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

In this paper, we investigate the effects of the non-singular stress (T stress) on the mode I near-tip fields lbr elastic perfectly plastic pressure-sensitive materials under plane-stress and small-scale yielding conditions. The T stress is the normal stress parallel to the crack faces. The yield criterion for pressure-sensitive materials is described by a linear combination of the effective stress and the hydrostatic stress. Plastic dilatancy is introduced by the normality flow rule. The results of our finite element computations based on a two-parameter boundary layer formulation show that the total angular span of the plastic sectors of the near-tip fields increases with increasing T stress for materials with moderately la~rge pressure sensitivity. The T stress also has significant effects on the sizes and shapes of the plastic zones. The height of the plastic zone increases substantially as the T stress increases, especially for materials with large pressure sensitivity. When the plastic strains are considered to be finite as lbr transformation toughened ceramics, the results of our finite element computations indicate that the phase transformation zones for strong transformation ceramics with large pressure sensitivity can be approximated by those lbr elastic-plastic materials with no limit on plastic strains. When the T stress and the stress intensity factor K are prescribed in the two-parameter boundary layer formulation to simulate the crack-tip constraint condition tor a single-edge notch bend specimen of zirconia ceramics, our finite element computation shows a spear shape of the phase transformation zone which agrees well with the corresponding experimental observation. 1. I n t r o d u c t i o n In classical Linear Elastic Fracture Mechanics (LEFM), it is assumed that the fracture processes occurring close to a crack tip are governed by the far-field stress intensity factor K . Then the elastic-plastic problem under small-scale yielding is replaced by a boundary layer formulation where the boundary condition is set to either the traction or displacement given by the elastic asymptot ic crack-tip field which is scaled by the stress intensity factor K . Larsson and Carlsson [ 1 ] investigated the plane-strain crack-tip fields for compact tension, bend, double edge-cracked, and center-cracked specimens by finite e lement computat ions. They found differences of 10 to 30 percent in fracture surface displacement, normal stress, and plastic zone size between the finite e lement solutions of the specimens. As suggested by Rice [2], Larsson and Carlsson [1] were able to eliminate the differences in the fields by introducing the non-singular stress term in their two-parameter boundary layer formulat ion where the remote traction or displacement is expressed in terms of K and the transverse T stress. The J integral (Rice [3]) and the HRR crack-tip fields (Hutchinson [4, 5]; Rice and Rosengren [6]) provide the basis for nonlinear fracture mechanics. An excellent review of the , /-based fracture mechanics can be found in Hutchinson [7]. McMeeking and Parks [8] and 224 Z.E.A. Ben-Aoun and J. Pan Shih and German [9] investigated the requirement of one parameter characterization and the .]-dominance of the crack-tip fields. Recently, Beteg6n and Hancock [ 10] attempted to characterize the crack-tip fields for hardening materials under mode I plane-strain conditions using J and T. Their results show that geometries which maintain J-dominance are characterized by zero or positive T stresses, while geometries with negative T stresses can be characterized by J and T under fully yielded conditions. Al-Ani and Hancock [11] showed that edge-cracked bars lose ,]-dominance for the crack length to width ratios less than 0.3 in bending and 0.5 in tension. The loss of J-dominance can be explained by negative T stresses, while geometries with positive T stresses retain J-dominance under fully yielded conditions. The T stress has been subsequently validated by experimental results as an appropriate parameter to characterize crack-tip constraint conditions even under fully yielded conditions (Hancock, Reuter, and Parks [ 12]). Also the angular variations of the mode I near-tip stresses as functions of the T stress were studied by Du and Hancock [13] for elastic perfectly plastic materials under plane strain conditions. Their results show that the crack-tip field appears to be an incomplete Prandtl field with elastic sectors bordering the crack faces. However, they did not attempt to characterize the angular variations of the near-tip stresses in their elastic sectors as in Dong and Pan [14] and Kim and Pan [15]. Kim and Pan [15] resolved the asymptotic structures of the mode I near-tip fields for pressure-sensitive materials under plane strain conditions. Their numerical results show that the solution of the elastic sector of finite stress given by Dong and Pan [ 14] are essential to obtain the complete angular variation of the near-tip stresses under small-scale yielding conditions. The numerical results of Kim and Pan [15] also show the effects of the T stress on the near-tip fields and the sizes and shapes of plastic zones for pressure-sensitive materials. O'Dowd and Shih [16, 17] investigated the J Q annulus concept using a two-parameter boundary layer formulation under plane strain conditions. Their work shows that J sets the size scale over which large stresses and strains develop and Q is the measurement of the stress triaxiality ahead of the tip. Negative (positive) Q values mean that the hydrostatic stress is decreased (increased) by Qcr0 from the Q = 0 reference state (here, o0 being a reference stress). Therefore Q provides a quantitative measure of the crack-tip constraint. The J Q characterization of the near-tip fields can be used from small-scale yielding to fully yielded conditions. Most of the research discussed above on the constraint conditions for pressure-insensitive Mises materials and on the near-tip fields for pressure-sensitive materials are under plane strain conditions. There are asymptotic analyses of near-tip fields under plane stress conditions. For example, Hutchinson [5] obtained the asymptotic crack-tip field for power-law hardening and perfectly plastic Mises materials. Pan and Shih [18] and Pan [191 obtained the asymptotic crack-tip fields for orthotropic materials. Li and Pan [20] obtained the asymptotic crack-tip fields for pressure-sensitive materials. Recently, Ben Aoun and Pan [21] obtained the neartip fields for elastic perfectly plastic pressure-sensitive materials under small-scale yielding conditions by both finite element computations and asymptotic analyses. In this paper, we study the effects of the non-singular stress on mode I near-tip fields for pressure-sensitive materials under plane-stress and small-scale yielding conditions. We first investigate the near-tip fields for elastic perfectly plastic pressure-sensitive materials by finite element methods using the two-parameter boundary layer formulation characterized by K and T under monotonically increasing proportional loading conditions. The asymptotic near-tip stress fields are assembled and compared with the finite element results. Then we Figure 1. The Coulomb-type and Mises Influences o f non-singular stress 225