1

A DUAL BOUNDARY ELEMENT PROCEDURE FORANALYSIS OF FRACTURE IN CONCRETE

S. PARVANOVA, G. GOSPODINOV

University of Architecture, Civil Engineering and Geodesy1, Christo Smirnenski Blvd., 1046, Sofia, Bulgariae-mail: slp_fce@uacg.bg, gospgosp_frf@uacg.bg

ABSTRACT. The present paper is concerned with the numerical implementation of the dualboundary element method (DBEM) in the linear elastic fracture mechanics (LEFM) context for crack propagation in plain concrete. The main advantage of the DBEM is that mixed-mode crack problems can be solved successfully with a single region formulation, thus avoiding the ineffective sub-region approach. The method is combined with a previously developed two-step subtraction singularity technique (TSSST) for evaluation of stress intensity factors (SIFs), which is generalized here for the case of multiple cracks. The crack growth condition requires a fracture criterion, which forms a simple numerical procedure along with the DBEM and TSSST, capable of treating the general problem of mixed-mode fracture. Examples of deep, shear-critical beams are numerically solved and analyzed with the present approach, and highly accurate results are obtained. The accuracy and efficiency of the implementation, described herein make this formulation very suitable to study the crack growth problems in plain concrete under a general Mode I/II conditions.

1. IntroductionIt is well known that the nonlinear response of concrete is mainly dominated by

progressive cracking which frequently results in local failure [1, 2]. In the case of plain concreteor reinforced concrete deep beams for example, a major diagonal tension crack might suddenly occur and dominate up to the final failure, as shown in Fig. 1 (a).

In general two principal approaches are used for the numerical modelling of cracks: discrete and smeared crack models. In the present study we employ the discrete crack modelling and some positive arguments for this choice are to be found in references [6, 7]. When the

crack sliding

crack opening

crack opening

flexural crack

shear crack

F/2F/2

F

Fig. 1 (a) Concrete shear beam and cracks development; (b) Opening mode of fracture due to central flexural crack; (c) Mixed-mode fracture due to diagonal crack propagation

(a) (c)

(b)

2

concrete shear beam is initially loaded, flexural cracks are expected to develop under pure opening Mode I – see Fig. 1 (b). It is relatively easy to analyse this type of cracking and calculate the stress intensity factor when the discrete crack model and LEFM are employed. The situation becomes more complex when a mixed-mode fracture develops, associated with a diagonal, usually known as critical or shear crack. The crack opening is combined with huge sliding and frictional forces, due to aggregate interlock over the both faces of the crack (Fig. 1 (c)). In this respect a reliable numerical method is needed to study the problem of multiple cracks propagation during the process of increasing loading.

The direct application of the classical BEM for 2D domain with edge or internal cracks is not possible, because the coincidence of the crack surfaces gives rise to a singular system of algebraic equations. Few techniques have been devised to overcome this difficulty, such as the sub-region method, the displacement discontinuity method, the dual boundary element method and others. It seems that the DBEM is most promising one, demonstrating many advantages. On the other hand, the method is very complex and requires unconditional application of discontinuous boundary elements and analytical treatment of the singular finite part integrals. In this study we use the dual boundary element method for linear, double node discontinuous boundary elements, already developed in paper [8]. In fact, the original idea of implementation of DBEM is dating back to 70-th for solving potential theory problems, but the method was refined by Aliabadi and others for the theory of elasticity and fracture mechanics applications – see references [2, 3, 4].

The successful implementation of the methods of LEFM for crack initiation and propagation involves a definition and application of certain criterion and we use the maximum circumferential stress criterion [1], which proves to be suitable for mixed-mode case of deformations. However, it requires very accurate calculation of IK and IIK SIFs for both fracture modes. We employ the two-step singularity subtraction technique, following the principal ideafrom paper [4], where this particular approach was employed for a single crack in Mode I, by means of the sub-region decomposition. First, the TSSST was adopted as a numerical procedure within the DBEM [9], and made it possible to consider an arbitrary deformation of the crack surfaces – details, examples and comparisons are given in paper [9]. In the present work a generalization is made accounting for cases of multiple cracks and arbitrary geometry. The crack path usually consists of number of linear segments with a chosen length. It is shown that such an approximation leads to very accurate results for SIFs required.

The main objectives of this study are to develop a numerical procedure, based on LEFM, for solving the problem of multiple crack propagation for shear-critical concrete beams. For this purpose the DBEM along with the TSSST, given in papers [8 and 9], are combined in a new numerical procedure. It requires additional equations for finding the crack direction and propagation conditions, so the maximum circumferential stress criterion was adopted to resolve the problem. Two examples of deep concrete beams are solved and analyzed, including the well known bench mark mixed-mode problem, given in [5, 6 and 7]. Some comparisons are made.

2. The dual boundary element method using linear, double-nodediscontinuous boundary elements

The term dual for this variant of the boundary element method steams from the fact that the displacement boundary integral equations are used on the one crack surface, whereas the traction equations are applied on the other. In this way, avoiding the singular set of algebraic equations, we can simply treat a domain containing several edge as well as internal cracks, as a single one. The DBEM allows to easily obtain the actual deformed shape of an arbitrary plane object, containing cracks, which can not be achieved by means of direct application of the boundary element method [2, 3, 8].

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Omitting details, which can be found in [8], the complete system of boundary integral equations are worked out to constitute the basis of the dual boundary element method at the crack point 0x′ (see Fig. 2) of a smooth boundary, as follows[2, 3, 8]:

(1)

* *0 0 0 0

* *0 0 0 0 0 0

1 1( ) ( ) ( , ) ( ) ( ) ( , ) ( ) ( ),2 2

1 1( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( ).2 2

j j ij i ij i

j j i kij k i kij k

u x u x u x x t x d x t x x u x d x

t x t x n x d x x t x d x n x s x x u x d x

Γ Γ

Γ Γ

′ ′′ ′ ′+ = Γ − Γ

′ ′′ ′ ′ ′ ′− = Γ − Γ

∫ ∫

∫ ∫

The indices i and j range from 1 to 2 and refer to the Cartesian coordinate directions;

ju ( x ) and jt ( x ) are displacement and traction functions on the boundary 㤰 respectively; *iju 䎀

*ijt represent the Kelvin displacement and traction fundamental solutions at a boundary point x ;

*kijd and *

kijs are third-order fundamental tensors, derived as linear combinations of derivatives of Kelvin’s tensors; 0in ( x )′ denotes the i -th coordinate of the unit normal to the boundary at point

0x′ ; the second integral of the first equation and the first integral of the second equation must beconsidered as Cauchy principal value integrals, whereas the last integral of the second integral equation is considered in a sense of Hadamard principal value integral [8].

It is important to note that the first integral equation in (1) is the usual displacement integral equation, while the second one is the traction equation. There is other important and specific feature of Eq. (1): when the collocation point 0x′ is on the crack surface (say 0( )ju x′ in displacement equation or 0( )jt x′ in traction equation), an additional jump term 0( )ju x′′ or 0( )jt x′′

will appear, due to the coincident node 0x′′on the other crack surface, detailed explanations for which are given in [8].

Fig. 2 (a) Contour discretization using continuous and discontinuous boundary elements.Utilization of the dual integral equations for both surfaces of the crack.(b) Discontinuous double-node boundary element and its shape functions N1(㯠) and N2(㯠)

N

N

N

D – displacement equationT – traction equation

C – continuous boundary elementN – discontinuous boundary element

N

N

N

N N N

䈠, N

䈠, N

D, N

D, N

CC

tips of the cracks 1, 2 and 3

N

T, N

D, N

T, ND, N

T, N

D, N

D, N 䈠, N

(a)

㯠

㯠=0.0

㯠=1

㯠=-1

I,J – begin and end points of the element1,2 – element’s nodes㯠 – natural coordinates (-1,+1)

I

J

n. 1

n. 2

㯠=-0.5I

J

n. 1

n. 2㯠=0.5

㭰

N1(㯠)

N2(㯠)

1

1

N1(㯠)=-㯠+1/2,N2(㯠)=㯠+1/2

(b)

L

L/4

4

In Fig. 2 (a) a rectangular plane domain is shown with three multi-linear edge cracks. The boundary discretization of the crack faces modeled by double node discontinuous elements is also shown together with the element itself and its shape functions (Fig. 2 (b)). On the boundary 㤰either continuous (C) and/or discontinuous (N) boundary elements could be employed. Also,there are no restrictions on the type of integral equations, displacement or traction, used. Operating on one of the crack surfaces two principal restrictions are to be met: (1) the dual system of integral equations must be used, namely displacement equations (denoted by D) on one side, and traction equations (denoted by T) on the other; (2) only discontinuous boundary elements must be employed for both crack faces. The above conditions are necessary for the existence of principal value integrals, assumed for the derivation of dual integral equations [8].

With the system of algebraic equations worked out and boundary functions available, one can find displacements and stresses for any point in the domain. In the present case the stresses

( )ij tipxσ at a particular crack tip point tipx could be found with the following expression

(2) * *( ) ( , ) ( ) ( ) ( , ) ( ) ( ),ij tip kij tip k kij tip kx d x x t x d x s x x u x d xΓ Γ

σ = Γ − Γ∫ ∫

where the related tensors are mentioned above.Having calculated the stresses at the crack tip, one can easily recalculate them in any

coordinate system of the same origin, or get the traction vector for a defined normal, if necessary. That will be described later on in the context of the formulation of a new numerical procedure for calculation of SIFs.

3. Application of two-step singularity subtraction technique for the calculation of stress intensity factors in case of multi-linear cracks

The main feature of the classical singularity subtraction (SS) technique is the combination of the singular solution due to first term of the Williams series expansion, and the solution of the real problem, applying superposition principle. For simplicity, we refer to first and second step solutions with details given in [9]. The elastic stress field near the crack tip vicinity is singular (Fig. 3). Because of the convergence difficulties arising in numerical modeling of the singular fields by the SS method these singularities are completely eliminated. As a result, the “regularization” process introduces the stress intensity factors ( IK and IIK for the mixed mode fracture) as additional unknowns for the original problem, therefore two extra conditions are needed to obtain a correct solution.

It is instructive to start the short description of the TSSST with some transformation formulas, concerning the stress field in the neighborhood of the singular point – the crack tip represented in Fig. 3. Suppose the element of the stress tensor mnσ at point a in the unprimed

1 2x x coordinate system (Fig. 3 (a)) is known. If a crack direction is defined at nonzero angle 㬐, one can calculate the stress tensor ijσ ′ in the primed coordinate system 1 2x x′ ′ (Fig. 3 (b)), as

(3) ,ij im jn mnl lσ σ′ =

where ijl denotes the matrix of direction cosines: cos( , )ij i jl x x′= , or in another form

(4)cos sin

.sin cosijl

α αα α

= −

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We assume that the crack propagates along the line of the last linear section pointing at a(Fig. 3 (c)). Then the tractions at point a of the imaginary crack surfaces can be calculated by means of the well known equation: i ij jt n ,σ′ ′ ′= where jn′ is the unit outward normal to the defined surface. Furthermore, it is convenient to calculate the two tractions it′ at point a, bearing in mind the obvious relationship I 22t σ′ ′= related to the definition of SIF for the opening Mode I and

II 12t σ′ ′= related to the definition of SIF for the sliding Mode II.In a linear elastic domain containing a discrete crack, the complete elastic solution is

assumed to be a sum of two components: a regular part (denoted by r superscript) and a singular part (denoted by s superscript) due to the presence of the crack tip. Therefore, the solution can be rearranged for the displacement and stress tensors, as follows

(5)r si i ir sij ij ij

u u u ,.σ σ σ

= −

= −

Since the complete solution ( iu , ijσ ) and the singular part of it ( siu , s

ijσ ) include the mentioned singularity, it follows from equations (5) that the singularity is cancelled out, and the remaining component of the solution ( r

iu , rijσ ) is therefore regular.

It is important to point out that for the TSSST in the first step we perform a solution by DBEM for the actual geometry, load and boundary conditions to get iu and ijσ fields. In the

second step, using the same boundary element discretization, the singular fields siu and s

ijσ are obtained for a reference value of SIF. The both solutions are “loaded” with error due to approximation of the boundary functions plus the presence of the crack tip, being a highly singular point. It follows therefore from equations (5), that as a result of subtraction the computational error is minimized - further details for the method and some results can be found in [9].

With both solutions available, we next introduce the traction it at a chosen point in the domain or at the boundary, for which the following equations hold:

crack tip

2x

1x

22σ

22σ12σ

11σ

point a at distance b

a

0b →

11σ ′2x′

1x′

crack tip

point a at distance b

22σ ′

12σ ′

α

IIt′2x′

1x′

crack tip

point a at distance b

It ′

α

IIt′It ′

Fig. 3 (a) Stresses at point 䌀 located at distance b→0 in front of the crack tip in 1 2x x coordinate system; (b) Stresses at point 䌀 in coordinate system 1 2x x′ ′ ; (c) Case (b) represented by means of tractions It ′ and IIt′ at point a governing the opening and sliding fracture modes

(a) (c)

(b)

6

(6) , ,s r r si i i i i it t t or t t t+ −= K = K

where K is the unknown SIF for Mode I or II, it is the relevant traction from the real first step solution, s

it is the respective traction from the singular second step solution, and rit denotes the

traction assumed a regular function free from crack tip singularity, which actually is not needed to find.

In order to get the values of SIFs K additional equations are needed – one for each different fracture mode. These are known as the “singularity” conditions for the regularization procedure. It is very natural to define them for the crack tip point, where the regular traction r

it is zero and from Eq. (6) follows that

(7) ( 0, 0) 0 ,r si i it r t tθ= = = − =K

yielding the final result

(8) .isi

tt

=K

The relationship (8) could be rewritten in expanded form, giving the required formulas for the two stress intensity factors:

(9) ,II s

I

tKt′

=′

(10) .IIII s

II

tKt′

=′

In what follows we postulate the assumptions for the application of TSSST, using LEFM,in the case of multiple mixed-mode cracks for concrete beams:

Ø The development of a particular crack is performed step by step, by choosing linear segments of specified length and direction, according to certain criterion; Ø Propagation of one crack is considered only, as far as the Williams field accounts for the presence of one crack tip. The other cracks are “frozen”, which means that in the firststep solution their real geometry has been accounted for, but the SIFs are calculated for the crack into consideration only;Ø If the singular solution step is performed for a given crack tip, the continuity of the Williams fields away from this crack line is maintained, even across the other cracks, if available. Hence the other cracks must remain closed which must be accounted for through specified boundary conditions.In order to achieve the above requirements a simple procedure was adopted in this paper

illustrated in Fig. 4. Suppose we perform the second step singular solution and the left crack is active while the right crack must remain passive. When calculating the singular stress and displacement fields we use the 1 2x x′ ′ coordinate system with the origin at the crack tip point i. We know that the DBEM requires that every discontinuous boundary element, set on the first surface of the crack (such as l-th element) has a counterpart (such as k-th element), put on the other crack surface with opposite normal. Since the nodal points of the two elements have identical coordinates and the distance ijr is the same, we use angle θ for the nodal point of the left boundary element and angle (2 )π θ− for the respective nodal point of the right boundary element, which leads to correct results. On the other hand, if we deal with a crack admittedlypassive (the right vertical crack in the Fig. 4), we use a single angle θ and distance ijr for both nodes with equal coordinates, thus treating the crack as closed.

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Further on, it is possible to carry out the same procedure by changing the status of the cracks in order to calculate the SIF for other crack tip. The correctness of the suggested procedure when multiple cracks are involved was demonstrated by number of numerical examples. They have shown that an interesting generalization is possible for the second step – we may consider only the active crack and do not perform a discretization on other cracks, making the presumptionthey do not exist. It turns out that the results for singular solution are very close, so we may reduce the number of unknowns substantially. Of course additional research is needed to justify this conclusion.

4. Linear elastic fracture mechanics criterion and procedures for crack initiation and propagation in the case of mixed-mode fracture

According to the principles of the LEFM under pure opening Mode I the crack will propagate when the stress intensity factor IK exceeds a material parameter or a constant called fracture toughness IcK . In the general case of mixed-mode crack propagation the maximum circumferential stress criterion constitutes that the fracture initiation starts in direction in which the circumferential stress, near the crack tip, is maximum. According to this criterion [1] the direction of crack propagation angle crθ is the solution of the following equation

(11) sin (3cos 1) 0 ,I cr II crK Kθ θ+ − =

where IK and IIK are the stress intensity factors, calculated at the corresponding instant.It is worth noting that equation (11) could be also written in the form

(12) sin (3cos 1) 0 ,Icr cr

II

KK

θ θ+ − =

i

2π θ−

the passive or“frozen” crack

this crack is closed in the singular solution

ijr

θ

nodal point of l-th opposite element with j coordinates

θ

nodal point j of k-th crack boundary element

2x′1x′

crack tip points

ijr

k-th boundary element

the contour 䄰 discretized with discontinuous boundary elements

the active crack

Fig. 4 Detail of a plate with multiple cracks: the left crack is active, whereas the right one is passive and remains closed when the singular solution is performed

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which suggests that the solution (the angle crθ ) can be found if the ratio of the two SIFs is known, not the actual values. In such a way an interesting simplification of the TSSST methodappears, having in mind that the values of tractions s

it , calculated for the crack tip, from the singular solution are equal. So in order to find the propagation angle, and therefore the crack pathtrajectory, we need one static solution from a reference load for every chosen crack length.

The crack initiation begins if the stress intensity factors and propagation angle crθ satisfy the following inequality [1]

(13) 3 2cos 3 cos sin ,2 2 2cr cr cr

I II IcK K Kθ θ θ− ≥

where IcK is the fracture toughness for fracture Mode I, which in case of plane stress can be derived from the fracture energy parameter fG using the following relationship

(14) 2 .Ic fK EG=

Here E is the elasticity modulus of the material. The both material constants IcK and fGwill be used in the examples.

A quasi-automatic algorithm and a software code are developed for a quasi-static crack propagation involving a number of successive analyses. Each analysis consists of the following steps:

1. Linear elastic numerical analysis is performed by the DBEM in order to find the strain and stress fields. A reference load F=1 is used in case of point load;

2. Stress intensity factors IK and IIK are calculated by means of the two-step singularity subtraction approach, given in section 3;

3. The direction of crack propagation angle crθ is then calculated from Eq. (12). Since this formula needs a ratio of SIFs, there is no need to obtain the crack propagation load, so a reference load value is usually used;

4. Once the crθ angle is calculated, the parametric equation (13) is satisfied in order to find the current crack propagation load, which helps to find the values of SIFs IK , IIK ; the respective displacements, stresses and tractions are updated accordingly;

5. For a user defined crack-length increment, the location of the new crack tip is determined and the crack geometry and mesh are updated;

6. A discretization is made along the new linear crack length performing consecutive static solution by the DBEM for a reference load;

7. The procedure is repeated between steps 1 to 6 until the user decides to stop the crack propagation, or if a certain desired location and crack length are arrived at.

5. Numerical results and conclusionsTwo mixed-mode multiple crack propagation problems were simulated by means of the

numerical methodology presented. In the first example we consider a symmetric, deep beamsimilar to that in Fig.1 in plane stress , subjected to uniform load q=10 units and thickness b=1 unit. Due to the symmetry, only one half of the beam is analysed – see Fig 5 (a), where two discrete cracks are shown. First one is vertical of length 0.5 units at the middle of the beam. Thesecond one is diagonal starting at distance 0.4 from the support and consists of five linear segments. The length and direction of these segments are chosen by the user. The idea of this purely academic example is to test the accuracy of DBEM+TSSST methodology with respect to SIFs calculations for both fracture modes.

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At first, the status of the middle crack was considered “frozen” and SIFs for the tip of first vertical segment of the left crack were calculated. Since there is no analytical solution for this case, another numerical solution is carried out for the same example, using FEM and ANSYS software package. A special singular quarter-point finite element is implemented in ANSYS for accurate calculation of SIFs, so employing a refined finite element mesh, we consider the results as reference once - see table 1. With the calculations for IK and IIK completed, we assume the current left crack portion “frozen” and calculate IK factor for the merely Mode I of the right crack, compared again with ANSYS calculations – table 1. The above steps are repeated for all five segments of the diagonal crack, thus simulating the crack growth. The results and subsequent comparisons are given in table 1, whereas in Fig. 5 (b) the displaced boundary element mesh is shown at the final stage of the diagonal crack development.

Table 1 Comparisons of stress intensity factors for both cracksDiagonal crack Vertical crack

Cracktip

KITSSST

KIANSYS

diff.%

KIITSSST

KIIANSYS

diff.%

KITSSST

KIANSYS

diff.%

1 3.2684 3.2246 1.36 0.2035 0.1945 4.61 12.5058 12.644 1.09

2 3.6041 3.5973 0.19 1.0074 1.0003 0.71 12.5005 12.638 1.09

3 4.0272 4.0157 0.29 1.5534 1.5326 1.36 12.4738 12.607 1.06

4 4.8763 4.8915 0.31 2.5533 2.4895 2.56 12.3494 12.467 0.94

5 5.1245 5.1717 0.91 3.4158 3.3494 1.98 12.1602 12.251 0.74

It can be concluded from table 1, that the difference between the two solutions is acceptable even in the case when the values of both SIFs are of the same order of magnitude. Since the load is kept always constant, the SIFs for both modes of the diagonal crack increase, as expected. The SIF IK for the vertical middle crack is almost constant during the diagonal quasi-crack growth, which indicates that the respective singular stresses do not change in magnitude.

h=1

0.10.15

0.10.05

1.10.4

L/2=1.5

middle vertical crack

the diagonal “shear”like crack

(a)

0.050.10.1

0.1

0.1

q=10

E=100㯐=0.3L/2=1.5h=1b=1

(b)

Fig. 5 Deep beam with two cracks: the diagonal left crack with five different tips; the right crack is vertical with one tip.(a) Geometrical and material data; (b) Final deformed shape

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As a second example a more interesting case of mixed-mode fracture in a single notched concrete beam, tested by Arrea and Ingraffea (the test set up and results are well illustrated in [6]), will be numerically solved and discussed. This experiment is considered as a benchmark case and has been numerically analyzed by a number of researchers [5, 6, 7]. Fig. 6 (a) shows the geometrical configuration and material parameters of the beam. In the experiment a displacement control was used in order to obtain the descending branch of the load-displacement relationship.The initial vertical crack is of 82 mm length and plane stress condition was assumed.

For this numerical test the complete DBEM plus TSSST were performed and the crack length increment was chosen 20 mm. Starting with the calculation of SIFs for the initial crack, the procedure for finding the direction of the new crack segment is performed by using Eq. (12). The next step requires correction of the reference load to be used in the static DBEM solution, by means of Eq. (13), followed by stress and displacement field update.

Adding the new crack segment and updating the geometry and boundary element mesh, the next static solution is performed with the reference (usually unit) load. The above procedure is repeated until the final stage is reached. The calculation process is made interactive, so the user is able to run, stop, correct data and continue the program execution. For this example 13 crack

Fig. 7 (a) Picture taken from the experiment; (b) The experimental envelope and numerical prediction of the crack path, according to the present solution

Numerical predictionExperimental envelope

(a) (b)

F 0.13F

E=24.8 GPa, 㯐=0.18KIC=1.65 MPa m1/2

Thickness=0.152 m

Fig. 6 Four-point single edge-notched shear beam: (a) Geometrical, material data, loading and boundary supports; (b) Deformed shape for final stage at F=10.98 kN

(a) (b)

458 mm458 mm

61

306

82

61

11

length increments have been realized – see the plot of deflected beam at the final stage of the crack propagation in Fig. 6 (b).

In Fig. 7 (a) a picture of the original experimental set up is shown along with a visualization of the crack at the final fracture stage. For the sake of better comparison the crack path, obtained by the present numerical prediction is scaled and put onto the picture.

The experimental envelope for crack growth, presented in Fig. 7 (b), is taken from reference [6], where the extreme values of the fracture energy parameter fG varies between 55-100 N/m. Again a very nice fitting is obtained with the present numerical crack path prediction. It should be pointed out that nonlinear fracture mechanics methods and procedureswere applied in papers [6, 7], whereas the methodology developed here is simple, based on the principles of the linear elastic fracture mechanics.

The diagrams given in Fig. 8 show the vertical load - crack mouth sliding displacement relationship. The experimental envelope for the extreme values of fracture energy parameter (Gf=55-100 N/m), taken from [6], and the present numerical predictions are compared. Apparently the peak load is better simulated for the values of Gf in the lower rang. Fairprediction for the descending curve is attained for the case of maximum value of fracture energy. It is expected that the methods of nonlinear fracture mechanics (for example the cohesive crack model) would improve the results in both directions.

Some extremely interesting results are plotted in Fig. 9, where the load-crack mouth sliding displacement and load-vertical deflection responses are given. A comparison is made with a FEM solution of Yang [5], where the same example has been numerically analyzed. There is non-coincidence about 15% for the peak load, while good fitting is observed in the descending branch of the diagram – see Fig. 9 (a). Similar conclusions may be drawn for the second graphic, plotted in Fig. 9 (b). What we emphasize on in this case however, is the fact that the snap-back phenomena, reported in the experiment, could be successfully reproduced by the present LEFM numerical approach.

0

20

40

60

80

100

120

140

160

180

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Fig. 8 Experimental envelope and the present numerical prediction for load-crack mouth sliding displacement for the extreme values of fracture energy parameter

Load

F (k

N)

CMSD (mm)

Gf=100 N/m

Gf=55 N/m

Experimental envelope

12

Based on the above numerical experiments, comparisons and analyses of the results, the following conclusions may be written:

Ø The dual boundary element method combined with the two-step subtraction singularity technique, constitute a good, accurate and reliable numerical procedure for finding the real crack path trajectories for beams, made of plain concrete;

Ø A unique feature of the above approach is that we perform the regularization processwith respect to crack tip point. That is exactly the point for which the calculations are correct. That is possible due to the fact, that a discontinuous boundary element is put there and the crack tip is not a nodal point but may be considered internal one;

Ø This technique is based on the principles of the LEFM and the formulated fracture criterion. It turns out being capable of predicting the phenomena of the crack growth in plain concrete, including the case of mixed-mode conditions;

Ø Although the main features of the deformation process and failure are portrayed, it is obvious that the problem is highly nonlinear and the real energy dissipation is not well reproduced. As a result there are some deviations in the critical peak loads.

It is author’s belief that the presented methodology is an important and necessary step for further development of the method. The results obtained for the crack path simulation of plain concrete beams under mixed-mode fracture are promising. They could serve as a good tool and basis for new developments of nonlinear fracture mechanics models.

REFERENCES

[1] BAZANT, Z., J. PLANAS. Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, LLC, 1998.[2] SALEH, A. L. Crack growth in concrete using boundary elements. Computational mechanics publications, Volum 30, Southampton, 1997.[3] PORTELA, A., M. ALIABADI AND D. ROOKE. The dual boundary element method: effective implementation of crack problems. Int. J. Num. Meth. Engng., 33(6), p. 1269-87, 1992. [4] DELLA-VENTURA, D., R.N.L. SMITH. Some application of singular fields in the solution of crack problems. Int. J. Num. Meth. Engng., 42, p. 927-942, 1998.

Crack increment=20 mm (Yang [5])

Crack increment=20 mm (DBEM)

(a) (b)

Fig. 9 (a) F-crack mouth sliding deflection curve; (b) F-load point deflection curve

CMSD (mm)

Load

F (k

N)

Deflection at loading point (mm)Lo

ad F

(kN

)

0

30

60

90

120

150

180

210

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

30

60

90

120

150

180

210

0 0.05 0.1 0.15 0.2 0.25 0.3

13

[5] YANG Z., Fully automatic modeling of mixed-mode crack propagation using scaled boundary element method. Engineering Fracture Mechanics, 73, p. 1711-1731, 2006.[6] ROTS, J., P. NAUTA, G. KUSTERS, J. BLAAUWENDRAAD. Smeared crack approach and fracture localization in concrete. HERON, Vol. 30, No 1, 1985.[7] ROTS, J., J. BLAAUWENDRAAD. Crack models for concrete: Discrete or smeared? Fixed, multi-directional or rotating? HERON, Vol. 34, No 1, 1989.[8] PARVANOVA, S., Dual boundary element method (in Bulgarian). Annual of UACEG, Sofia, Vol. XLII, 2005-2006.[9] PARVANOVA, S., Application of the subtraction singularity approach to dual boundary element method (in Bulgarian). Annual of UACEG, Sofia, Vol. XLIII, 2006-2007.

Acknowledgement . Funding for the research described in this paper was supplied by the National Science Fund under the contract № TH – 1406/04. This support is gratefully acknowledged.

䅐䅀䇐䄀 䇰䈀䇠䉠䅐䅀䈰䈀䄀 䇐䄀 䅀䈰䄀䆰䇐䆀䋰 䇀䅐䈠䇠䅀 䇐䄀 䄰䈀䄀䇐䆀䉰䆀䈠䅐 䅐䆰䅐䇀䅐䇐䈠䆀 䇰䈀䆀䆰䇠䅠䅐䇐䄀 䅰䄀 䈀䄀䅰䈀䈰䊀䅐䇐䆀䅐 䇐䄀 䄐䅐䈠䇠䇐䄀

䈐. 䇰䊠䈀䄠䄀䇐䇠䄠䄀, 䄰. 䄰䇠䈐䇰䇠䅀䆀䇐䇠䄠

䈰䏐䎀䌠䍐䐀䐐䎀䐠䍐䐠 䏰䏠 䄀䐀䑐䎀䐠䍐䎠䐠䐰䐀䌀, 䈐䐠䐀䏠䎀䐠䍐䎰䐐䐠䌠䏠 䎀 䄰䍐䏠䍀䍐䍰䎀䓰䌐䐰䎰. 䉐䐀䎀䐐䐠䏠 䈐䏀䎀䐀䏐䍐䏐䐐䎠䎀 1, 1046, 䈐䏠䑀䎀䓰, 䄐䒠䎰䌰䌀䐀䎀䓰

e-mail: slp_fce@uacg.bg, gospgosp_frf@uacg.bg

䈀䅐䅰䋠䇀䅐. 䄠 䏰䐀䍐䍀䐐䐠䌀䌠䍐䏐䌀䐠䌀 䐐䐠䌀䐠䎀䓰 䍐 䏐䌀䏰䐀䌀䌠䍐䏐䌀 䑰䎀䐐䎰䍐䏐䌀 䐀䍐䌀䎰䎀䍰䌀䑠䎀䓰 䏐䌀 䍀䐰䌀䎰䏐䎀䓰 䏀䍐䐠䏠䍀 䏐䌀 䌰䐀䌀䏐䎀䑰䏐䎀䐠䍐 䍐䎰䍐䏀䍐䏐䐠䎀 (䅀䇀䄰䅐) 䍰䌀 䐀䌀䍰䏰䐀䏠䐐䐠䐀䌀䏐䍐䏐䎀䍐 䏐䌀 䏰䐰䎠䏐䌀䐠䎀䏐䎀 䌠 䌐䍐䐠䏠䏐䌀, 䌠 䎠䏠䏐䐠䍐䎠䐐䐠䌀 䏐䌀 䎰䎀䏐䍐䎐䏐䌀䐠䌀 䏀䍐䑐䌀䏐䎀䎠䌀 䏐䌀 䐀䌀䍰䐀䐰䒀䍐䏐䎀䍐䐠䏠. 䇠䐐䏐䏠䌠䏐䏠䐠䏠 䏰䐀䍐䍀䎀䏀䐐䐠䌠䏠 䏐䌀 䅀䇀䄰䅐 䐐䍐 䐐䒠䐐䐠䏠䎀 䌠䒠䌠 䌠䒠䍰䏀䏠䍠䏐䏠䐐䐠䐠䌀 䍀䌀 䐐䍐 䐠䐀䍐䐠䎀䐀䌀 䐰䐐䏰䍐䒀䏐䏠 䐐䏀䍐䐐䍐䏐䌀䐠䌀 䑀䏠䐀䏀䌀 䏐䌀 䍀䍐䑀䏠䐀䏀䌀䑠䎀䎀 䌠 䐀䌀䏀䎠䎀䐠䍐 䏐䌀 䍐䍀䏐䌀 䏠䌐䎰䌀䐐䐠, 䎠䌀䐠䏠 䏰䏠 䐠䏠䍰䎀 䏐䌀䑰䎀䏐 䐐䍐 䎀䍰䌐䓰䌰䌠䌀 䎀䍰䏰䏠䎰䍰䌠䌀䏐䍐䐠䏠 䏐䌀 䏐䍐䍐䑀䍐䎠䐠䎀䌠䏐䎀䓰 䏀䍐䐠䏠䍀 䏐䌀 䏰䏠䍀䏠䌐䎰䌀䐐䐠䎀䐠䍐. 䇀䍐䐠䏠䍀䒠䐠 䐐䍐 䎠䏠䏀䌐䎀䏐䎀䐀䌀 䐐 䐀䌀䍰䌠䎀䐠䎀䓰 䌠 䏰䐀䍐䍀䎀䒀䏐䎀 䐀䌀䌐䏠䐠䎀 䏰䏠䍀䑐䏠䍀 䏐䌀 “䏠䐠䐐䐠䐀䌀䏐䓰䌠䌀䏐䍐” 䏐䌀 䐐䎀䏐䌰䐰䎰䓰䐀䏐䎀䐠䍐 䏰䏠䎰䍐䐠䌀 (䇀䇠䈐䇰) 䍰䌀 䎀䍰䑰䎀䐐䎰䍐䏐䎀䍐 䏐䌀 䎠䏠䍐䑀䎀䑠䎀䍐䏐䐠䎀䐠䍐 䍰䌀 䎀䏐䐠䍐䏐䍰䎀䌠䏐䏠䐐䐠 䏐䌀 䏐䌀䏰䐀䍐䍠䍐䏐䎀䓰䐠䌀, 䎠䌀䐠䏠 䌠 䏐䌀䐐䐠䏠䓰䒐䌀䐠䌀 䐀䌀䌐䏠䐠䌀 䏀䍐䐠䏠䍀䒠䐠 䍐 䏠䌐䏠䌐䒐䍐䏐 䍰䌀 䐐䎰䐰䑰䌀䓰 䏐䌀 䐐䎀䐐䐠䍐䏀䌀 䏠䐠 䏐䓰䎠䏠䎰䎠䏠 䏰䐰䎠䏐䌀䐠䎀䏐䎀. 䈀䌀䍰䏰䐀䏠䐐䐠䐀䌀䏐䍐䏐䎀䍐䐠䏠 䏐䌀 䏰䐰䎠䏐䌀䐠䎀䏐䎀䐠䍐 䎀䍰䎀䐐䎠䌠䌀 䍀䍐䑀䎀䏐䎀䐀䌀䏐䍐䐠䏠 䏐䌀 䎠䐀䎀䐠䍐䐀䎀䎐, 䎠䏠䎐䐠䏠 䍰䌀䍐䍀䏐䏠 䐐 䅀䇀䄰䅐 䎀 䇀䇠䈐䇰 䑀䏠䐀䏀䎀䐀䌀䐠 䏰䐀䏠䑠䍐䍀䐰䐀䌀 䍰䌀 䐀䍐䒀䌀䌠䌀䏐䍐 䏐䌀 䍰䌀䍀䌀䑰䌀䐠䌀 䌠 䏐䌀䎐-䏠䌐䒐䌀 䏰䏠䐐䐠䌀䏐䏠䌠䎠䌀. 䈀䍐䒀䍐䏐䎀 䐐䌀 䍀䌠䌀 䏰䐀䎀䏀䍐䐀䌀 䍰䌀 䌠䎀䐐䏠䎠䎀 䌰䐀䍐䍀䎀, 䐀䌀䌐䏠䐠䍐䒐䎀 䏐䌀 䐐䐀䓰䍰䌠䌀䏐䍐, 䎠䏠䎀䐠䏠 䏰䐀䎀 䐐䐀䌀䌠䏐䍐䏐䎀䍐䐠䏠 䏐䌀 䐀䍐䍰䐰䎰䐠䌀䐠䎀䐠䍐 䍀䍐䏀䏠䏐䐐䐠䐀䎀䐀䌀䐠 䏠䐠䎰䎀䑰䏐䌀 䐠䏠䑰䏐䏠䐐䐠 䏐䌀 䏀䍐䐠䏠䍀䌀. 䇠䎠䌀䍰䌠䌀 䐐䍐, 䑰䍐 䐠䌀䍰䎀 䑀䏠䐀䏀䐰䎰䎀䐀䏠䌠䎠䌀 䍐 䏀䏐䏠䌰䏠 䏰䏠䍀䑐䏠䍀䓰䒐䌀 䍰䌀 䎀䍰䐐䎰䍐䍀䌠䌀䏐䍐 䏐䌀 䐀䌀䍰䏰䐀䏠䐐䐠䐀䌀䏐䍐䏐䎀䍐 䏐䌀 䏰䐰䎠䏐䌀䐠䎀䏐䎀 䌠 䌐䍐䐠䏠䏐䌀 䏰䐀䎀 䐐䏀䍐䐐䍐䏐䌀 䑀䏠䐀䏀䌀 䏐䌀 䐀䌀䍰䐀䐰䒀䍐䏐䎀䍐.