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Abstract

This work is focused on polymorphic uncertainties in the framework of constitutive modeling for transversely isotropic materials. To this end, we propose a hybrid fuzzy-stochastic model, where the stochastic part accounting for aleatory uncertainties of material parameters is expanded with the multivariate polynomial chaos expansion. In order to account for epistemic uncertainties, polynomial chaos coefficients are treated as fuzzy variables. The underlying minimum and maximum optimization problem for the fuzzy analysis is approximated by α-level discretization, resulting in a separation of minimum and maximum problems. To become more universal, so-called quantities of interest are employed, which allow a general formulation for the target problem. Numerical examples with fuzzy, fuzzy-stochastic, and hybrid fuzzy-stochastic input demonstrate the versatility of the proposed formulation. Communicated by Paul Steinmann. MSC2010: 60A86. 99 100 EDUARD PENNER, ISMAIL CAYLAK, ALEX DRIDGER AND ROLF MAHNKEN numerically using a stochastic simulation, where the Monte Carlo (MC) method [Caflisch 1998; Hurtado and Barbat 1998] is widely used. Alternatively, spectral stochastic surrogate models, e.g., polynomial chaos expansion (PCE), are used in order to reduce the computational effort. Corresponding research areas are: linear elasticity of solids and mechanics [Ghanem and Spanos 1991], plasticity of solids and mechanics [Anders and Hori 1999; Rosić 2013], large deformations [Acharjee and Zabaras 2006; Acharjee 2006; Caylak et al. 2018], fluid flow [Le Maître et al. 2001; 2002], flow-structure interactions [Xiu and Karniadakis 2002; Xiu et al. 2001], and linear convection problems [Jardak et al. 2002].Contrary to aleatory uncertainties, epistemic uncertainties refer to subjectivity as a consequence of, e.g., incomplete scientific understanding or lack of measurements, which indicate a possible value range rather than a probability function. In addition, epistemic uncertainties are reducible by empirical effort, e.g., investing more in measurements. Methodologies for the modeling of epistemic uncertainties are, e.g., interval analysis and, increasingly applied over the last years, fuzzy analysis, which represents indistinct boundaries [Zadeh 1965]. In order to perform mathematical operations with fuzzy sets, the so-called α-level discretization method is applied. Here, the fuzzy response at each selected α-level is obtained by solving a minimum-maximum problem of a quantity of interest (QoI). In [Mahnken 2017] QoIs are employed within a variational formulation for fuzzy analysis in continuum mechanics.A realistic modeling of uncertainties requires a combination of different uncertainty types. Following [Graf et al. 2015], this is referred to as polymorphic uncertainties. Corresponding models are: Dempster-Shafer evidence theory [Dempster 1967], coherent lower prevision theory [Walley 1991], possibility theory [Dubois and Prade 2012], probability box (P-box) theory [Ferson et al. 2003], and fuzzy probability theory [Gudder 1998...