A simplified, nonlinear thermodynamic theory of beamshells

This paper develops a nonlinear thermodynamical theory for arbitrary elasti c beamshells (infinite cylindrical shells in plane strain) in which approxi mations are made only in the First Law of Thermodynamics (Conservation of E nergy) and in the associated constitutive relations. The basic approach is straightforward: the three-dimensional equations of motion and the Second L aw of Thermodynamics (Clausius-Duhem Inequality) for an infinite cylindrica l body subject to external loads and heating are written in integral-impuls e form and then specialized to beamshells. This requires neither formal exp ansions in a thickness coordinate nor a priori kinematic hypotheses such as those associated with the names of Kirchhoff or Cosserat. The resulting on e-dimensional, time-dependent equations involve a vector stress resultant N , a scalar stress couple, Al, a vector translational momentum L, a scalar r otational momentum R, an entropy resultant S, an average reciprocal tempera ture T, and an average transverse temperature gradient G. The unknowns N, M , L, R, and S are defined in terms of thickness-weighted integrals, but T a nd G are defined in terms of the surface values of the three-dimensional ab solute temperature. A power identity yields, automatically, definitions of a strain vector e and a scalar bending strain k whose local rates are conju gate, respectively, to N and M. Once an elastodynamic (kinetic plus strain) energy of the beamshell is defined, the introduction of a free energy intr oduces an additional unknown F, an entropy couple conjugate to G. Enforceme nt of the Second Law for all possible thermodynamic processes, a la Coleman and Noll [1], plus the key assumption that the time derivative of F is a f unction of the state variables only, leads to a complete and consistent set of simplified constitutive relations. In the present approach there is jus t one entropy inequality and just one energy equation, in contrast to that of Green and Naghdi [2] who introduce a hierarchy of such equations, essent ially one for each director they introduce.