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Methods for Complex Cracked Body Finite Element Assessments
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Procedia Structural Integrity 13 00 (2018) 1232–1237 Structural Integrity Procedia (2016) 000–000
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
Methods for Complex Cracked Body Finite Element Assessments Methods for Complex Cracked Body Finite Element Assessments
XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016,a,b Paço de Arcos, Portugal a a
Moritz Lessmanna*, John Sawyera, David Knowlesa,b Moritz Lessmann *, John Sawyer , David Knowles
Atkins, The Hub, 500 Park Avenue, Aztec West, Almondsbury, Bristol BS32 4RZ, UK Thermo-mechanical modeling of aWalk, high pressure turbine blade of an Atkins, The Hub, 500 Park Avenue, Aztec West, Almondsbury, BS32 4RZ, UK University of Bristol, University Clifton BS8Bristol 1TR, UK University of Bristol, University Walk, Clifton BS8 1TR, UK airplane gas turbine engine a a
b b
Abstract a b c Abstract P. Brandão , V. Infante , A.M. Deus * This paper presents approaches developed to allow consistent best practice assessments for complex 2/3D defects in engineering This paper presents approaches developed to allow consistent best practice assessments for complex 2/3Dwith defects in engineering structures subjected to primary loading and secondary thermal stresses. Methods are outlined and discussed reference to recent a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, structures subjected to primary loading secondary thermal stresses. Methods are outlined and discussedtolerable with reference to recent twoand three-dimensional cracked bodyand analyses of components which have been undertaken to determine defect sizes and Portugal b and twothree-dimensional cracked body analyses ofanalyses components which been to determine defect and sizes and inform sub-critical crackofgrowth calculations. TheInstituto considered ahave range of undertaken postulated semi-elliptical, through-wall fully IDMEC, Department Mechanical Engineering, Superior Técnico, Universidade de Lisboa, Av. Roviscotolerable Pais, 1, 1049-001 Lisboa, inform sub-critical crackingrowth calculations. The analyses a range of postulated through-wall fully circumferential defects pressure systems. Mixed element considered typePortugal meshing strategies with tiedsemi-elliptical, contact in combination withand multiple c CeFEMA, Department Mechanical Engineering, Instituto Técnico, Universidade Lisboa, Rovisco Pais, 1, 1049-001 Lisboa, circumferential defects inofpressure systems. Mixed element type meshing strategies withdetied contact in combination with multiple node transformation techniques around the defect front wereSuperior employed during mesh generation. ThisAv. facilitated highly refined meshes Portugal node around the defect front were employed during mesh generation.inThis highly refined meshes alongtransformation the defect fronttechniques which were required to accurately model extensive plastic deformation the facilitated region of interest. Displacement along the defectloads frontrequired which were required to accurately model extensive plastic deformation in since the region interest. Displacement driven thermal detailed assessment with multiple elasto-plastic material models, lowerofbound properties do not driven thermal loads required detailed assessment with multiple elasto-plastic material models, since lower bound properties doover not necessarily provide the most conservative results. The elasto-plastic J-Integral analyses were shown to provide significant benefit Abstractprovide the most conservative results. The elasto-plastic J-Integral analyses were shown to provide significant benefit over necessarily application of the more conservative Failure Assessment Diagram (FAD) approach. The undertaken assessments were validated against applicationsolutions of the more Failure Assessment Diagram (FAD) approach. TheInundertaken validated against analytical andconservative historic inspection and showed good agreement. summary assessments the adopted were modelling During their operation, modern aircraftevidence engine components are subjected to increasingly demanding operatingtechniques conditions, analytical solutions andand historic inspection evidence and of showed good agreement. In summary the adopted modelling techniques released conservatisms permitted detailed assessment complex geometries and load cases. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent released conservatisms and permitted detailed assessment of complex geometries and load cases. degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict © the 2018 The Authors. Published by Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation behaviour of HPT blades. Flight © 2018creep The Authors. Published by Elsevier B.V. © company, 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. wereresponsibility used to obtain and mechanical data for three different flight cycles. In order to create the 3D model Peer-review under of thethermal ECF22 organizers. Peer-review responsibility ECF22 needed forunder the FEM analysis,ofa the HPT bladeorganizers. scrap was scanned, and its chemical composition and material properties were Keywords: Crack Meshing, FAD, obtained.Crack The Body data Modelling, that was gathered was fed intoJ-Analysis the FEM model and different simulations were run, first with a simplified 3D Keywords: Crack Bodyshape, Modelling, Crack FAD, J-Analysis rectangular block in order to Meshing, better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a can be useful in the goal of predicting turbine blade life, given a set of FDR data. 1.model Introduction
1. Introduction tolerance assessments adopted across a broad range of safety critical industries, from aerospace through ©Defect 2016 The Authors. Published byare Elsevier B.V. Defect tolerance assessments are adopted across a broad industries, from aerospace through toPeer-review high pressure components in the energy industry. The aim 2016. ofof asafety defectcritical tolerance assessment is to demonstrate under responsibility of the Scientific Committee of range PCF to high pressure components energy industry. The aim a defect toleranceintegrity assessment to demonstrate integrity of a component whichinis the known to contain a defect, or toofassess the structural of a is component which integrity ofHigh a to component which is known a defect, assess the structural integrity of a component which Keywords: Pressure Blade; Creep; to Finite Method; or 3Dto Model; Simulation. is postulated containTurbine defects at present orcontain inElement the future. is postulated to contain defects at present or in the future. * Corresponding author. Tel.: +44 (0) 1454 662 077. * Corresponding Tel.: +44 (0) 1454 662 077. E-mail address:author. [email protected] E-mail address: [email protected]
2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby ECF22 organizers. Peer-review underauthor. responsibility the ECF22 organizers. * Corresponding Tel.: +351of 218419991. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.253
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
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In the past, defect tolerance assessments would generally follow analytical procedures, simplifying the assessed geometry to available hand-book solutions, or be conducted through elastic two-dimensional (2D) cracked-body finite element analysis (CBFEA). These approaches often require a range of simplifications, including, but not limited to, the component’s geometry, the precise defect location within the component, or the details of the loading conditions. Elastic assessments were then adjusted to allow for varying levels of plasticity which may occur due to the presence of primary and secondary stresses, which influence the local crack tip driving force. Necessarily, this led to conservatisms being introduced to allow for limitations in the generic methods applied. For industrial components this may result in significant over-estimates of crack tip driving forces. With improvements in computational capabilities, it is now possible to conduct more detailed elasto-plastic threedimensional (3D) CBFEA without recourse to separate consideration of elastic plastic interaction. This paper presents and discusses modelling techniques recently adopted to the assessment of industrial components, which have released conservatisms and permitted detailed assessment of complex geometries and load cases. The method for defect tolerance assessment through finite element cracked body modelling may be broken down into the following steps: 1. Construction of a geometrical model of the component which is to be assessed; 2. Specification of the shape, orientation and location of the defect in the geometrical model; 3. Mesh generation, paying particular attention to the region surrounding the defect front and any interaction with model discontinuities; 4. Appropriate selection of the loading conditions and material properties. The resulting models are then analysed to: 1. Extract Stress Intensity Factors (SIFs) and/or strain energy release rates (J-Integral) along the defect front; 2. Perform a limit load analysis for assessment against plastic collapse. The results (SIFs/J-Integrals/limit load) may be used in failure assessment diagram (FAD) or the assessment against a material allowable toughness. Combination of these results for a range of defect lengths/ orientations/locations and load cases permits the derivation of a tolerable defect size or informs subcritical crack growth calculations. 2. Three-Dimensional Meshing Techniques Meshes of 2D cracked body are generally constructed with a radial ring of elements refined at crack tip, Figure 1. The crack front may be modelled with a range of degenerate elements, with one such example consisting of the crack tip nodes constrained to each other and with the mid side nodes on the crack front side moved to the quarter points. Approaching the crack tip, this gives a crack tip singularity of σ→r-1/2 as r→0. This type of singularity is generally adopted for elastic analyses, but may also be adopted for inelastic analyses when combined with a refined mesh.
Figure 1 – Illustration of a 2D cracked-body mesh
For 3D models this radial ring of elements is required along the complete defect front, with planes of nodes forming the contour integrals which must remain normal to the defect front. Generation of such 3D cracked body meshes in geometrically complex regions can pose significant challenges. Refined mesh densities in the near-defect region may lead to difficulties in obtaining a sufficiently coarse mesh in the surrounding structure. When considering curved or nonplanar defects (i.e. semi-elliptical), commercial meshing algorithms may struggle to construct planar elements aligned normal to the defect front within refined near-defect region, especially in proximity to model discontinuities.
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2.1. Mixed Element Type Meshing By adopting a mixed element type meshing strategy of hexahedral elements in the near-defect region and secondorder tetrahedral elements in the remaining structure, it is possible to achieve a highly refined and structured radial mesh along the defect front, whilst permitting sufficient resolution of other features. Tied constraints are adopted to connect the two element type regions. Contours for calculation of J-Integral/SIFs should lie fully within hexahedrally meshed region, such that the size of this region is driven by the adopted mesh density and contour convergence. Experience of the authors in austenitic stainless steel has shown that a size of 0.5 – 1 mm in all directions from the crack tip is generally sufficient, within which 10-30 radial rings of planar elements are defined. However, this is dependent on the size of the assessed component, the adopted mesh density and the convergence of the J-Integral / SIFs, since the region must contain the complete contour region. As for any tied interfaces in finite element models, care should be taken to ensure that there isn’t a mismatch in the element size across the interface. Convergence checks on the mesh density within both regions is highly recommended. Examples of tied mixed element type regions are illustrated in Figure 2.
Figure 2 – Illustration of mixed element type meshing strategy on the surface (a) and along a defect front (b)
Figure 3 – Hexahedral (a) and mixed element type (b) meshing approaches to a semi-elliptical internal surface breaking defect in a cylinder. Comparable and smooth stress contours are observed across the transition in all cases
The effect of the mixed element type meshing strategy and tied constraint in proximity to a crack front has been assessed for a semi-elliptical internal surface breaking defect in a cylinder under axial, bending and pressure loading. Two different tetrahedral meshes with intentionally coarse elements in either direction of the transition are chosen and compared to a fully hexahedral mesh. In both cases smooth stress and strain contours are observed across the transition, see Figure 3. Differences in the computed J-Integral values between the three meshes and relative to the analytical solution are below 0.1% when considering elastic material properties. When an elasto-plastic material response is considered, additional attention to the mesh refinement must be paid. This is a general requirement and should always be addressed in the form of a mesh refinement sensitivity study. However, even for the extremely coarse tetrahedral meshing in proximity to the tied constraint in Figure 3 (c), a solution deviating by less than 0.3% is found. The use of a tetrahedral mesh, even with poor-quality elements, is therefore shown to not have a significant effect on the CBFEA results provided the mesh is sufficiently refined. This conclusion extends to more complex geometries
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
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and loading scenarios, with smooth stress and strain contours across the transition observed by the authors for a range of industrial components assessed with elastic and elasto-plastic material models. 2.2. Node Transformation Techniques For simple geometries and straight defect fronts, it is generally possible to obtain adequately structured hexahedral mesh qualities with commercially available software. The desired mesh structure is defined on the outer surfaces and then extruded along the defect front, as shown in Figure 2 (b). Challenges can arise for curved defects and straight defects positioned in proximity to geometrical features or transitions. In such cases, commercial mesh algorithms can struggle to produce a suitable mesh, often not achieving the required planes of nodes forming the contour integrals aligned normal to the defect front. For such cases, a multistaged approach to generating a suitable mesh has been developed, consisting of the transformation of a regular mesh into the required defect shape. The steps for this approach are outlined below: 1. The main model is meshed with a commercial software. A region surrounding the defect front, which would otherwise be meshed with hexahedral elements, is excluded from the model, such that a cut-out exists in the resulting mesh. Since this mesh is excluded from the contour integral calculation, tetrahedral elements may be used for this global mesh. The mesh seeding in proximity to the cut-out should be chosen to be compatible with the seed density adopted in step (2). 2. A mesh for a rectangular box is generated with a commercial meshing algorithm. The width and depth of this model should correspond to the size of the cut-out in (1), its length to the approximate length of the defect front. Focussed hexahedral elements as detailed in section 2.1 are extruded along the length of the box. Since this model is not curved, meshing algorithms will generate contour planes which remain normal along the length. 3. The mesh generated in (2) is exported. The nodal coordinates can then be modified via a transformation into the required defect shape (i.e. a semi-ellipse). Geometrical transformation functions for a range of geometries are readily available and may be implemented in an automated script. 4. For components with non-planar surfaces (e.g. pipes) at the extremities of the defect front, a further transformation is required of the end nodes. 5. A final modification of coordinates is required for mid-side nodes to ensure these lie mid-way between their respective corner nodes. Combining the modified nodal coordinates with the original element definition provides a local hexahedral mesh, its size and geometry corresponding to the cut-out in the global mesh. The above steps are illustrated in Figure 4. The local and global mesh are combined, with tied constraints defined on the mating surfaces. Since the position of the nodes is defined through the geometrical transformation, contour planes are guaranteed to lie normal to the defect front. An example of a mesh generated for a surface-breaking semielliptical defect in a toroidal pipe is illustrated in Figure 5.
Figure 4 – Illustration of the nodal transformation technique for obtaining the near-defect front mesh region
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
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Figure 5 – An example of a 3D cracked-body mesh for a semi-elliptical surface breaking defect in a toroidal pipe generated with mixed element types and nodal transformation
3. Failure Assessment Diagram vs. Elasto-Plastic Modelling 3.1. Failure Assessment Diagram Approach CBFEA conducted to determine a limiting defect size often consists of an elastic analysis to find the J-Integral / SIF values, combined with an elasto-perfectly-plastic limit load analysis. Assessment points for a range of defect lengths are derived and plotted on a FAD, with the limiting defect size defined when the failure assessment curve (FAC) is crossed, Figure 6. However, the problem with this lies in how to interpolate between the assessed defect lengths to determine what the limiting defect size would be (i.e. the defect length at which the FAC is crossed), as Kr and Lr do not linearly interpolate/extrapolate with defect length. The solution to this problem lies in understanding the meaning of the FAC. Regardless of how it is derived, the FAC is an approximation to the value:√(Je/J), based on the value of Lr [Anderson]. That is to say, how much higher would J be if the calculation included an elasto-plastic material model rather than the elastic assumptions made (Je). From this knowledge we can recast our problem by making an estimate of what J is based on the value of Je and Lr. In this case:
We also know that Kr is calculated as
𝐽𝐽���� � 𝐾𝐾� �
𝐽𝐽� 𝑓𝑓�𝐿𝐿� ��
𝐾𝐾� 𝐾𝐾���
So, if we recast using the elastic relation between J and K we can calculate 𝐽𝐽���� �
�𝐾𝐾� 𝐾𝐾��� �� 𝐸𝐸 � 𝑓𝑓�𝐿𝐿� ��
The material allowable if not already defined in terms of the energy release rate (Jmat) can also be calculated from Kmat. 𝐾𝐾��� � �𝐸𝐸 � 𝐽𝐽���
It is then possible to re-plot the value of Jopt1 verse crack length. Figure 7 provides a clearer illustration of the response of the defect with increasing defect size when compared to the FAD, assessing Jopt1 and Jep against the material allowable. The elasto-plastic assessment is shown to release conservatisms in the analysis as evidenced by a shift in the limiting defect size The limiting defect size (i.e. the point at which the material allowable is exceeded) may then be determined directly through a chosen interpolation scheme. This is in contrast to the FAD approach, where justification of a defect length at which the FAC is crossed is less clearly supported.
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
Figure 6 - Option 1 Failure Assessment Diagram
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Figure 7 – Alternative to the FAD approach
3.2. Elasto-Plastic Cracked Body Finite Element Modelling The above outlined Jopt1 approach simplifies the determination of a tolerable defect size and is recommended for load-controlled cases were an elasto-plastic analysis is not justified or possible. For displacement-controlled load cases in excess of yield, the FAD approach, informed by elastic CBFEA contains inherent conservatisms. Strain based assessment methods have been proposed as an alternative to the more conventional FAD [Budden]. However, it is also often possible to relieve significant conservatism by conducting a fully elasto-plastic analysis to determine Jep directly and compare this against the material allowable Jmat. Figure 7 presents a comparison of Jopt1 and Jep for a range of defect lengths under identical primary loading. The elasto-plastic analysis is shown to release significant conservatism as the defect length increases. A noticeable shift in the limiting defect size is observed as a result of the more detailed elasto-plastic analysis. Previous experience of the authors has shown that the benefits in conducting elasto-plastic analyses is greatest for components and loading scenarios where the response is strongly ductile and the failure assessment points would lie close to the Lr limit. When conducting elasto-plastic CBFEA, it is further recommended that sensitivities to the material model are assessed. For load controlled cases, a lower bound stress-strain model generally provides a conservative result. However, for displacement controlled load cases, such as thermal shock loading, where secondary stresses dominate, an upper bound material model may return higher J-Integral results.It is important to note that whilst an elasto-plastic CBFEA can reduce conservatisms and simplifies the interpretation of results, a limit load analysis remains necessary to ensure the limiting defect size is not bounded by plastic collapse. 4. Conclusions Finite element cracked-body meshing techniques have been outlined, which can significantly improve the ease of mesh generation and the mesh quality of complex three-dimensional components. The adoption of a mixed element type meshing strategy has been shown to not affect results, giving confidence in results obtained from such models. The advantage of assessing limiting defect sizes against a material allowable, as opposed through a failure assessment diagram approach, has been demonstrated. Combined, these techniques enable detailed assessments of industrial components, relieving conservatisms by permitting more accurate geometrical modelling. Acknowledgements The authors would like to thank EDF Energy for permission to present this paper. References Anderson T. L., 2005, Fracture Mechanics, Fundamentals and Applications - Third Edition. Budden P. J., 2005. Failure assessment diagram methods for strain-based fracture, Engineering Fracture Mechanics 73.
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ECF22 - Loading and Environmental effects on Structural Integrity ECF22 - Loading and Environmental effects on Structural Integrity
Methods for Complex Cracked Body Finite Element Assessments Methods for Complex Cracked Body Finite Element Assessments
XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016,a,b Paço de Arcos, Portugal a a
Moritz Lessmanna*, John Sawyera, David Knowlesa,b Moritz Lessmann *, John Sawyer , David Knowles
Atkins, The Hub, 500 Park Avenue, Aztec West, Almondsbury, Bristol BS32 4RZ, UK Thermo-mechanical modeling of aWalk, high pressure turbine blade of an Atkins, The Hub, 500 Park Avenue, Aztec West, Almondsbury, BS32 4RZ, UK University of Bristol, University Clifton BS8Bristol 1TR, UK University of Bristol, University Walk, Clifton BS8 1TR, UK airplane gas turbine engine a a
b b
Abstract a b c Abstract P. Brandão , V. Infante , A.M. Deus * This paper presents approaches developed to allow consistent best practice assessments for complex 2/3D defects in engineering This paper presents approaches developed to allow consistent best practice assessments for complex 2/3Dwith defects in engineering structures subjected to primary loading and secondary thermal stresses. Methods are outlined and discussed reference to recent a Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, structures subjected to primary loading secondary thermal stresses. Methods are outlined and discussedtolerable with reference to recent twoand three-dimensional cracked bodyand analyses of components which have been undertaken to determine defect sizes and Portugal b and twothree-dimensional cracked body analyses ofanalyses components which been to determine defect and sizes and inform sub-critical crackofgrowth calculations. TheInstituto considered ahave range of undertaken postulated semi-elliptical, through-wall fully IDMEC, Department Mechanical Engineering, Superior Técnico, Universidade de Lisboa, Av. Roviscotolerable Pais, 1, 1049-001 Lisboa, inform sub-critical crackingrowth calculations. The analyses a range of postulated through-wall fully circumferential defects pressure systems. Mixed element considered typePortugal meshing strategies with tiedsemi-elliptical, contact in combination withand multiple c CeFEMA, Department Mechanical Engineering, Instituto Técnico, Universidade Lisboa, Rovisco Pais, 1, 1049-001 Lisboa, circumferential defects inofpressure systems. Mixed element type meshing strategies withdetied contact in combination with multiple node transformation techniques around the defect front wereSuperior employed during mesh generation. ThisAv. facilitated highly refined meshes Portugal node around the defect front were employed during mesh generation.inThis highly refined meshes alongtransformation the defect fronttechniques which were required to accurately model extensive plastic deformation the facilitated region of interest. Displacement along the defectloads frontrequired which were required to accurately model extensive plastic deformation in since the region interest. Displacement driven thermal detailed assessment with multiple elasto-plastic material models, lowerofbound properties do not driven thermal loads required detailed assessment with multiple elasto-plastic material models, since lower bound properties doover not necessarily provide the most conservative results. The elasto-plastic J-Integral analyses were shown to provide significant benefit Abstractprovide the most conservative results. The elasto-plastic J-Integral analyses were shown to provide significant benefit over necessarily application of the more conservative Failure Assessment Diagram (FAD) approach. The undertaken assessments were validated against applicationsolutions of the more Failure Assessment Diagram (FAD) approach. TheInundertaken validated against analytical andconservative historic inspection and showed good agreement. summary assessments the adopted were modelling During their operation, modern aircraftevidence engine components are subjected to increasingly demanding operatingtechniques conditions, analytical solutions andand historic inspection evidence and of showed good agreement. In summary the adopted modelling techniques released conservatisms permitted detailed assessment complex geometries and load cases. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent released conservatisms and permitted detailed assessment of complex geometries and load cases. degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict © the 2018 The Authors. Published by Elsevier B.V.data records (FDR) for a specific aircraft, provided by a commercial aviation behaviour of HPT blades. Flight © 2018creep The Authors. Published by Elsevier B.V. © company, 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. wereresponsibility used to obtain and mechanical data for three different flight cycles. In order to create the 3D model Peer-review under of thethermal ECF22 organizers. Peer-review responsibility ECF22 needed forunder the FEM analysis,ofa the HPT bladeorganizers. scrap was scanned, and its chemical composition and material properties were Keywords: Crack Meshing, FAD, obtained.Crack The Body data Modelling, that was gathered was fed intoJ-Analysis the FEM model and different simulations were run, first with a simplified 3D Keywords: Crack Bodyshape, Modelling, Crack FAD, J-Analysis rectangular block in order to Meshing, better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a can be useful in the goal of predicting turbine blade life, given a set of FDR data. 1.model Introduction
1. Introduction tolerance assessments adopted across a broad range of safety critical industries, from aerospace through ©Defect 2016 The Authors. Published byare Elsevier B.V. Defect tolerance assessments are adopted across a broad industries, from aerospace through toPeer-review high pressure components in the energy industry. The aim 2016. ofof asafety defectcritical tolerance assessment is to demonstrate under responsibility of the Scientific Committee of range PCF to high pressure components energy industry. The aim a defect toleranceintegrity assessment to demonstrate integrity of a component whichinis the known to contain a defect, or toofassess the structural of a is component which integrity ofHigh a to component which is known a defect, assess the structural integrity of a component which Keywords: Pressure Blade; Creep; to Finite Method; or 3Dto Model; Simulation. is postulated containTurbine defects at present orcontain inElement the future. is postulated to contain defects at present or in the future. * Corresponding author. Tel.: +44 (0) 1454 662 077. * Corresponding Tel.: +44 (0) 1454 662 077. E-mail address:author. [email protected] E-mail address: [email protected]
2452-3216 © 2018 The Authors. Published by Elsevier B.V. 2452-3216 © 2018 Authors. Published Elsevier B.V. Peer-review underThe responsibility of theby ECF22 organizers. Peer-review underauthor. responsibility the ECF22 organizers. * Corresponding Tel.: +351of 218419991. E-mail address: [email protected]
2452-3216 © 2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Scientific Committee of PCF 2016. 2452-3216 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the ECF22 organizers. 10.1016/j.prostr.2018.12.253
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
1233
In the past, defect tolerance assessments would generally follow analytical procedures, simplifying the assessed geometry to available hand-book solutions, or be conducted through elastic two-dimensional (2D) cracked-body finite element analysis (CBFEA). These approaches often require a range of simplifications, including, but not limited to, the component’s geometry, the precise defect location within the component, or the details of the loading conditions. Elastic assessments were then adjusted to allow for varying levels of plasticity which may occur due to the presence of primary and secondary stresses, which influence the local crack tip driving force. Necessarily, this led to conservatisms being introduced to allow for limitations in the generic methods applied. For industrial components this may result in significant over-estimates of crack tip driving forces. With improvements in computational capabilities, it is now possible to conduct more detailed elasto-plastic threedimensional (3D) CBFEA without recourse to separate consideration of elastic plastic interaction. This paper presents and discusses modelling techniques recently adopted to the assessment of industrial components, which have released conservatisms and permitted detailed assessment of complex geometries and load cases. The method for defect tolerance assessment through finite element cracked body modelling may be broken down into the following steps: 1. Construction of a geometrical model of the component which is to be assessed; 2. Specification of the shape, orientation and location of the defect in the geometrical model; 3. Mesh generation, paying particular attention to the region surrounding the defect front and any interaction with model discontinuities; 4. Appropriate selection of the loading conditions and material properties. The resulting models are then analysed to: 1. Extract Stress Intensity Factors (SIFs) and/or strain energy release rates (J-Integral) along the defect front; 2. Perform a limit load analysis for assessment against plastic collapse. The results (SIFs/J-Integrals/limit load) may be used in failure assessment diagram (FAD) or the assessment against a material allowable toughness. Combination of these results for a range of defect lengths/ orientations/locations and load cases permits the derivation of a tolerable defect size or informs subcritical crack growth calculations. 2. Three-Dimensional Meshing Techniques Meshes of 2D cracked body are generally constructed with a radial ring of elements refined at crack tip, Figure 1. The crack front may be modelled with a range of degenerate elements, with one such example consisting of the crack tip nodes constrained to each other and with the mid side nodes on the crack front side moved to the quarter points. Approaching the crack tip, this gives a crack tip singularity of σ→r-1/2 as r→0. This type of singularity is generally adopted for elastic analyses, but may also be adopted for inelastic analyses when combined with a refined mesh.
Figure 1 – Illustration of a 2D cracked-body mesh
For 3D models this radial ring of elements is required along the complete defect front, with planes of nodes forming the contour integrals which must remain normal to the defect front. Generation of such 3D cracked body meshes in geometrically complex regions can pose significant challenges. Refined mesh densities in the near-defect region may lead to difficulties in obtaining a sufficiently coarse mesh in the surrounding structure. When considering curved or nonplanar defects (i.e. semi-elliptical), commercial meshing algorithms may struggle to construct planar elements aligned normal to the defect front within refined near-defect region, especially in proximity to model discontinuities.
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
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2.1. Mixed Element Type Meshing By adopting a mixed element type meshing strategy of hexahedral elements in the near-defect region and secondorder tetrahedral elements in the remaining structure, it is possible to achieve a highly refined and structured radial mesh along the defect front, whilst permitting sufficient resolution of other features. Tied constraints are adopted to connect the two element type regions. Contours for calculation of J-Integral/SIFs should lie fully within hexahedrally meshed region, such that the size of this region is driven by the adopted mesh density and contour convergence. Experience of the authors in austenitic stainless steel has shown that a size of 0.5 – 1 mm in all directions from the crack tip is generally sufficient, within which 10-30 radial rings of planar elements are defined. However, this is dependent on the size of the assessed component, the adopted mesh density and the convergence of the J-Integral / SIFs, since the region must contain the complete contour region. As for any tied interfaces in finite element models, care should be taken to ensure that there isn’t a mismatch in the element size across the interface. Convergence checks on the mesh density within both regions is highly recommended. Examples of tied mixed element type regions are illustrated in Figure 2.
Figure 2 – Illustration of mixed element type meshing strategy on the surface (a) and along a defect front (b)
Figure 3 – Hexahedral (a) and mixed element type (b) meshing approaches to a semi-elliptical internal surface breaking defect in a cylinder. Comparable and smooth stress contours are observed across the transition in all cases
The effect of the mixed element type meshing strategy and tied constraint in proximity to a crack front has been assessed for a semi-elliptical internal surface breaking defect in a cylinder under axial, bending and pressure loading. Two different tetrahedral meshes with intentionally coarse elements in either direction of the transition are chosen and compared to a fully hexahedral mesh. In both cases smooth stress and strain contours are observed across the transition, see Figure 3. Differences in the computed J-Integral values between the three meshes and relative to the analytical solution are below 0.1% when considering elastic material properties. When an elasto-plastic material response is considered, additional attention to the mesh refinement must be paid. This is a general requirement and should always be addressed in the form of a mesh refinement sensitivity study. However, even for the extremely coarse tetrahedral meshing in proximity to the tied constraint in Figure 3 (c), a solution deviating by less than 0.3% is found. The use of a tetrahedral mesh, even with poor-quality elements, is therefore shown to not have a significant effect on the CBFEA results provided the mesh is sufficiently refined. This conclusion extends to more complex geometries
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
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and loading scenarios, with smooth stress and strain contours across the transition observed by the authors for a range of industrial components assessed with elastic and elasto-plastic material models. 2.2. Node Transformation Techniques For simple geometries and straight defect fronts, it is generally possible to obtain adequately structured hexahedral mesh qualities with commercially available software. The desired mesh structure is defined on the outer surfaces and then extruded along the defect front, as shown in Figure 2 (b). Challenges can arise for curved defects and straight defects positioned in proximity to geometrical features or transitions. In such cases, commercial mesh algorithms can struggle to produce a suitable mesh, often not achieving the required planes of nodes forming the contour integrals aligned normal to the defect front. For such cases, a multistaged approach to generating a suitable mesh has been developed, consisting of the transformation of a regular mesh into the required defect shape. The steps for this approach are outlined below: 1. The main model is meshed with a commercial software. A region surrounding the defect front, which would otherwise be meshed with hexahedral elements, is excluded from the model, such that a cut-out exists in the resulting mesh. Since this mesh is excluded from the contour integral calculation, tetrahedral elements may be used for this global mesh. The mesh seeding in proximity to the cut-out should be chosen to be compatible with the seed density adopted in step (2). 2. A mesh for a rectangular box is generated with a commercial meshing algorithm. The width and depth of this model should correspond to the size of the cut-out in (1), its length to the approximate length of the defect front. Focussed hexahedral elements as detailed in section 2.1 are extruded along the length of the box. Since this model is not curved, meshing algorithms will generate contour planes which remain normal along the length. 3. The mesh generated in (2) is exported. The nodal coordinates can then be modified via a transformation into the required defect shape (i.e. a semi-ellipse). Geometrical transformation functions for a range of geometries are readily available and may be implemented in an automated script. 4. For components with non-planar surfaces (e.g. pipes) at the extremities of the defect front, a further transformation is required of the end nodes. 5. A final modification of coordinates is required for mid-side nodes to ensure these lie mid-way between their respective corner nodes. Combining the modified nodal coordinates with the original element definition provides a local hexahedral mesh, its size and geometry corresponding to the cut-out in the global mesh. The above steps are illustrated in Figure 4. The local and global mesh are combined, with tied constraints defined on the mating surfaces. Since the position of the nodes is defined through the geometrical transformation, contour planes are guaranteed to lie normal to the defect front. An example of a mesh generated for a surface-breaking semielliptical defect in a toroidal pipe is illustrated in Figure 5.
Figure 4 – Illustration of the nodal transformation technique for obtaining the near-defect front mesh region
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
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Figure 5 – An example of a 3D cracked-body mesh for a semi-elliptical surface breaking defect in a toroidal pipe generated with mixed element types and nodal transformation
3. Failure Assessment Diagram vs. Elasto-Plastic Modelling 3.1. Failure Assessment Diagram Approach CBFEA conducted to determine a limiting defect size often consists of an elastic analysis to find the J-Integral / SIF values, combined with an elasto-perfectly-plastic limit load analysis. Assessment points for a range of defect lengths are derived and plotted on a FAD, with the limiting defect size defined when the failure assessment curve (FAC) is crossed, Figure 6. However, the problem with this lies in how to interpolate between the assessed defect lengths to determine what the limiting defect size would be (i.e. the defect length at which the FAC is crossed), as Kr and Lr do not linearly interpolate/extrapolate with defect length. The solution to this problem lies in understanding the meaning of the FAC. Regardless of how it is derived, the FAC is an approximation to the value:√(Je/J), based on the value of Lr [Anderson]. That is to say, how much higher would J be if the calculation included an elasto-plastic material model rather than the elastic assumptions made (Je). From this knowledge we can recast our problem by making an estimate of what J is based on the value of Je and Lr. In this case:
We also know that Kr is calculated as
𝐽𝐽���� � 𝐾𝐾� �
𝐽𝐽� 𝑓𝑓�𝐿𝐿� ��
𝐾𝐾� 𝐾𝐾���
So, if we recast using the elastic relation between J and K we can calculate 𝐽𝐽���� �
�𝐾𝐾� 𝐾𝐾��� �� 𝐸𝐸 � 𝑓𝑓�𝐿𝐿� ��
The material allowable if not already defined in terms of the energy release rate (Jmat) can also be calculated from Kmat. 𝐾𝐾��� � �𝐸𝐸 � 𝐽𝐽���
It is then possible to re-plot the value of Jopt1 verse crack length. Figure 7 provides a clearer illustration of the response of the defect with increasing defect size when compared to the FAD, assessing Jopt1 and Jep against the material allowable. The elasto-plastic assessment is shown to release conservatisms in the analysis as evidenced by a shift in the limiting defect size The limiting defect size (i.e. the point at which the material allowable is exceeded) may then be determined directly through a chosen interpolation scheme. This is in contrast to the FAD approach, where justification of a defect length at which the FAC is crossed is less clearly supported.
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Moritz Lessmann et al. / Procedia Structural Integrity 13 (2018) 1232–1237 Author name / Structural Integrity Procedia 00 (2018) 000–000
Figure 6 - Option 1 Failure Assessment Diagram
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Figure 7 – Alternative to the FAD approach
3.2. Elasto-Plastic Cracked Body Finite Element Modelling The above outlined Jopt1 approach simplifies the determination of a tolerable defect size and is recommended for load-controlled cases were an elasto-plastic analysis is not justified or possible. For displacement-controlled load cases in excess of yield, the FAD approach, informed by elastic CBFEA contains inherent conservatisms. Strain based assessment methods have been proposed as an alternative to the more conventional FAD [Budden]. However, it is also often possible to relieve significant conservatism by conducting a fully elasto-plastic analysis to determine Jep directly and compare this against the material allowable Jmat. Figure 7 presents a comparison of Jopt1 and Jep for a range of defect lengths under identical primary loading. The elasto-plastic analysis is shown to release significant conservatism as the defect length increases. A noticeable shift in the limiting defect size is observed as a result of the more detailed elasto-plastic analysis. Previous experience of the authors has shown that the benefits in conducting elasto-plastic analyses is greatest for components and loading scenarios where the response is strongly ductile and the failure assessment points would lie close to the Lr limit. When conducting elasto-plastic CBFEA, it is further recommended that sensitivities to the material model are assessed. For load controlled cases, a lower bound stress-strain model generally provides a conservative result. However, for displacement controlled load cases, such as thermal shock loading, where secondary stresses dominate, an upper bound material model may return higher J-Integral results.It is important to note that whilst an elasto-plastic CBFEA can reduce conservatisms and simplifies the interpretation of results, a limit load analysis remains necessary to ensure the limiting defect size is not bounded by plastic collapse. 4. Conclusions Finite element cracked-body meshing techniques have been outlined, which can significantly improve the ease of mesh generation and the mesh quality of complex three-dimensional components. The adoption of a mixed element type meshing strategy has been shown to not affect results, giving confidence in results obtained from such models. The advantage of assessing limiting defect sizes against a material allowable, as opposed through a failure assessment diagram approach, has been demonstrated. Combined, these techniques enable detailed assessments of industrial components, relieving conservatisms by permitting more accurate geometrical modelling. Acknowledgements The authors would like to thank EDF Energy for permission to present this paper. References Anderson T. L., 2005, Fracture Mechanics, Fundamentals and Applications - Third Edition. Budden P. J., 2005. Failure assessment diagram methods for strain-based fracture, Engineering Fracture Mechanics 73.