$8.10 Introduction to Advanced Elasticity Theory 235 (7)With centre C2(centre of the o1o3 circle)draw arc TT3 which should then pass through S if all steps have been carried out correctly and the diagram is accurate.The construction is very much easier to follow if all steps connected with points P,and T are carried out in different colours. 8.10.Construction for the state of strain on a general strain plane The construction detailed above for determination of the state of stress on a general stress plane applies equally to the determination of strains when the symbols o1,o2 and o3 are replaced by the principal strain values 1,2 and 83. Thus,having constructed the three-dimensional Mohr representation of the principal strains as described in $8.4,the general plane is located as described above and illustrated in Fig.8.15. Fig.8.15.Graphical construction for the state of strain on a general strain plane. 8.11.State of stress-tensor notation The state of stress equations for any three-dimensional system of cartesian stress compo- nents have been obtained in $8.3 as: Pxn=oa·l+Oxy·m+ox双·n pm=0x·l+oy·m+0z·n Pn=0x·l+o2y·m+og·n The cartesian stress components within this equation can then be remembered conveniently in tensor notation as: x 0灯 Ux Oyy (general stress tensor) (8.33) 0
$8.10 Introduction to Advanced Elasticity Theory 235 (7) With centre C2 (centre of the a1 a3 circle) draw arc TI T3 which should then pass through S if all steps have been carried out correctly and the diagram is accurate. The construction is very much easier to follow if all steps connected with points P, Q and T are carried out in different colours. 8.10. Construction for the state of strain on a general strain plane The construction detailed above for determination of the state of stress on a general stress plane applies equally to the determination of strains when the symbols a~, a2 and 03 are replaced by the principal strain values EI , E:! and ~3. Thus, having constructed the three-dimensional Mohr representation of the principal strains as described in $8.4, the general plane is located as described above and illustrated in Fig. 8.15. Fig. 8.15. Graphical construction for the state of strain on a general strain plane. 8.11. State of stress-tensor notation The state of stress equations for any three-dimensional system of Cartesian stress components have been obtained in $8.3 as: The Cartesian stress components within this equation can then be remembered conveniently in tensor notation as: uxx uxy a, (general stress tensor) (8.33)
236 Mechanics of Materials 2 $8.12 For a principal stress system,i.e.no shear,this reduces to: 可1 0 01 0 02 (principal stress tensor) (8.34) 0 03 and a special case of this is the so-called "hydrostatic"stress system with equal principal stresses in all three directions,i.e.o=o2=o3=,and the tensor becomes: To 007 0 (hydrostatic stress tensor) (8.35) L00 G As shown in $23.16 it is often convenient to divide a general stress into two parts,one due to a hydrostatic stress=(+02+3),the other due to shearing deformations. Another convenient tensor notation is therefore that for pure shear,ie oxx=oyy=o2=0 giving the tensor: 0 Oxy 0y2 (pure shear tensor) (8.36) The general stress tensor(8.33)is then the combination of the hydrostatic stress tensor and the pure shear tensor. i.e.General three-dimensional stress state =hydrostatic stress state pure shear state. This approach is utilised in other sections of this text,notably:$8.16,$8.19 and $8.20. It therefore follows that an alternative method of presentation of a pure shear state of stress is,in tensor form: 「(01-6) 0 0 0 (02-) 0 (8.37) 0 0 (a3-)」 N.B.:It can be shown that the condition for a state of stress to be one of pure shear is that the first stress invariant is zero. i.e. I1=0x+0y+oz=0(see8.15) 8.12.The stress equations of equilibrium (a)In cartesian components In all the previous work on complex stress systems it has been assumed that the stresses acting on the sides of any element are constant.In many cases,however,a general system of direct,shear and body forces,as encountered in practical engineering applications,will produce stresses of variable magnitude throughout a component.Despite this,however, the distribution of these stresses must always be such that overall equilibrium both of the component,and of any element of material within the component,is maintained,and it is a consideration of the conditions necessary to produce this equilibrium which produces the so-called stress equations of equilibrium. Consider,therefore,a body subjected to such a general system of forces resulting in the cartesian stress components described in $8.2 together with the body-force stresses Fr
236 Mechanics of Materials 2 $8.12 For a principal stress system, i.e. no shear, this reduces to: 01 0 0 (principal stress tensor) (8.34) and a special case of this is the so-called “hydrostatic” stress system with equal principal stresses in all three directions, i.e. (TI = a2 = a3 = 5, and the tensor becomes: [ i] (hydrostatic stress tensor) (8.35) OOZ As shown in $23.16 it is often convenient to divide a general stress into two parts, one = $(al + 02 + a3), the other due to shearing deformations. Another convenient tensor notation is therefore that for pure shear, ie a, = aVy = a,? = 0 [ : u- ] (pure shear tensor) (8.36) The general stress tensor (8.33) is then the combination of the hydrostatic stress tensor and the pure shear tensor. i.e. General three-dimensional stress state = hydrostatic stress state + pure shear state. This approach is utilised in other sections of this text, notably: 58.16, 58.19 and 58.20. stress is, in tensor form: due to a hydrostatic stress giving the tensor: 0 axy ax2 It therefore follows that an alternative method of presentation of a pure shear state of (8.37) N.B.: It can be shown that the condition for a state of stress to be one of pure shear is that the first stress invariant is zero. i.e. 1, = a, +a,, +a,, = 0 (see 8.15) 8.12. The stress equations of equilibrium (a) In Cartesian components In all the previous work on complex stress systems it has been assumed that the stresses acting on the sides of any element are constant. In many cases, however, a general system of direct, shear and body forces, as encountered in practical engineering applications, will produce stresses of variable magnitude throughout a component. Despite this, however, the distribution of these stresses must always be such that overall equilibrium both of the component, and of any element of material within the component, is maintained, and it is a consideration of the conditions necessary to produce this equilibrium which produces the so-called stress equations of equilibrium. Consider, therefore, a body subjected to such a general system of forces resulting in the Cartesian stress components described in $8.2 together with the body-force stresses F
$8.12 Introduction to Advanced Elasticity Theory 237 Fy and F:.The element shown in Fig.8.16 then displays,for simplicity,only the stress components in the X direction together with the body-force stress components.It must be realised,however,that similar components act in the Y and Z directions and these must be considered when deriving equations for equilibrium in these directions:they,of course,have no effect on equilibrium in the X direction. dy dx az xy dy) y (Oo+ ax Oxx dx) Fig.8.16.Small element showing body force stresses and other stresses in the X direction only. It will be observed that on each pair of opposite faces the stress changes in magnitude in the following manner, e.g. stress on one face =ox stress on opposite face =ox+change in stress =xr+rate of change x distance between faces Now the rate of change of xx with x is given by ao/ax,partial differentials being used since ox may well be a function of y and z as well as of x. Therefore dog dx stress on opposite face=ax Multiplying by the area dy dz of the face on which this stress acts produces the force in the X direction. Thus,for equilibrium of forces in the X direction, (The body-force term being defined as a stress per unit volume is multiplied by the volume (dx dy dz)to obtain the corresponding force.)
$8.12 Introduction to Advanced Elasticity Theory 237 Fy and Fz. The element shown in Fig. 8.16 then displays, for simplicity, only the stress components in the X direction together with the body-force stress components, It must be realised, however, that similar components act in the Y and 2 directions and these must be considered when deriving equations for equilibrium in these directions: they, of course, have no effect on equilibrium in the X direction. X J Fig. 8.16. Small element showing body force stresses and other stresses in the X direction only. It will be observed that on each pair of opposite faces the stress changes in magnitude in the following manner, e.g. stress on one face = a, stress on opposite face = a, + change in stress = a, + rate of change x distance between faces Now the rate of change of CT,, with x is given by ao,/ax, partial differentials being used since a, may well be a function of y and z as well as of x. Therefore aaxx ax stress on opposite face = a,, + -dx Multiplying by the area dy dz of the face on which this stress acts produces the force in the X direction. Thus, for equilibrium of forces in the X direction, a 1 a dxdy+F,dxdydz=O (The body-force term being defined as a stress per unit volume is multiplied by the volume (dx dy dz) to obtain the corresponding force.)
238 Mechanics of Materials 2 $8.12 Dividing through by dxdydz and simplifying, doxx+十Fx=0】 ax ay Similarly,for equilibrium in the Y direction, (8.38) ax Oy +Fy=0 and in the Z direction, +g+F:=0 ax ay az these equations being termed the general stress equations of equilibrium. Bearing in mind the comments of $8.2,the symbol r in the above equations may be replaced by o,the mixed suffix denoting the fact that it is a shear stress,and the above equations can be remembered quite easily using a similar procedure to that used in $8.2 based on the suffices,i.e.first suffices and body-force terms are constant for each horizontal row and in the normal order x,y and z. X Z X y XZ 3x +Fx=0 Y x 影 z +Fy=0 EX zy 就 ay 疑 +Fz=0 The above equations have been derived by consideration of equilibrium of forces only, and this does not represent a complete check on the equilibrium of the system.This can only be achieved by an additional consideration of the moments of the forces which must also be in balance. Consider,therefore,the element shown in Fig.8.17 which,again for simplicity,shows only the stresses which produce moments about the Y axis.For convenience the origin of the cartesian coordinates has in this case been chosen to coincide with the centroid of the element.In this way the direct stress and body-force stress terms will be eliminated since the forces produced by these will have no moment about axes through the centroid. It has been assumed that shear stresses ty.ty and t act on the coordinate planes passing through G so that they will each increase and decrease on either side of these planes as described above. Thus,for equilibrium of moments about the Y axis, 8、dz1 dz dz Dividing through by (dx dy dz)and simplifying,this reduces to T红=tx
238 Mechanics of Materials 2 $8.12 Dividing through by dxdy dz and simplifying, I Similarly, for equilibrium in the Y direction, (8.38) I and in the Z direction, these equations being termed the general stress equations of equilibrium. Bearing in mind the comments of $8.2, the symbol t in the above equations may be replaced by o, the mixed suffix denoting the fact that it is a shear stress, and the above equations can be remembered quite easily using a similar procedure to that used in $8.2 based on the suffices, i.e. first suffices and body-force terms are constant for each horizontal row and in the normal order x, y and z. The above equations have been derived by consideration of equilibrium of forces only, and this does not represent a complete check on the equilibrium of the system. This can only be achieved by an additional consideration of the moments of the forces which must also be in balance. Consider, therefore, the element shown in Fig. 8.17 which, again for simplicity, shows only the stresses which produce moments about the Y axis. For convenience the origin of the Cartesian coordinates has in this case been chosen to coincide with the centroid of the element. In this way the direct stress and body-force stress terms will be eliminated since the forces produced by these will have no moment about axes through the centroid. It has been assumed that shear stresses tx,,, tyz and txz act on the coordinate planes passing through G so that they will each increase and decrease on either side of these planes as described above. Thus, for equilibrium of moments about the Y axis, dx - r, + -(t,)- dydz- - tu - -(tu)- dy&- = 0 [ ax a "1 2 dx 2 [ ax a "1 2 2 Dividing through by (dw dy dz) and simplifying, this reduces to txz =
$8.12 Introduction to Advanced Elasticity Theory 239 dx dz. dz'x dx G dz dx. dz Fig.8.17.Element showing only stresses which contribute to a moment about the Y axis. Similarly,by consideration of the equilibrium of moments about the X and Z axes, ℃y=z try tyx Thus the shears and complementary shears on adjacent faces are equal as in the simple two-dimensional case.The nine cartesian stress components thus reduce to six independent values, Ox Oxy 0x Oxx Txy ie. Oyy or Ovy Ozy 02 Tzy 02 (b)In cylindrical coordinates The equations of equilibrium derived above in cartesian components are very useful for components and stress systems which can easily be referred to a set of three mutually perpen- dicular axes.There are many cases,however,e.g.those components with axial symmetry, where other coordinate axes prove far more convenient.One such set of axes is the cylindrical coordinate system with variables r,6 and z as shown in Fig.8.18. Consider,therefore,the equilibrium in a radial direction of the element shown in Fig.8.19(a).Again,for simplicity,only those stresses which produce force components in this direction are indicated.It must be observed,however,that in this case the oae terms will also produce components in the radial direction as shown by Fig.8.19(b).The body-force stress components are denoted by FR.Fz and Fe. Therefore,resolving forces radially, s do -ore dr dzcos +[(oe+)-小(+)o -ooo dr dzsin 2
$8.12 Introduction to Advanced Elasticity Theory 239 Fig. 8.17. Element showing only stresses which contribute to a moment about the Y axis. Similarly, by consideration of the equilibrium of moments about the X and Z axes, tzr = ZYZ Gy = ryx Thus the shears and complementary shears on adjacent faces are equal as in the simple two-dimensional case. The nine Cartesian stress components thus reduce to sir independent values, 1.e. (6) In cylindrical coordinates The equations of equilibrium derived above in Cartesian components are very useful for components and stress systems which can easily be referred to a set of three mutually perpendicular axes. There are many cases, however, e.g. those components with axial symmetry, where other coordinate axes prove far more convenient. One such set of axes is the cylindrical coordinate system with variables r, 8 and z as shown in Fig. 8.18. Consider, therefore, the equilibrium in a radial direction of the element shown in Fig. 8.19(a). Again, for simplicity, only those stresses which produce force components in this direction are indicated. It must be observed, however, that in this case the terms will also produce components in the radial direction as shown by Fig. 8.19(b). The body-force stress components are denoted by FR, FZ and FQ. Therefore, resolving forces radially, de 2 drdzsin - + FR rdrdedz = 0