Crack Of Length 2a Research Articles

Cylindrical crack vis a vis parallel crack pair: shielding, energetics and asymptotics

Fibrous materials often contain cylindrical cracks due to delamination along the matrix-fiber interface. It is instructive to analyse a cylindrical crack of length 2a and diameter 2h in a homogeneous medium and compare the results with those for a pair of parallel cracks of length 2a and spacing 2h. The pair of parallel cracks mutually shielding each other is examined here with regard to the variation of stress intensity factors and energetics including the asymptotic limit of a pair of nearly coalescing parallel cracks. A unified formulation for parallel cracks/cylindrical crack based on crack opening displacement (COD) in terms of Chebyshev polynomials is developed. The characteristic variation of stress intensity factors as the cracks approach each other (h - 0) shows that the stress intensity factors vanish for the case of a vanishingly small cylindrical crack but not for the 2D parallel pair of cracks. The 2D case of a pair of collapsing parallel cracks ensures a finite energy release rate asymptoting to that of a single crack. Further research is needed to establish definitive asymptotic bounds for the case of extremely closely spaced cracks on the lines of Hutchinson and Suo (Adv Appl Mech 29:377-384, 1992), Kachanov (1993) and Gorbatikh et al. (Int J Fract (Lett Fract Micromech) 143:377-384, 2007). Results are presented for different values of Poisson's ratio.

The Plane Problem Solution of Piezoelectric Material with a Line Crack

A two-dimensional electromechanical analysis is performed on a transversely isotropic piezoelectric material containing a line crack of length 2a, which is subjected to uniform distributed forces with intensity q along the crack faces and uniform electric displacement fields at infinity. Based on the impermeable electric boundary conditions, the closed complex form expression for the electromechanical fields are deduced in the vicinity of the crack. Taking PZT-4 ceramic into consideration, electromechanical fields under different loads are illustrated through several examples. It can be seen that electromechanical fields show 1/2 order singularity at the crack tip, and the piezoelectric effect can’t be negelected.

Closed-form solution for a mode-III interface crack between two bonded dissimilar elastic layers

A mode-III crack at the interface between two bonded dissimilar elastic layers of equal thickness is considered. The closed-form solution for a crack of length 2a has been analytically derived for stress-free boundaries and clamped boundaries. Furthermore, the analytic solution for a semi-infinite interface crack in two bonded dissimilar layers has been directly determined without recourse to the Wiener-Hopf technique.

Crack tip fields of piezoelectric materials under antiplane impact

THE fracture mechanics of piezoelectric materials is extensively studied under static mechanical or electrical loads['-41. However, the failure of practical smart structure always happens under dynamic loads, so it is urgently needed to study the fracture dynamics of piezoelectric materials. The dynamic Green's functions for anisotropic piezoelectric materials were proposed by orris'^]. The dynamic representative formulas and the fundamental solutions were presented by Khutoryansky and ~osa'~] for piezoelectric materials. The present work was carried out on the dynamic fracture of piezoelectric materials under antiplane impact and a closed form solution was obtained. As the effect of time is omitted, i. e. t tends to infinity, the present results will be reduced to the static ones"]. 1 Dynamic antiplane fracture mechanics of piezoelectric materials The antiplane crack of length 2a in an electric material under remote stress and electric displacement impacts can be decomposed into the uniform antiplane stress field and inplane electric displacement and a central crack of length 2a under antiplane stress impact r, = - r" H( t ), and the inplane electric displacement impact D, = - Dm H( t ) . The singularities of the present problem coincide with the latter one, where the boundary condition can be written as rZy = - rmH(t), Dy = - DCOH(t), 1x1 a. (lb) The dynamic antiplane governing equations for piezoelectric material can be written as C,,V~W + e1sv2$ = pa2~/at2, (2a) els~2w - E~~v~# = 0. (2b) In the above equation, the body force other than inertia and the free charge are omitted. The antiplane constitutive relations for piezoelectric material arec6] rz, = ~,,a~/a~ + e,,a+/a~, r, = c44aw/ay + elsa#/ay, (3) n, = e,,aw/a~ - ~~~a$/a~, D, = elsawlay - ~~~a#/a~, (4) where c4,, e E are, respectively, the elastic modulus, piezoelectric constant and dielectric constant of the antiplane piezoelectricity ; r, , r, , D, , Dy are, respectively, the antiplane

The calculation of crack opening area and crack opening volume from stress intensity factors

This report describes a method which permits the calculation, directly from the elastic stress-intensity factors of the crack opening area of a two-dimensional crack or the crack opening volume of a penny-shaped crack when subjected to uniform or non-uniform stress distributions. Although stress intensity factors are well documented, especially for two-dimensional cracks (see e.g. [1-3]), it is sometimes necessary to obtain additional information pertaining to particular crack configurations or non-uniform loading systems. A quantity which can be calculated directly from the stress intensity factor is the crack opening area, as shown recently by M't)ller and Harris [4] for uniformly loaded two-dimensional crack configurations. The crack opening area may be of interest, e.g. in estimating leakage rates from cracks in fluid-containing components, such as thin-walled tubes in cooling circuits or chemical plant. Miiller and Harris [4] used the interrelation between the energy release rate and the stress intensity factor to derive an equation for the crack opening area of a two dimensional crack subjected to a uniform applied stress in terms of the corresponding stress intensity factor. In the present paper, starting from the weight function fomaulation, we derive a more general equation for computing the crack opening area of arbitrarily non-uniform loaded two-dimensional crack configurations from stress intensity factors. An analogous relationship for the crack opening volume of a penny-shaped crack subjected to radially symmetric stress distributions is also derived. Application of the results is demonstrated by selected examples. Two-dimensional cracks: Consider first a centre-synametric crack of length 2a in a finite sheet subjected to a non-uniform crack face pressure p(x)=-cy(x) corresponding to the centre-symmetric applied normal stress distribution ~(x) = cy(-x) at the prospective crack site. The coordinate x coincides with the crack line with its origin at the crack centre. Then the stress intensity factors Kin(a) and Kc2~(a) of the same cracked body subjected to two different load cases oo~(x) and ~2~(x), respectively, are interrelated according to Betti's reciprocal theorem and the superposition principle by (cf. e.g. [3])

SINGULAR FUNCTION APPROPRIATE FOR ANALYSIS OF BENDING PROBLEMS OF PLATES WITH CRACKS

In this paper two kinds of singular functions are derived to analyze bending problems of plates with cracks. Assume that an infinite thin elastic plate lies on the xy plane and that the plate contains a crack of length 2a along the y-axis and are loaded on the crack surfaces only. One of the singular functions W1 (x, y, a) is a deflection function in which the slope dW/dx is discontinuous on the crack opening line (-a, a) and deflection, bending moments and shearing forces are continuous. Another is the deflection W2 (x, y, a) of the plate which is discontinuous only on the crack opening. Substituting arbiterary values aj into a of the singular functions and superposing these functions, practical cracks in the plate are realized.

A refined analysis of drift flow of hydrogen to a crack tip

Diffusion of hydrogen to a crack tip is a fundamental process in hydrogen assisted cracking. In the initial stage of diffusion, the stress gradient is more effective than the concentration gradient, and thus, the drift flow is dominant. According to the analysis of the drift flow of solute atoms to a crack tip by lino [l], the number of hydrogen atoms n(t) which arrive at a mode I crack tip within a time t in a state of plane strain (per unit length of the crack front) is given by where Po is the initial uniform concentration in terms of the number of hydrogen atoms per unit volume, ~ Poisson's ratio, v H the partial atomic volume of hydrogen in the solid, D the diffusion coefficient, K I the mode I stress intensity factor; k and T have usual meanings. Eq. (i) is useful within a certain limit of t. In order to determine this limit quantitatively, it is necessary to evaluate (i) the influence of terms other than the singular term O//r term) of the stress field, as well as (ii) the influence of the concentration gradient. Consideration of the influence (i) is needed since only the singular term of the stress field has been taken into account in deriving (i). The purpose of this report is to calculate n(t) using the complete expression of the stress field and to evaluate the influence (i). The influence (ii) will be evaluated separately. Consider an infinite solid containing uniformly distributed hydrogen and an elastic crack of length 2a at y = 0, }x] ~ a. At time t = 0, tensile stress a is applied at infinity in the y direction, and hydrogen atoms begin to move to the crack tips. Neglecting the concentration gradient, the flow velocity V is given by V = -(D/kT) grad E, where E is the elastic interaction energy between a hydrogen atom and the stress field. Using the complete expression of the stress field of the crack [2] instead of only the singular term, we get i