1 Introduction and formulation of the problem
Abstract

The focus of the article is on the analysis of a semi-infinite crack at the interface between two dissimilar anisotropic elastic materials, loaded by a general asymmetrical system of forces acting on the crack faces. Recently derived symmetric and skew-symmetric weight function matrices are introduced for both plane strain and antiplane shear cracks, and used together with the fundamental reciprocal identity (Betti formula) in order to formulate the elastic fracture problem in terms of singular integral equations relating the applied loading and the resulting crack opening. The proposed compact formulation can be used to solve many problems in linear elastic fracture mechanics (for example various classic crack problems in homogeneous and heterogeneous anisotropic media, as piezoceramics or composite materials). This formulation is also fundamental in many multifield theories, where the elastic problem is coupled with other concurrent physical phenomena.
Keywords: Interfacial crack, Stroh formalism, Weight functions, Betty Identity, Singular integral.

Integral identities for a semi-infinite interfacial crack

in anisotropic elastic bimaterials

L. Morini, A. Piccolroaz, G. Mishuris and E. Radi

Dipartimento di Scienze e Metodi dell’Ingegneria, Universitá di Modena e Reggio Emilia,

Via Amendola 2, 42100, Reggio Emilia, Italy.

Department of Civil, Environmental and Mechanical Engineering, University of Trento,

Via Mesiano 77, 38123, Trento, Italy.

Institute of Mathematical and Physical Sciences, Aberystwyth University,

Ceredigion SY23 3BZ, Wales, U.K.

1 Introduction and formulation of the problem

The method of singular integral equations in linear elasticity was first developed for solving two-dimensional problems, (Mushkelishvili, 1953), and later extended to three-dimensional cases by means of multi-dimensional singular integral operators theory (Kupradze et al., 1979; Mikhlin & Prssdorf, 1980). Singular integral formulations for both two and three-dimensional crack problems have been derived by means of a general approach based on Green’s function method (Weaver, 1977; Budiansky & Rice, 1979; Linkov et al., 1997). As a result, the displacements and the stresses are defined by integral relations involving the Green’s functions, for which explicit expressions are required (Bigoni & Capuani, 2002). Although Green’s functions for many two and three-dimensional crack problems in isotropic and anisotropic elastic materials have been derived (Sinclair & Hirth, 1975; Weaver, 1977; Pan, 2000, 2003; Pan & Yuan, 2000), their utilization in evaluating physical displacements and stress fields on the crack faces implies, especially in the anisotropic case, challenging numerical estimation of integrals which convergence should be asserted carefully. Moreover, the approach based on Green’s function method works when the tractions applied on the discontinuity surface are symmetric, but not in the case of asymmetric loading acting on the crack faces.

Recently, using a procedure based on Betti’s reciprocal theorem and weight functions111Defined by Bueckner (1985) as singular non-trivial solutions of the homogeneous traction-free problem and later derived for general three-dimensional problems by Willis & Movchan (1995), and for interfacial cracks by Gao (1992) and Piccolroaz et al. (2009). an alternative method for deriving integral identities relating the applied loading and the resulting crack opening has been developed for two and three-dimensional semi-infinite interfacial cracks between dissimilar isotropic materials by Piccolroaz & Mishuris (2013). In the two-dimensional case, the obtained identities contain Cauchy type singular operators together with algebraic terms. The algebraic terms vanish in the case of homogeneous materials. This approach avoids the use of the Green’s functions without assuming the load to be symmetric.

The aim of this paper is to derive analogous integral identities for the case of semi-infinite interfacial cracks in anisotropic bimaterials subjected to two-dimensional deformations.

General expressions for symmetric and skew-symmetric weight functions for interfacial cracks in two-dimensional anisotropic bimaterials have been recently derived by Morini et al. (2013) by means of Stroh representation of displacements and fields (Stroh, 1962) combined with a Riemann-Hilbert formulation of the traction problem at the interface (Suo, 1990b). These expressions for the weight functions are used together with the results obtained for isotropic media by Piccolroaz & Mishuris (2013) in order to obtain integral formulation for interfacial cracks problems in anisotropic bimaterial solids with general asymmetric load applied at the crack faces.

We consider a two-dimensional semi-infinite crack between two dissimilar anisotropic elastic materials with asymmetric loading applied to the crack faces, the geometry of the system is shown in Fig.1. Further in the text, we will use the superscripts and to denote the quantities related to the upper and the lower elastic half planes, respectively. The crack is situated along the negative semi-axis . Both in-plane and antiplane stress and deformation, which in fully anisotropic materials are coupled (Ting, 1995), are taken into account. The symmetrical and skew-symmetrical parts of the loading are defined as follows:

(1)

where and denote the loading applied on the upper and lower crack faces, and , respectively (see Fig. 1).

In Section 2 preliminary results needed for the derivation of the integral identities and for the complete explanation of the proposed method are reported. In Section 2.1, the fundamental reciprocal identity and the weight functions, defined as special singular solution of the homogeneous traction-free problem are introduced. In Section 2.2, symmetric and skew-symmetric weight functions matrices for interfacial cracks in anisotropic bimaterials recently derived by Morini et al. (2013) are reported.

Section 3 contains the main results of the paper: integral identities (34), (35), (64) and (65) for two-dimensional crack problems between two dissimilar anisotropic materials are derived and discussed in details. The integral identities are derived for monoclinic-type materials, which are the most general class of anisotropic media where both in-plane and antiplane strain and in-plane and antiplane stress are uncoupled (Ting, 1995, 2000), and the Mode III can be treated separately by Mode I and II. By means of Betti’s formula and weight functions, both antiplane and plane strain fracture problems are formulated in terms of singular integral equations relating the applied loading and the resulting crack opening.

In Section 4, the obtained integral identities are used for studying cracks in monoclinic bimaterials loaded by systems of line forces acting on the crack faces. The proposed examples show that using the identities explicit expressions for crack opening and tractions ahead of the crack tip can be derived for both antiplane and in-plane problems. These simple illustrative cases demonstrate also that the proposed integral formulation is particularly easy to apply and can be very useful especially in the analysis of phenomena where the elastic behaviour of the material is coupled with other physical effects, as for example hydraulic fracturing, where both anisotropy of the geological materials and fluid motion must be taken into account.

Finally, in Appendix A, the Stroh formalism (Stroh, 1962), adopted by Suo (1990b) and Gao et al. (1992) in analysis of interfacial cracks in anisotropic bimaterials and recently used by Morini et al. (2013) for deriving symmetric and skew-symmetric weight functions, is briefly explained. In particular, explicit expressions for Stroh matrices and surface admittance tensor needed in weight functions expressions associated to monoclinic materials are reported.

Figure 1: Two-dimensional semi-infinite interfacial crack loaded by non necessarily symmetric forces applied on the crack faces.

2 Preliminary results

In this section relevant results obtained by several studies regarding interfacial cracks are reported. These results will be used further in the paper in order to develop an integral formulation for the problem of a semi-infinite interfacial crack in anisotropic bimaterials.

In Section 2.1, we introduce the Betti integral formula for a crack in an elastic body subjected to two-dimensional deformations with general asymmetric loading applied at the faces.

In Section 2.2, general matrix equations expressing weight functions in terms of the associate singular traction vectors, recently derived by Morini et al. (2013), and valid for interfacial cracks in a wide range of two-dimensional anisotropic bimaterials are reported.

2.1 The Betti formula

The Betti formula is generally used in linear elasticity in order to relate the physical solution to the weight function which is defined as special singular solution to the homogeneous traction-free problem (Bueckner, 1985; Willis & Movchan, 1995). Since the Betti integral theorem is independent of the specific elastic constitutive relations of the material, it applies to both isotropic and anisotropic media in the same form.

The notations and are introduced to indicate respectively the physical displacements and the traction vector acting on the plane . According to the fact that two-dimensional elastic deformations are here considered, both displacements and stress do not depend on the variable . Nevertheless, since both in-plane and anti-plane strain and stress are considered, non- zero components and are accounted for (Ting, 1995). The notations and are introduced to indicate the weight function, defined by Bueckner (1985) as a non-trivial singular solution of the homogeneous traction-free problem, and the associated traction vector, respectively. As it was shown by Willis & Movchan (1995), the weight function is defined in a different domain respect to physical displacement, where the crack is placed along the positive semi-axis . Following the procedure reported and discussed in Willis & Movchan (1995); Piccolroaz et al. (2009) and Piccolroaz & Mishuris (2013), from the application of the Betti integral formula to the physical fields and to weight functions for both the upper and the lower half-planes in Fig. 1, we obtain:

(2)

where is the rotation matrix:

and are the loading acting on the upper and on the lower crack faces, respectively, and is the physical traction at the interface ahead of the crack tip. The superscript denotes a function whose support is restricted to the positive semi-axis, . In eq. (2), denotes a shift of the weight function within the plane and the dot symbol stands for the scalar product.

Assuming perfect contact conditions at the interface, which implies displacement and traction continuity at the interface ahead of the crack tip, can be defined as follows:

(3)

Similarly, also the traction corresponding to the singular solution satisfies the continuity at the interface:

(4)

Using these definitions, (2) becomes:

(5)

where is the crack opening behind the tip, denotes that its support is restricted to the negative semi-axis, , and is known as the symmetric weight function (Willis & Movchan, 1995; Piccolroaz et al., 2009; Morini et al., 2013):

(6)

By expressing the loading acting on the crack faces in terms of the symmetric and skew-symmetric parts defined by (1), the Betti identity (5) finally becomes:

(7)

where is known as the skew-symmetric weight function (Willis & Movchan, 1995; Piccolroaz et al., 2009; Morini et al., 2013):

(8)

The integral identity (7) can be written in an equivalent form using the convolution respect to , denoted by the symbol (Arfken & Weber, 2005):

(9)

This integral identity relates physical traction and crack opening to weight functions and load applied at the crack faces, and will be used further in the text in order to formulate the interfacial crack problem between dissimilar anisotropic materials in terms of singular integral equations.

Note that, in order to simplify notations, in eq. (9) the scalar product between vectors and is replaced by the “row by column” product between the row vector and the column vector .

2.2 Symmetric and skew-symmetric weight functions for anisotropic bimaterials

Let us introduce the Fourier transform of a generic function with respect to the variable as follows:

(10)

In Morini et al. (2013), the following expressions for the Fourier transform of the singular displacements at the interface between two dissimilar anisotropic media have been derived:

(11)
(12)

where is the Fourier transform of the singular traction at the interface, which in the case of perfect contact condition is defined as in expression (4), and is the Hermitian definite positive surface admittance tensor (Gao et al., 1992), depending on the elastic properties of the materials and defined in details in Appendix A.

The superscripts and , used here and in the sequel, denote functions analytic in the upper and in the lower complex half-planes, respectively

Eqs. (11) and (12) represent general expressions relating the tractions applied on the bounding surfaces and the corresponding displacements , for the upper and lower half-planes, respectively.

The symmetric and skew-symmetric weight function matrices are derived by taking respectively the jump and the average of (Willis & Movchan, 1995; Piccolroaz et al., 2009):

(13)
(14)

Eqs. (13) and (14) can also be expressed in the compact form:

(15)
(16)

where and are the bimaterial matrices defined as follows (Suo, 1990b; Ting, 2000):

(17)
(18)

Expressions (15) and (16) are valid for interfacial cracks in general anisotropic two-dimensional media. Since in anisotropic materials in-plane and antiplane displacements and stresses are generally coupled (Ting, 1996, 2000), for the case of fully anisotropic media three linearly independent vectors and then must be defined for obtaining a complete basis of the singular solutions space. Nevertheless, there are several classes of anisotropic materials where in-plane and antiplane displacements and stress are uncoupled (Ting, 1995, 2000) and then Mode III deformation can be treated separately from Mode I and II as for the case of isotropic media (Piccolroaz et al., 2009, 2010; Piccolroaz & Mishuris, 2013). In the next Section integral identities are derived for interfacial crack between two media belonging to the most general of these classes, known as monoclinic materials.

3 Integral identities

In this Section, following the approach of Piccolroaz & Mishuris (2013), an integral formulation of the problem of a semi-infinite two-dimensional interfacial crack in anisotropic bimaterials is obtained. A particular class of anisotropic materials, where elastic properties are symmetrical with respect to a plane, is considered. These materials are known as monoclinic, and in the case in which the plane of symmetry coincides with both in-plane and antiplane displacements and in-plane and antiplane stress are uncoupled (Ting, 1995). Monoclinic having plane of symmetry at are the most general class of anisotropic materials where stress and strain are decoupled, and they include as subgroups all other classes having this property, such as orthotropic and cubic materials (Horgan & Miller, 1994; Ting, 2000). Explicit expressions for Stroh matrices and surface admittance tensor corresponding to these type of media are reported in Appendix A. These expressions have been used for evaluating bimaterial matrices (17) and (18).

In Sections 3.1 and 3.2, antiplane shear and plane strain interfacial cracks problems in monoclinic bimaterials are formulated in terms of singular integral equations by means of weight function expressions (15) and (16) and Betti integral identity (9).

3.1 Mode III

Considering antiplane deformations in monoclinic materials, as it is shown in Appendix A, constitutive relations reduce to scalar equations relating stresses and to , and then the traction and the displacements derivative for both upper and lower half-plane material become (Suo, 1990b):

(19)
(20)

where . The bimaterial matrices (17) and (18) reduce to:

(21)
(22)

where are elements of the reduced elastic compliance matrix (see Appendix A). According to general expressions (15) and (16), the Fourier transform of symmetric and skew-symmetric weight functions for an antiplane shear crack between two dissimilar monoclinic materials are:

(23)

where the following non-dimensional parameter has been introduced:

(24)

In the case of antiplane deformations, the Betti formula reduces to the scalar equation:

(25)

Applying the Fourier transform with respect to , defined by relation (10), to this identity, we obtain:

(26)

Multiplying both sides of (26) by , we obtain:

(27)

The factors in front of the unknown functions are given by:

(28)

If we apply the inverse Fourier transform to (27), we derive two distinct relationships corresponding to the two cases and :

(29)
(30)

It is important to note that the term cancels from (29) because it is a “” function, while and cancel from (30) because they are “” functions.

To proceed further, we need to evaluate the inverse Fourier transform of the function . Following the procedure illustrated by Piccolroaz & Mishuris (2013), we get:

(31)

Then we can define the singular operator and the orthogonal projectors acting on the real axis:

(32)
(33)

The operator is a singular operator of Cauchy type, and it transforms any function satisfying the Hlder condition into a new function which also satisfies this condition (Mushkelishvili, 1946). The properties of the operator in several functional spaces have been described in details in Prssdorf (1974).

The integral identities (29) and (30) for a Mode III interfacial crack between two dissimilar monoclinic materials become:

(34)
(35)

where is a singular integral operator, and is a compact integral operator (Gakhov & Cherski, 1978; Krein, 1958; Gohberg & Krein, 1958). These two operators look similar, but they are essentially different, in fact: , while , where is some functional space of functions defined on .

For explaining better this point, the integral identities (34) and (35) can be written in the extended form:

(36)
(37)

The integral in (36) is a Cauchy-type singular integral with a moving singularity, whereas the integral in (37) possesses a fixed point singularity (Duduchava, 1976, 1979).

In the case of a homogeneous monoclinic material, the integral identities (34) and (35) simplify, since , and thus there is no influence of the skew-symmetric loading.

Summarizing, the integral identities for Mode III interfacial cracks in monoclinic bimaterials are given by equations (34) and (35). The equation (34) in an invertible singular integral relation between the applied loading , and the corresponding crack opening . The equation (35) is an additional relation through which it is possible to define the behaviour of the solution . Since the operator is compact, it is not invertible, and thus for deriving the traction ahead of the crack tip one needs to evaluate by inversion of the equation (34) (see Mushkelishvili (1946) for details).

3.2 Mode I and II

For plane strain deformations in monoclinic materials, the surface admittance tensor is given by a matrix of the form (Ting, 1995):

(38)

where:

(39)
(40)
(41)

in which and are solutions of the eigenvalue problem associated to balance equations by means of Stroh representation of displacements and stresses (Stroh, 1962; Ting, 1996) (see Appendix A for more details). Since and are eigenvalues with positive imaginary part, is strictly positive, while the positive definiteness of the matrix , and consequently of , implies that (Ting, 1995):

(42)

Thus, bimaterial matrices and for an interfacial crack between dissimilar monoclinic materials under plane strain deformations can be decomposed into real and imaginary parts as follows:

(43)
(44)

where matrices and are defined as:

(45)

Note that and are real positive parameters defined similarly to those introduced by Suo (1990b) for orthotropic bimaterials:

(46)

Regarding matrix , two non-dimensional Dundurs-like parameters are defined (Ting, 1995; Suo, 1990b; Morini et al., 2013):

(47)

while the matrix depends by four non-dimensional Dundurs-like parameters (Ting, 1995; Suo, 1990b; Morini et al., 2013):

(48)
(49)

The Fourier transforms of the symmetric and skew-symmetric weight functions (15) and (16) for a plane monoclinic bimaterial assume the form:

(50)
(51)

Since in plane strain elastic bimaterials Mode I and Mode II are coupled, two linearly independent singular solutions and tractions are needed in order to define a complete basis of the singular solutions space (Piccolroaz et al., 2009). As a consequence, in this case symmetric and skew-symmetric weight functions [U] and , and the associate traction are represented by tensors which may be constructed by ordering the components of each singular solution in columns:

(52)

Correspondingly, the rotation matrix reduces to:

(53)

Applying the Fourier transform to the (9), we obtain:

(54)

Multiplying both sides by , the following identity is derived:

(55)

where and are given by:

(56)

Explicit expressions for these matrices can be computed using symmetric and skew-symmetric weight functions (50) and (51):

(57)
(58)

where and are:

(59)
(60)
(61)

Applying the inverse Fourier transform to the identity (55), for the two cases and , we get:

(62)
(63)

As for the case of antiplane deformations, illustrated in the previous Section, the term in eq. (55) cancels from the (62) because it is a “” function, while cancels from the (63) because it is a “” function.

Using the same inversion procedure of the previous Section the following integral identities for plane strain deformations in monoclinic bimaterials are derived:

(64)
(65)

where matrix operators , and are defined as follows:

(66)
(67)
(68)
(69)

Equations (64) (65), together with the definition of operators (66), (67), (68) and (69), form the system of integral identities for Mode I and II deformations in monoclinic bimaterials. The equation (64) is a system of two coupled singular integral equations, which decouples in the case where the Dundurs parameters and vanish. Observing expression (47), we can note that vanishes in the case of a homogeneous monoclinic material, while is zero only for some particular subclasses of materials, such as for orthotropic materials, where the quantity , defined by (40) and representing the imaginary part of the product of the eigenvalues, vanishes (Suo, 1990a; Gupta et al., 1992). As a consequence, for a homogeneous orthotropic material, the system (64) is reduced to the following decoupled equations: