). Studying the interaction of crack-like flaws using the MATLAB toolbox int_defects. Engineering

A


Introduction
Finite Element Analysis (FEA) can be used to investigate structures containing crack-like flaws.Typically, this type of investigation is motivated by a need to predict the initiation or progress of fracture, fatigue or ductile tearing from a pre-existing structural defect.The crack-driving force that results from a known structural loading state is determined using FEA and then compared with the material's fracture toughness properties to predict whether or not the crack will extend.
Compendia of functions which relate the load applied to a structure containing a crack-like flaw to the crack tip Stress Intensity Factor (SIF) or J-integral, or the structure's plastic limit load, are used widely in practical fracture-mechanics-based analysis of structural integrity [1].These pre-calculated functions are normally based on numerical modelling results, often with experimental validation for selected cases.In the Failure Assessment Diagram (FAD) approach, which forms the basis of general-purpose structural integrity assessment procedures such as BS 7910 [2] and R6 [3], compendia of SIF and limit load solutions are used to minimise the need to perform numerical modelling of structures containing crack-like flaws on a case-by-case basis.
Most modern FEA pre-processors such as Abaqus/CAE [4] can generate large-scale 3-dimensional meshes automatically, which makes it feasible to consider many different geometries and loading conditions in a parametric set of cracked-body analyses.Since the late 1980s, specialised packages for pre-and post-processing of FE models of fracture have also emerged, often incorporating specialpurpose meshing algorithms.These include FEACrack (Quest Integrity USA LLC, USA) [5], Zencrack (Zentech International Ltd., UK) [6] and FRANC3D (Fracture Analysis Consultants Inc., USA) [7].The Dual Boundary Element Method (DBEM) [8,9] has also frequently been used to determine the crack driving force in parametric studies which involve multiple crack geometries and loading conditions, sometimes using hybrid FEM-DBEM sub-modelling schemes [10].
One common complication in structural integrity analysis the possibility of multiple structural defects occurring close to one another [11].If the defects are close enough to interact then the effect of this interaction should be considered carefully when judging the structure's fitness-for-service.On the other hand, if it can be shown that each defect has a negligible effect on the other then each one can be considered in isolation.Consequently, integrity assessment procedures often contain criteria for predicting whether multiple defects will interact and rules specifying how the analysis should proceed if the interaction is judged to be significant, e.g. by re-characterising multiple flaws as a single, enclosing flaw [12][13][14].A comparison of several interaction criteria for co-planar flaws is given by Hasegawa & Miyazaki [15].
Formulating defect interaction criteria and recharacterisation rules is challenging because the range of possible situations that may occur in any individual assessment (in terms of defect sizes, shapes, spacings, loading states and failure mechanisms) is very broad.Any general interaction criterion must be valid for a wide range of different cases, so validation requires many experiments or models.Over the last three decades, schemes for parametric analysis of interacting flaws have been developed and refined [16][17][18].This technical note outlines a method for performing large-scale parametric analysis of interacting crack-like flaws.The method has been used in several previous studies [19][20][21][22][23], and has now been implemented in the free and open-source "int_defects" toolbox for MATLAB [24] which interfaces with the Abaqus FEA package [4,25].int_defects is designed for analysing the interaction of co-planar elliptical embedded flaws and semi-elliptical surface flaws in plates and thick-walled pipes.It supports arbitrary flaw dimensions, loading states and material stress-strain characteristics.

Description and capabilities
int_defects can determine SIFs, J-integrals and plastic limit loads for single semi-elliptical surface cracks and elliptical embedded cracks of arbitrary size/shape in a plate or pipe of finite thickness, as well as co-planar pairs of such cracks.It is designed to automate the generation, execution and post-processing of large sets of FE models which contain different crack geometries and loading states.int_defects includes functions for post-processing elastic-plastic model results to collate crack tip J-integral values.The J-integral is initially determined by the Abaqus/Standard FE solver using an equivalent domain integral [26] and in elastic analyses the Mode I SIF is calculated from the J-integral using the interaction integral method [27].Both J-integral and SIF results can be checked for path-independence automatically using the toolbox.int_defects also post-processes elastic → perfectly-plastic model results to determine plastic limit loads.For example, int_defects can automatically determine the Local Limit Load (LLL), i.e. the load at which a plastic ligament forms between a crack and the back face of the section.The LLL is significant for structural integrity assessment and is used in the FAD approach to conservatively estimate a flawed structure's proximity to plastic collapse [1,28].
In post-processing, int_defects can compare results from a set of linear elastic models of single cracks with those from models of paired cracks, to determine the degree of stress interaction between the cracks in proximity.The elastic interaction factor ( ) can be calculated as: where K I int is the SIF at a flaw that is part of an interacting pair and = K I is the SIF for the same flaw in isolation.all functions of position on the crack tip line, which is defined using the ellipse parametric angle .int_defects is designed for analysing co-planar pairs of cracks in simple structures; it cannot analyse structures containing volumetric defects or cases where more than two flaws exist (e.g.crack networks).It is also restricted to cracks which can be characterised as being either elliptical (for embedded flaws) or semi-elliptical (for surface flaws), which is the characterisation convention used in R6 [3] and BS 7910 [2].Since current fracture assessment procedures use relatively simple interaction rules, int_defects is a useful aid to procedure development despite these limitations.int_defects can also generate finite element models of single semielliptical surface defects and elliptical embedded defects.Therefore, in addition to its primary use for investigating defect interaction, int_defects can also be used for parametric studies of single cracks.This is useful for compiling compendia of pre-calculated SIF, limit load and strain energy release rate solutions for use in assessment.
int_defects is distributed as a MATLAB toolbox.It requires installations of the numerical computing environment MATLAB [24], the FE pre-processor Abaqus/CAE [4] and the FE solver Abaqus/Standard [25].The MATLAB Parallel Computing Toolbox [29] is recommended for processing large sets of models, but is not required.The distribution includes a user guide, a set of examples and commented MATLAB code.Validation of int_defects by comparison of results with well-known elastic, elastic-plastic and limit load solutions has been performed; some example validation cases are given in Appendix A.

Workflow
Fig. 1 shows the workflow used internally by int_defects.To analyse a set of models of flawed plates or pipes, a user performs the following steps in the MATLAB environment: 1. Decide on the set of cases to be analysed then define a data structure containing parameters which describe these cases.Save the data structure in a MATLAB .matfile.2. Create and execute the set of models which represent the cases defined in Step 1 using the functions int_defects_-write_input_parametric and int_defects_run_parametric_parallel, respectively.3. Extract the results.Depending on the type of analysis performed, these may be either contour integrals (K, J or elastic T-stress) or limit loads.
Additional actions subsequent to Step 3 will depend on the type of analysis being performed.For example, if the objective of the analysis is to determine elastic interaction factors (see Eq. ( 1)) for closely-spaced flaws then Steps 1-3 would be performed twice: once for a set of models of single flaws and once for models of interacting flaws.Then, the interaction factors would be calculated Fig. 1.Workflow elements used by the int_defects toolbox.Red boxes denote key high-level functions which loop over all models in a set.A set of models is formulated and solved (left-hand section), then the results are extracted and post-processed (right-hand sections).Determination of crack tip contour integrals and determination of limit loads require different post-processing methods.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)from the contour integral results (int_defects includes the function int_defects_calc_interaction_factor for this purpose).

Usage examples
The following examples illustrate how int_defects can be used and the range of problems that it can solve.Each problem is posed in the form of a question that int_defects is used to answer.Details of the computer hardware used and execution times for each set of models are given in Appendix B.

Usage example: Elastic interaction between twin surface flaws
Question: "A large cylindrical pressure vessel contains a two identical crack-like surface flaws in the axial-radial plane.It may be pressurised and/or subjected to thermal shock.To what degree will the flaws interact, according to linear-elastic fracture mechanics theory?" The elastic interaction between flaws, i.e. the amount by which the presence of one flaw affects the SIF at another if the material is assumed to be linear-elastic (see Equation 1), is a key factor in judging the significance of flaw interaction for structural integrity.and spacings in the range d 8 mm 32 mm have been used to in- vestigate the effect of these geometric parameters (see Fig. 2).The vessel may be subjected to internal pressurisation, a thermal shock involving a rapid internal quench or a combination of the two.The through-thickness stress profile resulting from the thermal shock is shown in Fig. 3: this approximates the thermal shock load imposed on a nuclear Reactor Pressure Vessel (RPV) experiencing a Medium Loss-Of-Coolant Accident (MLOCA) and internally quenched by emergency coolant injection [30].Using the assumption that r b i , pressurisation is simulated by a uniform through-wall tension.70 linear elastic FE models were used to determine the crack tip SIFs occurring in crack pairs with different combinations of aspect ratio, inter-flaw spacing and loading state.14 models of single cracks with different combinations of aspect ratio and loading state were also performed.Results from these two sets of models were used to determine the elastic interaction factor (see Equation 1) as a function of position on the crack tip line.A special-purpose function for calculating elastic interaction factors (in-t_defects_calc_interaction_factor) is included in the int_defects toolbox.Some of these results, for cracks with an aspect ratio of = 1 a c only, are shown in Fig. 4. For this geometry, the SIF is at a maximum close to the intersection between the crack tip line and the plate surface for both uniform stress and thermal shock loading modes (see Fig. 4a & c).Therefore, assuming a brittle fracture mechanism and ignoring crack-tip constraint effects, fracture would be expected to initiate at this location if the load was sufficient.For both Fig. 2. Model geometry used for investigating the interaction between a pair of identical surface flaws.(a.) Flaw geometry with dimensions in mm.(b.) Close-up of an Abaqus FE model generated using int_defects, with the calculated stress field (crack-normal component shown) resulting from crack-face loads used to represent the thermal shock loading state shown in Fig. 3. loading cases the SIF distribution which occurs for a pair of flaws is greater than for a single flaw.The SIF increases with decreasing inter-flaw spacing.Despite large differences in the SIF caused by each loading case (Fig. 4a & c), the proportional increase in SIF due to flaw proximity is similar (Fig. 4b & d) but not identical.Fig. 5 shows the elastic interaction factor at a location close to the near-side surface intersection ( °0 ) for pairs of identical cracks with various crack aspect ratios and spacings.Pairs of closely-spaced and wide (i.e.low aspect ratio) flaws under uniform Fig. 3. Through-wall distributions of stress resulting from uniform tension and thermal shock.The thermal shock stress profile was predicted via elastic analysis by González-Albuixech et al. [30] for a RPV 399 s into a MLOCA event.The stress at the internal surface is 480 MPa for both cases.) separated by different distances d and subjected to the two through-thickness loading conditions shown in Fig. 2. (a.) SIF for cracks subject to a uniform tensile stress and (b.) corresponding interaction factors.(c.) SIF for cracks subject to a thermal shock stress profile and (d.) corresponding interaction factors.stress interact most strongly.This interaction causes a maximum increase in near-side SIF of 20-25% with respect to a single crack for the closest, widest geometries considered (Fig. 5a).If necessary, FE results for the two loading cases could be linearly superimposed to determine the elastic interaction factors under combined pressurisation and thermal shock.

Usage example: Determination of single-flaw limit loads
Question: "What are the local and global plastic limit pressures for thick-walled pipes with an inner to outer radius ratio of 0.8, and containing single axial surface flaws of known size and shape?Are existing formulae for determining the limit pressures of axially flawed pipes accurate?" Plastic limit pressures are a key input in many structural integrity assessment procedures.In this example, int_defects was used to determine the limit pressures of thick-walled pipes containing surface-breaking axial flaws on the internal or external surface, for a wide range of flaw depths and aspect ratios.The analysis was used to determine the local limit pressure p LL (i.e. the pressure at which a plastic ligament forms connecting the flaw and the opposite wall of the pipe) and the global limit pressure p GL (i.e. the pressure at  .The pipe material was modelled as being elastic-plastic with a von Mises yield locus and no strain-hardening.The elastic properties were taken as E = 210 GPa and ν = 0.3 and the (arbitrary) elastic limit stress, or yield stress, is denoted Y .A monotonically-increasing pressure was applied to the pipe's internal surface and any internal crack faces.
An example of a model created by int_defects is shown in Fig. 6.The local limit pressures were determined using a function for this purpose included in the int_defects toolbox (int_defects_plastic_breakthrough).Pressurisation to the global limit causes plastic instability, so the global limit pressure was determined from the last completed model increment.Plastic limit pressures for internally and externally cracked pipes are shown in Fig. 7.As expected, the limit pressures are lowest for pipes containing deep, wide cracks.For very shallow and narrow cracks, the results converge towards the theoretical limit pressure for an unflawed thick-walled pipe p UL [31][32][33]: where r i and r o are the pipe inner and outer radii respectively, and Y is the material's elastic limit stress (which is equivalent to the yield stress for a non-strain-hardening material).The results in Fig. 7a and c were compared with the (semi-analytical) formulae for flawed pipe local limit pressures given by Lei [33] and included in Annex P of BS 7910:2013 [2].The BS 7910 formulae are calculated assuming a rectangular rather than semi-elliptical surface flaw profile and are given in Appendix C. Fig. 8 shows that the formula for external flaws gives a slightly lower local limit pressure than calculated via FEA for all cases, i.e. a slightly conservative estimate.On the other hand, the formula for external flaws predicts a higher local limit pressure than indicated by FEA for some cases: mainly for relatively deep cracks ( > 0.375 ).This result supports the continued use of the formula for internal flaw local limit pressure but suggests that the formula for external flaws could be reviewed.

Question: "A surface flaw and an embedded flaw occur close to one another in a wide steel plate under tension. How does the depth of the embedded flaw affect the load and crack tip location at which fracture will initiate?"
The steel plate has a thickness (b) of 80 mm and is acted upon by a remotely-applied tensile stress which is increased monotonically from 0 to 500 MPa.The plate contains two crack-like flaws in the same through-thickness plane, shown in Fig. 9.There is a surface-breaking flaw with a depth (a 1 ) of 20 mm and a total width ( c 2 1 ) of 40 mm, and an embedded flaw with a total size in the depth direction ( a 2 2 ) of 20 mm, a total size in the width direction ( c 2 2 ) of 40 mm, and a depth below the plate's surface denoted by d 2 .The material is relatively brittle, with a fracture initiation toughness of J 0.2 = 80 N mm −1 .
The plate material is modelled using incremental plasticity theory and its mechanical response is taken to follow a Ramberg-Osgood relationship [34]: where the Young's modulus E = 210 GPa, yield stress Y = 360 MPa, yield offset parameter = 1.667 and hardening coefficient n = 12.The material is assumed to exhibit a von Mises yield locus and isotropic strain-hardening behaviour.Although a fictional material which follows a Ramberg-Osgood relationship is used in this example, in general int_defects can use any monotonic true stress-strain curve.int_defects was used to generate and run a set of FE models of this crack pair to investigate the effect of the embedded crack depth (parameter d 2 , see Fig. 9) in the range 5 ≤ d 2 ≤ 20 mm.The J-integral as a function of position on each crack tip line was extracted from the results; this is shown in Fig. 10 for an applied tensile stress of 360 MPa.Two factors increase the J-integral over part of the crack tip lines of the interacting flaws: the proximity of the embedded flaw to the plate's surface and the proximity of the two flaws to each other.In this configuration, fracture would be expected to initiate from the surface flaw at the point closest to the embedded flaw ( °27 1 ), which is the location that experiences the greatest J-integral (see Fig. 10).However, if the embedded flaw is located only 5 mm deep then tearing-out of the embedded flaw may occur simultaneously at °268 2 .

Discussion and conclusions
The above examples illustrate the range of analyses that can be performed using the int_defects toolbox.The number of individual FE models used in each of these examples is relatively small, but the toolbox is designed for scalability and contains features for ensuring robust execution of large sets of models and for error-checking FE results.Model sets containing thousands of individual FE analyses are feasible [22].Although int_defects is limited in terms of the geometries that it can analyse, it provides a rapid and scalable way to investigate defect interactions.
Flaw interaction criteria are normally used at the start of fracture-mechanics-based structural integrity assessment procedures.Consequently, they affect the rest of the assessment and often have a decisive impact on the outcome.In most procedures, the criteria for flaw interaction are based on the geometry and spacing of the flaws.Other factors such as a loading state and material ductility are rarely taken into account, although a few procedures (such as SINTAP [35]) do consider these.int_defects can be used to examine different flaw geometries as well as factors such as loading state.It has been used to evaluate flaw interaction and recharacterisation rules in recent studies on the BS 7910 interaction criteria [36], the BS 7910 Annex E buried-to-surface recharacterisation rules [37], and the interaction criteria of various procedures including R6 [22].
In summary, int_defects is a useful tool for studying flaw interactions.It is simple to use, scalable and open-source.Although the range of 3D geometries that it supports is limited, its design instead emphasises features that are useful for formulating interaction criteria and recharacterisation rules, including automatic SIF and limit load analysis, easy parameterisation of the geometry and loading state, and robustness when handling large sets of models.

Data access statement
• The int_defects toolbox (v1.2.0) is available from the University of Bristol data repository: http://data.bris.ac.uk/data/dataset/2s1zavsbkctna2bnh6g6os9n2k • The underlying data for all examples shown in this paper can be downloaded from: http://data.bris.ac.uk/data/dataset/36wlnpm16muu52ppxhzaa0udz2 conditions.In most cases, pre-existing results for comparison will only exist for certain combinations of geometry/loading.
In addition to the examples shown below, comparisons of results from int_defects with the results of other authors are presented by Coules [22] (elastic analysis of twin surface cracks) and Coules & Bezensek [38] (limit load analysis of single surface cracks).

A.1. Validation case: Elastic analysis of single surface cracks
This example aims to validate int_defects' capability for elastic cracked-body analysis.Stress intensity factors for single semielliptical surface cracks in wide elastic plates of finite thickness were determined and compared with the well-known numerical results of Newman & Raju [39], as well as a wide range of results for this geometry from different sources.
Newman & Raju investigated cracks in the plane normal to the loading direction, with normalised depths in the range 0.2 0.8 and aspect ratios in the range 0.2 2 a c . The plate material was defined to have a Poisson's ratio = and an arbitrary elastic modulus was used (since this does not affect the stress intensity factor result).Two loading cases were investigated: pure tension resulting in a crack-transverse stress of unit magnitude, and pure bending resulting in a stress of unit magnitude at the plate's surface.A typical stress field for such cases determined using int_defects is shown in Fig. 11.

H.E. Coules and M.A. Probert
Engineering Fracture Mechanics 227 (2020) 106733 A comparison of results is shown in Fig. 12 (tension) and Fig. 13 (bending).The results are given in normalised form as where Q is a shape factor for elliptical cracks which is approximated by [39,40]: ) and are believed to be due to the low level of mesh refinement used by the earlier authors.In the bending case, some more recent results by Lei [41] are also presented -these also show good agreement Fig. 12. Normalised stress intensity factor as a function of position on the crack tip line for 20 semi-elliptical surface cracks with different depths (a/ b) and aspect ratios (a/c) in wide plates subjected to a unit tensile stress.Good agreement is observed between the results of int_defects and Newman & Raju [39].with int_defects.
A set of similar models was used to determine the SIF for surface cracks with an aspect ratio of = 0.  loaded in remote tension.The SIF at the deepest point in each crack was compared with results from a variety of other authors, as shown in Fig. 14.The int_defects result agrees well with those from other sources, lying close to where the highest density of results is seen.

A.2. Validation case: Limit loads of plates containing offset embedded cracks
In this example, global limit loads for finite plates containing embedded offset elliptical cracks are determined using int_defects and compared with results of FEA by Li et al. [51].The geometry is shown in Fig. 15: a flat plate is loaded in tension normal to the crack plane.The global limit loads for plates of this type were determined for all combinations of the following geometric parameters: Fig. 13.Normalised stress intensity factor as a function of position on the crack tip line for 20 semi-elliptical surface cracks with different depths (a/ b) and aspect ratios (a/c) in wide plates subjected to a pure bending, causing unit stress at the plate's surface.Good agreement is observed between int_defects results and those of Newman & Raju [39] and Lei [41].Fig. 14.Comparison of SIF results for semi-elliptical surface cracks in a plate under tension from different authors [18,[42][43][44][45][46][47][48][49][50].Normalised SIF at the deepest point in cracks with different depths and a constant aspect ratio of a/c = 0.6 is shown.After Isida et al. [42] and Yoshimura et al. [18].was defined as exhibiting incremental flow plasticity with a von Mises yield locus and no strain-hardening.The limit stress ( Y ) had a nominal value of 360 MPa, and elastic parameters E = 210 GPa and = 0.3 were used.Large-displacement (i.e.geometrically non- linear) spatial modelling was used for the analysis.An example of the finite element mesh used is shown in Fig. 16.Fig. 17 shows a comparison of global limit loads for this geometry (n GL ) determined using int_defects with FEA results from Li et al. [51].Good agreement is observed between the two sets of limit load predictions across the full range of geometric parameters: the global limit loads determined using int_defects differ from the results of Li et al. by a maximum of 3%.int_defects consistently predicts a slightly higher global limit load, which is probably due to differences in the modelling methods used.For example, int_defects determines the global limit load from the point at which the FEA solver is unable to continue due to global plastic instability whereas Li et al. determined the global limit load from the plate's load-displacement curve.The analysis with int_defects also used geometrically non-linear analysis, whereas Li et al. used a small-displacement formulation.Nevertheless, close agreement between the two sets of results demonstrates the accuracy of int_defects for limit load analysis.

A.3. Validation case: ASTM analytical round robin for elastic-plastic analysis
This example is based on an ASTM inter-laboratory round robin study concerning elastic-plastic analysis of a surface-cracked plate under tension, full details of which are reported by Wells & Allen [52].The single specimen used in Phase 1 of the study was a flat plate of 2219-T8 aluminium alloy containing a semi-elliptical surface crack in the plane normal to the loading direction.The plate was subjected to a monotonically-increasing tensile load.Experienced participants from 15 institutes were invited to predict the J- .

Table 1
Average pre-processing and execution times per finite element model for all analyses discussed in this article, and the total time taken to process each model set.All times are given as the elapsed real time ("wall-clock" time).For some sets of models, int_defects was used to execute models in parallel by employing multiple instances of the FE solver.This secondary parallelisation can significantly reduce the total time necessary to process the model set.Therefore, Table 1 Appendix C. Existing limit load solutions for axially-flawed pipes The following approximate solutions for the local limit pressure of thick-walled pipes containing single crack-like surface flaws in the axially-radial plane are given by Lei [33] and used in Annex P of BS 7910:2013 [2].The geometry is shown in Fig. 7b & d.These formulae assume a von Mises yield locus and a rectangular flaw shape.The solution for flaws on the internal pipe surface includes consideration of the internal pressure acting on the crack faces.Fig. 8 shows a comparison of limit loads determined by these formulae with finite element results from int_defects, for a pipe with a radius ratio = 0.8 The pressure vessel in this example has a thickness (b) of 166 mm and an internal radius r b i , such that the region containing flaws can be closely approximated by a flat plate in tension.The vessel wall contains a pair of identical internal surface flaws each with a depth (a) of 40 mm, both in the axial-radial plane.The aspect ratio of the flaws ( a c ) and the inter-flaw spacing (d) are undefined: multiple models with different aspect ratios in the range 4

Fig. 4 .
Fig. 4. SIF and elastic interaction factor as a function of position on the crack tip line for pairs of twin semi-circular cracks ( = 1

Fig. 5 .
Fig. 5. Stress intensity interaction factor for the near-side surface point ( °0 ) in twin surface crack pairs subjected to the through-wall loading cases shown in Fig. 3. (a.) uniform stress and (b.) thermal shock.

Fig. 6 .
Fig. 6.Elastic → perfect-plastic model of a pipe containing an internal surface flaw in the axial-radial plane at the local limit pressure.The flaw dimensions are = 0.5 a b

Fig. 8 .
Fig. 8. Accuracy of formulae used in Annex P of BS 7910:2013 for estimating the local limit pressure for pipes with surface-breaking axial cracks [2,33].Local limit pressures for 154 cases are shown, normalised using p LL Y .For some external cracks, the current BS 7910:2013 formula produces a

Fig. 9 .
Fig. 9.A pair of interacting coplanar crack-like flaws in a plate -one surface and one embedded.(a.) Flaw geometry.(b.) Close-up of an Abaqus FE model generated using int_defects, with the calculated stress field (crack-normal component shown) for a remotely applied tensile stress of 360 MPa.

Fig. 10 .
Fig. 10.J-integral as a function of position on each flaw in the interacting pair shown in Fig. 9, at an applied remote stress of 360 MPa (i.e.Y ).Results for three different embedded flaw depths are shown.Dashed lines show the J-integral for each flaw in isolation.

Fig. 11 .
Fig. 11.Typical mesh used for linear elastic analysis of a semi-elliptical surface crack in this validation case.The result for a crack with dimensions = 0.6 a c tension and bending cases, there is good agreement between the results of int_defects and those of Newman & Raju.The largest discrepancies occur for very wide cracks ( = 0.2a c

Fig. 15 .
Fig. 15.Flat plate in tension with an embedded semi-elliptical crack offset from the mid-thickness.

Fig. 16 ...
Fig. 16.Finite element model of a plate containing an offset elliptical crack under tension, generated using int_defects to determine plastic limit loads.The case shown has the geometric parameters = 0.3 c W

Fig. 17 .
Fig. 17.Normalised global limit loads of finite plates containing elliptical embedded flaws, loaded in tension (as shown in Fig. 15).Results for plates with different crack depths ( a b Article section Description of model set Models in set Mean pre-processing time (s) Mean execution time (s) Parallel solver instances CPUs per each of the examples presented in this article.
In these equations, 1 is the normalised flaw depth, i is a wall thickness parameter, and is the aspect ratio.For external flaws: also gives the total processing time for each set of models, for which the following relationship holds: