Modulus time dependent

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

Expanding Waves. As a further application we tmn to the expanding potential problem [261-263], where we shall work from the amplitude modulus to the phase. The time-dependent potential is of the form... 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

The situation is not so simple when these various parameters are time dependent. In the latter case, the moduli, designated by E(t)and G(t), are evaluated by examining the (time dependent) value of o needed to maintain a constant strain 7o- By constrast, the time-dependent compliances D(t) and J(t)are determined by measuring the time-dependent strain associated with a constant stress Oq. Thus whether the deformation mode is tension or shear, the modulus is a measure of the stress required to produce a unit strain. Likewise, the compliance is a measure of the strain associated with a unit stress. As required by these definitions, the units of compliance are the reciprocals of the units of the moduli m in the SI system. 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

Figure 3.8 Time-dependent shear modulus [as G(t)/Gol versus time (as t/r) (a) linear coordinates and (b) log-log coordinates.

A fully automated microscale indentor known as the Nano Indentor is available from Nano Instmments (257—259). Used with the Berkovich diamond indentor, this system has load and displacement resolutions of 0.3 N and 0.16 nm, respectively. Multiple indentations can be made on one specimen with spatial accuracy of better than 200 nm using a computer controlled sample manipulation table. This allows spatial mapping of mechanical properties. Hardness and elastic modulus are typically measured (259,260) but time-dependent phenomena such as creep and adhesive strength can also be monitored. 

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

For small shear strains we can define a time-dependent compliance (reciprocal modulus) by the equation... 

It has been also shown that when a thin polymer film is directly coated onto a substrate with a low modulus ( < 10 MPa), if the contact radius to layer thickness ratio is large (afh> 20), the surface layer will make a negligible contribution to the stiffness of the system and the layered solid system acts as a homogeneous half-space of substrate material while the surface and interfacial properties are governed by those of the layer [32,33]. The extension of the JKR theory to such layered bodies has two important implications. Firstly, hard and opaque materials can be coated on soft and clear substrates which deform more readily by small surface forces. Secondly, viscoelastic materials can be coated on soft elastic substrates, thereby reducing their time-dependent effects. 

Another manifestation of a time dependence to particle adhesion involves the phenomenon of total engulfment of the particle by the substrate. It is recognized that both the JKR and MP theories of adhesion assume that the contact radius a is small compared to the particle radius R. Realistically, however, that may not be the case. Rather, the contact radius depends on the work of adhesion between the two materials, as well as their mechanical properties such as the Young s modulus E or yield strength Y. Accordingly, there is no fundamental reason why the contact radius cannot be the same size as the particle radius. For the sake of the present discussion, let us ignore some mathematical complexities and simply assume that both the JKR and MP theories can be simply expanded to include large contact radii. Let us further assume that, under conditions of no externally applied load, the contact and particle radii are equal, that is a(0) = R. Under these conditions, Eq. 29 reduces to... 

The only unknown on the right hand side is a value for modulus E. For the plastic this is time-dependent but a suitable value may be obtained by reference to the creep curves in Fig. 2.5. A section across these curves at the service life of 1 year gives the isochronous graph shown in Fig. 2.13. The maximum strain is recommended as 1.5% so a secant modulus may be taken at this value and is found to be 347 MN/m. This is then used in the above equation. 

A plastic is stressed at a constant rate up to 30 MN/m in 60 seconds and the stress then decreases to zero at a linear rate in a further 30 seconds. If the time dependent creep modulus for the plashc can be expressed in the form... 

A plastic with a time dependent creep modulus as in the previous example is stressed at a linear rate to 40 MN/m in 100 seconds. At this time the stress in reduced to 30 MN/m and kept constant at this level. If the elastic and viscous components of the modulus are 3.5 GN/m and 50 x 10 Ns/m, use Boltzmann s Superposition Principle to calculate the strain after (a) 60 seconds and (b) 130 seconds. 

In this method appropriate values of such time-dependent properties as the modulus are selected and substituted into the standard equations. It has been found that this approach is sufficiently accurate if the value chosen for the modulus takes into account the projected service life of the product and/or the limiting strain of the plastic, assuming that the limiting strain for the material is known. Unfortunately, this is not just a straightforward value applicable to all plastics or even to one plastic in all its applications. This type of evaluation takes into consideration the value to use as a safety factor. If no history exist a high value will be required. In time with service condition inputs, the SF can be reduced if justified. 

A time dependent modulus is then calculated using the extreme fiber stress level for each of the materials at the initial stress value level using the loading-time curve developed. If the deflection at the desired life is excessive, the section is increased in size and the deflection recalculated. By iteration the second can be made such that the creep and load deflection is equal to the maximum allowed at the design life of the chair. This calculation can be programmed for a computer solution. 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

Dynamic oscillatory shear measurements of polymeric materials are generally performed by applying a time dependent strain of y(t) = y0sin(cot) and the resultant shear stress is a(t) = y0[G sin(a)t) + G"cos(cot)], with G and G" being the storage and loss modulus, respectively. 

The modulus-time or modulus-frequency relationship (or, graphically, the corresponding curve) at a fixed Temperature is basic to an understanding of the mechanical properties of polymers. Either can be converted directly to the other. By combining one.of these relations (curves) with a second major response curve or description which gives the temperature dependence of these time-dependent curves, one can cither predict much of the response of a given polymer under widely varying conditions or make rather... 

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