Stochastic hyperelastic modeling considering dependency of material parameters

Caylak, Ismail; Penner, Eduard; Dridger, Alex; Mahnken, Rolf

doi:10.1007/s00466-018-1563-z

Cited by 15 publications

(15 citation statements)

References 23 publications

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“…Then, in Section 3, for the stochastic sphere, after we review the elastic solution to the cavitation problem under uniformly applied tensile dead load, we recast the problem in the stochastic setting, and find the probabilistic solution. Concluding remarks are provided in Section 4. by the mean value and the variance is the most commonly used in many practical applications [9,20,29]. Here, we combine finite elasticity and information theory, and rely on the following general hypotheses [35][36][37]: (A1) Material objectivity: The principle of material objectivity (frame indifference) states that constitutive equations must be invariant under changes of frame of reference.…”

Section: Introductionmentioning

confidence: 99%

Likely Cavitation in Stochastic Elasticity

Mihai

Fitt

Woolley

et al. 2018

J Elast

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We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation postbifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as 'likely cavitation'. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.We recall that a homogeneous hyperelastic model is described by a strain-energy function W (F) that depends on the deformation gradient tensor, F, with respect to a fixed reference configuration, and is characterised by a set of deterministic model parameters [14,39,57]. In contrast, a stochastic homogeneous hyperelastic model is defined by a stochastic strain-energy function, for which the model parameters are random variables that satisfy standard probability laws [35,[52][53][54]. In this case, each model parameter is described in terms of its mean value and its variance, which contains information about the range of values about the mean value. While it is rarely possible if ever to obtain complete information about a random quantity in an elastic sample of material, the partial information provided

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Section: Introductionmentioning

confidence: 99%

Likely Cavitation in Stochastic Elasticity

Mihai

Fitt

Woolley

et al. 2018

J Elast

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“…Observe, that the uncorrelated Gaussian random variables X i , i = 1,…, m , can be exactly represented by the PCE with their corresponding PC coefficients

\begin{matrix} alignleft \\ align-1 {\hat{X}}_{i k} = \{\begin{matrix} μ_{X_{i}} & for k = 0, \\ σ_{X_{i}} & for k = i, \\ 0 & otherwise, \end{matrix} & align-2 \end{matrix}

where

μ_{X_{i}} : = double-struckE ([], X_{i})

and

σ_{X_{i}}

describe the population mean value and the standard deviation, respectively, of the Gaussian random variables X i . In the case of correlated random variables X i , i = 1,…, m the reader is referred to Caylak et al, where correlations between hyperelastic material parameters for rubber materials at large deformations are investigated.…”

Section: Polymorphic Uncertainty Modelingmentioning

confidence: 99%

“…where X i ∶= E [X i ] and X i describe the population mean value and the standard deviation, respectively, of the Gaussian random variables X i . In the case of correlated random variables X i , i = 1, … , m the reader is referred to Caylak et al, [12] where correlations between hyperelastic material parameters for rubber materials at large deformations are investigated.…”

Section: Order Pmentioning

confidence: 99%

Optimization with constraints considering polymorphic uncertainties

et al. 2019

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In this contribution, a numerical design strategy for the optimization under polymorphic uncertainty is introduced and applied to a self‐weight minimization of a framework structure. The polymorphic uncertainty, which affects the constraint function of the optimization problem, is thereby modeled in terms of stochastic variables, fuzzy sets, and intervals to account for variability, imprecision and insufficient information. The stochastic quantities are computed using polynomial chaos expansion resulting in a purely fuzzy‐valued formulation of the constraint functions which can be computed using α‐cut optimization. Afterward, the constraint function can be interpreted in a possibilistic manner, resulting in a flexible formulation to include expert knowledge and to achieve a robust design.

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“…As mentioned before in Section 2.2, the MC simulation, based on a relatively large number of samples, could be used for numerical evaluation of the QoI in (6). However, if only individual empirical moments in (7)-(9) are sought, a discrete surrogate model, e.g., in terms of the polynomial chaos expansion (PCE) [Ghanem and Spanos 1991;Caylak et al 2018] can be used in order to reduce the numerical effort. This expansion involves a basis of known random functions with deterministic PC coefficients.…”

Section: Stochastic Analysismentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

A polynomial chaos expanded hybrid fuzzy-stochastic model for transversely fiber reinforced plastics

Penner

Caylak

Dridger

et al. 2019

Math. Mech. Compl. Sys.

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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.

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Stochastic hyperelastic modeling considering dependency of material parameters

Cited by 15 publications

References 23 publications

Likely Cavitation in Stochastic Elasticity

Likely Cavitation in Stochastic Elasticity

Optimization with constraints considering polymorphic uncertainties

A polynomial chaos expanded hybrid fuzzy-stochastic model for transversely fiber reinforced plastics

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