Elastomer-based composites embedded with thermally-responsive material (TRM) and a liquid-phase Joule heater are capable of reversibly changing their elastic rigidity by up to four orders of magnitude. At room temperature, the TRM layer is rigid and prevents the surrounding elastomer from elastically bending or stretching. When activated, the embedded Joule heater softens or melts the TRM, which leads to a dramatic reduction in the elastic rigidity of the composite. In this manuscript, we examine the activation of these composites by performing analytical, numerical, and experimental studies of the temperature distribution, thermal history, and phase transition. We consider both low melting point (LMP) metal alloys (e.g. Field's metal) and shape memory polymer (SMP). An analytical solution using the Galerkin Based Integral (GBI) method is derived for the cases where no phase change is involved, while a numerical scheme using the Latent Heat Accumulation (LHA) method is utilized to probe scenarios where phase change has a central role in the elastic rigidity change. The analytical and numerical studies predict a temperature history that is in good agreement with experimental measurements obtained with an IR thermometer. Analysis of the internal temperature distribution leads to scaling laws for determining the required activation time and allowable input power rate for composites containing either LMP alloys or SMP. These scaling laws could potentially be used to inform the design of rigidity tunable composites (RTC) used in assistive wearable technologies and biologically-inspired soft-matter robotics. (C) 2013 Elsevier Ltd. All rights reserved.