Stress and strain partition in deformation of multiphase alloys

A new theory for the inhomogeneous deformation of multiphase alloys was proposed. The theory was based on the concept that the strain energy density of a composite body increased along the path on which the strain energy density had an extreme value. A special form of strain energy density was formulated by means of the modified rules of mixtures for stress and strain. A first order differential equation was derived from the strain energy density through variational principle or Lagrangian multiplier method. The differential equation was regarded as a general governing equation for the stress and strain partition of two phase alloys. The validity of the new theory was demonstrated by applying the governing equation to several practical alloys. In chapter 3 the theory was applied to the plastic deformation of two spheroidized carbon steels consisted of soft ferrite matrix and rigid carbide (cementite) phase. The partitioned stresses and strains were calculated by the theory, and the in situ stress-strain curves of the ferrite matrix were obtained from them. Also, it was possible to evaluate the back stress and unrelaxed plastic incompatibility. In chapter 4 the expressions for the average elastic constants of equiaxed composites were derived from the governing equation. Then the average elastic constants of WC-Co alloys were evaluated by the expressions. All the calculated elastic constants, Young's modulus, shear modulus, bulk modulus and Poisson's ratio, fell-within the Hashin and Shtrikman's upper and lower bounds. By inspecting the governing equation for plastic deformation of equiaxed two phase alloys, it was found that the governing equation described an intermediate deformation between the iso-stress condition and the iso-strain condition. The stress and strain partitions of a duplex stainless steel and a spheroidized carbon steel were also calculated by the theory, and the results were compared with each other. Aspects of the stress and strain partition in these two representative alloys were largely different from one another. In chapter 5 the tensile deformation behaviours of coarse ferrite-martensite dual phase steels containing 17-50 percent martensite were analyzed, and the stress and strain partition was evaluated by the newly developed theory. And then, it was attempted to find the relations between the deformation behaviours and the stress and strain partition. At various strains the dependence of the stress of dual phase steels on the volume fraction of martensite(f) had a transition at f=0.3;the strengthening effect by the martensite phase was much reduced after f=0.3. According to the theoretical analysis of stress and strain partition, the transition was caused by the plastic deformation of martensite. When f was less than 0.3, three stage deformation appeared in the log-log plots of strain hardening rate and strain, however, the phenomenon disappeared at larger f. This was also because of the plastic deformation of the martensite phase. The calculation for the stress and strain partition showed that the martensite phase yielded after uniform strain when the martensite content was less than about 25 percent, but it deformed plastically before the uniform strain with larger amount of martensite because the martensite phase was softened by dilution of carbon content. The in situ stress-strain curves of the ferrite matrix were slightly dependent on the geometric slip distance only at early deformation, while those of the martensite were largely dependent on its volume fraction. The internal stress in the dual phase steels increased rapidly in small strain region, however, the increasing rate was also reduced by the plastic relaxation at ferrite and plastic deformation of martensite