Asymptotic Fracture Modes in Strain-Gradient Elasticity: Size Effects and Characteristic Lengths for Isotropic Materials

Abstract

Size-effects characterize the fracture process of many engineering materials. Their modelling calls for material constitutive relations which are not indifferent, as standard elasticity, to variations of scale: strain-gradient elasticity or plasticity have often served the purpose. The three classical crack opening problems of fracture mechanics are here solved within the framework of linear strain-gradient elasticity for the most general isotropic material. Apart from the Lame constants, this is completely identified by five additional moduli, modelling its micro-structural characteristics. This general setting allows for recovering, as particular cases, previous analyses in (Gourgiotis and Geogiadis in J. Mech. Phys. Solids 57(11): 1898-1920, 2009), relative to modes I and II for the so-called Simplified Mindlin materials (17), and in (Radi in Int. J. Solids Struct. 45(10): 3033-3058, 2008), relative to mode III for couple-stress materials. More importantly, having a rather refined material description allows for understanding how the energy release rate is affected by the actual values of the characteristic lengths. Hence we demonstrate the strengthening effect of suitable microstructures and their optimality in order to provide cohesive like actions on the crack lips. Despite being limited to the linear elastic hypothesis, the solutions found constitute an useful insight to develop more complex, possibly nonlinear, constitutive relationships for the strain-gradient terms as for plastic or viscoelastic materials

Trefftz collocation method for two‐dimensional strain gradient elasticity

“…Correctness of (12) can be checked by its direct substitution into equilibrium equations (8). Note that both gradient potentials 1 , 2 in (12) are needed to provide that the gradient part of displacements contains the nonzero divergence (equals to ∇ ⋅ 1 ) and the nonzero curl (equals to ∇ × 2 ).…”

“…3 A peculiarity of SGET is its assumption that the strain energy density depends not only on the strain but also on the strain gradients. Such an assumption is essential for problems where the gradients of the field variables become relatively high, for example, in small-scale structures, [4][5][6] in domains with stress concentrated around cracks, corners, and inclusions, [7][8][9][10] in high frequency processes 5,11 and in the metamaterials with nonlocal type of internal interactions. 12,13 The pioneering work on the development of numerical methods in SGET was undertaken by Oden et al in their paper in 1970, 14 where they adopted the finite element method (FEM).…”

The Boussinesq problem in dipolar gradient elasticity

“…Cook and Weitsman [19], Eshel and Rosenfeld [20,21]). More recently, this approach and related extensions have been employed to analyze various problems involving, among other areas, fracture (Chen et al [22], Shi et al [23], Georgiadis [24], Radi and Gei [25], Grentzelou and Georgiadis [26,27], Gourgiotis and Georgiadis [28], Aravas and Giannakopoulos [29], Aslan and Forest [30], Gourgiotis et al [31], Piccolroaz et al [32], Sciarra and Vidoli [33]), wave propagation (see e.g. Maugin and Miled [34], Engelbrecht et al [35], dell'Isola et al [36], Polyzos and Fotiadis [37], Gourgiotis et al [38]), poroelasticity (Madeo et al [39]), mechanics of defects (Lazar and Maugin [40,41], Forest [42]), and stress concentration due to discrete loadings (Georgiadis and Anagnostou [14], Exadaktylos [43], Lazar and Maugin [44]).…”

Vibration of cracked functionally graded microplates by the strain gradient theory and extended isogeometric analysis

“…The choice of the crack-tip enrichment functions depends on the original displacement description of the problem and how well the function can capture the near tip asymptotic field [47]. In this specific problem of the small-scale plates in which the RPT and SGT are employed, the enrichment functions should also account for the inclusion of highorder strain gradient terms [60].…”