Plastic bending

Abstract: Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, bh. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, bh, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.

Abstract: Summary. Previous attempts to deduce the stress distribution in the bending lithosphere near a consuming plate margin have relied on the observed bathymetry and an assumed constitutive relation for lithospheric behaviour, eg. perfectly elastic, viscous/perfectly plastic, or elastic perfectly plastic. From the point of view of rock mechanics, each of these approximations fails to describe one or more of several basic phenomena, including brittle failure of rock, temperature dependence of elasticity, and temperature and/or strain rate dependence of ductile behaviour. In order to formulate a more realistic constitutive relation, a limiting yield strength curve, which is primarily a function of temperature, is constructed from data from brittle failure and ductile flow experiments. The moments which can be supported by plates with this constitutive behaviour are compared to the moments calculated from bathymetric profiles. The comparison indicates that moments required by the bathymetric data are consistent with moments supported by plates with experimentally determined constitutive laws as extrapolated to geo- logically reasonable temperatures and strain rates. The stresses developed in such models are required to reach values greater than 100 MPat in the depth range 25-45 km. Geotherms necessary for strength curves consistent with moments calculated from the bathymetric data match those derived from heat flow data for the Aleutian, Bonin, Mariana and Tonga trenches. Of the trenches studied, only the geotherm inferred from the Kuril trench data is significantly different, perhaps implying that the Kuril plate is weaker than the others. The strength curves show that as a first approximation it is better to assume that bending moment is independent of curvature of the plate than to assume that bending moment and curvature are linearly related.