3 BasicDefinitionsofStressandStrain 3.1INTRODUCTION Themechanicalpropertiesofmaterialsdescribetheircharacteristic responsestoappliedloadsanddisplacements.However,mosttextsrelate themechanicalpropertiesofmaterialstostressesandstrains.Itis,therefore, importantforthereadertobecomefamiliarwiththebasicdefinitionsof stressandstrainbeforeproceedingontotheremainingchaptersofthis book.However,thewell-preparedreadermaychoosetoskip/skimthis chapter,andthenmoveontoChap.4inwhichthefundamentalsofelas- ticityareintroduced. Thebasicdefinitionsofstressandstrainarepresentedinthischapter alongwithexperimentalmethodsforthemeasurementandapplicationof strainandstress.Thechapterstartswiththerelationshipsbetweenapplied loads/displacementsandgeometrythatgiverisetothebasicdefinitionsof strainandstress.Simpleexperimentalmethodsforthemeasurementof strainandstressarethenpresentedbeforedescribingthetestmachines thatareoftenusedfortheapplicationofstrainandstressinthelaboratory. 3.2BASICDEFINITIONSOFSTRESS Theforcesappliedtothesurfaceofabodymayberesolvedintocompo- nentsthatareperpendicularorparalleltothesurface,Figs3.1(a)–3.1(c).In Copyright © 2003 Marcel Dekker, Inc. FIGURE 3.1 Different types of stress: (a)] uniaxial tension; (b) uniaxial com- pression; (c) twisting moment. (After Ashby and Jones, 1996. Courtesy of Butterworth Heinemann.) Copyright © 2003 Marcel Dekker, Inc. caseswhereuniformforcesareappliedinadirectionthatisperpendicularto thesurface,i.e.,alongthedirectionnormaltothesurface,Figs3.1(a)and 3.1(b),wecandefineauniaxialstress,,intermsofthenormalaxialload, P n ,dividedbythecross-sectionalarea,A.Thisgives ¼ Appliedload(normaltosurface) cross-sectionalarea ¼ P n A ð3:1Þ ItisalsoapparentfromtheaboveexpressionthatstresshasSIunitsof newtonspersquaremeter(N/m 2 )orpascals(Pa)].Someoldertextsand mostengineeringreportsintheU.S.A.mayalsousetheoldEnglishunits ofpoundspersquareinch(psi)torepresentstress.Inanycase,uniaxial stressmaybepositiveornegative,dependingonthedirectionofapplied loadFigs3.1(a)and3.1(b).Whentheappliedloadissuchthatittendsto stretchtheatomswithinasolidelement,thesignconventiondictatesthat thestressispositiveortensile,Fig.3.1(a)].Conversely,whentheapplied load is such that it tends to compress the atoms within a solid element, the uniaxial stress is negative or compressive, Fig. 3.1(b). Hence, the uniaxial stress may be positive (tensile) or negative (compressive), depending on the direction of the applied load with respect to the solid element that is being deformed. Similarly, the effects of twisting [Fig. 3.1(c)] on a given area can be characterized by shear stress, which is often denoted by the Greek letter, , and is given by:  ¼ Applied load (parallel to surface) cross-sectional area ¼ P s A ð3:2Þ Shear stress also has units of newtons per square meter square (N/m 2) or pounds per square inch (psi ). It is induced by torque or twisting moments that result in applied loads that are parallel to a deformed area of solid, Fig. 3.1(c). The above definitions of tensile and shear stress apply only to cases where the cross-sectional areas are uniform. More rigorous definitions are needed to describe the stress and strain when the cross-sectional areas are not uniform. Under such circumstances, it is usual to define uniaxial and shear stresses with respect to infinitesimally small elements, as shown in Fig 3.1. The uniaxial stresses can then be defined as the limits of the following expressions, as the sizes, dA, of the elements tend towards zero:  ¼ lim dA!0 P n dA  ð3:3Þ Copyright © 2003 Marcel Dekker, Inc. and ¼lim dA!0 P s dA  ð3:4Þ whereP n andP s aretherespectivenormalandshearloads,anddAisthe areaoftheinfinitesimallysmallelement.Theabovetermsareillustrated schematicallyinFigs3.1(a)and3.1(c),withFbeingequivalenttoP. Unlikeforce,stressisnotavectorquantitythatcanbedescribed simplybyitsmagnitudeanddirection.Instead,thegeneraldefinitionof stressrequiresthespecificationofadirectionnormaltoanareaelement, andadirectionparalleltotheappliedforce.Stressis,therefore,asecond ranktensorquantity,whichgenerallyrequiresthespecificationoftwodirec- tionnormals.AnintroductiontotensornotationwillbeprovidedinChap.4. However,fornow,thereadermaythinkofthestresstensorasamatrixthat containsallthepossiblecomponentsofstressonanelement.Thisconcept willbecomeclearerasweproceedinthischapter. Thestate of stress on a small element may be represented by ortho- gonal stress components within a Cartesian co-ordinate framework (Fig. 3.2). Note that there are nine stress components on the orthogonal faces of FIGURE3.2(a)StatesofStressonan Element,(b)positiveshearstress and (c) Negative shear stress. Courtesy of Dr. Seyed M. Allameh. Copyright © 2003 Marcel Dekker, Inc. thecubeshowninFig.3.2.Hence,a3Â3matrixmaybeusedtodescribe allthepossibleuniaxialandshearstressesthatcanactonanelement.The readershouldnotethataspecialsignconventionisusedtodeterminethe suffixesinFig.3.2.Thefirstsuffix,i,inthe ij or  ij terms corresponds to the direction of normal to the plane, while the second suffix, j, corresponds to the direction of the force. Furthermore, when both directions are posi- tive or negative, the stress term is positive. Similarly, when the direction of the load is opposite to the direction of the plane normal, the stress term is negative. We may now describe the complete stress tensor for a generalized three-dimensional stress state as ½¼  xx  xy  xz  yx  yy  yz  zx  zy  zz 2 4 3 5 ð3:5Þ Note that the above matrix, Eq. (3.5), contains only six independent terms since  ij ¼  ji for momen t equilibrium. The generalized state of stress at a point can, therefore, be described by three uniaxial stress terms ( xx ,  yy ,  zz ) and three shear stress terms ( xy ,  yz ,  zx ). The uniaxial and shear stresses may also be defined for any three orthogonal axes in the Cartesian co-ordinate system. Similarly, cylindrical (r;;L) and spherical (r;;L) co-ordinates may be used to describe the generalized state of stress on an element. In the case of a cylindrical co-ordinate system, the stress tensor is given by ½¼  rr  r  rL  r    L  Lr  L  LL 2 4 3 5 ð3:6Þ Similarly, for a spherical co-ordinate system, the stress tensor is given by ½¼  rr  r  r  r      r     2 4 3 5 ð3:7Þ It should be apparent from the above discussion that the generalized three-dimensional states of stress on an element may be described by any of the above co-ordinate systems. In general, however, the choice of co-ordi- nate system depends on the geometry of the solid that is being analyzed. Hence, the analysis of a cylindrical solid will often utilize a cylindrical co- ordinate system, while the analysis of a spherical solid will generally be done within a spherical co-ordinate framework. Copyright © 2003 Marcel Dekker, Inc. Inalltheaboveco-ordinatesystems,sixindependentstresscompo- nentsarerequiredtofullydescribethestateofstressonanelement.Luckily, mostproblemsinengineeringinvolvesimpleuniaxialorshearstates[Fig. 3.1).Hence,manyofthecomponentsoftheabovestresstensorsareoften equaltozero.Thissimplifiesthecomputationaleffortthatisneededforthe calculationofstressesandstrainsinmanypracticalproblems.Nevertheless, thereadershouldretainapictureofthegeneralizedstateofstressonan element,aswedevelopthebasicconceptsofmechanicalpropertiesinthe subsequentchaptersofthisbook.Wewillnowturnourattentiontothe basicdefinitionsofstrain. 3.3BASICDEFINITIONSOFSTRAIN Appliedloadsordisplacementsresultinchangesinthedimensionsorshape ofasolid.Forthesimplecaseofauniaxialdisplacementofasolidwitha uniformcross-sectionalarea[Fig.3.3(a)],theaxialstrain,",iscanbedefined simplyastheratioofthechangeinlength,u,totheoriginallength,l.Thisis givenby[Fig.3.3(a)]: "¼u=lð3:8Þ Notethatuniaxialstrainisadimensionlessquantitysinceitrepresents theratiooftwolengthterms.Furthermore,strainasdescribedbyEq.(3.8), isoftenreferredtoastheengineeringstrain.Itassumesthatauniform displacementoccursacrossthegaugelength[Fig.3.3(a)].However,it doesnotaccountfortheincrementalnatureofdisplacementduringthe deformationprocess.Nevertheless,theengineeringstrainisgenerallysatis- factoryformostengineeringpurposes. Similarly,forsmalldisplacements,ashearstrain,,canbedefinedas theangulardisplacementinducedbyanappliedshearstress.Theshear strain,,isgivenby ¼w=l¼tanð3:9Þ where,w,landareshownschematicallyinFig.3.3(b).Theanglehas unitsofradians.However,theshearstrainisgenerallypresentedasadimen- sionlessquantity. Itisimportanttonoteherethattheaboveequationsfortheengineer- ingstrainassumethatthestressesareuniformacrosstheareaelementsor uniformcross-sectionsthatarebeingdeformed.Theengineeringshearand axialstrainsmustbedistinguishedfromtheso-called‘‘truestrains’’which willbedescribedinChap.5. Similartostress,theengineeringstrainmayhavethreeuniaxial(" xx , " yy ," zz )andshear( xy , yz , zx )components,Fig.3.4(a).Thethree-dimen- Copyright © 2003 Marcel Dekker, Inc. sional strain components may also be perceived in terms of the simple definitions or axial and shear strains presented earlier (Fig. 3.3). However, the same shape change may also be resolved as an axial or shear strain, depending on the choice of co-ordinate system. Also, note that the displace- ment vectors along the (x, y, z) axes are usually described by displacement co-ordinates (u, v, w). The uniaxial strain, " xx , due to displacement gradient in the x direction is given by " xx ¼ u þ @u @x  dx ! À u dx ¼ @u @x ð3:10Þ FIGURE 3.3 Definitions of strain: (a)] uniaxial strain; (b) shear strain. (After Ashby and Jones, 1996. Courtesy of Butterworth-Heinemann.) Copyright © 2003 Marcel Dekker, Inc. Similarly, the shear strain due to relative displacement gradient in the y- direction is given by Fig. 3.4 to be: " xy ¼ v þ @v @x  dx ! À v dx ¼ @v @x ð3:11Þ It should be clear from the above equations that nine strain components can be defined for a generalized state of deformation at a point. These can be presented in the following strain matrix: FIGURE 3.4 Definitions of strain and rotation: (a) components of strain; (b) rotation about the x –y plane. (After Hearn, 1985—courtesy of Elsevier Science.) Copyright © 2003 Marcel Dekker, Inc. ½"¼ " xx " xy " xz " yx " yy " yz " zx " zy " zz 2 6 4 3 7 5 ¼ @u @x @u @y @u @z @v @x @v @y @v @z @w @x @w @y @w @z 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð3:12Þ Notethatsometextsmayusethetransposedversionofthedisplace- mentgradientmatrixgivenabove.Ifthisisdone,thetransposedversions shouldbemaintainedtoobtaintheresultsofthestrainmatrixthatwillbe presentedsubsequently.Also,theaboveformofthedisplacementgradient strainmatrixisoftenavoidedinproblemswherestraincanbeinducedasa resultofrotationwithoutastress.Thisisbecauseweareoftenconcerned withstrainsinducedasaresultofappliedstresses.Hence,forseveralpro- blemsinvolvingstress-induceddeformation,wesubtractouttherotation termstoobtainrelativedisplacementsthatdescribethelocalchangesin theshapeofthebody,Fig.3.4(b). The rotation strains may be obtained by considering the possible rota- tions about any of the three orthogonal axes in a Cartesian co-ordinate system. For simplicity, let us start by considering the special case of defor- mation by rotation about the z axis, i.e., deformation in the x–y plane. This is illustrated schematically in Fig. 3.4(b). The average rotation in the x–y plane is given by ! xy ¼ 1 2 @v @x À @u @y  ð3:13Þ Similarly, we may obtain expressions for ! yz and ! zx by cyclic permu- tations of the x, y and z position terms and subscripts, and the correspond- ing (u, v, w) displacement terms. This yields: ! yz ¼ 1 2 @w @y À @v @z  ð3:14Þ and ! zx ¼ 1 2 @u @z À @w @x  ð3:15Þ The components of the rotation matrix can thus be expressed in the following matrix form: Copyright © 2003 Marcel Dekker, Inc. ½! ij ¼ 0 ! xy ! xz ! yx 0 ! yz ! zx ! zy 0 2 6 4 3 7 5 ¼ 0 1 2 @u @y À @v @x  1 2 @u @z À @w @x  1 2 @v @x À @u @y  0 1 2 @v @z À @w @y  1 2 @w @x À @u @z  1 2 @w @y À @v @z  0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ð3:16Þ Subtracting Eq. (3.16) from Eq. (3.12) yield s the following matrix for the shape changes: ½" ij ¼ " xx " xy " xz " yx " yy " yz " zx " zy " zz 2 6 4 3 7 5 ¼ @u @x 1 2 @u @y þ @v @x  1 2 @u @z þ @w @x  1 2 @v @x þ @u @y  @v @y 1 2 @v @z þ @w @y  1 2 @w @x þ @u @z  1 2 @w @y þ @v @z  @w @z 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ð3:17Þ Note that the sign convention is similar to that described earlier for stress. The first suffix in the e ij term corresponds to the direction of the normal to the plane, while the second suffix corresponds to the direction of the displacement induced by the applied strain. Similarly, three shear strains are the strain components with the mixed suffixes, i.e., " xy , " yz ,and " zx . It is also important to recognize the patterns in subscripts (x, y, z)and the displacements (u, v, w) in Eqs (3.16) and (3.17). This makes it easier to remember the expressions for the possible components of strain on a three- dimensional element. Note also that the factor of 1/2 in Eq. (3.17) is often not included in several engineering problems where only a few strain com- ponents are applied. The tensorial strains (" ij terms) are then replaced by corresponding tangential shear strain terms,  ij which are given by  ij ¼ 2" ij ð3:18Þ The strain matrix for stress-induced displacements is thus given by ½ ij ¼ 0  xy  xz  yx 0  yz  zx  zy 0 2 6 4 3 7 5 ¼ 0 @u @y þ @v @x @u @z þ @w @x @v @x þ @u @y 0 @v @z þ @w @y @w @x þ @u @z @w @y þ @v @z 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð3:19Þ Copyright © 2003 Marcel Dekker, Inc. [...]... application of strain and stress BIBLIOGRAPHY Ashby, M F and D R H Jones, D R H (1996) Engineering Materials 1: An Introduction to Their Properties and Applications 2nd ed ButterworthHeinemann, Oxford, UK Courtney, T H (1990) Mechanical Behavior of Materials, McGraw-Hill, New York Dieter, G E (1986) Mechanical Metallurgy 3rd ed McGraw-Hill, New York Dowling, N (1993) Mechanical Behavior of Materials, ... space for the storage of the acquired load–displacement data In any case, electromechanical and servohydraulic testing machines are generally suitable for the testing of all classes of materials Electromechanical testing machines are particularly suitable for tests in which the loads are increased continuously (monotonic loading) or decreased continuously with time Also, stiff electromechanical testing... scientists and engineers to control the interfacial properties of composites by the careful engineering of interfacial dimensions and interfacial phases to minimize the levels of interfacial residual stress in different directions 3 .4 MOHR’S CIRCLE OF STRESS AND STRAIN Let us now consider the simple case of a two-dimensional stress state on an element in a bar of uniform rectangular cross sectional area subjected... Behavior of Materials, Prentice Hall, Englewood Cliffs, NJ Hearn, E J (1985) Mechanics of Materials vols 1 and 2, 2nd ed Pergamon Press, New York Hertzberg, R W (1996) Deformation and Fracture Mechanics of Engineering Materials 4th ed Wiley, New York McClintock, F A and A S Argon, A S (eds) (1965) Mechanical Behavior of Materials Tech Books, Fairfax, VA Copyright © 2003 Marcel Dekker, Inc ... example of a stress measurement technique 3.7.1 Strain Gauge Measurements The strain gauge is essentially a length of wire of foil that is attached to a nonconducting substrate The gauge is bonded to the surface that is being strained The resistance of the wire, R, is given by R¼ l A ð3 :40 Þ where R is the resistivity, l is the length of the wire, and A is the crosssectional area Hence, the resistance of. .. q Therefore, the amount of interference is related to the maximum shear, which is given by 1 max ¼ ðp À q Þ 2 ð3 :44 Þ The fringe patterns therefore provide a visual indication of the variations in the maximum shear stress However, in the case of stresses along a free unloaded boundary, one of the principal stresses is zero The fringe patterns therefore correspond to half of the other principal stress... the wrong signs or magnitudes of stresses The actual construction of the Mohr’s circle is a relatively simple process once the magnitudes of the radius, R, and center position, C, have been computed using Eqs (3. 24) and (3.25), respectively Note that the locus of the circle describes all the possible states of stress on the element at the point, P, for various values of  between 08 and 1808 It is... stress tensor Hence, I1 can be determined from: I1 ¼ xx þ yy þ zz ð3:28Þ Similarly, I2 , the second invariant of the stress tensor can be obtained from the algebraic sum of the cofactors of the three terms in any of the three rows or columns of the stress tensor This gives the same value of I2 , which may also be computed from 2 2 2 I2 ¼ Àxx yy À yy zz À zz xx þ xy þ yz þ zx ð3:29Þ Note the... to cause separation of bonds, in the absence of shear stress components that may induce plastic flow In any case, the hydrostatic stress can be calculated from the first invariant of the stress tensor This gives h ¼ I1 xx þ yy þ zz ¼ 3 3 ð3: 34 Hence, the hydrostatic stress is equal to the average of the leading diagonal terms in the stress tensor, Eqs (3.5) and (3. 34) A state of pure hydrostatic... 0 determinant of the ij tensor Upon substitution of the appropriate parameters, it is easy to show that J1 ¼ 0 It is particularly important to discuss the parameter J2 since it is often encountered in several problems in plasticity In fact, the conventional theory of plasticity is often to referred to as the J2 deformation theory, and plasticity is often observed to occur when J2 reaches a critical . 3 BasicDefinitionsofStressandStrain 3.1INTRODUCTION Themechanicalpropertiesofmaterialsdescribetheircharacteristic responsestoappliedloadsanddisplacements.However,mosttextsrelate themechanicalpropertiesofmaterialstostressesandstrains.Itis,therefore, importantforthereadertobecomefamiliarwiththebasicdefinitionsof stressandstrainbeforeproceedingontotheremainingchaptersofthis book.However,thewell-preparedreadermaychoosetoskip/skimthis chapter,andthenmoveontoChap.4inwhichthefundamentalsofelas- ticityareintroduced. Thebasicdefinitionsofstressandstrainarepresentedinthischapter alongwithexperimentalmethodsforthemeasurementandapplicationof strainandstress.Thechapterstartswiththerelationshipsbetweenapplied loads/displacementsandgeometrythatgiverisetothebasicdefinitionsof strainandstress.Simpleexperimentalmethodsforthemeasurementof strainandstressarethenpresentedbeforedescribingthetestmachines thatareoftenusedfortheapplicationofstrainandstressinthelaboratory. 3.2BASICDEFINITIONSOFSTRESS Theforcesappliedtothesurfaceofabodymayberesolvedintocompo- nentsthatareperpendicularorparalleltothesurface,Figs3.1(a)–3.1(c).In Copyright. invariant of the stress tensor can be obtained from the algebraic sum of the cofactors of the three terms in any of the three rows or columns of the stress tensor. This gives the same value of I 2 ,. Inc. temperature.Undersuchconditions,theresidualstresses, i ,canbe estimatedfromexpressionsoftheform:  i $EÁÁT¼E i ð 1 À 2 ÞðTÀT 0 Þð3:22Þ whereE i istheYoung’smodulusintheidirection,isthethermalexpan- sioncoefficientalongthedirectioni,subscripts1and2denotethetwo materialsincontact,Tistheactual/currenttemperature,andT 0 isthe referencestress-freetemperaturebelowwhichresidualstressescanbuild up.Abovethistemperature,residualstressesarerelaxedbyflowprocesses. Interfacialresidualstressconsiderationsareparticularlyimportantin thedesignofcompositematerials.Thisisbecauseofthelargedifferences thataretypicallyobservedbetweenthethermalexpansioncoefficientsof differentmaterials.Compositesmust,therefore,beengineeredtominimize thethermalresidualstrains/stresses.Failuretodosomayresultincracking iftheresidualstresslevelsaresufficientlylarge.Interfacialresidualstress levelsmaybecontrolledincompositesbythecarefulselectionofcomposite constituentsthathavesimilarthermalexpansioncoefficients.However,this isoftenimpossibleintherealworld.Itis,therefore,morecommonfor scientistsandengineerstocontroltheinterfacialpropertiesofcomposites bythecarefulengineeringofinterfacialdimensionsandinterfacialphasesto minimizethelevelsofinterfacialresidualstressindifferentdirections. 3.4MOHR’SCIRCLEOFSTRESSANDSTRAIN Letusnowconsiderthesimplecaseofatwo-dimensionalstressstateonan elementinabarofuniformrectangularcrosssectionalareasubjectedto uniaxialtension,[Fig.3.6].Ifwenowtakeasliceacrosstheelementatan angle,,thenormalandshearforcesontheinclinedplanecanberesolved usingstandardforcebalanceandbasictrigonometry.Thedependenceofthe stresscomponentsontheplaneangle,