Plastic Exfoliation of a Periodic System of Thin Near-Boundary Inclusions
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PLASTIC EXFOLIATION OF A PERIODIC SYSTEM OF THIN NEAR-BOUNDARY INCLUSIONS V. А. Kryven,1,2 А. R. Boiko,1 V. B. Valyashek,1 and L. І. Tsymbalyuk1
UDC 539.375
We obtain a numerical-analytic solution of an antiplane problem of the stress-strain state of elastoplastic half space containing a periodic system of thin rigid tunnel inclusions parallel to the boundary of the half space. Prior to loading, these inclusions were in unilateral mechanical contact with the medium. We determine the stress-strain state and study the phenomenon of plastic exfoliation of the inclusions. The diagrams of the critical load are plotted on the coordinates of the geometric parameters of the system of inclusions under the condition of constancy of the critical length of interface plastic strips. Keywords: unilaterally exfoliated inclusions, interface plastic strips, antiplane deformation, periodic system of inclusions.
Introduction The investigation of the stress-strain states (SSS) of bodies with systems of inclusions interacting with each other and with the boundary of the body are of significant interest both in the theory of strength of composites and reinforced materials and for the prediction and optimization of their deformation characteristics [1–4]. These problems are especially important in the elastoplastic formulation with regard for the imperfect contact of inclusions with the main medium. In what follows, we study the phenomenon of plastic exfoliation of a system of thin equidistant rectilinear inclusions located in a plane parallel to the boundary of an ideal elastoplastic half space. We consider the case of unilateral contact of inclusions with the main medium: inclusions do not contact with the medium on the boundary but are in perfect contact on the opposite side. The SSS is caused by a monotonically increasing shear load
τ xz = 0 ,
τ yz = τ ∞
applied at infinity (Fig. 1: a is the distance between the inclusions and the boundary of the half space, 2c is the width of the inclusions, 2b is the distance between the centers of neighboring inclusions, and d is the length of the strips of interface plastic exfoliation). The problem can be reformulated as a boundary-value problem in the domain D (half strip x > 0 , 0 < y < b , containing a cut made along the segment x = a , 0 ≤ y ≤ c ) for an analytic one-sheeted function τ(ζ) = τ yz (x, y) + iτ xz (x, y) , ζ = x + iy : 1 2
Pulyui Ternopil National Technical University, Ternopil, Ukraine. Corresponding author; e-mail: [email protected].
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 56, No. 1, pp. 89–93, January–February, 2020. Original article submitted February 21, 2019. 90
1068-820X/20/5601–0090
© 2020
Springer Science+Business Media, LLC
PLASTIC EXFOLIATION OF A PERIODIC SYSTEM OF THIN NEAR-BOUNDARY INCLUSIONS
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Fig. 1. Cross section of the body.
(ζ ∈{x + ib, 0 < x < + ∞} ∪ {iy, 0 < y < b} ∪ {x, 0 < x < a}
Im τ(ζ) = 0
∪ {a − 0, 0 < y < c} ∪ {x, a < x < + ∞}) , τ(ζ) = k
(ζ ∈{a + 0, c − d < y < c}) ,
Re τ(ζ) = 0,
(1)
(ζ ∈{a + 0, 0 < y < c − d})
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