Viscoelasticity mechanical models

We have previously introduced viscoelastic mechanical models, but we may also use electronic models as well. The mechanical spring can be replaced by a capacitor, which stores electronic energy, according to Coulomb s law ... 

The over-stress idea can be incorporated into the viscoelastic mechanical models by means of adding sliding elements to describe yield or termination points. With such an application, equations containing viscosity terms are obtained in a form similar to (24.114). For example, Brinson et al. (1975), Renieri et al. (1976), Sancaktar and Brinson (1979, 1980), Sancaktar (1981), Sancaktar and Padgilwar (1982), Sancaktar et al. (1984), and Sancaktar and Schenck (1985) used the modified Bingham model developed by Brinson (1974) to describe the shear and tensile material behavior of structural adhesives in the bulk and bonded forms ... 

Based on the theoretical analysis about the morphology and the etfeet of crystallinity on the development of residual stresses for injection molded crystalline polymers, the residual stresses have been simulated by means of a new four elements viscoelastic mechanical model. Considering the crystalline orientation of injection molded parts, we view the parts as orthotropic solids to simulate the development of residual stresses in both longitudinal and transverse directions of the parts. 

Our objectives in this section are twofold to describe and analyze a mechanical model for a viscoelastic material, and to describe and interpret an experimental procedure used to study polymer samples. We shall begin with the model and then proceed to relate the two. Pay attention to the difference between the model and the actual observed behavior. 

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). 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

The basic viscoelastic theory assumes a timewise linear relationship between stress and strain. Based on this assumption and using mechanical models thought to represent the behavior of a plastic material, it can be shown that the stress, at any time t, in a plastic held at a constant strain (relaxation test), is given by ... 

In consideration of the clinical importance of being able to visualize any implant material within the body, radio-opaque hydrogels for NPR have been formulated [69]. Copolymers of iodobenzoyl-oxo-ethyl methacrylate (4IEMA) and hydrophilic PVP or hydroxylethyl methacrylate (HEMA) exhibit appropriate swelling characteristics, viscoelastic mechanical properties, and excellent cytocompatibility [69]. Moreover, inclusion of the covalently attached iodine molecules allowed for hydrogel visualization via X-ray in a porcine cadaveric spine model [69],... 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

Real polymers are more complex than these simple mechanical models. Qualitatively, when a real polymer is forced to flow through a contraction or expansion in an extrusion screw, it will exhibit viscoelastic behaviour. The polymer molecules will be elongated if forced through a contraction, or they will retract when they flow into an expansion. The effect of viscoelastic behavior in a capillary rheometer is observed in the form of recirculation flow just before the polymer enters the... 

The second assumption is based on contact mechanics models in which viscoelastic effects that might influence the instability point (pull-off) and adhesion are negligible or can be allowed for. The third assumption is based on representing the cantilever with a point mass model. Simulations using a distributed mass model indicate that ultrasonic vibration of the cantilever is relatively small and in many cases less than 0.05 of the UFM normal deflection (Hirsekorn et al. 1997). 

Figure 8.1. Diagram showing Maxwell mechanical model of viscoelastic behavior of connective tissues. In this model an elastic element (spring) with a stiffness Em is in series with a viscous element (dashpot) with viscosity T m. This model is used to represent time dependent relaxation of stress in a specimen bold of fixed length.

A plastic material is defined as one that does not undergo a permanent deformation until a certain yield stress has been exceeded. A perfectly plastic body showing no elasticity would have the stress-strain behavior depicted in Figure 8-15. Under influence of a small stress, no deformation occurs when the stress is increased, the material will suddenly start to flow at applied stress a(t (the yield stress). The material will then continue to flow at the same stress until this is removed the material retains its total deformation. In reality, few bodies are perfectly plastic rather, they are plasto-elastic or plasto-viscoelastic. The mechanical model used to represent a plastic body, also called a St. Venant body, is a friction element. The... 

A more common body is the plasto-viscoelastic, or Bingham body. Its mechanical model is shown in Figure 8-16C. When a stress is applied that is below the yield stress, the Bingham body reacts as an elastic body. At stress values beyond the yield stress, there are two components, one of which is constant and is represented by the friction ele-... 

Figure 8-16 Mechanical Models for a Plastic Body. (A) St. Venant body, (B) plasto-elastic body, and (C) plasto-viscoelastic or Bingham body.

Stoner et al. (1974) have proposed a mechanical model for postmortem striated muscle it is shown in Figure 8-28. The model is a combination of the Voigt model with a four-element viscoelastic model. The former includes a contractile element (CE), which is the force generator. The element SE is a spring that is passively elongated by the shortening of the CE and thus develops an... 

Figure 8-33 Mechanical Model Proposed by Morrow and Mohsenin. (A) Viscoelastic foods, (B) the strain time, (C) stress time, characteristics of the system.

Figure 12.10. Simple mechanical models representing the structure of a viscoelastic material (top), their response with time (center) and the expression that relates structure to behavior (bottom). To = jj/E is a relaxation time.

Fig. 35a-c. Mechanical models of creep and recovery, (d) Modified model for viscoelastic creep... 

Fig. 11-16. Simple mechanical models of viscoelastic behavior, (a) Voigt or Kelvin element and (b) Maxwell element.

Mechanical models are useful tools for selecting appropriate mathematical functions to describe particular phenomena. The models have no physical relation to real materials, and it should be realized that an inhnite number of dilTerenl models can be used to represent a given phenomenon. Two models are mentioned here to introduce the reader to such concepts, which are widely used in studies of viscoelastic behavior. 

Limitations to the effectiveness of mechanical models occur because actual polymers are characterized by many relaxation times instead of single values and because use of the models mentioned assumes linear viscoelastic behavior which is observed only at small levels of stress and strain. The linear elements are nevertheless useful in constructing appropriate mathematical expressions for viscoelastic behavior and for understanding such phenomena. 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

Time constants are related to the relaxation times and can be found in equations based on mechanical models (phenomenological approaches), in constitutive equations (empirical or semiempirical) for viscoelastic fluids that are based on either molecular theories or continuum mechanics. Equations based on mechanical models are covered in later sections, particularly in the treatment of creep-compliance studies while the Bird-Leider relationship is an example of an empirical relationship for viscoelastic fluids. 

The mechanical response of viscoelastic materials to mechanical excitation has traditionally been modeled in terms of elastic and viscous components such as springs and dashpots (1-3). The corresponding theory is analogous to the electric circuit theory, which is extensively described in engineering textbooks. In many respects the use of mechanical models plays a didactic role in interpreting the viscoelasticity of materials in the simplest cases. However, it must be emphasized that the representation of the viscoelastic behavior in terms of springs and dashpots does not imply that these elements reflect the molecular mechanisms causing the actual relaxation... 

Mechanical models that represent the evolution of the viscoelastic functions in the time and frequency domain are described in Chapter 10. Chapter 11 discusses experimental scaling laws for viscosity, equiUbrium recovery... 

In practice, viscoelastic properties can be determined by static and dynamic tests. The typical static test procedure is the creep test. Here, a constant shear stress is applied to the sample over a defined length of time and then removed. The shear strain is monitored as a function of time. The level of stress employed should be high enough to cause sample deformation, but should not result in the destruction of any internal structure present. A typical creep curve is illustrated in Fig. 13A together with the four-element mechanical model that can be used to explain the observations. The creep compliance represents the ratio between shear strain rate and constant stress at any time t. 

Roscoe, R. Mechanical models for the representation of viscoelastic properties. Br. J. Appl. Phys. 1950, 1,... 

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