Stationary variational estimates for the effective response and field fluctuations in nonlinear composites |
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| Institution: | 1. Computational Mechanics Lab., Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India;2. Advanced Computational Mechanics Lab., Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA;3. Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA;1. Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China;2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland;1. School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA;2. Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854, USA;1. Department of Materials Science and Engineering, Monash University, Clayton, Victoria 3800, Australia;2. John A. Paulson School of Engineering and Applied Sciences, Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA |
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| Abstract: | This paper presents a variational method for estimating the effective constitutive response of composite materials with nonlinear constitutive behavior. The method is based on a stationary variational principle for the macroscopic potential in terms of the corresponding potential of a linear comparison composite (LCC) whose properties are the trial fields in the variational principle. When used in combination with estimates for the LCC that are exact to second order in the heterogeneity contrast, the resulting estimates for the nonlinear composite are also guaranteed to be exact to second-order in the contrast. In addition, the new method allows full optimization with respect to the properties of the LCC, leading to estimates that are fully stationary and exhibit no duality gaps. As a result, the effective response and field statistics of the nonlinear composite can be estimated directly from the appropriately optimized linear comparison composite. By way of illustration, the method is applied to a porous, isotropic, power-law material, and the results are found to compare favorably with earlier bounds and estimates. However, the basic ideas of the method are expected to work for broad classes of composites materials, whose effective response can be given appropriate variational representations, including more general elasto-plastic and soft hyperelastic composites and polycrystals. |
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| Keywords: | Composites Polycrystals Viscoplasticity Nonlinear homogenization Variational approximations |
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