Unified characterization of failure surfaces and golden-ratio ductile-to-brittle classification for isotropic materials

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The failure surface, often referred to as the failure angle, defines the specific planar orientation within a material that reaches its load-carrying capacity, represented by material strengths. Analyzing this critical surface is elemental for material characterization, providing profound insights into ductility and brittleness. Despite the diversity of methodologies employed for determining the failure plane from various criteria, a universally accepted theory that systematically governs this characteristic across a broad spectrum of isotropic materials remains elusive. Therefore, this paper aims to develop a unified framework for predicting the failure surface for all homogeneous isotropic solids by considering the convergence of three key material constitutive models: elasticity, failure, and plasticity. A universal energy-based failure criterion is utilized to determine failure angles under fundamental loading scenarios, including uniaxial tension, uniaxial compression, pure shear, and biaxial tension–compression. The sliding, splitting, and crushing behaviors are obtained from the direct and shear strain increments, while the ratio of the two strain increments elucidates the dominant roles in ductile and brittle failure modes. For the first time, the developed theory links the failure angle to Poisson’s ratio, and uniaxial strength properties, unveiling a connection between intrinsic material parameters and extrinsic ductility and brittleness induced by external loadings. The failure angle representing ductile-to-brittle transition under the applied stresses in the principal stress coordinates is shown to be directly related to the golden ratio and independent of loading types. This research addresses longstanding mysteries by providing a deeper understanding of the physics of solids and suggesting potential applications with a phase-field model for predicting the evolving fracture direction.