Linear stress-strain behavior

Young s Modulus. Young s moduli, E, for several resins are plotted vs. temperature in Fig. 7. Young s moduli were determined from stress-strain diagrams. At 4K, their values are within 10%. Therefore, the low-temperature values of E do not depend markedly on the detailed chemical structure. It must be emphasized that epoxy resins are energy-elastic and have a nearly linear stress-strain behavior to fracture at low temperatures. No rate dependence was found over several decades. This is not true for many high polymers, such as polyethylene (PE), which are not cross-linked. PE behaves viscoelastically, even at 4 K [%... 

Figure 8.69 For a fully bridged crack, the matrix can undergo multiple cracking and the final failure can involve fiber pull-out. These eflects give rise to non-linear stress-strain behavior even though both components are brittle. (

Clearly, equation (12.22) can be valid only for systems exhibiting linear stress-strain behavior. In addition, impact strength in real polymers is not equal to the area under a low-strain-rate stress-strain curve. 

In any case, it should be noted that the predictions of simple models such as discussed above assume linear stress-strain behavior. In practice, the presence of a filler may not decrease ductility, as mentioned above (Sahu and Broutman, 1972 Wambach et ai, 1968), but may increase it... 

A is the Proportional Limit which is the end of the region in which the resin exhibits linear stress—strain behavior... 

Soil has highly non-linear stress-strain behavior and consequently soil stiffness is dependent on its stress state, as shown in Fig. 23.1. It can be seen that at low stress in a small scale model, a soil is significantly softer than would be the case for the same soil at higher stress in the prototype. It is therefore important when modelling soil to replicate the stress level of the prototype in the model. Without doing so, the soil stiffness would not be correct and hence test results would have no quantifiable relation to the prototype scenario. There are two possible methods available to ensure the stress state in the soil model is correct. The first method is to carry out tests at nearly full prototype scale proportions, hence accurately replicating the stress state in the soil. However this method is both time consuming and expensive. 

Non-Linear Stress-Strain Behavior and True Flexural Strength... 

Based on the load-strain and load-deflection measurements, PSZT exhibits non-linear stress-strain behavior. A plot of linear-elastically computed stress (or engineering stress) versus strain for poled-depoled specimens tested at room temperature, 75, 86, 105 and 120°C is shown in Fig. 3. Deviations from linear-elastic behavior initiate at a nominal stress level of approximately 20 30 MPa for specimens tested at room temperature and 10 20 MPa for specimens tested at an elevated temperature. Furthermore, the extent of non-linearity increases as the testing temperature increases. Conversion of the load-strain data to the true stress-strain behavior was achieved by implementing the approach first described by Nadai and adapted by Chen et al. ° The true compressive (o-c) and tensile stresses (at) were calculated as follows ... 

Aiming at linear stress-strain behavior, consider Pso/ where E denotes a characteristic modulus. To retain linearity assume that all retardation times do not depend on 0-. Hence, in (6.15) and (6.18), Xy = (m,T,V ) and Xp = (m,T,V ),p = I,..., N. Furthermore, to establish linear stress-strain relations, consider the expansion in (6.10) truncated after three terms, which, resorting to the diagonalized form, reads... 

The proposed constitutive law modeling the concrete in monotonic compression for the cases of confined and unconfined concrete is the Kent-Park model as described in Park and Paulay (1975). As it is shown in Fig. 4, the material follows a parabolic stress-strain curve up to a maximum stress equal to the cylinder s strength, after which it decays linearly with strain until the residual strength is reached. In tension, the model assumes a linear stress—strain behavior up until the tensile limit of the material is reached, and then the stiffness and strength decays with increasing strain (Fig. 5). 

Table 15.4 shows the work recoveiy of some fibers under different strains and in different humidities. Inorganic fibers, such as glass, have very high work recovery at a low strain of 1%. However, they break at moderate and high strains, and no work recoveiy can be measured. The work recovery of polymer fibers is lower than their elastic (strain) recoveiy due to the non-linear stress-strain behavior. In addition, the work recovery also is affected by both strain and humidify. 

All of the above models represent linear viscoelastic behavior in that they are all assumed to be made up of combinations of linear springs and linear dashpots. In general, all such linear stress (strain) behavior can be described by a linear differential equation of the type... 

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