Sommario
Si considera il problema piano di due «craks» simmetrici all'interfaccia tra una inclusione circolare rigida e la circostante matrice, in regime di carico biassiale. Facendo uso della tecnica delle variabili complesse, viene risolto il problema dei valori al contorno, ricavando altresi le espressioni delle tensioni e degli spostamenti lungo il contorno dell'inclusione. Inoltre, applicando un criterio tensionale, già proposto dagli autori, che consente di studiare sia la estensione del crack all'interfaccia che la sua deviazione nella matrice, si analizza la risposta del sistema considerato. I valori critici dei carichi applicati, nonchè l'angolo di frattura, risultano espressi in funzione dell'angolo al centro sotteso dai cracks.
Infine, vengono illustrati graficamente e discussi gli effetti del carico laterale.
Summary
The problem of two symmetric arc cracks lying between a rigid circular inclusion embedded in an infinite matrix under biaxial loading at infinity is considered. By using the complex variable tecnique, the boundary value problem is solved and stress and displacement components are calculated along the inclusion boundary. Moreover, investigating the local stress field, a stress criterion, already proposed by authors, allowing either the crack extension at the interface or its deviation into the matrix to be taken into account, is applied to study the fracture response of the elastic system. The critical applied loads as well as the angle of the incipient crack extension are expressed in terms of the central angle subtended by the crack arcs.
Finally the biaxial load effects are graphically shown and discussed.
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Abbreviations
- S +,S − :
-
regions occupied by the matrix and inclusion respectively
- r 0 :
-
radius of the inclusion
- Γ:
-
boundary of the inclusion
- Γ S :
-
arc cracks
- Γ D :
-
remainder arcs of Γ
- μ, k :
-
elastic properties of the matrix
- ν :
-
Poisson's ratio
- λ :
-
lnk/2π
- z :
-
complex variable
- x, y :
-
cartesian coordinates
- r, ϕ :
-
polar coordinates
- ρ, α :
-
local polar coordinates
- u r,u ϕ :
-
polar components of the displacement
- u * r ,u *ϕ :
-
dimensionless displacements
- u, v :
-
cartesian components of the displacement
- 2ϑ :
-
central angle subtended by the arc crack
- ψ :
-
angle betweenT-direction andx-axis
- σ :
-
point of the circumference Γ
- σ 1,σ 2,σ 3,σ 4 :
-
terminal points of the crack arcs
- ()+, ()− :
-
limit values of () asz →σ fromS + andS − respectively
- α 0 :
-
angle of crack extension
- D :
-
cartesian complex displacement
- Φ(z), Ψ(z):
-
complex potentials
- x(z):
-
Plemelj function
- A 1,A 2 :
-
Constants appearing in the solution of the
- C 0,C 1,C 2 :
-
Hilbert problem
- ε :
-
angle of rotation of the inclusion
- ∈∞ :
-
rotation at infinity
- P(z):
-
polynomial appearing in the solution of the Hilbert problem
- K (sinϑ),E (sinϑ):
-
complete elliptic integrals
- E (β, sinϑ),E (β, sinϑ):
-
incomplete elliptic integrals
- m (ϑ):
-
parameter appearing in potential function
- Km:
-
fracture toughness of the matrix
- K b :
-
bond strength parameter
- K 1,K 2 :
-
stress intensity factors
- K :
-
complex stress intensity factor
- N′:
-
real constant
- M :
-
moment of the stress applied to the boundary Γ
- σ rr,σ ϕϕ,σ rϕ :
-
polar components of stress
- σ ρρ,σ αα,σ αρ :
-
local polar components of stress
- σ *αα :
-
dimensionless circumferential stress
- N, T :
-
principal stresses at infinity
- s :
-
biaxial load parameter
- N*:
-
critical value of the dimensionless compression
References
England A. H.,An Arc Crack Around a Circular Elastic Inclusion. J. Appl. Mech., Vol. 33, 1966, pp. 637–640.
Perlman, A. B., Sih, G. C.,Elastostatic Problems of Curvilinear Cracks in Bonded Dissimilar Materials. Int. J. Engng. Sci., Vol. 5, 1967, pp. 845–867.
Sendeckyj, G. P.,A class of Interface Crack Problems, ASTM, STP 560, 1974, pp. 92–104.
Toya, M.,A Crack Along the Interface of a Rigid Circular Inclusion Embedded in an Elastic Solid. Int. J. Fracture, Vol. 9, 1973, pp. 463–470.
Toya, M.,A Crack Along the Interface of a Circular Inclusion Embedded in an Infinite Solids. J. Mech. Phys. Solids, Vol. 22, 1974, pp. 325–348.
Toya, M.,Debonding along the Interface of an Elliptic Rigid Inclusion. Int. J. Fracture, Vol. 11, 1975, pp. 989–1002.
Testa, R. B. andStubbs, N.,Bond Failure and Inelastic Response of Concrete. J. Engng. Mech. Division, EM2, Vol. 103, 1977, pp. 295–310.
Muskhelishvili, N. I.,Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff Ltd, Groningen, Holland, 1953.
Piva, A., Viola, E.,Biaxial Load Effects on Crack Between Dissimilar Media. Eng. Fract. Mech., Vol. 13, 1980, pp. 143–174.
Viola, E., Piva, A.,Plane Strain Interfacial Fracture Analysis of a Bimaterial Incompressible Body. Eng. Fract. Mech., 1980 (submitted for publication).
Sendeckyj, G. P.,Debonding of Rigid Curvilinear Inclusion in Longitudinal Shear Deformation. Eng. Fract. Mech., Vol. 6, 1974, pp. 33–45.
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Financial support of the National Research Council (C.N.R.) (research contribution N. 79.01625.07) is gratefully acknowledged.
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Viola, E., Piva, A. Two arc cracks around a circular rigid inclusion. Meccanica 15, 166–176 (1980). https://doi.org/10.1007/BF02128927
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DOI: https://doi.org/10.1007/BF02128927