Three‐dimensional stress fields due to notches in plates under linear elastic and elastic–plastic conditions

Abstract

The paper deals with a 3D multi-parametric stress field representation ahead of notches in plates of finite thickness under different loading conditions. Under certain hypotheses, the 3D governing equations of elasticity can be reduced to a system where a bi-harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, and the latter provides the solution of the corresponding out-of-plane shear notch problem. The solution is valid close the notch edge, through the plate thickness, with exclusion of a very limited zone close to the free surface of the plates. Also, under elastic-plastic conditions there are circumstances where it is possible to separate the in-plane problem from the out-of-plane problem. A new analytical frame is proposed, and the results are compared with those obtained from 3D finite element models of a plate with a square notch. In the presence of remote applied tensile stress, the slope of the induced, out-of-plane, shear stress component matches that of the mode III plastic problem

Determination of mode III fracture toughness for different materials using a new designed test configuration

“…K II and K III terms) is seen in the crack front. This phenomenon which occurs due to the three-dimensional effects of stress field along the crack front of a thick plate has also been reported for other cracked or notched specimens [37][38][39][40][41][42][43]. For example, Pook et al [40,41], He et al [42], Lazzarin and his coworkers [43] have demonstrated the existence of coupled mode II and III components in the crack front of 3D plates subjected to anti-plane loading.…”

On the stress state in rectilinear anisotropic thick plates with blunt cracks

“…• To show that the approximate 3D solution derived by Zappalorto and Carraro 46 can be extended also to radiused cracks, provided that the notch tip radius is sufficiently small; • To show that the in plane stress fields, σ xx , σ yy and σ xy in a thick 3D anisotropic notched plate can be accurately assessed on the basis of the plane solution, as derived by Zappalorto and Carraro 22 ; • to determine the exact close form solution to the pure antiplane problem for a blunt crack in an anisotropic body; • to prove the existence of coupled modes, as highlighted by previous papers for isotropic bodies, [34][35][36][37][38][39][40][41][42][43][44][45] also for monoclinic plates with blunt cracks. • To provide analytical expressions of the Generalised Stress Intensity Factors for orthotropic plates with blunt cracks under mode I, II and III.…”

“…[34][35][36][37][38][39][40][41][42][43][44][45] On parallel tracks, a new threedimensional theory to be applied to thick anisotropic plates with pointed V-notches was developed by Zappalorto and Carraro 46 and Zappalorto et al 47 According to this solution, the 3D governing equations of elasticity can be reduced to two uncoupled equations in the two-dimensional space; the former is linked to the solution of the corresponding plane notch problem, the latter to the solution of the corresponding out-of-plane shear notch problem. [34][35][36][37][38][39][40][41][42][43][44][45] On parallel tracks, a new threedimensional theory to be applied to thick anisotropic plates with pointed V-notches was developed by Zappalorto and Carraro 46 and Zappalorto et al 47 According to this solution, the 3D governing equations of elasticity can be reduced to two uncoupled equations in the two-dimensional space; the former is linked to the solution of the corresponding plane notch problem, the latter to the solution of the corresponding out-of-plane shear notch problem.…”

Antiplane contact problems of a half-space with a semielliptic hole

“…In defective or weak surfaces in contact there may also be preexisting holes or cracks. Considering solutions for notches, we find important works of Lazzarin et al [11,12] and Berto and Lazzarin [13]. A discussion on stress in piezoelectric media with an elliptic hole is given in [14].…”