Application of Force and Energy Approaches to the Problem of a One-Dimensional, Fully Connected, Nonlinear-Spring Lattice Structure
Final rept. Jun 2015
ARMY RESEARCH LAB ABERDEEN PROVING GROUND MD WEAPONS AND MATERIALS RESEARCH DIRECTORATE
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In this report, force and energy methods are applied to the problem of 1-dimensional, fully connected, nonlinear-spring lattices. The report confirms that energy and force methods produce equivalent results, even when nonlinear, non-local effects come into play. Demonstrating this compatibility is slightly complicated for the problem of fully connected lattices, because the boundary conditions of simple loading and uniform spacing are, in general, incompatible. In this report, both boundary conditions are studied separately and it is indeed shown that the force approach involving free-body diagrams and the energy approach involving potential spring energy produce compatible measures of lattice force, if the conceptualization is properly formulated. Two interesting revelations ensued from employing an energy approach 1 the lattice force a derivative quantity is not equal to the net applied external load, but rather it equals the average internal force across the lattice and 2 the global lattice force, being a sum of internodal forces, nonetheless requires a multiplier on each local internodal force proportional to the internodal separation. The multiplier arises from an application of the chain rule, when taking the spatial derivative of energy.
- Numerical Mathematics