The engineering tensile stress-strain curve is obtained by static loading of a standard specimen, that is, by applying the load slowly enough that all parts of the specimen are in equilibrium at any instant. The curve is usually obtained by controlling the loading rate in the tensile machine. ASTM Standards require a loading rate not exceeding 100,000 lb/in^{2} (70 kgf/mm^{2})/min. An alternate method of obtaining the curve is to specify the strain rate as the independent variable, in which case the loading rate is continuously adjusted to maintain the required strain rate. A strain rate of 0.05 in/in/(min) is commonly used. It is measured usually by an extensometer attached to the gauge length of the specimen. Figure 1 shows several stress-strain curves.

**Figure 1.** Comparative stress-strain diagrams. (1) Soft brass; (2) low carbon steel; (3) hard bronze; (4) cold rolled steel; (5) medium carbon steel, annealed; (6) medium carbon steel, heat treated.

For most engineering materials, the curve will have an initial linear elastic region (Figure 2) in which deformation is reversible and time independent. The slope in this region is Young’s modulus E. The proportional elastic limit (PEL) is the point where the curve starts to deviate from a straight line. The elastic limit (frequently indistinguishable from PEL) is the point on the curve beyond which plastic deformation is present after release of the load. If the stress is increased further, the stress-strain curve departs more and more from the straight line. Unloading the specimen at point X (Figure 2), the portion XX’ is linear and is essentially parallel to the original line OX’’. The horizontal distance OX’ is called the permanent set corresponding to the stress at X. This is the basis for the construction of the arbitrary yield strength. To determine the yield strength, a straight line XX’ is drawn parallel to the initial elastic line OX’’ but displaced from it by an arbitrary value of permanent strain. The permanent strain commonly used is 0.20 percent of the original gauge length. The intersection of this line with the curve determines the stress value called the yield strength. In reporting the yield strength, the amount of permanent set should be specified. The arbitrary yield strength is used especially for those materials not exhibiting a natural yield point such as nonferrous metals; but it is not limited to these. Plastic behavior is somewhat time-dependent, particularly at high temperatures. Also at high temperatures, a small amount of time-dependent reversible strain may be detectable, indicative of anelastic behavior.

**Figure 2.** General stress-strain diagram

The ultimate tensile strength (UTS) is the maximum load sustained by the specimen divided by the original specimen cross-sectional area. The percent elongation at failure is the plastic extension of the specimen at failure expressed as (the change in original gauge length x 100) divided by the original gauge length. This extension is the sum of the uniform and non-uniform elongations. The uniform elongation is that which occurs prior to the UTS. It has an unequivocal significance, being associated with uni-axial stress, whereas the non-uniform elongation which occurs during localized extension (necking) is associated with triaxial stress.

The non-uniform elongation will depend on geometry, particularly the ratio of specimen gauge length L_{0} to diameter D or square root of cross sectional area A. ASTM Standards specify test-specimen geometry for a number of specimen sizes. The ratio L_{0}/√A is maintained at 4.5 for flat and round-cross-section specimens. The original gauge length should always be stated in reporting elongation values.

The specimen percent reduction in area (RA) is the contraction in cross-sectional area at the fracture expressed as a percentage of the original area. It is obtained by measurement of the cross section of the broken specimen at the fracture location. The RA along with the load at fracture can be used to obtain the fracture stress, that is, fracture load divided by cross-sectional area at the fracture.

The type of fracture in tension gives some indications of the quality of the material, but this is considerably affected by the testing temperature, speed of testing, the shape and size of the test piece, and other conditions. Contraction is greatest in tough and ductile materials and least in brittle materials. In general, fractures are either of the shear or of the separation (loss of cohesion) type. Flat tensile specimens of ductile metals often show shear failures if the ratio of width to thickness is greater than 6:1. A completely shear-type failure may terminate in a chisel edge, for a flat specimen, or a point rupture, for a round specimen. Separation failures occur in brittle materials, such as certain cast irons. Combinations of both shear and separation failures are common on round specimens of ductile metal. Failure often starts at the axis in a necked region and produces a relatively flat area which grows until the material shears along a cone-shaped surface at the outside of the specimen, resulting in what is known as the cup-and-cone fracture. Double cup-and-cone and rosette fractures sometimes occur. Several types of tensile fractures are shown in Figure 3.

**Figure 3.** Typical metal fractures in tension

Annealed or hot-rolled mild steels generally exhibit a yield point (see Figure 4). Here, in a constant strain-rate test, a large increment of extension occurs under constant load at the elastic limit or at a stress just below the elastic limit. In the latter event the stress drops suddenly from the upper yield point to the lower yield point. Subsequent to the drop, the yield-point extension occurs at constant stress, followed by a rise to the UTS. Plastic flow during the yield-point extension is discontinuous; successive zones of plastic deformation, known as Luder’s bands or stretcher strains, appear until the entire specimen gauge length has been uniformly deformed at the end of the yield-point extension. This behavior causes a banded or stepped appearance on the metal surface. The exact form of the stress-strain curve for this class of material is sensitive to test temperature, test strain rate, and the characteristics of the tensile machine employed.

**Figure 4.** Yielding of annealed steel

The plastic behavior in a uni-axial tensile test can be represented as the true stress-strain curve. The true stress σ is based on the instantaneous cross section A, so that σ = load/A. The instantaneous true strain increment is -*d*A/A, or *d*L/L prior to necking. Total true strain ε is ln(L/L_{0}) prior to necking. The true stress-strain curve or flow curve obtained has the typical form shown in Figure 5. In the part of the test subsequent to the maximum load point (UTS), when necking occurs, the true strain of interest is that which occurs in an infinitesimal length at the region of minimum cross section. True strain for this element can still be expressed as ln(A_{0}/A), where A refers to the minimum cross section. Methods of constructing the true stress-strain curve are described in the technical literature.

In the range between initial yielding and the neighborhood of the maximum load point the relationship between plastic strain εp and true stress often approximates

σ = *k* (ε_{p})^{n}

where *k* is the strength coefficient and *n* is the work-hardening exponent. For a material which shows a yield point the relationship applies only to the rising part of the curve beyond the lower yield. It can be shown that at the maximum load point the slope of the true stress-strain curve equals the true stress, from which it can be deduced that for a material obeying the above exponential relationship between ε_{p} and *n*, ε_{p} = *n* at the maximum load point. The exponent strongly influences the spread between YS and UTS on the engineering stress-strain curve. A point on the flow curve indentifies the flow stress corresponding to a certain strain, that is, the stress required to bring about this amount of plastic deformation. The concept of true strain is useful for accurately describing large amounts of plastic deformation. The linear strain definition (L - L_{0})/L_{0} fails to correct for the continuously changing gauge length, which leads to an increasing error as deformation proceeds.

During extension of a specimen under tension, the change in the specimen cross-sectional area is related to the elongation by Poisson’s ratio μ, which is the ratio of strain in a transverse direction to that in the longitudinal direction.

**Figure 5.** True stress-strain curve for 20°C annealed mild steel

The general effect of increased strain rate is to increase the resistance to plastic deformation and thus to raise the flow curve. Decreasing test temperature also raises the flow curve. The effect of strain rate is expressed as strain-rate sensitivity. Its value can be measured in the tension test if the strain rate is suddenly increased by a small increment during the plastic extension. The flow stress will then jump to a higher value.

**Compression Testing** - The compressive stress-strain curve is similar to the tensile stress-strain curve up to the yield strength. Thereafter, the progressively increasing specimen cross section causes the compressive stress-strain curve to diverge from the tensile curve. Some ductile metals will not fail in the compression test. Complex behavior occurs when the direction of stressing is changed, because of the Bauschinger effect, which can be described as follows: If a specimen is first plastically strained in tension, its yield stress in compression is reduced and vice versa.

**Combined Stresses** - This refers to the situation in which stresses are present on each of the faces of a cubic element of the material. For a given cube orientation the applied stresses may include shear stresses over the cube faces as well as stresses normal to them. By a suitable rotation of axes the problem can be simplified: applied stresses on the new cubic element are equivalent to three mutually orthogonal principal stresses σ_{1} , σ_{2} , σ_{3} alone, each acting normal to a cube face.

Prediction of the conditions under which plastic yielding will occur under combined stresses can be made with the help of several empirical theories. In the maximum-shear-stress theory the criterion for yielding is that yielding will occur when

σ_{1} – σ_{3} = σ_{ys}

in which σ1 and σ3 are the largest and smallest principal stresses, respectively, and σ_{ys} is the uniaxial tensile yield strength. This is the simplest theory for predicting yielding under combined stresses. A more accurate prediction can be made by the distortion-energy theory, according to which the criterion is

(σ_{1} – σ_{2})^{2} + (σ_{2} – σ_{3})^{2} + (σ_{2} - σ_{1})^{2} = 2(σ_{ys})^{2}

Stress-strain curves in the plastic region for combined stress loading can be constructed. However, a particular stress state does not determine a unique strain value. The latter will depend on the stress-state path which is followed.

**Plane strain** is a condition where strain is confined to two dimensions. There is generally stress in the third direction, but because of mechanical constraints, strain in this dimension is prevented. Plane strain occurs in certain metalworking operations. It can also occur in the neighborhood of a crack tip in a tensile loaded member if the member is sufficiently thick. The material at the crack tip is then in triaxial tension, which condition promotes brittle fracture. On the other hand, ductility is enhanced and fracture is suppressed by triaxial compression.

**Stress Concentration** in a structure or machine part having a notch or any abrupt change in cross section, the maximum stress will occur at this location and will be greater than the stress calculated by elementary formulas based upon simplified assumptions as to the stress distribution. The ratio of this maximum stress to the nominal stress (calculated by the elementary formulas) is the stress-concentration factor Kt. This is a constant for the particular geometry and is independent of the material, provided it is isotropic. The stress-concentration factor may be determined experimentally or, in some cases, theoretically from the mathematical theory of elasticity. The factors shown in Figures 6 to 13 were determined from both photoelastic tests and the theory of elasticity. Stress concentration will cause failure of brittle materials if the concentrated stress is larger than the ultimate strength of the material. In ductile materials, concentrated stresses higher than the yield strength will generally cause local plastic deformation and redistribution of stresses (rendering them more uniform). On the other hand, even with ductile materials areas of stress concentration are possible sites for fatigue if the component is cyclically loaded.

**Figure 6.** Flat plate with semicircular fillets and grooves or with holes; I, II, and III are in tension or compression; IV and V are in bending

**Figure 7. **Flat plate with grooves, in tension

**Figure 8.** Flat plate with fillets, in tension

**Figure 9.** Flat plate with grooves, in bending

**Figure 10.** Flat plate with fillets, in bending

**Figure 11. **Flat plate with angular notch, in tension or bending

**Figure 12. **Grooved shaft in torsion

**Figure 13. **Filleted shaft in torsion

**Reproduced from Mechanical Properties of Materials by John Symonds*

References:

Davis et al., ‘‘Testing and Inspection of Engineering Materials’’.

Timoshenko, ‘‘Strength of Materials’’ pt. II.

Richards, ‘‘Engineering Materials Science’’.

Nadai, ‘‘Plasticity’’.

Tetelman and McEvily, ‘‘Fracture of Structural Materials’’.

ASTM STP-833, ‘‘Fracture Mechanic’’.

McClintock and Argon, ‘‘Mechanical Behavior of Materials’’.

Dieter, ‘‘Mechanical Metallurgy’’.

Blaznynski, ‘‘Plasticity and Modern Metal Forming Technology’’.

ASME and ASTM Standards.