PSI - Issue 32

Valery Vasiliev et al. / Procedia Structural Integrity 32 (2021) 124–130 Vasiliev and Lurie/ Structural Integrity Procedia 00 (2019) 000–000

125

2

elasticity theory, provides regularization of singular solutions of differential equations of elasticity theory (see Lazar and Polyzos (2015), Lurie and Belov (2014) and etc) and allows one to describe size effects and provides regularization of singular solutions of differential equations of elasticity theory (see Sciarra and Vidoli (2013); Gourgiotis and Georgiadis (2009) and etc). These results show significant departure from the predictions of standard fracture mechanics and indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. To solve applied problems, as a rule, one-parameter models are used where only one additional parameter is introduced respect to classical elasticity . Using the gradient model leads to high order equilibrium equation for displacement with operator, which is product the Lame operator and generalized Helmholtz operator. For example, for one of the most popular the strain gradient elasticity theory (SGET) the tensors of stresses and moments are determined by the formulas (see 2 2 , , , 2 , ( 2 ) ij ij ij ijk k ij ij k ij kk s s              , ij ij     and equilibrium equations have the form: , 0 ij j   , where , ij ij ijk k      2 , , (1 )( 2 ) ij k ij ij k s         here ,   are the Lame coefficients, s is the scale parameter, ij  is the Kronecker delta. The applied variant of the generalized elasticity (AVGE) proposed by the authors of the work Vasiliev and Lurie (2018). This model are formulated for the static boundary conditions in the terms the local stresses ij  and displacements i U , which are defined as solutions of the nonhomogeneous Helmholtz equations 2 i i i U s U u    , 2 ij ij ij s      . The right sides of these equations are the classical nonlocal solutions i u and ij  solutions of theory of elasticity. To construct the local solutions it is first proposed to solve the classical elasticity problem for generalized stresses and displacements, and then find local fields by solving the corresponding Helmholtz equations. It was shown in Vasiliev and Lurie (2018). that above Helmholtz equations are arisen in the result averaging procedure of the nonlocal solutions over a representative fragment of material taking into account not only the local values of the corresponding functions but also their local derivatives. In the proposed generalized theory, Hooke's law is written on nonlocal generalized stresses ij  and deformations / ij i j u x     , which formally coincide with the classical solution. Equilibrium equations are formulated in the terms of the generalized stresses ij  the same as for classical elasticity. As for stresses ij  , their importance is not negated. Moreover, it is assumed that it is these stresses are important for assessing the strength and that they are involved in the strength criteria. These stresses are found after the definition of generalized stresses (“classical”) ij  from the equations, 2 ( ) ij ij ij l      , so that they specify a nonsingular stress field. The main peculiarity of the presented paper is the proposed approach for the assessment of the material fracture. For the considered brittle and quasi-brittle materials we propose to use the failure criteria formulated with respect to the local stresses estimated within of AVGE. These stresses have the finite values in the whole domain and also at the crack tip. Then we identify the length scale parameters for the known experimental data with pre-cracked brittle and quasi-brittle materials and show that identified parameters allow us to predict the failure loads for the experimental samples with different type of cracks by using maximum principal local stress criterion. The paper is organized as follows. In Section 2 we briefly describe the regular solutions for isotropic and orthotropic plates with cracks. In Section 3 we describe the concept of the stress concentration and show the application of its for the assessment of the material fracture. 2. Gradient solutions for isotropic and orthotropic planes with finite and infinite cracks 1. Let us consider at the first a plane problem for an infinite strip loaded by stresses  in the direction of the axis ОУ and containing a finite crack of length 2 l , , 0 l x l y     for the classical cracks mechanics. The classical solution to this problem is written in a complex-valued form through harmonic and biharmonic potentials ( ) w  and ( ) w  , w x i y   . Then we can write the following equations for the stresses

(1)

11 yy xy                     22 Re( ), Re( ), Im xx

0 0 ,       , ) / [2( ) ] l w w w l     . To receive the nonclassical nonsingular solution for the AVGE model, we find regular solutions for the harmonic 2 2 1/ 2 / ( ) , w w l    0 2 2 2 3 2 0 (

here