NONLINEAR ELASTIC AND VISCOELASTIC 1–D STRESS – DISPLACEMENT WAVE PROPAGATION MODELING AND ANALYSIS

Abstract

NONLINEAR ELASTIC AND VISCOELASTIC 1–D STRESS – DISPLACEMENT WAVE PROPAGATION MODELING AND ANALYSIS 

Two separate sets of nonlinear 3–D constitutive relations are modeled in terms of (1) elastic strain invariants and (2) viscoelastic strain and strain rate invariants each assembled in their own multi-dimensional Maclaurin series. Characterization of material properties is formulated and evaluated. Fundamental difficulties associated with the nature of the nonlinear functions defining the stress-strain relations are examined. Several protocols for joint analytical and experimental viscoelastic material characterization and wave motion analyses are presented. As an illustrative problem, the 1–D wave propagation phenomenon is studied in detail. It is shown that the wave front velocities in linear elasticity and viscoelasticity are invariant with respect to the magnitude and character of the forcing function. In nonlinear elastic and viscoelastic media, the specific characteristics of the constitutive relation nonlinearities also do not influence the wave front velocity, but impact stresses and strains. Both must be evaluated on a case by case basis for each specific material properties and impact load-time functions.

The nonlinear analyses are carried out using Poincaré’s successive approximation method (SAM). The use of SAM successively leads to linear elastic PDEs and to linear viscoelastic IPDEs respectively for each and every unknown approximation term of the stress and displace- ment wave series as well as the series representing the propagation velocity. The influences of the impact force loading pattern and of the nonlinear elastic and viscoelastic contributions on responding wave motions and wave front velocities and amplitudes are analyzed and evaluated. 

In particular, both the classical and distributed Dirac delta function impact loads are analyzed in detail and found to be incompatible models compared to the real world in certain areas in both linear and nonlinear elastic and viscoelastic media. An actual impact force pattern is then modeled and analyzed in the same medium and it was found to also work well with the SAM.