A general model of one-dimensional body with a scalar microstructure is introduced. Field equations are obtained via a variational principle, as Euler-Lagrange equations of a suitable energetic functional. The evolution of finite amplitude strain solitary waves is studied, taking into account both micro and macro dissipations. The formation, propagation and attenuation/amplification of bell-shaped and kink-shaped waves is proved. For a very simple form of the modal equation, the nonlinearity in the microlevel leads to a complicated term in the equation of motion and opens up direct ways for determining material constants characterizing the microstructure.