Schematic of elastic stress waves von Mises propagating in an infinite plate with a semi-infinite edge crack at different steps using the present formulation with the linear ramp functio
INTRODUCTION AND OBJECTIVE
Statement of crack problems
Many catastrophic accidents have been reported to be related to the initiation and propagation of cracks such as the collapse of buildings during earthquake, failure of working mechanical components and damage of transport facilities These accidents caused a great loss both in terms of economic properties and human lives In cases where cracks were detected early, long before collapse occurs, lots of efforts and money still have to be spent for maintenance Some examples of cracks appearing in engineering structures in Vietnam involve the Song Tranh hydropower dam and the Thang Long bridge (see Figure 1.1 and Figure 1.2) Most recently, in June 2016, a domestic flight from Ha Noi to Can Tho had to make an emergency landing in Tan Son Nhat Airport because the airplane windshield was cracked (see Figure 1.3)
Figure 1.1 Cracks observed in Song Tranh hydropower dam
Nowadays, fracture analysis has become more and more important in various fields, in order to predict and prevent failure events of structures Crack opening in structures subjected to arbitrary loading conditions could be described as a combination of the three following modes (illustrated in Figure 1.4)
- Mode I: Opening mode (in-plane tension) - Mode II: Sliding mode (in-plane shearing) - Mode III: Tearing mode (out-of-plane shearing) a) Opening mode b) Sliding mode c) Tearing mode
Figure 1.4 The three basic modes of fracture Figure 1.2 Cracks observed on road surface of Thang Long Bridge
Figure 1.3 Cracks in an airplane windshield
Early studies have introduced fundamental concepts of fracture mechanics such as stress intensity factor and energy release rate Once a crack is opened, new free surfaces are created The energy release rate determines the energy dissipated per unit of newly created fracture surface area, which should be equal to the work done to create the fracture surfaces That is the idea of the Griffith criteria for crack opening in theory of linear elastic fracture mechanics (LEFM) for brittle materials High stress value is usually observed in region near the crack tip It has been proved theoretically that for the case of a small circular hole inside an infinite plate subject to remote tensile stress, one could expect the stress near the hole could be expected to be three times the remote tensile stress Determination of stress intensity factor provides a finite evaluation of the stress state near the crack tip LEFM theory infers that stress field is singular at the crack tip, i.e stress tends to infinity, which is in fact not realistic Later authors like Irwin and Dugdale supplement the LEFM theory by adding a local plastic zone surrounding the crack tip Hence, the stress at the crack tip is not infinity but has a finite value Further details on theory could be referred to the book by Gdoutos [1]
In cases where the crack propagates rapidly, such that the material inertia is significant and cannot be ignored, one should consider dynamic fracture The situation is much more complicated than that of static cracks The feature that distinguishes dynamic crack behavior from the static or quasi-static one is the stress waves Stress waves can arise by two ways: they are emitted at the crack tip or from the external load applied on domain boundaries The waves can be reflected or scattered as they encounter any free surfaces When stress waves reflected from domain boundaries are back to crack tip, the stress state at crack tip is altered and the crack speed could be changed If the spatial sizes are small, a number of reflective waves arrive at the crack tip successively, increasing the complexity.
Advanced functional composite materials
Orthotropic composite materials and their structures are used widely in various fields in engineering One of the most preeminent property of composite is the high strength to weight ratio in comparison with conventional engineering materials In many cases, orthotropic composites are fabricated in thin plate forms which are so susceptible to fault A typical fault in composite structure is cracking due to imperfection in fabrication process or hard working conditions such as overload, fatigue, corrosion and so on
Functionally graded materials (FGMs) are advanced composites, which have been manufactured based on the concept of continuous variation of the material properties in one or more specified directions [2] In recent years, the FGMs have been applied in the manufacture of structure parts that subjected to non-uniform working requirements [3]
For instance, in a thermal protection system, FGMs evolve the advantage of typical ceramics such as corrosion and heat resistance and of typical metal such as mechanical strength and stiffness FGMs can be applied to make thermal barrier coating for space applications, transport system, energy conversion system, thermal-electric and piezoelectric devices, dental and medical implants, and many others
Figure 1.5 Cracks observed on an aircraft body made from composite material
Figure 1.6 Crack growth in a FGM specimen [4]
Along with the applications of these new advanced materials in engineering, their structural behavior such as vibration, buckling and fracture are under question For such reason, crack behavior of orthotropic composite materials and functionally graded materials has become an interesting study subject (see Figure 1.7 and Figure 1.8).
Literature review
Studying fracture mechanics, including quantitative analysis of the initiation and propagation of cracks, is essential in analyzing the durability of engineering structures
However, the fracture problems are usually complicated Analytical solution is only available for basic problems with simple geometry and simple boundary conditions, which are mostly served for studying purpose For engineering problems, numerical solutions seem to be more suitable Thanks to the development of technology such as high-performance computing, various computational methods and algorithms have been developed and intensively investigated for crack problems
The currently most popular numerical method is the Finite Element Method (FEM)
Based on the variational principle and Galerkin method to numerically approximate solution for partial differential equations, FEM has been introduced since the 50s of twentieth century which still be favored in both academic and industrial communities
The main idea of FEM is discretizing the problem domain into non-overlapping sub- domains, namely elements [5] An element usually has a simple geometric shape such as a line (1D element), a triangle or a quadrilateral (2D element), a tetrahedron or a hexahedron (3D element) Equilibrium is enforced locally at nodes, which are the vertices of elements The to-be-solved unknowns, termed by degrees of freedom (DOFs) are associated with nodes The field variables, which could be displacement, pressure, temperature, etc., depending on the specific problems, are approximated as a linear combination of the corresponding nodal values, i.e the DOFs Usually, the numerical HUURUEHWZHHQDSSUR[LPDWHGUHVXOWVZLWKWKH³WUXH´RQHUHGXFHVZKHQPRUHHOements (more nodes) are used to discretize the domain FEM attracts lots of attention due to many desirable properties, such as simple in implementation, rather fast computation and reasonable accuracy
In crack problems, cracks could be directly modeled as discontinuities in geometry
When crack propagation is considered, the geometric discontinuities have to be updated, following by an update on domain discretization, i.e re-meshing [6] In practice, re- meshing is a challenging and burdensome task due to the arbitrary crack path and require lots of computational efforts Therefore, approaches that can model crack propagation without re-meshing have been introduced and gained favors [7] Instead of directly create new surfaces to model cracks, the effect of cracks as a jump in displacement and discontinuous stress fields are mathematically induced in the formulation of numerical solution, namely enrichments, with the help of the principle of partition of unity
Depending on how the enrichments are taken into account, the approaches can be classified into two categories: intrinsic and extrinsic In the class of intrinsic enriched methods, the enrichment terms are added by replacing the standard shape functions by some special shape functions, at least at some specific regions, to capture the jumps caused by a discontinuity, which could be a crack, a void or an inclusion Using the intrinsic enrichments, the number of degrees of freedom does not change, with the price that the approximation space is changed In contrast, the extrinsic enrichment approach does not change the standard shape functions Instead, the jumps are induced by adding additional degrees of freedom locally in region surrounding the discontinuities
Besides modeling crack as sharp discontinuities, there are approaches that attempt to GHYHORS³smeared crack´LQZKLFKDGDPDJH]RQHLVFRQVLGHUHGSome representatives of smeared crack models can be listed as the gradient-enhanced damage model [8] and the phase field model [9, 10] In smeared crack models, a damage parameter is used to simulate the state of the material The damage parameter is usually ranged between 0 and 1, in which value 0 stands for intact material and value 1 stands for totally broken material Hence, instead of a sharp crack, a damage zone is obtained The size of damage zone is controlled by a length-parameter whose physical meaning is still being LQYHVWLJDWHG 7KH ³VPHDUHG FUDFN´ FRQYHUJHV WR WKH ³VKDUS FUDFN´ ZKHQ WKLV OHQJWK- parameter is close to zero One advantage of damage model is that the problem can be formulated as continuum problem and damage parameter can be solved as a field variable From computational point of view, the damage model is rather simple and easy an empirical function for description of damage evolution The recently introduced phase field model for fracture is formulated from energy conservation, hence it alleviates the limitation of empirical functions A disadvantage of phase field model is that it requires very fine element mesh and therefore it is time-consuming and computationally costly
1.3.1 Extended Finite Element method (XFEM)
Extended Finite Element method (XFEM) [11] is a well-known computational method LQPRGHOLQJFUDFNVZKLFK³H[WHQGV´WKHVWDQGDUG)LQLWH(OHPHQWPHWKRGE\H[WULQVLF enrichments to represent the jumps at discontinuities Specific functions should be used as enrichments for each type of discontinuities, i.e cracks, voids or inclusions For simulation of cracks, the Heaviside (step) functions are usually employed to capture the jump in displacement, while the so-called branch functions model the singularities at crack tip The tip-enriched functions are usually chosen based on the asymptotic solutions of displacement and stress fields when crack exists Since the introduction, XFEM has been investigated and applied in various types of problems, such as crack propagation in different kinds of material like isotropic materials [11], orthotropic materials [12] and functionally graded materials [13, 14]; crack propagation in different types of structures like solid structures [15] shell/plate structures [16, 17]; dynamic crack propagation analysis [18-20]
In XFEM model, a propagating crack is often described by a series line segments The crack topology is realized by the aid of level set method [21] Usually, two types of level set functions are needed: a normal level set, which is defined by the signed-distance from a given point to the crack segment, and a tangent level set, which is defined by the signed-distance to the line perpendicular to the crack segment at crack tip One tangent level set must be defined for each crack tip As crack propagates, the level set functions must be updated In order to determine the direction of propagating crack, appropriate criterion have to be taken Usually, this criterion is calculated from the stress intensity factors, which in turn can be computed by the so-called J-integral
Although FEM is powerful and convenient, it is not without shortcomings The existence of the finite element mesh is actually a disadvantage in many cases One example is when large deformation cannot be ignored The standard element has to be convexed in shape in order to provide good approximated solution When large deformations occur, some elements could be distorted such that convexity is lost and numerical errors add up considerably Another example is the case where the mesh has to be updated, as already mentioned above Hence, the meshfree method has been proposed to alleviate the difficulties related to finite element mesh [22] As the term VXJJHVWV WKH FODVV RI ³PHVKIUHH´ PHWKRGV GR QRW UHTXLUH GLVFUHWL]LQJ WKH SUREOHP domain into finite elements Instead, the problem domain is represented only by nodes, including nodes on boundaries and nodes inside the domain When there is a change in domain geometry, nodes could be added or removed without difficulties
In terms of XFEM, re-meshing is not required However, one has to track the position of crack relatively to the finite element mesh One of three scenarios may happen for each element: i) the element is entirely cut into two parts by the crack, ii) the element is partially cut by the crack and thus contains the crack tip, and iii) the element is not cut by the crack Determination which case happens could be tricky due to the complicated of the domain geometry and the arbitrary of crack path [23] As mention above, mesh is not be required in meshfree methods but scattered nodes are used for discretion of the problem domain These nodes play the key roll of approximation or interpolation without any connection between them as "element" in mesh-based method So meshfree methods are good at dealing with problems with discontinuities such as crack propagation problems On the other hand, applying the enrichment functions into a meshfree approximation scheme results in an approach which do not need update in domain geometry and the challenge of tracking the relative position of crack with respect to the finite element mesh can be avoided The use of meshfree methods to these fracture problems, however, lags a little behind
There are several meshfree methods, based on several types of basis functions used in meshfree approximation such as the Element-Free Galerkin method (EFG) [24], the meshless local Petrov-Galerkin method (MLPG) [25], the Reproducing Kernel Particle method (RKPM) [26], the Radial Point Interpolation method (RPIM) [27] and the Moving Kriging method (MK) [28, 29] Most of the meshfree basis functions (e.g EFG, MLPG, and RKPM) do not possess Kronecker-delta property as in standard Finite Element basis functions Hence, essential boundary conditions cannot be enforced directly In such situation, further techniques like Lagrange multipliers or penalty method are usually employed to impose the boundary conditions The RPIM and MK basis functions are in contrast, satisfy the Kronecker-delta property, thus they allow direct imposition of essential boundary conditions Interestingly, these two meshfree approaches share similarities in formulation, although being developed from different sources, and can be classified into one category
To apply meshless methods to fracture problems, there are two types of enrichment techniques In the first type, the basics functions are augmented to include functions similar to the near tip asymptotic fields (intrinsic enrichment) as per [29-31] The second type is more efficient that the displacement field is enriched by using the Heaviside
This approach is known as an extrinsic enrichment meshfree method and only the nodes surrounding the crack are taken into account As a consequence, meshfree methods using the extrinsic enrichment are thus suitable for crack growth simulation The new approaches are formed in a way that making use of not only the advantages of the RPIM and MK shape functions [32, 33] and but also the versatility of the vector level set method [34] Additionally, as compared with lower-order finite elements that are commonly applied and can be captured only the linear crack opening, the meshfree methods however are dominated and have the great advantage to capture more realistic crack openings due to the higher-order continuity and non-local interpolation character [35].
Fundamental of Fracture Mechanics
In linear elastic fracture mechanics, a macro cracked problem is usually considered in which the model contents an initial crack The model can be subjected to various kinds of load such as static, harmonic and dynamic The main tasks of such problems are to answer these following questions:
- Does the crack propagate under the given load?
- If the crack grows, what is the direction of the propagation?
- What is he maximum length of the crack that cannot destroy the structure seriously?
- What is the maximum load that the structure can work well with"$QGVRRQô
The principle difficulties in cracked problems are the discontinuity at the crack path (2- D) or crack face (3-D) and the singularity at the crack tip Generally, there are three types of loading that can make a crack develop (see Figure 1.4) Mode I occurs when the principle load is applied normal to the crack plane, tends to open the crack Mode II happens when the crack faces are subjected to an in-plane shear loading and tends to slide the crack faces on each other Mode III appears when the crack faces are subjected to an out-of-plane shear loading and tends to tear the two crack faces
Because of the singularity of stress at the crack tip, the behavior of material at this special location cannot be calculated by normal elasticity theorem In general, the stress, strain and displacement fields for the zone ahead of crack tip are given in term of special factors called stress intensity factors (SIFs) There are three SIFs (K K I , II ,K III ) with respect to three modes of fracture Stress fields ahead of a crack tip in an isotropic linear elastic material can be written as
II II II ij ij r
(1.1) where randT are illustrated in Figure 1.7; f ij is tensor of non-dimensional values, function of angle T and depend on crack mode
Figure 1.7 Definition of the coordinate axis ahead of a crack tip
In linear elastic, isotropic material, stress and displacement fields ahead of a crack tip for Mode I and Mode II are given below
2 0 for planestress for planestrain cos sin cos 3
V W ư Đ ãă áâ ạêôơ Đ ãă áâ ạ Đăâ ãáạºằẳ ê º Đ ã Đ ã Đ ã ă áô ă á ă áằ â ạơ â ạ â ạẳ ® ®¯ Đ ã Đ ã Đ ã ă á ă á ă á â ạ â ạ â ạ °° °° °° °° °° °°¯
P S ư Đ ãê Đ ãº ° ă áâ ạô ă áâ ạằ ° ơ ẳ đ Đ ãê Đ ãº ° ă áô ă áằ ° â ạơ â ạẳ ¯
2 0 for planestress for planestrain cos 1 sin sin 3
V V ê º Đ ã Đ ã Đ ã ă áâ ạôơ ă áâ ạ ăâ áạằẳ Đ ã Đ ã Đ ã ă á ă á ă á â ạ â ạ â ạ ® ¯ ê º Đ ã Đ ã Đ ã ă áô ă á ă áằ â ạơ â ạ â ạẳ 0 ° °° °° °® °° °° °° ¯
P S ư Đ ãê Đ ãº ° ă áâ ạô ă áâ ạằ ° ơ ẳ đ Đ ãê Đ ãº ° ă áô ă áằ ° â ạơ â ạẳ ¯
(1.5) where P is shear modulus; N 3 4Q (plane strain); N 3 Q / 1 Q (plane stress)
In the linear elastic fracture mechanics, the evaluation of the stress intensity factors under static or dynamic loading condition is so essential The SIF values play a key role in estimating the residual strength of cracked structures, predicting the direction of crack growth and so on However, analytical solution for stress intensity factor is given in very few simple cases of model and load
1.4.2 Crack behavior in orthotropic materials
Assume an orthotropic crack body subjected to arbitrary forces with general boundary conditions as shown in Figure 1.8 The global Cartesian coordinate system is denoted by X 1, X 2 , local Cartesian coordinate system is denoted by x x 1, 2 and local polar coordinate system at the crack-tip is denoted by r , T A following characteristic equation can be obtained using equilibrium and compatibility conditions [36]
C P C P C C P C PC (1.6) where C ij are elements of the compliance tensor of orthotropic model [36]
Figure 1.8 2D orthotropic body with crack
Lekhnitskii [36] proved that the roots of Eq (1.6) are always complex or purely imaginary and come in conjugate pairs as P P 1 , 1 and P P 2 , 2 , they have the form:
The stress and displacement fields ahead of a crack tip for Mode I and Mode II are given below
2 Re cos sin cos sin
2 Re cos sin cos sin
S P P T P T T P T ư ê Đ ãº ° ô ăă ááằ ° ôơ â ạằẳ ° ê Đ ãº ° ô ă áằ đ ô ă áằ ° ơ â ạẳ ° ê Đ ãº ° ô ă áằ ° ôơ ăâ áạằẳ ¯
Re cos sin cos sin
Re cos sin cos sin
2 Re cos sin cos sin
V S P P T P T T P T ư ê Đ ãº ° ô ăă ááằ ° ôơ â ạằẳ ° ê Đ ãº ° ô ă áằ đ ô ă áằ ° ơ â ạẳ ° ê Đ ãº ° ô ă áằ ° ôơ ăâ áạằẳ ¯
Re cos sin cos sin
Re cos sin cos sin
ZKHUH³5H´GHQRWHVWKHUHDOSDUWRIWKHFRPSOH[QXPEHUDQGp q i , i are defined as
1.4.3 Crack behavior in functionally graded materials
Consider a 2-D functionally graded material (FGM) body with a crack as shown in Figure 1.9(ODVWLFPRGXOXVDQG3RLVVRQảVUDWLRDUHIXQFWLRQVRIORFDWLRQThe global and local coordinate systems are defined similarly to section 1.4.1 and 1.4.2 The stress fields ahead of the crack tip are the same with the case of isotropic homogenous material
However, the displacement fields at crack tip zone are modified in which the shear modulus parameter P and bulk modulus N are calculated at the position of the crack tip, denoted by P TIP and N TIP , respectively
(1.14) where P TIP is shear modulus and N TIP is the bulk modulus at the crack tip They are determined as PTIP E TIP / 2 1 QTIP ; TIP TIP
Q (plane stress); N TIP 3 4Q TIP (plane strain)
Figure 1.9 2-D FGM body with crack
Objective of the dissertation
Together with the innovations in computer technology, computational fracture mechanics have been intensively investigated in recent years with many reports available in literatures discussing a wide range of aspects However, development of advanced numerical methods and tools is a constant requirement to satisfy higher demand in structural safety and durability Based on the advantages and disadvantages of various trends in crack modeling, this thesis focuses on the following points:
- Develop a computational model for fracture analysis based on meshfree approach and enrichment functions for mathematical description of discontinuities The class of radial basis functions is employed due to its advantage of possessing the Kronecker delta property Currently, there are models based on enriched EFG method for isotropic materials [29], orthotropic composites [37] and thermos-mechanical fracture problems x 1
[38] Nevertheless, there is still little available literatures on meshfree model based on point interpolation for fracture problems
- Extend the model to investigate fracture problems in different types of materials: isotropic, orthotropic and functionally graded materials (FGM) The behaviors of materials have to be incorporated into the numerical model
- Consider dynamic fracture The inertia effect in most of practical engineering problems cannot be neglected, so dynamic fracture analysis is also investigated on various types of materials
In terms of numerical implementations, nodes are selected into three groups: i) non- enriched nodes, ii) nodes enriched by the step function and (iii) nodes enriched by the branch functions This procedure is in general simpler than those in XFEM which involves also the geometry of element The role of enrichment functions associated with the VWUHVV VLQJXODULW\ DW FUDFN WLS LH ³WLS-HQULFKHG IXQFWLRQV´ DUH LQYHVWLJDWHG Typically, four-fold branch functions have to be considered based on the analytical solution of stress field near the crack tip Recently, ramp function has been proposed as alternative to the branch functions [39] The interesting point is that only one ramp function needed for each tip-enriched node, instead of four branch functions as usual
Thus the number of additional degrees of freedom is reduced Details on aspects related to the implementation of the proposed method are discussed in the thesis
The proposed computational model is validated through many problems, in order to investigate its accuracy, efficiency and applicability Comparison is conducted between numerical results and reference results, obtained from analytical solution or from other author in form of experiments or numerical simulations.
Outline of the thesis
The thesis is organized as follows:
Chapter 1 shows the overview of fracture mechanics on three types of materials, i.e isotropic materials, orthotropic materials and functionally graded materials; some related studies and objectives of the thesis
Chapter 2 is dedicated for review of meshfree methods using the class of radial basis functions, including some advanced knowledge in construction of meshfree shape functions Enrichment methods, fundamental equations, J-integral for crack tip behavior and numerical implementation procedure are also presented in this chapter
Chapter 3 shows the application of extended meshless radial interpolation method (X- RPIM) for static stress intensity factors calculation and quasi-static crack growth simulation of 2D isotropic solid
Chapter 4 presents the dynamic form of J-integral and interaction integral method
Several transient dynamic crack analyses of isotropic and orthotropic composite materials, quasi-static crack growth simulation of orthotropic materials are performed to verify the accuracy of the extended meshless approach (X-RPIM) The linear ramp function is first introduced in meshless context to model the singularity at crack tip, applied for static and dynamic crack analyses of isotropic
Chapter 5 shows the application of X-RPIM for static and dynamic stress intensity factors computation of functionally graded materials
Chapter 6 are reserved for application of the improved extended meshless moving Kriging methods (X-MK) for crack modeling of isotropic and functionally graded materials Three types of correlation functions are applied to improve the moving Kriging shape functions, two of them are first presented in this thesis
Finally, some conclusions and outlooks are extracted in Chapter 7.
EXTENDED MESHFREE GALERKIN METHODS FOR
Enrichment methods
In terms of fracture modeling using local enriched partition-of-unity approaches, the extended meshfree methods consist of enriching the conventional approximation for the discontinuity and singularity induced by crack Inherit from the XFEM model for crack faces, the Heaviside step functions H f x defined in Eq (2.19), have found to be particularly suitable for capturing the discontinuity at the crack faces [7, 12, 13, 15, 21, 39, 41] Figure 2.1 shows the description for the signed distance f x
! ® ¯ x x x (2.19) where f x is the signed distance from the crack line
Figure 2.1 Schematic representation of the distance r and angle T at a crack-tip
2.2.2 Standard enrichment for crack tip using branch functions
To further improve the accuracy of the solution especially at the crack-tip, other functions, which are derived from known analytical solutions, have also been incorporated into the approximation The asymptotic displacement fields derived from linear fracture mechanics, which are defined in Eq (2.20) for isotropic materials and are known as the fourfold enrichment functions \ j x (j = 1, 2, 3, 4), are often taken to represent the singular behaviors of the crack-tip in fracture mechanics problems
4 1 sin , cos , sin sin , cos sin
\ ă Đ T Tã á â ạ x (2.20) where r is the distance from the crack-tip x TIP to x, and T is the angle between the tangent to the crack-line and the segment xx TIP as depicted in Figure 2.1
The methodology being presented here allows the insertion of the discontinuities independently from the given discretized scattered nodes, and it is frequently integrated with the level set methods for an efficient way of tracking the crack path [34, 41] x TIP x I
In this study, apart from the standard enrichment for crack-tip, another recently developed enrichment scheme based on the linear ramp function is also adopted The main motivation of employing this ramp function [39, 42, 43] for crack-tip enrichment in our meshfree X-RPIM approach is to enhance the efficient computation, reducing the number of extra DOFs, simplifying implementation, consequently saving the computational effort
Figure 2.2 Definition of the sets of nodes W b and W S , respectively
According to [34, 41], the crack is modeled by the crack-tip position and a vector level set function In this fashion, the vector level set function is defined through the signed distance function f given by the closet point projection to the crack surface and its gradient To this end, the enriched form to capture the crack by using the signed distance function and the distance from the crack-tip is expressed as
(2.21) where I I are the RPIM shape functions The detail of the generation of these shape functions are given in many papers, see e.g., [41] and references therein, and no repetition is necessary to present here W is the set of all scattered nodes of the domain,
W b indicates the set of nodes whose support contains the point x and is bisected by the crack-line (see Figure 2.2) and W S is the set of nodes whose support contains the point x and is split by the crack-line and contains the crack-tip D I and E Ij are additional x I
0 f x TIP x x I unknowns in the approximation formulation, which appear in the global mass and stiffness matrices as well as the loading vector [41]
2.2.3 New enrichment for crack tip using ramp function
The ordinary fourfold branch functions in Eq (2.20) are often taken as crack-tip enrichments in the local enriched partition-of-unity methods An alternative scheme based on the ramp functions along with the jump Heaviside functions for modeling strong discontinuities like crack in solids using XFEM was developed in [44] The ramp functions were later applied to the crack modeling of for instance general inelastic materials [43], ductile fracture [42], and crack in bimaterials [39] In this work, the linear ramp functions associated with the Heaviside functions are first integrated into meshfree X-RPIM in a dynamical fracture context in order to capture the singular behavior of crack-tip effectively and accurately
It is worth noting also that since the jump Heaviside function in Eq (2.19) itself is generally incapable of accurately representing the singularity at crack-tip It is because the jump functions itself cannot capture exact location of the crack-tip, and hence its integration with the linear ramp function is developed to overcome such drawback
Theoretically, the Heaviside functions could be used as a tip enrichment function in terms of the XFEM to model crack-tip fields, see e.g., [45, 46], however by describing the crack-tip behaviors in such a way the numerical tip, as stated in [43], must always be propagated to the edge of the element, which is not the true position of the crack-tip in general, leading to inconsistency in simulation
Basically, the main goal of handling the linear ramp functions is to linearly ramp functions down the displacement jump to the crack-tip, located at some point within the region of tip nodes [39, 42-44], while the Heaviside functions are utilized to capture the strong discontinuity induced by the crack faces Therefore, this combination allows one to accurately model the crack-tip behavior, by capturing exact values at the location of crack-tip In other words, the underlying idea behind that combination is to ensure that the enrichment disappears exactly at the crack-tip, or to smoothly vanish the enrichment to zero at the crack-tip [39, 43] To this end, the linear ramp functions R x 1 are introduced in terms of a crack length (l c ) inside the support domain and crack-tip coordinate ( ,x x 1 2 ) as depicted in Figure 2.3
The enriched approximation for the displacement field in presence of cracks using the new linear ramp functions can now be rewritten as
Figure 2.3 Schematic of a crack tip coordinates in terms of linear ramp function
Crack is represented by a black curve (X 1 , X 2 ) is the global coordinate system whereas (x 1 , x 2 ) is the local coordinate system at the crack-tip In practice, for simplicity the value of l c is taken as the radius of the support domain Green nodes are the split nodes enriched by the Heaviside function only, the red nodes enriched by the linear ramp function associated with Heaviside function
Once again, the linear ramp function is used to reduce the number of additional DOFs required to model the crack-tip behaviors, while it is expected to obtain a comparable x 1 x 2 l c
T method Obviously, the difference between two schemes in Eq (2.21) and Eq (2.23) is the number of additional DOFs that are required to model crack-tip For each tip node, it is clearly that only two additional DOFs are needed for the new linear ramp function whereas eight additional DOFs are for the standard enrichment scheme As a result the present enrichment using the linear ramp functions is effectively and simply implemented compared to that using the standard enrichment in modeling crack-tip behaviors
A graphical representation of the linear ramp functions, Heaviside functions and their combination depicted in Figure 2.4 shows the enrichment for Gauss points behind the crack-tip and in front of the crack-tip, it decreases linearly to zero
Meshfree Galerkin method for fracture problems and solution procedure28 1 Fundamental equations of elastic problems
This section presents about fundamental equations of elastic problems Generally, the formulations are given in dynamic terms in which the field variables such as displacement, stress, strain, etc are functions of time In static state, those variables are independent from time
Consider a 2-D solid domain : 2 bounded by *, boundary * C denotes the initial crack face, the body is subjected to a traction t on * t and body force b as shown in Figure 2.7 The displacement, stress and strain fields satisfy the momentum conservation law of elastodynamics given by [47]
Uw w ı E u in : (2.27) where is the divergence operator, ı is the Cauchy stress tensor, U is the mass density, t indicates time, and u is the vector of the displacements
Figure 2.7 Notation representation of a cracked model
The Dirichlet and Neumann boundary conditions are given by [47]
0 ı Q on * C (2.30) in which u is the vector of prescribed displacements on the displacement boundary, n is the unit outward normal vector and the Eq (2.30) implies a traction free condition at the crack-faces The initial conditions are specified as the following equations:
* t t b with u and u are the initial displacements and velocities, respectively The relationship between stress and strain defined through the generalized HookeảVODZLVJLYHQDV ı &İ (2.33) where C is the matrix of elastic constanWV GHSHQGLQJ RQ