Synopsis We extend methods from the kinetic theory of gases to obtain a general constitutive relation for the collisional source of second moment in dilute flows of identical, smooth, highly inelastic spheres. In the derivation, we base all statistical averaging on an anisotropic Maxwellian distribution function, which is sensitive to all components of the full second moment of fluctuation velocity and is not based on the assumption that the fluctuations are nearly isotropic. In the case of

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... geneous shear flow, we combine the constitutive relation with the balance equation for full second moment to determine, for prescribed values of shear rate, coefficient of restitution, and solid fraction, both exact numerical and approximate closed-form solutions for the second moment and pressure tensor. Most striking are the resulting normal pressure differences, which are predicted by this theory but not by kinetic theories for nearly elastic particles.