Effective classical representations of heterogeneous systems fail to have an effect on the overall response of components on the patial scale of heterogeneity. This effect may be critical if the
effective continuum subjects' scale differs from the material's microstructure scale and then leads to
size-dependent effects and other deviations from conventional theories. This paper is concerned with the thermoelastic behavior of rotating nanoscale beams subjected to thermal loading under mechanical thermal loads based on the non-local strain gradient theory (NSGT). Also, a new mathematical model
and governing equations were constructed within the framework of the extended thermoelastic theory with phase delay (DPL) and the Euler-Bernoulli beam theory. In contrast to many problems, it was
taken into account that the thermal conductivity and specific heat of the material are variable and linearly dependent on temperature change. A specific operator has been entered to convert the
nonlinear heat equation into a linear one. Using the Laplace transform method, the considered problem
is solved and the expressions of the studied field variables are obtained. The numerical findings demonstrate that a variety of variables, such as temperature change, Coriolis force due to rotation, angular velocity, material properties, and nonlocal length scale parameters, have a significant influence on the mechanical and thermal waves.