Crack Opening Displacement Concept Research Articles

Determination of opening stresses for railway steel under low cycle fatigue using digital image correlation

Crack closure phenomenon is important to study as it provides an estimation to fatigue life of the components. It becomes even more complex under low cycle fatigue (LCF), since under LCF high amount of plasticity is induced within the material near notches or defects, as a result the assumptions used by linear elastic fracture mechanics (LEFM) approach become invalid. Evaluation of opening stresses for mechanical components undergoing LCF phenomenon requires a robust methodology to correctly predict the fatigue life. In this study, an experimental campaign was carried out for determination of opening stresses of railway steels (25CrMo4 and 30NiCrMoV12) subjected to LCF using digital image correlation (DIC) technique. The concept of crack opening displacement (COD) was used for the analysis. Two different methodologies were introduced to analyze experimental data for the identification of opening levels. Experimental results were then compared with crack closure prediction model, Newman model. Results from Newman model agreed well with the experimental analysis. Newman model provided very good prediction for strain ratio Rε = −1, however, for the materials undergoing strain ratio Rε = 0, stress ratio must be considered rather than strain ratio, because Newman model can’t predict stress relaxation behaviour.

Numerical Comparison Research on the Solution of Stress Intensity Factors of Multiple Crack Problems

A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization; 2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix; 3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.

Numerical solution of stress intensity factors of multiple cracks in great numbers with eigen crack opening displacement boundary integral equations

Based on the eigen crack opening displacement (COD) boundary integral equations (BIEs), a newly developed computational approach is proposed for the analysis of multiple crack problems. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity BIEs in an iterative fashion with a small size of system matrix. The interactions among cracks are dealt with by two parts according to the distances of cracks to the current crack. The strong effects of cracks in adjacent group are treated with the aid of the local Eshelby matrix derived from the traction BIEs in discrete form. While the relatively week effects of cracks in far-field group are treated in the iteration procedures. Numerical examples are provided for the stress intensity factors of multiple cracks, up to several thousands in number, with the proposed approach. By comparing with the analytical solutions in the literature as well as solutions of the dual BIEs, the effectiveness and the efficiencies of the proposed approach are verified.

Porosity dependence of strength in brittle solids

The effect of pore size and pore volume fraction on strength in brittle solids is evaluated. The analysis considers that the strength degradaton of a solid containing a large number of spherical pores is due to a strong effect of porosity on Young's modulus. Each pore is assumed to possess radial or annular flaws emanating from the pore surface whose lengths are considered to be independent of pore size. The effect of stress concentration induced by the presence of the pore is included in the equation for strength through the Young's modulus dependence of porosity originally developed using the concept of crack opening displacement. It is shown that the strength of a solid containing spherical pores is controlled by the pore size, pore volume fracton and the radial (or annular) crack size to pore size ratio. Predicted variation of strength with pore volume fraction is tested against experimental data for glass and polycrystalline alumina.

Effect of porosity and stress concentration associated with pores on elastic creep by crack growth in brittle solids

The solution for the creep rate of a porous solid as a function of pore volume fraction, pore size and the radial or annular flaw size, is presented. The analysis is based on the concept of crack opening displacement which assumes that the displacement of a solid under stress is solely due to the opening of all radial or annular cracks emanating from the surface of the cavities. It is shown that the strain rate by crack growth is a strong function of pore size and preexisting flaw size. A linear relationship was found to exist between the strain rate by crack growth and the porosity volume fraction.

A model for the porosity dependence of Young's modulus in brittle solids based on crack opening displacement

A crack opening displacement concept has been introduced to model the porosity dependence of Young's modulus in polycrystalline and single phase solids. In developing the theoretical model, it is assumed that each cylindrical cavity possesses radial cracks and spherical pores possess annular flaws. When an external stress is applied on such a solid, its elastic response is shown to be governed by the pore size, the width of an annular flaw, the number of pores (or pore volume fraction) and the flaw to pore size ratio. The validity of the present approach is tested against a number of experimental data.[/p]

Comparative Evaluation of Fracture Mechanics Methodologies as Applied to Offshore Structural Design and Integrity Analyses

A comparative study was performed to evaluate the three most widely used elastic-plastic fracture mechanics methodologies. The three methods are the crack opening displacement (COD) design curve, which is widely used in the offshore industry, the British Central Electricity Generating Board (CEGB) R6 method, and the J-resistance curve method. Analyses of various flawed geometries based on these three methods were performed using consistent material fracture properties. The results of both the COD and R6 methods were compared with those of the J-resistance curve, which is the most rigorous of the three methods. The results of the COD concept, following BSI PD 6493 procedures, appear to be conservative at applied stress levels below 60 percent of the material yield strength and unconservative above this value. However, the results of the R6 method appear to be consistently conservative. The behavior of COD design curve solutions can be improved when the structural geometry and the actual stress-strain relationship are properly considered in the analyses.

The application of elastic-plastic fracture mechanics parameters in fracture safe design

The report summarizes some of the methods which are currently used for assessing the fracture toughness of materials under elastic and elastic-plastic conditions. The main parameters which are considered are (1) plane strain fracture toughness ( K Ic), (2) equivalent energy ( K Icd), (3) contour integral ( J) and (4) crack opening displacement (COD). Gross strain crack tolerance and stress concentration methods are also discussed. It is concluded that of these parameters, the contour integral and the crack opening displacement have most potential for future development. These two parameters are shown to be equivalent, however, at the present stage of development the COD concept has several advantages over the J concept. Firstly, the COD concept is able to take into account, secondary stresses, such as welding residual stresses. Because these stresses are in equilibrium, they do not appear in energy measurements to evaluate J. Secondly, the COD value is a physical measure of the crack tip conditions which includes the effect of stress state and thickness. It is, therefore, possible to measure and calculate COD levels for cracks in real structures. It is not possible to evaluate J for real structures since J methods are appropriate only to in-plane problems. This also means that partial wall (thumbnail) flaws are better characterized by the COD concept. The COD concept has been developed to a stage where it is possible to estimate the significance of flaws in welded structures provided the toughness of the material and the acting stresses or strains are known. This development is described and the method used to analyze tests on model pressure vessels with 6″ thick walls. A comparison is made with other methods, and it is concluded that although the COD analysis gives conservative estimates of the flaw size to cause failure, further work is necessary to be able to predict vessel burst conditions when failure is preceded by extensive plasticity and stable ductile tearing. A simple nomogram to determine COD levels to ensure leak before break conditions is also developed.

Deformation Behaviors and Fracture Characteristics in Plate Containing Two Collinear Notches

A series of research so far carried out, has clarified that among various kinds of multiple notches existing in steel structures, the collinear notch has the largest influence upon brittle fracture initiations. In this research, therefore, the collinear notch is taken up as the subject of study, and examination is made on deformation behaviors ahead of notch tip and on fracture characteristics by elastic-plastic analysis based on the finite element method (FEM) and by experiments.The results are summarized as follows : (1) A crack opening displacement (COD) obtained by FEM is approximately equal to the theoretical value by E. Smith and a measured value by Moire method, which enables the coefficient of interaction effects of notches to be estimated by FEM.(2) When notches are contiguous very closely to each other, fracture stress is decreased and at the same time the critical COD is decreased. Therefore the concept of COD is not simply applicable to that case. The reason for the occurrence of such phenomenon is clarified through examination of the relation between the plastic strain in notch tip zone and the COD which are respectively calculated by FEM.

Growth of fatigue cracks in metals and polymers

Empirical data on the propagation of tensile fatigue cracks in metals and thermoplastics have been examined. It was found that a cyclic crack propagation relationship, based on the stress intensity factor concept, exists which can be successfully utilised for both types of materials. The proposed equation has a form /.a x = MA n where A is a function of ΔK and mean K. The analysis of results suggests that this equation incorporating the influence of mean stress intensity factor provides an excellent fit to the investigated data. The possible modified forms of such a relationship in terms of strain energy release rate, the crack tip yielding and the crack opening displacement concepts are also indicated.

Compatibility of linear elastic ( k1 c) and general yielding (COD) fracture mechanics

An investigation of the applicability of the general yielding fracture mechanics concept of crack opening displacement is made in relation to the well established concepts of linear elastic fracture mechanics. The nature of the relationship between crack opening displacement and stress intensity factor is explored using the concepts of linear elasticity and model analyses proposed by Dugdale. Tests on C/Mn steel, a low alloy steel and a manual metal arc weld deposit define the nature of this relationship in the stress region for which the theoretical compatibility is no longer to be expected. Based on this relationship a single fracture test procedure may be used to determine COD or K 1 c (depending on the stress conditions at failure). The results of the tests analysed in terms of J, the contour integral, determine the significance of J as a fracture characterising parameter. Defect tolerance parameters for the assessment of the significance of flaws in welded structures are also introduced.

An Investigation of static COD and dynamic COD

It is generally accepted that KIC criterion is usefull for initiation of brittle fracture, but this method has a weak point that the linear elastic fracture mechanics approach must be applied only when a yield zone arround a crack tip is very small compared to crack length. Brittle fracture of a structure after large scale yielding, however, is observed sometimes at comparative high temperature.As an approach to this phenomenon, the crack opening displacement (COD) concept available in large scale yielding was proposed by A. A. Wells about ten years ago. After that, many papers about mathematical theory of COD concept or engineering criterion of fracture have been published. Nevertheless some problems to be solved have been left still. One of them is how to decide the critical COD for brittle fracture at comparative high temperature, since at such temperature a ductile crack mostly propergates before initiation of cleavage fracture.For this reason, in this paper the initiation of ductile crack was investigated precisely using single-notched and double-notched specimens of 60 kg/mm2 high strength steel in static and dynamic pure bending COD test. As a result, it was made clear that the critical COD for the initiation of ductile crack is independent of temperature and strain rate. Moreover, an influence of notch distance of double-notched specimen on critical COD and distorsion behavior of specimen was studied experimentally and numerically, since double-notched specimen have been used sometimes in COD test. And the most suitable notch distance was obtained, of which the double-notched specimen shows small interaction of notches and distorsion behavior close to single-notched specimen.

On the sharpness of cracks compared with Wells's COD

Crack opening displacement concept in fracture mechanics, considering crack tip radius as measure of extension potential

Crack opening displacements from elastic-plastic analyses of externally notched tension bars

Ten elastic-plastic solutions of externally sharply notched tension bars, produced by digital computation using the Marcal-King finite element method, have been assessed with respect to notch stress intensifications and crack opening displacements, following an initial definition of the limitations of finite mesh size. These cases embrace plane stress, plane strain and axial symmetry, with changes of notch depth and flank angle. It is shown in all the cases studied that first yield, and maximum plastic stresses, occur always at points removed from the crack plane, and that the crack plane plastic zone stress, which is less than the maximum, is also approximately constant from the instant of first loading to general yield. It thus determines the fully plastic constraint factor in many cases, and confirms and simplifies the use of the crack opening displacement concept of crack extension force. In order to create a basis for comparison with the elastic-plastic solutions, the strip yielding COD model is modified to accommodate plane strain conditions. A simple correction procedure is then established, on the basis of linear fracture mechanics, in order to predict COD values at points within the crack near to its tip. Acceptable correlations are then established between COD distributions near to the crack tip, as determined by the elastic-plastic and strip yielding models. A further correlation is attempted for the slit notch case, between values of G, determined on the one hand from the products of crack plane plastic zone stresses, and opening displacements at nodal points corresponding with the elastic-plastic boundary within the notch, and crack extension forces on the other hand, arising from the Bowie solution for the same specimen geometry, modified with a plastic zone crack length factor. The correspondence is close, both for plane stress and plane strain, and also with the accepted elastic solution for axial symmetry. The potential significance of these findings is discussed in terms of COD measurements in plane strain fracture toughness testing, experimental study of the plane strain/plane stress transition in specimens of finite thickness, and computational development of the strip yielding model.

The crack opening displacement approach to fracture mechanics in yielding materials

An introduction is given to the fracture mechanics approach employing the concept of crack opening displacement for application to situations in which linear elastic fracture mechanics is invalidated by yielding. The hypothesis of a critical crack opening displacement to fracture has been examined experimentally using mild steel specimens of vastly differing dimensions. Subsidiary experiments have been carried out to define the factors responsible for the apparent effect of absolute size on the results. A theoretical analysis simulates elastic-plastic conditions to give a relationship between applied stresses and strains, crack length, and crack opening displacement.