Beam elastic response modeling

Materials are modeled classically as either viscous, such as water or molasses, or elastic in nature, such as steel beams or metal springs. In general, polymers are complex materials that behave in a combined response to strain with both viscous and elastic characteristics. Under conditions where the material exhibits both viscous and elastic rheological behavior, the polymers are described as viscoelastic. That is, polymers have substantial viscous and elastic characteristics when strained. 

The unique cellular morphologies of foams play a key role in determining their deformation mechanisms [51. They also allow the development of very simple alternative equations based on the mechanical models of beam theory (a branch of civil engineering) combined with scaling concepts, to estimate both the thermoelastic properties and the strengths of foams. Such simple relationships have been developed for foams manifesting elastomeric, elastic-plastic and elastic-brittle responses to mechanical defonnation. While much of this work has focused on the responses of foams to compressive defonnation because of the special importance of this deformation mode in many applications of foams, the responses of foams to tensile and shear deformation have also been considered within this theoretical framework. 

Plastic hinges are modeled by nonlinear torsional springs mounted over revolute joints. The nonlinear response of the torsional springs are described in terms of a quasi-static moment-angle relationship based on the curves for elastic-plastic behavior, where stiffness of the elastic part is obtained from the elementary beam theory [19] as... 

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