Hookian springs

The constitutive equation of elasticity is represented by the Hookian spring (Fig. 16). Hook s law states that the stress is proportional to the strain... 

Rheological and elastic properties under flow and deformations are highly characteristic for many soft materials like complex fluids, pastes, sands, and gels, viz. soft (often metastable) solids of dissolved macromolecular constituents [1]. Shear deformations, which conserve volume but stretch material elements, often provide the simplest experimental route to investigate the materials. Moreover, solids and fluids respond in a characteristically different way to shear, the former elastically, the latter by flow. The former are characterized by a shear modulus Go, corresponding to a Hookian spring constant, the latter by a Newtonian viscosity r]o, which quantifies the dissipation. 

The extension from the three atom, one-dimensional case to that of an isolated polymer chain of finite length is straightforward. As before, the linear ehain is composed only of equal masses, m, separated by a constant distance, d (the repeat distance). Hookian springs with force constants, f, join the masses. Again, the chain is assumed to vibrate in only one dimension, say the x-axis. The equation of motion for the n mass, when it is displaced from its equilibrium position, is (analogous to equation 6.9)... 

Consider a one dimensional approximation of a triatomic linear molecule represented by identical mass points with mass, m, linked by weightless Hookian springs with force constants, (Figure 6.2). The equations of motion for this system is given (equation 6.9) by... 

Figure 13.2 Behavior of a Hookian spring and a Newtonian dashpot under excitation of constant ioad.

Note that the simple Hooke s law behavior of the stress in a solid is analogous to Newton s law for the stress of a fluid. For a simple Newtonian fluid, the shear stress is proportional to the rate of strain, y (shear rate), whereas in a Hookian solid, it is proportional to the strain, y, itself. For a fluid that shares both viscous and elastic behavior, the equation for the shear stress must incorporate both of these laws— Newton s and Hooke s. A possible constitutive relationship between the stress in a fluid and the strain is described by the Maxwell model (Eq. 6.3), which assumes that a purely viscous damper described by Eq. 6.1 and a pure spring described by Eq. 6.2 are connected in series (i.e., the two y from Eqs. 6.1 and 6.2 are additive). 

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