A nonlinear theory is developed to calculate the amount of thermal stress relieved by plastic deformation in the ductile matrix of a two-phase, transversely isotropic composite. The theory is intended for the class of composites containing aligned spheroidal inclusions at moderate concentration. Based on an energy criterion, the effective stress of the heterogeneously deformed matrix is first established as a function of thermal strain, and by this, the critical inclusion concentration required to induce plastic flow in the ductile matrix is determined. Such a critical concentration is found to be low, typically below 10% for boron/aluminum regardless of the inclusion shape and for graphite/aluminum when the aspect ratio of inclusions is greater than one, but can go up to 80% when graphite inclusions take the form of thin discs. Beyond the critical concentration the deformation in the matrix is no longer elastic, and the ensuing plastic flow will result in a stress reduction. The amount of thermal stress relieved in the matrix is demonstrated by comparing its average stress with that derived by assuming it to be ideally elastic, and it is found that the stress reductions for all inclusion shapes are quite significant, in most cases exceeding 50% of the total elastic stress. The quantitative accuracy of the theory is partially assessed by a comparison with some exact solutions when the inclusions take the forms of spherical particles and circular fibers. A side product of this study is the overall elastoplastic coefficient of thermal expansion as a function of the aspect ratio and volume fraction of inclusions.
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