High Performance Fracture Approach to Fatigue Crack ...web.mscsoftware.com/events/aero2002/partner/pdf/p02701.pdf · Paper Reference Number: 2001 -27 1 High Performance Fracture Approach

Paper Reference Number: 2001-27

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High Performance Fracture Approach to Fatigue Crack Analysis & Life Prediction Javier Viñals (*) – SENER - Av. Zugazarte 56 - 48930 Las Arenas (SPAIN) Tel. +34-944817768 - [email protected] (*) Coordinator of HiPER-CRACK project (funding by the EU Fifth Framework, Contract number G5RD-CT-2000-00204), on behalf of CRF Fiat (IT), ITSAK (GR), UPCO (SP), JRC-Ispra (IT), MSC (GE), SAMTECH (BE), nCODE (UK) and VOLVO AERO (SW). ABSTRACT Fracture and fatigue issues are subject to concerted efforts by designers, production and maintenance engineers to ensure components safe operation. The purpose of the work reported here is to develop and validate improved numerical methods for damage tolerant analysis (life prediction) and to integrate them in high performance computer simulation environments. The analysis is a direct improvement of current practices where the limitations are that stress redistribution due to crack advance cannot be taken into account, zones newly becoming critical due to stress redistribution are missed and multiple site damage cannot be addressed. The project incorporates the laws of fracture mechanics into life prediction calculations through the development of cohesive interface elements, which are able to capture the decohesion taking place in the fracture zone. The cohesive interface elements will be integrated in the environment of standard finite element codes. This will allow a new approach to finite element modelling, which will permit combined stress and damage tolerance analysis in a single step, taking into account stress redistribution and crack interaction effects. The project will afford significant improvements in fatigue crack growth design methodology and will result in measurable safety and reliability improvements. Important potential users include automobile manufacturers and aeronautics industry. The project is at the mid term point (Duration: April 2000 – March 2003). The main work (Cohesive Interface Elements) has been done by one educational centre (UPCO) and two research institutes (ITSAK and EC-JRC-ISIS). The three software developers (MSC, SAMTECH and nCode) are in the commercial code implementation stage, and, the three end-users (SENER, CRF and VOLVO AERO) in aerospace and automotive fields expect to begin the verification and validation process by January 2002. A subcontractor in the civil engineering field will be employed by ITSAK. All companies involved represent a cross section of available expertise and potential industrial application at the European level. Seven EU countries are involved. Emphasis has been placed on the multisectorial character of the project and three key industries are involved for demonstration and evaluation. INTRODUCTION The project aim centres on the increased safety of aerospace, automotive and civil structures. Damage tolerance and life prediction analysis contribute directly to the avoidance of catastrophic failure and its associated losses. The project affords significant improvements in the design methodology against fatigue crack growth and


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will therefore result in measurable improvements in safety and reliability. By providing seamless analysis that bridges the gap between finite element and life prediction calculations, the project will allow for interfaces with optimisation modules. Reduction of up to 20% in calculation time and computer costs, due to a more efficient method of crack analysis, will allow more rapid and reliable design of critical components. The project work will contribute to a better understanding of damage tolerance analysis and will thus assist in developing unified standards related to design and structural safety. Codes and guidelines related to damage tolerance analysis are currently mostly based on American standards and almost no European codes for fatigue analysis exist. Furthermore existing standards for damage tolerant approach to fatigue design, e.g. MIL-A-83444 (USAF), Military specification – Airplane Damage Tolerance Requirements, July 1974, are partly invalidated by multiple site damage phenomena and the subsequent stress redistribution due to interaction of multiple cracks, to be actively studied under the proposed work. Applications of this technology are virtually unlimited; potential users include, for instance, automobile manufacturers and the aeronautics industry. These industries must account for questions related to reliability and life assessment of vehicles and aircraft components. Problems related to the fatigue resistance of these structures and their capacity to tolerate required fatigue damage levels between two inspections are of paramount importance. The aircraft industry (both commercial and military aircraft) is subject to damage tolerance requirements that have to be met by their products, which may include airframes as well as turbine engines. The automobile industry has to account for questions related to crash worthiness and fatigue related optimisation of automotive components, for which realistic assessment of fatigue life of structural components at the early design stage still needs improved and high performance numerical tools. Other important applications include civil structures, which are continuously submitted to varying loads, welded storage tanks, cracks in railroad rails, underwater pipelines, as well as oil and natural gas pipelines, all of which have exhibited spectacular catastrophic failures in recent years. Design of new such structures or analysis, aiming to extend their life through retrofitting, heavily relies on accurate fatigue damage assessment. The project is directly related to the goals of FP5 in the field of RTD on Generic Technologies in that it helps develop the scientific and technological base in one of the critical areas of the Programme: Materials and their Production and Transformation. It specifically addresses the requirements for expanding the limits and durability of structural materials through the introduction of advanced state-of-the-art life prediction methodologies. The research is of medium to long-term nature and it applies to more than one product or sector. It improves safety and reliability by understanding deterioration and failure mechanisms (fatigue) of advanced metals and alloys and by focusing on ‘full life-cycle approach’ promotes the efficient use of materials. It also supports the characterisation, modelling and testing for advanced materials applications resulting in improved quality of life through safety (safer transport and construction), economy (lighter weight, lower emissions) and sustainable use of materials (longer life).


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The project creates interfaces and is relevant to other FP5 directions as improving aircraft safety through the development and validation of improved theoretical tools for the simulation of structural behaviour (thematic priority 1.1.3-4.2.2) and large-scale simulation and visualisation technologies (thematic priority 1.1.2-4.4). It is important to remark that several users group will be set-up (at end user locations) during the course of the project consisting of companies, which have already expressed an interest in the cohesive element approach to fracture, and others that will be invited to join. One of the partners, VOLVO AERO, is considered the first member of this group. The users group meetings are scheduled on March 2002 in Italy (CRF Fiat) and on September 2002 in Sweden (VOLVO AERO). A final short course for the methodology dissemination is also planned in March 2003 in Spain (UPCO). In addition to providing feedback and further test cases, the users group will be instrumental in the dissemination of project results. The project co-ordinator SENER has extensive experience in project management and will be responsible for the administrative and financial coordination of the project. PROBLEM DEFINITION With the proposed capability, a finite element code will be able to predict crack initiation, subsequent growth and net loss of structural strength, thus completing the entire phase of damage tolerance analysis. Of particular interest is the prediction of the rate of growth and shape of a surface crack in components or structures of arbitrary geometry subjected to general - possibly mixed-mode - loading histories. Inevitably, this requires advanced computational capabilities for the tracking of two- and three-dimensional crack fronts. Additionally, many applications of engineering interest, specially those concerned with low-strength high-toughness materials, violate the small-scale yielding assumption of linear elastic fracture mechanics and require explicit consideration of plastic deformations in the component or structure. This combination of circumstances favours the use of finite element methods for stress analysis. In the past twenty years, finite element analyses have been extensively used in the design and failure analysis of structures. Traditionally, such techniques are used for the first step preceding the damage tolerance analysis stage (i.e. for the identification of the most likely location of failure). This is typically done by performing a stress analysis of the uncracked structure and identifying regions where some arbitrarily chosen measure of stress attains a critical value under the expected service loads elevated by appropriate safety factors. After this is completed, the stresses calculated by the finite element simulation are used as a boundary condition to a fictitious fracture/fatigue problem (e.g. S-N analysis) involving hypothetical flaws of arbitrarily chosen lengths and orientations. This problem is then solved to determine the growth rate (growth history) and final lengths of these initial flaws at a given period of time. It is information that is eventually used in completing the loop of damage tolerance analysis. Such an approach is unsatisfactory for many reasons. First, it is a well-known fact that cracks will not necessarily grow in parts of the structure where a single stress measure is maximum. Indeed, they may nucleate and grow at sites where a complex combination


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of stress components is present and their location and initial size cannot be predicted by simple stress analysis. This fact immediately renders the common practice of postulating an initial crack flaw length and orientation at a particular location highly inaccurate. Secondly, the solution of the ensuing fictitious fracture/fatigue problem is done in a fashion uncoupled to the structure itself. Indeed in the real structure the presence of the growing crack would redistribute the stresses and will thus affect further crack growth in a coupled way to the structural response (multi-site cracking). The root of the problem comes from the current incapability of codes to incorporate the principles of fracture mechanics. The project will focus on development of new numerical methods for fatigue fracture analysis which specifically take into account the physics of the decohesion process. It will also provide a feasibility study for the proposed methodology, based on cohesive interface finite elements, through validation against experimental results and end-user applications. ANALYSIS 1. COHESIVE THEORY BACKGROUND The starting issue is the viewpoint - Dugdale (1960), Rice (1972) and others - that regards fracture as a gradual phenomenon in which separation takes place across an extended crack `tip', or cohesive zone, and is resisted by cohesive forces. This theory of fracture permits the incorporation into the analysis of fracture parameters such as the spall strength - the peak cohesive traction - and the fracture energy - the area under the cohesive law - of the material. The cohesive model is based on micromechanical aspects related to voids and dislocations. Cohesive laws have been built into finite element analyses as mixed boundary conditions, Needleman (1994). Here, following Ortiz and Suresh (1993) and Camacho and Ortiz (1996), the cohesive laws will be embedded directly into cohesive finite elements. These elements are line- or surface-like and are compatible with general bulk finite element discretizations of the solid, including those that account for plasticity and large deformations. Cohesive elements bridge nascent surfaces and govern their separation in accordance with a cohesive law. Camacho and Ortiz (1996) have shown that mesh size-independent results are obtained when the mesh adequately resolves the cohesive zone. Examples of cohesive law to be considered during the project are: Elastic reversible law (mostly on brittle materials), Dugdale law and Smith-Rose-Ferrante law. In this context, the tracking of ductile cracks in solids undergoing large-scale plasticity has received scant attention in the computational literature. Marusich and Ortiz (1995) have developed a method of crack tracking based on continuous and adaptive remeshing. While the approach permits the competition between ductile and brittle fracture mechanisms, its generalisation to three dimensions is not straightforward. Perhaps a more fundamental difficulty concerns the formulation of fatigue crack growth laws in the presence of large-scale plasticity and arbitrary mixed-mode loading. While generalisations of Paris' law based on the J-integral have been proposed, their validity


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is not always born out by testing. The concept of cohesive interface elements, as proposed in this project, shows promise of successful application to fatigue crack growth and life prediction, as, for example, evidenced by the analysis of three-dimensional fatigue crack growth in aluminium shafts subject to axial loading (Perez et al. 1997, De-Andres et al. 1999). Attention will also focus on work on free energy formulation for fatigue crack initiation (Mura and Nakasone 1990, Venkataraman et al. 1991, Mura 1994). Extension of this formulation to fatigue crack propagation will be attempted using the cohesive zone formulation following an angle of attack, similar to the one used by Rice (1966) in his plastic work fatigue crack growth model. Conventional fatigue modelling can take one of three forms: total life, life to crack initiation and fatigue life consumed in crack propagation until fracture. In all three methods pseudo elastic nominal or local stresses are obtained from conventional finite element analysis. In the S-N method, simple nominal elastic stress cycles are used to characterise damage. In the E-N method, notch correction procedures are used to handle the local plasticity at the surface location where cracking will initiate. The fracture mechanics approach again relies on a nominal elastic stress at the crack location but, unlike the current project, cracks are not explicitly modelled in FE, rather a numerical compliance function handles the change in loading on the crack as growth proceeds. One key ingredient for the successful use of cohesive elements in fracture is the appropriate choice of cohesive traction laws describing the decohesion process. As has been shown by the work of Needleman (1994), Tvergaard and others, the exact shape of the cohesive law does not greatly influence the predicted fracture behaviour. However, the choice of the area enclosed by this law as well as either its height (maximum sustainable traction) or the chosen value of critical displacement for complete separation may have pronounced effects on the predicted behaviour. Two out of these three quantities should be chosen on the basis of experimental measurements for crack initiation and growth. One of the important issues will be the extrapolation algorithm in fatigue simulations, and creep. 2. COHESIVE INTERFACE ELEMENTS (CIE) During the initial phases of the project, the work done has been focused on the development of the 2D and 3D Cohesive Interface Elements (CIE) by UPCO, ITSAK and EC-JRC-ISIS. The cohesive model has been incorporated in the general purpose finite element code FEAP in the form of a cohesive element. Subroutines for two and three- dimensional elements has been developed. At this point cohesive elements for infinitesimal deformation have been implemented. The cohesive law is chosen as Smith-Ferrante’s universal binding law. The definition of the two-dimensional 4-noded and 6-noded elements is showed in Figure 1. The three-dimensional elements are completely analogous. Several verification examples have been performed for 2D and 3D Elements.

a) 2D elements. EXAMPLE 1: Three-point bending (Figure 2) and EXAMPLE 2: Double cantilever beam (Figure 3).


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• Table 1 and 2 show the energy release rate values calculated from numerical simulations and percent error from the analytical results for both aluminium and steel for double cantilever beam example (Figure 4) using 2D cohesive element.

• Table 3 and 4 show the stress intensity factors calculated from numerical simulations and percent error from the analytical results for both aluminium and steel for three point bending example (Figure 5) using 2D cohesive element. Analytical solution for the problem is K= 37.874 MPa mm1/2.

b) 3D elements. EXAMPLE 3: Determination of the J Integral for a 3 – Point

Bending specimen in the fully plastic regime and EXAMPLE 4: Determination of the Stress Intensity factor KIII for a shaft subjected to torsion with a circumferential crack. Quadrilateral cohesive element is used in these examples.

• 3 – Point Bending specimen: The computational mesh is shown in Figure 6.

The crack plane is meshed with cohesive elements, while the bulk is modeled with classical elastoplastic 8-noded isoparametric elements. Three different specimens have been chosen with different mesh sizes from the finest to the coarsest mesh in order to check the evolution of the error compared with the known analytical solution. Figure 7 shows contours of the Von-Mises equivalent stress for one of the specimens (specimen 2) where it can be appreciated the presence of the fully-plastic regime in the uncracked ligament. Figure 8 shows a log-plot where a comparison between analytical and numerical results for the specimens studied is made. It also represents the relative error between the numerical and the analytical solution (considered here as the correct one) for both the elastic & plastic part of the J integral against the number of elements in the mesh.

• Shaft subjected to torsion. The assumed test configuration is shown in Figure 9. A shaft is subjected to a torsional moment and a small axial load in order to avoid friction between crack faces. The specimen has an initial pre-crack within its mid-section sharpened by fatigue, as in the case of the 3 - point bending test. The mesh is similar to that for the 3–point bending specimen, i.e., the crack plane is meshed with cohesive elements and for the bulk classical elasto-plastic 8 nodes isoparametric elements are employed. The mesh has been refined close to the crack front in order to get more accuracy in the zone of interest. Three different specimens have been chosen with different mesh sizes in order to check the evolution of the error compared with the known analytical solution. The mesh is shown in Figure 10.

Figure 11 shows contours of the Von-Mises equivalent stress for one of the specimens at two different perspectives. It can be appreciated the circumferential symmetry of the solution, as expected. Figure 12 shows contours of the damage variable for one of the specimens in the study. It is also observed the circumferential symmetry of the solution, as expected. Figure 13 shows a log-plot with the comparison between analytical and


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numerical results for the specimens proposed and a table representing the relative error against the number of elements in the mesh. As it can be seen for both specimens, when the cohesive zone is captured by at least two elements, convergence is very fast, otherwise it is slower. In the “J” Integral specimen, in which the fully plastic condition is achieved, the cohesive zone is larger than in the torsional one and is captured by a higher number of cohesive elements, obtaining better convergence.

3. FATIGUE ALGORITHM The objective is to develop algorithms for the use of cohesive interface elements for fatigue and life prediction analysis. This work is on going by UPCO, ITSAK and EC-JRC-ISIS, and will be an important input for the commercial code implementation. The extrapolation scheme have been defined as a procedure which allows to evaluate the fatigue life of a specimen without reproducing the entire fatigue life by a tedious and time consuming one cycle-to-cycle or pick-by-pick procedure. Based on the definition aforementioned, any proposed extrapolation scheme involves the following steps not completely independent: A) Identify the “blocks” to be extrapolated, B) Define the number of “blocks” which are going to be extrapolated, and C) Calculate the amount of life consumed in these “N-blocks”. The step A) is not properly part of the extrapolation scheme but it is necessary in a real case because involves the approximation of the more complex actual service history by assuming that it is equivalent to repeated applications of a simple loading sequence of finite length (what it is called “block”). In the step B) and due to the impossibility to reproduce hundreds, thousands or even millions of blocks it is computed pick-by-pick the effect of one or a few single blocks and from these it must be determined a number “N” of the following blocks in which it is possible to predict the subsequent degradation of the model with a minimum error. In the step C) it is exposed the degradation of the model in terms of one or several internal variables (what it is called “damage variables”) which are the ones which are going to be calculated or whose values after the “N” blocks” are going to be predicted from their growth in the few blocks previously reproduced computationally in a pick-by-pick manner. The algorithms defined to implement the mentioned Extrapolation Scheme are shown in Figures 14 and 15. These algorithms have been fully implemented into the Finite Element Code FEAP but Numerical Verification tests required are still been developed. 4. ADAPTATION OF COHESIVE INTERFACE ELEMENTS INTO COMMERCIAL CODES The objective is to make the cohesive interface elements and associated fatigue algorithms available through major commercial software, using a common platform for all partners. Mainly, the software partners are involved (MSC, SAMTECH and nCode). The 2D and 3D elements have been introduced in Samcef/Mecano. Several tests are being performed. Figure 16 shows the initial mesh used for the crack propagation.


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Figure 17 shows the curve load–displacement found by ITSAK and the one found by SAMTECH. Figure 18 is the stress distribution calculated. The pre and post processes are being developed by MSC. The initial work performed has been focused on the understanding of the SAMTECH user scenario, the crack elements requirements, the implementation method and the development phases. In relation with the user scenario, the idea is to leave to the user the selection of the crack zones from existing mesh. The generation of the crack elements will be taking into account the original mesh with the new interface elements. The envision elements to be supported are: 2D elements (Quad 4 and Quad 6), and 3D elements (Wedge 6, Hex 8, Wedge 12 and Hex 16). The implementation method (see Figures 19 and 20) will be based on MSC/PATRAN interface, introducing a call HiPER-CRACK in the top menu bar and element generation menu. Help on line shall be also available. 5. VERIFICATION AND VALIDATION The end users shall be in charge of the verification and validation of the methodology developed. The fields to be considered are aeronautical (SENER and VOLVOAERO), civil (ITSAK) and automotive (CRF Fiat). SENER is being performed the classical analysis. Two components have been chosen: an external fairing (see figure 21) submitted to mechanical loading, and the exhaust diffuser submitted to thermal loading, that includes thermal fatigue and creep effects (figure 22). The external fairing crack analysis has already been finished. Two methodologies have been used: ESACRACK software (standard and analytical code, where the crack mode is chosen and the complete component geometry is not taken into account) and the BEASY code (Boundary Element methodology). As pre and post processor, MSC/PATRAN has been used. CRF Fiat has chosen an automobile component (a twist beam with central part made of Fe510D steel, figure 23), and has begun the activities relative to material characterisation on standard specimens. DISCUSSIONS Expected technical achievements of the proposed research are: • The development of computational tools that will bridge the gap between finite

element analysis and life prediction analysis. These will predict crack initiation subsequent growth and net loss of structural strength, thus completing the entire phase of damage tolerance analysis.

• Comparison of above tools with conventional or alternative methodologies. Verification and validation.

• Integration of above tools within high performance parallelised software environments.


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• Application to end-users problems in three key industries: aerospace, automotive and civil engineering.

• Dissemination of results through the creation of a users group and other measures as well as through the efforts of the participating software companies as relates to the commercialisation and marketing of their products.

Technical risks are the novelty of the proposed methods and algorithms and performance risks within commercial software. As regards the first, the methods are new to the industrial world, but have a solid research base, already exhibiting excellent performance in preliminary studies in the research environment. Given the previous experience of ITSAK, UPCO and EC-JRC-ISIS, and the collaboration of world experts in the project, the migration of the research methods to a general-purpose capability, is believed to entail little to medium risk. The performance-associated risk is also believed to be medium to low, given the solver performance of the selected commercial software and performance improvements due to its parallelisation. An additional constraint is the implicit time integration algorithm of the selected codes, which will impose limits to highly dynamic applications. These issues will form the basis for future collaboration between the partners and will need to be resolved before full use can be made of the capability in the industrial environment. This is estimated to be a 3 – 5 year process after completion of the project. CONCLUSIONS The major development effort for the project is to create a high-performance simulation capability for fatigue crack initiation and propagation analysis. • Methods and software development: This represents the effort of the research

organisations to create cohesive interface elements and fatigue algorithms that will serve as input to the software companies for the creation of a final exploitable product. The elements and methods will be tested against established benchmarks.

• Verification and Validation: The methods will be verified against ‘exact’ solutions.

Comparisons will be made with classical methods of analysis – i.e. fatigue post-processing – for the purpose of a) duplicating the results of those methods in simple cases and b) establishing the ability of the proposed methodology to capture effects that the traditional methods cannot – e.g. multi-site damage and irregular cycles. Material characterisation, both constitutive and decohesion will be based mostly on available experimental data as well as on experiments which may be performed as deemed appropriate. The simulations of these experiments also serve as calibration and verification studies of the cohesive elements. They will guide the selection of fracture parameters in the industrial applications. The results will establish the range of applicability of the method and provide guidelines for its application in the commercial software environment, through the materials data handling module. Available component data will serve in validation studies and will provide the final proof of concept for the proposed methodology.

• Industrial applications: The end-user organisations will actively follow the early stages of the project, contributing with an industrial perspective on the software


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specification, and giving input on requirements for functionality, adaptability, versatility and foreseen applications. Later, when reasonably useable implementation exists, the end-user organisations will apply it to a real or realistic engineering problem. The software will then be evaluated in terms of user friendliness and accuracy both against classical methods of analysis and against experiments. During the end-users’ applications, coupled algorithms will be developed for such diverse multi-physics applications as creep fracture in thermal fields which will be fed back to the software developers and will form the basis for future collaboration between the partners.

The major benefit for the MSC software users shall be to have an additional and important tool for fatigue life analysis, integrated in MSC/PATRAN, which means a very friendly work environment. ACKNOWLEDGEMENTS HiPER-CRACK project (funding by the EU Fifth Framework, Contract number G5RD-CT-2000-00204), and partners: CRF Fiat (IT), ITSAK (GR), UPCO (SP), JRC-Ispra (IT), MSC (GE), SAMTECH (BE), nCODE (UK) and VolvoAero (SW). FIGURES AND PLOTS Figure 1: Element in x-y. (a) 4-node element, (b) 6-node element Figure 2: Three-point bending beam specimen geometry Figure 3: Double cantilever beam specimen geometry Figure 4: A sample mesh for 2D double cantilever problem Figure 5: A sample mesh for 2D three point bending problem Figure 6: Computational mesh (specimen 1) Figure 7: Von-Misses stress (specimen 2) Figure 8: Numerical vs. Analytical results Figure 9: Schematic tested specimen Figure 10: Computational mesh (specimen 3) Figure 11: Von-Mises stress (specimen 1) Figure 12: Damage Plot (specimen 1) Figure 13: Numerical vs. Analytical results Figure 14: Stragegy for the Extrapolation algorithm Figure 15: Implementation of the extrapolation algorithm Figure 16: Initial mesh used for the crack propagation Figure 17: Curve load –displacement found by ITSAK and the one found by SAMTECH Figure 18: Stress distribution at time 0.7 Figure 19: Implementation method Figure 20: Help on line Figure 21: External fairing Figure 22: Exhaust diffuser Figure 23: Twist beam


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Figure 1

P

w t

ah

Figure 2

h

P

P t w

a

Figure 3

Crack tip

Figure 4

Table 1: Energy release rate values for aluminium for double cantilever beam. Analytical solution is G= 2.083 MPa mm.

Material 1 (Aluminium)

Crack tip cohesive element size (mm)

G (MPa mm)

% Error

Mesh 1 2.5 1.876 9.95 Mesh 2 1 1.918 7.91 Mesh 3 0.5 1.932 7.23 Mesh 4 0.25 1.938 6.97

Table 2: Energy release rate values for steel for double cantilever beam. Analytical solution is G= 0.746 MPa mm.

Material 2 Crack tip cohesive element size (mm)

G (MPa mm)

% Error

Mesh 1 2.5 0.197 73.60 Mesh 2 1 0.476 36.23 Mesh 3 0.5 0.751 0.67 (?) Mesh 4 0.25 0.899 20.57 (?)

Crack tip

Figure 5

Table 3: Energy release rate values for aluminium for three point bending.


K % Error

Mesh 1 1.67 39.22 3.54 Mesh 2 1.33 38.89 2.69 Mesh 3 1 38.60 1.91 Mesh 4 0.67 38.35 1.27 Mesh 5 0.33 38.14 0.70

Table 4: Energy release rate values for steel for three point bending.


K (MPa mm1/2)

% Error

Mesh 1 1.67 66.43 75.40 Mesh 2 1.33 61.65 62.78 Mesh 3 1 55.27 45.95 Mesh 4 0.67 47.46 25.30 Mesh 5 0.33 39.99 5.61


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Figure 6

Figure 7

Error vs Total Number of Elements

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Figure 8

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Figure 12

Error vs Total number of Elements

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Figure 13

Calculate 2φ∆ in N/2 + N/2

Calculate 1φ∆ in N blocks

N = N/2

FINSI

21 φφ ∆−∆

< tolerance

Figure 14

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Figure 17


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Figure 23

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High Performance Fracture Approach to Fatigue Crack ...web.mscsoftware.com/events/aero2002/partner/pdf/p02701.pdf · Paper Reference Number: 2001 -27 1 High Performance Fracture Approach