Path-independent J-integral for cracks in decagonal quasicrystals

The path-independent J-integral is derived for fracture mechanics analysis of decagonal quasicrystals (QCs). The gradient theory of quasicrystals is developed here to consider large strain gradients at the crack tip vicinity. The constitutive equations contain phonon and phason stresses, and the higher-order stress tensor. The higher-order elastic material parameters are proportional to the internal length material parameter and the conventional elastic coefficients. The FEM equations are derived to solve general boundary value problems for the strain gradient theory of the QCs.


Introduction
The quasicrystals (QCs) have a structure of atoms as something between crystals and amourphous materials. It is observed a long-range quasiperiodic translational and orientational orders. They have special properties with effective engineering application. They were discovered in 1984 by Shechtman et al. [1]. The decagonal quasicrystals have ten-fold rotational symmetries and they belong to the class of two-dimensional (2-d) quasicrystals, with a quasiperiodic atomic arrangement in a plane, and a periodic one in the third direction.
In literature there are utilized three various models for a reliable description of elastodynamics of quasicrystals. Bak`s model [2] considers phasons for a particular structure disorders in quasicrystals. Then, phonons and phasons play similar roles in the dynamics and they are described by the balance of momentum. Lubensky et al. [3] consider that the phason field is described by a diffusion equation with a very large diffusion time. It follows from properties that phasons are insensitive to spatial translations and oppositely, phason modes represent the relative motion of the constituent density waves. Recently, Agiasofitou et al. [4] have utilized the wavetelegraph type equations for the elastodynamic model. However, a unique opinion on governing equations for phason fields is still missing. A comprehensive state of the art of investigations on the mechanical analyses of QCs can be found in monographs [5][6][7]. One can find there that QCs are generally considered as brittle materials. Therefore, crack analyses are very important to understand the effect of cracks on the mechanical behavior of a quasicrystal material. Up to now cracks are analyzed by classical elasticity theory [8][9][10].
It is well known that large strain gradients are occurred in the crack-tip vicinity. Therefore, the gradient elasticity theory appears most suited for studying the strain and stress fields near the crack-tip for crystalline materials [11][12][13][14][15]. In this study, the strain gradient theory is extended to quasicrystal materials. Constitutive equations in gradient theory of quasicrystals for phonon and phason stresses, and the higher-order stress tensor are given by the phonon and phason strains and the gradient of phonon strains in this study. The phason displacements represent already the atomic rearrangement and they are significantly smaller than phonon ones [16]. The gradients of phason strains are negligible in comparison to gradients of phonon strains, since the phason fields are smooth.
In this study, the higher-order J-integral and the energy release rate are derived for cracks in quasicrystals described by the strain-gradient elasticity. It is shown the path independence of this integral.

Gradient theory of quasicrystals
The elasticity theory of quasicrystals can be based on the phonon and phason displacements [5]. The phonon displacement () while the phason strains () The constitutive equations for phonon and phason stresses in classical elasticity theory of quasicrystals have the following form [5,16] ij ijkl kl ijkl kl where ijkl c , ijkl K and klij A denote phonon elastic tensor, phason elastic tensor and phononphason coupling elastic constant tensor, respectively. Since the phason strains are significantly smaller than phonon strains, the gradients of phason strains can be neglected in comparison with gradients of phonon strains in the gradient theory of quasicrystals. Then, the constitutive equations are written as where g represents the higher-order elastic coefficients. The symbols ijk  are used to denote the higher-order stress tensor components. The phonon strain-gradient tensor η is defined as The constitutive equations for plane elasticity of decagonal QC are given by [ where material coefficients ij c , A and i K denote the classical phonon elastic coefficients, the phonon-phason coupling parameter and the phason elastic coefficients, respectively.
In gradient theory, these constitutive equations can be extended according to (4) and rewritten as 11 In equation (9) it is assumed that higher-order elastic parameters jklmni g are proportional to the conventional elastic stiffness coefficients klmn c as 2 jklmni li jkmn g l c  = and l is the internal length material parameter [17].
Then, the variation of deformation energy in a solid with volume V is given by where : : with i  being the Cartesian components of the unit tangent vector on  .
The external work of the applied external "forces" From the principle of virtual work, viz., 0 UW  −= , the following governing equations is obtained:

J -integral for quasicrystals
At a virtual extension of a crack along its plane by da, the change of potential energy Π in cracked quasicrystal body Ω is given by Futhermore it is assumed that crack grows along 1 x axis of the coordinate system and its origin moves together with the crack-tip. The following identity is valid That identity is utilized in eq. (16) , and utilizing the principle of virtual work, equation (18) is reduced to Applying the Gauss divergence theorem to domain integral in eq. (19), one can write (20) in which the governing equations (15) are utilized. is chosen [15].
If the ratio / ( / ) l a w is decreasing both phonon and phason crack displacements are reduced with respect to ones obtained by classical theory. A similar phenomenon is observed for 0 J integral. A reduction of the 0 J integral value with growing the size effect parameter is observed in Fig. 3. Similar behaviour has been observed also for crystal material [15].

Conclusion
The J -integral defined in a global coordinate system vanishes when the integration contour chosen to calculate the J -integral encloses all irregularities in considered domain. The Jointegral obtained in this paper will provide a useful way to study fracture problems in quasicrystal materials.