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Figure 1: View of Drucker–Prager yield surface in 3D space of principal stresses for =, =. The Drucker–Prager yield criterion [1] is a pressure-dependent model for determining whether a material has failed or undergone plastic yielding.
The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as + (+) (+) + = where the principal stresses , are assumed to be aligned with the axes of anisotropy with in the rolling direction and perpendicular to the rolling direction, =, is the R-value in the rolling direction, and is the R-value perpendicular to the rolling direction.
The Tsai-Wu criterion predicts failure when the failure index in a laminate reaches 1. This failure criterion is a specialization of the general quadratic failure criterion proposed by Gol'denblat and Kopnov [ 2 ] and can be expressed in the form
The mode I fracture toughness for plane strain is defined as K I c = Y σ c π a {\displaystyle K_{\rm {Ic}}=Y\sigma _{c}{\sqrt {\pi a}}} where σ c {\displaystyle \sigma _{c}} is a critical value of the far field stress and Y {\displaystyle Y} is a dimensionless factor that depends on the geometry, material properties, and loading condition.
Prior to yield, material response can be assumed to be of a linear elastic, nonlinear elastic, or viscoelastic behavior. In materials science and engineering , the von Mises yield criterion is also formulated in terms of the von Mises stress or equivalent tensile stress , σ v {\displaystyle \sigma _{\text{v}}} .
However, the possibility of negative values of and the resulting imaginary makes the use of these quantities problematic in practice. Another related set of widely used invariants is ( ξ , ρ , θ {\displaystyle \xi ,\rho ,\theta \,} ) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates).
The debt ceiling returned on January 2, but Congress has several months to address it before the nation could default on its obligations. (Jemal Countess/Getty Images)
The plane stress, anisotropic, Hosford yield surface for four values of n and R=2.0. The Logan-Hosford yield criterion for anisotropic plasticity [2] [3] is similar to Hill's generalized yield criterion and has the form