_{1}

^{*}

A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization; 2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix; 3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.

Solid materials that contain large number of cracks can be the fracture mechanical models for many structural materials, especially for lots of brittle and quasi-brittle materials, such as cementitious materials in ref. [

In 1957, Irwin first postulated the concept of stress intensity factor (SIF). SIF became one of the main parameters which need to be obtained, depending on the stress field around the crack tips. Based on the value of SIFs, crack behaviors such as fatigue propagation can be investigated. The analysis of SIFs plays a greatly important role and real concern in linear elastic fracture mechanisms (LEFM), since solids with multiple cracks become one class of the most important problems in fracture mechanics. However, as mentioned earlier, the exact crack sizes, shapes, locations as well as orientations are unable to know in advance, it is an enormous challenge for researchers to seek an appropriate numerical model for LEFM problems, leading to huge complexities such as the computational efficiency and storage of solids with multiple cracks. As a result, most of them are incredibly difficult to solve by analytical procedures in ref. [

The finite element method (FEM), now widely available in many fields, not only has achieved a great reputation after decades of development in the applications of fracture mechanics, but also has many mature commercial software tools, such as ANSYS and ABAQUS,etc. Despite of its widely spread popularity, solids with cracks in large number are probably one class of the most difficult problems to simulate since the crack tips need to be discretized. To the author’s best knowledge, the advantage of the FEM is to model the solids as a whole, not enough to understand the local information.

Owing to the advantage of boundary discretization only, the boundary element method (BEM) is subsequently considered as an efficient numerical method to deal with multiple crack problems. In other words, the BEM reduces the dimension of original problems compared with the FEM, enjoyed absolute advantages in improving the speed and accuracy of calculation. Inevitably, the conventional BEM also has its own weaknesses in solving large-scale cracks due to its dense and asymmetric of the final system matrix. It is well known there are generically two difficulties for numerical modeling in ref. [

A combination or dual boundary integral equations (BIEs) in refs. [

The analytical solutions for most of multiple crack problems are restricted to obtain, except a few regular cracks. In order to eliminate the unknowns appearing on the crack surfaces, an analytical Green’s functions, known as the Erdogan function in ref. [

In 1995, Telles et al. put forward the numerical Green’s function (NGF) procedure, which can be modeled as the superposition of a basic plus a complementary solution problem in refs. [

To overcome these weaknesses of the above mentioned algorithms, the concept of the eigen crack opening displacement (COD) are firstly proposed in author's previous paper in ref. [

In order to efficiently and numerically simulate the interaction effects between the cracks, a superposition technique as in NGF is applied and the local Eshelby matrix derived from the traction BIE in discretized form is introduced subsequently. Due to limited space, more details will be discussed in Section 3.2. The eigen strain BIEs for multiple inclusion problems can be also found in refs. [

In the present paper, two dimensional multiple crack problems in finite/infinite plates are numerically considered and discretized by using the boundary point method (BPM) in refs. [

Without loss of generality, one two dimensional elastic domain Ω with outer boundary Гwhich contains multiple N_{C} cracks is considered, where N_{C} is the total number of cracks. Suppose there is a source point y, the displacements at y can be expressed with the unknown displacement discontinuities as follows in refs. [

γ u i ( y ) = ∫ Γ τ j ( x ) u i j * ( x , y ) d Γ ( x ) − ∫ Γ u j ( x ) τ i j * ( x , y ) d Γ ( x ) − ∑ m = 1 N C ∫ A m + Δ u j ( x ) τ i j * ( x , y ) d A ( x ) , y ∈ ( Γ ∪ Ω ) \ A m ( m = 1 , ⋯ , N C ) (1)

where u i j * and τ i j * are respectively the displacement and traction fundamental solutions in ref. [_{m} represents the crack number m. x and y represent the field point and the source point respectively. γis the free term coefficient, which depends on the location of the source point and the geometry of the boundary. γ = 1 when the source point y is inside in the domain Ω, γ = 0.5 if it is on the outer boundary Гand only if the outer boundary is smooth and continuous. Δu_{i} are the displacement discontinuities, or the CODs at the crack surfaces, which can be defined as follows:

Δ u i ( x ) = u i ( x ) | x ∈ A + − u i ( x ) | x ∈ A − (2)

where A^{+} represents the upper surface of one crack, A^{-} represents the lower surface, respectively. It is obvious from Equation (1) that if all the unknown displacement discontinuities (CODs) are obtained in advance, the multiple crack problems can be solved by Equation (1) in a discrete form as in conventional BEM in ref. [

γ σ i j ( y ) = ∫ Γ τ k ( x ) u i j k * ( x , y ) d Γ ( x ) − ∫ Γ u k ( x ) τ i j k * ( x , y ) d Γ ( x ) − ∑ m = 1 N C ∫ A m + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) , y ∈ ( Γ ∪ Ω ) \ ( ∪ A m ) (3)

γ τ i ( y ) = n j ( y ) ∫ Γ τ k ( x ) u i j k * ( x , y ) d Γ ( x ) − n j ( y ) ∫ Γ u k ( x ) τ i j k * ( x , y ) d Γ ( x ) − n j ( y ) ∑ m = 1 N C ∫ A m + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) , y ∈ ( Γ ∪ Ω ) \ ( ∪ A m ) (4)

where n_{j} represents the unit outer direction cosine at the source point y. Equations (1), (3) as well as (4) are designated as the eigen COD BIEs in the present paper. The CODs in those equations are unknown and should be computed with the concept of eigen COD, which will be shown in Section 3.1. Anyway, as it mentioned before, because there are general two difficulties in conventional BEM, i.e., accurately describe the stress field around the crack tips and the degeneration of the boundary integral equations resulting from crack surfaces coincide, Equation (1) cannot be employed alone to solve multiple crack problems.

For the numerical solution of multiple crack problems with dual formulations, the boundary integral equations used are as follows for the crack surfaces, together with the use of Equation (1) for the outer boundary:

1 2 { u i ( y + ) + u i ( y − ) } = ∫ Γ τ j ( x ) u i j * ( x , y ) d Γ ( x ) − ∫ Γ u j ( x ) τ i j * ( x , y ) d Γ ( x ) − ∑ m = 1 N C ∫ A m + Δ u j ( x ) τ i j * ( x , y ) d A ( x ) , y ∈ A m ( m = 1 , ⋯ , N C ) (5)

τ i ( y ) = n j ( y ) ∫ Γ τ k ( x ) u i j k * ( x , y ) d Γ ( x ) − n j ( y ) ∫ Γ u k ( x ) τ i j k * ( x , y ) d Γ ( x ) − n j ( y ) ∑ m = 1 N C ∫ A m + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) , y ∈ A m ( m = 1 , ⋯ , N C ) (6)

which means that both the displacement and the traction equations should be employed when the source point y is placed on the crack surface, thus avoiding the degeneration of the system matrix and getting rid of interfaces between the sub-regions into the modeling scheme in refs. [

Since the analytical solutions for most of multiple crack problems are restricted to obtain, Telles et al. put forward the numerical Green’s function in 1995, which can be modeled as the superposition of a basic plus a complementary solution problem in refs. [

γ u i ( y ) = ∫ Γ τ j ( x ) u i j G ( x , y ) d Γ ( x ) − ∫ Γ u j ( x ) τ i j G ( x , y ) d Γ ( x ) (7)

where the superscript G means the Green's function fundamental solution as follows:

u i j G ( x , y ) = u i j * ( x , y ) + u i j C ( x , y ) (8)

τ i j G ( x , y ) = τ i j * ( x , y ) + τ i j C ( x , y ) (9)

The complementary parts of the NGF, u i j C and τ i j C need to be determined by numerical means by using the traction BIE in refs. [

In Section 2.1, the eigen COD BIEs have been defined. The CODs are the unknowns and need to be computed step by step with the concept of eigen COD, which will be discussed in what follows.

For convenience, we consider an infinite domain containing a traction free crack A under far-field loading σ as shown in

this part. The second part is that an opened crack under the fictitious tractions which applied on the upper and lower surface of the crack but without far-field loading as shown in _{i}, redefined as the crack opening displacements of current crack A under the fictitious tractions, τ_{i}, acting on the surface.

To better explain the following processes, an infinite domain contains only one crack is considered. By making the source point y on the crack surface, the expression of the traction, a hypersingular traction integral equation, can be derived from Equation (4) in the global coordinate after a limiting process as follows:

n j ( y ) HFP ∫ A + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) = − τ i ( y ) , y ∈ A + (10)

where the capital HFP in the left side means the Hadamard finite part when the source y and the field point x coincide, which has hypersingularity. Without loss of generality, the single crack can be modeled as the case of a straight crack in refs. [

μ 2 π a ( 1 − ν ) HFP ∫ − 1 + 1 δ i r 2 ( x , y ) d A ( x ) = − τ i ( y ) (11)

where μ and ν are the shear modulus and Poisson’s ratio of the solid material, respectively. r represents the distance between field point x and source point y. δ_{i} stands for the COD for convenience.a in denominator means the half length of the crack. It can be seen from Equation (11) that the COD, δ_{i}, has to be solved numerically.

To eliminate the subscript i, the explicitly analytical expression of the HFP integral in ref. [

μ 2 π a ( 1 − ν ) { ∑ j = 1 , j ≠ k N G w j ( ξ j − ξ k ) 2 δ j − [ ∑ j = 1 , j ≠ k N G w j ( ξ j − ξ k ) 2 + 2 1 − ξ i 2 ] δ k − [ ∑ j = 1 , j ≠ k N G w j ξ j − ξ k − ln ( 1 − ξ k 1 + ξ i ) ] ∂ δ k ∂ ξ + w k 2 ∂ 2 δ k ∂ ξ 2 } = − τ k , k = 1 , ⋯ , N G (12)

where the subscript j and k are respectively the collocation points and the Gauss stations. w_{j} and ξ_{j} are respectively the Gauss weight functions and station functions. N_{G} is the total discrete Gauss number. It is noticed that in Equation (12) there exists the derivatives of the COD, ∂ δ k , which can be computed by using a Lagrange polynomial interpolation as follows:

δ = ∑ k = 1 N G l k δ k (13)

where l_{k} represent the coefficients of the Lagrange polynomials interpolation with an order number of (N_{G} + 2), making the Equation (13) satisfy that δ ( + a ) = δ ( − a ) = 0 .

Finally, by embedding Equation (13) into Equation (12), an equation in matrix form can be written for both opening mode and sliding mode of the CODs as follows:

a − 1 S 0 δ = τ , S 0 = [ S 0 0 S ] (14)

where the vector form δ and fictitious tractions τ have a size of (2N_{G} × 1). S is discretized from Equation (12) in matrix form with a size of (N_{G}×N_{G}), which can be defined as the basic matrix for multiple crack problems in the present paper.

In this section, an infinite domain contains multiple cracks is considered as shown in

The main idea is to first select a crack as the current crack A (research object), and then divide all cracks into two groups according to a distance of other cracks to the current crack, defined as the adjacent group and the far-field group. It needs to be explained that the intermediate crack is usually selected as the current crack

to facilitate the calculation in the present paper. The adjacent group (interior of the dashed circle) is consisted of the cracks with relatively small distances while the far-field group (exterior of the dashed circle) is made up of those with relatively large distances. Correspondingly, the adjacent group has strong interaction effects while the far-field group has week interaction effects to the current crack, respectively.

In order to better explain the derivation of local Eshelby matrix, only the adjacent group is considered, that is, the dashed circle around the current crack A contains all the cracks in infinite domain while the far-field group just contains crack number of zero. The number of cracks in the adjacent group being denoted as N_{L} (N_{L} = N_{C}). By placing the source point y on the crack surfaces, Equation (10) can be rewritten as follows:

n j ( y ) HFP ∫ A l + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) + n j ( y ) ∑ m = 1 , m ≠ l N L ∫ A m + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) = − τ i ( y ) , y ∈ A l (15)

The first integral in the left side of Equation (15) has hypersingularity with the same structure as that of Equation (11), which can be discretized and computed by using Equation (12) and Equation (14), respectively. The second integral in the left side of Equation (15) is regular, which is easy to compute without any difficulty. After discretization and arrangement, Equation (15) can be rewritten in matrix form as follows:

[ a 1 − 1 S 0 S 1 , 2 ⋯ S 1 , N L S 2 , 1 a 2 − 1 S 0 ⋯ S 2 , N L ⋮ ⋮ ⋱ ⋮ S N L , 1 S N L , 2 ⋯ a N L − 1 S 0 ] { δ 1 δ 2 ⋮ δ N L } = { τ 1 τ 2 ⋮ τ N L } (16)

where a_{k} in the left side represent the half length of the kth crack. S k m represent the sub-matrices which derived from the discrete form corresponding to the second regular integral in the left side of Equation (15).

By inversing the total square matrix to the right side, the eigen COD, δ_{k}, of the kth crack can be obtained as follows:

δ ( k ) = S k τ ( k ) , k = 1 , 2 , ⋯ , N C (17)

where the right vector τ ( k ) = { τ 1 , τ 2 , ⋯ , τ N L } k T are the fictitious tractions of all cracks in the adjacent group with a size of ((2N_{G} × N_{L}) × 1). The left vector δ ( k ) are the eigen COD of the kth crack with a size of (2N_{G} × 1). S k are the inverse matrix of Equation (16), which is named as the local Eshelby matrix with a size of (2N_{G} × (2N_{G} × N_{L})) in the present paper. Since the discrete Gauss numberN_{G} and the adjacent number N_{L} are always a limit number, the matrix S k have a small size.

It should be noted that the above local Eshelby matrix S k are distinctly different for each crack. Once the radius of the dashed circle is defined in

In order to avoid directly compute stresses or displacements around the crack tips due to its singularity, the SIFs are computed by using a polynomial expansion of fictitious tractions in this section.

Under normal circumstance, we can define the normal direction (opening mode) as n and the tangential direction (sliding mode) as t, respectively, with respect to the crack in the local coordinate as shown in _{P} as follows:

τ = c 0 + c 1 η a + c 2 ( η a ) 2 + ⋅ ⋅ ⋅ + c N P ( η a ) N P , η ∈ [ − a , + a ] (18)

where c_{i} represent the coefficients of the polynomial.

According to the analytical solution of fracture mechanics [

K I R = 1 π a ∫ − a a p ( η ) ( a − η ) a 2 − η 2 d η (19)

K I L = 1 π a ∫ − a a p ( η ) ( a + η ) a 2 − η 2 d η (20)

where p is the loading on the crack surfaces, and p is the same with the fictitious tractions τ in the present paper.

Then embedded Equation (18) into Equation (19) and (20), respectively, the SIFs at the two crack tips of a crack can be easily obtained as follows:

{ K R K L } = { c 0 + 1 2 ( ± c 1 + c 2 ) + 1 ⋅ 3 2 ⋅ 4 ( ± c 3 + c 4 ) + ⋯ + 1 ⋅ 3 ⋅ ⋅ ⋅ ( N P − 1 ) 2 ⋅ 4 ⋅ ⋅ ⋅ N P ( ± c N P − 1 + c N P ) } π a (21)

where K^{R} and K^{L} represent the SIFs of right and left tips of the crack, respectively. It is important to note that the troubles of computing stresses or displacements around the crack tips can be avoided by using Equation (21), which only related to the coefficients of the polynomial expansion of the fictitious tractions.

It is obvious that all the unknown CODs and the fictitious tractions are related to the outer loading mode, the geometries, the quantities as well as the distributions of cracks in finite/infinite domain, which having mutual interactions among them. The solution procedures of the eigen COD BIEs are mainly divided into four stages as follows, i.e., the initiation stage, the iteration stage, the convergence check stage and the post process stage, respectively.

1) Initializing all the data related to the finite/infinite domain, such as boundary conditions and geometric information of cracks, etc.;

2) Computing the local Eshelby matrices, S k ;

3) Computing the boundary unknowns by using the displacement eigen COD boundary integral Equation (1);

4) Computing the fictitious tractions of all cracks by using the traction eigen COD boundary integral Equation (4);

5) Computing the initial SIFs by using Equation (21).

Since all cracks are divided into two groups, adjacent group and far-field group, the eigen CODs of all cracks are correspondingly calculated with respect to the fictitious tractions by two parts as follows:

1) The first part is calculated by using the fictitious tractions of the current crack and other cracks in the adjacent group via the local Eshelby matrix Equation (17);

2) The second part is calculated by using the fictitious tractions of all the cracks in the far-field group. A modified traction boundary integral equation of Equation (4) can be obtained as follows:

τ i ( y ∈ A l + ) = n j ( y ) ∫ Γ τ k ( x ) u i j k * ( x , y ) d Γ ( x ) − n j ( y ) ∫ Γ u k ( x ) τ i j k * ( x , y ) d Γ ( x ) − n j ( y ) ∑ m = 1 , m ∉ N L l N C ∫ A m + Δ u k ( x ) τ i j k * ( x , y ) d A ( x ) , l = 1 , ⋯ , N C (22)

where N L l represent the cracks of adjacent group centered at the current crack l. Then computing the corresponding SIFs of all cracks using Equation (21).

In this block, we defined the maximum iteration error as follows:

K max = max | K ( k ) − K ( k − 1 ) | (23)

where K_{max} represents the maximum difference of the SIFs between the two iterations. The superscript k is the iteration count.

The convergence criterion of the present eigen COD BIEs is chosen as follows:

K max / σ π a ≤ 10 − 3 (24)

1) If the convergence criterion is not satisfied Equation (24), it should be solved the boundary unknowns again and then return to the previous iteration stage;

2) If the convergence criterion is satisfied Equation (24), then go to the next post process stage in what follows.

The post process stage can be carried out according to the computational needs or the interests of the researchers such as the overall properties and the fracture properties of these solid materials, as well as the local stresses or strain field around the crack tips, etc.

In conclusion, the solution procedures of the present eigen COD BIEs can be described in the flow chart as shown in

In this section, solutions of stress intensity factors of multiple crack problems in finite/infinite plates are presented to demonstrate the accuracy and the efficiency

by using the above three kinds of boundary integral equations. The numerical examples are solved by using a desk-top computer Dell with Intel Core Dual CPU, 2.50 GHz, 8 Gb of memory.

The first example is a square plate (W = H) in tension contains four cracks (N_{C} = 4) with the same length (a = 0.2 W) as shown in _{L} = 4, the same as that of the total cracks (N_{L} = N_{C}) in the eigen COD algorithm.

The stress intensity factors are computed with the variation of angle θ of the left and the right cracks symmetrically while the upper and the bottom cracks are kept stationary. The computed results are presented and compared in

the stress intensity factors at the crack tips A, B and C. The results by Chen & Chang in ref. [

The second example is a square plate in tension with multiple cracks, and two typical examples are as illustrated in _{L} = 9 in the eigen COD algorithm.

The effects of the number adopted for crack discretization, N_{G}, are compared in _{C} = 9 (the tilting angle θ = 55.86˚) and compared in _{C} = 25 (the tilting angle θ = -55.75˚) for the three algorithms. It is seen from the _{G}, which become stable when the number for crack discretization is equal to and greater than 13 (N_{G} ≥ 13) in all these examples.

The CPU time are compared as listed in _{C} = 121 (the tilting angle θ = 60.50˚). In Dual BIEs, the final system matrix contains all unknown information both on boundaries and crack surfaces, the size will be grow large with the total crack number increases, while in the NGF algorithm, the size of final system matrix to determine the complementary solution also grow large with the total crack number increases.

In this section, infinite plates contain multiple rows of periodical cracks are

Algorithm | N_{G} | K 1 , R / σ π a | K 2 , R / σ π a | K 1 , L / σ π a | K 2 , L / σ π a |
---|---|---|---|---|---|

Dual BIEs | 5 | 0.34038 | 0.46616 | 0.37789 | 0.47206 |

9 | 0.33831 | 0.46603 | 0.37281 | 0.47155 | |

13 | 0.33769 | 0.46597 | 0.37127 | 0.47139 | |

17 | 0.33737 | 0.46595 | 0.37049 | 0.47131 | |

21 | 0.33701 | 0.46593 | 0.36958 | 0.47121 | |

NGF | 5 | 0.33859 | 0.46567 | 0.36993 | 0.47118 |

9 | 0.33837 | 0.46563 | 0.36940 | 0.47111 | |

13 | 0.33831 | 0.46563 | 0.36927 | 0.47109 | |

17 | 0.33829 | 0.46562 | 0.36922 | 0.47108 | |

21 | 0.33828 | 0.46562 | 0.36920 | 0.47108 | |

Eigen COD | 5 | 0.33753 | 0.46584 | 0.37135 | 0.47135 |

9 | 0.33732 | 0.46582 | 0.37080 | 0.47129 | |

13 | 0.33727 | 0.46582 | 0.37067 | 0.47128 | |

17 | 0.33724 | 0.46582 | 0.37062 | 0.47127 | |

21 | 0.33723 | 0.46582 | 0.37059 | 0.47127 |

Algorithm | N_{G} | K 1 , R / σ π a | K 2 , R / σ π a | K 1 , L / σ π a | K 2 , L / σ π a |
---|---|---|---|---|---|

Dual BIEs | 5 | 0.35240 | −0.42233 | 0.35648 | −0.40404 |

9 | 0.34964 | −0.42487 | 0.35343 | −0.40808 | |

13 | 0.34882 | −0.42564 | 0.35251 | −0.40931 | |

17 | 0.34840 | −0.42604 | 0.35205 | −0.40994 | |

21 | 0.34811 | −0.42632 | 0.35173 | −0.41037 | |

NGF | 5 | 0.32574 | −0.41434 | 0.32749 | −0.39836 |

9 | 0.32568 | −0.41470 | 0.32741 | −0.39887 | |

13 | 0.32567 | −0.41479 | 0.32739 | −0.39900 | |

17 | 0.32566 | −0.41483 | 0.32738 | −0.39905 | |

21 | 0.32566 | −0.41484 | 0.32738 | −0.39908 | |

Eigen COD | 5 | 0.35019 | −0.42246 | 0.35326 | −0.40629 |

9 | 0.34988 | −0.42278 | 0.35292 | −0.40677 | |

13 | 0.34980 | −0.42286 | 0.35284 | −0.40689 | |

17 | 0.34977 | −0.42289 | 0.35280 | −0.40694 | |

21 | 0.34976 | −0.42291 | 0.35279 | −0.40696 |

N_{C}/n | θ | Algorithm | CPU time (s) |
---|---|---|---|

121/11 | 60.50˚ | Dual BIEs | 810.367 |

NGF | 405.612 | ||

Eigen COD | 20.548 |

considered. In order to make the problem being much convenient, the crack problems are composed of cracks with the same size, orientation and spacing. The cracks in arrays considered are placed periodically in a number of configurations. Since all the cracks in arrays are under the same loading condition, one crack in the array is designated as the representative crack while the influences of all the other cracks, serving as an infinite series, on the representative crack can be summed up.

The first example, as shown in _{C} = 501) are taken into consideration instead of using an infinite number. It should be noted that the total number of crack is set to be an odd number so that the current crack can be centrally arranged for convenience.

The calculated results for I-mode SIFs are expressed as shown in _{C} increasing, the absolute error between computed results and theoretical solutions is decreasing. It can be seen that the computing results are matched well by using the three algorithms, even though at a/b = 0.9, the error is still less than 1%. It is so trivial that we could assume N_{C} = 501 is enough to describe infinite array of periodically spaced collinear cracks problem. If the local number of cracks N_{L} and the gauss points N_{G} increase, the accuracy will be even more satisfactory.

The last example in this block is an infinite plate containing double periodical collinear cracks with the same length under far-field uniform tension perpendicular to the crack surfaces as shown in

Algorithm | a/b N_{C } | 0.10 | 0.50 | 0.80 | 0.90 |
---|---|---|---|---|---|

Dual BIEs | 101 | 1.00408 | 1.12737 | 1.55942 | 2.09185 |

201 | 1.00409 | 1.12762 | 1.55988 | 2.09276 | |

301 | 1.00411 | 1.12791 | 1.56165 | 2.09452 | |

401 | 1.00411 | 1.12810 | 1.56232 | 2.10257 | |

501 | 1.00411 | 1.12825 | 1.56319 | 2.10299 | |

NGF | 101 | 1.00411 | 1.12765 | 1.56021 | 2.09685 |

201 | 1.00412 | 1.12803 | 1.56205 | 2.10075 | |

301 | 1.00412 | 1.12816 | 1.56266 | 2.10206 | |

401 | 1.00412 | 1.12823 | 1.56297 | 2.10271 | |

501 | 1.00412 | 1.12827 | 1.56316 | 2.10311 | |

Eigen COD | 101 | 1.00412 | 1.12769 | 1.56025 | 2.09689 |

201 | 1.00414 | 1.12808 | 1.56209 | 2.10079 | |

301 | 1.00414 | 1.12821 | 1.56270 | 2.10210 | |

401 | 1.00414 | 1.12828 | 1.56301 | 2.10275 | |

501 | 1.00414 | 1.12832 | 1.56320 | 2.10315 |

problem is located at the center. The total number of crack is (N × N).

The normalized stress intensity factors of the central crack at different number of rows N are calculated and compared as listed in

The CPU time are also compared as listed in

N_{C} | Algorithm | h/d = 1.0 | |||||
---|---|---|---|---|---|---|---|

Present | Wang [ | ||||||

a/b = 0.5 | 0.8 | 0.99 | a/b = 0.5 | 0.8 | 0.99 | ||

9 × 9 | Dual BIEs | 1.10683 | 1.55238 | 6.36674 | 1.11330 | 1.55774 | 6.36983 |

NGF | 1.12183 | 1.56247 | 6.37052 | ||||

Eigen COD | 1.11325 | 1.55773 | 6.36917 | ||||

11 × 11 | Dual BIEs | 1.10746 | 1.56023 | 6.36794 | 1.11330 | 1.55774 | 6.36974 |

NGF | 1.12357 | 1.56514 | 6.37742 | ||||

Eigen COD | 1.11325 | 1.55773 | 6.36917 | ||||

13 × 13 | Dual BIEs | 1.10822 | 1.56437 | 6.36815 | 1.11330 | 1.55773 | 6.36964 |

NGF | 1.12483 | 1.56776 | 6.37981 | ||||

Eigen COD | 1.11326 | 1.55773 | 6.36982 |

N_{C} | Algorithm | CPU time (s) |
---|---|---|

9 × 9 | Dual BIEs | 226.529 |

NGF | 137.630 | |

Eigen COD | 15.419 | |

11 × 11 | Dual BIEs | 668.137 |

NGF | 445.818 | |

Eigen COD | 18.521 | |

13 × 13 | Dual BIEs | 1692.356 |

NGF | 653.128 | |

Eigen COD | 22.069 |

guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks using ordinary desk-top computers.

A new algorithm based on the eigen COD BIEs is proposed to simulate solid materials with cracks in large number, where the eigen COD is defined as the crack opening displacements of a crack under the fictitious tractions loading on the crack surfaces. With the concept of eigen COD, multiple crack problems can be easily solved in an iterative fashion with a relatively small size of final system matrix compared with the Dual BIEs and the NGF, showing the practical significance of the present approach. Through the division of adjacent group and far-field group, the local Eshelby matrix, reflecting the strong interactions among cracks and avoiding numerical iteration divergence in the case of dense cracks, derived from the traction boundary integral equation in discrete form is introduced. Numerical examples verify the feasibility of the eigen COD BIEs for the simulation of multiple crack problems from two aspects of the numerical calculation accuracy and efficiency of computation. The numerical results of stress intensity factors show that the eigen COD algorithm has intrinsic scientific and rationality. Not only ensures the accuracy, also greatly improves the efficiency. The overall properties such as the rigidities and the local details such as the stresses around the crack tips can be achieved in future research, which have important theoretical significance and engineering application value.

This work is supported by the Funds Area of the National Natural Science Foundation of China (Grant No. 11662005), and the Science Foundation of Jiangxi Province (Grant No. 20202BABL201015).

The first author would like to respectively gratefully acknowledge Prof. Yijun Liu of the University of Cincinnati and his doctorial advisor Prof. Hang Ma of Shanghai University, for their advices in the research on boundary element method in modeling multiple crack problems.

The author declares no conflicts of interest regarding the publication of this paper.

Zhao, G. (2020) Numerical Comparison Research on the Solution of Stress Intensity Factors of Multiple Crack Problems. Advances in Pure Mathematics, 10, 706-727. https://doi.org/10.4236/apm.2020.1012044