We consider a recently proposed model in which dark matter interacts with a thermal background of dark radiation. Dark radiation consists of relativistic degrees of freedom which allow larger values of the expansion rate of the universe today to be consistent with CMB data ($H_0$-problem). Scattering between dark matter and radiation suppresses the matter power spectrum at small scales and can explain the apparent discrepancies between $Lambda$CDM predictions of the matter power spectrum and direct measurements of Large Scale Structure LSS ($sigma_8$-problem). We go beyond previous work in two ways: 1. we enlarge the parameter space of our previous model and allow for an arbitrary fraction of the dark matter to be interacting and 2. we update the data sets used in our fits, most importantly we include LSS data with full $k$-dependence to explore the sensitivity of current data to the shape of the matter power spectrum. We find that LSS data prefer models with overall suppressed matter clustering due to dark matter - dark radiation interactions over $Lambda$CDM at 3-4 $sigma$. However recent weak lensing measurements of the power spectrum are not yet precise enough to clearly distinguish two limits of the model with different predicted shapes for the linear matter power spectrum. In two Appendices we give a derivation of the coupled dark matter and dark radiation perturbation equations from the Boltzmann equation in order to clarify a confusion in the recent literature, and we derive analytic approximations to the solutions of the perturbation equations in the two physically interesting limits of all dark matter weakly interacting or a small fraction of dark matter strongly interacting.
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