Simple constitutive equations for steel at high temperature
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I.
BACKGROUND
THE recent increases in computational speed and availability of finite element software are making stress analysis of casting processes feasible and desirable. A major obstacle to accurate mathematical analysis is finding and evaluating material constitutive equations that adequately describe the complex relationship that exists between stress, strain, and time at elevated temperatures. These equations should characterize mechanical behavior of the metal under the conditions encountered during the process. Choices include time-independent elastoplastic relationships, elastoplastic models with creep, unified models with evolving internal state variables, and elasto-viscoplastic models. Each of these approaches has both merits and problems. Assuming stress, tr, is caused solely by elastic strain, ee, the rate forms of the constitutive equations for an isotropic material in uniaxial loading are 6" = E~e = ~e "~ Ep -~- ~T
[1] [2]
where the symbols are defined in the Nomencl~iture at the end of the article. Integrating these equations under the appropriate boundary conditions produces the deformation history (stress-strain-time response) of the material under any arbitrarily chosen loading conditions. These loading conditions could range widely and include the limiting cases defined in standard tensile tests (imposed Constant total strain rate), creep tests (imposed constant stress), and stress relaxation tests (imposed constant total strain). In a typical casting, every point in the material experiences a different complex loading history, which usually varies greatly as time progresses. To perform the
PATRICK F. KOZLOWSKI, formerly Graduate Student, University of Illinois at Urbana-Champaign, is Manufacturing and Systems Engineer, Caterpillar, Inc., Aurora, IL. BRIAN G. THOMAS, Associate Professor, and HAO WANG, Graduate Student, are with the Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801. JEAN A. AZZI, formerly Graduate Student, University of Illinois at Urbana-Champaign, is Research Engineer, Woodward Governor Company, Rockford, IL 61101. Manuscript submitted February 5, 1991. METALLURGICAL TRANSACTIONS A
required multidimensional analysis, the effective (uniaxial) applied stress and strain needed in Eqs. [1] and [2] are usually determined from their spatial components using a von Mises criterion, m The equations are then solved using the finite element method, which simultaneously calculates incremental changes in stress and strain at each location in the solid and at each time step during the simulation, under these complex loading conditions.t2] In doing this, the increments of inelastic strain are decomposed into their spatial components in proportion to the deviatoric stresses using the Prandtl-Reuss equations.tl] Inelastic strains are induced by the stresses, or elastic strains, which arise in response to the loading conditions. In many casting processes, small differences in the thermal strains, er, constitute a significant
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