The Stress-Strain Diagram

Young s modulus of elasticity, or the tensile modulus, is the ratio of the stress applied to the strain within this linear region. It provides an indication of stiffness or how much a material or part will stretch under a given load. For example, a material that has a high tensile modulus is rigid and resists stretching. 

The tensile modulus can be determined from the slope of the linear portion of this stress-strain curve. If the relationship between stress and strain is linear to the yield point, where deformation continues without an increased load, the modulus of elasticity can be calculated by dividing the yield strength (pascals) by the elongation to yield  

When a linear relationship between the stress and strain is no longer present, the proportional limit is reached. On the diagram this is the highest point on the linear portion of the graph or where the curve no longer is a straight line. The material at this point is still elastic. The proportional limit is sometimes called the yield point. 

At some stress level above the proportional limit the material will no longer return to its original shape it will be permanently deformed. This region beyond the yield point is called the plastic range. 

For many polymers and metals the yield point is not clearly defined. In these cases the offset yield strength is used. To obtain this value, a line parallel to the linear portion of the curve is drawn such that 

---

As discussed in Section 2.0 (Exploration), the earth s crust is part of a dynamic system and movements within the crust are accommodated partly by rock deformation. Like any other material, rocks may react to stress with an elastic, ductile or brittle response, as described in the stress-strain diagram in Figure 5.5. 

Figure 5.5 The stress - strain diagram for a reservoir rock...

Both % El and % RA are frequendy used as a measure of workabifity. Workabifity information also is obtained from parameters such as strain hardening, yield strength, ultimate tensile strength, area under the stress—strain diagram, and strain-rate sensitivity. 

As an example, for room-temperature applications most metals can be considered to be truly elastic. When stresses beyond the yield point are permitted in the design, permanent deformation is considered to be a function only of applied load and can be determined directly from the stress-strain diagram. The behavior of most plastics is much more dependent on the time of application of the load, the past history of loading, the current and past temperature cycles, and the environmental conditions. Ignorance of these conditions has resulted in the appearance on the market of plastic products that were improperly designed. Fortunately, product performance has been greatly improved as the amount of technical information on the mechanical properties of plastics has increased in the past half century. More importantly, designers have become more familiar with the behavior of plastics rather than... 

The secant modulus measurement is used during the designing of a product in place of a modulus of elasticity for materials where the stress-strain diagram does not demonstrate a linear proportionality of stress to strain or E is difficult to locate. 

These stress-strain diagrams may be applied, for example, to the investigation of a rod of which has its total volume is glass fiber and half plastic. If the glass fibers are laid parallel to the axis of the rod, at any cross-section, half the total cross-sectional area is glass and half plastic. If the rod is stretched 0.5%, reference to the stress-strain diagrams... 

On the stress-strain diagram, what does the elastic range indicate The plastic range ... 

Figure 15.4 gives the stress-strain diagrams for a typical fiber, plastic, and elastomer and the average properties for each. The approximate relative area under the curve is fiber, 1 elastomers, 15 thermoplastics, 150. Coatings and adhesives, the two other types of end-uses for polymers, will vary considerably in their tensile properties, but many have moduli generally between elastomers and plastics. They must have some elongation and are usually of low crystallinity. 

Finally, the modulus of elasticity E (Young s modulus), which is a measure of the stiffness of the polymer, can be calculated from the stress-strain diagram. According to Hooke s law there is a linear relation between the stress o and the strain e ... 

We turn our attention now to some practical aspects of mechanical property determinations. The important quantities such as modulus, strength, and ductility are typically summarized in graphical form on a stress-strain diagram. The details of how the experiment is performed and how the stress-strain diagram is generated are described for some common types of applied forces below. 

Look up the stress-strain diagram for naval brass and determine the CW% where a stress of 400 MPa is applied. Cite the source of your information. 

Fig. 23a. Stress-strain-diagram of a Polyethylene (Vestolen A 6042) film (stretching velocity 0,26 mm/s) b) Experimental (row I), synthesized (row II), and resolved (row III) bands of the CHj-rocking bands. The experimental spectra were scanned at the indicated positions (circled numbers) of the stress-strain-diagram (a).

Tensile properties (tensile strength, elongation, modulus) were measured on an Instron tensile tester (ASTM D882-61T Method A). The tensile modulus was the slope of the initial straight portion of the stress-strain diagram. The heat-distortion... 

Figure 1.9. Deformation of typical elastoplastic materials (hardened metal and polymer) in the stress-strain diagram (after Guy, 1976).

For practical applications empirically determined creep data are being used, such as D(t) or, more often, E(t) curves at various levels of stress and temperature. The most often used way of representing creep data is, however, the bundle of creep isochrones, derived from actual creep curves by intersecting them with lines of constant (log) time (see Figure 7.7). These cr-e-curves should be carefully distinguished from the stress-strain diagram discussed before, as generated in a simple tensile test ... 

The strength properties of solids are most simply illustrated by the stress-strain diagram, which describes the behaviour of homogeneous brittle and ductile specimens of uniform cross section subjected to uniaxial tension (see Fig. 13.60). Within the linear region the strain is proportional to the stress and the deformation is reversible. If the material fails and ruptures at a certain tension and a certain small elongation it is called brittle. If permanent or plastic deformation sets in after elastic deformation at some critical stress, the material is called ductile. 

A very important diagram for fibres and yams is the stress—strain diagram, where the specific stress is plotted as a function of the elongation (extensional strain) in %. The curve starts at an elongation of zero and ends in the breaking point at the ultimate specific stress (=tensile strength or tenacity) and the ultimate elongation (=strain at break). 

FIG. 13.88 Diagram of the specific tenacity (ffb/p) versus the initial specific modulus (Ea/p) for conventional man made fibres. 0 is the limiting tangential slope in the stress-strain diagram for strain tending to zero. The diagonal lines show the indicated ffbr/ 0-ratio this varies from = 1 for elastomeric filaments and 0.2 for tyre yarns (ty) to 0.03 for yarns such as polyacrylonitrile. 

Fig. 13.94 shows that if the stress-strain diagram of the undrawn yam is determined, the stress-strain diagrams of all the drawn yams, obtained at definite draw ratios, can be estimated. 

PTFE requires a sealing compression which is noticeably above the stress strain diagram (Figure 3b). A reliable sealing is only possible by using special labyrinth geometries matching the PTFE features. 

In rigid polyethylene foam (y = 32 kg/m, cell diameter between 0.5 and 1.5 mm) the anisotropy of the macrostructure is particularly reflected by the shape of the stress-strain diagram (Fig. 14). When a load is applied normally to the foaming direction, the deformation of the material increases perceptibly. In contrast, the resistance of the structure to compressive stress applied in the direction parallel to foaming increases since unit surface area of the material contains more rigid GSE struts in the latter case than in the former. 

The stress-strain diagram gives considerably more information about the product tested than the single-value result, which is all that can be obtained from the present unit ... 

Another example of the use of polarized radiation in imaging studies is the analysis of poly(vinylidene fluoride)(PVDF) films, which have been uniaxially elongated at different temperatures. Depending on the thermal, mechanical and electrical pretreatment, PVDF can exist in different modifications [59]. The crystal structure of the cmmpled 11(a) modification can be converted into the aU-tra s 1(P) form by tensile stress below 140°C (see Figure 9.27a). Figure 9.27b shows the stress-strain diagrams of PVDF films in the 11(a) form which have been elongated to 400 % strain at 100 and 150°C. The observed decrease in stress upon elevation of the... 

As shown above, the ZrO/Ni composites examined by disk-bend testing are found to deform in a nonlinear manner, so that composition-dependent fracture strengths cannot be obtained directly from the stress-strain diagram in Fig. 3. Under the circumstances, we now make a micromechanical analysis to estimate actual stresses to be developed by plastic deformation of the ductile constituent on the basis of an established "mean-field" model [12]. In the following, the macrostress a) is related to the microstresses and (o) such... 

The basic deformation modes are represented by regions in the stress-strain diagram and exhibit very different response slopes. 

The appearance of a permanent set is said to mark a yield point, which indicates the upper limit of usefiilness for any material. Unlike some metals, in particular, the ferrous alloys, the drop-of-beam effect and a sharp knee in the stress-strain diagram are not exhibited by plastics. An arbitrary yield point is usually assigned to them. Typical of these arbitrary values is the 0.2% or the 1% offset yield stress (Figure 3.3a). 

---