Continuum Theories of Mixtures: Basic Theory and Historical Development

Atkin, R. J.; Craine, R. E.

doi:10.1093/qjmam/29.2.209

Cited by 643 publications

(292 citation statements)

References 3 publications

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“…In the derivation of the balance laws we shall consider the motion of the solid and fluid phases separately. Then, we will use the mixture theory [32][33][34][35][36][37][38] to combine the field equations, choosing the intrinsic motion of the solid phase as the reference motion relative to which the motion of the fluid phase is described.…”

Section: Balance Lawsmentioning

confidence: 99%

“…5d be the motion of the fluid, which could be distinct from 4> t if seepage takes place in the saturated region Si C S3 W . Now, let a s and tr w be the Cauchy partial stress tensors [32][33][34][35][36][37][38] arising from the intergranular and fluid stresses, respectively, and let n denote the unit normal vector to the surface d(f>,(^l) of the deformed region $,(^0-The Cauchy total stress tensor a is obtained from the sum £ = o-s + <r w .…”

Section: Balance Of Linear Momentummentioning

confidence: 99%

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A mathematical framework for finite strain elastoplastic consolidation Part 1: Balance laws, variational formulation, and linearization

Borja

Alarcón

1995

Computer Methods in Applied Mechanics and Engineering

109

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A mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented. Strong and weak forms of the boundary-value problem are derived using both the material and spatial descriptions. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. Balance laws are written for the soil-water mixture following the motion of the soil matrix alone. It is shown that the motion of the fluid phase only affects the Jacobian of the solid phase motion, and therefore can be characterized completely by the motion of the soil matrix. Furthermore, it is shown from energy balance consideration that the effective, or intergranular, stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil-water mixture. Finally, it is shown that the mathematical model is amenable to consistent linearization, and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method.

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Section: Balance Lawsmentioning

confidence: 99%

Section: Balance Of Linear Momentummentioning

confidence: 99%

A mathematical framework for finite strain elastoplastic consolidation Part 1: Balance laws, variational formulation, and linearization

Borja

Alarcón

1995

Computer Methods in Applied Mechanics and Engineering

109

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“…The equations governing the flow of a fluid through a porous solid can be derived systematically within the context of the theory of mixtures (see Truesdell [29,30], Atkin and Craine [31,32], Bowen [33], Rajagopal and Tao [34]). We shall not derive the equation from scratch, here, we refer the reader to Rajagopal [3] where a derivation is provided.…”

Section: Governing Equationsmentioning

confidence: 99%

Flow of a Fluid Through a Porous Solid Due to High Pressure Gradients

2013

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Abstract. It is well known that the viscosity of fluids could vary by several orders of magnitude with pressure. This fact is not usually taken into account in many important applications involving the flow of fluids through a porous media, like the problems of enhanced oil recovery or carbon dioxide sequestration where very high pressure differentials are involved. Another important technical problem where such high pressure differentials are involved is that of extracting unconventional oil deposits such as shale which is becoming ever so important now. In this study, we show that the traditional approach that ignores the variation of the viscosity and drag with the pressure greatly over-predicts the mass flux taking place through the porous structure. While taking the pressure dependence of viscosity and drag leads to a ceiling flux, the traditional approaches lead to a continued increase in the flux with the pressure difference. In this study, we consider a generalization of the classical Brinkman equation that takes the dependence of the viscosity and the Drag coefficient on pressure. To our knowledge, this is the first study to carry out such an analysis.

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“…Using equations (3-2), (3-4)-(3-6), (3)(4)(5)(6)(7), and (4-1)-(4.-5) we can rewrite (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) in the non-dimensional form. as…”

Section: Conservation Of Linear Momentum For the Fluid Constituentmentioning

confidence: 99%

Stability of flows in fluidized beds

Rajagopal¹

1992

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DE92 017669 DISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United Stat_,s Government.Neither the Unit¢_t States Government nor any agency th,;r¢of, nor any of their cmp[oy_s, makes any warrant)', express or implied, or assum¢:s any legal liability or r_sponsi-bility for the accuracy, completeness, or usef'ulne.ss of' any information, apparatus, product, or process disclo_d, or rcpr¢_nts that ks use would not infringe privately own_ rights. Reference: herein to any specific _mmercial product, proc,css, or service by trade name, trademark, rna)_ufacturer, or otherwise does not r_e,_saril), constitute or imply its endorsement, r¢com-mc)_dation, or favoring by th.¢ United States Government or any agency thereof. The q¢ws an, d opinions of authors exprcss_ herein do not r_c_,_.ssarily state or rcfl_t tho_ of the United States Oov©rnmcnt or any agency thc,rc_f.

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Continuum Theories of Mixtures: Basic Theory and Historical Development

Cited by 643 publications

References 3 publications

A mathematical framework for finite strain elastoplastic consolidation Part 1: Balance laws, variational formulation, and linearization

A mathematical framework for finite strain elastoplastic consolidation Part 1: Balance laws, variational formulation, and linearization

Flow of a Fluid Through a Porous Solid Due to High Pressure Gradients

Stability of flows in fluidized beds

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