PSI - Issue 32

ScienceDirect Available online at ww.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2021) 000–000 Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2021) 000–000 Available online at www.sciencedirect.com Procedia Structural Integrity 32 (2021) 124–130

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2452-3216 © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” 10.1016/j.prostr.2021.09.018 2452-3216 © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” * Corresponding author. Tel.: +7-903-794-72-79; fax: +7-499-135-61-90. E-mail address: salurie@mail.ru 2452-3216 © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the XXIIth Winter School on Continuous Media Mechanics” Abstract We consider the fracture mechanics problem for the finite and semi-infinite cracks in the gradient elasticity. Local stress fields that define the fracture the strength of materials are found as solutions of the inhomogeneous Helmholtz equations in which the inhomogeneity is determined by classical stresses. To construct solutions, the radial multipliers method and the Papkovich Neuber representation are used. We show that the use of the gradient theory of elasticity is carried out simultaneously with the identification of the scale parameter and an indication of its physical meaning and fundamental role in fracture mechanics. We established that the local stresses in the vicinity of crack tips are regular, have the form characteristic of stress concentration, and depend only on the level of acting stresses and the scale parameter, which is found as a result of mechanical testing of material sample Keywords: cracks mechanics; gradient elasticity; complex potentials; radial multipliers, stress concentration 1. Introduction The problem of singularities in the theory of elasticity and fracture mechanics is widely discussed in the works Carpinteri and Paggi (2009), Sih and Tang (2005) and etc. The singularity of solutions for stresses in the linear theory of elasticity at the crack tip excludes the use of traditional criteria for the strength of bodies with stress concentration. The formally obtained singular solutions contradict not only the physical meaning, but also the postulates of the theory of elasticity. Such paradoxes still require their explanation. Gradient elasticity allows one to describe size effects (see Aifantis (2014), Askes and Aifantis (2011)]), provides regularization of singular solutions of differential equations of Abstract We consider the fracture mechanics problem for the finite and semi-infinite cracks in the gradient elasticity. Local stress fields that define the fracture the strength of materials are found as solutions of the inhomogeneous Helmholtz equations in which the inhomogeneity is determined by classical stresses. To construct solutions, the radial multipliers method and the Papkovich Neuber representation are used. We show that the use of the gradient theory of elasticity is carried out simultaneously with the id ntification of the scale parameter and an indication of its physical meaning nd fundamental rol in fracture mechanics. We established that the local stresses in the vicinity of crack tip are regular, have t e f rm characteristic of stress concentration, and depend only on the level of acting stresses and the scale parameter, which is found as a result of mechanical testing of material sample Keywords: cracks mechanics; gradient elasticity; complex potentials; radial multipliers, stress concentration 1. Introduction The problem of singularities in the theory of elasticity and fracture mechanics is widely discussed in the works Carpinteri and Paggi (2009), Sih and Tang (2005) and etc. The singularity of solutions for stresses in the linear theory of elasticity at the crack tip excludes the use of traditional criteria for the strength of bodies with stress concentration. The formally obtained singular solutions contradict not only the physical meaning, but also the postulates of the theory of elasticity. Such paradoxes still require their explanation. Gradient elasticity allows one to describe size effects (see Aifantis (2014), Askes and Aifantis (2011)]), provides regularization of singular solutions of differential equations of XXIIth Winter School on Continuous Media Mechanics On the failure analysis of cracked plates within the strain gradient elasticity in terms of the stress concentration Valery Vasiliev a , Sergey Lurie a,b* * XXIIth Winter School on Continuous Media Mechanics On the failure analysis of cracked plates within the strain gradient elasticity in terms of the stress concentration Valery Vasiliev a , Sergey Lurie a,b* * a Instituite for Problem in Mechanics of the Russian Acad. of Scs. Pr.Vernaadskogo 101,119526, M scow, Russia b Instituite of Applied Mechanics of the Russian Acad. of Scs, Leningradskii pr, 7, 125040, Moscow, Russia a Instituite for Problem in Mechanics of the Russian Acad. of Scs. Pr.Vernaadskogo 101,119526, Moscow, Russia b Instituite of Applied Mechanics of the Russian Acad. of Scs, Leningradskii pr, 7, 125040, Moscow, Russia * Corresponding author. Tel.: +7-903-794-72-79; fax: +7-499-135-61-90. E-mail address: salurie@mail.ru