Non-singular antiplane fracture theory within nonlocal anisotropic elasticity

Materials and Design 88 (2015) 854–861

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Non-singular antiplane fracture theory within nonlocal anisotropic elasticity S. Mahmoud Mousavi a,⁎, Alexander M. Korsunsky b a b

Department of Civil and Structural Engineering, Box 12100, Aalto University, FI-00076 Aalto, Finland Multi-Beam Laboratory for Engineering Microscopy, Department of Engineering Science, University of Oxford, OX1 3PJ, UK

a r t i c l e

i n f o

Article history: Received 14 July 2015 Received in revised form 11 September 2015 Accepted 12 September 2015 Available online 15 September 2015 Keywords: Cracks Anisotropy Fracture mechanics Dislocations Nonlocal elasticity Integral equations

a b s t r a c t In the present paper, the distributed dislocation technique is applied for the analysis of anisotropic materials weakened by cracks. Eringen's theory of nonlocal elasticity of Helmholtz type is employed. The non-singular screw dislocation within anisotropic elasticity is distributed to model cracks of mode III. The corresponding dislocation density functions are evaluated using the proper crack-face boundary conditions. The nonlocal stress field within a plane weakened by cracks is determined. The crack opening displacement is also discussed within the framework of nonlocal elasticity. The stress singularity of the classical linear elasticity is removed by the introduction of the nonlocal theory of elasticity. The general anisotropic case and the special case of orthotropic material are studied. The effect of material orthotropy is presented for a crack which is not necessarily aligned with the principal orthotropy direction. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Simulation of fracture is a challenging issue in solid mechanics and is generally addressed based on either the continuum-based methods or the discontinuous approaches. Avoiding singularity related issues when dealing with discontinuity is the main motivation in developing new methods and frameworks. The discontinuous approaches, such as lattice models [7,8,29], peridynamics [31] and molecular dynamics [28] are efficient in the presence of discontinuities such as cracks or interfaces. In fact, these discontinuous approaches are nonlocal models accounting for the effects of long-range forces having similar discrete computational structures. Peridynamics model can be cast as an upscaling of molecular dynamics [30,34]. Here, in order to study fracture mechanics, we are particularly interested in continuum-based methods. The classical continuum mechanics lacks any internal length scale and cannot capture size effects. This provides the motivation to extend the classical theory towards generalized continuum mechanics which enrich the classical elasticity with the capability to describe size effects by incorporating internal length scales in the model. The internal length scale originates from the inner structure of the material. Nonlocal elasticity is an extension of the classical elasticity which offers an appropriate framework for studying long⁎ Corresponding author. E-mail addresses: mahmoud.mousavi@aalto.fi (S.M. Mousavi), [email protected] (A.M. Korsunsky).

http://dx.doi.org/10.1016/j.matdes.2015.09.068 0264-1275/© 2015 Elsevier Ltd. All rights reserved.

range (nonlocal) interactions [12]. Within this framework, various nonlocal kernels have been proposed. Among those is Eringen's theory of nonlocal elasticity [12] which leads to a Helmholtz-type governing equation. In this framework, a nonlocal stress field is introduced while no nonlocal strain appears and the strain and the displacement fields are identical to those in classical elasticity. Gradient elasticity is another extension of classical continuum mechanics in which the strain energy depends on the elastic strain and the strain gradients [23] and can be related to nonlocal elasticity [1]. Dislocations are a subject of common interest in different disciplines in solid mechanics. Within fracture mechanics, crystal dislocations play a role of the fundamental object that governs the physics of plastic deformation and fracture, while they can also be macroscopically treated within continuum mechanics as basic elementary objects involved in mathematical fracture modeling [5]. In other words, the dislocation solution (that falls into a broader class of continuum elasticity solutions called the nuclei of strain, or eigenstrains) [6] may be considered as the fundamental building block (kernel) used in constructing an integral equation formulation of a crack problem. An arbitrary configuration of cracks can be modeled by distributing the dislocations. The distributed dislocation technique (DDT) has been extensively applied within classical elasticity [4,13,17,25,33]. In line with classical fracture mechanics, DDT gives rise to the classical singular solutions for cracks. This is due to the singular stress fields of dislocations within classical elasticity. Interestingly, generalized continua models such as nonlocal and gradient elasticity provide nonsingular solutions to dislocations [20]. This motivates the extension of the DDT towards generalized elasticity to formulate a nonsingular fracture theory.

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

Recently, the distributed dislocation technique has been employed to investigate defects within generalized continua. An extension of the classical DDT towards couple stress elasticity was given by Gourgiotis and Georgiadis [16]. However, the calculated stress fields for cracks of mode I, mode II and mode III in couple-stress elasticity are more singular than in the classical elasticity. Other candidate theories for the generalization of DDT towards generalized elasticity are gradient elasticity and nonlocal elasticity. Within nonlocal elasticity and without using DDT, Eringen [11,12] found a nonsingular stress solution of a mode III crack. Mousavi and Lazar [26] formulated the distributed dislocation technique for cracks based on nonsingular dislocations in Eringen nonlocal isotropic elasticity (of Helmholtz type). They studied cracks of modes I, II and III and reported finite stress fields for cracks within nonlocal elasticity. Mousavi et al. [27] also generalized the classical DDT to the framework of (first and second) gradient elasticity, in which the stress fields of dislocations are nonsingular. However, they considered only the classical boundary conditions and reported the nonsingular stress fields of single crack as well as multiple cracks of mode III. Later, Mousavi and Aifantis [24] generalized this solution considering the non-standard boundary conditions of gradient elasticity of Helmholtz type (Gradela). In this paper, we employ the DDT to analyze cracks within nonlocal anisotropic elasticity. Such general form of anisotropy is also simplified for an orthotropic material. The effect of material properties as well as crack orientation is studied and corresponding nonlocal stress fields are illustrated. This paper is organized as follows. In Section 2, the solution for the screw dislocation in nonlocal elasticity of Helmholtz type is reviewed from the literature. In Section 3, the distributed dislocation technique is employed to construct the integral equations for representing the cracks of mode III. Numerical examples are presented in Section 4. In Section 5, the conclusions are given. 2. Screw dislocations in nonlocal anisotropic elasticity of Helmholtz type In this section, we present the framework of nonlocal anisotropic elasticity of Helmholtz type, and within this framework, the solution for the screw dislocation.

855

summation. In the absence of body forces, the nonlocal stress tensor satisfies the equilibrium condition t ij; j ¼ 0;

ð5Þ

which means that the stress is self-equilibrated. The natural boundary condition reads t ij n j ¼ ^t i ;

ð6Þ

where nj and ^t i represent the vector normal to the external boundary and the prescribed boundary tractions, respectively. In this formulation of nonlocal elasticity, no nonlocal strain appears. One can consider the differential operator L as an elliptic differential operator of the generalized Helmholtz-type. For two-dimensional problems and in particular, for anti-plane strain problems, it reduces to [19] 2

2

2

L ¼ 1−L11 ∂xx −L12 ∂xy −L22 ∂yy ;

ð7Þ

while L11, L12 and L22 are the three parameters incorporating the length scale effects of anisotropy for two-dimensional problems. By applying the differential operator L to Eq. (1), we obtain the following differential equation for tij Lt ij ¼ σ ij :

ð8Þ

Considering Eqs. (5) and (8) and with L being the Green function of a linear differential operator (Eq. (7)), we have σ ij; j ¼ 0:

ð9Þ

Consequently, both nonlocal and classical stress tensors fulfill the equilibrium Eqs. (5) and (9). In view of Eq. (2), the nonlocal kernel of the nonlocal anisotropic elasticity in 2D reads [19] α ðjx−x0 jÞ ¼

1 K0 2πc

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  L22 ðx−x0 Þ2 −2L12 ðx−x0 Þðy−y0 Þ þ L11 ðy−y0 Þ2 =c ;

ð10Þ

2.1. Nonlocal anisotropic elasticity of Helmholtz type In the theory of nonlocal anisotropic elasticity (e.g., [10,12]), the nonlocal stress tensor tij is defined at any point x of the analyzed domain of volume V as the following convolution integral: Z t ij ðxÞ ¼

V

α ðjx−x0 jÞσ ij ðx0 Þ dV ðx0 Þ ;

α ðjx−x0 jÞ ¼

ð2Þ

In Eq. (1), σij is the stress tensor of classical anisotropic elasticity related to the local classical elastic strain tensor eij at point x′ ∈ V as σ ij ðx0 Þ ¼ cijkl eij ðx0 Þ ;

ð3Þ

where cijkl is the tensor of the local elastic moduli. The classical elastic strain tensor is the symmetric part of the classical elastic distortion tensor eij ¼

 1 β þ βji : 2 ij

ð4Þ

In the presence of dislocations, both tensors (eij and βij ) are incompatible. We employ a comma to indicate the partial derivative with respect to rectilinear coordinates, and repeated indices imply

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L11 L22 −L212 . Here K0 is the modi-

fied Bessel function of the second kind and order n = 0. In the isotropic limit, we have L11 = L22 = ‘2 and L12 = 0, and the isotropic twodimensional nonlocal kernel reduces to [3,10,12,21]

ð1Þ

where α(| x − x′| ) is a nonlocal kernel, and here is assumed to be the Green function of a linear differential operator L, i.e. [19] Lα ðjx−x0 jÞ ¼ δðx−x0 Þ:

while x = (x, y), x′ = (x′, y′) and c ¼

1 K0 2π‘2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðx−x0 Þ2 þ ðy−y0 Þ2 =‘ ;

ð11Þ

where ‘ is the characteristic material length. 2.2. Screw dislocation in the classical anisotropic elasticity A straight screw dislocation is considered whose line coincides with the z-axis of a Cartesian coordinate system in an infinitely extended medium. In this case, the problem reduces to an anti-plane strain case. In classical elasticity, the classical stress reads σ zx ¼

bz C e c45 x−c55 y bz C e c44 x−c45 y ; σ zy ¼ : 2π c44 x2 −2c45 xy þ c55 y2 2π c44 x2 −2c45 xy þ c55 y2 ð12Þ

where bz is the Burgers vector of the screw dislocation, and c44 = c3232, c45 = c3231 and c55 = c3131 are the elastic constants in the Voigt notaqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion. The parameter C e ¼ c44 c55 −c245 is known as the energy

856

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

coefficient [18]. Due to the positive definiteness of the strain energy [18], we have Ce N 0 and c44 N 0. 2.3. Screw dislocation in nonlocal anisotropic elasticity of Helmholtz type Within nonlocal anisotropic elasticity, the selection of the length scales (L11, L12 and L22) is not arbitrary. Different selections result in different nonlocal stress fields. However, in order to obtain analytical solution for the screw dislocation, simplicity in the constitutive structure of the nonlocality is required. The accuracy of such assumption can to be validated with experimental observation. It is assumed that the length scales L11, L12 and L22 depend on the local elastic constants of the antiplane strain problem (c44, c45 and c55) and the characteristic material length ‘. If these length scales are considered as L11 ¼

c44 c55 C 2e

‘2 ; L12 ¼

c44 c45 C 2e

‘2 ;

L22 ¼

c244 C 2e

‘2 ;

ð13Þ

using the nonlocal kernel (Eq. (10)), the nonlocal stress (Eq. (1)) reads [19] t zx ¼

bz C e c45 x−c55 y 2π c44 x2 −2c45 xy þ c55 y2 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19 < c44 x2 −2c45 xy þ c55 y2 @ c44 x2 −2c45 xy þ c55 y2 A= ; K1  1− ; : c44 ‘2 c44 ‘2 ð14aÞ

t zy

bz C e c44 x−c45 y ¼ 2π c44 x2 −2c45 xy þ c55 y2 8 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < c44 x2 −2c45 xy þ c55 y2 c44 x2 −2c45 xy þ c55 y2 A= ; K1@  1− ; : c44 ‘2 c44 ‘2 ð14bÞ

where K1 is the modified Bessel function of order 1. In contrast to the classical stress (Eq. (12)) with 1/r-singularity at the dislocation core, the nonlocal stress tensor (Eq. (14)) is no longer singular. In fact, since limξ → 0ξK1(ξ) = 1 + O(ξ2), the modified Bessel function K1 in Eq. (14) regularizes the classical 1/r-singularity of the dislocation. In the classical limit ‘ → 0, the solutions (Eq. (14)) reduce to the classical case (Eq. (12)). A nonlocally elastic plane assumed to contain a screw dislocation located at a point with coordinates (η, ζ), the nonlocal stress field within the plane may be deduced from Eq. (14) by replacing (x,y) with (x − η , y − ζ). These nonlocal dislocation stresses serve as the kernel of the crack integral equation. 3. Distributed dislocation technique in nonlocal anisotropic elasticity of Helmholtz type The dislocation solution may be considered fundamental in constructing an integral equation formulation of a crack problem. An arbitrary configuration of cracks can be modeled by distributing the dislocations. The distributed dislocation technique (DDT) has been extensively used in classical elasticity [17,33]. Within classical fracture mechanics, the elastic solution for a crack pffiffiffi contains the 1= r-singularity at the crack tips. This arises as the consequence of the 1/r-singularity that exists at the classical dislocation core: the DDT formulation based on the singular dislocation solution leads to the corresponding singularity in the kernel of the integral equation for the equilibrium of a crack. Recently this technique has been used within nonlocal elasticity. Mousavi and Lazar [26] generalized the distributed dislocation technique towards nonlocal isotropic elasticity of Helmholtz type. They

reported finite stress fields for cracks within nonlocal elasticity. In extension of that work, here, the DDT is applied to the nonlocal anisotropic elasticity of Helmholtz type. It is formulated to analyze cracks of mode III. For the detail of the formulation of distributed dislocation technique within nonlocal elasticity, see [26]. Application of the DDT in nonlocal elasticity can be interpreted in two approaches. The first approach is to evaluate the classical stress using DDT, and then determine the nonlocal stress field using (Eq. (1)), i.e. a convolution with kernel α. The second approach is that we define a socalled nonlocal dislocation density and apply it to the nonlocal stress of screw dislocations. Since the nonlocal stress field of the screw dislocation is available [19], we follow the second approach to evaluate the nonlocal stress field which is more convenient. On the other hand, since no nonlocal displacement and strain fields are defined, we can simply use the classical dislocation density to evaluate the displacement and strain fields. Consider a plane weakened by a curved crack described in parametric form as x ¼ α ðsÞ; −1 b s b 1 :

ð15aÞ

y ¼ βðsÞ;

ð15bÞ

−1 b s b 1:

The local orthogonal coordinate system (t,n) is chosen such that the origin moves along the crack while the t-axis remains tangent to the crack surface and the n-axis is perpendicular to it. The anti-plane traction on the surface of the crack in terms of the nonlocal stress components in the Cartesian coordinates (x,y) becomes t nz ðx; yÞ ¼ t yz cosðθÞ−t xz sinðθÞ;

ð16Þ

where   θðsÞ ¼ arctan β0 ðsÞ=α 0 ðsÞ

ð17Þ

is the angle between x- and t-axes, and prime denotes differentiation with respect to the argument. A crack is represented by an unknown continuous distribution of dislocations. Suppose that dislocations with dislocation density Bz(ξ) are distributed over an infinitesimal segment dl = m(ξ)dξ along the crack, while mðξÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 ðα 0 ðξÞÞ þ β0 ðξÞ ; −1 b ξ b 1:

ð18Þ

Employing the principle of superposition, the anti-plane traction at the crack line due to the presence of the above-mentioned distribution of dislocations is Z t nz ðα ðsÞ; βðsÞÞ ¼

1 −1

mðξÞK z ðs; ξÞBz ðξÞ dξ;

ð19Þ

in which the kernel of the integral is K z ðs; ξÞ ¼ K z ðα ðsÞ; βðsÞ; α ðξÞ; βðξÞÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( Ce c44 x−c45 y c44 x2 −2c45 xy þ c55 y2 1− ¼ 2 2 2π c44 x −2c45 xy þ c55 y c44 ‘2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1) c44 x2 −2c45 xy þ c55 y2 A  K1@ cosðθðsÞÞ c44 ‘2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( Ce c45 x−c55 y c44 x2 −2c45 xy þ c55 y2 1− − 2 2 2π c44 x −2c45 xy þ c55 y c44 ‘2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1) c44 x2 −2c45 xy þ c55 y2 A  K1@ sinðθðsÞÞ; c44 ‘2 ð20Þ

where x = α(s) −α(ξ), y= β(s) − β(ξ). As described in the previous section, the modified Bessel function K1 regularizes the kernels (Eq. (20)).

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

Thus the integral Eq. (19) is non-singular. For an isotropic material (Ce =c44 = c55 = μ,c45 = 0), the kernel (Eq. (20)) is simplified to [26] K z ðs; ξÞ ¼

μ xn r r o μ yn r r o 1− K 1 1− K 1 cosðθðsÞÞ þ sinðθðsÞÞ: 2 2 2π r ‘ ‘ 2π r ‘ ‘ ð21Þ

where r 2 = x2 + y2. Consider a plane with the far-field uniform loading t ∞yz ¼ t yz0 ;

t ∞xz ¼ 0:

ð22Þ

According to Bueckner's principle [6], the solution of the crack problem can be obtained by the superposition of two constituent problems [17]. The first problem deals with the stresses arising in an un-cracked body under remote shear field, and the second one deals with the stresses within an un-cracked body, due to the application of equal and opposite tractions present along the line of the crack in the first problem, but in the absence of remote loading. This will enforce the traction-free condition along the crack-face. Now by inducing the stress field arising in the second problem via distribution of dislocations, the equations for determining the unknown dislocation density can be written down. In the first problem, the un-cracked body under loading (Eq. (22)) is in a state of anti-plane shear with following uniform stress field t yz ðx; yÞ ¼ t yz0 ;

t xz ðx; yÞ ¼ 0:

ð23Þ

Therefore, the traction at the location of the crack in the un-cracked plane reads t nz ðx; yÞ ¼ t yz0 cosðθÞ:

ð24Þ

Considering the second problem and using the Bueckner's superposition principle [6], the left-hand side of the integral Eq. (19) is identical to the traction in Eq. (24) with the opposite sign, Z −t yz0 cosðθÞ ¼

1 −1

mðξÞK z ðs; ξÞBz ðξÞ dξ:

ð25Þ

1 −1

mðξÞBz ðξÞ dξ ¼ 0

fracture behavior of zirconia has been analyzed in various investigations recently [32,35]. This material and its alloys are used in applications ranging as widely as oxygen sensors, solid oxide fuel cells, aerospace thermal barrier coatings, prosthodontics, and diamond imitation jewelry. At room temperature the thermodynamically stable phase of pure zirconia possesses monoclinic crystal structure. A peculiar property of zirconia doped with yttria or ceria is the formation of a metastable tetragonal phase normally present at elevated temperatures, that may transform to monoclinic structure under applied stress. Since this transformation is accompanied by the increase in the volume of a unit cell, this leads to the reduction in the driving force for fracture, leading to the phenomenon known as transformation toughening [14]. We choose monoclinic zirconia with diad (twofold) symmetry parallel to the z-axis as the material for our study of antiplane fracture. The relevant stiffness parameters of this material are [22,36]: c44 ¼ 100  109 N=m2 ; c55 ¼ 81  109 N=m2 ; c45 ¼ −23  109 N=m2 :

ð27Þ Here, the singular and non-singular integral equations are solved by the method of singular value decomposition. For details of this method, see Golub and Van Loan [15]. The comparison of the Gauss–Chebyshev quadrature scheme [9] and the singular value decomposition method demonstrates that the two techniques work equally well and are in agreement [27]. In what follows, in order to solve the integral equations, the crack line is discretized into 100 segments. Consider a plane weakened by a crack of length 2a inclined by the angle θ0 to the x-axis under uniform anti-plane shear loading (Eq. (22)) (Fig. 1). The crack faces are assumed to be traction free and the boundary conditions along the crack faces are t nz ¼ 0 on crack line:

ð28aÞ

t ∞yz →t yz0 ; t ∞xz →0;

ð28bÞ

at r→∞:

The geometric configuration of the crack is described by

The geometric crack closure requirement, i.e., Z

857

ð26Þ

x1 ¼ α 1 ðsÞ ¼ as cosðθ0 Þ;

−1 b s b 1 :

y1 ¼ β1 ðsÞ ¼ as sinðθ0 Þ; −1 b s b 1:

should be applied to the integral equation of embedded crack to ensure single-valued displacement field around the crack. The dislocation density function is the unknown scalar-valued line distribution function which can be obtained by solving the integral Eq. (25) together with the closure condition (26). Once the dislocation density function Bz(ξ) is determined, the nonlocal stress distribution in the plane can be evaluated. The solution derived using DDT satisfies the equilibrium equations and incompatibility and boundary conditions [26]. 4. Numerical results Here, we apply the distributed dislocation technique for the analysis of material weakened by a crack. General anisotropic as well as orthotropic materials are studied within the nonlocal elasticity of Helmholtz type. 4.1. Anisotropic elastic plane weakened by an inclined crack In this section, the numerical results are provided for the analysis of an anisotropic medium, where the xy-plane is one of the planes of crystal symmetry. Zirconia is an example of extremely interesting ceramic material that has been extensively studied due to its remarkable functional, structural and mechanical properties [22]. In particular, the

Fig. 1. Plane weakened by an inclined crack.

ð29aÞ ð29bÞ

858

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

In this case, the integral Eq. (25) reduces to Z −t yz0 cosðθ0 Þ ¼

1 −1

aK z ðs; ξÞBz ðξÞ dξ

ð30Þ

where the kernel K z(s, ξ) is obtained from Eq. (20) after employing θ = θ0, x = a(s − ξ) cos (θ0) and y = a(s − ξ) sin (θ0). In classical elasticity, the integral Eq. (30) reduces to the classical singular integral equation −

2π t yz0 cosðθ0 Þ ¼ aC e

Z

1

(

c44 x−c45 y ð31Þ cosðθ0 Þ c44 x2 −2c45 xy þ c55 y2 ) c45 x−c55 y sinðθ0 Þ Bz ðξÞ dξ s; − c44 x2 −2c45 xy þ c55 y2 −1

where x = a(s − ξ) cos (θ0) and y =a(s − ξ) sin (θ0). Integral Eq. (30) must be solved to evaluate the dislocation density function Bz(ξ) in nonlocal elasticity. For embedded cracks, the displacement field is single-valued away from the crack line and there are no net dislocations over the interval −1 b ξ b 1. Hence, in order to determine the dislocation density functions, the integral Eqs. (31) and (30) are solved together with the closure requirement (Eq. (26)). In order to study the effect of the characteristic material length ‘, a specific inclined crack with θ0 = π/6 (Fig. 1) is considered in an anisotropic plane with elastic constants (Eq. (27)). For definiteness, it is assumed that the plane is under loading σyz0 = tyz0 = Ce. Fig. 2 depicts the dislocation density Bz obtained by solving integral Eqs. (26) and (30) for different values of the internal length. According to Fig. 2, the dislocation density in both classical and nonlocal anisotropic elasticity possess singularity at the crack tips. However, the sign of this singularity in nonlocal anisotropic elasticity is opposite the classical case. In other words, in the vicinity of the crack tips, the nonlocal effect is dominant even for very small characteristic length ℓ. Away from the crack tips, the effect of nonlocality vanishes and the results for nonlocal elasticity approach those in classical elasticity. This effect has also been reported for isotropic materials [26]. In the case of a horizontal crack (θ0 = 0), the crack opening displacement (COD) reads g ðxÞ ¼ 2wcrack ðx; 0Þ:

ð32Þ

In the form of nonlocal elasticity chosen in this study, no nonlocal elastic strain is defined: the strain is identical to the classical strain

(Eq. (3)). In other words, the elastic strain that arises in this nonlocal elasticity formulation is related to classical stress through the classical constitutive equations, i.e. Hooke's law (Eq. (3)). Only once the classical stress is computed in this way, its nonlocal counterpart can be evaluated. Consequently, it is not meaningful to consider Hooke's law as the underlying relationship between strain and the nonlocal stress tensor. For the same reason, evaluating strain using nonlocal dislocation density is not a correct operation. Instead, the nonlocal dislocation density should be used to determine the nonlocal stress field (Eq. (1)). In order to satisfyEq. (3), we need to use the classical dislocation density to evaluate the strain tensor. Consequently, the displacement field within nonlocal elasticity (wcrack) is identical to the classical displacement field ). (wcrack 0 Assuming the screw dislocation with branch cut along the negative y-axis, the solution for the classical as well as nonlocal displacement field read w0 ðx; 0Þ ¼ wðx; 0Þ ¼ −

bz sgnðxÞ: 4

ð33Þ

The distribution of dislocations results in the displacement field ðx; 0Þ ¼ wcrack ðx; 0Þ ¼ − wcrack 0

1 4

Z

1 −1

sgnðx−aξÞBclz ðξÞdξ;

ð34Þ

where Bcl z denotes the classical deislocation density. Substituting the displacement field Eq. (34) into Eq. (32) gives the COD within classical as well as nonlocal elasticity g ðxÞ ¼ −

1 2

Z

1 −1

sgnðx−aξÞBcl z ðξÞdξ:

ð35Þ

The crack opening displacement (Eq. (35)) may be simplified to [17] Z g ðxÞ ¼ −

x −a

Bclz ðxÞdx:

ð36Þ

In other words, one may use the classical dislocation density to determine the crack opening displacement in nonlocal elasticity. Consequently, the difference of the classical and nonlocal dislocation density (Fig. 2) does not influence the crack opening profile. Once the dislocation density function Bz(ξ) is determined, employing the superposition principle, the expression for the nonlocal stress distribution in the plane weakened by an inclined crack under uniform anti-plane shear loading reads t xz ðX; Y Þ ¼

Z 1 aC e c45 x−c55 y  2 2 2π −1 c44 x −2c45 xy þ c55 y 8 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < c44 x2 −2c45 xy þ c55 y2 c44 x2 −2c45 xy þ c55 y2 A= B ðξÞ dξ  1− K1@ ; z : c44 ‘2 c44 ‘2

ð37aÞ Z 1 aC e c44 x−c45 y  t yz ðX; Y Þ ¼ t yz0 þ 2 2 2π −1 c44 x −2c45 xy þ c55 y 8 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 2 c44 x −2c45 xy þ c55 y c44 x2 −2c45 xy þ c55 y2 A= B ðξÞ dξ;  1− K1@ : ; z c44 ‘2 c44 ‘2

ð37bÞ where x = X − aξ cos (θ0) and y = Y − aξ sin (θ0). In the case of isotropic material (Ce = c44 = c55 = μ, c45 = 0), the nonlocal stresses (Eq. (37)) are simplified to those reported in [26], i.e.

Fig. 2. Screw dislocation density for mode III, σyz0 =tyz0 =Ce and θ0 =π/6.

t xz ðX; Y Þ ¼ −

aμ 2π

Z

yn r r o 1− K 1 Bz ðξÞ dξ: 2 ‘ ‘ −1 r 1

ð38aÞ

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

t yz ðX; Y Þ ¼ t yz0 þ

aμ 2π

Z

xn r r o 1− K 1 Bz ðξÞ dξ: 2 ‘ ‘ −1 r 1

ð38bÞ

In classical elasticity, the stress components (Eq. (37)) reduce to the classical stress for anisotropic medium, t xz ðX; Y Þ ¼

aC e 2π

Z

1

c45 x−c55 y 2 2 −1 c44 x −2c45 xy þ c55 y

t yz ðX; Y Þ ¼ t yz0 þ

aC e 2π

Z

Bz ðξÞ dξ

1

c44 x−c45 y B ðξÞ dξ: 2 2 z −1 c44 x −2c45 xy þ c55 y

ð39aÞ

ð39bÞ

As expected, the nonlocal stress components (Eq. (37)) as well as the classical stress components (Eq. (39)) fulfill the equilibrium condition. The nonlocal stress components (Eq. (37)) are finite everywhere while in classical elasticity, the stress components (Eq. (39)) are singular at the crack tips. In order to investigate this behavior, an inclined crack with θ0 = π/6 (Fig. 1) is considered in an anisotropic plane with elastic constants (Eq. (27)). The plane is assumed to be under loading σyz0 = tyz0 = Ce. The characteristic material length is considered to be equal to ‘ = 0.1a. A system of polar coordinates (r, θ) is employed to express the stress field as:

859

pffiffiffiffi while the energy coefficient is simplified to C e ¼ c44 ω . For an orthotropic material, the dislocation density as well as the stress components can be obtained by substituting c45 = 0 in Eqs. (30) and (37), respectively. In order to see the effect of the non-dimensional orthotropy parameter ω, consider a horizontal crack (θ0 = 0). In this case, the principal orthotropy direction is along the crack faces. The integral Eq. (30) reduces to −

2π pffiffiffiffi t yz0 ¼ c44 ω

Z

 

1 ajs−ξj ajs−ξj 1− K1 Bz ðξÞdξ: ‘ ‘ −1 s−ξ 1

ð42Þ

Consider the far-field loading to be independent of material orthotropy parameter. Then Eq. (42) together with the closure requirement Eq. (26) demonstrates that the dislocation density function is inpffiffiffiffi versely proportional to ω. In fact, we have Bz ðs; a=l; ωÞ ¼

t yz0 pffiffiffiffi f ðs; a=lÞ c44 ω

ð43Þ

pffiffiffiffi and the dislocation density is inversely proportional to ω and is a function of a/l. The nonlocal stress field (Eq. (37)) along the crack line reads t yz ðx; 0Þ ¼ t yz0 pffiffiffiffi Z  

ac44 ω 1 1 jx−aξj jx−aξj þ 1− K1 Bz ðξÞ dξ; 2π ‘ ‘ −1 x−aξ

t rz ¼ t yz sinðθÞ þ t xz cosðθÞ:

ð40aÞ

t θz ¼ t yz cosðθÞ−t xz sinðθÞ:

ð40bÞ

ð44Þ

Fig. 3 depicts the tθz stress component and its classical companion σθz along the crack faces (θ0 = π/6). As given in Eq. (37), Fig. 3 depicts that the nonlocal stress tθz is nonsingular. As required due to the boundary conditions, tθz is zero at the crack face −a b r b a , θ = 0 and in particular, at the crack tip. tθz reaches its maximum value near the crack tip ahead of the crack.

while txz = 0 on x-axis. Considering Eq. (43), the nonlocal stress (Eq. 44) is simplified to

ð45Þ

4.2. Orthotropic elastic plane weakened by a horizontal crack In this section, a cracked orthotropic material is studied using DDT. The general formulation for anisotropic materials can be simplified to orthotropic materials (c45 = 0) by setting L12 = 0. The nondimensional orthotropy parameter is defined as ω ¼ L11 =L22 ¼ c55 =c44 ;

t yz ðx; 0Þ ¼ Φðx=l; a=lÞ t yz0  

Z 1 a 1 jx−aξj jx−aξj ¼1þ 1− K1 f ðs; a=lÞ dξ; 2π −1 x−aξ ‘ ‘

ð41Þ

Fig. 3. Nonlocal (tθz) and classical stresses (σθz) at θ=θ0 =π/6, ‘ =0.1a, σyz0 =tyz0 =Ce.

According to Eq. (45), once an orthotropic material is weakened by a crack along the principal orthotropy direction (y = 0) and is under the far-field uniform traction (tyz0) perpendicular to the crack line, the nonlocal stress field on y = 0 is a function of x/l and a/l and is independent of material orthotropy ω. Considering the far-field loading to be σyz0 = tyz0 =c44, Fig. 4 depicts the nonlocal stress tyz on the x-axis which is independent of ω. In the same Figure, classical stress σyz is also depicted.

Fig. 4. Nonlocal (tyz) and classical stresses (σyz) along x-axis, ‘ =0.1a, σyz0 =tyz0 =c44.

860

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

Allegri and Scarpa [2] studied the case of a crack subjected to antiplane shear in nonlocal continuum using a Helmholtz type kernel. The study of [2] is based on asymptotic analysis and gives the nonlocal stress as K III t yz ðx; 0Þ ¼ pffiffiffiffiffi Ψðx=l; ωÞ ¼ t yz0 π‘

rffiffiffi a Ψðx=l; ωÞ: ‘

ð46Þ

They reported that the peak value of the nonlocal stress tyz depends on the material orthotropy and provided a table for the peak values of the nonlocal stress. In contrast, the current results based on DDT (Eq.(45) and Fig. 4) demonstrates that once an orthotropic material is weakened by a crack along the principal orthotropy direction (y = 0) and is under the far-field uniform traction (σyz0 = tyz0 = c44) perpendicular to the crack line, the nonlocal stress field is independent of value of the orthotropy parameter. This contradiction might be due to the fact that, in asymptotic solution, the order of the asymptotic expansion is decisive. For instance and as stated in [2], for a particular case of a mode I Griffith crack, the compressive T-stress term dominates the nonlocal stress field for cracks shorter than four times the nonlocal length scale. Thus, an asymptotic approach provides approximate solution while the DDT is constructed based on the analytical solution of dislocations and gives exact full-field expressions for the stress field. A higherorder expansion of the asymptotic solution can be derived to clarify the relationship between the asymptotic method and the present approach (the distributed dislocation technique). Furthermore, the nonlocal stress (Eq. (46)) derived in [2] suggests pffiffiffiffiffi that t yz π‘=K III is independent of the crack length (a) and is a function of x/‘ and ω. The nonlocal stress based on DDT (Eq. (45)) is a function of pffiffiffiffiffi x/‘ and a/‘. Fig. 5 depicts t yz π‘=K III obtained from DDT versus (x −a)/‘. It is observed that variations of characteristic length (‘) is effective and consequently, the nonlocal stress based on DDT does not follow the form suggested in Eq. (46). Fig. 5 depicts the variation of the peak value of normalized nonlocal stress for different values of characteristic length. The location of the peak nonlocal stress is ‘ b x−a b 1:5‘. 4.3. Orthotropic elastic plane weakened by an inclined crack For an orthotropic material, the dislocation density as well as the stress components can be obtained by substituting c45 = 0 in Eqs. (30) and (37), respectively. To explore the nonlocal and material orthotropy effects, now an inclined crack (θ0 = π/4) is considered. In

pffiffiffiffiffi Fig. 5. Normalized nonlocal stress (t yz ¼ t yz π‘=K III ) along x-axis versus (x−a)/‘, K III ¼ pffiffiffiffiffiffi t yz0 πa.

this case, the crack is not along the principal orthotropy direction of the material anymore. The plane is assumed to be under the far-field uniform traction σyz0 = tyz0 = c44. Considering the polar coordinates (r,θ), nonlocal (tθz) and classical stresses (σθz) at θ = π/4 is demonstrated in Fig. 6 for three different values of orthotropy parameter ω = 0.5,1 , 2. It is observed that by reducing ω, the nonlocal stress concentration is reduced. Fig. 6 shows that the result from DDT is in agreement with the boundary condition that prescribes traction-free crack faces. In order to investigate the effect of the crack orientation, the maxiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mum shear stress, i.e. t 2xz þ t 2yz, is plotted in Fig. 7 for cracks of different angles. The orthotropy parameter is considered to be ω = 0.5 and the characteristic material length is ‘ = 0.1a. The far-field uniform traction σyz0 = tyz0 = c44 is applied to the plane. It is observed that, among cracks with different orientations, the highest critical shear stress belongs to the horizontal crack (θ0 = 0).

5. Conclusion In this paper, the analysis of anisotropic materials weakened by a general inclined crack within nonlocal elasticity of Helmholtz type was presented. A dislocation-based approach was used to model cracks of mode III. This distributed dislocation technique (DDT) demonstrated that the nonlocal stress tensor is non-singular. Consequently, stressbased criteria can be employed to predict fracture in anisotropic materials (Eringen [12]). The dislocation density and nonlocal stress field were determined for anisotropic plane with an inclined mode III crack. As required, the solution proposed by DDT satisfied the equilibrium equations as well as the incompatibility and boundary conditions. It is concluded that the crack opening displacement within nonlocal elasticity is identical to the one in classical elasticity. Nonlocal elasticity gives regularized nonlocal stress field while the singularities of the dislocation density and the strain fields are not removed. The case of an orthotropic plane with horizontal and inclined crack was also studied. It was observed that for the case of a horizontal crack in an orthotropic plane, the nonlocal stress concentration along and in front of crack does not depend on the orthotropy parameter. Once the orthotropic material is weakened by an inclined crack, higher values of the orthotropy lead to higher nonlocal stress concentration along and in front of the crack. The present paper deals with mode III fracture. Mode I, mode II and mixed mode solutions can be obtained in similar fashion, but are more laborious to derive, and will be studied separately.

Fig. 6. Nonlocal (tθz) and classical stresses (σθz) at θ=θ0 =π/4,‘ =0.1a, σyz0 =tyz0 =c44.

S.M. Mousavi, A.M. Korsunsky / Materials and Design 88 (2015) 854–861

Fig. 7. Nonlocal maximum shear stress along θ=θ0, ℓ=0.1a,ω=0.5,σyz0 =tyz0 =c44.

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