Series Elastic Component

It is implicit in the idea of a series elastic component that there should be a lag between the activation of the contractile apparatus and the rise of force measured between the ends of the muscle. The observed time lag is also commensurate with the idea that the series component is largely due to the crossbridges themselves. 

The Maxwell Model. The first model of viscoelasticity was proposed by Maxwell in 1867, and it assumes that the viscous and elastic components occur in series, as in Figure 5.60a. We will develop the model for the case of shear, but the results are equally general for the case of tension. The mathematical development of the Maxwell model is fairly straightforward when we consider that the applied shear stress, r, is the same on both the elastic, Xe, and viscous, Xy, elements. 

The model represents a liquid (able to have irreversible deformations) with some additional reversible (elastic) deformations. If put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components as per the Maxwell Model. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. It is important to note limitations of such a model, as it is unable to predict creep in materials based on a simple dashpot and spring connected in series. The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time [23-26],... 

Figure 16.4 illustrates the mechanical components of the oculomotor plant for horizontal eye movements, the lateral and medial rectus muscle, and the eyeball. The agonist muscle is modeled as a parallel combination of an active state tension generator Fag> viscosity element Bag> and elastic element TlT) connected to a series elastic element Rse- The antagonist muscle is similarly modeled as a parallel combination of an active state tension generator Tant> viscosity element Rant> and elastic element TlT) connected to a series elastic element Rse- The eyeball is modeled as a sphere with moment of inertia /p, connected to viscosity element Bp and elastic element Kp. The passive elasticity of each muscle is included in spring Kp for ease in analysis. Each of the elements defined in the oculomotor plant is ideal and linear. 

The paradox of Maxwell s model. A popular representation of models in rheology mimics the equivalent electrical circuits with dipolar components. The elastic component is naturally symbolized by a spring and the viscous component by a damper or dashpot (a piston filled with a viscous fluid able to circulate). The viscoelastic relaxation is thus represented with these two components mounted in series, as shown in Figure 11.12a and is known as Maxwell s model (Oswald 2005). (In this representation, the customary notation is used for facilitating comparison with the literature.)... 

Guiu and Pratt [59] have shown how a model consisting of an Eyring dashpot in series with an elastic element leads to a simple equation to describe stress relaxation curves in tension. Suppose that there is a total strain e on the system of Figure 11.16, consisting of an elastic component es and an Eyring or viscous component ev, such that... 

Skeletal muscles are the only actuators in the human body. A skeletal muscle is composed of two types of structural components active contractile elements and inert compliant materials. The contractile elements are contained within the muscle fibers. The fibers vary in length from a few millimeters to more than 40 cm, and their width is between 1 and 150 im. Approximately, 85% of the mass of a muscle consists of the muscle fibers composed from sarcomeres, while the remaining 15% is largely composed of the connective tissue, which contains variable proportions of collagen, reticular, and elastic fibers. The connective tissues provide an arrangement of simple, spring-like elements (elastic components of the muscle) that exist both in series and in parallel with the contractile elements. 

Figure 25.9 illustrates the mechanical components of the updated oculomotor plant for horizontal eye movements, the lateral and medial rectus muscle, and the eyeball. The agonist muscle is modeled as a parallel combination of viscosity B2 and series elasticity connected to the parallel combination of... 

In the Maxwell model, the two units are connected in series, so that each bears the full stress individually and the deformations of the elastic and viscous components are additive ... 

A plastic component was subjected to a series of step changes in stress as follows. An initial constant stress of 10 MN/m was applied for 1000 seconds at which time the stress level was increased to a constant level of 20 MN/m. After a further 1000 seconds the stress level was decreased to 5 MN/m which was maintained for 1000 seconds before the stress was increased to 25 MN/m for 1000 seconds after which the stress was completely removed. If the material may be represented by a Maxwell model in which the elastic constant = 1 GN/m and the viscous constant rj = 4000 GNs/m, calculate the strain 4500 seconds after the first stress was applied. 

The usual way in which the deformation changes with time, has been dealt with in 6.1. The best representation appeared to be a Maxwell element with a Kelvin-Voigt element in series the deformation is then composed of three components an immediate elastic strain, which recovers spontaneously after removal of the load, a delayed elastic strain which gradually recovers, and a permanent strain. Moreover, we noticed that a single retardation time (a single Kelvin-Voigt element) is not sufficient we need to introduce a spectrum ... 

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