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 substantial 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 test1 in which one end of a rod or wire specimen is clamped in a loading frame and the other subjected to a controlled displacement see Fig 1 A transducer connected in series with the specimen provides an electronic reading of the load P 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 P would be monitored as a function of load The engineering measures of stress and strain denoted in this module as e and e respectively are determined from the measured the load and de ection using the original specimen cross sectional area A0 and length L0 as e P A0 e L0 1 When the stress e 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 1 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 reasonable approximation so that stress is proportional to strain with the constant of proportionality being the modulus of elasticity or Young s modulus denoted E e 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 associated with stress induced plastic ow 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 mobility 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 ow 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 ow are usually not reversed 2 when the load is removed so the proportional limit is often the same as or at least close to the materials s elastic limit Elasticity is the property of complete and immediate recovery from an imposed displacement on release of the load and the elastic limit is the value of stress at which the material experiences a permanent residual strain that is not lost on unloading The residual strain induced by a given stress can be determined by drawing an unloading line from the highest point reached on the se ee curve at that stress back to the strain axis drawn with a slope equal to that of the initial elastic loading line This is done because the material unloads elastically there being no force driving the molecular structure back to its original position A closely related term is the yield stress denoted Y in these modules this is the stress needed to induce plastic deformation in the specimen Since it is often di cult to pinpoint the exact stress at which plastic deformation begins the yield stress is often taken to be the stress needed to induce a speci ed amount of permanent strain typically 0 2 The construction used to nd this o set yield stress is shown in Fig 2 in which a line of slope E is drawn from the strain axis at e 0 2 this is the unloading line that would result in the speci ed permanent strain The stress at the point of intersection with the e e curve is the o set yield stress Figure 3 shows the engineering stress strain curve for copper with an enlarged scale now showing strains from zero up to specimen fracture Here it appears that the rate of strain hardening2 diminishes up to a point labeled UTS for Ultimate Tensile Strength denoted f in these modules Beyond that point the material appears to strain soften so that each increment of 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 plotting procedure however as is the maximum observed in the curve at the UTS Beyond the yield point molecular ow causes a substantial reduction in the specimen cross sectional area A so the true stress t P A actually borne by the material is larger than the engineering stress computed from the original cross sectional area e P A0 The load must equal the true stress times the actual area P t A and as long as strain hardening can increase t enough to compensate for the reduced area A the load and therefore the engineering stress will continue to rise