In this paper, a method for determining the limit load surface of fiber-reinforced nonlinear composites such as elasto-plastic composite is proposed. Using the stress approach of the homogeneous theory and the linear matching method (LMM), the limit load surface of the fiber-reinforced composite is numerically evaluated in the π-plane, and at the same time, two limit analyses determine the approximate Hill's anisotropic yield criterion for the limit load surface of the fiber-reinforced composite. The Hill's yield criterion determined by 2 limit analyses becomes the inscribed ellipse of the limit load surface evaluated numerically in the π- plane, and the limit load surface can be evaluated more accurately by the two tangent lines perpendicular to the minor axis of the inscribed ellipse and the circumscribed circle of the inscribed ellipse. This means that the limit load surface of fiber-reinforced nonlinear composite can be completely determined by only 2 limit analyses. In addition, it is found that the limit load surface is related to the equivalent strength surface of composite, and that it satisfies the Reuss and Voigt bounds.

In this paper, a smoothed weighted function with the compact support is suggested to improve the numerical accuracy of blending elements for the extended finite element method (XFEM) applied to the crack problem. Such a weight function consists of the shape functions used for the meshless method, reflecting the characteristics of the enriched bases inside the blending element. The effectiveness of this weight function is validated through the numerical examples.

Linear matching method (LMM) is one of the effective numerical methods for the limit and shakedown analysis, which computes the converging upper series by solving iteratively linear elastic analysis with Young’s modulus varying in space. LMM, which is capable of implementing by using the commercial finite element packages such as ABAQUS and ANSYS has been widely used in the practical applications including fatigue, creep and composite analysis. Nevertheless, LMM could not ensure numerical accuracy for discontinuous problems such as crack analysis since it computes the mechanical quantities including stress, strain and displacement on the base of conventional finite element method, likewise other numerical methods for the limit and shakedown analysis. Meanwhile, extended finite element method (XFEM), recently proposed, is an attractive numerical method for the analysis of discontinuous problems which enriches finite element approximate space by some special functions. In this paper, authors proposed a very straightforward method for the limit analysis by the combination of XFEM with LMM. Numerical validation is done for two types of typical fracture specimens. Numerical examples show that the limit analysis by combining XFEM with LMM gives more accurate result compared with the one by combining of conventional finite element method with LMM. Furthermore, we demonstrated that the choice of enrichment region plays an important role in the improvement of numerical accuracy of our proposed method.