In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.

BackgroundAn osteon consists of a multi-layered bone matrix and interstitial fluid flow in the lacunar–canalicular system. Loading-induced interstitial fluid flow in the lacunar–canalicular system is critical for osteocyte mechanotransduction and bone remodelling.MethodsTo investigate the effects of the lamellar structure and heterogeneous material properties of the osteon on the distributions of interstitial fluid flow and seepage velocity, an osteon is idealized as a hollow two-dimensional poroelastic multi-layered slab model subjected to cyclic loading. Based on poroelastic theory, the analytical solutions of interstitial fluid pressure and seepage velocity in lacunar–canalicular pores were obtained.ResultsThe results show that strain magnitude has a greater influence on interstitial fluid pressure than loading frequency. Interestingly, the heterogeneous distribution of permeability produces remarkable variations in interstitial fluid pressure and seepage velocity in the cross-section of cortical bone. In addition, interstitial fluid flow stimuli to osteocytes are mostly controlled by the value of permeability at the surface of the osteon rather than at the inner wall of the osteon.ConclusionInterstitial fluid flow induced by cycling loading stimuli to an osteocyte housed in a lacunar–canalicular pore is not only correlated with strain amplitude and loading frequency, but also closely correlated with the spatial gradient distribution of permeability. This model can help us better understand the fluid flow stimuli to osteocytes during bone remodelling.

The fracture behavior of a functionally graded layered structure (FGLS) with an interface crack under thermal loading is investigated. Considering new boundary conditions, it is assumed that interface crack is partly insulated, and the temperature drop across the crack surfaces is the result of the thermal resistance due to the heat conduction through the crack region. The problem is formulated in terms of a system of singular integral equations. Numerical results are presented to show the influence of the material nonhomogeneity parameters and the dimensionless thermal resistance on the thermal stress intensity factors (TSIFs).Keywords Functionally graded layered structure (FGLS) · Thermal resistance · Thermal stress intensity factors (TSIFs) IntroductionOne of the purposes for developing functionally graded materials (FGMs) is to replace the conventional homogeneous materials to reduce the magnitude of residual and thermal stresses and thus to increase the bonding strength where the materials are subjected to extremely high thermal loading. In designing components involving FGMs, an important aspect of the problem is the question of mechanical failure, specifically the fracture failure [1]. Fatigue and fracture characterization of materials and related analysis require the solution of certain standard crack problems. So far, many crack problems in FGMs have been solved to obtain the crack tip thermal stress intensity factors (TSIFs) by assuming the elastic properties varying due to functional forms [2][3][4][5][6][7].Assuming the perfect thermal insulation of the crack surfaces, Noda and Jin [8] studied the crack problem for an infinite functionally graded material (FGM) subjected to a steady-state heat flux over the crack surfaces by assuming continuously varying thermal properties. Choi et al. [9] studied the crack problems in FGM-like nonhomogeneous materials under thermal loading. The properties of practical FGMs, nevertheless, may vary in an arbitrary manner. Chen et al. [10] investigated the mode I surface crack problem for an orthotropic-graded strip under transient loading.Considering the partial insulation along the crack surfaces, Borgi and Erdogan [11,12] considered the problem of a functionally graded coating bonded to a homogeneous substrate with a partially insulated interface crack between the two materials subject to both thermal and mechanical loading. The problem of an embedded partially insulated crack in a graded coating bonded to a homogeneous substrate under thermal and mechanical loading is considered by Borgi et al [13]. In their studies, the continuity conditions of the temperature field and heat flux along the crack axis, outside the crack and along the interface are considered.In this paper, consider new boundary conditions, assuming that the interface crack is partially insulated and the temperature drop across the crack surfaces is the result of the thermal resistance due to the heat conduction

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