Energy Approach to the Evaluation of the Limiting Equilibrium State of Orthotropic Cracked Bodies
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ENERGY APPROACH TO THE EVALUATION OF THE LIMITING EQUILIBRIUM STATE OF ORTHOTROPIC CRACKED BODIES M. M. Hvozdyuk and O. V. Hembara
UDC 620.193.01:539.379
We propose an energy criterion of crack propagation in anisotropic bodies based on the equation of energy balance at the onset of crack initiation. In this case, unlike the other well-known energy approaches, parallel with the equation specifying the limiting equilibrium of an anisotropic cracked body, we also use an additional equation for finding the direction of initial crack growth. The formulated criterion is applied to establish the limiting equilibrium state of an infinite orthotropic plate containing a rectilinear crack.
The analysis of the regularities of fracture of reinforced anisotropic materials (including orthotropic composites) is an urgent problem in the development of new structural materials with prescribed properties. In what follows, we formulate an energy criterion of crack growth in orthotropic bodies based on the equation of energy balance at the onset of crack propagation. Unlike the existing criteria [1], in this case, parallel with the equation specifying the limiting equilibrium of anisotropic cracked bodies, we use the equation for the determination of the initial direction of crack propagation. This equation follows from the hypothesis that the initial direction of crack propagation coincides with the direction in which the ratio of the potential energy to the fracture energy in the elementary volume near the crack tip is maximum. Computational Model of Crack Growth in Anisotropic Bodies Consider an elastic anisotropic plane Ω bounded by a line D and containing a crack of length L. According to the law of conservation of energy, the work A of the surface forces in the line D and bulk forces in the plane Ω per unit time is equal to the sum of the growth rate of energy dissipated in the region of plastic deformations W˙ and energy Γ˙ spent per unit time to increase the crack length by Δ L , namely, A˙ = W˙ + Γ˙ .
(1)
The overdot denotes the total derivative of the corresponding quantity with respect to time t. The crack tip moves according to the law L = L ( t ), where dL(t ) > 0. dt Since the quantities A, W, and Γ are functions of L and t and ∂L ≠ 0, ∂t Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 5, pp. 53–56, September–October, 2004. Original article submitted August 12, 2004. 1068–820X/04/4005–0629
© 2005
Springer Science+Business Media, Inc.
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M. M. H VOZDYUK
AND
O. V. HEMBARA
we have ∂( A − W ) ∂Γ = , ∂L ∂L
(2)
By using the definition of the derivative, we rewrite Eq. (2) in the form A( L + ΔL) − W ( L + ΔL) − A( L) + W ( L) ⎤ lim ⎡ = ⎢ ⎥⎦ Δ L→0 ⎣ ΔL
Γ( L + ΔL) − Γ( L) ⎤ lim ⎡ . ⎢ ⎥⎦ Δ L→0 ⎣ ΔL
(3)
Since Γ( L + ΔL) − Γ( L) ⎤ lim ⎡ = γc , ⎣⎢ ⎦⎥ ΔL
Δ L→0
where γc is the specific fracture energy spent to form a crack of unit area, and A( L + ΔL) − A( L) W ( L) − W ( L + ΔL) ⎤ + lim ⎡ = γs , ⎢ ⎥⎦ Δ L→0 ⎣ ΔL ΔL w
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