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Forward and inverse problems for the standard linear solid viscoelastic model using a mixed displacement-stress finite element formulation

In this study, a mixed formulation of stress and displacement equations was used to simulate behavior of materials described by the standard linear solid (SLS) viscoelastic model for homogeneous materials. The SLS viscoelastic model contains a Maxwell element and an elastic spring that are assembled in parallel. In the current work, viscous damping was included in the deviator stress while the hydrostatic stress remained un-damped. For the forward problem, mixed displacement-stress equations using the SLS model in the time-domain were discretized using the finite element method. Both displacements and stresses were unknown parameters. The implementation of this problem was relatively straightforward, and no iterative process was needed to solve the equations. Both responses (displacements and stresses) can be obtained simultaneously which is the major advantage of this method. Additionally, the stability of the system of equations is good, and no instability issues were observed for even the longest dynamic simulations. To solve the inverse problem, an algorithm has been developed to estimate the viscoelastic properties in a finite domain by comparing measured and simulated dynamic responses. To this end, the Nelder-Mead simplex optimization technique was used to update the viscoelastic parameters.
To illustrate the forward problem, example problems for 2D-plane strain have been prepared. First, the results of mixed formulations were compared with displacements from finite element analyses of elastic models that did not include the viscous damping parameter. Then, a parametric study was conducted to study the effects of varying the viscous damping parameter over a wide range of values. For the inverse problem, the viscoelastic parameters of a homogeneous concrete domain with excitation provided by an impact force history were estimated using the proposed rheological model (SLS). Noise in the measured responses was also investigated to study its effect on reconstruction of the viscoelastic parameters.

Author(s):

Alireza Pakravan    
New Mexico State University
United States

Craig Newtson    
New Mexico State University
United States