Let V denote a material body and let P=∫v ∑ a=1 N J αX αν denote the internal production of entropy of V, where J a are the thermodynamic fluxes and X a are the thermodynamic forces. A constitutive theory is obtained which determines the thermodynamic fluxes in terms of variational derivatives of a convex functional Φ[ W; X] which assumes its absolute minimum at X a = 0. It is shown that 0 ⩽ P ⩽ Φ[ W;2 X]− Φ[ W; X] for all X a and for all additional parameters W α , and hence we have identical satisfaction of the Second Law of Thermodynamics. If Φ = ∝ v φ( W; X) dv, then J a = ∂φ ∂X a and we obtain the reciprocity relations ∂J a ∂X b = ∂J b ∂x a . The constitutive theory is not restricted to linear relations since the only requirements on φ( W; X) are convexity in X a and ( ∂ ∂X a )φ¦ x a=0 = J a¦ x a=0 = 0 . Specific constitutive relations are obtained for the heat flux vector and the dissipative stress tensor. Since φ can depend on the parameters W α as well as on the X a 's, the dissipative stress and the heat flux can depend on the strain and temperature as well as on the rate of strain and the temperature gradient.