Constant stress applications

For constant stress applications, the isochronous stress-strain curve can be used with standard equations by choosing the appropriate effective... 

From Fig. 2.35 it may be seen that for the Maxwell model, the strain at any time, t, after the application of a constant stress, Cg, is given by... 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

Time characterizing the response of a viscoelastic material to the instantaneous application of a constant stress. 

Elastomers exhibit this behavior due to their unique, crosslinked structure (cf. Section 1.3.2.2). It has been found that as the temperatme of an elastomer increases, so does the elastic modulus. The elastic modulus is simply a measme of the resistance to the uncoiling of randomly oriented chains in an elastomer sample under stress. Application of a stress eventually tends to untangle the chains and align them in the direction of the stress, but an increase in temperatme will increase the thermal motion of the chains and make it harder to induce orientation. This leads to a higher elastic modulus. Under a constant force, some chain orientation will take place, but an increase in temperatme will stimulate a reversion to a randomly coiled conformation and the elastomer will contract. 

Some applications require the material to remain under constant stress for years, yet it is often not reasonable to conduct such extended time measurements. One approach which circumvents this employs time-temperature superposition. Measurements are obtained over a shorter time span at differing temperatures. A master curve of C as a function of a reduced time tl a where a is a shift factor, is generated, and this allows the results to be extended to longer times. The shift factor is obtained by employing the Williams, Landel, and Ferry (WLF) relationship... 

The application of force to a stationary or moving system can be described in static, kinematic, or dynamic terms that define the mechanical similarity of processing equipment and the solids or liquids within their confines. Static similarity relates the deformation under constant stress of one body... 

The application of force to a stationary or moving system can be described in static, kinematic, or dynamic terms that define the mechanical similarity of processing equipment and the solids or liquids within their confines. Static similarity relates the deformation under constant stress of one body or structure to that of another it exists when geometric similarity is maintained even as elastic or plastic deformation of stressed structural components occurs [53], In contrast, kinematic similarity encompasses the additional dimension of time, while dynamic similarity involves the forces (e.g., pressure, gravitational, centrifugal) that accelerate or retard moving masses in dynamic systems. The inclusion of tune as another dimension necessitates the consideration of corresponding times, t and t, for which the time scale ratio t, defined as t = t It, is a constant. 

Experimentally, the dynamic shear moduli are usually measured by applying sinusoidal oscillatory shear in constant stress or constant strain rheometers. This can be in parallel plate, cone-and-plate or concentric cylinder (Couette) geometries. An excellent monograph on rheology, including its application to polymers, is provided by Macosko (1994). 

Creep measurements involve the application of a constant stress (usually a shearing stress) to the sample and the measurement of the resulting sample deformation as a function of time. Figure 9.6 shows a typical creep and recovery curve. In stress-relaxation measurements, the sample is subjected to an instantaneous predetermined deformation and the decay of the stress within the sample as the structural segments flow into more relaxed positions is measured as a function of time. 

The elastic properties discussed so far relate to stresses applied at relatively low rates. When forces are applied at rapid rates, then dynamic moduli are obtained. The energy relationships and the orders of magnitude of the data are much different [570]. Because of the experimental difficulties, only little work at rapid rates has been carried out with cotton fiber compared to that done with testing at low rates of application of stress. In contrast, cotton also responds to zero rate of loading, i.e., the application of a constant stress. Under this condition the fiber exhibits creep that is measured by determining fiber elongation at various intervals of time after the load has been applied. Creep is time-dependent and may be reversible upon removal of the load. However, even a low load applied to a fiber for a long period of time will cause the fiber to break. 

The creep characteristic of plastic foams is important in structural applications. Creep, either short-term or long-term, is the change in dimensions caused by constant stress. The deformation of polystyrene foam under various static loads... 

Molecular dynamics simulations have been used in a variety of ways. They can be used to compute mechanical moduli by studying the response of a model of the bulk polymer to a constant stress or strain, and to study the diffusion of molecules in membranes and polymers.There are numerous biomolecular applications. Structural, dynamic, and thermodynamic data from molecular dynamics have provided insights into the structure-function relationships, binding affinities, mobility, and stability of proteins, nucleic acids, and other macromolecules that cannot be obtained from static models. 

For smaller particles, smaller stresses are exerted. Thus, in order to predict sedimentation it is necessary to measure the viscosity at very low stresses (or shear rates). These measurements can be carried out using a constant stress rheometer (Carrimed, Bohlin, Rheometrics, Haake or Physica). Usually, a good correlation is obtained between the rate of creaming or sedimentation, v, and the residual viscosity rj 0), as will be described in Chapter 21. Above a certain value of ri(0), v becomes equal to 0. Clearly, in order to minimize sedimentation it is necessary to increase rj 0) an acceptable level for the high shear viscosity must be achieved, depending on the application. In some cases, a high rj[0) may be accompanied by a high rj (which may not be acceptable for apphcation, for example if spontaneous dispersion on dilution is required). If this is the case, the formulation chemist should seek an alternative thickener. 

The inability to maintain a constant deformation following application of a constant stress. Such systems display a phenomenon referred to as creep, in which sample deformation continues as a function of time. This is illustrated in Figure 10.3 (16,19). 

There are, however, three other types of important and qualitatively different mechanical tests, representing deformation modes which occur often during the practical use of engineering materials. These tests, which will be discussed briefly below, can be used to anticipate the performance of a plastic part as a function of time under some deformation states which are encountered very often in technological applications namely, constant stress for a long time,... 

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