A weakest link theory for macroscopically homogeneous isotropic materials containing randomly oriented microcracks uniformly distributed in location is developed under the assumption that fracture depends only on the macroscopic stress normal to a crack plane. The function representing the number of cracks per unit volume failing at each value of normal stress is expanded as a Taylor series with coefficients determined from tensile test data. This function is used without additional assumptions to determine the probability of fracture under arbitrary (but not predominantly compressive) stress conditions. The results can be readily incorporated into a finite-element code to predict the failure probability of any structure to which the code applies.
constitution and alteration of the surface layers. Calcite, aragonite, Ca(OH),, and unhydrated material can be detected. ' S. Goto, M. Dairnon, G . Hosaka, and R. Kondo, "Composition and Morphology ofHydrated Tricalcium Silicate," J . A m . Cclrrim. So<.Tricalcium Silicate Pastesl 2 Bemardo Mafchese, "Scanning Electron Microscopy Contribution to the Study of Bernardo Marchese, "Scanning Electron Microscopy Contribution to thc Study of I' Bemardo Marckse, "Scanning Electron Microscopy Contribution to the Study of Bemardo Marchcse, "Discussion of the Paper 'Studies on the Hydration of I6 Bemardo Marchese. "SEM Toweraohv ofTwin Fracture Surfaccs of Alite Pastes 1." Cpm. C'orzcr, RPS., 3 [2] 149-61 (1973).
A frequently used approximate treatment of fracture statistics for polyaxial stress states assumes that the probability of survival is the product of the probabilities of survival of the structure for the principal stresses applied individually. The present paper shows that this assumption is generally unconservative and therefore the approximation serves as a lower bound to the failure probability. A simple technique is given for finding an upper bound in cases of biaxial tension prox~ded the uniaxial fracture behavior is described satisfactorily by Weibull's two-parameter formula. The upper bound is a good approximation when in high stress regions the stresses are equibiaxial, or nearly so, as in laterally loaded or spinning disks.
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