CHAPTER 8 INTRODUCTION TO ADVANCED ELASTICITY THEORY 8.1.Type of stress Any element of material may be subjected to three independent types of stress.Two of these have been considered in detai previously,namely direct stresses and shear stresses, and need not be considered further here.The third type,however,has not been specifically mentioned previously although it has in fact been present in some of the loading cases considered in earlier chapters;these are the so-called body-force stresses.These body forces arise by virtue of the bulk of the material,typical examples being: (a)gravitational force due to a component's own weight:this has particular significance in civil engineering applications,e.g.dam and chimney design; (b)centrifugal force,depending on radius and speed of rotation,with particular significance in high-speed engine or turbine design; (c)magnetic field forces. In many practical engineering applications the only body force present is the gravitational one,and in the majority of cases its effect is minimal compared with the other applied forces due to mechanical loading.In such cases it is therefore normally neglected.In high- speed dynamic loading situations such as the instances quoted in (b)above,however,the centrifugal forces far exceed any other form of loading and are therefore the primary factor for consideration. Unlike direct and shear stresses,body force stresses are defined as force per unit volume, and particular note must be taken of this definition in relation to the proofs of formulae which follow. 8.2.The cartesian stress components:notation and sign convention Consider an element of material subjected to a complex stress system in three dimensions. Whatever the type of applied loading the resulting stresses can always be reduced to the nine components,i.e.three direct and six shear,shown in Fig.8.1. It will be observed that in this case a modified notation is used for the stresses.This is termed the double-suffix notation and it is particularly useful in the detailed study of stress problems since it indicates both the direction of the stress and the plane on which it acts. The first suffix gives the direction of the stress. The second suffix gives the direction of the normal of the plane on which the stress acts. Thus,for example, 220
CHAPTER 8 INTRODUCTION TO ADVANCED ELASTICITY THEORY 8.1. Type of stress Any element of material may be subjected to three independent types of stress. Two of these have been considered in detai previously, namely direct stresses and shear stresses, and need not be considered further here. The third type, however, has not been specifically mentioned previously although it has in fact been present in some of the loading cases considered in earlier chapters; these are the so-called body-force stresses. These body forces arise by virtue of the bulk of the material, typical examples being: (a) gravitational force due to a component’s own weight: this has particular significance in (b) centrifugal force, depending on radius and speed of rotation, with particular significance (c) magnetic field forces. civil engineering applications, e.g. dam and chimney design; in high-speed engine or turbine design; In many practical engineering applications the only body force present is the gravitational one, and in the majority of cases its effect is minimal compared with the other applied forces due to mechanical loading. In such cases it is therefore normally neglected. In highspeed dynamic loading situations such as the instances quoted in (b) above, however, the centrifugal forces far exceed any other form of loading and are therefore the primary factor for consideration. Unlike direct and shear stresses, body force stresses are defined as force per unit volume, and particular note must be taken of this definition in relation to the proofs of formulae which follow. 8.2. The Cartesian stress components: notation and sign convention Consider an element of material subjected to a complex stress system in three dimensions. Whatever the type of applied loading the resulting stresses can always be reduced to the nine components, i.e. three direct and six shear, shown in Fig. 8.1. It will be observed that in this case a modified notation is used for the stresses. This is termed the double-suffix notation and it is particularly useful in the detailed study of stress problems since it indicates both the direction of the stress and the plane on which it acts. Thefirst suffix gives the direction of the stress. The second suffix gives the direction of the normal of the plane on which the stress acts. Thus, for example, 220
$8.3 Introduction to Advanced Elasticity Theory 221 0码 Fig.8.1.The cartesian stress components. or is the stress in the X direction on the X facing face (i.e.a direct stress).Common suffices therefore always indicate that the stress is a direct stress.Similarly,ory is the stress in the X direction on the Y facing face (i.e.a shear stress).Mixed suffices always indicate the presence of shear stresses and thus allow the alternative symbols osy or tx.Indeed,the alternative symbol t is not strictly necessary now since the suffices indicate whether the stress o is a direct one or a shear. 8.2.1.Sign conventions (a)Direct stresses.As always,direct stresses are assumed positive when tensile and nega- tive when compressive. (b)Shear stresses.Shear stresses are taken to be positive if they act in a positive cartesian (X.Y or Z)direction whilst acting on a plane whose outer normal points also in a positive cartesian direction. Thus positive shear is assumed with direction and facing face. Alternatively',positive shear is also given with-direction and-facing face (a double negative making a positive,as usual). A careful study of Fig.8.1 will now reveal that all stresses shown are positive in nature. The cartesian stress components considered here relate to the three mutually perpendicular axes X,Y and Z.In certain loading cases,notably those involving axial symmetry,this system of components is inconvenient and an alternative set known as cylindrical components is used.These involve the variables,radius r.angle 6 and axial distance z.and will be considered in detail later. 8.3.The state of stress at a point Consider any point O within a stressed material,the nine cartesian stress components at O being known.It is now possible to determine the normal,direct and resultant stresses which act on any plane through O whatever its inclination relative to the cartesian axes. Suppose one such plane ABC has a normal n which makes angles nx.ny and nz with the YZ.XZ and XY planes respectively as shown in Figs.8.2 and 8.3.(Angles between planes
58.3 Introduction to Advanced Elasticity Theory -% Y 22 1 Fig. 8.1. The Cartesian stress components. a,, is the stress in the X direction on the X facing face (i.e. a direct stress). Common suffices therefore always indicate that the stress is a direct stress. Similarly, a,, is the stress in the X direction on the Y facing face (i.e. a shear stress). Mixed suffices always indicate the presence of shear stresses and thus allow the alternative symbols oxy or xx,. Indeed, the alternative symbol x is not strictly necessary now since the suffices indicate whether the stress a is a direct one or a shear. 8.2.1. Sign conventions (a) Direct stresses. As always, direct stresses are assumed positive when tensile and negative when compressive. (b) Shear stresses. Shear stresses are taken to be positive if they act in a positive Cartesian (X, Y or Z) direction whilst acting on a plane whose outer normal points also in a positive Cartesian direction. Thus positive shear is assumed with + direction and + facing face. Alternatively’, positive shear is also given with -- direction and - facing face (a double A careful study of Fig. 8.1 will now reveal that all stresses shown are positive in nature. The cartesian stress components considered here relate to the three mutually perpendicular axes X, Y and Z. In certain loading cases, notably those involving axial symmetry, this system of components is inconvenient and an alternative set known as cylindrical components is used. These involve the variables, radius r, angle 6 and axial distance z, and will be considered in detail later. negative making a positive, as usual). 83. The state of stress at a point Consider any point Q within a stressed material, the nine Cartesian stress components at Q being known. It is now possible to determine the normal, direct and resultant stresses which act on any plane through Q whatever its inclination relative to the Cartesian axes. Suppose one such plane ABC has a normal n which makes angles nx, ny and nz with the YZ, XZ and XY planes respectively as shown in Figs. 8.2 and 8.3. (Angles between planes
222 Mechanics of Materials 2 $8.3 B asn关 Bany y=nz Fig.8.2. A Z (normol to ABC) an B Body-force stresses Surface area ABC.△s introduced at centroid (shown removed from elemert for C simplification】 Fig.8.3.The state of stress on an inclined plane through any given point in a three-dimensional cartesian stress system. ABC and YZ are given by the angle between the normals to both planes n and x,etc.) For convenience,let the plane ABC initially be some perpendicular distance h from so that the cartesian stress components actually acting at can be shown on the sides of the tetrahdedron element ABCO so formed(Fig.8.3).In the derivation below the value of h will be reduced to zero so that the equations obtained will relate to the condition when ABC passes through O. In addition to the cartesian components,the unknown components of the stress on the plane ABC,i.e.Pn,Py and pin,are also indicated,as are the body-force field stress components which act at the centre of gravity of the tetrahedron.(To improve clarity of the diagram they are shown displaced from the element.) Since body-force stresses are defined as forces/unit volume,the components in the X,Y and Z directions are of the form F×△S年 where ASh/3 is the volume of the tetrahedron.If the area of the surface ABC,i.e.AS,is assumed small then all stresses can be taken to be uniform and the component of force in
222 Mechanics of Materials 2 $8.3 I f 7.1-12 Fig. 8.2 A 2 t ;2 I FX Body- farce stresses introduced at centroid (shown rernced frwn element for simpificatm) L AK2 AS Fig. 8.3. The state of stress bn an inclined plane through any given point in a three-dimensional Cartesian stress system. ABC and YZ are given by the angle between the normals to both planes n and x, etc.) For convenience, let the plane ABC initially be some perpendicular distance h from Q so that the Cartesian stress components actually acting at Q can be shown on the sides of the tetrahdedron element ABCQ so formed (Fig. 8.3). In the derivation below the value of h will be reduced to zero so that the equations obtained will relate to the condition when ABC passes through Q. In addition to the Cartesian components, the unknown components of the stress on the plane ABC, i.e. pxn, pvn and pzn, are also indicated, as are the body-force field stress components which act at the centre of gravity of the tetrahedron. (To improve clarity of the diagram they are shown displaced from the element.) Since body-force stresses are defined as forces/unit volume, the components in the X, Y and Z directions are of the form F x AS; where Ash13 is the volume of the tetrahedron. If the area of the surface ABC, i.e. AS, is assumed small then all stresses can be taken to be uniform and the component of force in
§8.3 Introduction to Advanced Elasticity Theory 223 the X direction due to o is given by oa△S cosnx Stress components in the other axial directions will be similar in form. Thus,for equilibrium of forces in the X direction, PiAS+F,AS=u AS cosnx+tyAS cosny+iAS cosnz As h0(i.e.plane ABC passes through o),the second term above becomes very small and can be neglected.The above equation then reduces to Pxn =Oxx cos nx txy cos ny txz cos nz (8.1) Similarly,for equilibrium of forces in the y and z directions, Pyn =oyy cos ny +tyx cos nx tyz cos nz (8.2) Pan =Ou cos nz +tur cos nx tay cos ny (8.3) The resultant stress pn on the plane ABC is then given by pm=Vp层n+p品+p) (8.4) The normal stress on is given by resolution perpendicular to the face ABC, i.e. On Pxn cos nx Pyn cos ny Pin cos nz (8.5) and,by Pythagoras'theorem (Fig.8.4),the shear stress tn is given by tn =V(pi-2) (8.6) Ammmmro n C Fig.8.4.Normal,shear and resultant stresses on the plane ABC. It is often convenient and quicker to define the line of action of the resultant stress p by the direction cosines I'=cos(Pnx)=Pxn/Pn (8.7) m'=cos(Pny)=Pyn/pn (8.8) n'=cos(Pnz)=Pin/Pn (8.9)
58.3 Introduction to Advanced Elasticity Theory 223 the X direction due to a, is given by a,AS cos nx Stress components in the other axial directions will be similar in form. Thus, for equilibrium of forces in the X direction, h 3 pxnAS + F,AS- = cxxAScosnx + t,!,AScosny + t,,AScosnz As h + 0 (i.e. plane ABC passes through Q), the second term above becomes very small and can be neglected. The above equation then reduces to pxn = a, COS nx + txy COS ny + r,, COS nz (8.1) Similarly, for equilibrium of forces in the y and z directions, (8.2) i (8.3) pYn = uyy cos ny + tyx cos nx + tyz cos nz pm = a, cos nz + tu cos nx + tzy cos ny The resultant stress pn on the plane ABC is then given by Pn Jb:n + P;n + Pzn) (8.4) The normal stress a, is given by resolution perpendicular to the face ABC, i.e. and, by Pythagoras’ theorem (Fig. 8.4), the shear stress sn is given by = pxn COS nx + Pyn COS ny + Pzn COS nz rn = J(Pi-4) Fig. 8.4. Normal, shear and resultant stresses on the plane ABC. It is often convenient and quicker to define the line of action of the resu-.ant stress pn by the direction cosines 1’ = cos(pnx) = PxnIpn (8.7)
224 Mechanics of Materials 2 $8.4 The direction of the plane ABC being given by other direction cosines I cos nx,m cos ny,n cos ny It can be shown by simple geometry that 2+m2+n2=1and('2+(m)2+(n')2=1 Equations (8.1),(8.2)and (8.3)may now be written in two alternative ways. (a)Using the common symbol o for stress and relying on the double suffix notation to discriminate between shear and direct stresses: Pxn =Oxx cos nx Oxy cos ny +oxz cos nz (8.10) Pyn dyx cos nx oyy cos ny +oyz cos nz (8.11) Pan =xx cos nx oxy cos ny +oz cos nz (8.12) In each of the above equations the first suffix is common throughout,the second suffix on the right-hand-side terms are in the order x,y,z throughout,and in each case the cosine term relates to the second suffix.These points should aid memorisation of the equations. (b)Using the direction cosine form: Prm=oxxl+Oym+ox红n (8.13) Pyn dyxl +oyym oyzn (8.14) Pin Oul ogym +oun (8.15) Memory is again aided by the notes above,but in this case it is the direction cosines,l,m and n which relate to the appropriate second suffices x,y and z. Thus,provided that the direction cosines of a plane are known,together with the cartesian stress components at some point o on the plane,the direct,normal and shear stresses on the plane at o may be determined using,firstly,eqns.(8.13-15)and,subsequently, eqns.(8.4-6). Alternatively the procedure may be carried out graphically as will be shown in $8.9. 8.4.Direct,shear and resultant stresses on an oblique plane Consider again the oblique plane ABC having direction cosines /m and n,i.e.these are the cosines of the angle between the normal to plane and the x,y,z directions. In general,the resultant stress on the plane p will not be normal to the plane and it can therefore be resolved into two alternative sets of components. (a)In the co-ordinate directions giving components px P and pn,as shown in Fig.8.5, with values given by eqns.(8.13).(8.14)and (8.15). (b)Normal and tangential to the plane as shown in Fig.8.6.giving components,of on (normal or direct stress)and n(shear stress)with values given by eqns.(8.5)and (8.6). The value of the resultant stress can thus be obtained from either of the following equations: p后=品+品 (8.16)
224 Mechanics of Materials 2 88.4 The direction of the plane ABC being given by other direction cosines I = cosnx, tn = cosny, n = cosny It can be shown by simple geometry that l2 + m2 + n2 = 1 and (l’)’ + (m’)2 + (n’)* = 1 Equations (8.1), (8.2) and (8.3) may now be written in two alternative ways. (a) Using the cummon symbol cr for stress and relying on the double suffix notation to pxn = axx cos nx + uxy cos ny + axz cos nz (8. IO) pYn = uyx cos nx + ayy cos ny + uyz cos nz (8.1 1) pm = a, cos nx + azy cos ny + a, cos nz (8.12) In each of the above equations the first suffix is common throughout, the second suffix on the right-hand-side terms are in the order x, y, z throughout, and in each case the cosine term relates to the second suffix. These points should aid memorisation of the equations. discriminate between shear and direct stresses: (b) Using the direction cosine form: Pxn = uxxl + uxym + axzn pyn = uyxl+ uyym + uyzn pzn = uzrl + u,m + u,n (8.13) (8.14) (8. IS) Memory is again aided by the notes above, but in this case it is the direction cosines, 1, rn and n which relate to the appropriate second suffices x, y and z. Thus, provided that the direction cosines of a plane are known, together with the Cartesian stress components at some point Q on the plane, the direct, normal and shear stresses on the plane at Q may be determined using, firstly, eqns. (8.13-15) and, subsequently, eqns. (8.4-6). Alternatively the procedure may be carried out graphically as will be shown in $8.9. 8.4. Direct, shear and resultant stresses on an oblique plane Consider again the oblique plane ABC having direction cosines I, m and n, i.e. these are In general, the resultant stress on the plane pII will not be normal to the plane and it can the cosines of the angle between the normal to plane and the x, y. z directions. therefore be resolved into two alternative sets of Components. (a) In the co-ordinate directions giving components pI,, , pyn and p:,, , as shown in Fig. 8.5, with values given by eqns. (8.13), (8.14) and (8.15). (b) Normal and tangential to the plane as shown in Fig. 8.6, giving components, of a,, (normal or direct stress) and r,, (shear stress) with values given by eqns. (8.5) and (8.6). The value of the resultant stress can thus be obtained from either of the following equations: p,Z=d+d (8.16)