The classic nonlinear fracture problem of a fully yielded, mixed mode stationary crack in a power law hardening material for conditions of plane stress under small-scale yielding is reconsidered. It has been determined that two different asymptotic solutions are required to represent the full range of mixed mode loading. Neither asymptotic solution has the double root of the linear elastic counterpart, i.e., the nonlinear plane stress problem does not have a mixed mode asymptotic solution. The mode II dominant asymptotic solution consists of two terms. The leading term is the pure mode II HRR term, while the second term is symmetric with an eigenvalue slightly weaker than the HRR eigenvalue. This two-term solution applies to a relatively large range of mixed mode loading. The mode I dominant asymptotic solution also consists of a symmetric and an antisymmetric term with different eigenvalues, and has a limited range of applicability near mode I. The pure mode I HRR term is the symmetric term. Contrary to expected behavior based on energy considerations and experience with higher order solutions, the antisymmetric term has an eigenvalue that is stronger than the HRR eigenvalue. This antisymmetric asymptotic solution, which cannot exist without the presence of the mode I HRR term, depends on two small parameters: the distance from the crack tip, r, and the ratio of mode II to mode I loading, K₂/K₁. The interpretation is that this two-term asymptotic solution exists for small r in the limit as K₂/K₁ approaches zero. An unusual feature of the second term is that it does not exist in the limit as r approaches zero, and therefore from a mathematical point of view this term does not cause the J-integral to be infinite. The asymptotic results are confirmed with full-field finite element analysis by using the J₂ deformation theory of plasticity using a computational domain that covers eleven decades of radial detail. This validates the asymptotic solutions and shows that a two-parameter fracture theory can be used very near mode I and near mode II. The transition from one asymptotic solution to the other, which is demonstrated to occur near mode I, gives rise to a loss of dominance of these two-term asymptotic solutions. The hardening exponent, “n”, plays an important role in the ranges of validity of the two asymptotic solutions. Finally, the asymptotic solutions are shown to agree with solutions from the non-hardening limit, and the comparisons are consistent with those of the full-field results.

考虑了小屈服下平面应力条件下幂律硬化材料中完全屈服的混合模式固定裂纹的经典非线性断裂问题。已经确定需要两种不同的渐近解来表示混合模式加载的整个范围。两种渐近解都不具有线性弹性对应物的双根，即非线性平面应力问题不具有混合模渐近解。模式II主导渐近解由两个项组成。前项是纯模式II HRR项，而第二项是对称的，其特征值比HRR特征值稍弱。这两个解决方案适用于较大范围的混合模式加载。模式I主导渐近解还由具有不同特征值的对称项和反对称项组成，并且在模式I附近具有有限的适用范围。纯模式I HRR项是对称项。与基于能量考虑和具有高阶解决方案的经验的预期行为相反，反对称项的特征值强于HRR特征值。这种不对称渐近解在没有模式I HRR项的情况下是不存在的，它取决于两个小参数：距裂纹尖端的距离r和模式II与模式I载荷的比，与基于能量考虑和具有高阶解决方案的经验的预期行为相反，反对称项的特征值强于HRR特征值。这种不对称渐近解在没有模式I HRR项的情况下是不存在的，它取决于两个小参数：距裂纹尖端的距离r和模式II与模式I载荷的比，与基于能量考虑和具有高阶解决方案的经验的预期行为相反，反对称项的特征值强于HRR特征值。这种不对称渐近解在没有模式I HRR项的情况下是不存在的，它取决于两个小参数：距裂纹尖端的距离r和模式II与模式I载荷的比，K ₂ / K ₁。解释是，当K ₂ / K ₁接近零时，对于极限中的小r存在该两项渐近解。第二项的一个不寻常的特征是，当r接近零时，第二项不存在于极限中，因此从数学角度来看，该项不会导致J积分为无穷大。使用J ₂通过全场有限元分析来确认渐近结果可塑性变形理论，使用的计算域涵盖了十一个径向细节。这验证了渐近解，并表明可以在非常接近模式I和非常接近模式II的情况下使用两参数断裂理论。从一种渐近解到另一种渐近解的过渡（已证明发生在模式I附近）会导致这些两项渐近解的支配性丧失。硬化指数“ n”在两个渐近解的有效性范围内起着重要作用。最后，渐近解显示与非硬化极限的解一致，并且比较与全场结果一致。