An infinite row of collinear cracks in an infinite elastic sheet

Koiter, W. T.

doi:10.1007/bf00536108

Cited by 140 publications

(77 citation statements)

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“…The expressions of K 0 I and K 0 II here may be found in Tada et al (2000)'s handbook, although they were not given in the original sources (Westergaard, 1939;Koiter, 1959). The expressions of T 0 and B 0 , on the other hand, seem to be new.…”

Section: Expressions Of the Stress Intensity Factors Non-singular Stmentioning

confidence: 92%

“…• In Section 4, we briefly recall the analytical solution of the problem at order 0 in α, identical to Westergaard (1939)'s andKoiter (1959)'s solution for an infinite array of cracks aligned on a straight line.…”

Section: Dmentioning

confidence: 99%

“…The solution to this classical problem was provided by Westergaard (1939) and Koiter (1959) by different methods, and compiled in Tada et al (2000)'s handbook. For completeness and as a guide to the solution of the first-order problem, we briefly recall here (with some slight improvements) Koiter (1959)'s treatment, based on Muskhelishvili (1953)'s method.…”

Section: Zeroth-order Solutionmentioning

confidence: 99%

“…Combining equations (16) and (22) and the expressions (15) 3,4 of the constants Γ and Γ ′ , one gets the final expressions of Φ 0 (z) and Ω 0 (z) (Koiter, 1959):…”

Section: Determination Of Muskhelishvili's Potentialsmentioning

confidence: 99%

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Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem

Leblond

Frelat

2014

Journal of the Mechanics and Physics of Solids

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. Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem. Journal of the Mechanics and Physics of Solids, Elsevier, 2014, 64, pp.133 -153. 10.1016/j.jmps.2013 Development of fracture facets from a crack loaded in mode I+III: solution and application of a model 2D problem AbstractIt is experimentally well-known that a crack loaded in mode I+III propagates through formation of discrete fracture facets inclined at a certain tilt angle on the original crack plane, depending on the ratio of the mode III to mode I initial stress intensity factors. Pollard et al. (1982) have proposed to calculate this angle by considering the tractions on all possible future infinitesimal facets and assuming shear tractions to be zero on that which will actually develop. In this paper we consider the opposite case of well-developed facets; the stress field near the lateral fronts of such facets becomes independent of the initial crack and essentially 2D in a plane perpendicular to the main direction of crack propagation.To determine this stress field, we solve the model 2D problem of an infinite plate containing an infinite periodic array of cracks inclined at some angle on a straight line, and loaded through uniform stresses at infinity. This is done first analytically, for small values of this angle, by combining Muskhelishvili (1953)'s formalism and a first-order perturbation procedure. The formulae found for the 2D stress intensity factors are then extended in an approximate way to larger angles by using another reference solution, and finally assessed through comparison with some finite element results.To finally illustrate the possible future application of these formulae to the prediction of the stationary tilt angle, we introduce the tentative assumption that the 2D mode II stress intensity factor is zero on the lateral fronts of the facets. An approximate formula providing the tilt angle as a function of the ratio of the mode III to mode I stress intensity factors of the initial crack is deduced from there. This formula, which slightly depends on the type of loading imposed, predicts somewhat smaller angles than that of Pollard et al. (1982).

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Section: Expressions Of the Stress Intensity Factors Non-singular Stmentioning

confidence: 92%

Section: Dmentioning

confidence: 99%

Section: Zeroth-order Solutionmentioning

confidence: 99%

“…Combining equations (16) and (22) and the expressions (15) 3,4 of the constants Γ and Γ ′ , one gets the final expressions of Φ 0 (z) and Ω 0 (z) (Koiter, 1959):…”

Section: Determination Of Muskhelishvili's Potentialsmentioning

confidence: 99%

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Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem

Leblond

Frelat

2014

Journal of the Mechanics and Physics of Solids

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“…When we perform the same analysis for systems of different size, we find that strength is overestimated in the regime of small L. This can be corrected by taking into account the periodic boundary conditions in our simulations which lead to self-interactions across the lateral system boundaries, which enhance the stress concentration associated with the "crack phantoms." For a periodic array of mode III cracks with spacing L and length a, the fracture toughness modifies to K = K ∞ /κ(a/L) [17] where…”

Section: B Relation To the Failure Stressmentioning

confidence: 99%

Emergent patterns of localized damage as a precursor to catastrophic failure in a random fuse network

et al. 2013

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We study the failure of disordered materials by numerical simulations of the random fuse model. We identify emergent patterns of localized damage prior to catastrophic failure by statistically averaging the density of damage around the eventual failure nucleation point. The resulting pattern depends on fracture density and obeys the same scaling relations as would be expected for the stress field generated by a critical crack nucleating in a finite, disorder-free effective medium of varying size. The growth of this critical crack absorbs preexisting clusters according to a well-defined scaling relation. Unfortunately, in single model runs such precursory signals are not obvious. Our results imply that reliable and accurate prediction of failure in time-independent, microscopically brittle random materials in a real case is inherently problematic, and degrades with system size.

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System of Cracks in Elasticity Theory

Sharfuddin¹

1975

Z Angew Math Mech

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General methods for solving punch and crack problems in elasticity theory have been developed by Muskhelishvili and others. In this paper general equations concerning a system of cracks have been established making use of Muskhelishvil's method. A closed form solution to the infinite row of collinear cracks has been found and an explicit solution of the problem of an infinite row of parallel cracks has been obtained after reducing it to a Fredholm type of integral equation.

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An infinite row of collinear cracks in an infinite elastic sheet

Cited by 140 publications

References 0 publications

Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem

Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem

Emergent patterns of localized damage as a precursor to catastrophic failure in a random fuse network

System of Cracks in Elasticity Theory

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