The 2011 Koiter Lecture: The Simple Logic of Classical Nonlinear Thermodynamic Shell Theory

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Energy Density ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Virtual Power ◽  
Ill Conditioning ◽  
Simple Logic ◽  
Reduced Forms

A classical nonlinear thermodynamic theory of elastic shells is derived by specializing the three-dimensional equations of motion and the second law of thermodynamics to a very general, shell-like body. No assumptions are made on how unknowns vary through the thickness. Extensional and bending strains are derived from the equations of motion via the principle of virtual power. The Coleman-Noll procedure plus the second law applied to an assumed form of the first law leads to constitutive relations plus reduced forms of the first and second laws. To avoid potential ill conditioning, a Legendre-Fenchel transformation is used to define a mixed-energy density, the logical place to impose the constitutive Kirchhoff hypothesis, if desired, because such an energy density rests, ultimately, on experiments. The Ladevèze-Pécastaings treatment of three-dimensional edge effects to obtain accurate two-dimensional solutions is discussed.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

A Displacement Equivalence-Based Damage Model for Brittle Materials—Part I: Theory

2003 ◽  
Vol 70 (5) ◽  
pp. 681-687 ◽  
Author(s):  
C. K. Soh ◽  
Y. Liu ◽  
Y. Yang ◽  
Y. Dong
Keyword(s):  
Crack Opening ◽  
Damage Model ◽  
Three Dimensional ◽  
Brittle Materials ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Stress And Strain ◽  
Onsager Relations ◽  
Normal Deformation ◽  
3D Effect

In this paper, a displacement equivalence-based damage model for brittle materials is proposed. A new damage deactivation criterion, which depends on both the stress and strain states of the materials, is adopted. Based on the concept of effective stress, the virtual undamaged configuration is introduced, and the assumption of displacement equivalence is proposed to correlate the damaged and the virtual undamaged configurations. Then, an additional crack-opening-induced normal deformation is introduced, and the three-dimensional (3D) effect of these opened cracks is also considered. The evolution rule of damage is deduced using the Onsager relations, which also ensure that the second law of thermodynamics is satisfied.

scholarly journals Thermodynamic properties of modified gravity theories

2016 ◽  
Vol 13 (06) ◽  
pp. 1630007 ◽  
Author(s):  
Kazuharu Bamba
Keyword(s):  
Thermodynamic Properties ◽  
Modified Gravity ◽  
Equations Of Motion ◽  
Apparent Horizon ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Modified Gravity Theories ◽  
Gravity Theories ◽  
Clausius Relation ◽  
The Universe

We review thermodynamic properties of modified gravity theories, such as [Formula: see text] gravity and [Formula: see text] gravity, where [Formula: see text] is the scalar curvature and [Formula: see text] is the torsion scalar in teleparallelism. In particular, we explore the equivalence between the equations of motion for modified gravity theories and the Clausius relation in thermodynamics. In addition, thermodynamics of the cosmological apparent horizon is investigated in [Formula: see text] gravity. We show both equilibrium and nonequilibrium descriptions of thermodynamics. It is demonstrated that the second law of thermodynamics in the universe can be met, when the temperature of the outside of the apparent horizon is equivalent to that of the inside of it.

scholarly journals Coupled constitutive relations: a second law based higher-order closure for hydrodynamics

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180323 ◽  
Author(s):  
Anirudh Singh Rana ◽  
Vinay Kumar Gupta ◽  
Henning Struchtrup
Keyword(s):  
Boundary Conditions ◽  
Constitutive Relations ◽  
Second Law Of Thermodynamics ◽  
Navier Stokes ◽  
Thermodynamic Force ◽  
Benchmark Problems ◽  
Second Law ◽  
System Of Equations ◽  
Thermodynamic Forces ◽  
Rarefaction Effects

In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.

Method of Virtual Power Applied to Cosserat Surfaces With Deformable Directors

1993 ◽  
Vol 46 (11S) ◽  
pp. S266-S278
Author(s):  
Boris Krajnc Alves ◽  
Jacob Lubliner
Keyword(s):  
Direct Method ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Momentum Balance ◽  
Power Method ◽  
Virtual Power ◽  
Balance Equations ◽  
Dimensional Theory ◽  
Momentum Balance Equation ◽  
Cosserat Surface

Following a brief outline of the method of virtual power, the local equations of motion for a Cosserat surface with inextensible directors are derived by means of this method. The model obtained coincides with the results derived from three-dimensional theory by Simo and Fox. Subsequently the model is extended so as to account for deformable directors. Besides the linear-momentum and moment-of-momentum balance equations, one additional scalar equation is derived. This equation replaces the director-momentum balance equation of Naghdi and therefore eliminates the necessity of introducing constitutive restrictions. The equivalence between the model derived by the virtual-power method and the results from the direct method of Naghdi are finally noted.

scholarly journals Microscopic foundation of the second law of thermodynamics within nonunitary Newtonian gravity

2019 ◽  
Vol 17 (08) ◽  
pp. 1941006
Author(s):  
Sergio De Filippo ◽  
Filippo Maimone ◽  
Adele Naddeo ◽  
Giovanni Scelza
Keyword(s):  
Numerical Calculation ◽  
Thermal Equilibrium ◽  
Energy Levels ◽  
Three Dimensional ◽  
Second Law Of Thermodynamics ◽  
Initial Energy ◽  
Von Neumann Entropy ◽  
Second Law ◽  
Newtonian Gravity ◽  
Von Neumann

The quest for a microscopic foundation of thermodynamics is addressed within the Nonunitary Newtonian Gravity (NNG) model through the study of a specific closed system, namely, a three-dimensional harmonic nanocrystal. A numerical calculation of the nanocrystal von Neumann entropy as a function of time is performed, showing a sharp monotonic increase, followed by a stabilization at late times. This behavior is consistent with the emergence of a microcanonical ensemble within the initial energy levels, signaling, in this way, the establishment of a nonunitary gravity-induced thermal equilibrium.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

First and second laws of thermodynamics: relationship, “inconsistency”, hidden effects

2020 ◽  
pp. 63-73
Author(s):  
A. M. Savchenko ◽  
Yu. V. Konovalov ◽  
A. V. Laushkin
Keyword(s):  
Free Energy ◽  
Internal Energy ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Entropy Of Mixing ◽  
Ideal Mixing ◽  
Laws Of Thermodynamics ◽  
Relationship Of ◽  
The Relationship

The relationship of the first and second laws of thermodynamics based on their energy nature is considered. It is noted that the processes described by the second law of thermodynamics often take place hidden within the system, which makes it difficult to detect them. Nevertheless, even with ideal mixing, an increase in the internal energy of the system occurs, numerically equal to an increase in free energy. The largest contribution to the change in the value of free energy is made by the entropy of mixing, which has energy significance. The entropy of mixing can do the job, which is confirmed in particular by osmotic processes.

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