Elastic spring model

E. Boggs Interplay of Experiment and Theory in Determining Molecular Geometries. B. Theoretical Methods. -A. Y. Meyer Molecular Mechanics alias Mass Points and Elastic Springs Model of Molecules. - K. B, Wiberg Atoms in Molecular Environments. - Z. B. Maksic The Modelling of Molecules as Collections of Modified Atoms. 

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

Bueche (16,152) had earlier proposed a related theory based on a spring-bead model (springs with a rubberlikc elasticity spring constant coupled in a linear chain by beads whose friction factor supplies the viscous resistance). This theory as extended by Fox and co-workers (28,153) gives... 

The cantilevers can be modelled as elastic springs (Fig. 4), generally characterised by spring constants calculated from the resonant frequency of the spring as follows ... 

It is convenient to use a simple weightless Hookean, or ideal, elastic spring with a modulus G and a simple Newtonian (fluid) dashpot or shock absorber having a liquid with a viscosity of 17 as models to demonstrate the deformation of an elastic solid and an ideal liquid, respectively. The stress-strain curves for these models are shown in Figure 14.1. 

Models are used to describe the behavior of materials. The fluid or liquid part of the behavior is described in terms of a Newtonian dashpot or shock absorber, while the elastic or solid part of the behavior is described in terms of a Hookean or ideal elastic spring. The Hookean spring represents bond flexing, while the Newtonian dashpot represents chain and local segmental movement. 

It is interesting to examine the bead-spring models to see what flow-induced configurational changes would be required in order to develop N2 values of the proper magnitude and sign. In the Rouse model, the components of the stress tensor are related directly to averages of the internal coordinates of the beads. For the simplest case of the elastic dumbbell ... 

Figure H3.3.4 Mechanical models are often used to model the response of foods in creep or stress relaxation experiments. The models are combinations of elastic (spring) and viscous (dashpot) elements. The stiffness of each spring is represent by its compliance (J= strain/stress), and the viscosity of each dashpot is represent by a Newtonian viscosity (ri). The form of the arrangement is often named after the person who originally proposed the model. The model shown is called a Burgers model. Each element in the middle—i.e., a spring and dashpot arranged in parallel—is called a Kelvin-Voigt unit.

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. The model can be represented by the following equation ... 

The Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the stress relaxation behaviors of polymers. 

Single-molecule theories originated in early polymer physics work (45) to describe the flow behavior of very dilute polymer solutions, which are free of interpolymer chain effects. Most commonly, the macromolecular chain, capable of viscoelastic response, is represented by the well-known bead-spring model or cartoon, shown in Fig. 3.8(a), which consists of a series of small spheres connected to elastic springs. 

Figure 1.41. Potential energies for the bead-spring model LJ1—Lennard-Jones potential LJ2—van der Waals potential EXP1, EXP2—short-range polar potential FENE—finitely extensible nonlinear elastic potential.

The first model which took into account the swelling properties of the polymeric resins was Gregor s model, which is based on the concept of inter-phase distribution [125] and considers the matrix of the resin as a network of elastic springs [4], When the resin swells, the network is expanded and exerts pressure on the internal pore (see Figure 7.23) thereafter, the swelling pressure developed in the resin influences the ion-exchange equilibrium. 

In order to incorporate both tendencies, Lazare, Sundheim, and Gregor developed an improved model, where the elastic and electrostatic interactions were included. In this model, the resin was regarded as a set of charged planar capacitors, with the plates interconnected by elastic springs (Figure 7.24) [126], The balance between forces is attained when the elastic forces provided by the polymeric resin stabilize the dissolution propensity. 

The simplest of these approaches includes Gaussian Network Models (GNM) or Elastic Network Models (ENM) which assume that the native state represents the minimum energy configuration. A structure is represented as a network of beads connected by harmonic springs.12,13 One bead represents one residue and is usually centered on the position of the Ca carbon. Single parameter harmonic interactions are assigned to bead pairs which fall within a certain cutoff distance Rc. In case of proteins, Rc is usually around 8-10 A. The representation of the molecule in the... 

Understanding of the mechanism of creep failure of polymeric fibres is required for the prediction of lifetimes in technical applications (Northolt et al., 2005). For describing the viscoelastic properties of a polymer fibre use is made of a rheological model as depicted in Fig. 13.103. It consists of a series arrangement of an "elastic" spring representing the chain modulus ech and a "shear" spring, yd with viscoelastic and plastic properties... 

In Fig. 3-5a, the polymer coil is modeled as a series of beads equally spaced along the polymer backbone and connected to each other by springs. The beads account for the viscous forces and the springs the elastic forces in the molecule the portion of the chain represented by a single spring is called a submolecule. The bead-spring model is... 

This equation describes Hookean elasticity, and Po = G (G is the modulus of rigidity). In Fig. 9, the classical mechanical spring model representing Eq. (14) is illustrated. If, however, it is assumed that jSi is the only nonzero constant in Eq. (13), then ... 

Such circuits are constructed on the basis of three elementary units a spring, a dashpot and a slider, which are sketched in fig. 3.50. Following computer language, we call these pictures icons. Icon (a) mimics a purely elastic spring, icon (b) the purely viscous movement of a piston in a viscous liquid. The slider (c) represents a system with a yield stress, i.e. where a minimum force is required to achieve flow. Here, we shall only consider icons (a) and (b). In mechanical models we construct circuits consisting of a number of springs and a number of dashpots, arranged in such a way that the experimental observations are optimally accounted for. The two simplest circuits are sketched in fig. 3.51a and b. 

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