Stress-strain curves viscous

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

When a plastic material is subjected to an external force, a part of the work done is elastically stored and the rest is irreversibly (or viscously) dissipated hence a viscoelastic material exists. The relative magnitudes of such elastic and viscous responses depend, among other things, on how fast the body is being deformed. It can be seen via tensile stress-strain curves that the faster the material is deformed, the greater will be the stress developed since less of the work done can be dissipated in the shorter time. 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

Although PBT fiber also has a plateau region in the stress-strain curve [4], the crystalline chains do not respond to external strain in the first few percent of deformation. They increased in length only when the strain is above 4% (see Figure 11.13). Therefore, initial macroscopic deformation involved viscous flow of the amorphous phase. Furthermore, PBT undergoes strain-induced crystal transformation at moderately low strains of 15-20% [75], The differences in their microscopic crystalline chain deformation explained why PTT has a better elastic recovery than PBT even though both have contracted chains and knees in their stress-strain curves [4, 69],... 

Figure 7.3. Determination of elastic and viscous components. Incremental stress-strain curve constructed by stretching a specimen in strain increments of 2 to 5% and allowing the specimen to relax to an equilibrium stress before an additional strain increment is added. The elastic fraction is defined as the equilibrium stress divided by the initial stress. (Adapted from Silver, 1987.)...

A strain with increment of 2% is then applied axially to the tendon and the motion of the crosshead is stopped at that point. The initial (total) stress is then recorded as well as the stress after a period of time when it (the stress) no longer decreases with increasing time. This final stress value is the time-independent stress termed the elastic stress. The stress lost to viscous slippage (viscous stress) is the difference between the initial stress and the final stress. The elastic stress is just the stress at equilibrium and is plotted versus strain to get an elastic stress-strain curve. The elastic stress, as pointed out in Chapter 6, is the stress stored at the molecular level as a change in conformation of a helical or extended macromolecule. In theory, the slope of the elastic stress-strain curve is proportional to the molecular stiffness of the molecule being stretched. 

Elastic and viscous stress-strain curves can be experimentally determined from incremental stress-strain curves measured on samples of different tendons. Typical elastic and viscous stress-strain curves for rat tail and turkey tendons are shown in Figures 7.4 and 7.5. For both types of tendons the curves at high strains are approximately linear. As we discuss in Chapter 8, the elastic modulus can be calculated for collagen, because most of the tendon is composed of collagen and water, by dividing the elastic slope by the collagen content of tendon. When this is done the value of the elastic modulus of collagen in tendon is somewhere between 7 and 9 GPA. 

From the viscous stress-strain curve using Equations (4.1), (4.2), and (8.2) we can calculate the collagen fibril length. The collagen fibril lengths in tendon range from about 20 pm for during tendon development to in excess... 

Figure 7.4. Total, elastic, and viscous stress-strain curves for collagen fibers from rat tail tendon. The total stress-strain curve (open boxes) was obtained by collecting all the initial, instantaneous, force measurements at increasing time intervals and then dividing by the initial cross-sectional area. The elastic stress-strain curve (closed diamonds) was obtained by collecting all the force measurements at equilibrium and then dividing by the initial cross-sectional area. The viscous component curve (closed squares) was obtained as the difference between the total and the elastic stresses. Error bars represent one standard deviation of the mean.

Determination of the Elastic and Viscous Stress-Strain Curves for Model Collagen Fiber Systems... 

Figure 7.5. Representative total elastic and viscous stress-strain curves for unmineralized and mineralized avian tendons. The curves show the total, elastic, and viscous stress-strain relations for gastrocnemius tendon segments proximal to the bifurcation point, B, in Figure 3.29 for animals after (A) and prior to (B), the onset of mineralization. Note the different scales in A and B and the increased slope of the elastic stress-strain curve and decreased strain to failure for mineralized (A) compared to unmineralized (B) tendons.

Figure 7.6. Effective mechanical fibril length versus fibril segment length. Plot of effective fibril length in pm determined from viscous stress-strain curves for rat tail tendon and self-assembled collagen fibers versus fibril segment length. The correlation coefficient (R2) for the line shown is 0.944 (see Silver et al., 2003).

Figure 7.7. Total, elastic, and viscous stress-strain curves for uncrosslinked self-assembled type I collagen fibers.Total (open squares), elastic (filled diamonds), and viscous (filled squares) stress-strain curves for self-assembled uncrosslinked collagen fibers obtained from incremental stress-strain measurements at a strain rate of 10%/min. The fibers were tested immediately after manufacture and were not aged at room temperature. Error bars represent one standard deviation of the mean value for total and viscous stress components. Standard deviations for the elastic stress components are similar to those shown for the total stress but are omitted to present a clearer plot. The straight line for the elastic stress-strain curve closely overlaps the line for the viscous stress-strain curve. Note that the viscous stress-strain curve is above the elastic curve suggesting that viscous sliding is the predominant energy absorbing mechanism for uncrosslinked collagen fibers.

Elastic and viscous stress-strain curves have been measured for human aorta as well as other arteries. The curves are all similar in that the stress is much lower than that for skin (see Figure 7.11). The lower stress values are consistent with a different network structure of vessel wall compared to skin, which is reflected by the smooth muscle content of aortic tissue. Smooth muscle is absent from skin. The curves for different vessels have similar shapes, however, on careful review the curves have much lower values for the high strain moduli. This relates to the differences in the structure of the media from each of these vessels and potential crosslinking differences. 

Figure 7.12. Elastic and viscous stress-strain curves for normal articular cartilage. Elastic (top) and viscous (bottom) stress-strain curves were obtained by plotting the equilibrium (elastic) and the total-equilibrium (viscous) stresses for visibly normal cartilage.

Elastic and viscous stress-strain curves for unmineralized and mineralized turkey tendons are plotted in Figure 7.5. In general, the elastic stress-strain curves for tendons with low mineral content (0.029 weight fraction of mineral) are lower than those that are seen for mineralized tendons (mineral content about 0.3). 

From the stress value at a particular strain found on the viscous stress-strain curve (see the second term found in Equation (8.3)) we can calculate the viscosity from the stress if we know the strain rate. Because... 

Figure 8.5. Plots of elastic modulus versus mineral content (b) and days of mineralization (a) from incremental stress-strain tests performed on mineralized self-assembled type I collagen fibers. Slopes were obtained from the straight portions of the elastic and viscous stress-strain curves.

Figure 10.2. Mean elastic and viscous stress-strain curves for cartilage. Plot of elastic (A) and viscous (B) stress-strain curves for cartilage as a function of visual grade. The visual grade used was 1, shiny and smooth 2, slightly fibrillated 3, mildly fibrillated 4, fibrillated 5, very fibrillated and 6, fissured. The equation for the linear approximation for the stress-strain curve for each group is given, as well as the correlation coefficient. Note the decreased slope with increased severity of osteoarthritis. This data is consistent with down-regulation of mechanochemical transduction and tissue catabolism.

Figure 7.12. Stress-strain curves showing yield stress (

Butter, and other unctuous materials, may be qualitatively described by a modified Bingham body (Elliott and Ganz, 1971 Elliott and Green, 1972), which consists of viscous, plastic and elastic elements in series. The stress-strain behavior for the model proposed by Elliot and Ganz (1971) is shown in Figure 7.12B. Diener and Heldman (1968) proposed a more complex model to describe how butter behaves when a low level of strain is applied. The model consists of plastic and viscous elements in parallel, coupled in series with a viscous element in parallel with a combination of a viscous and an elastic element. Figure 7.12C shows the stress-strain curve for... 

The equations can be generalized for both shear and tension, and G can be replaced by E. The mechanical analogue for the Maxwell unit can be represented by a combination of a spring and a dashpot arranged in series so that the stress is the same on both elements. This means that the total strain is the sum of the strains on each element as expressed by Equation 13.19. A typical stress-strain curve predicted by the Maxwell model is shown in Figure 13.12(a). Under conditions of constant stress, a Maxwell body shows instantaneous elastic deformation first, followed by a viscous flow. 

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