# Chapter 47. The Multiaxial Equivalent of Stressed Volume

- John B. Wachtman Jr.

Published Online: 28 MAR 2008

DOI: 10.1002/9780470314180.ch47

Copyright © 1993 The American Ceramic Society

Book Title

## Proceedings of the 17th Annual Conference on Composites and Advanced Ceramic Materials, Part 1 of 2: Ceramic Engineering and Science Proceedings, Volume 14, Issue 7/8

Additional Information

#### How to Cite

Tucker, W. T. and Johnson, C. A. (2008) The Multiaxial Equivalent of Stressed Volume, in Proceedings of the 17th Annual Conference on Composites and Advanced Ceramic Materials, Part 1 of 2: Ceramic Engineering and Science Proceedings, Volume 14, Issue 7/8 (ed J. B. Wachtman), John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9780470314180.ch47

#### Publication History

- Published Online: 28 MAR 2008
- Published Print: 1 JAN 1993

#### Book Series:

#### ISBN Information

Print ISBN: 9780470375266

Online ISBN: 9780470314180

- Summary
- Chapter

### Keywords:

- probabilistic;
- integrated;
- multiaxial stress;
- probability;
- uniaxial model

### Summary

A comprehensive probabilistic fracture analysis methodology should allow data to be combined or pooled from multiple specimen sizes and geometries such that all available fracture data can be integrated into strength/probability estimates and such that size-scaling aspects of the model can be tested. Moreover, this capability must, among other things, allow for multiaxial stress states, and be capable of providing a measure of the statistical uncertainty in estimates of strength/probability (confidence and tolerance bounds). Previous efforts of the authors have resulted in estimators for combined data, confidence and tolerance bounds on estimates, bias and bias correction of estimates, and statistical efficiency of the estimators. These developments have been based on the uniaxial model [equation] of where *k* is a dimensionless “load factor” or “structure factor” and s̀_{max} is the maximum stress in the structure at the time of failure. For the case of uniform uniaxial tension, *k* is unity and Eq. (1) reduces to the form of the two-parameter Weibull distribution commonly encountered in the statistical literature. For all other loading geometries, *k* is a function of *m* and, when evaluated, is always less than unity. The product of *k* × *V* is often termed the “effective volume” and, as the term implies, is the volume of material that is effectively under uniform uniaxial tension.

The results presented here show that similar techniques are applicable to more comprehensive models of multiaxial failure. Building on previous work by the first author (reported elsewhere) and generalizing results recently reported in the literature, it is shown that the Batdorf-Heinisch's (B-H) flaw density distribution and the Lamon-Evans' (L-E) elemental strength approaches to weakest-link fracture statistics for multiaxial loading give equivalent probability predictions for equivalent failure criteria. A generalization is also given detailing necessary and sufficient conditions for the B-H and L-E approaches to be equivalent. This allows a general size factor to be defined that simultaneously takes into account geometry, loading, and multiaxial stresses. Thus, the form of Eq. (1) is quite general. This factor can be applied on an elemental basis or to a component structure as a whole, which has consequences in determining probability predictions. Also employing these results, it is indicated how a contradictory conclusion, also reported in the literature, was reached. All in all, there appear to be no obvious roadblocks in incorporating the effects of multiaxial stresses into current analysis methods for combining data, determining limits, etc. Also, the form of the general size factor indicates that it may be possible to obtain generalizations that cover time and/or temperature effects. The key is to generalize Eq. (1) by adding another dimension, say time, and then integrate over this dimension in obtaining a generalization to Eq. (1). Thus, the equivalence of B-H and L-E has broad-reaching implications.