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The results in Ref. [1] are not suitable for the cases of β≥2. For this reason, by using the methods in Ref [1] and Ref [2], we derive the general expressions of anisotropic plastic fields at a rapidly propagating plane-stress crack-tip for both the cases of β=2 and β>2.

On condition that any perfectly plastic stress component at a crack tip is nothing but the function of η, by making use of equilibrium equations, anisotropic plastic stress-strain-rate relations, compatibility equations and Hill anisotropic plastic yield condition, in the present paper, we derive the generally analytical expressions of the anisotropic plastic stress field at a mixed-mode crack tip under plane and anti-plane strain. Applying these generally analytical expressions to the mixed-model cracks, we can obtain the analytical expressions of anisotropic plastic stress fields at the tips of mixed-mode I–III, II–III and I–II–III cracks.

The results in Ref. [1] are not suitable for the cases of α≥2. For this reason, we use the method in Ref. [1] to derive the general expressions of the anisotropic plastic stress fields at a stationary plane-stress crack-tip for both of the cases of α=2 and α>2. As an example, we give the analytical expressions of the anisotropic plastic stress fields at the stationary tips of mode I and mode II plane-stress cracks for the case of α=2.

Under the condition that all the stress components at a crack-tip are the functions of θ only, making use of the equations of steady-state motion, stress-strain relations and Hill anisotropic yield conditions, we obtained the general solutions at a crack-tip in both the cases of anti-plane and in-plane strains. Applying these general solutions to the concrete cracks, the anisotropic plastic fields at the rapidly propagating tips of mode III and mode I cracks are derived.

Under the condition that all the stress components at a crack-tip are the functions of θ only, making use of the equations of steady-state motion. Hill anisotropic yield condition and stress-strain relations, we obtain the general solution of anisotropic plastic field at a rapidly propagating plane-stress crack-tip. Applying this general solution to four particular cases of anisotropy, the general solutions of these four particular cases are derived. Finally, we give the anisotropic plastic field at the rapidly propagating plane-stress mode I crack-tip in the case of X=Y=Z

Under the condition that all the stress components at a crack-tip are the functions of θ only, making use of equilibrium equations and hydrostatic stress-dependent yield condition, in this paper, we derive the generally analytical expressions of the hydrostatic stress-dependent perfectly-plastic stress fields at a stationary plane-stress crack-tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of hydrostatic stress-dependent perfectly-plastic stress fields at the tips of mode I and mode II cracks are obtained.

Under the condition that all the perfectly plastic stress components at a crack tip are the functions of ϑ only, making use of equilibrium equations and Von-Mises yield condition containing Poisson ratio, in this paper, we derive the generally analytical expressions of perfectly plastic stress field at a stationary plane-strain crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the stationary tips of Mode I, Mode II and Mixed-Mode I–II plane-strain cracks are obtained. These analytical expressions contain Poisson ratio.

All the stress components at a rapidly propagating crack-tip in elastic perfectly-plastic material are the functions of θ only. Making use of this condition and the equations of steady-state motion, plastic stress-strain relations, and Mises yield condition with Poisson ratio, in this paper, we derive the general expression of perfectly plastic field at a rapidly propagating plane-strain crack-tip. Applying this general expression with Poisson ratio to Mode I crack, the perfectly plastic field at the rapidly propagating tip of Mode 1 plane-strain crack is obtained. This perfectly plastic field contains a Poisson ratio, and thus, we can obtain the effect of Poisson ratio on the perfectly plastic field at the rapidly propagating tip of Mode I plane-strain crack.

Under the condition that all the perfectly plastic stress components at a crack tip are the functions of θ only, making use of the Mises yield condition, steady-state moving equations and elastic perfectly-plastic constitutive equations, we derive the generally analytical expressions of perfectly plastic fields at a rapidly propagating plane-stress crack tip. Applying these generally analytical expressions to the concrete crack, we obtain the analytical expressions of perfectly plastic fields at the rapidly propagating tips of modes I and II plane-stress cracks.

In this paper, we proposed a new criterion of mixed-mode brittle fracture, i.e., the strain energy criterion, which can be stated as (KI/KIc)2+(KI/KIc)2+(KI/KIc)2=1. This criterion is shown to be in good agreement with known experimental data.

On condition that any perfectly plastic stress component near a singular point is nothing but the function of θ only, making use of equilibrium equations and Hill anisotropic yield condition, we derive the general analytical expressions of the anisotropic plastic stress field near a singular point in both the cases of anti-plane and in-plane strains. Applying these general analytical expressions to the concrete cracks and the plane-strain bodies with a singular point, the anisotropic plastic stress fields at the tips of Mode I, Mode II, Mode III and mixed mode I–II cracks, and the limit loads of anisotropic plastic plane-strain bodies with a singular point are obtained.

Under the condition that any perfectly plastic stress components at a crack tip are nothing but the functions of 0 only making use of equilibrium equations. Hill anisotropic yield condition and unloading stress-strain relations, in this paper, we derive the general analytical expressions of anisotropic plastic stress fields at the slowly steady propagating tips of plane and anti-plane strain. Applying these general analytical expressions to the concrete cracks, the analytical expressions of anisotropic plastic stress fields at the-slowly steady propagating tips of Mode I and Mode III cracks are obtained. For the isotropic plastic material, the anisotropic plastic stress fields at a slowly propagating crack tip become the perfectly plastic stress fields.

Under the hypothesis that all the perfectly plastic stress components at a crack tip are the functions of θ only, making use of yield conditions and equilibrium equations, we derive the generally analytical expressions of the perfectly plastic stress field at a crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the tips of Mode I. Mode II, Mode III and Mixed Mode I–II cracks are obtained.

It is well-known that the present mixed mode brittle fracture criteria are all the opening mode fracture criterion. We consider that mixed mode brittle fracture of sliding mode fracture exists too. Hence we propose three criteria of mixed mode brittle fracture of sliding mode fracture;: the radial shearing stress criterion, the maximum shearing stress criterion and the distortional strain-energy-density criterion. Thus, we can overall explain the phenomena of brittle fracture in the structural elements with cracks.

Under the condition that all the perfectly plastic stress components at a crack tip are the functions of θ only, making use of the Treasca yield condition, steady-state moving equations and elastic perfectly-plastic constitutive equations, we derive the generally analytical expressions of perfectly palstic stress field at a rapidly propagating plane-stress crack tip. Applying these generally analytical expressions to the concrete crack, we obtain the analytical expressions of perfectly plastic stress field at the rapidly propagating tips of models I and II plane-stress cracks.