Stress-Strain Behavior - Linearity

Note that the conditions above can be deduced from the requirement that the creep compliance is only a function of time (D(t)), and not a function of stress level (D(t,a)), for a linear material  

Relaxation tests may be used in the same manner to determine linearity. The above discussion focuses on stress linearity. For viscoelastic materials there is another important linearity which will be discussed at length in Chapter 6, that of translational linearity with time. 

Before discussing mechanical models or other mathematical representation of viscoelastic behavior, it is very important to note that the preceding section deals only with observed behavior or the experimental response of polymers under laboratory conditions. That is, the viscoelastic properties are defined from observations of real behavior and need not be defined by a particular mathematical model. Mathematical models are developed for the simple purpose of understanding and describing observed behavior. Also, as will be evident later, other loading modes such as constant strain rate and steady state oscillation, etc. can be used to determine viscoelastic properties. 

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The properties of the lamina constituents, the fibers and the matrix, have been only briefly discussed so far. Their stress-strain behavior is typified as one of the four classes depicted in Figure 1-8. Fibers generally exhibit linear elastic behavior, although reinforcing steel bars in concrete are more nearly elastic-pertectly plastic. Aluminum, as well as... 

Several experiments will now be described from which the foregoing basic stiffness and strength information can be obtained. For many, but not all, composite materials, the stress-strain behavior is linear from zero load to the ultimate or fracture load. Such linear behavior is typical for glass-epoxy composite materials and is quite reasonable for boron-epoxy and graphite-epoxy composite materials except for the shear behavior that is very nonlinear to fracture. 

Consequently, changing the temperature or the strain rate of a TP may have a considerable effect on its observed stress-strain behavior. At lower temperatures or higher strain rates, the stress-strain curve of a TP may exhibit a steeper initial slope and a higher yield stress. In the extreme, the stress-strain curve may show the minor deviation from initial linearity and the lower failure strain characteristic of a brittle material. 

Brittleness Brittle materials exhibit tensile stress-strain behavior different from that illustrated in Fig. 2-13. Specimens of such materials fracture without appreciable material yielding. Thus, the tensile stress-strain curves of brittle materials often show relatively little deviation from the initial linearity, relatively low strain at failure, and no point of zero slope. Different materials may exhibit significantly different tensile stress-strain behavior when exposed to different factors such as the same temperature and strain rate or at different temperatures. Tensile stress-strain data obtained per ASTM for several plastics at room temperature are shown in Table 2-3. 

Fig. 6.2 Room temperature stress-strain behavior of a woven 0°/90° Q/SiC composite. Because of processing-related matrix cracking and progressive fracture near the crossover points of fiber bundles, Stage II behavior (non-linear stress-strain response) is observed from the onset of loading. Above a strain of approximately 0.5% the composite exhibits Stage III (linear) behavior.

Figure 10.2. Stress-strain behavior. With elastic (reversible) deformation, stress and strain are linearly proportional in most materials (exceptions include polymers and concrete). With plastic (permanent) deformation, the stress-strain relationship is nonlinear.

Thermosets are polymeric materials which when heated form permanent network structures via the formation of intermolecular crosslinks. Whether the final product has a glass transition temperature, Tg, above or below room temperature, and therefore normally exists as an elastomer or a glass, it is, strictly speaking, a thermo-set. In practice, however, thermosets are identified as highly crosslinked polymers that are glassy and brittle at room temperature. These materials typically exhibit high moduli, near linear elastic stress-strain behavior, and poor resistance to fracture. 

The fact that thermosets are typically brittle and generally exhibit linear elastic stress-strain behavior suggests that linear elastic fracture mechanics (LEFM) and test methods may be applicable. In fact, these approaches have proven very popular, as is evidenced by the successful use of a number of LEFM-based fracture... 

If a material exhibits linear-elastic stress-strain behavior prior to rupture (an ideal behavior approximated by many thermosets), then a simple relationship exists between the material s fracture toughness and its fracture surface energy, J (or G),... 

The nano modified PDMS systems discussed have properties that enable a more confident prediction of ageing trends. The properties of the carborane modified system change in a linear fashion with radiation dose as opposed to the nonlinear trend observed for conventional particulate filled PDMS. The silica and carbon nano tubular systems display simplified mechanical properties. The reduction in Mullins effect or move to a more linear, less complex stress strain behavior, allows increased accuracy in property measurements. 

The influence of temperature on the stress-strain behavior of polymers is generally opposite to that of straining rates. This is not surprising in view of the correspondence of time and temperature in the linear viscoelastic region (Section I l.5.2.iii). The curves in Fig. 11-23 are representative of the behavior of a partially crystalline plastic. 

Polymers which yield extensively under stress exhibit nonlinear stress-strain behavior. This invalidates the application of linear elastic fracture mechanics. It is usually assumed that the LEFM approach can be used if the size of the plastic zone is small compared to the dimensions of the object. Alternative concepts have been proposed for rating the fracture resistance of tougher polymers, like polyolelins, but empirical pendulum impact or dart drop tests are deeply entrenched forjudging such behavior. 

Clarke and co-workers studied the effect of chain configurational properties on the stress—strain behavior of glassy linear polymers. They examined the relationship between chain structure and strain hardening by employing controlled stress molecular dynamics on a polyethylenelike chain. Variation of the sample preparation history produces chemically identical materials with vastly different responses to applied stress. 

Effects of Chain Configurational Properties on the Stress-Strain Behavior of Glassy Linear Polymers. 

A re-examination of the data in Fig. 3.13 reveals a slight s-shape, indicating a behavior of PMMA-crazes similar to those in PC . A complete unloadingreloading loop of a PMMA-craze, as shown in Fig. 3.14 exhibits a rubber-like stress-strain behavior, taking into account the proportionalities between 2v and e as well as between K, and for the stable craze investigated. This observation differs somewhat from other results on the same material showing a linear... 

The load-displacement curves for C(T) tests of the neat EpoxyH were almost linear until the final unstable fracture. The fracture toughness value in 77K-LNj was 210 J/m and that in RT-air was 120 J/m. Thus the toughness increased by 1.8 times by changing the test environment from RT-air to 77K-LN. Brown and co-workers have found that amorphous polymers crazed in 77K-LNj, but not in a helium or vacuum at about 78K [20-22]. They have also reported that the stress-strain behavior of all polymers, amorphous and crystalline, is affected by at low temperatures [22]. Kneifel has reported that the fracture toughness of epoxy in 77K-LNj is higher than that in RT-air and 5K, and that the reason for this is the reduced notch effect by plastic deformation [23]. Then, the increase of the fracture toughness of the neat EpoxyH in this study is probably caused by the similar effect. 

Figure 5. Stress-strain behavior of the linear polyethylene in Figure 3. The yield stress increases with oxidation.

When compared to URPs, the analysis and design of RPs is simpler in some respects and more complicated in others. Simplifications are possible since the stress-strain behavior of RPs is frequently linear to failure and they are less time-dependent. For high performance applications, they have their first damage occurring at stresses just below their high ultimate strength properties. They are also much less temperature-dependent, particularly RTSs (reinforced TSs). 

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 5.16 Stress-strain behavior for standard linear solid subjected to a sinusoidal stress. The system is elastic with a high modulus at very high frequencies and a lower modulus at low frequencies. At intermediate frequencies, hysteresis develops and the loss passes through a maximum.

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. (

Ductile properties such as crack pattern and deformations prefiguring the nearing failure are important characteristics regarding the fracture behavior of structural concrete members. The tests demonstrated that in general TRC members have a distinctive ductile behavior although the stress-strain-behavior of the fabrics is linear-elastic until a brittle tensile failure. While the deformations under service loads (SLS) are rather small, the load-bearing behavior of the specimens is characterized by a distinctive stabilized crack pattern as well as high deformations in ultimate limit state (ULS) of L/30 - L/20. 

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