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STRESS-STRAIN CURVES
David Roylance
Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
August 23, 2001
Introduction
Stress-strain curves are an extremely important graphical measure of a material’s mechanical properties, and all students of Mechanics of Materials will encounter them often. However, they are not without some subtlety, especially in the case of ductile materials that can undergo sub- stantial geometrical change during testing. This module will provide an introductory discussion of several points needed to interpret these curves, and in doing so will also provide a preliminary overview of several aspects of a material’s mechanical properties. However, this module will not attempt to survey the broad range of stress-strain curves exhibited by modern engineering materials (the atlas by Boyer cited in the References section can be consulted for this). Several of the topics mentioned here — especially yield and fracture — will appear with more detail in later modules.
“Engineering” Stress-Strain Curves
Perhaps the most important test of a material’s mechanical response is the tensile test^1 , in which one end of a rod or wire specimen is clamped in a loading frame and the other subjected to a controlled displacement δ (see Fig. 1). A transducer connected in series with the specimen provides an electronic reading of the load P (δ) corresponding to the displacement. Alternatively, modern servo-controlled testing machines permit using load rather than displacement as the controlled variable, in which case the displacement δ(P ) would be monitored as a function of load. The engineering measures of stress and strain, denoted in this module as σe and e respec- tively, are determined from the measured the load and deflection using the original specimen cross-sectional area A 0 and length L 0 as
σe =
P
A 0
, e =
δ L 0
When the stress σe is plotted against the strain e, an engineering stress-strain curve such as that shown in Fig. 2 is obtained. (^1) Stress-strain testing, as well as almost all experimental procedures in mechanics of materials, is detailed by standards-setting organizations, notably the American Society for Testing and Materials (ASTM). Tensile testing of metals is prescribed by ASTM Test E8, plastics by ASTM D638, and composite materials by ASTM D3039.
Figure 1: The tension test.
Figure 2: Low-strain region of the engineering stress-strain curve for annealed polycrystaline copper; this curve is typical of that of many ductile metals.
In the early (low strain) portion of the curve, many materials obey Hooke’s law to a reason- able approximation, so that stress is proportional to strain with the constant of proportionality being the modulus of elasticity or Young’s modulus, denoted E:
σe = Ee (2) As strain is increased, many materials eventually deviate from this linear proportionality, the point of departure being termed the proportional limit. This nonlinearity is usually as- sociated with stress-induced “plastic” flow in the specimen. Here the material is undergoing a rearrangement of its internal molecular or microscopic structure, in which atoms are being moved to new equilibrium positions. This plasticity requires a mechanism for molecular mo- bility, which in crystalline materials can arise from dislocation motion (discussed further in a later module.) Materials lacking this mobility, for instance by having internal microstructures that block dislocation motion, are usually brittle rather than ductile. The stress-strain curve for brittle materials are typically linear over their full range of strain, eventually terminating in fracture without appreciable plastic flow. Note in Fig. 2 that the stress needed to increase the strain beyond the proportional limit in a ductile material continues to rise beyond the proportional limit; the material requires an ever-increasing stress to continue straining, a mechanism termed strain hardening. These microstructural rearrangements associated with plastic flow are usually not reversed
is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. At the UTS the differential of the load P is zero, giving an analytical relation between the true stress and the area at necking:
P = σtA → dP = 0 = σtdA + Adσt → − dA A
dσt σt
The last expression states that the load and therefore the engineering stress will reach a maxi- mum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress. Even though the UTS is perhaps the materials property most commonly reported in tensile tests, it is not a direct measure of the material due to the influence of geometry as discussed above, and should be used with caution. The yield stress σY is usually preferred to the UTS in designing with ductile metals, although the UTS is a valid design criterion for brittle materials that do not exhibit these flow-induced reductions in cross-sectional area. The true stress is not quite uniform throughout the specimen, and there will always be some location - perhaps a nick or some other defect at the surface - where the local stress is maximum. Once the maximum in the engineering curve has been reached, the localized flow at this site cannot be compensated by further strain hardening, so the area there is reduced further. This increases the local stress even more, which accelerates the flow further. This localized and increasing flow soon leads to a “neck” in the gage length of the specimen such as that seen in Fig. 4.
Figure 4: Necking in a tensile specimen.
Until the neck forms, the deformation is essentially uniform throughout the specimen, but after necking all subsequent deformation takes place in the neck. The neck becomes smaller and smaller, local true stress increasing all the time, until the specimen fails. This will be the failure mode for most ductile metals. As the neck shrinks, the nonuniform geometry there alters the uniaxial stress state to a complex one involving shear components as well as normal stresses. The specimen often fails finally with a “cup and cone” geometry as seen in Fig. 5, in which the outer regions fail in shear and the interior in tension. When the specimen fractures, the engineering strain at break — denoted f — will include the deformation in the necked region and the unnecked region together. Since the true strain in the neck is larger than that in the unnecked material, the value of f will depend on the fraction of the gage length that has necked. Therefore, f is a function of the specimen geometry as well as the material, and thus is only a
crude measure of material ductility.
Figure 5: Cup-and-cone fracture in a ductile metal.
Figure 6 shows the engineering stress-strain curve for a semicrystalline thermoplastic. The response of this material is similar to that of copper seen in Fig. 3, in that it shows a proportional limit followed by a maximum in the curve at which necking takes place. (It is common to term this maximum as the yield stress in plastics, although plastic flow has actually begun at earlier strains.)
Figure 6: Stress-strain curve for polyamide (nylon) thermoplastic.
The polymer, however, differs dramatically from copper in that the neck does not continue shrinking until the specimen fails. Rather, the material in the neck stretches only to a “natural draw ratio” which is a function of temperature and specimen processing, beyond which the material in the neck stops stretching and new material at the neck shoulders necks down. The neck then propagates until it spans the full gage length of the specimen, a process called drawing. This process can be observed without the need for a testing machine, by stretching a polyethylene “six-pack holder,” as seen in Fig. 7. Not all polymers are able to sustain this drawing process. As will be discussed in the next section, it occurs when the necking process produces a strengthened microstructure whose breaking load is greater than that needed to induce necking in the untransformed material just outside the neck.
Figure 8: Comparison of engineering and true stress-strain curves for copper. An arrow indicates the position on the “true” curve of the UTS on the engineering curve.
σt = Ant → log σt = log A + n log t (8)
Figure 9 is a log-log plot of the true stress-strain data^3 for copper from Fig. 8 that demonstrates this relation. Here the parameter n = 0.474 is called the strain hardening parameter, useful as a measure of the resistance to necking. Ductile metals at room temperature usually exhibit values of n from 0.02 to 0.5.
Figure 9: Power-law representation of the plastic stress-strain relation for copper.
A graphical method known as the “Consid`ere construction” uses a form of the true stress- strain curve to quantify the differences in necking and drawing from material to material. This method replots the tensile stress-strain curve with true stress σt as the ordinate and extension ratio λ = L/L 0 as the abscissa. From Eqn. 6, the engineering stress σe corresponding to any
(^3) Here percent strain was used for t; this produces the same value for n but a different A than if full rather than percentage values were used.
value of true stress σt is slope of a secant line drawn from origin (λ = 0, not λ = 1) to intersect the σt − λ curve at σt.
Figure 10: Consid`ere construction. (a) True stress-strain curve with no tangents - no necking or drawing. (b) One tangent - necking but not drawing. (c) Two tangents - necking and drawing.
Among the many possible shapes the true stress-strain curves could assume, let us consider the concave up, concave down, and sigmoidal shapes shown in Fig. 10. These differ in the number of tangent points that can be found for the secant line, and produce the following yield characteristics:
(a) No tangents: Here the curve is always concave upward as in part (a) of Fig. 10, so the slope of the secant line rises continuously. Therefore the engineering stress rises as well, without showing a yield drop. Eventually fracture intercedes, so a true stress-strain curve of this shape identifies a material that fractures before it yields.
(b) One tangent: The curve is concave downward as in part (b) of Fig. 10, so a secant line reaches a tangent point at λ = λY. The slope of the secant line, and therefore the engineering stress as well, begins to fall at this point. This is then the yield stress σY seen as a maximum in stress on a conventional stress-strain curve, and λY is the extension ratio at yield. The yielding process begins at some adventitious location in the gage length of the specimen, and continues at that location rather than being initiated elsewhere because the secant modulus has been reduced at the first location. The specimen is now flowing at a single location with decreasing resistance, leading eventually to failure. Ductile metals such as aluminum fail in this way, showing a marked reduction in cross sectional area at the position of yield and eventual fracture.
(c) Two tangents: For sigmoidal stress-strain curves as in part (c) of Fig. 10, the engineering stress begins to fall at an extension ration λY , but then rises again at λd. As in the previous one-tangent case, material begins to yield at a single position when λ = λY , producing a neck that in turn implies a nonuniform distribution of strain along the gage length. Material at the neck location then stretches to λd, after which the engineering stress there would have to rise to stretch it further. But this stress is greater than that needed to stretch material at the edge of the neck from λY to λd, so material already in the neck stops stretching and the neck propagates outward from the initial yield location. Only material within the neck shoulders is being stretched during propagation, with material inside the necked-down region holding constant at λd, the material’s “natural draw ratio,” and material outside holding at λY. When all the material has been drawn into the necked region, the stress begins to rise uniformly in the specimen until eventually fracture occurs.
The increase in strain hardening rate needed to sustain the drawing process in semicrys- talline polymers arises from a dramatic transformation in the material’s microstructure. These materials are initially “spherulitic,” containing flat lamellar crystalline plates, perhaps 10 nm
Figure 12: Energy associated with increments of strain
Table 1: Energy absorption of various materials.
Material Maximum Maximum Modulus of Density Max. Energy Strain, % Stress, MPa Toughness, MJ/m^3 kg/m^3 J/kg Ancient Iron 0.03 70 0.01 7,800 1. Modern spring steel 0.3 700 1.0 7,800 130 Yew wood 0.3 120 0.5 600 900 Tendon 8.0 70 2.8 1,100 2, Rubber 300 7 10.0 1,200 8,
will return to its original shape. But when the strain exceeds the yield point, the material is deformed irreversibly, so that some residual strain will persist even after unloading. The modulus of resilience is then the quantity of energy the material can absorb without suffering damage. Similarly, the modulus of toughness is the energy needed to completely fracture the material. Materials showing good impact resistance are generally those with high moduli of toughness.
Figure 13: Moduli of resilience and toughness.
Table 1^4 lists energy absorption values for a number of common materials. Note that natural and polymeric materials can provide extremely high energy absorption per unit weight.
During loading, the area under the stress-strain curve is the strain energy per unit volume absorbed by the material. Conversely, the area under the unloading curve is the energy released by the material. In the elastic range, these areas are equal and no net energy is absorbed. But
(^4) J.E. Gordon, Structures, or Why Things Don’t Fall Down, Plenum Press, New York, 1978.
if the material is loaded into the plastic range as shown in Fig. 14, the energy absorbed exceeds the energy released and the difference is dissipated as heat.
Figure 14: Energy loss = area under stress-strain loop.
Compression
The above discussion is concerned primarily with simple tension, i.e. uniaxial loading that increases the interatomic spacing. However, as long as the loads are sufficiently small (stresses less than the proportional limit), in many materials the relations outlined above apply equally well if loads are placed so as to put the specimen in compression rather than tension. The expression for deformation and a given load δ = P L/AE applies just as in tension, with negative values for δ and P indicating compression. Further, the modulus E is the same in tension and compression to a good approximation, and the stress-strain curve simply extends as a straight line into the third quadrant as shown in Fig. 15.
Figure 15: Stress-strain curve in tension and compression.
There are some practical difficulties in performing stress-strain tests in compression. If excessively large loads are mistakenly applied in a tensile test, perhaps by wrong settings on the testing machine, the specimen simply breaks and the test must be repeated with a new specimen. But in compression, a mistake can easily damage the load cell or other sensitive components, since even after specimen failure the loads are not necessarily relieved. Specimens loaded cyclically so as to alternate between tension and compression can exhibit hysteresis loops if the loads are high enough to induce plastic flow (stresses above the yield stress). The enclosed area in the loop seen in Fig. 16 is the strain energy per unit volume released as heat in each loading cycle. This is the well-known tendency of a wire that is being
Prob. 1
- Using the relations of Eqn. 6, plot the true stress-strain curve for aluminum (using data from Prob.1) up to the strain of neck formation.
- Replot the the results of the previous problem using log-log axes as in Fig. 9 to determine the parameters A and n in Eqn. 8 for aluminum.
- Using Eqn. 8 with parameters A = 800 MPa, n = 0.2, plot the engineering stress-strain curve up to a strain of e = 0.4. Does the material neck? Explain why the curve is or is not valid at strains beyond necking.
- Using the parameters of the previous problem, use the condition (dσe/de)neck = 0 to show that the engineering strain at necking is e,neck = 0.221.
- Use a Consid`ere construction (plot σt vs. λ, as in Fig. 10 ) to verify the result of the previous problem.
- Elastomers (rubber) have stress-strain relations of the form
σe =
E
( λ −
λ^2
) ,
where E is the initial modulus. Use the Consid`ere construction to show whether this material will neck, or draw.
- Show that a power-law material (one obeying Eqn. 8) necks when the true strain t becomes equal to the strain-hardening exponent n.
- Show that the UTS (engineering stress at incipient necking) for a power-law material (Eqn. 8) is
σf = Ann en
- Show that the strain energy U =
∫ σ d can be computed using either engineering or true values of stress and strain, with equal result.
- Show that the strain energy needed to neck a power-law material (Eqn.8) is
U =
Ann+ n + 1
