Determination of the Sizes of Plastic Zones in a Double-Curvature Orthotropic Shell with Surface Crack with Regard for t
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DETERMINATION OF THE SIZES OF PLASTIC ZONES IN A DOUBLE-CURVATURE ORTHOTROPIC SHELL WITH SURFACE CRACK WITH REGARD FOR THE HARDENING OF THE MATERIAL K. M. Dovbnya1 and N. D. Er’omina1,2 The problem of surface cracks in orthotropic shells made of hardened materials is considered within the framework of the δ c -model. The analytic and numerical solutions of the problem are presented. We also study the influence of the curvature of the shell, hardening, loading, and crack length on the sizes of plastic zones. Keywords: hardening, surface crack, orthotropic shell, δ c -model.
At present, the requirements to structural materials permanently increase. The strength, i.e., the ability to resist fracture, is the most important among their properties. The presence of cracks significantly decreases the load-bearing ability of shell structures. Therefore, numerous researchers study the stresses formed in the vicinities of cracks within the framework of the theory of brittle fracture. However, in the major part of materials, the zones of plastic strains are formed in narrow strips on the continuations of the cracks. The isotropic shells were considered by using an analog of the δ c -model in [1–3]. In this case, the zone of plastic strains is modeled by the lines of discontinuity of displacements and the angles of rotation. Later, the δ c -model was used to study the orthotropic shells of any curvature containing cracks of various configurations (through, surface, and internal) in perfectly elastoplastic materials [4–6]. However, in practice, the materials can be deformed beyond the limit of plasticity, i.e., can be hardened. A generalization of the Leonov–Panasyuk–Dugdale δ c -model that takes into account this effect of hardening [10, 11] was proposed in [7–9]. The isotropic shells were also investigated [10]. In the present work, we consider a surface crack in an orthotropic shell modeled by using an analog of the δ c -model generalized for materials of this kind. Statement of the Problem We study a surface crack located in a shallow elastoplastic orthotropic shell of constant thickness h (Fig. 1). The crack is oriented along one of the lines of principal curvatures in the median surface of the shell. Assume that the material of the shell is linearly hardened in the process of deformation and that the crack depth is equal to d . Then the thickness of the shell under the crack is d1 = h – d . Suppose that the crack lips are not in contact and that they are loaded by moments and forces symmetric about the crack line. We also assume that the crack sizes are large as compared with the thickness of the shell and are small as compared with the other linear dimensions of the shell. 1 2
Donets’k National University, Donets’k, Ukraine.
Corresponding author; e-mail: [email protected].
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 52, No. 2, pp. 124–129, March–April, 2016. Original article submitted September 20, 2013. 1068-820X/16/5202–0287
© 2016
Springer Science+Business Media New York
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K. M. DOVBNYA
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