The stresses in a face-centered cubic single crystal under uniaxial tension
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I.
INTRODUCTION
W H E N a specimen is deformed in a test machine, it will usually be subjected to a constraint effect provided by grips. This end effect produces a strong localized nonuniform deformation. It has often been tacitly assumed, however, that this localized deformation does not extend throughout a single-crystal specimen. Because o f elastic anisotropy and slip characteristics, it is quite possible that such an assumption is not valid. To ascertain whether localized constraints would produce stress nonuniformities and how these stress nonuniformities would be influenced by orientation and specimen dimensions, an elastic and elasto-plastic deformation analysis was undertaken for small deformations. In this study, the p r o b l e m is first approached by considering the behavior o f a specimen strained without constraints and subsequently with constraints. II.
T H E O R E T I C A L BACKGROUND
To study the deformation behavior o f a single-crystal specimen, it was necessary to develop a theoretical background to take the elastic anisotropy and, especially, the specific slip characteristics into account. This has been done in a previous article, m and here we give an abbreviated summary. In the elastic deformation region, the elastic constitutive relation for an anisotropic crystal is readily given by the general Hooke's law. {O'} : [Gel {ee} [1] where {~r} and {ee} are the stresses and elastic strains, respectively. The term [Ce] is the elastic rigidity matrix with a 6 by 6 size. When deformation exceeds the elastic limit by a small amount o f crystallographic slip, the mechanics o f elastoplastic deformation can be simply developed by the elastoplastic increment theory. {do'} =
[Cep] {deep}
[2]
XIAOYU HU, Postdoctoral Researcher, is with the Department of Materials Science and Engineering, T h e Ohio State University, Columbus, OH 43210-1179. CHAO WEI, Graduate Student, HAROLD MARGOLIN, Professor, and SAID NOURBAKHSH, Associate Professor, are with the Department o f Materials S c i e n c e and Engineering, Polytechnic University, Brooklyn, NY 11201. Manuscript submitted July l , 1992. METALLURGICAL TRANSACTIONS A
where {deep} is the total elasto-plastic strain increment. The term [Cep] is called the elasto-plastic rigidity matrix with the same 6 by 6 size as the elastic rigidity matrix (Eq. [1]) and is expressed as
[Cep] = [Ce] - [Ce] [(I)1 = ( [ H I t --~ [TIt
[Ce]
[T] [O]
[T]) -1 [TIt
[3]
ICe]
[41
In Eqs. [4] and [5], [T] is the transformation matrix with a 6 by N size, and N is the n u m b e r o f active slip systems. In this matrix, each element, t0, is written by ti) = l / 2 (nibj + njbi)
[5]
where ni and b i represents the slip plane normal and the slip direction vector, respectively. The term [T]t is the transpose matrix o f [T]; [H]' is called the hardening matrix and is applied to active slip systems when plastic deformation occurs. In the [H]' matrix, each element ho is designated the hardening rate and denotes the increment o f flow stress on system i due to an increment o f sh
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