Finite Element Method (FEM) Calculations of Stress-Strain behavior of alpha-beta Ti-Mn alloys: Part II. Stress and strai
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I.
INTRODUCTION
W H E N E V E R a material containing two or more phases is subjected to stress, there will be constraints to the resulting deformation because of the requirements of compatibility. These constraints produce interaction stresses and strains, which result in an inhomogeneous distribution of both stress and strain. The nature of these inhomogeneities will depend on the mechanical properties of the phases present, their volume fractions, morphology, and size. Nonuniformity of strains in two ductile phase alloys have been reported by UnckeP and a number of subsequent investigators. 2-12 None of these studies involved direct measurement of strains, although estimates of strain differences have been made indirectly by shape changes, ~ recrystallization behavior, 2"3"4 hardness measurements, 6.9.12 residual stress measurements, 8 and metallographic observations." FEM has been used to calculate strain distributions 6'9'~3and stress and strain distributions 4,ts for two ductile phase materials. Since most mechanical properties depend on the nature of stress and strain distributions, it is important to evaluate this distribution theoretically and, if possible, experimentally. The preceding calculations of Part 116 have provided data from which the theoretical stress and strain in each phase, which correspond to a given overall strain in the alloy, can be determined. Also presented in the following account are the results of experiments to measure directly the strain distributions, at a given strain, as a function of volume fraction. II.
T H E O R E T I C A L AND E X P E R I M E N T A L PROCEDURE
The procedure for obtaining the theoretical data for stress and strain distributions by the FEM has been described in Part I. 16 The average stress and strain distributions in the y-direction in each phase have been obtained in the following way. SREERAMAMURTHY ANKEM is Research Engineer with RMI Company, 1000 Warren Avenue. Niles. OH 44446. HAROLD MARGOLIN is Professor in the Department of Physical and Engineering Metallurgy, Polytechnic Institute of New York, 333 Jay Street. Brooklyn, NY 11201. Manuscript submitted May 1, 1980. METALLURGICAL TRANSACTIONS A
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For a given applied load the NASTRAN program used by the authors gives the stresses in the y and x directions for each element and the displacements in the y and x directions of all the nodes. To calculate the average stresses in the y-direction in each phase, the stresses in the y-direction in all the elements of each phase are added and divided by the total number of elements of that phase. From the knowledge of the displacement in the y-direction of two successive nodes on a given line, the true strain can be calculated between these two nodal points. To obtain the average true strain in the y-direction of each phase, the strains of all the lines in each phase are added and divided by the number of lines in each phase. When the line is shared by both phases, it is counted as belonging half to one phase and the other half to the other phase. The nodal li
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