Plastic deformation of a bilayer

The issue of substrate curvature in an elastic bilayer system with arbitrary layer thicknesses was considered in Section 2.2.1, where results were presented for the variation of equi-biaxial stress through the thickness of isotropic film-substrate systems for any ratio of thicknesses and any ratio of elastic moduli. In this section, extensions of these results into the realm of plastic response are pursued whereby useful insights into the conditions governing the onset and spread of plastic flow in the bilayer can be extracted. 

If one or both of the materials are modeled as being elastic-ideally plastic, then the methods adopted for study of elastic response can be extended to establish conditions for onset of plastic deformation and the progression 

The discussion here follows directly from that in Section 2.2. A film of thickness h is bonded to a substrate of thickness hs, with no restrictions on the thickness ratio. The stress and deformation fields are referred to a cylindrical coordinate system with polar coordinates in the plane of the system and with the z—direction normal to the interface the origin of coordinates lies in the substrate midplane. The equi-biaxial stress components are referred to polar coordinates, but they could equally well be expressed in rectangular coordinates. As long as the response is in the range of geometrically linear behavior, the common shape of the film and substrate in plan view is immaterial. As in Section 2.2, the biaxial elastic moduli of the film and substrate are Mf and Mg, respectively, and the corresponding coefficients of linear thermal expansion are af and dg, respectively. 

Some state of the film-substrate system is identified as the initial state. The temperature in this state is the reference temperature Tq, and T represents the current temperature. All components of stress and plastic strain are presumed to be equal to zero in the reference state minor modifications are required to consider any other initial state. The reference substrate curvature is also zero at Tq. A change in temperature from the reference state is denoted as T = T — Tq in all subsequent discussion. 

The in-plane extensional strain and curvature in the film-substrate system arises as a consequence of thermal expansion mismatch between the film and substrate materials during temperature excursion from the reference temperature. Conditions are assumed to be such that the temperature is uniform throughout the film and substrate at all times during thermal cycling. Some particular temperatures at which distinct transitions occur in the elastoplastic deformation of the film material are first identified by adopting the assumption that, over the range of temperature, the properties of the film and substrate materials remain essentially unchanged effects of temperature dependence of plastic yield or flow behavior on the evolution of film stress and substrate curvature are examined subsequently. 

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