Mechanism of Development of the Area of Passive Deformation in a Nonlinear Elastic Orthotropic Body with a Crack
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International Applied Mechanics, Vol. 56, No. 4, July, 2020
MECHANISM OF DEVELOPMENT OF THE AREA OF PASSIVE DEFORMATION IN A NONLINEAR ELASTIC ORTHOTROPIC BODY WITH A CRACK
A. A. Kaminsky and E. E. Kurchakov
A nonlinear thin-walled elastic orthotropic body with a mode 1 crack and a fracture process zone near its tip is considered. An equilibrium boundary-value problem is stated in terms of the components of the displacement vector. The equations relating the components of the stress vectors at points on the opposite boundaries of the fracture process zone with the components of the vector of displacement of these points relative to each other are used. The mechanism of development of the area of passive deformation around the fracture process zone is established using the solution of the boundary-value problem. The strain state at some points of the area of passive deformation is analyzed. The evolution of the area of passive deformation under loading of the body is studied. The area of passive deformation is compared with the nonlinear zone near the crack tip. Keywords: nonlinear elastic orthotropic body, mode 1 crack, fracture process zone, passive deformation area Introduction. As experiments testify, a fracture process zone in the form of a narrow area with microcracks, pores, and laminations occurs near the crack front [12]. This zone should be taken into account in stating boundary-value problems of fracture mechanics, which involves certain difficulties. They can be avoided by representing the boundaries of the fracture process zone as the surfaces of an open cut acted upon by opposite stress vectors [3, 10, 17, 18]. It is necessary to consider the components of the stress vectors at points on the opposite boundaries of the fracture process zone dependent on the components of the vector of displacement of these points relative to each other and to use appropriate equations. Moreover, it is necessary that the strength criterion be satisfied at the end of the fracture process zone [3, 14]. It is shown in [11] that the length of the fracture process zone increases with loads, while the stresses pm its boundaries decrease. As a result, a certain area of passive deformation arises near the fracture process zone. In what follows, we will study the mechanism of development of the area of passive deformation in a nonlinear elastic orthotropic body with a mode 1 crack. We will restrict oneself to small deformations, while the appropriate boundary-value problem will be stated in terms of the components of the displacement vector. 1. Constitutive Equations. Let us consider the following tensor linear constitutive equations relating the components of the stress tensor S and the components of the strain tensor D [13]: S ab =
E ab Ô æ abgd E ö g + çG Dgd - g ab ÷ , Z Wè Z ø
(1.1)
where F= K-
E2 , Z
W = X-
E2 , Z
(1.2)
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov St., Kyiv, Ukraine 03057; e-mail: [email protected]. Translated from Prikladnaya Mekhanika, Vol. 56,
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