A thermoelastic mechanism that manifests itself as nonlinear dissipation of acoustic waves without implying either hysteresis or nonlinear viscous-like losses is considered. This idea was earlier considered at rheological level by the present authors for microinhomogeneous solids with defects possessing elastic nonlinearity and linear viscous-like dissipation. In the present paper, this mechanism is discussed first from a more general point of view in terms of relaxators with parameters varied due to purely reactive nonlinearity. Then it is considered in details for a practically important special case of such relaxators in the form of cracks containing inner contacts. It is shown that, due to strong stress concentration at the contacts and due to enhanced temperature gradients, the thermoelastic dissipation at these relaxators can be very efficient. Implications for linear small-amplitude losses as well as nonlinear effects of induced transparency and induced dissipation are discussed. The two latter effects may coexist and be observed at different frequency ranges (below and above the relaxation maximum). The considered amplitude-dependent effects are typical of the same defects that may also exhibit hysteretic properties due to adhesion or friction mechanisms. Thus the discussed non-hysteretic and hysteretic mechanisms may be activated simultaneously, which should be taken into account in theoretical interpretation of observations.