Saturated porous continua in the frame of hybrid description

Brazgina, O.V.; Иванова, Е. А.; Иванова, Е. В.

doi:10.1007/s00161-016-0495-0

Cited by 8 publications

(3 citation statements)

References 44 publications

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“…As shown in [4, 28, 54], the differential equations relating the strain measures and the velocity gradients have the form

\nabla v = - \frac{δ g}{δ t} \cdot {boldg}^{- 1}, \nabla bold-italicω = \frac{δ boldΘ}{δ t} + boldΘ \times bold-italicω + false(\nabla boldv false) \cdot boldΘ .

We note that Equation () follows from Equations () and (). The first equation in () is well known.…”

Section: Basic Equations Of the Cosserat Continuummentioning

confidence: 96%

Modeling of electrodynamic processes by means of mechanical analogies

Иванова

2020

Z Angew Math Mech

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This study continues the line of earlier research in mechanical models of electrodynamic processes suggested in previous works. The basic steps we take to construct these models are: to formulate equations of a special type Cosserat continuum, and then to suggest analogies between quantities characterizing the stress–strain state of the continuum and quantities characterizing electrodynamic processes. In addition to the previously introduced mechanical analogies of the electric field vector and the magnetic induction vector, in this paper we provide mechanical analogies of the magnetic field vector, the electric induction vector, the electric current density and the electric charge density. In the framework of the suggested model, we obtain a set of differential equations that coincide with the first Maxwell's equation (the one with the displacement current term), the Gauss law for electric field and the charge conservation law. As a debatable question, we discuss the possibility of modifying the Maxwell–Faraday equation and the Gauss law for magnetic field.

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“…As shown in [4, 28, 54], the differential equations relating the strain measures and the velocity gradients have the form

\nabla v = - \frac{δ g}{δ t} \cdot {boldg}^{- 1}, \nabla bold-italicω = \frac{δ boldΘ}{δ t} + boldΘ \times bold-italicω + false(\nabla boldv false) \cdot boldΘ .

We note that Equation () follows from Equations () and (). The first equation in () is well known.…”

Section: Basic Equations Of the Cosserat Continuummentioning

confidence: 96%

Modeling of electrodynamic processes by means of mechanical analogies

Иванова

2020

Z Angew Math Mech

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“…Note that we allow the possibility of the vacancies' occurrence by considering the proppant density as a function of space coordinates and time. Some information on continuum mixture theory relevant to this paper can be found in [Brazgina et al 2016].…”

Section: Physicomathematical Modelmentioning

confidence: 99%

Untitled

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. 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...

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“…More details regarding the differences of the material time derivative in material and in spatial description can be found in[18], where also the importance of the quantity w in context with the ALE-formulation is mentioned, or in[3]. The Poisson relations (3.2) and (3.3) extensively discussed in[41], pp.…”

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confidence: 99%

Extended micropolar approach within the framework of 3M theories and variations thereof

Vilchevskaya

Müller

Eremeyev

2022

Continuum Mech. Thermodyn.

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As part of his groundbreaking work on generalized continuum mechanics, Eringen proposed what he called 3M theories, namely the concept of micromorphic, microstretch, and micropolar materials modeling. The micromorphic approach provides the most general framework for a continuum with translational and (internal) rotational degrees of freedom (DOF), whilst the rotational DOFs of micromorphic and micropolar continua are subjected to more and more constraints. More recently, an “extended” micropolar theory has been presented by one of the authors: Eringen’s 3M theories were children of solid mechanics based on the concept of the indestructible material particle. Extended micropolar theory was formulated both ways for material systems as well as in spatial description, which is useful when describing fluid matter. The latter opens the possibility to model situations and materials with a continuum point that on the microscale consists no longer of the same elementary units during a physical process. The difference culminates in an equation for the microinertia tensor, which is no longer a kinematic identity. Rather it contains a new continuum field, namely an independent production term and, consequently, establishes a new constitutive quantity. This makes it possible to describe processes of structural change, which are difficult if not impossible to be captured within the material particle model. This paper compares the various theories and points out their communalities as well as their differences.

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Saturated porous continua in the frame of hybrid description

Cited by 8 publications

References 44 publications

Modeling of electrodynamic processes by means of mechanical analogies

Modeling of electrodynamic processes by means of mechanical analogies

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Extended micropolar approach within the framework of 3M theories and variations thereof

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