## 2 Theory  
### Thermal Heating  
This section develops a model of the photothermal heating's contribution to the photomechanical stress response, which depends on two parameters; the stress change, \(\delta\sigma\), when the temperature is increased by \(\delta T\) and the response time of the photomechanical response. These parameters depend on the thermo-mechanical properties of the material and its geometry. \(\delta\sigma/\delta T\) is measured in the fiber sample by monitoring the force required to keep its length fixed as a function of temperature in what we call the clamped boundary condition. The response time of the photomechanical effect is determined from the photomechanical experiment. As a first iteration, we estimate the photothermal contribution by assuming that the measured response time of the fiber is due to heating. The observation of only one time constant implies that only a single mechanism is at play. If the model's prediction of the photomechanical response matches the measured value, then it is highly likely that photothermal heating dominates. The dynamics of the photothermal heating process is complex but can be numerically modeled [43]. Rather than doing so, we seek, here, to estimate the effect under the assumption that the observed dynamics is solely due to a temperature change, then to compare the magnitude of the measured response with that determined from the model. Then, as we describe in the next section, we develop a model that relates the stress response of both the photothermal heating and hole burning/reorientational mechanisms (see Section 2.2) to the tensor components of the stress response, which provides a second independent measurement. The heat equation gives the rate of temperature change \(dT/dt\)  
\[\frac{dT}{dt}=-\frac{1}{t_{N}}(T-T_{a})+\frac{I_{0}A}{mc}, \tag{1}\]  
where \(t_{N}\) is the Newton cooling time constant, which we will get from the data; assuming that the photothermal process is the only mechanism, \(T\) is the elevated sample temperature due to heating, and \(T_{a}\) is the ambient temperature. The last term in Equation (1) is the heating term, due to the intensity, \(I_{0}\), absorbed by the sample from the light, wherein \(A\) is the area of illumination, \(m\) is the mass of the illuminated part of the sample and \(c\) is the specific heat. In the derivation of Equation (1), we have used the fact that \(\delta T\) is given by the energy deposited \(\delta E=I_{0}A\delta t\) in time interval \(\delta t\), or  
\[\delta T=\frac{\delta E}{mc}. \tag{2}\]  
The temperature increase above ambient temperature at infinite time defined as \(\Delta T=T_{\infty}-T_{a}\) is reached when \(dT/dt=0\), so Equation (1) yields  
\[\Delta T\equiv(T_{\infty}-T_{a})=\frac{I_{0}At_{N}}{mc}. \tag{3}\]  
Using the fact that the heated mass \(m=\rho v\) is related to the illuminated volume \(v\) and polymer density \(\rho\)[44; 45], Equation (3) can be expressed as  
\[\Delta T=\frac{I_{0}t_{N}}{\rho cv}, \tag{4}\]  
where the volume is given by \(v=Aw\) with \(w\) the sample thickness. The linear photomechanical stress response \(\Delta\sigma/I_{0}\) from photothermal heating is given by  
\[\kappa_{\sigma}^{(1)}=-\frac{d\sigma}{dT}\cdot\frac{\Delta T}{I_{0}}. \tag{5}\]  
As we will show later, the stress, as a function of temperature for a glassy polymer sample that is mounted between a fixed clamp and a stress sensor, is well approximated by the function  
\[\sigma=\sigma_{0}+\frac{\sigma_{1}}{(1+(\frac{T}{I_{0}})^{n})}, \tag{6}\]  
so we will use this as the working model for the materials studied here.