Mechanics of Composite Materials. Recent Advances by Zvi Hashin

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We shall not go into more details. Although derived for a situation with no plastic flow eq. (55) remains useful during plastic flow. σΩ in (55) must then be interpreted as the (average) actual stress in the grains of the group Ω minus the internal stresses due to the inhomogeneous plastic flow. The latter stresses follow from the mentioned theory (Kröner 1961). Instead of eq. (47) that was derived for aggregates with constant elastic parameters we now have to use *Α-«*>--ΑΑε£ + <ΑΛ£>. (56) Α^ depends on elastic moduli and the grain shape statistics.

In order to keep our dis­ cussion simple we shall not consider this here. (3) is now rewritten as θ(ϊ>» J G s l V> dV + J G q V > dS' (3'> G(r,r') is the Green's function of the heat conduction Neumann problem. It is known for simple geometries, for instance half-space and sphere, provided the con­ duction properties are homogeneous and isotropic. ^- W*V\\r-r'\ ) A^cohst. (4) The difference in the appearance of eqs. (1') and (3') explains why we had to in­ troduce different operation symbols (o and o) in eqs.

A routine calculation leads to where 69 = $ — §. By Δ we denote generally a new modified Green's function that is related to the function Γ by Ale = \ ? (r> - ^ij(t) f^cr,r')AKe(r'). (45) Formally, (44) has the same appearance as (25). e. to the problem of finite thickness of the layer. Here we are interested in the heat flux through an infinitely thin layer. This heat flux is in the x-direction. , are relevant. Therefore we may replace 3 , § , 6g and Δ in (44) by the (xx)-components. From now on we interprete eq.