Abstract
A variational method is developed to estimate the macroscopic constitutive response of composite materials consisting of aggregates of viscoplastic single-crystal grains and other inhomogeneities. The method derives from a stationary variational principle for the macroscopic stress potential of the viscoplastic composite in terms of the corresponding potential of a linear comparison composite (LCC), whose viscosities and eigenstrain rates are the trial fields in the variational principle. The resulting estimates for the macroscopic response are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy known bounds. In addition, unlike earlier ‘second-order’ methods, the new method allows optimization with respect to both the viscosities and eigenstrain rates, leading to estimates that are fully stationary and exhibit no duality gaps. Consequently, the macroscopic response and field statistics of the nonlinear composite can be estimated directly from the suitably optimized LCC, without the need for difficult-to-compute correction terms. The method is applied to a simple example of a porous single crystal, and the results are found to be more accurate than earlier estimates.
Highlights
Macroscopic samples of metals and minerals usually appear in the form of aggregates of large numbers of one or more types of single-crystal grains and other inhomogeneities, such as voids and cracks, which are distributed with random positions and orientations in the sample
We have developed a variational method for estimating the macroscopic properties of nonlinear composites with viscoplastic crystalline phases in terms of the corresponding properties of suitably designed linear comparison composite (LCC)
The phases of the LCC are characterized by certain slip viscosities and eigenstrain rates, which in turn play the role of trial fields in a variational statement for the stress potential of the viscoplastic composites
Summary
Macroscopic samples of metals and minerals usually appear in the form of aggregates of large numbers of one or more types of single-crystal grains and other inhomogeneities, such as voids and cracks, which are. The ‘second-order’ method makes use of more general LCCs incorporating suitably selected eigenstrain rates, leading to an improved ‘generalized secant’ approximation of the nonlinear constitutive relations and ensuring that the resulting estimates are exact to second order in the heterogeneity contrast, and in agreement with the perturbation expansions of Suquet & Ponte Castañeda [17]. We propose a new variational method, where both the viscosities and eigenstrain rates of the constituent phases of the LCC are generated by consistent optimization procedures This leads to ‘full stationarity’ for the resulting estimates, which are still exact to second order in the contrast, but have all the advantages of the earlier ‘variational’ estimates in that the macroscopic constitutive relation and fields statistics of the nonlinear composite can be conveniently expressed in terms of the corresponding quantities for the suitably optimized. Roman letters (b), second-order tensors by boldface italic Roman or boldface Greek letters (C, α) and fourth-order tensors by double-struck letters (P)
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