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The influence of biaxiality of loading on the form of caustics in cracked plates

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Summary

The optical method of reflected caustics was applied up-to-now to problems of cracked plates under uniaxial loading. Only the problem of the biaxial tension of the plate has been considered for the particular case where the crack is transverse to the longitudinal axis of the plate which coincided with the loading axis. In this paper the influence of a biaxial loading of the plate on the form and orientation of the caustic was studied in connection with the orientation of the crack. New modified relations were given for the evaluation of the complex stress intensity factorK=K I −iK II in terms of the angle ϕ of the angular displacement of the caustic axis. For the accurate evaluation ofK I andK II nomograms of correction factorsδ max y′ ,δ max x′ andδ max x′ were given in terms of the angle of inclination of the crack ω=(90−β) and the biaxiality factork. Experimental evidence with PMMA internally cracked plates corroborated the results of theory.

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Abbreviations

Φ(z), Ω(z):

complex-stress function of Muskhelishvili

σ xx ,σ yy ,τ xy :

crack tip stress referred to Cartesian coordinate system

r, ϑ:

polar coordinate system centered at crack tip

K I ,K II :

stress intensity factors for ModeI andII loading, respectively

ω:

angle of inclination of the crack

β:

90°-ω

k :

ratio of stresses at infinity

σ 1,σ 2 :

principal stresses at crack tip

a :

crack length

σ:

stress applied at infinity along the transverse boundaries of the plate

X ′r,f ,Y ′r,f :

parametric equations of the reflected caustics referred to the Cartesian systemO′X′Y′ on the reference screen: (r) reflected caustics from rear face of the specimen and (f) reflected caustics from the front face of the specimen

r 0 :

radius of the generatrix curve on the specimen around the crack tip (initial curve)

c r,f :

optical constants of the material for reflections from the rear and front faces of the specimen respectively

Λ m :

magnification ratio of the optical set-up

z 0 :

distance between the reference-screen and the middle plane of the specimen

z i :

distance between the focus of the light beam and the middle plane of the specimen

d :

thickness of specimen

ε:

2 for the reflected caustics from the rear face of the specimen and 1 for the reflected caustics from the front face of the specimen

C r,f :

εz 0 dc r,f /λ m (2π)1/2

v :

Poisson's ratio

E :

elastic modulus of the material

A :

(1+k)+(1−k) cos 2ω

B :

(1−k) sin 2ω

C :

1+k 2+(1−k 2) cos 2ω 2 tan−1 (B/A)=2 tan−1 (K II/KI)

D max y′ ,D max x′ ,D min x′ :

the maximum and the ninimum diameter of caustics along the axisO′y′ andO′x′ of the crack respectively

δ max y′ ,δ max x′ ,δ min x′ :

the correction factors forD max y′ ,D max x′ andD min x′ respectively

D max t ,D max l :

the maximum transverse and longitudinal diameters of the caustics respectively

δ max t ,δ max l :

the correction factors forD max t ,D max l respectively

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Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, The National Technical University of Athens, 5, K. Zographou Street Zographou, 624, Athens, Greece

    P. S. Theocaris & G. A. Papadopoulos

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  1. P. S. Theocaris
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  2. G. A. Papadopoulos
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Theocaris, P.S., Papadopoulos, G.A. The influence of biaxiality of loading on the form of caustics in cracked plates. Acta Mechanica 44, 201–222 (1982). https://doi.org/10.1007/BF01303338

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