Mechanical response and simulation for constitutive equations with distributed order derivatives

Mechanical response and simulation for constitutive equation with distributed order derivatives were considered. We investigated the creep compliance, creep recovery, relaxation modulus, stress–strain behavior under harmonic deformation for each case of two constitutive equations. We express these responses and results as easily computable forms and simulate them by using MATHEMATICA 8. The results involve the exponential integral function, convergent improper integrals on the infinite interval [Formula: see text] and the numerical integral method for the convolution integral. For both equations, stress responses to harmonic deformation display hysteresis phenomena and energy dissipation. The two constitutive equations characterize viscoelastic models of fluid-like and solid-like, respectively.

Harmonic Deformation of Planar Curves

We establish a principle of deformation of an arbitrary planar curve, so that the integral of a harmonic function over this curve does not change. The equations of deformation can be derived from a specific “potential.” Several applications are presented.