| dc.description.abstract | Can the mathematical framework developed for constitutive nonlinearities be extended to arbitrary dimensions and capture the structural relationship between forces and displacements? In this work, the mathematics developed to capture plasticity (i.e., the nonlinear relationship between the six components of stress and strain) are leveraged to describe general nonlinear force-displacement responses. Plasticity constitutive models seek to describe the nonlinear relationship between stress and strain, usually via 1) an additive decomposition of strain, 2) some assumption regarding the existence of an elastic domain, and 3) evolution equations that govern internal state variables, most importantly the inelastic strain.
Herein we develop nonlinear substructures to provide a method to describe structural relation-ships between force and displacement associated with various degrees of freedom essential for prediction of global response, these being analogous to stresses and strains. We draw inspiration from linear substructure analysis, a historical structural model order reduction method. Linear sub-structure analysis reduces the computational order of a discretized structural component from the full set of degrees of freedom needed to solve a boundary value problem (e.g., the displacements of all nodes in an FEA mesh) to a predefined and much smaller set of retained degrees of freedom, usually via a linear transformation. Given only one initial analysis considering all degrees of freedom, this technique reduces the computational cost associated with subsequent analyses of the same component by eliminating degrees of freedom, usually internal to the body, which are not essential for interfacing the component with a larger system/assembly and/or for providing essential engineering performance information.
In this work, linear substructure analysis is extended to consider general nonlinear responses by leveraging the mathematical framework developed for computational plasticity. While the latter provides nonlinear constitutive relationships between six independent stress and strain components, we show that the same mathematical formulation can capture similar relations between an arbitrary number of forces and displacements (i.e., the retained degrees of freedom). The developed nonlinear substructure method is then demonstrated by analyzing a sweep morphing wing comprised of an array of multi-material unit cells at reduced computational cost with less than 15% error. | |
