compact tension specimens, taken from beam sections of modern and older ordinary C-Mn structural steels, were tested at intermediate loading rates at room temperature and -30°C. The experimental work, except the loading rates used, was performed according to ASTM E-813. In order to investigate transferability of data, full-scale beam sections were also tested at intermediate loading rates. The fracture toughness of C-Mn structural steels depends strongly on the loading rate, and decreases rapidly with increasing loading rate at and just above the maximum prescribed in ASTM E-813. Fracture toughness data for structures exposed to intermediate loading rates indicate the requirement for testing at appropriate loading rates. The behaviour of full-scale structural elements subjected to intermediate loading rates can, provided certain conditions are fulfilled, be predicted from data obtained from small laboratory specimens

For a cracked structural component the crack tip loading rate very often exceeds the maximum rate prescribed in current fracture toughness standards, e.g., ASTM E813. In a previous work, Lorentzon and Eriksson, it was found that loading rates at and just above the ASTM limit significantly affect the fracture toughness of ordinary structural steels. The effect of the loading rate upon the stress and strain distribution around a mode I plane strain crack tip has been calculated with the finite element method. A boundary layer formulation based upon Westergaard's exact analytical solution for an infinite plate and a Perzyna visco-plastic material model has been used. The results show that viscous effects are significant only very close to the crack tip.

Numerous cracks have been discovered in the Vårby Bridge near Stockholm, Sweden. All cracks are found at the junctions between cross girders and the main girders, more specifically, at the welds connecting the vertical web stiffeners to the top flanges of the main girders. In order to identify the reason for the observed cracks, an ongoing investigation under the commision of the bridge owner was started in the spring of 2009. One conclusions so far is that the observed cracks are to 100 % certainty a result of fatigue.FEM-modeling is currently going on as a part of a master thesis. As the fatigue process is distortional the propagation phase of the observed cracks might slow down or even stop. One task will thus be to determine the stress intensity factor versus crack length relationship in order to model a growing crack. Suggestions of possible of methods of refurbishment, based upon the results, will then be presented to the bridge owner.

Numerous cracks have been discovered in the Varby Bridge near Stockholm, Sweden. All cracks are found at the junctions between the cross girders and the main girders, more specifically, at the welds connecting the vertical web stiffeners to the top flanges of the main girders. The cracks might possibly be causing serious problems if they are allowed to propagate through the entire length of the weld, thereby permitting out-of-plane bending of the main girder web. In order to identify the reason for the observed cracks, an ongoing investigation under the commision of the bridge owner was started in the spring of 2009. One conclusion so far, is that the observed cracks conclusively are a result of fatigue As a part of a master thesis, FEM-modelling is currently under way. As the fatigue process is distortional, the propagation phase of the observed cracks might slow down or even stop. One task will be to determine the stress intensity factor versus crack length relationship in order to model a growing crack. The final chosen method of refurbishment will be based upon the results of the study and will be implemented in cooperation with the bridge owner

In this work, the finite element method is employed to gain an understanding of the behaviour of a cracked bridge roller bearing in service. The cracked roller is considered as an edge-cracked disk (two-dimensional plane strain system) subjected to a radial compressive line load. The crack parameters K_I and K_II are calculated for the relevant load configuration and angle of disk rotation. The calculated data are also used to check the accuracy of approximate SIF solutions reported earlier [1] and [2]. For plain Mode I loading very good agreement is found between the obtained results and data presented in Schindler and Morf (1994). 

The paper is aimed at finding the likely failure mechanism of a bridge roller bearing made of high strength martensitic stainless steel. Spectroscopy and finite element stress analysis of the roller indicated that an initial radial surface crack, found at an end face of the roller and close to the contact region, was induced by stress corrosion cracking (SCC). The initial crack subsequently changed shape and increased in size under growth through fatigue and finally formed a quarter-circle radial crack centred on the end face corner of the roller. Numerically computed stress intensity factors for the final crack showed that crack loading was predominantly in Mode II. For a crack size as observed on the fracture surface, the maximum service load, as specified by the manufacturer, enhanced by a certain roller bearing misalignment effect, was sufficient for failure through fracture.

With the innovation of elastomeric bearings in the mid-1950s steel bearings lost their interest and significance both in research and development and subsequently even in application. Steel bearings were gradually abandoned in bridges, followed by the technical literature and design standards. However, a great number of steel bearings remain today in service world-wide and will pose their particular challenges in the future. To the author’s knowledge, just in Sweden, high strength stainless steel bearings still exist in no less than some 650 bridges. In recent years, a large number of such bearings have failed with an alarmingly high frequency in Sweden during a period of six to twenty years after installation making them a serious maintenance cost issue.

After a brief summary of the history of high strength stainless steel bearings, the paper reviews service experience of such bearings in Sweden and elsewhere. Accompanying finite element analyses were performed in order to gain insight into the likely failure mechanism. Finally, this comprehensive review leads to a conclusion that identifies the causes of the failures occurred and makes some recommendations.

Although previous investigations of the stainless steel bearings have not been able to clearly identify the cause(s) of the failures occurred, it is found that the failures primarily occurred due to initiation of cracks through stress corrosion cracking followed by fatigue crack growth requiring a certain stress range and a sufficiently large number of cycles until final failure ensued through sudden and instable fracture after fatigue growth to a critical crack size.

In integral abutment bridges clamped abutment piles are in addition to a compressive normal force subjected to bending load cycles from daily and yearly temperature variations. Through experiments with full-scale specimens representing a clamped pile it is shown that a steel pipe pile loaded in bending can withstand several hundred load cycles at strain ranges greater than 6 times the yield strain with almost full load bearing capacity. By means of an example it is shown that by permitting pile strains greater than the yield strain, in contrast to most present design codes, integral abutment bridges can be erected with a span length up to 500m and a prospected service life of 120 years.

In integral abutment bridges clamped abutment piles are in addition to a compressive normal force subjected to bending load cycles from daily and yearly temperature variations. Through experiments with full-scale specimens representing a clamped pile it is shown that a steel pipe pile loaded in bending can withstand several hundred load cycles at strain ranges greater than 6 times the yield strain with almost full load bearing capacity. By means of an example it is shown that by permitting pile strains greater than the yield strain, in contrast to most present design codes, integral abutment bridges can be erected with a span length up to 500m and a prospected service life of 120 years.

Peridynamics is a nonlocal formulation of solid mechanics capable of unguided modelling of crack initiation, propagation and fracture. Peridynamics is based upon integral equations, thereby avoiding spatial derivatives, which are not defined at discontinuities, such as crack surfaces. Rice’s J-contour integral is a firmly established expression in classic continuum solid mechanics, used as a fracture characterizing parameter for both linear and nonlinear elastic materials. A corresponding nonlocal J-integral has previously been derived for peridynamic modelling, which is based on the calculation of a set of displacement derivatives and force interactions associated with the contour of the integral. In this paper, we present an alternative calculation of the classical linear elastic J-integral for use in peridynamics, by writing Rice’s J-integral as a function entirely of displacement derivatives. The accuracy of the proposed J-integral on displacement formulation is investigated by applying it to the exact analytical displacement solution of an infinite specimen with a central crack and comparing the exact analytical expression of its J-integral. Further comparison with a well-known peridynamic crack problem shows very good agreement. The suggested method is computationally efficient and further allows testing of the accuracy of a peridynamic model as such.

The J-integral is in its original formulation expressed as a contour integral. The contour formulation was, however, found cumbersome early on to apply in the finite element analysis, for which method the more directly applicable J-area integral formulation was later developed. In a previous study, we expressed the J-contour integral as a function of displacements only, to make the integral directly applicable in peridynamics (Stenström and Eriksson in Int J Fract 216:173–183, 2019). In this article we extend the work to include the J-area integral by deriving it as a function of displacements only, to obtain the alternative method of calculating the J-integral in peridynamics as well. The properties of the area formulation are then compared with those of the contour formulation, using an exact analytical solution for an infinite plate with a central crack in Mode I loading. The results show that the J-area integral is less sensitive to local disturbances compared to the contour counterpart. However, peridynamic implementation is straightforward and of similar scope for both formulations. In addition, discretization, effects of boundaries, both crack surfaces and other boundaries, and integration contour corners in peridynamics are considered.

In this work, the essential work of fracture (EWF) method is introduced for a peridynamic (PD) material model to characterize fracture toughness of ductile materials. First, an analytical derivation for the path-independence of the PD J-integral is provided. Thereafter, the classical J-integral and PD J-integral are computed on a number of analytical crack problems, for subsequent investigation on how it performs under large scale yielding of thin sheets. To represent a highly nonlinear elastic behavior, a new adaptive bond stiffness calibration and a modified bond-damage model with gradual softening are proposed. The model is employed for two different materials: a lower-ductility bainitic-martensitic steel and a higher-ductility bainitic steel. Up to the start of the softening phase, the PD model recovers the experimentally obtained stress-strain response of both materials. Due to the high failure sensitivity on the presence of defects for the lower-ductility material, the PD model could not recover the experimentally obtained EWF values. For the higher-ductility bainitic material, the PD model was able to match very well the experimentally obtained EWF values. Moreover, the J-integral value obtained from the PD model, at the absolute maximum specimen load, matched the corresponding EWF value.

Indenters in the slip planes of a bolted lap joint increase its load bearing capacity. In an experimental study, conducted at the Luleå University of Technology, Sweden, a part of the European R&D project PROLIFE, RFCS 2015-00025, indenters between two plates a) were loaded in compression and b) shear loaded in a lap joint. The load to press a 2.5 mm diameter stainless steel indenter 2.3 mm into the plates was 11 kN and the effective friction of the joint was improved. In a reference test with two shear planes and plain as rolled plates, no indenters and an M30 bolt pre-loaded to 320 kN, the joint slip resistance force was 54.5 kN and the effective friction coefficient μeff=0.09. For an identical arrangement but with 29 indenters per shear plane, the slip resistance was close to 250 kN and μeff was increased to 0.40, at the current Eurocode acceptable joint slip of 0.15 mm.