University of Illinois Urbana-Champaign

Resolved shearing stress energy theory of failure by slip of polycrystalline metal with face centered cubic lattice crystals

Smith, James O.

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https://hdl.handle.net/2142/111901

Description

Title
Resolved shearing stress energy theory of failure by slip of polycrystalline metal with face centered cubic lattice crystals
Author(s)
Smith, James O.
Issue Date
1961-05
Keyword(s)
  • Shearing Stress Energy Theory
  • Failure
  • Slip
  • Polycrystalline Metal
  • Cubic Lattice Crystals
Abstract
In this report a resolved shearing stress energy theory of failure of ductile polycrystalline metal by slip has been developed. It is based on the generally accepted concept that the first slip occurs in a crystal when the maximum value of the resolved shearing stress reaches a critical value τrs. The theory is developed for a metal of face-centered cubic lattice in which each crystal has 12 slip systems and 12 resolved shearing stresses. The lattice directions of the crystals are assumed to be randomly oriented at angles φ and ψ with the directions of the principal stresses σ1, σ2, σ3 of a homogeneous state of three dimensional stress. The general equations for the twelve resolved shearing stresses are derived in terms of σ1, σ2, σ3 φ, and ψ. Each crystal in a polycrystalline metal is surrounded by neighboring crystals with different orientation angles φ and ψ. These neighbors restrain the movement by slip of the crystal when the critical resolved shearing stress τrs is first reached. Consequently, each crystal is held in equilibrium by a highly statically in determinate system of forces (stresses). Before unrestrained slip can occur it is necessary for the resolved shearing stress on several slip planes to reach the critical value of τrs. The actual crystal is replaced by a model in order to study the development of slip. The model is a plane structure consisting of twelve pin-connected two force members that have a yield point stress of τrs. The model is statically indeterminate to the fourth degree. The dimensions of each member are chosen so that the stress in the member is equal to the actual resolved shearing stress on the corresponding slip plane in the crystal. The fully plastic load is determined for each arrangement of the stresses in the model. The analysis of the results obtained from the model leads to the conclusion that slip starts in a polycrystalline metal when the average value of the elastic strain energy Wrs per unit volume stored in the crystals by the twelve resolved shearing stresses reaches a certain constant value W1rs. A general expression for the value of Wrs is derived. Stereographic charts are given for determining the values of Wrs for any orientation angles φ and ψ for six different states of stress. Analysis of the results obtained from these six states of stress confirm the fact that slip starts in a polycrystalline metal in the crystals for which the orientation angles φ = ψ = 0° or 90° when the energy Wrs reaches the constant value W1rs and that all the crystals will have started to slip when the energy Wrs reaches the value 1.56 W1rs. The general equation for the energy Wrs has the same form as the equation for the energy distortion. Hence, the result can also be stated in the same form as the octahedral shearing stress.
Publisher
Department of Theoretical and Applied Mechanics. College of Engineering. University of Illinois at Urbana-Champaign
Series/Report Name or Number
  • TAM R 189
  • 1967-0485
ISSN
0073-5264
Type of Resource
text
Language
eng
Permalink
http://hdl.handle.net/2142/111901
Copyright and License Information
Copyright 1961 Board of Trustees of the University of Illinois

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Technical Reports - Theoretical and Applied Mechanics (TAM) PRIMARY
TAM technical reports include manuscripts intended for publication, theses judged to have general interest, notes prepared for short courses, symposia compiled from outstanding undergraduate projects, and reports prepared for research-sponsoring agencies.