Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials, Part 1: Plane strain case

Abstract

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.

Effect of second non‐singular term of mode I near the tip of a V‐notched crack

“…Furthermore, the size and shape of the plastic zone under plane strain conditions for a zero opening crack should be determined. The mode I stress fields are defined by substituting the stress intensity factors of ( K n I /σ o ) 2 = 0.112, 0.132 and 0.323 and the V‐stresses 4 V /σ o = 0.5, 0 and −0.5 used by Kim and Pan 18 into ; the plastic zone is determined by substituting these stresses into the Mises yield criterion 18 : where σ eq is the Mises equivalent stress and σ zz is the out‐of‐plane normal stress, and under plane stress condition, σ zz = 0, and in plane strain condition, σ zz =ν (σ rr +σ θθ ). If σ eq becomes equal to the uniaxial tensile yield stress, σ o , , the yielding occurs.…”

Influences of non-singular stresses on plane-stress near-tip fields for pressure-sensitive materials and applications to transformation toughened ceramics

“…It turns out that the T stress has significant effects on the stresses and hydrostatic tension directly ahead of the tip for materials with small pressure sensitivity. However, as the pressure sensitivity increases, the effects on the stresses and hydrostatic tension directly ahead of the tip decreases, as shown in [15]. In general, when the value of T decreases, the stresses and hydrostatic tension directly ahead of the tip decrease.…”

“…On the other hand, for perfectly plastic materials under plane strain conditions, a constant stress sector is located ahead of the tip. The angular span of the front constant stress sector is determined for a given pressure sensitivity [15]. Therefore the T stress has no influence on the angular span of the front constant stress sector.…”

“…In general, when the value of T decreases, the stresses and hydrostatic tension directly ahead of the tip decrease. The T stress can also affect the sizes and shapes of plastic zones as shown in [15] under plane strain conditions.…”

“…In this study, we account for pressure-sensitive yielding or phase transformation by a Coulombtype criterion [23] which is a linear combination of the mean stress a,~ (= crkk/3) and the tensile effective stress cr~ (= (38ijsij/2) 1/2 where 8ij = aij -a~ij ), as in [15]. Here, ~Sij is the Kronecker delta and subscripts i, j, and k have the range of 1 to 3.…”