The Strainstress Curves From Tensile Tests For A Ductile Metal E.g. Mild Steel
1.4. Sign convention for direct stress and strain
Tensile stresses and strains are considered POSITIVE in sense producing an increase in length. Compressive stresses and strains are considered NEGATIVE in sense producing a decrease in length.
1.5. Elastic materials โ Hooke's law
A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed. A particular form of elasticity which applies to a large range of engineering materials, at least over part of their load range, produces deformations which are proportional to the loads producing them. Since loads are proportional to the stresses they produce and deformations are proportional to the strains, this also implies that, whilst materials are elastic, stress is proportional to strain. Hooke's law, in its simplest form*, therefore states that stress (er) oc strain (e) stress i.e. โ = constant*
strain
It will be seen in later sections that this law is obeyed within certain limits by most ferrous alloys and it can even be assumed to apply to other engineering materials such as concrete, timber and nonferrous alloys with reasonable accuracy. Whilst a material is elastic the deformation produced by any load will be completely recovered when the load is removed; there is no permanent deformation.
Other classifications of materials with which the reader should be acquainted are as follows:
A material which has a uniform structure throughout without any flaws or discontinuities is termed a homogeneous material. Nonhomogeneous or inhomogeneous materials such as concrete and poorquality cast iron will thus have a structure which varies from point to point depending on its constituents and the presence of casting flaws or impurities.
If a material exhibits uniform properties throughout in all directions it is said to be isotropic; conversely one which does not exhibit this uniform behaviour is said to be nonisotropic or anisotropic.
An orthotropic material is one which has different properties in different planes. A typical example of such a material is wood, although some composites which contain systematically orientated "inhomogeneities" may also be considered to fall into this category.
1.6. Modulus of elasticity โ Young's modulus
Within the elastic limits of materials, i.e. within the limits in which Hooke's law applies, it has been shown that stress
This constant is given the symbol E and termed the modulus of elasticity or Young's modulus.
_ stress a
strain e
P bL PL
* Readers should be warned that in more complex stress cases this simple form of Hooke's law will not apply and misapplication could prove dangerous; see ยง14.1, page 361.
Young's modulus E is generally assumed to be the same in tension or compression and for most engineering materials has a high numerical value. Typically, E = 200 x 109 N/m2 for steel, so that it will be observed from (1.1) that strains are normally very small since
In most common engineering applications strains do not often exceed 0.003 or 0.3 % so that the assumption used later in the text that deformations are small in relation to original dimensions is generally well founded.
The actual value of Young's modulus for any material is normally determined by carrying out a standard tensile test on a specimen of the material as described below.
1.7. Tensile test
In order to compare the strengths of various materials it is necessary to carry out some standard form of test to establish their relative properties. One such test is the standard tensile test in which a circular bar of uniform crosssection is subjected to a gradually increasing tensile load until failure occurs. Measurements of the change in length of a selected gauge length of the bar are recorded throughout the loading operation by means of extensometers and a graph of load against extension or stress against strain is produced as shown in Fig. 1.3; this shows a typical result for a test on a mild (low carbon) steel bar; other materials will exhibit different graphs but of a similar general form see Figs 1.5 to 1.7.
Elastic
Elastic
For the first part of the test it will be observed that Hooke's law is obeyed, i.e. the material behaves elastically and stress is proportional to strain, giving the straightline graph indicated. Some point A is eventually reached, however, when the linear nature of the graph ceases and this point is termed the limit of proportionality.
For a short period beyond this point the material may still be elastic in the sense that deformations are completely recovered when load is removed (i.e. strain returns to zero) but
Hooke's law does not apply. The limiting point B for this condition is termed the elastic limit. For most practical purposes it can often be assumed that points A and B are coincident.
Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable. There will thus be some permanent deformation or permanent set when load is removed. After the points C, termed the upper yield point, and D, the lower yield point, relatively rapid increases in strain occur without correspondingly high increases in load or stress. The graph thus becomes much more shallow and covers a much greater portion of the strain axis than does the elastic range of the material. The capacity of a material to allow these large plastic deformations is a measure of the socalled ductility of the material, and this will be discussed in greater detail below.
For certain materials, for example, high carbon steels and nonferrous metals, it is not possible to detect any difference between the upper and lower yield points and in some cases no yield point exists at all. In such cases a proof stress is used to indicate the onset of plastic strain or as a comparison of the relative properties with another similar material. This involves a measure of the permanent deformation produced by a loading cycle; the 0.1 % proof stress, for example, is that stress which, when removed, produces a permanent strain or "set" of 0.1 % of the original gauge lengthsee Fig. 1.4(a).
Fig. 1.4. (a) Determination of 0.1 % proof stress. Fig. 1.4. (b) Permanent deformation or "set" after straining beyond the yield point.
The 0.1 % proof stress value may be determined from the tensile test curve for the material in question as follows:
Mark the point P on the strain axis which is equivalent to 0.1 % strain. From P draw a line parallel with the initial straight line portion of the tensile test curve to cut the curve in N. The stress corresponding to AT is then the 0.1 % proof stress. A material is considered to satisfy its specification if the permanent set is no more than 0.1 % after the proof stress has been applied for 15 seconds and removed.
Beyond the yield point some increase in load is required to take the strain to point E on the graph. Between D and E the material is said to be in the elasticplastic state, some of the section remaining elastic and hence contributing to recovery of the original dimensions if load is removed, the remainder being plastic. Beyond E the crosssectional area of the bar begins to reduce rapidly over a relatively small length of the bar and the bar is said to neck. This necking takes place whilst the load reduces, and fracture of the bar finally occurs at point F.
The nominal stress at failure, termed the maximum or ultimate tensile stress, is given by the load at E divided by the original crosssectional area of the bar. (This is also known as the tensile strength of the material of the bar.) Owing to the large reduction in area produced by the necking process the actual stress at fracture is often greater than the above value. Since, however, designers are interested in maximum loads which can be carried by the complete crosssection, the stress at fracture is seldom of any practical value.
If load is removed from the test specimen after the yield point C has been passed, e.g. to some position S, Fig. 1.4(b), the unloading line ST can, for most practical purposes, be taken to be linear. Thus, despite the fact that loading to S comprises both elastic (OC) and partially plastic (CS) portions, the unloading procedure is totally elastic. A second load cycle, commencing with the permanent elongation associated with the strain OT, would then follow the line TS and continue along the previous curve to failure at F. It will be observed, however, that the repeated load cycle has the effect of increasing the elastic range of the material, i.e. raising the effective yield point from C to S, while the tensile strength is unaltered. The procedure could be repeated along the line PQ, etc., and the material is said to have been work hardened.
In fact, careful observation shows that the material will no longer exhibit true elasticity since the unloading and reloading lines will form a small hysteresis loop, neither being precisely linear. Repeated loading and unloading will produce a yield point approaching the ultimate stress value but the elongation or strain to failure will be much reduced.
Typical stressstrain curves resulting from tensile tests on other engineering materials are shown in Figs 1.5 to 1.7.
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