Microsoft Word - 2297 T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 503 Focussed on “Crack Paths” Microcrack interaction with circular inclusion and interfacial zone Tomas Profant, Miroslav Hrstka Brno University of Technology, Technická 2896/2, 616 69 Brno, Czech Republic profant@fme.vutbr.cz, mira.hrstka@gmail.com Jan Klusák CEITEC IPM, Institute of Physics of Materials; Zižkova 22, 616 62, Brno, Czech Republic klusak@ipm.cz ABSTRACT. A geometrically simplified plane elasticity problem of a finite small crack emanating from a thin interfacial zone surrounding the circular inclusion situated in the finite bounded domain is investigated. The crack is arbitrarily oriented and modelled using the distribution dislocation technique. This model represents the inner solution of the studied problem. The corresponding fundamental solution is based on the application of Muskhelishvili complex potentials in the form of the Laurent series. The coefficients of the series are evaluated from the compatibility conditions along the interfaces of the inclusion, the interfacial zone and the enclosing matrix. The fundamental solution is also used in the solution of the boundary integral method approximating the stress and strain relations of the so-called outer solution. The asymptotic analysis at the point of the crack initiation combines the inner and the outer solution and results in the evaluation of the stress intensity factors of the crack tip, which lies in the matrix. The topological derivative is subsequently used to approximate the energy release rate field associated with the perturbing crack in the matrix. The extreme values of the energy release rate allow one to assess the crack path direction of the initiated microcrack. KEYWORDS. Crack path assessment; Complex potentials; Interfacial zone; Circular inclusion; Fundamental solution; Topological derivative. Citation: Profant, T., Hrstka, M., Klusák, J., Microcrack interaction with circular inclusion and interfacial zone, Frattura ed Integrità Strutturale, 48 (2019) 503-512. Received: 29.11.2018 Accepted: 22.01.2019 Published: 04.01.2019 Copyright: © 2019 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION icrocracking is a phenomenon appearing in the vicinity of some stress rising features, where relatively short cracks rise and grow. An inclusion whose material properties differ from the ones of the surrounding matrix can be taken into account as a potential generator of such stresses [1], [2], [3]. The stress distribution generated by the presence of the inclusion can be influenced by its interfacial zone between the inclusion and the matrix. In spite of M http://www.gruppofrattura.it/VA/48/2297.mp4 T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 504 the idealization of the problem geometry, the analytical mathematical tools can propose an energetic balance at the crack tip initiated in this stress field. The energy consideration is a basic concept widely used not only in the fracture mechanics. The energy release rate (ERR) associated with a finite small and arbitrary oriented crack initiation at the interface between the interfacial zone of the circular inclusion and the matrix is considered to assess the direction of the crack path influenced by the presence of the inclusion and its interfacial zone. The topological derivative field indicates the variation of a response functional when an infinitesimal hole is introduced into the body, where the response functional is the total potential energy of the fractured matrix. As the first, it is introduced the fundamental solution in the form of the unit point force or edge dislocation interacting with the inclusion and the interfacial zone, [4]. The fundamental solution is found in the form of the Laurent series whose coefficients are evaluated from the compatibility conditions between the inclusion, interfacial zone and infinity matrix. The fundamental solution is applied to the boundary integral method and the continuously distributed dislocation method [5]. Under the assumption of a relatively short crack with respect to the inclusion and matrix dimensions, the asymptotic analysis [6] introduces the outer and inner solution of the problem represented by the uncracked finite matrix and cracked infinity matrix, respectively. The asymptotic analysis also demonstrates that the mismatch between the stress values along the outer boundary of the uncracked matrix and the stresses at the same points of the unbounded cracked matrix is proportional to the crack length. The energy momentum tensor [7], [8] and the approximation of the energy release rate for any crack size and orientation by means of a topological derivative can be evaluated from the inner solution and the corresponding stress intensity factors at the crack tip lying in the matrix. Figure 1: The problem formulation. The cracked matrix under the statically equilibrated external load and containing the inclusion with thin zone. The length of the crack is small with respect to the dimension of the inclusion. Figure 2: The point force f and the dislocation with Burgers vector b at the circular inclusion with the interfacial zone. FUNDAMENTAL SOLUTION he derivation of the fundamental solution, see Fig. 2, describing the interaction between the unit point force f or dislocation with Burgers vector b and the circular inclusion with an interfacial zone is based on [4], but contrary to its results it is supplemented by a rigid displacement of the body, because the fundamental solution is also to be T T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 505 applied in the method of the boundary integral equations as the outer solution of the problem. The basis for the expression of the fundamental solution is the complex analysis in the plane elasticity represented by Muskhelishvili’s complex potentials, e.g. [10], which for the matrix 1W can be written in the form (the series of complex potential for domains 2W and 3W can be found in [3], [4]) ( ) ( ) ( ) ( ) ( )1 0 0 ; ; , ; ln , ;s n nn n n n z z h k z z h k zf z f z g z g z g g z ¥ ¥ - - - - = = = + =- - +å å , (1) ( ) ( ) ( ) ( ) ( )1 0 0 ; ; , ; ln , ;s n nn n n n z z p k z k z p k z z z y z y z g z g z g g g z z ¥ ¥ - - - - = = = + =- - + + - å å , (2) where ( )1 1 1 1 , for point force, 2 1 i 1, for dislocation. 1 f k b k k g p k m g k ìïï =- =ïï +ïí ïï = =ïï +ïî and 1m is shear modulus and 1k is Kolosov’s constant of the material of the domain 1W . The functions ( ); s zf z and ( );s zy z are the well-known complex potentials giving at the point iz x y  the response of the point force ix yf f f  or edge dislocation ix yb b b  situated at point iz x h= + in the infinite complex plane. The coefficients  , ;nh k   and  , ;np k   are determined from the conditions of the continuity of displacements and the resultants of the tractions along the interfaces among the matrix 1W , interfacial zone 2W and inclusion, see [3], [4]. ASYMPTOTIC ANALYSIS AND STRESS INTENSITY FACTOR EVALUATION he purpose of using the methods analyzing the path of the crack initiated from interface between the interfacial zone 2W and the matrix 1W follows from the stress composite expansion applied at the point of the crack initiation represented by the parameter a , see Fig. 1. The composite expansion, i.e. an asymptotic expansion with respect to the small parameter e , is introduced under the assumption that the crack eg is small with respect to the dimensions of the inclusion 3W and can be written as follows, see [3], ( ) ( ) ( ) ( ) ( )0( 0 ) ( 0 )ˆ, / ,e e= + +T x T x T x T x (3) where [ ],x yºx ,  ( 0 )T x is the outer stress,  ( 0 ) /T x  is the inner stress and  ( 0 )T̂ x is an error when the members associated with ne , where 0n  , are neglected in the asymptotic expansion of the stress  ,T x  . The outer stress  ( 0 )T x is defined in the crack-free domain 1W as the solution of the following boundary problem ( )( 0 ) 0⋅ =T x for Î Wx (4) ( )( 0 ) P⋅ =T x n t for Î ¶Wx (5) where ( ) ( ) ( ) ( ) 1 2 , 0 for 1, 0 , 0, for 0, 1 P T T ì = ïï=í ï = ïî n t n (6) T T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 506 and 1 2 3W = W ÈW ÈW with outward normal n . The boundary problem (4), (5) and (6) can only be solved using some numerical methods - the boundary element method in this case. This method ensures that the boundary conditions in (6) are fulfilled at discrete points along the boundary of the domain W . On the other hand, the conditions of the stress and displacement compatibility at the interfaces of 1W , 2W and 3W are satisfied automatically via the fundamental solution described in previous section. The fundamental solution also makes it possible to express the boundary conditions for the inner problem represented in (3) by the stress tensor    0 /T x  , which must be independent of the crack length parameter e . The inner problem can be formulated in such a way that a crack of the unit length 1 1e eg g =º is initiated in arbitrary direction along the scaled ˆ ˆ /X x e= -axis from the interface between the unloaded matrix 1 ¥W and the interfacial zone 2W . The domain 1 ¥W is a domain 1W with an outer boundary extended to infinity. The crack faces are loaded with radial and tangential stresses ( )( 0 )Tqq x and ( ) ( 0 ) rT q x from the outer problem (4), (5) and (6), which are generated in the uncracked matrix 1W at points of the crack eg . Thus, ( )( 0 ) 0⋅ =T X for e ¥ÎWX (7) ( )( 0 ) ˆe= ⋅t T X n for 1gÎX (8) ( )( 0 ) 0T X for  ¥X (9) where e ¥W is a cracked domain 1 2 3 ¥ ¥W = W ÈW ÈW . The inner problem is due to the scaled coordinate /X x e= independent of the parameter e . Similarly to [9], the crack in (7), (8) and (9) is modelled using the continuously distributed dislocation technique [3], [5]. If the dislocation with Burgers vector ix yb b b= + is situated at the point 1/z e ¥Z = ÎW , then the stress tensor components YYT and XYT appearing at the point 1/z e ¥= ÎWZ can be written as follows using the potentials (1) and (2), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 0 2 1 i 1 1 ; i ; 2 1 , ; , ; 1 , ; , ; YY XY n n n n n n n n n T T b b b n bh k bh k b n h k bp k m k g g g g ¥ - - - - - - = - - - - - - é -ZêZ + Z = - + + +ê+ -Z -Z-Zêë é+ - Z + Z +êë ùù+ + Z - Z úúûû å Z Z Z Z ZZ Z Z ZZ Z (10) The dislocations will be distributed along the ˆ ˆ /X x e= -axis, i.e. for ˆ ˆ / 0Y y e= = , see Fig. 1. Hence the standard transformations of vectors and tensors with respect to the rotations through angles a and q in XY and ˆ ˆXY planes, respectively, must be applied to the Eqn. (10), see [5]. The singular part represented by the first three expressions in (10) is simply transformed to the Cauchy type singular kernel. The transformed regular part of (10) represented by the infinite sum can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 ˆ ˆ ˆ 3 2 ˆˆ ; ; cos sin ; cos ; cos sin 2 ; sin ; cos sin ; sin sin 2 , XXX XYY XXYXYY YXX YYY YXY H X H H H H H H q q q q q q q q q q X = + - + + + - X Ξ X Ξ X Ξ X Ξ X Ξ X Ξ (11) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 ˆ ˆ ˆ 2 3 ˆˆ ; ; sin ; cos sin ; sin sin 2 ; cos sin ; cos ; cos sin 2 , XXX XYY XXYYYY YXX YYY YXY H X H H H H H H q q q q q q q q q q X =- - + + + + - X Ξ X Ξ X Ξ X Ξ X Ξ X Ξ (12) T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 507 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 ˆ ˆ ˆ 2 2 ˆˆ ; ; cos sin ; cos sin ; cos cos 2 ; cos sin ; cos sin ; cos 2 sin , XXX XYY XXYXXY YXX YYY YXY H X H H H H H H q q q q q q q q q q q q X =- + - - - + - X Ξ X Ξ X Ξ X Ξ X Ξ X Ξ (13) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 ˆ ˆ ˆ 2 2 ˆˆ ; ; cos sin ; cos sin ; cos 2 sin ; cos sin ; cos sin ; cos cos 2 , XXX XYY XXYYXY YXX YYY YXY H X H H H H H H q q q q q q q q q q q q X = - - + - + - X Ξ X Ξ X Ξ X Ξ X Ξ X Ξ (14) where 1 1 1 1 ˆ ˆˆ ˆcos cos , sin sin , cos cos , sin sinR X R X R Ra q a q a q a qé ùé ù= + + = + X + Xê ú ê úë û ë ûX Ξ (15) and ( ) ( ) ( ){ ( ) ( ) ( ) } 1 1 1 2 1 ; , ; , ; 1 , ; , ; , n n XXX n n n n n n n H n h k h k n h k p k g g g g ¥ - - - - - - = - - - - - - = Á Z - Z + + + Z - Z åX Ξ Z Z ZZ Z (16) ( ) ( ) ( ){ ( ) ( ) ( ) } 1 1 1 2 1 ; , ; , ; 1 , ; , ; , n n XYY n n n n n n n H n h k h k p n h k p k g g g g ¥ - - - - - - = - - - - - - = Á Z - Z - - + Z + Z åX Ξ Z Z ZZ Z (17) ( ) ( ) ( ){ ( ) ( ) ( ) } 1 1 1 2 1 ; , ; , ; 1 , ; , ; , n n XXY n n n n n n n H n h k h k n h k p k g g g g ¥ - - - - - - = - - - - - - =  - Z + Z + + + Z - Z åX Ξ Z Z ZZ Z (18) ( ) ( ) ( ){ ( ) ( ) ( ) } 1 1 1 2 1 ; , ; , ; 1 , ; , ; , n n YXX n n n n n n n H n h k h k n h k p k g g g g ¥ - - - - - - = - - - - - - =  Z + Z + + + Z - Z åX Ξ Z Z ZZ Z (19) ( ) ( ) ( ){ ( ) ( ) ( ) } 1 1 1 2 1 ; , ; , ; 1 , ; , ; , n n YYY n n n n n n n H n h k h k n h k p k g g g g ¥ - - - - - - = - - - - - - =  Z + Z - - + Z + Z åX Ξ Z Z ZZ Z (20) ( ) ( ) ( ){ ( ) ( ) ( ) } 1 1 1 2 1 ; , ; , ; 1 , ; , ; . n n YXY n n n n n n n H n h k h k n h k p k g g g g ¥ - - - - - - = - - - - - - = Á Z + Z - - + Z + Z åX Ξ Z Z ZZ Z (21) The symbols { }. and {}.Á mean real and imaginary value of the complex expression. The dislocation is continuously distributed along the ˆ /x e -axis in the interval ( )ˆ 0,1X Î . This process is modelled by introducing the density of the Burgers vector ( )ˆ ˆYB X and ( )ˆ ˆXB X , which is in relation with Burgers vector ( )ˆ ˆ,X Yb b=b as follows ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆd d ˆ ˆ, , ˆ ˆd d Y X Y X b b B B X X X = X = X X (22) T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 508 where ˆ ˆ cos sin , sin cos . X YX X YY b b b b b b q q q q = + = - + (23) Figure 3: The stress tensor component xxT of the fundamental solution ( )* , ; xT x h compared with the FEM solution along the positive x - axis for the unit point force 1f fh= = situated at the point [ ],x h . Figure 4: The stress tensor components xyT of the fundamental solution ( )* , ; xT x h compared with the FEM solution along the positive x - axis for the unit point force 1f fh= = situated at the point [ ],x h . Figure 5: The stress tensor components yyT of the fundamental solution ( )* , ; xT x h compared with the FEM solution along the positive x - axis for the unit point force 1f fh= = situated at the point [ ],x h . Figure 6: The energy release rate Ge as a function of the crack inclination q and the outer stress 1T when 0a =  . T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 509 Figure 7: The energy release rate Ge as a function of the crack inclination q and the crack initiation angle a . Figure 8: The energy release rate Ge as a function of the crack inclination q , the crack initiation angle a and elastic modulus of the interfacial zone 2E . Finally, the boundary conditions (8) and (9) in the inner problem formulation (7), lead to the solution of the following system of the singular integral equations following from the above mentioned transformations of (10) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( 0 ) 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ 1 0 2ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ; ; d ˆˆ1 yy X XYY Y YYY T X B H X B H X X m e p k é ùæ ö÷çê ú= X X + X + X X÷ç ÷çê úè ø+ -Xë û ò , (24) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( 0 ) 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ̂ 1 0 2ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ; ; d ˆˆ1 xy X XXY Y YXY T X B H X B H X X m e p k é ùæ ö÷çê ú= X + X + X X X÷ç ÷çê úè ø+ -Xë û ò (25) and integral equations ensuring the closed crack tips ( ) ( ) 1 1 ˆ ˆ 0 0 ˆ ˆ ˆ ˆd 0, d 0. X Y B BX X = X X =ò ò (26) The outer stresses ( )( 0 )ˆˆ ˆyyT x , ( ) ( 0 ) ˆˆ ˆxyT x are the solution of (4), (5) and (6) at the points ˆx̂ Xe= of the crack eg in the case of the loaded but uncracked matrix 1W and also linearly transformed with respect to the local coordinate system ˆ̂xy . The unknown Burgers vector densities ( )ˆ ˆXB X and ( )ˆ ˆYB X are found in the form ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 , 1 Y Y X X B Bl lc c - - X = X -X X X = X -X X , (27) where the value of the exponent l is different from the value -0.5 due to the mismatch of the materials of the matrix 1W and interfacial zone 2W . The functions ( )ˆ ˆYc X and ( )ˆ ˆXc X are continuous in the interval ( )ˆ 0,1XÎ and approximated by Jacobi polynomials. The stress intensity factors IK and IIK at the crack tip lying in the matrix 1W can be evaluated from the dislocation densities ( )ˆ ˆYB X and ( )ˆ ˆXB X as follows ( ) 1 12 ˆ ˆ, , 1 2 1 1 I II Y X K l m p c k + = + , (28) T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 510 ENERGY RELEASE RATE EVALUATION or the purpose of the assessment of the crack path direction, the knowledge of the stress intensity factors at the tips of the just existing crack is insufficient and the change of the energy state of the unit crack advance is necessary. The shape optimization methods tender an interesting way to eliminate the problem of the complexity of the geometry of the presented problem. The shape functional ( )U eW plays an important role in the following method, because represents the potential energy of the fracture problem (3). The change of the shape functional with respect to the small perturbation de of the crack eg at its tip x̂ e= can be expressed by its asymptotic expansion, see [3], [8] and [9], ( ) ( ) ( ) ( ) ( ) ˆ ˆ , x U U g J x oe de e ede de+ =W = W + + (29) where J is the topological derivative of U on the perturbed domain e de+W . It is supposed that ( )g e is the positive function such that ( ) 0g de  with 0de  . The additional requirements about the remainder of (29) can be found in [8]. In [8] it is also shown the relation between the shape derivative of  U  and the topological one in the sense to the domain variation produced by an uniform expansion of some perturbation. This method, the so-called topological-shape sensitivity method, uses the concept of shape sensitivity analysis as an intermediate step in the topological derivative calculation. From the combination of the topological and shape sensitivity one gets the expression of the energy release rate, [3], [9], ( ) ( ) ( ) 2 2ˆˆ ˆ 0 ˆ ˆ lim dx I IIx x B G x J x S K K E Σ n e de e ee e de e = =  ¶ é ù= =- ⋅ ⋅ = +ë ûò (30) where 1 1/ (1 )E E n= - for plane strain, 1E E= for plane stress, ( )ˆJ x is a derivative of  ˆJ x with respect to de , Σe is the Eshelby’s momentum tensor [7], x̂e is a basis vector of x̂ -axis and n is an unit normal to the boundary Bde¶ of the stress-free circular shaped perturbation Bde at the crack tip x̂ e= . NUMERICAL RESULTS correct description of the stress field surrounding the inclusion 3W with the interfacial zone 2W situated in the matrix 1W is conditioned by the fundamental solution (1) behavior at the inclusion, interfacial zone and matrix interfaces. The Figs. 3-5 show a numerical example of the convergence and the fulfilling of the compatibility conditions of the stress component xxT , xyT and yyT of the stress tensor   * , ;T x  along the positive x -axis ( [ ], 0x=x , 0x > ) generated by the fundamental solution (1) for the domain 1W and its additional forms for the domain 2W and 3W along their interfaces. The applied values of elastic moduli, geometry and degree of Laurent series are also added. The unit point force f is parallel with respect to y hº -axis and situated at a point [ ],x h . The graphs show the matching of the Laurent approximation of the fundamental solution and the FEM solution along the positive x -axis for series degree  10n (especially in the interval   10,11x mm, where the interfacial zone is situated). In the next numerical example, see Fig. 6, the case of the assessment of the crack path direction 0q of the crack initiated from the point on the interface between the matrix and interfacial zone corresponding to 0a =  is considered. The elastic moduli of the used materials ( )1 1,E n , ( )2 2,E n , ( )3 3,E n , the finite crack size e , geometry 1R , 2R and degree of Laurent series are also given in the graph. The graph shows the dependence of the corresponding energy release rate Ge on q , where the external load 1T varies from zero value to the negative value of 2T to model the shear loading of the cracked matrix. The crack initiation direction corresponding to the local maximum value of Ge is of course fairly F A T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 511 influenced by the sheer stresses and mode II loading. It is worth mentioning that the maximum value of the angle 50q  is due to the value of the stress singularity exponent l , which becomes complex for the 50q >  . The complex value of l requires different approach of the stress field description. The graph also shows that for some combination of the external stresses 1T getting some specific value from the range ( )1 80, 60T Î - - MPa it is difficult to assess the crack path direction, because the dependence of the energy release rate Ge is rather flat without the clear local extremal value. The Fig. 7 shows the position of the crack initiation at the interface between the interfacial zone and matrix represented by the angle a and the crack inclination q . The elastic moduli of the used materials ( )1 1,E n , ( )2 2,E n , ( )3 3,E n , the finite crack size e , geometry 1R , 2R and degree of Laurent series are again given in the graph. The external load 1T is zero in this case, while 2 100 MPaT  . The local maximum value of Ge is for 40a  and the crack path direction 0q is near the zero value, i.e. the crack growth under the mode I loading. Similarly to the previous example, the maximum values of Ge are limited due to the real values of the stress singularity exponent l . The influence of the interfacial zone on the crack initiation position and potentially crack growing direction shows Fig. 8. The width of the interfacial zone 2W plays an unimportant role in the crack behavior and it is not illustrated there. On the other hand, the material properties of 2W influence the point of the crack initiation as well as the crack path direction. The grey curves represent the material of the interfacial zone with elastic moduli similar to those of the stiffer particle 3W , the black ones similar to the material softer then the matrix 1W . On can see that the position of the point of the crack initiation represented by the angle a tends to the lower values for the stiffer material of 2W and the crack path direction 0q moves to the positive values. CONCLUSIONS he energy release rate associated with the arbitrary oriented finite small crack initiating at the interface between the interfacial zone of the circular inclusion and the matrix was evaluated. The scheme of the energy release rate evaluation was adopted from [9] but modified to the presented problem. The proposed procedure combines traditional as well as modern analytical approaches with a numerical one. The most important part of the analysis is the establishing of the fundamental solution given by the interaction between the edge dislocation or point force and circular inclusion with an interfacial zone. A numerical example of the convergence of the approximating series of the fundamental solution was shown. The asymptotic analysis, where the crack perturbs the matrix only and the crack has to have a finite small length, allows one to apply the fundamental solution to model as the cracked infinity as the uncrack finite loaded matrix containing the inclusion. The numerical results showed the influence of the external loads on the crack path direction but also the position of its point of initiation along the interface between the interfacial zone and matrix. The influence of the material properties of the interfacial zone on the cracking of the matrix was also shown. Even that the character of the presented results is theoretical only and their validation is desirable, the applied procedure offers an interesting tools to solve the fracture problems of the composites with sound mathematical background. ACKNOWLEDGMENTS his research has been financially supported by the Czech Science Foundation through the Grant 16-18702S and by the Ministry of Education, Youth and Sports of the Czech Republic under the project CEITEC 2020 [LQ1601]. REFERENCES [1] Krepl, O., Klusák, J. (2017). The influence of non-singular terms on the precision of stress description near a sharp material inclusion tip, Theor. Appl. Fract. Mech., 90, pp. 85–99, DOI: 10.1016/J.TAFMEC.2017.03.007. [2] Krepl, O., Klusák, J. (2016). Reconstruction of a 2D stress field around the tip of a sharp material inclusion, Procedia Struct. Integr., 2, pp. 1920–7, DOI: 10.1016/J.PROSTR.2016.06.241. T T T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 512 [3] Profant, T., Hrstka, M., Klusák, J. (2019). An asymptotic analysis of crack initiation from an interfacial zone surrounding the circular inclusion, Compos. Struct., 208, pp. 479–97, DOI: 10.1016/J.COMPSTRUCT.2018.10.020. 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