Stress vs Strain Curve Explanation PDF Free: A Comprehensive Guide for Engineers and Students

hence the region between A to B stress and strain are not proportional. The point B in the curve is known as yield point (also known as elastic limit) and the corresponding stress is known as yield strength (σy) of the material. below this point the body can retain its original shape and size but if yield stress goes beyond (σy) the body will start deforming permanently and is said to be permanently set

Interstitial Free steel (IF) steel is often termed as a clean steel that refers to very low to negligible level of interstitial solute atoms that may be present in the steel and hence the Fe lattice is practically free of strains caused by the interstitials, thus resulting in high formability and high strain rate sensitivity. Microalloying elements Nb and Ti added to the steel combine with C and N atoms and hence, make the steel essentially free of interstitials. Recently, IF steels are being intended for applications in various structural parts viz., cross members, B-pillars, longitudinal beams etc. [1, 2], besides regular applications in automobile industry. These steels are normally characterized with low yield strength and hence high formability and deep drawability. Thus, new scientific approaches should be explored to optimize the processing parameters in order to achieve optimized properties in IF steels with high strength and toughness without compromising on formability. Recently, analysis of flow stress versus strain curves describing the hot/warm working characteristics of metals and alloys has received great attraction and several works have since been executed to characterize the flow behavior using physical simulations as well as mathematical modeling, cf. [3,4,5,6]. However, the constitutive models reported by Fang et al. [3]; Phaniraj et al. [4] and Wei et al. [6] are considered unsuitable as they did not consider the effect of applied strain on the flow stress modeling. Basically, their models predict that either the flow stress does not change with strain under a steady-state situation or it is considered as the peak stress. Hence, these constitutive models are considered inappropriate when the material exhibits dynamic recovery (DRV), dynamic recrystallization (DRX) or work hardening (WH) during deformation.

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Deformation process generally involves both the dislocation nucleation (strain hardening) as well as annihilation (dynamic softening). These two conflicting activities may occur concurrently during deformation process and directly affect the resultant dislocation density which is signified by the flow stress curves. The homogenization annealing (at 1200 C for 2 min) of the steel prior to the deformation process, develops a uniform structure having a low dislocation density. Therefore, during deformation, a large dislocation density is accumulated, rising rapidly initially as the deformation progresses, facilitating strain hardening. This is contested by annihilation of dislocations and hence, there is an appearance of peak in the stress versus strain curve. As the dynamic restoration process is well-adjusted by strain hardening, the flow stress decreases and grasps a steady state during further straining. On the other hand, DRX encountered during deformation process when the true strain overcomes the critical strain required for DRX. Prior to the stress value reaching the peak stress, the work hardening in respect of dislocation density enhancement occurs simultaneously. Hence, the flow stress rises up to a peak value, following which the rising rate gradually drops as the dynamic softening rate becomes greater than the work hardening rate. There is a significant effect of temperature on the appearance of the peak stress. It can be seen that the critical strain for the peak stress gradually increases with increasing strain rate and/or decreasing temperature. Moreover, at higher temperatures and lower strain rates, both the ferrite and austenite phases exhibit relatively low critical strains to achieve a steady state stress. This suggests that the rate of dislocation generation is balanced by the dynamic restoration process [33].

a Flow stress versus deformation temperatures at different strain rates and b flow stress versus logarithm of strain rate at a particular temperature for the IF steel at a true strain of 0.6

Constitutive equations have been developed to predict the flow stress behavior during isothermal hot compression testing and finally a relationship has been established between total strain, strain rate, temperature and flow stress of the IF steel. The detailed analysis is presented in following sections.

All the above-mentioned equations and the constitutive equations do not contemplate the influence of applied strain on the flow stress. Basically, this model predicts that either the flow stress does not change with strain under a steady-state condition or it happens to be the peak stress [30, 38]. However, the flow stress of the investigated steel differs continuously with rising strain and/or at higher strain rate deformation, and the peak stress behavior is observed only for low strain rate deformation. Also, many authors have suggested that the strain has a prominent effect on the flow stress (among other parameters) of the material [6]. Hence, the effect of applied strain on the hot deformation flow stress should be taken into consideration in order to derive a complete constitutive equation to be able to predict the flow stress correctly.

Comparisons between experimental and predicted flow stresses from the constitutive equations (considering compensation of strain) at temperatures a 650 C, b 700 C, c 750 C, d 800 C, e 850 C, f 900 C, g 950 C, h 1000 C, i 1050 C and j 1100 C

The apparent Q for warm/hot deformation of the IF steel is found to decrease continuously from 324 to 285 kJ/mol in the ferritic phase region (α) and 365 to 342 kJ/mol in the austenite phase region (γ) with the increase of true strain from 0.05 to 0.6. The activation energies of deformation differ greatly from the corresponding self-diffusion activation energies of α (239 kJ/mol) and γ (270 kJ/mol). This is due to the work hardening effect dominated in the early stage of deformation of the IF steel and the DRX/DRV dominating the deformation with increase in the amount of strain. Furthermore, the average value of stress exponent (n) is calculated to be 6.7 and 5.5 for α and γ phase regions, respectively. It indicates that the hot/warm deformation is controlled by the mechanisms of dislocation glide and dislocation climb. TEM analysis also confirms that the plastic deformation is controlled by dislocations motion through bypassing of the precipitates by glide/climb mechanisms.

where each of the materials constants, viz., Q, A,\(\alpha_1\) and n is a function of strain as described above and \(Z = \dot\varepsilon \exp \left( \fracQRT \right)\). The modified constitutive equation is found to predict the flow stress precisely with the experimentally obtained data both in γ and α phase regions showing an excellent fitting with high correlation coefficient, Rcc (0.982 and 0.936, respectively) and extremely low value of AREE (7% and and 11%, respectively).

The engineering tension test is widely used to provide basic design information on the strength of materials and as an acceptance test for the specification of materials. In the tension test a specimen is subjected to a continually increasing uniaxial tensile force while simultaneous observations are made of the elongation of the specimen. The parameters, which are used to describe the stress-strain curve of a metal, are the tensile strength, yield strength or yield point, percent elongation, and reduction of area. The first two are strength parameters; the last two indicate ductility.

• The engineering tension test is widely used to provide basic design information on images/the strength of materials and as an acceptance test for the specification of materials. In the tension test a specimen is subjected to a continually increasing uniaxial tensile force while simultaneous observations are made of the elongation of the specimen. An engineering stress-strain curve is constructed from the load elongation measurements (Fig. 1).Figure 1. The engineering stress-strain curveIt is obtained by dividing the load by the original area of the cross section of the specimen.(1)The strain used for the engineering stress-strain curve is the average linear strain, which is obtained by dividing the elongation of the gage length of the specimen, d, by its original length.(2)Since both the stress and the strain are obtained by dividing the load and elongation by constant factors, the load-elongation curve will have the same shape as the engineering stress-strain curve. The two curves are frequently used interchangeably.The shape and magnitude of the stress-strain curve of a metal will depend on its composition, heat treatment, prior history of plastic deformation, and the strain rate, temperature, and state of stress imposed during the testing. The parameters, which are used to describe the stress-strain curve of a metal, are the tensile strength, yield strength or yield point, percent elongation, and reduction of area. The first two are strength parameters; the last two indicate ductility.The general shape of the engineering stress-strain curve (Fig. 1) requires further explanation. In the elastic region stress is linearly proportional to strain. When the load exceeds a value corresponding to the yield strength, the specimen undergoes gross plastic deformation. It is permanently deformed if the load is released to zero. The stress to produce continued plastic deformation increases with increasing plastic strain, i.e., the metal strain-hardens. The volume of the specimen remains constant during plastic deformation, AL = A0L0 and as the specimen elongates, it decreases uniformly along the gage length in cross-sectional area. Initially the strain hardening more than compensates for this decrease in area and the engineering stress (proportional to load P) continues to rise with increasing strain. Eventually a point is reached where the decrease in specimen cross-sectional area is greater than the increase in deformation load arising from strain hardening. This condition will be reached first at some point in the specimen that is slightly weaker than the rest. All further plastic deformation is concentrated in this region, and the specimen begins to neck or thin down locally. Because the cross-sectional area now is decreasing far more rapidly than strain hardening increases the deformation load, the actual load required to deform the specimen falls off and the engineering stress likewise continues to decrease until fracture occurs.Tensile StrengthThe tensile strength, or ultimate tensile strength (UTS), is the maximum load divided by the original cross-sectional area of the specimen.(3)The tensile strength is the value most often quoted from the results of a tension test; yet in reality it is a value of little fundamental significance with regard to the strength of a metal. For ductile metals the tensile strength should be regarded as a measure of the maximum load, which a metal can withstand under the very restrictive conditions of uniaxial loading. It will be shown that this value bears little relation to the useful strength of the metal under the more complex conditions of stress, which are usually encountered. For many years it was customary to base the strength of members on the tensile strength, suitably reduced by a factor of safety. The current trend is to the more rational approach of basing the static design of ductile metals on the yield strength. However, because of the long practice of using the tensile strength to determine the strength of materials, it has become a very familiar property, and as such it is a very useful identification of a material in the same sense that the chemical composition serves to identify a metal or alloy. Further, because the tensile strength is easy to determine and is a quite reproducible property, it is useful for the purposes of specifications and for quality control of a product. Extensive empirical correlations between tensile strength and properties such as hardness and fatigue strength are often quite useful. For brittle materials, the tensile strength is a valid criterion for design.Measures of YieldingThe stress at which plastic deformation or yielding is observed to begin depends on the sensitivity of the strain measurements. With most materials there is a gradual transition from elastic to plastic behavior, and the point at which plastic deformation begins is hard to define with precision. Various criteria for the initiation of yielding are used depending on the sensitivity of the strain measurements and the intended use of the data.True elastic limit based on micro strain measurements at strains on order of 2 x 10-6 in in. This elastic limit is a very low value and is related to the motion of a few hundred dislocations.Proportional limit is the highest stress at which stress is directly proportional to strain. It is obtained by observing the deviation from the straight-line portion of the stress-strain curve.Elastic limit is the greatest stress the material can withstand without any measurable permanent strain remaining on the complete release of load. With increasing sensitivity of strain measurement, the value of the elastic limit is decreased until at the limit it equals the true elastic limit determined from micro strain measurements. With the sensitivity of strain usually employed in engineering studies (10-4in in), the elastic limit is greater than the proportional limit. Determination of the elastic limit requires a tedious incremental loading-unloading test procedure.The yield strength is the stress required to produce a small-specified amount of plastic deformation. The usual definition of this property is the offset yield strength determined by the stress corresponding to the intersection of the stress-strain curve and a line parallel to the elastic part of the curve offset by a specified strain (Fig. 1). In the United States the offset is usually specified as a strain of 0.2 or 0.1 percent (e = 0.002 or 0.001).(4)

A good way of looking at offset yield strength is that after a specimen has been loaded to its 0.2 percent offset yield strength and then unloaded it will be 0.2 percent longer than before the test. The offset yield strength is often referred to in Great Britain as the proof stress, where offset values are either 0.1 or 0.5 percent. The yield strength obtained by an offset method is commonly used for design and specification purposes because it avoids the practical difficulties of measuring the elastic limit or proportional limit.Some materials have essentially no linear portion to their stress-strain curve, for example, soft copper or gray cast iron. For these materials the offset method cannot be used and the usual practice is to define the yield strength as the stress to produce some total strain, for example, e = 0.005.Measures of DuctilityAt our present degree of understanding, ductility is a qualitative, subjective property of a material. In general, measurements of ductility are of interest in three ways:To indicate the extent to which a metal can be deformed without fracture in metalworking operations such as rolling and extrusion. To indicate to the designer, in a general way, the ability of the metal to flow plastically before fracture. A high ductility indicates that the material is "forgiving" and likely to deform locally without fracture should the designer err in the stress calculation or the prediction of severe loads. To serve as an indicator of changes in impurity level or processing conditions. Ductility measurements may be specified to assess material quality even though no direct relationship exists between the ductility measurement and performance in service. The conventional measures of ductility that are obtained from the tension test are the engineering strain at fracture ef (usually called the elongation) and the reduction of area at fracture q. Both of these properties are obtained after fracture by putting the specimen back together and taking measurements of Lf and Af .(5)(6)Because an appreciable fraction of the plastic deformation will be concentrated in the necked region of the tension specimen, the value of ef will depend on the gage length L0 over which the measurement was taken. The smaller the gage length the greater will be the contribution to the overall elongation from the necked region and the higher will be the value of ef. Therefore, when reporting values of percentage elongation, the gage length L0 always should be given.The reduction of area does not suffer from this difficulty. Reduction of area values can be converted into an equivalent zero-gage-length elongation e0. From the constancy of volume relationship for plastic deformation A*L = A0*L0, we obtain(7)This represents the elongation based on a very short gage length near the fracture.Another way to avoid the complication from necking is to base the percentage elongation on the uniform strain out to the point at which necking begins. The uniform elongation eu correlates well with stretch-forming operations. Since the engineering stress-strain curve often is quite flat in the vicinity of necking, it may be difficult to establish the strain at maximum load without ambiguity. In this case the method suggested by Nelson and Winlock is useful. 2ff7e9595c