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It needs to be emphasized that the choice of particular expansion variables, here ∇ u and ∇ T , affects the solutions of nonlinear problems like shockwave structure. Garc´ıa- Col´ın and Green emphasized that the description of nonequilibrium continuum mechanics is ambiguous whenever the choice of “equilibrium” variables – energy or longitudinal temperature or transverse temperature in this case – is ambiguous $^{17}$. The numerical value of a Taylor’s series in the deviations from equilibrium, truncated after the first nonlinear term, is clearly sensitive to the choice of independent variable. 

In the nonequilibrium pressure tensor the superscript t indicates the transposed tensor and I is the unit tensor: 

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η is the shear viscosity, and λ = η$_{v}$ − η is defined by the bulk viscosity η$_{v}$ . In the shockwave problem the pressure-tensor definitions give 

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For a two-temperature continuum model it is necessary to formulate the “equilibrium pressure” P eq as a function of the (nonequilibrium) energy, density, and the two temper- atures. The viscosities and conductivity could likewise depend upon these state variables and κ can be a tensor, as we show later, with an example. 

When we define T$_{xx}$ and T$_{yy}$ as continuum state variables it becomes necessary for us to formulate constitutive relations for their evolution. The simplest such models begin by separating the energy into two parts: a density-dependent “cold curve” e $^{cold}$( ρ ) and an additional kinetic or “thermal” part, proportional to temperature: 

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where ck is a scalar heat capacity. The functional form of the cold curve produces a corresponding contribution to the pressure: 

Gr¨ uneisen’s γ defines a corresponding thermal pressure: 

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