Microcantilever Bending Experiments and Measurement of the Elastic Size Effect Based on Gradient Elasticity

Classical continuum theory fails to predict micron-sized structural behavior due to an elastic size effect. Various higher-order deformation theories, such as couple stress theory, have been developed to simulate stiffening of structures with decreasing size. In this work, micro-scale cantilever bending experiments are conducted and an increase in the effective elastic modulus is observed with decrease in beam thickness.<br/>Microstructural and mechanical characterization of the polycrystalline copper plate revealed that the crystal consists of non-textured equiaxed grains with an average grain size of 2.3 µm, and the measured elastic modulus was 112 GPa. Micro-scale cantilevers with thicknesses ranging from 1.6 µm to 8.6 µm were fabricated within the copper plate using femtosecond laser machining followed by focused ion beam milling. A nanoindenter equipped with a wedge tip was utilized to apply line load on the fabricated samples. Finite element analysis was performed to exclude the effect of substrate deformation for calculating pure beam stiffnesses. The measured effective elastic modulus increases from 94 GPa to 215 GPa with decreasing thickness.<br/>Couple stress theory is used to model the size effect. In the couple stress theory, a length scale parameter is introduced as an additional material property in addition to Lamé constants. An optimization problem was formulated to find the length scale parameter that minimizes the deformation difference between the experiment result and finite element analysis. The length scale parameter determined by solving the optimization problem is 0.78 µm.<br/>The physical origin of the length scale parameter is discussed by considering mobile dislocations escaping via the free surfaces. Based on experiment results, it is suggested that the length scale parameter in metals is dependent on the microstructure of the crystal. As the couple stress theory is the simplest higher-order deformation theory, it has the advantage of low computational cost for analyzing structures with arbitrary geometry. Due to this advantage, FEA simulations based on the couple stress theory have the potential to bridge between micro/nano-scale (e.g. molecular dynamics, dislocation dynamics) simulations and bulk simulations, and can be widely used in the design and analysis of micro-scale devices.