PSI- Issue 9

Andrea SPAGNOLI et al. / Procedia Structural Integrity 9 (2018) 159–164 A. Spagnoli et al. / Structural Integrity Procedia 00 (2018) 000–000

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an infinite medium, subjected to a remote mode II stress field, has been considered by Ballarini and Plesha (1987), who have used an analytical method to compute the crack tip Stress Intensity Factors (SIFs) and the direction of crack propagation under monotonic loading. A similar approach has also been adopted by Tong et al. (1995), but here, di ff erently, the object of the investigation was the behaviour of cracks under cyclic loading. In particular, their purpose was to quantify the sliding mode crack closure which occurs under shear fatigue loading and is responsible for a reduction of the mode II SIF. An intriguing study on the e ff ect of mixed mode and partial crack closure due to sinusoidal crack profile can be found in Xiaoping and Comninou (1989). The aforementioned models often di ff er in the method of solution and computation of the stress intensity factors. In general, the geometrical complexity of real cracks and material models would require a purely numerical technique. However, we would like to remark the importance of analytical methods in fracture mechanics, since they provide the correct form of singularities and serve as a benchmark for the numerical procedures (Erdogan, 2000). In this spirit, we turned our attention to analytical methods, particularly those derived from the application of the complex function theory; among them, a very flexible technique for the solution of crack problems in di ff erent geometric configurations is the Distributed Dislocation Technique (DDT) (Hills et al., 2013a). In the present paper, the model presented in Spagnoli et al. (2018) is applied to analyse the experimental results related to the shear mode fracture toughness of di ff erent types of natural stones (Backers et al., 2002). In particular, the e ff ect of crack shielding due to roughness and frictional interface along the crack path is taken into account to predict the fracture resistance for di ff erent relative crack sizes and various levels of confinement pressure. In order to consider the e ff ects of friction and roughness, we make use of an interface model formulated as a consti tutive relationship between opposing points along the crack. Globally, the crack surfaces are smooth and frictionless, a hypothesis which allows us to obtain a straightforward implementation within the technique of the distributed dislo cation. The surface interference is modelled by means of bridging stresses, which are added to the stress distribution determined by the remote loads, and computed at a discrete number of points. Let us define the relative displace ment increments between two opposing points as additively composed of a recoverable elastic part dw e i and a non recoverable plastic part dw p i , which accounts for frictional sliding and dilatancy. The stresses on the crack interface are related to the displacement increments by means of interface sti ff nesses E EP i j : σ i = E EP i j dw j i , j = t , n (1) where t , n denote, respectively, the tangential and normal directions with respect to the nominally flat crack surface (Fig. 1a). The interface sti ff nesses are dependent on a slip function F , a slip potential G and their derivative, that is: 2. Description of the model E i j is the elastic interface sti ff ness, which has to be calibrated to avoid interpenetrability between the crack surfaces. The surface roughness is described through a saw-tooth model, characterised by a constant angle α , mean length of the asperities equal to 2 L and height h (equal to twice the arithmetical mean deviation, which is a widely used roughness parameter of the crack profile). The coe ffi cient of Coulomb friction f is constant everywhere. With the previous assumptions, the functions F and G have the following formulations: F = | σ 1 | + f σ 2 = | σ n sin α + σ t cos α | + f ( σ n cos α − σ t sin α ) , G = | σ 1 | = | σ n sin α + σ t cos α | (3) We can define a crack size parameter by considering the ratio c / L , where c / 2 L approaching the unity ideally identifies the case of a short crack. The parameters used to describe the surface roughness are somehow related to a specific material, and the scale length might di ff er of several orders of magnitude. The stress state in an elastic body with a crack can be determined by introducing a suitable distribution of disloca tions along the crack line (dislocations are commonly used in the theory of elasticity as kernel of integral equations, E EP i j = E i j if F < 0 or dF = 0 , E EP i j = E i j − ∂ F ∂σ p E iq E p j ∂ G ∂σ q ∂ F ∂σ p E pq ∂ G ∂σ q if F = dF = 0 (2)