Development of a Fracture Model
Chapter 5 Development of a Fracture Model
In the development of an integrated model for evaluating low-temperature brittle fracture, a proper fracture model is essential for the determination of stresses and forces acting on a pipe wall during a low-temperature brittle fracture development in pressurised pipelines. Previous research that was conducted by Atti (2004) employed an analytical stress intensity fracture shape function in calculation stress intensity factor. Sheibani and Olson (2013, p.741) noted that the determination of stress intensity factor plays an important role in elastic fracture mechanics (LEFM) problems. The propagation of fracture is controlled by the established stress field near the tip the crack (or simply crack tip). Due to the fact that this stress field is inherently singular or asymptotic dominant, it is mainly characterised by the nature of stress intensity factor (SIF). The only difference in our cases is that we are dealing with a curved surface when in normal conditions; the shape function is only valid for an infinite plate with a hole and a crack. When the above function is applied on a pipeline puncture, the curvity of the pipe wall will bring significant error (Rahman et al., 1998). That’s why there is a need for a more complex fracture model to be developed. The development of the improved fracture model for this study will involve four distinct steps. These steps are;
Developing a finite element model in ABAQUS for calculating the stress intensity factor.
Deriving the weight function parameters from results of the ABAQUS model
Implementing the failure assessment diagram (FAD) to account for the plastic collapse
Generalisation of the weight function parameters by curve fitting and neural network
The completed model can then be used to assess the fracture condition of a given geometry.
FEM Modelling using ABAQUS
What is Finite Element Analysis?
According to Huebner (2001), the finite element technique is a special numerical analysis technique that is used in obtaining various approximate solutions to a wide variety of engineering problems. The technique was originally designed for studying stresses in various airframe structures and was later adapted to a wide range of mechanics problems. Engineering problems were initially solved by deriving complex differential equations relating to the problems. Other principles and laws of physics were then used or applied to these differential equations. Equations such as Newton’s second law of motion, laws of equilibrium, laws of conservation of energy and mass were applied but the outcome or resulting equations were difficult to solve. The solution to these problems was the development of finite element method (FEM).
Finite Element Analysis (FEA) has become a most popular method of solving real life problems in mechanical, heat transfer, magnetic, and other field of analysis. According to Reddy (1988), finite element technique is one of the most robust numerical techniques ever invented for use in solving integration and differential equations of initial as well as boundary-value problems in regions that are geometrically complicated. There exists data that can simply not be ignored when we are analyzing any given element using the finite element technique. This input data is used in defining the domain (boundary and initial conditions) as well as the physical properties of the material. After gaining deep knowledge of this data and if the analysis is carefully done, satisfactory results are bound to be outcome. The process involved in a finite element analysis is quite methodological and makes it quite easy for the results to be applied to real situations. The finite element analysis of a given problem is further noted by Reddy (1988) to be so systematic in a manner that it can be divided into a series of logical steps that are easy to implement on a computer and the results be used in solving a wide range of problems by a mere change of the input data into a special computer program (ABAQUS in this case).
Finite element analysis can be conducted for one, two and even three-dimensional situations (problems). The easiest ones are the problems that involve one or two dimensional situations as they can be solved without the help of a computer. Finite element as a numerical technique can help us in finding the approximate solutions to the various boundary value problems involved in an element of a pipeline under stress from thermodynamic forces due to anthropogenic carbon dioxide transportation.
In this case, ABAQUS is our Finite Element Analysis (FEA) software of choice and we used it in analysing a pipeline material and to find out how applied stresses would affect the material.
FEA analysis helped us in determining most points of weaknesses in pipeline design after low-temperature induced brittle fracture (LTBF) has kicked into action. The analysis was conducted by creation of a mesh of points in the shape of a pipeline that carried information about the pipeline material and the object at every point for analysis. In addition to the ability to determine reactions to the stresses acting upon the material, Finite Element Analysis can also allow us to analyze the effects of fatigue as well as heat transfer on the pipeline material.
The Finite Element Analysis (FEA) involved the modelling of a section of a pipeline used for transporting anthropogenic CO2 for the sole purpose of locating as well as solving potential as well as existing performance or structural issues that result due to low temperature-induced fracture (LTBF). The method would allow us to numerically and mathematically solve quite complex structural problems.
A finite element (FE) model is made up of a system of points known as “nodes” that forms the shape of a pipe. Connected to these specific nodes are the “finite elements” that create the finite mesh as well as contain the structural and material properties of the pipeline model- these define how the material would react to various thermodynamic conditions and forces. The density of the created finite element mesh may not be uniform throughout the material. These are however dependent on the expected changes in stress levels of a given area. The areas that experience greater fluctuations in stress levels would require a higher mesh density that the areas that experience reduced stress variations as would be seen later in this chapter. Points of interest included fracture points or cracks on previously tested pipeline materials, holes and other high-stress regions.
The main advantages of FEA that motivated our choice for the technique are numerous. FEA can be employed on a new product design or in refining an already existing product (such as an existing pipeline material) in order to ensure that the given design can perform to the required standards before manufacturing. FEA can in most cases be used in predicting and improving product performance as well as reliability, reducing testing and physical prototyping costs, evaluating various materials and designs and optimizing designs as well as reducing material usage (Roylance, 2001; Chaskalovic,2008; Zienkiewicz et al., 2008; Bathe,1976). A general purpose Finite Element package provides necessary tools to perform such analysis for a wide variety of problems without compromising accuracy. The commercial FEA package ABAQUS is applied in this study.
Abaqus is noted by Hibbitt et al, (2004) to be a collection of specialized engineering simulation programs that are based on the principles of Finite Element Method (FEM). This quite powerful software application is capable of solving relatively simple linear engineering problems to the most complex nonlinear simulations. Abaqus has an extensive library of elements and this allows it to model virtually any form of geometry. The behaviour of pipeline materials can be simulated by means of Abaqus.
The ABAQUS is one of the most powerful commercial finite element software packages. It has strong capabilities for solving linier and non-linear problems. The process of solving a general problem using ABAQUS involves three stages (Hibbitt et al, 2004):
ABAQUS pre-processor which provides a compatible input file which includes all model data to the solver.
ABAQUS Solver which solves the problem based on implicit algorithm for static problems or explicit algorithm for dynamic problems.
ABAQUS post-processor which can be used for displaying the results of the problem.
The three stages of analysis using ABAQUS (Hibbitt et al, 2004)
In the pre-processing stage, the problem at hand is defined as an accurate model and then an ABAQUS input file created. The model can either be created in an ABAQUS/CAE environment or using any other appropriate pre-processors like Solidworks. Alternatively, an input file can be created directly using an available text program file like notepad.
The simulation process can be run as a background process in which either ABAQUS/Explicit or ABAQUS/Standard numerically solves the given problem. The simulations may take just seconds to several days to execute depending on the complexity of the situation as well as the computational power of the computer being used. Once the simulation has been completed, the variables such as stress and strain are calculated and then viewed by means of the Visualization module of ABAQUS/CAE. This specific module allows us a chance of displaying the output file in various ways including animations, X-Y plots, colour contour as well as deformed shape plots (Hibbitt et al, 2004).
According to Cristea (2012) ABAQUS can affectively be used in crack modeling of ductile /fracture using the Abaques/ Explicit as well as the new eXtended Finite Element Method
(XFEM) functionality in Abaqus/Standard. By means of an Abaqus/Explicit users can take advantage of pipe or tube symmetry in order to come up with 1/8th models as well as reduce simulation time (Cristea,2012). Pressure can effectively be applied to the pipe’s inner surface of the ABAQUS model and then ramped up from zero (0 bar) to burst (over 1600 bars). Crack occurrence in pressurized pipelines can be simulated by making use of ‘element deletion’ to effectively signal when plastic strain in a given part of the model has achieved a critical value (that had previously been identified via real-world burst/rupture tests. Whenever a failure criterion is achieved by a given element, it is effectively dropped from the model and thereby leaving a space behind. A series of connected and yet deleted-elements spaces is what creates the characteristics appearance of a crack. This Abaqus/Explicit simulation behavior is basically static until such a time that the crack opens at which point the analysis shifts dramatically to a dynamic mode. eXtended Finite Element Method (XFEM) enables very advanced crack modeling ability. eXtended Finite Element Method (XFEM) is a relatively new feature to the Abaqus Unified FEA suite. It is currently accessible in the Abaqus/Standard version and allows us to simulate crack initiation, and propagation on the test model on an arbitrary path that even cuts across the FEA model’s element boundaries. On the basis of a solution-dependent crack initiation as well as propagation path, the eXtended Finite Element Method (XFEM) feature does not require the formulated mesh to strictly conform to the object’s geometric discontinuities. This effectively requires reduced levels of mesh refinement in the areas lying in close proximity to the crack tip. This is by all means, a more superior methodology to the ones used in earlier simulations.
The ABAQUS/CAE environment provides a simple, consistent interface for creating, submitting, monitoring and evaluating results for specific problems. It is divided into modules where each module defines a logical aspect of the modelling process, such as geometry, material properties, mesh, boundary conditions, and many more (Hibbitt et al, 2004)
This section presents the development of the fracture model in ABAQUS used to predict the propagation of cracks along the surface of a pressurised pipeline carrying anthropogenic carbon dioxide. The FEM model calculates the mode I stress intensity factor at the crack tip of the pressurised pipeline in response to several loading conditions.
Geometry
The problem in this study consists of a cylindrical tube of defined diameter and thickness, a puncture hole of given diameter on the tube wall, and a longitudinal crack starts from the edge of the puncture, as showed in figure 5.1.
Figure 5.1 Geometry of the problem in this study
In the ABAQUS, only half of the tube is modelled due to symmetry of the geometry, as shown in such the computational runtime is reduced. The model geometry in ABAQUS is showed in figure 5.2, the length of the tube is significantly larger than the puncture, as compared to figure 5.1, to minimise the effect of fixed boundary at both end of the tube.
Mesh
Mesh generation is one of the most important procedures of finite element analysis (FEA). Finite Element Method (FEM) involves the division of the analysis region into numerous sub-regions. The small regions are the actual elements that are connected with other adjacent elements at various nodes. Mesh generation is the process of generating various geometric data of a material’s elements as well as their nodes and entails the computation of nodes coordinates, defining the nodes’ connectivity as well as construction of the elements. Mesh in this case designates the aggregates of a material’s elements, lines and nodes representing their overall connectivity. The geometric characteristics of the generated material elements have a great effect on the accuracy and performance of a finite element analysis. This makes mesh generation to be one of the most important procedures in a finite element analysis and modelling (Bui & Vahn , 1990). Lewis et al., (1995, p.47) noted that in finite element analysis, generation of an appropriate mesh is quite a great challenge. They noted that it is important for a mesh to be generated on the basis of a suitable density distribution for the numerical analysis to provide results that are as optimal as possible and with the lowest computational costs.
5.1.1.1 Characteristics of a good mesh
According to Yu et al., (2008), a good mesh must be able to possess certain qualities. The first quality is that it must be able to preserve the features of the material under study. The second quality is that it must be highly adaptable. The third on is that it must be of high quality. The two initial properties demands that the mesh generation process captures all the important features of a material and at the same time have control of seamlessly producing very dense meshes at the regions of interest as well as coarse ones elsewhere. Quality of a give mesh can be gauged by either it means of a solution-dependent (Brandt, 1977) or geometry-dependent criteria (Pebay & Baker, 2003). It is important for the quality of mesh to be guaranteed in both simulation and molecular modelling because “”skinny triangles may cause poor quality of approximations of finite and boundary element methods (Jollife,1986).
As crack tip can cause stress concentration, the stress and strain gradients are large as a crack is approached. Therefore, the finite element mesh must be refined in the vicinity of the crack to get accurate stress and strains. Also, for mesh convergence, the singularity at the crack tip is modelled by collapsing element side into single node. The final mesh is shown in figure 5.3 using 20-node quadratic brick with reduced integration element (C3D20R). Figure 5.4 shows the mesh refinement in the vicinity of the crack tip. The red line in figure 5.4 represents the position of the crack.
One of the most common difficulties in the modelling of Linear Elastic Fracture Mechanics
(LEFM) problems by employing finite element technique is related to the fact that the polynomial basis functions applied for most conventional elements can never characterize the singular crack-tip stress as well as strain fields that are predicted by the guiding theory and principles. The implication of this is that as the mesh gets refined then the finite element solution would initially commence to converge towards the theoretical solution but would diverge eventually. This kind of difficulty was identified and demonstrated during the early development of the concept of finite element method as noted by Chan (1970). Several other researchers studied special finite element formulations that take into consideration stress intensity factors and singular basis functions as nodal variables (Benzley, 1974;Tracey, 1971; Papaioannou, 1974). Even though their works were highly successful, the special purpose elements that they crafted are not readily available in most commercial general-purpose finite element programs like ABAQUS and thus are rarely used. A major advancement in the application of finite element method (FEM) for solving LEFM problems was the independent and simultaneous development of “quarter-point” elements in the 1970s by Henshell and Shaw (1975) and their fellow researcher at that time Barsoum (1976). These authors indicated that the right or appropriate crack-tip displacement, strain and stress fields can be modeled by means of standard quadratic order isoparametric finite elements in cases where an element’s mid-side node is moved to a potion that is a quarter of the way from the determined crack-tip to the element’s far end. It is this procedure that introduces a singularity into the intricate process of mapping between an element’s Cartesian space and its parametric coordinate space. The works of Henshell and Shaw (1975) presented a clear description of a quadrilateral quarter-point element. Barsoum (1976) proposed the collapsing of one edge of a given element at the crack tip. This is at the point where the crack-tip nodes are inherently constrained to shift together. The development of quarter-point elements was therefore a major milestone in the advancement of finite element methods for LEFM. It is these developments and standards that allowed us to accurately use ABAQUS in modeling the crack tip fields more accurately with quite minimal preprocessing requirements.
Figure 5.3 Mesh of the complete model
Figure 5.4 Mesh in the vicinity of the crack tip
Boundary Conditions
The derivation of accurate and realistic boundary conditions is essential to the accurate modelling of any given phenomenon (Feigley et al., 2011; Inoue, 2008). In this study, it is simply not possible and it’s totally unnecessary to simulate the entire pipeline system. Therefore, we generally chose a particular region of interest (section of the pipeline having a hole and a propagating crack) in which to carry out the simulation. The area of interest certainly has a boundary with the surrounding material and environment. While carrying out the numerical simulation, we must consider all the physical processes that are active in the c boundary region. In certain cases, the boundary conditions are crucial for the simulation of the area of interest’s physical processes. It is worth noting that different boundary conditions would result in different simulation outcomes. The use of improper set of simulation boundary conditions may have a great negative influence on the simulation system. On the contrary, the use of a proper set of simulation boundary conditions can help in avoiding all that trouble. In this study, we arranged the appropriate boundary conditions for different scenarios involved. In order for our calculations to yield meaningful figures throughout the domain, each of our equations required equally meaningful values for boundary of the domain calculation. The boundary conditions that would give us meaningful values are presented in the next section.
In ABAQUS three dimensional analysis, each node has six degrees of freedom, which are displacements and rotations in x, y, z axis. All nodes on the symmetrical plane except for those within the puncture and crack are set to be fixed in x-axis displacement and y-z-axis rotation. All nodes on both end of the tube are set to be fixed in all degree of freedom. Figure 5.5 shows the boundary condition at the puncture and the crack. The red rectangle area represents the crack face.
Figure 5.5 Boundary condition at the puncture and crack
Evaluation of Fracture Parameters
According to Sladeng et al (2006, p.604), cracking of structures is a very important phenomenon in various engineering applications. The application of linear elastic fracture mechanics in the prediction of crack behaviour in various solids under direct loading is very well established in various engineering applications. Within the conventional linear elastic fracture mechanics, the analysis is often done in two dimensions with a primary focus on the determination of Stress Intensity Factors (SIF) – KI and KII that characterizes cases of near-tip stress. Several studies have been conducted over the years and they have indicated the general need to take into considerations non-singular terms in the Williams’ series expansion so as to realize better predictions of the crack propagation path as well as stability as well as fracture toughness in solids that are elastic and exposed to conditions of very low crack tip stress traxiality (Sumpter and Hancock, 1991; Cotterell and Rice, 1980).
The leading non-singular term is usually referred to as the T-stress. This T-stress represents the stress that acts parallel to the existing crack plane. Both of these crack parameters (T-stress and stress intensity factor) are quite important for the sake of obtaining a more precise characterization of the existing crack fields within the crack tip vicinity. The accurate computational techniques for the evaluation of these crack parameters are therefore needed. Because of the mathematical complexity of initial and boundary value problems, most investigations on stress analysis are mainly restricted to isotropic materials (Nemat-Alla and Noda, 1996). The work of Ozturk and Erdogan (1999) for instance employed singular integral equation in the investigation of mode I as well as mixed-mode crack problems in a medium of infinite nonhomogeneous orthotropic character having a crack that is aligned to one of the existing principal material axes as well as a constant Poisson’s ratio.ABAQUS/Standard provides users with the following techniques for performing fracture mechanics analyses.
Crack propagation
The crack propagation function or capability allows for the quasi-static crack growth of the crack along a series of predefined paths to be analyzed in 2-D cases. The cracks would then debond along surfaces that are user-defined. A total of 3 crack propagation criteria are included in ABAQUS. Contour integrals (J integral) can be utilized in crack propagation problems like in our study.
Line spring elements
This is used for modelling part-through cracks in material shells that can otherwise be inexpensively modelled by means of line spring elements using a static procedure. The use of J-integral, a widely accepted parameter in quasi-static fracture mechanics involves linear material responses. It however has but with limitations for largely nonlinear material responses.
Stress intensity factors are employed in linear elastic fracture mechanics for the sole purpose of measuring the strength of localizes crack-tip fields. ABAQUS also provided us with evaluation techniques for fracture mechanics. They include the crack propagation direction as well as T-stress that represents cracks that run parallel to the crack faces.
In the ABAQUS, modules of evaluating fracture parameters such as J-integral, stress intensity factor and T-stress are provided. In this study, the mode I stress intensity factor, KI, is applied. As described in previous chapters, the stress intensity factors play an important role in linear fracture mechanics. They characterise the influence of load or deformation of the crack-tip stress and strain field. In ABAQUS, the stress intensity factor is extracted from simulation result by build-in interaction integral method ADDIN Mendeley Citation{988f4c4c-331a-4ed9-a3db-33bb301455f4} CSL_CITATION { “citationItems” : [ { “id” : “ITEM-1”, “itemData” : { “author” : [ { “family” : “Shih”, “given” : “C. F.” }, { “family” : “Asaro”, “given” : “R.J.” } ], “container-title” : “Journal of Applied Mechanics”, “id” : “ITEM-1”, “issued” : { “date-parts” : [ [ “1988” ] ] }, “note” : “u003cm:noteu003eu003c/m:noteu003e”, “title” : “Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part Iu2014Small Scale Yielding”, “type” : “article-journal” }, “uris” : [ “http://www.mendeley.com/documents/?uuid=988f4c4c-331a-4ed9-a3db-33bb301455f4” ] } ], “mendeley” : { “previouslyFormattedCitation” : “(Shih u0026#38; Asaro, 1988)” }, “properties” : { “noteIndex” : 0 }, “schema” : “https://github.com/citation-style-language/schema/raw/master/csl-citation.json” } (Shih & Asaro, 1988).The evaluation of stress intensity factor is an important step in the prediction of crack size that would ultimately propagate under various loading conditions in a given material under study at certain pressure, temperature, and environment (Zafošnik et al., 2002). The association between crack sizes is quite dependent on the various conditions existing near the crack-tip, a region where non-elastic effects are quite significant. Provided that such a region is relatively small when compared to the crack dimensions, a perfectly linear elastic stress field can be assumed to occur around the tip of the crack. In such a case, the onset of a fracture is very much controlled by the concentration of the stress intensity factors KI, KII as well as KIII (these relate to tensile, object in-plane shear as well as out-of-plane shear in that order).
In the case of geometries that are arbitrary cracked, it is quite important for us to determine the stress intensity factors for the geometries of interest in order for us to establish the relationship the critical parameters of KI, KII and KIII as well as establish the direction of crack propagation. In most cases, mode III is quire separable and can therefore be dealt with independently. However, Zafošnik et al., (2002) noted that the combined effects of both modes I and II in cases of shear and tensile loading could present major challenges in the analysis stage. It is these calculations that must be conducted by means of finite element methods employing virtual crack extension (VCE) (Hellen, 1975).In order evaluate the stress intensity factor, a contour integral mush be defined with proper contour region. In figure 5.6, the red line shows the region of the contour integral at the crack tip and crack extension direction.
Figure 5.6 Region of contour integral and crack extension direction
Loading
In order to use multiple reference method to evaluate the weight function, the stress intensity factors of the geometry under two independent loadings are required. A uniform and a linear loading are applied to the crack surface as surface traction. The corresponding stress intensity factors are then calculated. Figure 5.6 shows the two independent loading profiles.
Figure 5.6. Schematic diagram of the two independent crack face loading, the bold line indicates the crack surface.
According to Wagner and Millwater (2012 ,p.23), weight functions are vital components of any damage tolerance control plan due to the fact that they allow for the quick computation of stress intensity factor for stress acting along the uncracked plane or crack line. The traditional technique for computing weight functions is by means of multiple (between two to four ) reference stress solutions or by means of auxiliary conditions to come up with the coefficients in a set of solutions. Even though this technique has been indicated to provide relatively good results in various scenarios and loading conditions, the truncated series acts as a source of error that is quite difficult to accurately determine and quantify while the methods also requires several high-quality reference solutions plus other auxiliary data. A usual alternative is to employ a WCTSE (see Wagner and Millwater,2012) technique since it provides atechniqu for accurate and efficient development of the weight function for use in scenarios of arbitrary geometry while also supporting loading scenarios of single complex variable finite element solution minus the need for other reference solutions or auxiliary data. Wagner and Millwater (2012) suggested the application of complex Taylor series expansion method within the formulation of the finite element formulation in order to obtain the derivatives of the displacements of the crack opening with respect to crack length straight from the finite element analysis. These special derivatives allow us to directly evaluate the weight function. However, this technique requires a little perturbation of the crack length along the assumed / imaginary axis; while the real coordinate mesh remain unchanged. Due to the fact that the real coordinate mesh is unchanged, standard finite element meshes as well as meshing algorithms can be applied. Due to the fact that the error in the weight function is mainly controlled by the mesh’s accuracy, typical convergence tests can be employed in order to obtain high confidence in the material’s weight functions. In such a case, a number of numerical examples are calculated and then compared to other well known and published weight function solutions and/ or finite element (J integral) or the boundary element solutions.
Optimisation of Crack Tip Mesh Size
Mesh optimization is important for the accuracy and robustness of computational algorithms employed in the calculation of fracture parameters as well as accurate prediction of crack path patterns (Khoei et al., 2008). Gustavo et al (1996) noted that even though mathematical theory of classical linear elasticity is sound and well established, there exists a general lack of certain “ingredients” towards the achievement of numerical solutions of most real-life technological /engineering problems. In their work, they mentioned that the automatic construction of 3-D meshes in geometries that arbitrary is one of these critical ingredients. They noted that several techniques of mesh construction are in existence but there is a general need for further improvement to be made in order for the required level of generality and robustness to be achieved. Mesh optimization has several effects on the modeling process. For instance, it effectively reduces the total number of gradient iterations by between twenty and thirty percent (Gustavo et al 1996,p.10). The number of errors also decreases while the number of elements is increased. In the long run, the mesh quality is highly improved. Just as Khoei et al., (2008) and Gustavo et al (1996) mentioned, automated mesh construction is important for all sorts of shape crack growth determination in finite element mesh. Khoei et al., (2008) employed a Zienkiewicz-Zhu error estimator in conjunction with a modified SPR technique that is based on the recovery of gradients employing analytical crack tip fields so as to obtain a more precise estimation of errors.. The optimization of crack-tip singular finite element mesh size can be realized via the adaptive mesh strategy.
As described in previous section, the accuracy of the result from ABAQUS depends on the level of mesh refinement in the vicinity of the crack tip. However, finer mesh requires
