Thermomechanics of Shells with Sigular Curves

• Autorzy: Makowski J. , Pietraszkiewicz W.
• Języki publikacji: EN

Abstrakty

EN

We formulate rigorously the global and local laws of mechanics and thermodynamics for shells with singularities at some stationary or moving curves in the shell base surface (itself not necessarily smooth). The laws are derived in an exact manner from underlying laws of continuum thermomechanics written in the integral - impulse form for the shell-like body. Our formulation is sufficiently general to include not only traditional applications to reversible problems of regular shells, but also those modeling irreversible and non-smooth processes in irregular shells. We assume that the shell-like body is represented in the physical space by the base surface' which in a reference configuration is only Lipschitz continuous with almost smooth boundary. By a moving singular curve we mean a one-parameter family of piecewise smooth surface curves which transverse the reference configuration of the shell base surface and across which various thermomechanical field variables may suffer jump discontinuities. However, all the fields are assumed to be regular enough for the generalized surface transport and gradient-divergence theorems to be applicable. As a result of complex transformations presented in the report, at regular points of the reference base surface and for almost all time instants we obtain five local laws of shell thermomechanics: the balance of mass, linear momentum, angular momentum, and energy as well as the entropy inequality. From the transformations we also obtain, corresponding to the laws of shell thermomechanics, five continuity conditions at regular points of every singular surface curve. Additionally, we discuss exact 2D shell kinematics and exact 2D shell strain measures. The principal features of the derived field equations and side conditions are: 1) the classical expressions for the linear and angular momenta are not assumed from the outset (they must be given by appropriate constitutive equations), 2) there is no classical splitting of the total energy into the sum of internal and kinetic energies (such a splitting is considered as a part of constitutive theory), 3) the entropy source and the entropy influx are not directly related to temperature, 4) there are two additional terms in the equation of energy balance which represent the interstitial working (they require a suitable constitutive prescription). We show that within the general shell thermomechanics the constitutive equations are needed for the surface stress tensor, the surface couple tensor, the specific total energy, the specific entropy, and the heat influx vector fields. But additionally we need the constitutive prescription for the linear and angular momenta vectors, as well as possibly for several other supplementary field variables. General expressions for the constitutive equations are given through response functionals of the histories of motion and temperature fields. For spatially first-grad "simple" shells we propose reduced forms of constitutive equations in the spatial and material representations. We also discuss additional constitutive assumptions which would allow us to eliminate temperatures, heats and entropy influxes at the upper and lower shell faces, as well as fields describing the interstitial working, the extra entropy source and the extra entropy flux. We derive the reduced dissipation inequality for shells and use it to develop thermodynamically consistent constitutive equations appropriate for heat conducting and thermo-visco-elastic shells. Particular forms of constitutive equations for thermoelastic, isothermal or higher-grad shells are proposed. By introducing thermodynamic potentials we also discuss constitutive nature of representing the total shell energy density as the sum of potential, kinetic and interstitial energy densities. Finally, we propose general and some specific forms of the kinetic constitutive equations for the linear and angular momenta. The results presented in this report may be considered as an introduction to a variety of thermomechanical problems of the regular and irregular shells, which might be formulated and solved already in the near future.

Słowa kluczowe

EN

thermomechanics; thermodynamics

• Wydawca: Wydawnictwo Instytutu Maszyn Przepływowych PAN
• Czasopismo: Zeszyty Naukowe Instytutu Maszyn Przepływowych Polskiej Akademii Nauk w Gdańsku
• Rocznik: 2002
• Tom: nr 528/1487
• Strony: 1--100
• Opis fizyczny: Bibliogr. 59 poz., rys.

Twórcy

autor

Makowski J.

autor

Pietraszkiewicz W.