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As a further illustration of this approach, we consider the behavior of viscoelastic dampers.
Constitutive models based upon fractional derivative (Makris and Constantinou, 1991; Makris et al.,
1993) and generalized Maxwell representations have been proposed in the literature. Here we employ
the latter representation to study device response. A finite element method is used to solve the
associated coupled thermomechanical problem for a two-dimensional model of solid viscoelastic
dampers. Again, numerical results are compared with data from physical experiments and some
interesting aspects of the cyclic response are revealed.
Finally, based upon the results of these case studies, arguments are given in favor of an increasingly
important role for computational continuum mechanics in the development and design of seismic
protective devices.
METALLIC DAMPERS
Constitutive Model
Practical two-surface phenomenological models, based primarily on the work of Krieg (1975), have
found wide application in the computational mechanics literature, and are adopted here for the cyclic
analysis of metallic dampers. The specific formulation selected is a modified version of the model
developed in Banerjee et al. (1987) and Chopra and Dargush (1994). The stress space behavior of the
model is depicted in Fig. la, which shows two distinct, but nested, cylindrical yield surfaces. The inner
or loading surface separates the elastic and inelastic response regimes. It is characterized by its center
and radius represented by the back stress and inner yield strength, respectively. On the other hand, the
outer or bounding surface, which completely contains the smaller inner surface, is always centered at
the origin of stress space with radius equal to variable outer yield strength. Translation of the inner
surface corresponds to kinematic hardening, while expansion of the outer surface produces isotropic
hardening. This separation of kinematic and isotropic hardening mechanisms proves to be quite useful
in representing the stabilized cyclic response of structural steel. The yield criteria, flow rules, and
hardening rules are established to ensure that the state of stress always lies on or within both surfaces,
that all transitions during loading are smooth, and that infinitesimal strain cycles do not cause
anomalous behavior. The present model requires the determination of five inelastic material
parameters, including the yield strength of the loading surface, the initial yield strength of the
bounding surface and three hardening parameters. (Details of the model are recorded in Sant, 2002).
The model has been implemented as a user-defined material model within the general-purpose finite
element program ABAQUS (1998). Three-dimensional, plane stress and uniaxial versions of the model
have been developed. Both explicit and implicit integration schemes are available.
This two-surface model was used to represent ASTM A36 structural steel. Material parameters were
established as follows. The elastic modulus and Poisson ratio were equated with the usual handbook
values, while the remaining five parameters were established from the stabilized cyclic data presented
by Cofie and Krawinkler (1985). Parameter values, obtained from the Marquardt (1963) algorithm for
nonlinear least squares curve fitting, are presented Dargush et al. (2002). A comparison of the stressstrain
response obtained from the two-surface model and the experimental data is displayed in Fig. Ib.