NOTES OF INTEREST 27 Now since ds?+de2+de de?+de?+(-de1 -de2)2=2(de?+de2de1 de),(2.30) equation 2.29 becomes d-(2/3)(dei+de+de?)n (2.31) This derivation also holds where o3 0,since this is equivalent to a stress state 0=01-03,=02-3,0}=03-03=0. NOTES OF INTEREST Otto Z.Mohr(1835-1918)worked as a civil engineer designing bridges.At 32,he was appointed a professor of engineering mechanics at Stuttgart Polytecknium.Among other contributions,he also devised the graphical method of analyzing the stress at a point. He then extended Coulomb's idea that failure is caused by shear stresses into a failure criterion based on maximum shear stress,or diameter of the largest circle.He proposed the different failure stresses in tension,shear,and compression could be combined into a single diagram,in which the tangents form an envelope of safe stress combinations. This is essentially the Tresca yield criterion.It may be noted that early workers used the term"failure criteria,"which failed to distinguish between fracture and yielding. In 1868,Tresca presented two notes to the French Academy (Comptes Rendus Acad.Sci.Paris,1864,p.754).From these,Saint-Venant established the first theory of plasticity based on the assumptions that 1)plastic deformation does not change the volume of a material, 2)directions of principal stresses and principal strains coincide, 3)the maximum shear stress at a point is a constant. The Tresca criterion is also called the Guest or the "maximum shear stress" criterion. In letters to William Thompson,John Clerk Maxwell(1831-1879)proposed that "strain energy of distortion"was critical,but he never published this idea and it was forgotten.M.T.Huber,in 1904,first formulated the expression for"distortional strain energy,.” U=[1/12G][(o2-a3)2+(a3-o1)2+(a1-o22]whereU=am/6G). The same idea was independently developed by von Mises (Gottinger:Nachr. Math.Plys.,1913,p.582)for whom the criterion is generally called.It is also referred to by the names of several people who independently proposed it:Huber,Hencky,as well as Maxwell.It is also known as the "maximum distortional energy"theory and the"octahedral shear stress"theory.The first name reflects that the elastic energy in an isotropic material,associated with shear(in contrast to dilatation)is proportional to (02-03)2+(o3-1)2+(o1-02)2.The second name reflects that the shear terms, (o2-o3),(o3-1),and (o1-o2),can be represented as the edges of an octahedron in principal stress space
NOTES OF INTEREST 27 Now since dε2 1 + dε2 2 + dε2 3 = dε2 1 + dε2 2 + (−dε1 − dε2) 2 = 2 dε2 1 + dε2dε1 + dε2 3 , (2.30) equation 2.29 becomes d¯ε = (2/3) dε2 1 + dε2 2 + dε2 e 1/2 . (2.31) This derivation also holds where σ3 = 0, since this is equivalent to a stress state σ 1 = σ1 − σ3, σ 2 = σ2 − σ3, σ 3 = σ3 − σ3 = 0. NOTES OF INTEREST Otto Z. Mohr (1835–1918) worked as a civil engineer designing bridges. At 32, he was appointed a professor of engineering mechanics at Stuttgart Polytecknium. Among other contributions, he also devised the graphical method of analyzing the stress at a point. He then extended Coulomb’s idea that failure is caused by shear stresses into a failure criterion based on maximum shear stress, or diameter of the largest circle. He proposed the different failure stresses in tension, shear, and compression could be combined into a single diagram, in which the tangents form an envelope of safe stress combinations. This is essentially the Tresca yield criterion. It may be noted that early workers used the term “failure criteria,” which failed to distinguish between fracture and yielding. In 1868, Tresca presented two notes to the French Academy (Comptes Rendus Acad. Sci. Paris, 1864, p. 754). From these, Saint-Venant established the first theory of plasticity based on the assumptions that 1) plastic deformation does not change the volume of a material, 2) directions of principal stresses and principal strains coincide, 3) the maximum shear stress at a point is a constant. The Tresca criterion is also called the Guest or the “maximum shear stress” criterion. In letters to William Thompson, John Clerk Maxwell (1831–1879) proposed that “strain energy of distortion” was critical, but he never published this idea and it was forgotten. M. T. Huber, in 1904, first formulated the expression for “distortional strain energy,” U = [1/(12G)][(σ2 − σ3) 2 + (σ3 − σ1) 2 + (σ1 − σ2) 2 ] where U = σ2 yp/(6G). The same idea was independently developed by von Mises (Gottinger. Nachr ¨ . Math. Phys., 1913, p. 582) for whom the criterion is generally called. It is also referred to by the names of several people who independently proposed it: Huber, Hencky, as well as Maxwell. It is also known as the “maximum distortional energy” theory and the “octahedral shear stress” theory. The first name reflects that the elastic energy in an isotropic material, associated with shear (in contrast to dilatation) is proportional to (σ2 − σ3) 2 + (σ3 − σ1) 2 + (σ1 − σ2) 2. The second name reflects that the shear terms, (σ2 − σ3), (σ3 − σ1), and (σ1 − σ2), can be represented as the edges of an octahedron in principal stress space
28 PLASTICITY REFERENCES F.A.McClintock and A.S.Argon,Mechanical Behavior ofMaterials,Addison Wesley, 1966. W.F.Hosford,Mechanical Behavior of Materials,Cambridge University Press, 2005. PROBLEMS 2.1.a)If the principal stresses on a material with a yield stress in shear are o1= 175 MPa and o2 =350 MPa,what tensile stress o3 must be applied to cause yielding according to the Tresca criterion? b)If the stresses in a)were compressive,what tensile stress o3 must be applied to cause yielding according to the Tresca criterion? 2.2.Consider a 6-cm-diameter tube with 1-mm-thick wall with closed ends made from a metal with a tensile yield strength of 25 MPa.After applying a com- pressive load of 2,000 N to the ends,what internal pressure is required to cause yielding according to a)the Tresca criterion and b)the von Mises criterion? 2.3.Consider a 0.5-m-diameter cylindrical pressure vessel with hemispherical ends made from a metal for which k 500 MPa.If no section of the pressure vessel is to yield under an internal pressure of 35 MPa,what is the mini- mum wall thickness according to a)the Tresca criterion?b)the von Mises criterion? 2.4.A thin-wall tube is subjected to combined tensile and torsional loading.Find the relationship between the axial stress,o,the shear stress,t,and the tensile yield strength,Y,to cause yielding according to a)the Tresca criterion,and b)the von Mises criterion. 2.5.Consider a plane-strain compression test with a compressive load,Fy,a strip width,w,an indenter width,b,and a strip thickness,t.Using the von Mises criterion,find: a)as a function of y. b)as a function of o,. c)an expression for the work per volume in terms of gy and oy. d)an expression in the form of oy=f(K,Ey,n)assuming G =KEm 2.6.The following yield criterion has been proposed:"Yielding will occur when the sum of the two largest shear stresses reaches a critical value."Stated mathemat- ically, (o1-03)+(o1-02)=Cif(o1-02)>(o2-03)0r (o2-03)+(σ1-02)=Cif(σ1-02)≤(o2-03) whereo1>o2>03,C=2Y,and Y=tensile yield strength.Plot the yield locus with o3 =0 in o1 -02 space
28 PLASTICITY REFERENCES F. A. McClintock and A. S. Argon, Mechanical Behavior of Materials, Addison Wesley, 1966. W. F. Hosford, Mechanical Behavior of Materials, Cambridge University Press, 2005. PROBLEMS 2.1. a) If the principal stresses on a material with a yield stress in shear are σ1 = 175 MPa and σ2 = 350 MPa, what tensile stress σ3 must be applied to cause yielding according to the Tresca criterion? b) If the stresses in a) were compressive, what tensile stress σ3 must be applied to cause yielding according to the Tresca criterion? 2.2. Consider a 6-cm-diameter tube with 1-mm-thick wall with closed ends made from a metal with a tensile yield strength of 25 MPa. After applying a compressive load of 2,000 N to the ends, what internal pressure is required to cause yielding according to a) the Tresca criterion and b) the von Mises criterion? 2.3. Consider a 0.5-m-diameter cylindrical pressure vessel with hemispherical ends made from a metal for which k = 500 MPa. If no section of the pressure vessel is to yield under an internal pressure of 35 MPa, what is the minimum wall thickness according to a) the Tresca criterion? b) the von Mises criterion? 2.4. A thin-wall tube is subjected to combined tensile and torsional loading. Find the relationship between the axial stress, σ, the shear stress, τ, and the tensile yield strength, Y, to cause yielding according to a) the Tresca criterion, and b) the von Mises criterion. 2.5. Consider a plane-strain compression test with a compressive load, Fy, a strip width, w, an indenter width, b, and a strip thickness, t. Using the von Mises criterion, find: a) ε¯ as a function of εy. b) σ¯ as a function of σy. c) an expression for the work per volume in terms of εy and σy. d) an expression in the form of σy = f (K, εy, n) assuming ¯σ = Kε¯n. 2.6. The following yield criterion has been proposed: “Yielding will occur when the sum of the two largest shear stresses reaches a critical value.” Stated mathematically, (σ1 − σ3) + (σ1 − σ2) = C if (σ1 − σ2) > (σ2 − σ3) or (σ2 − σ3) + (σ1 − σ2) = C if (σ1 − σ2) ≤ (σ2 − σ3) where σ1 > σ2 > σ3, C = 2Y, and Y = tensile yield strength. Plot the yield locus with σ3 = 0 in σ1 − σ2 space
PROBLEMS 29 2.7.Consider the stress states 15 3 O 01 1103 3 10 0 and 3 5 0 0 05 000 a)Find om for each. b)Find the deviatoric stress in the normal directions for each. 2.8.Calculate the ratio of /max for a)pure shear,b)uniaxial tension,and c) plane-strain tension.Assume the von Mises criterion. 2.9.A material yields under a biaxial stress state,o3 =-(1/2)o1. a)Assuming the von Mises criterion,find ds/ds2. b)What is the ratio of tmax/Y at yielding? 2.10.A material is subjected to stresses in the ratio,o1,2=0.301 and o3 =-0.501. Find the ratio ofo1/Y at yielding using the a)Tresca and b)von Mises criteria. 2.11.Plot g versus 82 for a constant level of 0.10,according to a)von Mises. b)Tresca 2.12.A proposed yield criterion is that yielding will occur if the diameter of Mohr's largest circle plus half of the diameter of the second largest Mohr's circle reaches a critical value.Plot the yield locus with o3 =0 in o1-o2 space
PROBLEMS 29 2.7. Consider the stress states 15 3 0 3 10 0 0 05 and 10 3 0 3 50 0 00 . a) Find σm for each. b) Find the deviatoric stress in the normal directions for each. 2.8. Calculate the ratio of ¯σ/τmax for a) pure shear, b) uniaxial tension, and c) plane-strain tension. Assume the von Mises criterion. 2.9. A material yields under a biaxial stress state, σ3 = −(1/2)σ1. a) Assuming the von Mises criterion, find dε1/dε2. b) What is the ratio of τmax/Y at yielding? 2.10. A material is subjected to stresses in the ratio, σ1, σ2 = 0.3σ1 and σ3 = −0.5σ1. Find the ratio of σ1/Y at yielding using the a) Tresca and b) von Mises criteria. 2.11. Plot ε1 versus ε2 for a constant level of ¯ε = 0.10, according to a) von Mises. b) Tresca 2.12. A proposed yield criterion is that yielding will occur if the diameter of Mohr’s largest circle plus half of the diameter of the second largest Mohr’s circle reaches a critical value. Plot the yield locus with σ3 = 0 in σ1 − σ2 space.
3 Strain Hardening When metals are deformed plastically at temperatures lower than would cause recrys- tallization,they are said to be cold worked.Cold working increases the strength and hardness.The terms work hardening and strain hardening are used to describe this. Cold working usually decreases the ductility. Tension tests are used to measure the effect of strain on strength.Sometimes other tests,such as torsion,compression,and bulge testing are used,but the tension test is simpler and most commonly used.The major emphasis in this chapter is the dependence of yield (or flow)stress on strain. 3.1 THE TENSION TEST The temperature and strain rate influence test results.Generally,in a tension test,the strain rate is in the order of 10-2 to 10-3/s and the temperature is between 18 and 25C. These effects are discussed in Chapter 5.Measurements are made in a gauge section that is under uniaxial tension during the test. Initially the deformation is elastic and the tensile force is proportional to the elongation.Elastic deformation is recoverable.It disappears when the tensile force is removed.At higher forces the deformation is plastic,or nonrecoverable.In a ductile material,the force reaches a maximum and then decreases until fracture.Figure 3.1 is a schematic tensile load-extension curve. Stress and strain are computed from measurements in a tension test of the tensile force,F,and the elongation,Ae.The nominal or engineering stress,s,and strain,e, are defined as s=F/Ao (3.1) and e=△e/eo. (3.2) where Ao is the initial cross-sectional area,and Co is the initial gauge length.Since Ao,and Co are constants,the shapes of the s-e and F-Ae curves are identical.That 30
3 Strain Hardening When metals are deformed plastically at temperatures lower than would cause recrystallization, they are said to be cold worked. Cold working increases the strength and hardness. The terms work hardening and strain hardening are used to describe this. Cold working usually decreases the ductility. Tension tests are used to measure the effect of strain on strength. Sometimes other tests, such as torsion, compression, and bulge testing are used, but the tension test is simpler and most commonly used. The major emphasis in this chapter is the dependence of yield (or flow) stress on strain. 3.1 THE TENSION TEST The temperature and strain rate influence test results. Generally, in a tension test, the strain rate is in the order of 10−2 to 10−3/s and the temperature is between 18 and 25◦C. These effects are discussed in Chapter 5. Measurements are made in a gauge section that is under uniaxial tension during the test. Initially the deformation is elastic and the tensile force is proportional to the elongation. Elastic deformation is recoverable. It disappears when the tensile force is removed. At higher forces the deformation is plastic, or nonrecoverable. In a ductile material, the force reaches a maximum and then decreases until fracture. Figure 3.1 is a schematic tensile load-extension curve. Stress and strain are computed from measurements in a tension test of the tensile force, F, and the elongation, . The nominal or engineering stress, s, and strain, e, are defined as s = F/A0 (3.1) and e = /0. (3.2) where A0 is the initial cross-sectional area, and 0 is the initial gauge length. Since A0, and 0 are constants, the shapes of the s−e and F− curves are identical. That 30
3.1 THE TENSION TEST 31 Su IFracture Uniform plastic Yield- deformation Non-uniform n Elastic plastic behavior deformation below this level 0 0 △Rore (a) (b) Figure 3.1.(a)Load-extension and engineering stress-strain curve of a ductile metal.(b)Expansion of initial part of the curve. is why the axes of Figure 3.1 have double notation.The stress at which plastic flow begins,sy,is called the yield strength,Y,and is defined as Y=Fy/Ao=Sy. (3.3) The tensile strength or ultimate tensile strength,s,is defined as Su Fmax/A0. (3.4) Ductility is a measure of the amount that a material can be deformed before failure. Two common parameters are used to describe ductility:elongation and reduction of area. elongation=100e-to) (3.5) Co %reduction of area=100o-4) (3.6 Ao where Af and ef are the cross-sectional area and gauge length at fracture.Although standard values of 4r and er are usually used,the elongation depends on the ratio of the gauge length-to-diameter because the elongation after necking depends on the diameter and the uniform elongation depends on the gauge length.The reduction of area is much less dependent on specimen dimensions.The reduction of area in a tension test should not be confused with the reduction of area,r,in a metal working process, r=4-A) (3.7) where 4 is the cross-sectional area after forming. Figure 3.2 shows the yielding behavior of an annealed brass and a low-carbon steel.Brass is typical of most ductile metals.Yielding is so gradual that it is difficult
3.1 THE TENSION TEST 31 Figure 3.1. (a) Load–extension and engineering stress–strain curve of a ductile metal. (b) Expansion of initial part of the curve. is why the axes of Figure 3.1 have double notation. The stress at which plastic flow begins, sy, is called the yield strength, Y, and is defined as Y = Fy/A0 = sy . (3.3) The tensile strength or ultimate tensile strength, su, is defined as su = Fmax/A0. (3.4) Ductility is a measure of the amount that a material can be deformed before failure. Two common parameters are used to describe ductility: % elongation and % reduction of area. % elongation = 100 ( f − 0) 0 (3.5) % reduction of area = 100(A0 − A f ) A0 , (3.6) where A f and f are the cross-sectional area and gauge length at fracture. Although standard values of A f and f are usually used, the % elongation depends on the ratio of the gauge length-to-diameter because the elongation after necking depends on the diameter and the uniform elongation depends on the gauge length. The % reduction of area is much less dependent on specimen dimensions. The % reduction of area in a tension test should not be confused with the reduction of area, r, in a metal working process, r = (A0 − A) A0 , (3.7) where A is the cross-sectional area after forming. Figure 3.2 shows the yielding behavior of an annealed brass and a low-carbon steel. Brass is typical of most ductile metals. Yielding is so gradual that it is difficult