230 Mechanics of Materials 2 $8.7 Fig.8.11.Mohr circle equivalent procedure to that of Fig.8.10. 8.7.The combined Mohr diagram for three-dimensional stress and strain systems Consider any three-dimensional stress system with principal stresses o1,o2 and o3(all assumed tensile).Principal strains are then related to the principal stresses as follows: 1=E(a1 -vo2 -v3),etc. Ee1=01-v(o2+03) =01-(01+02+3)+G1 (1) Now the hydrostatic,volumetric or mean stress o is defined as ō=(1+02+03) Therefore substituting in (1), EE1=01(1+v)-3vo (2) But the volumetric stress o may also be written in terms of the bulk modulus, volumetric stress ie. bulk modulus K= volumetric strain and volumetric strain sum of the three linear strains =E1+e2+E3=△ 水-⊙ but E=3K(1-2) E G=△K=△ 3(1-2v)
230 Mechanics of Materials 2 $8.7 C M A Fig. 8.1 1. Mohr circle equivalent procedure to that of Fig. 8.10. 8.7. The combined Mohr diagram for three-dimensional stress and strain systems Consider any three-dimensional stress system with principal stresses 01, a2 and a3 (all assumed tensile). Principal strains are then related to the principal stresses as follows: 1 E ~1 = -(a1 - ua2 - ua3), etc. EEI 1 CTI - 402 +~3) = 01 - u(a1 + 02 + a3) + Val Now the hydrostatic, volumetric or mean stress 5 is defined as - 0 = + 02 + a3) Therefore substituting in (l), EEI = al(l + U) - 3~5 But the volumetric stress 5 may also be written in terms of the bulk modulus, i.e. volumetric stress volumetric strain bulk modulus K = and volumetric strain = sum of the three linear strains = E1 4- E2 -I- &3 = A .. but - a K=- A E = 3K(l - 2~) E- .. - I4 a=AK=A 3(1 - 2~)
§8.7 Introduction to Advanced Elasticity Theory 231 Substituting in(2), 3v△E E81=01(1+)- 3(1-2v) and,since E=2G(1+v), v△2G(1+v) 2G(1+v)E1=01(1+v)- (1-2v) 01=2G e1+0-20J But,mean strain E=(e1+e2+e3)=3△ 1=2G 61+1-2] (8.30) Alternatively,re-writing eqn.(8.16)in terms of g1, 01 3v e1= -E 2G-(1-2 But E=3=3 But E=2G(1+v)=3K(1-2v) ie. 3K=2G (1+) (1-2v) (1-2) E= 2G(1+v) ō(1-2v) e1= 01 2G+ 2G(1+) 1「 3va ie. 1= 2G1-1+ (8.31) In the above derivation the cartesian stresses o,oy and o could have been used in place of the principal stresses o,o2 and o3 to yield more general expressions but of identicial form.It therefore follows that the stress and associated strain in any given direction within a complex three-dimensional stress system is given by egns.(8.30)and (8.31)which must satisfy the three-dimensional Mohr's circle construction. Comparison of eqns.(8.30)and (8.31)indicates that 3v 2Ge1=1-(1+) Thus,having constructed the three-dimensional Mohr's stress circle representations,the equivalent strain values may be obtained simply by reference to a new axis displaced a distance (3v/(1+v))o as shown in Fig.8.12 bringing the new axis origin to O
$8.7 Introduction to Advanced Elasticity Theory 23 1 Substituting in (2), 3uAE EEI = a1(l + v) - 3(1 - 2~) and, since E = 2G(1 + v), But, mean strain .. Alternatively, re-writing eqn. (8.16) in terms of EI, 01 &I=------- 2G (1 - 2~) 3u - E But But i.e. .. .. i.e. - A5 3 3K E=--=- E = 2G(1+ U) = 3K(l - 2~) (1 + v) 3K = 2G- (1 -2u) - a(1 - 2u) 2G(1+ u) 01 a (1 - 2u) - + -- 2G 2G (1+u) E= (8.30) (8.31) In t.c above derivation the Cartesian stresses a, aYY an^ nZz could ..ave been used in place of the principal stresses a1, a2 and c73 to yield more general expressions but of identicial form. It therefore follows that the stress and associated strain in any given direction within a complex three-dimensional stress system is given by eqns. (8.30) and (8.31) which must satisfy the three-dimensional Mohr’s circle construction. Comparison of eqns. (8.30) and (8.31) indicates that Thus, having constructed the three-dimensional Mohr’s stress circle representations, the equivalent strain values may be obtained simply by reference to a new axis displaced a distance (3u/(l + u))5 as shown in Fig. 8.12 bringing the new axis origin to 0
232 Mechanics of Materials 2 $8.8 Strain origin 4一02 00g 02 2GE 3u (1+四ō +20e2 2Ge1 85 2 Fig.8.12.The "combined Mohr diagram"for three-dimensional stress and strain systems. Distances from the new axis to any principal stress value,e.g.o1,will then be 2G times the corresponding e principal strain value, i.e. Oo1÷2G=E1 Thus the same circle construction will apply for both stresses and strains provided that: (the shear ifetaeto the of he hear (b)a scale factor of 2G,[=E/(1+v)],is applied to measurements from the new axis. 8.8.Application of the combined circle to two-dimensional stress systems The procedure of $14.13 uses a common set of axes and a common centre for Mohr's stress and strain circles,each having an appropriate radius and scale factor.An alternative procedure utilises the combined circle approach introduced above where a single circle can be used in association with two different origins to obtain both stress and strain values. As in the above section the relationship between the stress and strain scales is stress scale E =2G strain scale (1+v) This is in fact the condition for both the stress and strain circles to have the same radius and should not be confused with the condition required in $14.13f of the alternative approach for the two circles to be concentric,when the ratio of scales is E/(I-v). EJ.Hearn.Mechanics of Materials /Butterworth-Heinemann.1997. For equal radii of both the stress and strain circles (01-02) (e1-E2) 2x stress scale2x strain scale stress scale (a1-02)(1-02)E E strain scale (eI-62)(a1 -2)(1+)(1v)
232 Mechanics of Materials 2 $8.8 Fig. 8.12. The “combined Mohr diagram” for three-dimensional stress and strain systems. Distances from the new axis to any principal stress value, e.g. (TI, will then be 2G times the corresponding principal strain value, i.e. 0’01 t 2G = E] Thus the same circle construction will apply for both stresses and strains provided that: 3v - (a) the shear strain axis is offset a distance ___ (T to the right of the shear stress axis; (b) a scale factor of 2G, [= E/(1 + u)], is applied to measurements from the new axis. (1 + v) 8.8. Application of the combined circle to two-dimensional stress systems The procedure of $14.13? uses a common set of axes and a common centre for Mohr’s stress and strain circles, each having an appropriate radius and scale factor. An alternative procedure utilises the combined circle approach introduced above where a single circle can be used in association with two different origins to obtain both stress and strain values. As in the above section the relationship between the stress and strain scales is stress scale E strain scale (1 + v) - - 2G This is in fact the condition for both the stress and strain circles to have the same radius$ and should not be confused with the condition required in §14.13? of the alternative approach for the two circles to be concentric, when the ratio of scales is E/(1 - u). t E.J. Hearn, Mechanics of Materiuls I, Butterworth-Heinemann, 1997. For equal radii of both the stress and strain circles (@I -02) - (El -E?) - 2 x stress scale 2 x strain scale stress scale (ui - a2) (01 - uz) E E strain scale (el - e2) (ui - 02) (I + u) (1 + v) - - -
s8.8 Introduction to Advanced Elasticity Theory 233 Strom origin g 2y (+2) y12 Stress origin Fig.8.13.Combined Mohr diagram for two-dimensional stress and strain systems. With reference to Fig.8.13 the two origins must then be positioned such that (01+02) OA= 2 x strees scale (e1+e2) O'A= 2 x strain scale OA (01+02) strain scale O'A (1+82) stress scale (o1+02),.(1+v) (E1+E2) E 1 But 81=Eo1-o2） 1 e2=E(o-o4) 1 e1+82=Ea1+m1-W 0A=o+2)E1+四=1+” OA-(a1+2)(1-)E (1-v) Thus the distance between the two origins is given by 00=0A-0A=0A- -20A (1+v) (o1+2) 2 [1- (1-) (1+v) (o1+02)2) 2(1+v) (01+02) (1+v) 2v 6 =1+) (8.32)
$8.8 Introduction to Advanced Elasticity Theory Fig. 8.13. Combined Mohr diagram for two-dimensional stress and strain systems. With reference to Fig. 8.13 the two origins must then be positioned such that (01 + 02) 2 x strees scale OA = (El +E2) 2 x strain scale O‘A = .. OA - (01 + 02) strain scale O’A (&I + ~2) stress scale (01 +02> (1 + u> (El + E2) E -- - - x- 1 E But E] = -(01 - ua2) Thus the distance between the two origins is given by 233 (8.32)
234 Mechanics of Materials 2 $8.9 where is the mean stress in the two-dimensional stress system=(+02)=position of centre of stress circle. The relationship is thus identical in form to the three-dimensional equivalent with 2 replacing 3 for the two-dimensional system. Again,therefore,the single-circle construction applies for both stresses and strain provided that the axes are offset by the appropriate amount and a scale factor for strains of 2G is applied. 8.9.Graphical construction for the state of stress at a point The following procedure enables the determination of the direct(o)and shear(n)stresses at any point on a plane whose direction cosines are known and,in particular,on the octa- hedral planes (see $8.19). The construction procedure for Mohr's circle representation of three-dimensional stress systems has been introduced in $8.4.Thus,for a given state of stress producing principal stress o1,o2 and 03,Mohr's circles are as shown in Fig.8.8. For a given plane S characterised by direction cosines I,m and n the remainder of the required construction proceeds as follows (Fig.8.14).(Only half the complete Mohr's circle representation is shown since this is sufficient for the execution of the construction procedure.) 0 Fig.8.14.Graphical construction for the state of stress on a general stress plane (1)Set off angle a=cos-/from the vertical at o to cut the circles in 2 and 23. (2)With centre C(centre of o2,o3 circle)draw arc Q203. (3)Set off angle y=cos-n from the vertical at o3 to cut the circles at P and P2. (4)With centre C3 (centre of a,oz circle)draw arc P1P2. (5)The position S representing the required plane is then given by the point where the two arcs Q203 and PiP2 intersect.The stresses on this plane are then os and ts as shown.Careful study of the above'construction procedure shows that the suffices of points considered in each step always complete the grouping 1,2,3.This should aid memorisation of the procedure. (6)As a check on the accuracy of the drawing,set off angles B=cos-1 m on either side of the vertical through o2 to cut the o2o3 circle in T and the o1o2 circle in T3
234 Mechanics of Materials 2 $8.9 where 5 is the mean stress in the two-dimensional stress system = :(a1 + 02) = position of centre of stress circle. The relationship is thus identical in form to the three-dimensional equivalent with 2 replacing 3 for the two-dimensional system. Again, therefore, the single-circle construction applies for both stresses and strain provided that the axes are offset by the appropriate amount and a scale factor for strains of 2G is applied. 8.9. Graphical construction for the state of stress at a point The following procedure enables the determination of the direct (a,) and shear (rn) stresses at any point on a plane whose direction cosines are known and, in particular, on the octahedral planes (see 58.19). The construction procedure for Mohr’s circle representation of three-dimensional stress systems has been introduced in 58.4. Thus, for a given state of stress producing principal stress a’, a;? and a3, Mohr’s circles are as shown in Fig. 8.8. For a given plane S characterised by direction cosines I, m and n the remainder of the required construction proceeds as follows (Fig. 8.14). (Only half the complete Mohr’s circle representation is shown since this is sufficient for the execution of the construction procedure.) Fig. 8.14. Graphical construction for the state of stress on a general stress plane. (1) Set off angle a! = cos-’ 1 from the vertical at a’ to cut the circles in Q2 and Q3. (2) With centre CI (centre of a2, a3 circle) draw arc QzQ3. (3) Set off angle y = cos-’ n from the vertical at a3 to cut the circles at PI and P2. (4) With centre C3 (centre of 01, a2 circle) draw arc P1P2. (5) The position S representing the required plane is then given by the point where the two arcs Q2Q3 and PIP^ intersect. The stresses on this plane are then a, and r, as shown. Careful study of the above’construction procedure shows that the suffices of points considered in each step always complete the grouping 1, 2, 3. This should aid memorisation of the procedure. (6) As a check on the accuracy of the drawing, set off angles /? = cos-’ m on either side of the vertical through a2 to cut the 0253 circle in TI and the 0102 circle in T3