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Mixed boundary conditions for FFT-based homogenization at finite strains. (English) Zbl 1359.74356
Summary: In this article we introduce a Lippmann-Schwinger formulation for the unit cell problem of periodic homogenization of elasticity at finite strains incorporating arbitrary mixed boundary conditions. Such problems occur frequently, for instance when validating computational results with tensile tests, where the deformation gradient in loading direction is fixed, as is the stress in the corresponding orthogonal plane. Previous Lippmann-Schwinger formulations involving mixed boundary can only describe tensile tests where the vector of applied force is proportional to a coordinate direction. Utilizing suitable orthogonal projectors we develop a Lippmann-Schwinger framework for arbitrary mixed boundary conditions. The resulting fixed point and Newton-Krylov algorithms preserve the positive characteristics of existing FFT-algorithms. We demonstrate the power of the proposed methods with a series of numerical examples, including continuous fiber reinforced laminates and a complex nonwoven structure of a long fiber reinforced thermoplastic, resulting in a speed-up of some computations by a factor of 1000.

MSC:
 74Q05 Homogenization in equilibrium problems of solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 74E30 Composite and mixture properties 45G10 Other nonlinear integral equations 65T50 Numerical methods for discrete and fast Fourier transforms 74B20 Nonlinear elasticity
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