e vs. engineering strain ε - john.maloney.org

1 Notes on true strainevs. engineering strain john MaloneySept. 20, 2006 strain is normalized deformation. We can express this relationship in differential form fora bar undergoing axial deformation asd( strain )=dLL, or an infinitesimal deformationdLnormalized to lengthL. We properly find the total axial straine(known as thetrue strain ) by integrating this expres-sion from the initial lengthL0to the final lengthLF:e LFL0dLL=ln(L)|LFL0=ln(LFL0)=ln(L0+ LL0)=ln(1+ LL0)where Lis the amount of deformation (positive for elongation). The lengthLis kept insidethe integral because it changes as the bar deforms from lengthL0to lengthLF. If Lis small compared to the length of the bar, thenL L0at all stages of the deformationprocess and the1 Lterm can be taken outside the integral. The resulting approximate strain is known as theengineering strain : 1L0 LFL0dL=1L0L|LFL0=LF L0L0= LL0 The same result is acquired through a Taylor series expansion ofe=ln(1+ LL0).

2 The Taylorseriesf(x0+ x)=f(x0)+ f(x) x x0 x+12! 2f(x) x2 x0 x2+13! 3f(x) x3 x0 x3+..is especially useful for estimatingf(x0+ x) whenf(x0) is known and xis small. In thiscase,f(x)=ln(x),x0=1, and x= LL0. So we havee=ln(1+ LL0) ln(1)+(1) 1( LL0)=( LL0)= The relationship between true straineand engineering strain is exactlye=ln(1+ ). Bothstrains, as normalized quantities, are unitless. Small strains are sometimes described as apercent ( , ), or in , or micros ( , 2000 =2000 parts per million= ). How close are the the values ofeand ? Plotted below are the engineering and true strainsfor values up to 1. The agreement is quite good for strains of less than (see inset).Note that, by convention, an engineering material is considered to have yielded deformedbeyond recovery at an engineering strain of , or At this value, the differencebetween engineering and true strain is less than one part in a Characteristics of true straine:1.

3 It s the exact value, not an Sequential strains can be added: if two strainse1ande2are executed sequentially, the totalstrain ise1+e2=ln(L1L0)+ln(L2L1)=ln(L1L0 L2L1)=ln(L2L0)This is not the case with engineering strain , where the total strain isL2 L0L0, 1+ 2=L1 L0L0+L2 L1L13. It s used to characterize materials that deform by large amounts (considerable fractionsof their length up to many times their length). A quick look at the literature shows that truestrain has been recently used to characterize materials like polyamide yarn, epoxy, rubber,and It s geometrically symmetric: that is, if the strain associated with being stretched tontimesthe original length ise, then the strain associated with being compressed to1nthe originallength is e. Characteristics of engineering strain :1. It s easier to It s overwhelmingly preferred in engineering analyses of materials that experience onlysmall strains (including the common construction materials concrete, wood, and steel, forexample, under normal use).

4 3. It s symmetric in terms of displacements: that is, if the strain associated with beingstretched a distance Lis , then the strain associated with being compressed a distance Lis .2