AUTOMATED METHODOLOGY FOR MODELING CRACK EXTENSION IN FINITE ELEMENT MODELS

Transcription

1 AUTOMATED METHODOLOGY FOR MODELING CRACK THEME Structural Analysis - Durability, Fatigue & Fracture. James J. Kosloski Senior Engineering Manager - CAE Associates Dr. Michael Bak Senior Engineering Manager - CAE Associates Thomas Meyer Associate - CAE Associates KEYWORDS fracture mechanics, crack growth, mesh morphing, life prediction, fatigue, stress intensity factor, finite element. SUMMARY The durability of aging aerospace components is compromised by the existence and continued growth of cracks due to cyclic loading conditions. Fracture mechanics provides accepted analytical and numerical approaches for predicting the remaining fatigue life for a known existing crack in a known cyclic loading environment. Fracture mechanics can be used to predict how the cracks will grow as a function of cyclic loading, the path they will take, and at which point the crack will propagate to failure. In this paper we present an automated method for modelling the extension of a crack in a finite element analysis. Typical procedures for calculating crack growth life involve determining the stress intensity factor (K) at the crack tip as a function of crack length and applied loading. This data can then be used with an initial flaw size, the material fracture toughness, and Paris law constants to determine how many cycles it will take to grow a crack to failure. To obtain the stress intensity factor as a function of crack length, the finite element model must be updated to include the extended crack. The typical procedure for

2 EXTENSION IN FINITE ELEMENT extending a crack involves manually modifying the underlying geometry using CAD tools, re-meshing the model, and reapplying boundary conditions and loads. This procedure is tedious and time consuming. An automated method for extending a crack in a finite element model has been developed using User Programmable Features (UPF) in the ANSYS finite element code. In this method the existing mesh of the model is morphed so that the edge of an element lies along the predicted crack growth direction. This element is then separated from its neighbouring element, thus incrementally extending the crack by one element length. The model is reanalyzed and a new stress intensity factor is calculated. The procedure is automatically repeated, extending the crack by one element length and recalculating K until the desired crack length is obtained. Upon completion of a series of such analyses of the structure with increasingly large cracks, the fracture life-time can be obtained by using material crack growth data.

3 1: Introduction Understanding how cracks will grow in a complicated structure is a major challenge. This must be well understood for safe operation and for setting safe inspection intervals to detect the presence and growth of unexpected cracks. There are two parts in such predictions: 1) determining the Stress Intensity Factor (SIF) for a crack in a complex part & 2) determining the rate of cyclic crack growth which is controlled by the SIF. This paper reports a new innovative method for the first part; obtaining SIFs for an arbitrary structure. It has the advantage of using existing CAD or finite element files and eliminates the need to include crack geometry in the finite element mesh. Currently there are several techniques available to predict SIF (KI) for a given cracked geometry using general purpose finite element codes such as ANSYS. These include special crack-tip elements, J-integral calculations around the crack tip and curve-fitting to theoretical infinite stresses at crack tip. Each of these techniques requires that the crack be modelled and included in the finite element mesh. Two existing methods for including a crack in a finite element model are: 1. Model distinct element edges along the entire crack path using coupled, coincident nodes. A crack can then be extended by de-coupling nodes along the crack boundary. The disadvantage here is that the crack path must be fully known before hand, as no update of the crack direction can be included after meshing. Load redistribution must be done by hand if the crack affects the overall compliance of the structure. 2. Regenerate the model after each crack growth increment. The existing model must be modified, usually by changing the CAD geometry, to include a longer crack. This method has the advantage of not having to predetermine the crack path and can more easily account for load redistribution. However, considerable effort is required on each crack increment to develop new CAD file with a crack, re-mesh the model and, re-apply boundary conditions.. This paper describes a method developed by CAE Associates which eliminates the aforementioned disadvantages. In this method, a crack is inserted and automatically extended without changing element numbering in the finite element model. There is no need to include the crack in the CAD or FEA file, no need to predetermine the crack path, and no need to re-mesh or reapply loads or boundary conditions. The output of the analysis is a file of SIF versus crack length which can then be used in crack growth utility codes such as ncode DesignLife (TM), NASGRO and AFGROW. Figure 1 illustrates the overall method, which will be discussed below. This new method will greatly increase the efficiency of crack propagation analyses. It is particularly relevant for application in overhaul and repair procedures as well as setting subsequent

4 EXTENSION IN FINITE ELEMENT inspection intervals. We refer to this new method as the Automated Crack Extension or ACE. The desired output of the automated procedure is to obtain the stress intensity factor (SIF) at the crack tip as a function of the crack length, a. Such information can subsequently be used as input for cyclic crack growth codes. Figure 1: Pictorial Representation of Automated Crack Extension (ACE) Method Key features of the ACE method. The ACE procedure automatically predicts crack growth direction, extends the crack in the finite element model, and performs analysis until a specified crack length is reached. o By default the crack grows perpendicular to the maximum principal stress, but other crack growth directions can easily be incorporated. o The crack path is calculated and advanced internally. There is no need to pre-determine the crack path. o KI is automatically calculated based on the J integral approach and output to a text file. The output is SIF versus the crack length which is compatible with fracture mechanics codes to calculate cyclic crack life. Only one baseline CAD geometry is needed. There is no need to model any crack explicitly in the CAD model.

5 Only one finite element mesh is needed. There is no need for manual remesh of the structure as the crack advances. Node and element numbers of the original mesh are maintained. ACE includes the effect of load redistribution due to the crack. The procedure is automated and efficient - no manual interaction is required. Mesh-morphing and crack extension add negligible time to analysis procedure. The initial crack location, orientation and length can be arbitrarily defined without explicitly including it in the mesh. The procedure allows the same global mesh and boundary conditions to be used with cracks at any location. It can be used to evaluate the severity of cracks found during inspection. 2: Overall Methodology ACE utilizes the ANSYS general purpose finite element code to perform stress analyses of a cracked structure. The ACE process consists of the following four steps: 1. Definition of the un-cracked structure and loading. The model of structure is created in the normal manner using either a CAD file or an existing FE model. At this step, the structure has no crack and the FE mesh need not anticipate the location or path of the crack. 2. Initial Crack Definition The starting and ending locations of the initial crack are specified by coordinate location. ACE inserts the crack in the un-cracked FE mesh. A straight line is assumed between the start and end points of the crack. Figure 2 shows the original mesh and the mesh with the initial crack inserted.

6 EXTENSION IN FINITE ELEMENT Figure 2: Initial Crack Insertion The element connectivity is modified and new nodes are added on one side of the crack so that elements no longer share nodes on the crack faces. The coordinate locations of the crack face nodes are modified so they lie directly along the initial crack. The node location modification and element redefinition is performed using the user programmable feature (UPF) capabilities of ANSYS. A user routine was developed to perform these modifications quickly, without requiring a full re-mesh of the geometry. 3. Structural analysis & SIF calculation Structural analysis of the cracked body is performed to obtain and save the SIF versus crack length. This analysis includes the crack and will capture load redistributions due to its presence. The ANSYS CINT command is used to calculate the J-integral at the crack tip. The crack tip node number and a coordinate system representing the crack normal direction is stored by the routine and the J-integral calculation is done without any user intervention. The SIF (K) is calculated from the J-integral and the element material modulus. The SIF and present crack length is saved to a file for post processing in a cyclic crack growth code. 4. Crack Extension & mesh morphing along the new crack front. The cracked-body stress field is evaluated and the crack is advanced by a prescribed amount in a direction perpendicular to the cracked body maximum principal stress. As with the initial crack, the elements are separated along the crack boundary by generating duplicate nodes and redefining the element connectivity on one side of the crack. The nodes are moved so that they lie exactly on the new crack face. The routine checks the new element shapes. If an element will become too distorted by moving its nodes to the crack face, that element is automatically split into 2 triangles. Figure 3 illustrates the crack advance algorithm after the first increment.

7 Figure 3: First Crack Extension Increment The process is repeated during each increment of crack extension until the final defined crack length is reached. Figure 4 shows the crack advance algorithm after the final increment. Figure 4: Final Crack Extension Increment During each increment of crack advance, the SIF and crack length are appended to a text file. This file of crack length vs. SIF can then be used to calculate the number of cycles it takes to grow a crack using a crack growth code. 3: Theory The crack extension routine in ANSYS consists of two parts. An APDL macro [1] and a user programmable feature (UPF) of a custom ANSYS command (usr4) [2]. The macro sets up the model for the call to the custom command. It determines the nodes and element edges that will represent the crack face, defines a coordinate system that points in the crack growth direction, splits elements into 2 triangles if required, and performs additional node smoothing if required. The macro also tracks the crack tip nodes, calls

8 EXTENSION IN FINITE ELEMENT the ANSYS CINT command [3] to calculate the J-integral, and outputs the crack length and SIF to a text file. The crack direction for the initial crack is determined by the input locations for the crack start and crack end. For subsequent increments the macro calculates the principal stress direction a small distance ahead of the crack tip and l creates a new coordinate system aligned with this principal direction. If desired, the macro can generate a series of plots of the crack extension. The custom command takes the next node along the crack face, as determined by the macro, and moves it so that it lies directly on the newly calculated crack surface. Duplicate nodes are created and elements are redefined based on these newly created nodes. Smoothing of nearby elements is performed to make sure the elements retain reasonable shapes. Any temperatures that were applied on the original nodes are copied to the newly created nodes by this routine. The inputs to the macro are: 1. Node number for crack starting point. 2. Coordinates of initial crack end point. 3. Desired crack extension length. 4. Radius around crack for smoothing of nodes. 5. Number of contours to be used with the CINT command. 6. Decay coefficient, that determines how quickly the transition from moved nodes to non-moved nodes occurs. 7. Number of additional smoothing passes to use to improve element shapes. The custom command is called by the macro with the following inputs: 8. Radius from input number 4 above. 9. Node at the start of crack at this increment. This node will be duplicated and attached elements on one side redefined. 10. Node at the end of crack for this increment. This node will be moved by the custom command. 11. Crack increment length.

9 The macro will determine how many elements must be modified to achieve the desired input crack extension and it will call the custom command once for each element. 4: Examples The automated procedure was verified using a model of a plate with a hole under tension, which has a known theoretical solution for KI vs. a [4]. See Figure 5. Figure 5: Verification analysis of a plate with cracked hole. A half symmetry model was built in ANSYS for use in validating the ACE method. The loading was applied and an initial mesh was generated. As with any finite element analysis, a reasonably fine mesh is required for accurate results. Therefore, local mesh refinement was used to generate a fine mesh in the expected crack growth region. Figure 6 shows the original un-cracked mesh. Figure 6: Example Model Mesh

10 EXTENSION IN FINITE ELEMENT For comparison, an additional 1/4 symmetry model was created in which the crack was advanced manually by incrementally deleting symmetry constraints along the boundary. Stress intensity factor versus crack length data calculated from these two analyses were compared to a theoretical value. The theoretical solution is valid only for a << width of plate. Figure 7 shows that ACE matches the theoretical solution, as well as the manual method Stress Intensity (K) Theory Manual method Automated Method Crack Length (a) Figure 7: SIF vs. Crack Length for Example Problem. Airframe structural beam example The problem of a crack in structure similar to an airframe beam was used as a second example of the ACE method. Such problems are of critical importance when cracks are discovered during maintenance operations on aircraft. Figure 8 shows a prototypical airframe beam section that was used to demonstrate the new ACE method. Figure 8: Prototypical Airframe Beam Section

11 Using ANSYS, the un-cracked geometry was meshed, loads and boundary conditions applied, and analyzed as a 2D plane stress representation. Again a refined mesh was used in the region of expected crack propagation. Figure 9 shows the 2D plane stress model used for this example. Figure 9: 2D Finite Element Model of Airframe Example The ACE procedure was used to advance the crack more than 20 times. This is equivalent to conducting 20 independent analyses if the previously mentioned classical methods were used. Total run time for a crack advance from 0.15 inches to 2 inches was about 30 seconds. There was no user intervention during the analysis. Figure 10 shows the initial crack. Figure 11 shows the final crack extension. In each of these figures the crack is annotated in red. Figure 10: Initial Crack in Airframe Example

12 EXTENSION IN FINITE ELEMENT Figure 11: Final Crack in Airframe Example The redistribution of loads during the crack advance can be seen in Figure 12. Figure 12: Redistribution of Stress During Crack Extension. Figure 13 shows the resulting SIF along the crack path which can be used in a crack growth simulation program.

13 KI a Figure 13: Stress Intensity vs. Crack Length for Airframe Example 5: Conclusions and Future Work When a crack is discovered in a service part during routine inspections or otherwise, it is necessary to understand how, and at what rate, the crack will grow and how its presence affects the compliance of the structure. The time required to provide these answers using the traditional methods is often very long and can significantly delay the part s return to service. The new ACE method offers the ability to quickly evaluate such cracks by using existing CAD or FE models. The use of the ACE method provides an automated technique for inserting and extending a crack in an existing FE mesh. The starting crack can be inserted at any location in a model without prior definition in the CAD file or re-meshing. J-integral fracture mechanics calculations are used to extract stress intensity factors at the crack tips at each increment. No special crack tip elements are required for the J-integral calculations. The ACE method has been developed for 2D finite element models with a single crack. Future enhancements to the ACE method would include extending the capability to 3D, where the crack boundary would be a surface, and the capability to have multiple cracks growing simultaneously. Additional enhancements would involve automatically passing the SIF vs. a data into a fracture life code such as ncode DesignLife (TM), NASGRO, or AFGROW.

14 EXTENSION IN FINITE ELEMENT REFERENCES [1] ANSYS Parametric Design Language Guide, ANSYS Release 12.1, November ANSYS, Inc, Canonsburg, PA. [2] ANSYS Programmer's Manual for Mechanical APDL, ANSYS Release 12.1, November ANSYS, Inc, Canonsburg, PA. [3] ANSYS Commands Reference, ANSYS Release 12.1, November ANSYS, Inc, Canonsburg, PA. [4] Dowling, Norman E., 1993, Mechanical Behavior of Materials : Engineering Methods for Deformation, Fracture & Fatigue, Prentice Hall, Englewood Cliffs, NJ.