Fractional Burgers wave equation

Abstract

Thermodynamically consistent fractional Burgers constitutive models for viscoelastic media, divided into two classes according to model behavior in stress relaxation and creep tests near the initial time instant, are coupled with the equation of motion and strain forming the fractional Burgers wave equations. Cauchy problem is solved for both classes of Burgers models using integral transform method and analytical solution is obtained as a convolution of the solution kernels and initial data. The form of solution kernel is found to be dependent on model parameters, while its support properties implied infinite wave propagation speed for the first class and finite for the second class. Spatial profiles corresponding to the initial Dirac delta displacement with zero initial velocity display features which are not expected in wave propagation behavior.

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

“…Relaxation modulus σ sr (creep compliance ε cr ) is the stress (strain) history function obtained as a response to the strain (stress) assumed as the Heaviside step function H. According to the material behaviour in stress relaxation and creep tests at the initial time-instant, one differentiates the materials having either finite or infinite glass modulus σ (g) sr = σ sr (0), implying the finite or zero value of the glass compliance ε (g) cr = ε cr (0). The wave propagation speed, obtained as [4] for the distributed-order constitutive model (1.3) and in [8] for the fractional Burgers models (1.4) and (1.5), is the implication of these material properties. On the other hand, according to the material behaviour in stress relaxation and creep tests for large time, one differentiates fluid-like materials, having equilibrium compliance ε (e) cr = lim t→∞ ε cr (t) infinite and therefore the equilibrium modulus σ (e) sr = lim t→∞ σ sr (t) zero, from solid-like materials, having both equilibrium compliance and finite equilibrium modulus.…”

“…The fractional Burgers wave equation, represented by the governing equations (1.1), (1.2), and either (1.4) or (1.5), is solved for the Cauchy problem in [8]. The wave propagation speed is found to be infinite for models belonging to the first class, given by (1.4), contrary to the case of models of the second class (1.5), that yield finite wave propagation speed.…”