Constitutive Equations for an Induced-Strain Bending Actuator with a Variable Substrate

The constitutive equations for an induced-strain bimorph are derived for a transducer with a substrate that has a variable thickness and variable modulus. The constitutive equations for sensing and actuation are derived using an energy analysis for the case of a tip force, constant moment, and distributed pressure on the bimorph. The expression for each of the constitutive terms is equivalent to the expression for an ideal bimorph multiplied by a nondimensional modifier. For this analysis we define an ideal bimorph as one which has a negligible substrate thickness. The nondimensional modifier is only a function of two parameters: the ratio of substrate thickness to active layer thickness and the ratio of elastic moduli between the active material and the substrate layer. This allows the results to be easily compared to previous solutions for an ideal bimorph. Expressions for the blocked force, free deflection, and stored electrical energy are derived from the constitutive equations. The results demonstrate that stored energy and the effective coupling coefficient of the actuator do not vary appreciably for thickness ratios less than 1, indicating that actuator performance can be maintained even when the substrate thickness is an appreciable percentage compared to the thickness of the active layer. Blocked force and free deflection do change appreciably above thickness ratios on the order of 0.5, indicating that varying the thickness and modulus of the substrate layer is an additional design freedom in achieving tradeoffs in blocked force and free deflection with bimorph actuators. The results of this paper are useful for the design of bimorph transducers with specific performance requirements. The results also provide insight into some of the limitations exhibited by polymeric bimorphs actuated by an electric field. Our analysis demonstrates that the work output of a bimorph can be one to two orders of magnitude smaller than the stored mechanical energy when the active layer thickness is small compared to the overall bimorph thickness. This result is consistent with other models that suggest that the induced strain layer in certain ionic polymer materials is only a small portion of the overall transducer thickness. Experimental verification is provided on a series of bimorphs ranging from thickness ratios of approximately 0.1 to 10. Experimental results closely agree with the model predictions, although discrepancies become more significant as the thickness ratio approaches 10.