Stress state in the vicinity of an inclined elliptical defect and stress intensity factors for biaxial loading of a plate

Abstract

The problem of determining the stress state of a plate with an inclined elliptical notch under biaxial loading is considered. The Kolosov-Muskhelishvili method is used to obtain an expression for the stress near the vertex of an inclined ellipse, whose particular case are expressions for the stress in the case of an inclined crack. The stress intensity factors K I and K II were determined experimentally by holographic interferometry in the case of extension of a plate with an inclined crack-like defect. The calculation results are compared with experimental data.Key words: fracture mechanics, stress intensity factors, Kolosov-Muskhelishvili method, stress state, plate with an inclined elliptical notch, holographic interferometry.Introduction. The fracture stress criteria can be classified as simple and complex. The simple criteria take into account only the singular stress component and are expressed in terms of stress intensity factors (SIFs). The complex criteria are obtained by a refined analysis of the stress state (SS) near the tip a crack-like defect and take into account the regular stress component [1]. In the case of slant cracks, a defect is modeled by an elliptical notch. The most widely used and best validated approach is the one in which the SS characteristics should be determined not at the crack tip but at a certain distance r 0 from it [1,2]. Holographic interferometric studies [3] using complex criteria of the limiting equilibrium of an inclined crack [1] have provided experimental data which are in better agreement with the results of theoretical studies [1]. The error was 4.7%. Eftis and Subramonian [1] obtained a more exact expression for the Westergaard function contained in the singular solution for an inclined crack in a plate under biaxial loading [1]. This approximate two-component solution is fairly exact. Theocaris and Michopoulos [2] obtained an exact solution of the problem for the stress tensor in a plane with an inclined crack under biaxial loading using expressions for the complex potentials Φ(z) and Ω(z) [4].The relative error of the sum σ 1 + σ 2 and the difference σ 1 − σ 2 in the principal stresses obtained using both methods was determined in [2]. The effect of the biaxial loading ε on the stress tensor components σ x , σ y , and τ xy and the maximum shear stress τ max has been studied using the Kolosov-Muskhelishvili method [4]. Theocaris and Spyropoulos [5] presented isochromatic patterns and gave a review of photoelasticity methods for a plate with an inclined crack. In the present paper, the results [2] for a plate with an elliptical notch are extended to the biaxial loading of the plate. Expressions for the stress tensor components are obtained, the effect of the ellipse parameter m for the biaxial loading of a plate with an inclined elliptical notch is studied, and experimental data obtained using holographic interferometry are given.Methods of fracture mechanics allow one to determine the fracture resistance of metal welded joints in the presence of cracks. However, a f...

Stress-strain state of an inclined elliptical defect in a plate under biaxial loading

“…The expressions for the stresses in polar coordinates are obtained by substituting the expressions for the stress tensor components σ x , σ y , and τ xy [5] into the well-known formula σ r = σ x cos 2 θ + σ y sin 2 θ + τ xy sin 2θ,…”

“…Using the formulas obtained in [5] for the complex potentials Φ(z) and Ψ(z) and the formulas given in [3] we can calculate the displacements:…”

“…Using formula (4) and the expressions for the potentials Φ(z) and Ψ(z) [5], after several transformations we obtain exact formulas for the displacements:…”

“…and the formulas for the stresses σ x , σ y , and τ xy [5], after a series of transformations (7) we express τ max as…”

“…Direct use of fracture mechanics to study the stress concentration characteristic of welds is not valid [4]. In [5], exact formulas for the stress tensor components, the sum σ x + σ y , and the difference σ x − σ y of stresses in Cartesian coordinates were obtained using the Kolosov-Muskhelishvili method, and stress intensity factors for a plate with an inclined elliptical defect under uniaxial tension were determined by holographic interferometry.…”