Mathematical Modeling in Mechanics of Granular Materials

Sadovskaya, O. V.; Sadovskii, V. M.; Altenbach, Holm

doi:10.1007/978-3-642-29053-4

Cited by 63 publications

(31 citation statements)

References 6 publications

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“…Rheological models of uniaxial stress-strain state of porous materials are constructed in [5]. A simple rheological scheme taking into account the main features of deformation of porous metals is shown in Fig.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Series connection of an elastic spring and a plastic hinge in the scheme corresponds to the Prandtl-Reuss theory of elastic-plastic flow. Constitutive relationships of this theory can be represented in the form of principle of maximum of the energy dissipation rate [5]:…”

Section: Governing Equationsmentioning

confidence: 99%

“…Constitutive relationships of a rigid contact are formulated in the form of variational inequality [5]:…”

Section: Governing Equationsmentioning

confidence: 99%

“…Third variant, when the projection belongs to the conical surface, is realized under the fulfillment of two conditions: s / ∈ K and ε + ε 0 / ∈ C. For an isotropic material the elastic compliance tensor b is characterized by two independent parameters: the bulk modulus k and the shear modulus µ. Formulas for calculating the projection onto the conical surface are as follows [5]:…”

Section: Governing Equationsmentioning

confidence: 99%

“…For a more general situation such estimates were obtained in the monograph [5]. Dissipativity of boundary conditions in this case means that in each point of the boundary the inequality…”

Section: Mathematical Modelmentioning

confidence: 99%

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Phenomenological Modeling of Deformation of Porous and Cellular Materials Taking Into Account the Increase in Stiffness Because of the Collapse of Pores

Sadovskii¹,

Sadovskaya²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. Mathematical model for the description of deformation of a porous material with a random distribution of pores is constructed on the basis of generalized rheological method taking into account the different resistance of a material to tension and compression. Phenomenological parameters of the model are determined by means of the approximate computations for the problem of static loading of a cubic periodicity cell with spherical cavities. Within the framework of this theory the fields of displacements and stresses around expanding cavity of spherical shape in a homogenious space are constructed. It is shown that the porosity does not change at the stage of elastic deformation. When the pressure increases, a zone of plastic compaction is formed in the vicinity of a cavity, and in a part of this zone the collapse of pores takes place. Engineering formulas for calculation of critical pressures of the elastic limit state and the limit state of a medium with open pores, and also the formulas for determining the radiuses of interfaces of zones of plasticity and compaction are obtained. A complex of parallel programs for numerical analysis of the dynamic deformation of porous materials on multiprocessor computer systems is worked out. By means of this complex a series of computations for deformation of the protection elements, made of metal foams, under the action of localized impulsive loads was performed. The random nature of distribution of the pores size is taken into account with the help of computer modeling. Computations allow one to estimate plastic dissipative part of the energy of impact.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

“…Constitutive relationships of a rigid contact are formulated in the form of variational inequality [5]:…”

Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

“…For a more general situation such estimates were obtained in the monograph [5]. Dissipativity of boundary conditions in this case means that in each point of the boundary the inequality…”

Section: Mathematical Modelmentioning

confidence: 99%

See 3 more Smart Citations

Phenomenological Modeling of Deformation of Porous and Cellular Materials Taking Into Account the Increase in Stiffness Because of the Collapse of Pores

Sadovskii¹,

Sadovskaya²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Stationary longitudinal thermoelastic waves and the waves of the rotation type in the non-linear micropolar medium

Ерофеев

Leontieva

Malkhanov

2017

Z. Angew. Math. Mech.

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Key wordsCosserat continuum, the micropolar medium, Riemann wave, bending-torsion tensor, longitudinal thermoelastic wave, elastic rotational wave, evolution of a wave.In this paper we consider the nonlinear thermoelastic plane longitudinal waves and plane nonlinear elastic rotational waves in the model of a geometrically nonlinear micropolar continuum (Cosserat medium). The equation for longitudinal thermoelastic waves is studied in two extreme cases: we consider the medium with low and high thermal conductivity. It has been shown that in these cases the equation for longitudinal thermoelastic waves reduces to the known equations describing the classical Riemann waves and Riemann waves with damping. The peculiarities of propagation of these waves have been analyzed. The equation for rotational waves in the absence of thermoelastic effects, is reduced to a nonlinear ordinary differential equation of the second order for stationary waves. It has been demonstrated that the equation has solutions in the supersonic case in the form of periodic waves. The dependences of the amplitude on the wave number and the relative rotational angle of the stationary rotation wave has been obtained.

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Variational Inequalities in the Dynamics of Elastic-Plastic Media: Thermodynamic Consistency, Mathematical Correctness, Numerical Implementation

Sadovskii

Sadovskaya

2020

Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy

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Mathematical Modeling in Mechanics of Granular Materials

Cited by 63 publications

References 6 publications

Phenomenological Modeling of Deformation of Porous and Cellular Materials Taking Into Account the Increase in Stiffness Because of the Collapse of Pores

Phenomenological Modeling of Deformation of Porous and Cellular Materials Taking Into Account the Increase in Stiffness Because of the Collapse of Pores

Stationary longitudinal thermoelastic waves and the waves of the rotation type in the non-linear micropolar medium

Variational Inequalities in the Dynamics of Elastic-Plastic Media: Thermodynamic Consistency, Mathematical Correctness, Numerical Implementation

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