Part III
Time Dependent Deformation - Viscous Dissipation
104
Chapter 9
1-D Thought Models of Viscoelasticity
"Everything flows" [ (panta rhei)] - The aphorism of the Greek Philosopher Heraclitus (435-475 BCE) insists on ever-present change as be ing the fundamental essence of the universe. It is also a hallmark of solid materials that "flow" in time when subjected to loading. In contrast to elas- tic, fracture and plastic deformation that occur instantaneously in response to loading, solid materials can dissipate externally supplied work in time at constant load or constant deformation. The time dependent deformation at constant load is called "creep", and the time dependent change in force or stresses at constant deformation is called "relaxation". Creep and relaxation are two opposite phenomena that have at its source viscous phenomena at molecular, mesoscale and macroscale. The focus of viscoelasticity is to quan- titatively assess this time dependent behavior of solids.
9.1 Creep Compliance and Relaxation Mod-
ulus
The simplest rheological model of viscoelasticity is a spring of stiffness E in series with a dashpot defined by a viscosity n (see Fig. 9.1); a system that goes by the name of Maxwell model. Denoting by & and e' the (normal- ized) displacement at the point of load application and of the dashpot, the
105
4EY fE
E
t-t
Figure 9.1: 1-D Toy Model of Viscoelasticity: Maxwell Model with creep (strain) response ((t) = o imposed) and (stress) relaxation response ((t) = &o imposed).
elasticity of the 1-D system is defined by:
o=E(e-e)
(9.1)
While e is controlled from the outside, the evolution in time of the viscous strain is defined by a rate law: dev - - (9.2) u-3P Consider that a constant stress is applied, in the form o = ooH (t - to): where H (t - to) is the Heaviside function (H (x < 0) = 0; H (x > 0) = 1) Such a test corresponds to a creep test. Integrating Eq. (9.2) and using this result in the stress equation of state (9.1) provides a means to determine the measurable creep strain:
(t,to= 0-7 T L =OoJ(t,to)
(9.3) where T = n/E is a charact&ristic tim, whereas J(t,to) = e(t,to)/o, is called the creep compliance, which in the case of the Maxwell model reads as J(t,to)= 1+ t-to 10