STRESS-STRAIN CURVESDavid RoylanceDepartment of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridge, MA 02139August 23, 2001IntroductionStress-strain curves are an extremely important graphical measure of a material’s mechanicalproperties, and all students of Mechanics of Materials will encounter them often. However, theyare 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 discussionof several points needed to interpret these curves, and in doing so will also provide a preliminaryoverview of several aspects of a material’s mechanical properties. However, this module willnot attempt to survey the broad range of stress-strain curves exhibited by modern engineeringmaterials (the atlas by Boyer cited in the References section can be consulted for this). Severalof the topics mentioned here — especially yield and fracture — will appear with more detail inlater modules.“Engineering” Stress-Strain CurvesPerhaps the most important test of a material’s mechanical response is the tensile test1,inwhichone end of a rod or wire specimen is clamped in a loading frame and the other subjected toa controlled displacement δ (see Fig. 1). A transducer connected in series with the specimenprovides 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 thecontrolled variable, in which case the displacement δ(P ) would be monitored as a function ofload.The engineering measures of stress and strain, denoted in this module as σeand erespec-tively, are determined from the measured the load and deflection using the original specimencross-sectional area A0and length L0asσe=PA0,e=δL0(1)When the stress σeis plotted against the strain e,anengineering stress-strain curve such asthat shown in Fig. 2 is obtained.1Stress-strain testing, as well as almost all experimental procedures in mechanics of materials, is detailed bystandards-setting organizations, notably the American Society for Testing and Materials (ASTM). Tensile testingof metals is prescribed by ASTM Test E8, plastics by ASTM D638, and composite materials by ASTM D3039.1Figure 1: The tension test.Figure 2: Low-strain region of the engineering stress-strain curve for annealed polycrystalinecopper; 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 proportionalitybeing the modulus of elasticity or Young’s modulus, denoted E:σe= Ee(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 undergoinga rearrangement of its internal molecular or microscopic structure, in which atoms are beingmoved 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 alater module.) Materials lacking this mobility, for instance by having internal microstructuresthat block dislocation motion, are usually brittle rather than ductile. The stress-strain curvefor brittle materials are typically linear over their full range of strain, eventually terminating infracture without appreciable plastic flow.Note in Fig. 2 that the stress needed to increase the strain beyond the proportional limitin a ductile material continues to rise beyond the proportional limit; the material requires anever-increasing stress to continue straining, a mechanism termed strain hardening.These microstructural rearrangements associated with plastic flow are usually not reversed2when the load is removed, so the proportional limit is often the same as or at least close to thematerials’s elastic limit. Elasticity is the property of complete and immediate recovery froman imposed displacement on release of the load, and the elastic limit is the value of stress atwhich the material experiences a permanent residual strain that is not lost on unloading. Theresidual strain induced by a given stress can be determined by drawing an unloading line fromthe highest point reached on the se - ee curve at that stress back to the strain axis, drawn witha slope equal to that of the initial elastic loading line. This is done because the material unloadselastically, there being no force driving the molecular structure back to its original position.A closely related term is the yield stress, denoted σYin these modules; this is the stressneeded to induce plastic deformation in the specimen. Since it is often difficult to pinpoint theexact stress at which plastic deformation begins, the yield stress is often taken to be the stressneeded to induce a specified amount of permanent strain, typically 0.2%. The construction usedto find this “offset yield stress” is shown in Fig. 2, in which a line of slope E is drawn from thestrain axis at e=0.2%; this is the unloading line that would result in the specified permanentstrain. The stress at the point of intersection with the σe− ecurve is the offset yield stress.Figure 3 shows the engineering stress-strain curve for copper with an enlarged scale, nowshowing strains from zero up to specimen fracture. Here it appears that the rate of strainhardening2diminishes up to a point labeled UTS, for Ultimate Tensile Strength (denoted σfinthese modules). Beyond that point, the material appears to strain soften, so that each incrementof additional strain requires a smaller stress.Figure 3: Full engineering stress-strain curve for annealed polycrystalline copper.The apparent change from strain hardening to strain softening is an artifact of the plottingprocedure, however, as is the maximum observed in the curve at the UTS. Beyond the yieldpoint, molecular flow causes a substantial reduction in the specimen cross-sectional area A,sothetruestressσt= P/A actually borne by the material is larger than the engineering stresscomputed from the original cross-sectional area (σe= P/A0). The load must equal the truestress times the actual area (P = σtA), and as long as