A simplified, nonlinear thermodynamic theory of beamshells

Author:
J. G. Simmonds

Journal:
Quart. Appl. Math. **59** (2001), 401-412

MSC:
Primary 74K25; Secondary 74A15, 74B05

DOI:
https://doi.org/10.1090/qam/1848525

MathSciNet review:
MR1848525

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Abstract: This paper develops a nonlinear thermodynamical theory for arbitrary elastic beamshells (infinite cylindrical shells in plane strain) in which approximations are made only in the First Law of Thermodynamics (Conservation of Energy) and in the associated constitutive relations. The basic approach is straightforward: the three-dimensional equations of motion and the Second Law of Thermodynamics (Clausius-Duhem Inequality) for an infinite cylindrical body subject to external loads and heating are written in integral-impulse form and then specialized to beamshells. This requires neither formal expansions in a thickness coordinate nor a priori kinematic hypotheses such as those associated with the names of Kirchhoff or Cosserat. The resulting one-dimensional, time-dependent equations involve a vector stress resultant **N**, a scalar stress couple , a vector translational momentum **L**, a scalar rotational momentum , an entropy resultant , an average reciprocal temperature , and an average transverse temperature gradient . The unknowns **N**, , **L**, , and are defined in terms of thickness-weighted integrals, but and are defined in terms of the surface values of the three-dimensional absolute temperature. A power identity yields, automatically, definitions of a strain vector **e** and a scalar bending strain whose local rates are conjugate, respectively, to **N** and . Once an elastodynamic (kinetic plus strain) energy of the beamshell is defined, the introduction of a free energy introduces an additional unknown , an entropy couple conjugate to . Enforcement of the Second Law for all possible thermodynamic processes, à la Coleman and Noll [1], plus the key assumption that the time derivative of is a function of the state variables only, leads to a complete and consistent set of simplified constitutive relations. In the present approach there is just one entropy inequality and just one energy equation, in contrast to that of Green and Naghdi [2] who introduce a hierarchy of such equations, essentially one for each director they introduce.

**[1]**B. D. Coleman and W. Noll,*The thermodynamics of elastic materials with heat conduction and viscosity*, Arch. Rat. Mech. Anal.**51**, 1-53 (1963) MR**0153153****[2]**A. E. Green and P. M. Naghdi,*On thermal effects in the theory of shells*, Proc. Roy. Soc. London**A365**, 161-190 (1979) MR**527033****[3]**J. G. Simmonds,*The nonlinear thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions*Flexible Shells (E. L. Axelrad and F. A. Emmerling, eds.), Springer-Verlag, Berlin, 1984, pp. 1-11 MR**765868****[4]**A. Libai and J. G. Simmonds,*Nonlinear elastic shell theory*, Adv. Appl. Mech.**23**, 271-371 (1983)**[5]**A. Libai and J. G. Simmonds,*The Nonlinear Theory of Elastic Shells: One Spatial Dimension*, Academic Press, Boston, 1988 MR**931535****[6]**A. Libai and J. G. Simmonds,*The Nonlinear Theory of Elastic Shells*, 2nd ed., Cambridge Univ. Press, Cambridge, U.K., 1998 MR**1610214****[7]**C. A. Truesdell,*Rational Thermodynamics*, 2nd ed., Springer-Verlag, New York, 1984 MR**766401****[8]**S. P. Timoshenko,*On the correction for shear of the differential equation for transverse vibrations of prismatic bars*, Philos. Mag.**41**, 744-746 (1921)

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DOI:
https://doi.org/10.1090/qam/1848525

Article copyright:
© Copyright 2001
American Mathematical Society