Properties for Design of Composite Structures. Neil McCartney

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Название Properties for Design of Composite Structures
Автор произведения Neil McCartney
Жанр Техническая литература
Серия
Издательство Техническая литература
Год выпуска 0
isbn 9781118789780



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      2.8.1 Conservative Body Forces

      Consider now the special case when the heat source per unit mass r = 0 and the body force is derivable from a scalar potential function ζ as follows

      The effects of the Earth’s gravitational field can then be taken into account. The substitution of (2.46) into (2.41) leads to the relation

      As

      rho v period nabla zeta equals nabla period left-parenthesis rho v zeta right-parenthesis minus zeta nabla period left-parenthesis rho v right-parenthesis equals nabla period left-parenthesis rho v zeta right-parenthesis plus StartFraction partial-differential left-parenthesis rho zeta right-parenthesis Over partial-differential t EndFraction comma(2.48)

      it then follows from (2.47) that the local form (2.44) for the total energy balance equation may also be expressed

      StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis rho e plus rho zeta plus one-half rho v squared right-parenthesis equals minus nabla period left-bracket h minus sigma period v plus rho v left-parenthesis e plus zeta plus one-half zero width space v squared right-parenthesis zero width space zero width space right-bracket period(2.49)

      The energy balance equation may, therefore, be written in the more compact form

      which is the sum of the internal energy, potential energy and kinetic energy per unit volume, and where the total energy flux Jχ is given by

      On using (2.51), relation (2.52) may also be written in the form

      upper J Subscript chi Baseline equals chi v plus h minus sigma period v comma(2.53)

      which identifies the mechanisms whereby energy can enter or leave the system, namely advection (the term χv), as heat (the term h) and as external work (the term −σ.v).

      The energy balance equation (2.50) implies energy conservation as the energy within any region V always remains fixed during any nonequilibrium process provided that there is no energy flow across the surface S bounding the region V. This result is easily established by integrating (2.50) over the region V, and then making use of the divergence theorem.

      2.9 Equations of State for Hydrostatic Stress States

      The balance equations of mass, momentum and energy introduce a variety of physical variables, but they do not involve in any way the properties of materials. An equation of state must now be introduced, implying the existence of a variety of material properties, depending upon the complexity of the material. If elastic effects are to be included in models, then it is through an equation of state that they must be introduced. Before using strain as a thermodynamic state variable, it is useful first to introduce some key thermodynamic relationships that can be applied when the stress field is hydrostatic, i.e. shear stresses are absent, characterised by a pressure p, such that the stress tensor has the form σ=−pI where I is the second-order unit tensor.

      2.9.1 Global Thermodynamic Relations

      It is essential to understand fully the nature of thermodynamic principles that underpin modelling procedures. It is assumed now that the system studied is multi-component and uniform, and that the temperature and pressure are uniform as would be the case for many equilibrium states of the system. At equilibrium, the classical thermodynamic relations for a uniform multi-component mixture, where Mk,k=1,…,n denote the total masses of the various species, are given by

      where U is the internal energy of the system and F and G are the corresponding Helmholtz and Gibbs energies defined by

      upper F equals upper U minus upper T upper S comma upper G equals upper F plus p upper V equals upper U minus upper T upper S plus p upper V period(2.55)

      The brackets {} in (2.54) denote a set of state variables associated with the various species defined for k=1,…,n. The state variables S, T, V and p are the entropy, thermodynamic (i.e. absolute) temperature, volume and pressure, respectively, and the parameters μk,k=1,…,n, are the mass-based chemical potentials. Each of the relations (2.54) defines, for media having uniform state variables and hydrostatic stress states (i.e. in the absence of shear stresses), an equation of state for three different but equivalent