The effective stress-strain function is defined such that the incremental work per volume is dw=σ1dε1+σ2dε2+σ3dε3=σ¯d¯ε. For simplicity consider a stress state with σ3=0.Then
σ¯d¯ε=σ1dε1+σ2dε2=σ1dε1(1+αρ), (2.26)
NOTES OF INTEREST 27
whereα=σ2/σ1andρ=dε2/dε1.Then
d¯ε=dε1(σ1/σ¯)(1+αρ). (2.27) From the flow rules, ρ=dε2dε1=[σ2−(1/2)σ1]/[σ1−(1/2)σ2]=(2α−1)/(2− α) or
α=(2ρ+1)/(2+ρ). (2.28)
Combining equations2.27and2.28,
d¯ε=dε1(σ1/σ¯)[2(1+ρ+ρ2)/(2+ρ)]. (2.29) Withσ3 =0,the von Mises expression for ¯σ is
σ¯ =
σ12+σ22−σ1σ2 1/2 =(1−α+α2)1/2σ1. (2.30) Combining equations2.28and2.29,
σ1
σ¯ =
2+ρ
√3
(1+ρ+ρ2)1/2. (2.31)
Sinceρ=dε2/dε1,
d¯ε= 2
√3 dε12+dε1dε2+dε22
1/2. (2.32)
Now since with constant value
dε12+dε22+dε32=0 dε12+dε22+(−dε1−dε2)2=2
dε12+dε2dε1+dε22 , (2.33) equation2.32becomes
d¯ε= (2/3)
dε12+dε22+dε23
1/2. (2.34)
This derivation also holds whereσ3 =0,since this is equivalent to a stress state σ1 =σ1−σ3, σ2=σ2−σ3, σ3=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 Polytechnikum. 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.∗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 theGuestor 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.” The same idea was independently developed by von Mises†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.
REFERENCES
W. F. Hosford,Mechanical Behavior of Materials, Cambridge University Press, 2005.
F. A. McClintock and A. S. Argon,Mechanical Behavior of Materials, Addison-Wesley, 1966.
PROBLEMS
2.1. a) If the principal stresses on a material with a yield stress in shear areσ1 = 175 MPa andσ2=350 MPa what tensile stressσ3must be applied to cause yielding according to the Tresca criterion?
b) If the stresses in (a) were compressive, what tensile stressσ3must 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. Apply a compressive load of 2000 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 whichk=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 stressσ, the shear stressτ, and the tensile
∗ Tresca,Comptes Rendus Acad. Sci. Paris(1864), p. 754.
† von Mises,G¨ottinger Nachr.Math. Phys.(1913), p. 582.
PROBLEMS 29
yield strengthYto cause yielding according to (a) the Tresca criterion, (b) the von Mises criterion.
2.5. Consider a plane-strain compression test with a compressive loadFy, a strip width w, an indenter width b, and a strip thickness t. Using the von Mises criterion, find
a) ε¯ as a function ofεy, b) σ¯ as a function ofσy,
c) an expression for the work per volume in terms ofεyandσy, d) an expression in the form ofσy= 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 mathemat- ically
(σ1−σ3)+(σ1−σ2)=C if (σ1−σ2)>(σ2−σ3) or (σ2−σ3)+(σ1−σ2)=C if (σ1−σ2)≤(σ2−σ3) whereσ1 > σ2> σ3,C =2Y, andY=tensile yield strength.
2.7. Consider the stress states
15 3 0
3 10 0
0 0 5
and
10 3 0
3 5 0
0 0 0
. a) Findσmfor each.
b) Find the deviatoric stress in the normal directions for each c) What is the sum of the deviatoric stresses for each?
2.8. A thin wall tube with closed ends is made from steel with a yield strength of 250 MPa. The tube is 2 m long with a wall thickness of 2 mm and a diameter of 8 cm. In service it will experience an axial load of 8 kN and a torque of 2.7 Nm. What is the maximum internal pressure it can withstand without yielding according to (a) the Tresca criterion and (b) the von Mises criterion?
2.9. Calculate the ratio of ¯σ /τmax for (a) pure shear, (b) uniaxial tension, and (c) plane-strain tension. Assume the von Mises criterion.
2.10. A material yields under a biaxial stress state,σ3= −(1/2)σ1. a) Assuming the von Mises criterion, finddε1/dε2.
b) What is the ratio ofτmax/Y at yielding?
2.11. A material is subjected to stresses in the ratioσ1, σ2=0.3σ1, andσ3= −0.5σ1. Find the ratio ofσ1/Y at yielding using (a) the Tresca criterion and (b) the von Mises criterion.
2.12. A proposed yield criterion is that yielding will occur when the diameter of the largest Mohr’s circle plus half the diameter of the second largest Mohr’s circle reaches a critical value. Plot the yield locus inσ1vs. σ2inσ3=0 space.
2.13. Make plot of ε1 versus ε2 for a constant level of ¯ε = 0.10 according to (a) von Mises and (b) Tresca.
3 Strain Hardening
When metals are deformed plastically at temperatures lower than would cause recrys- tallization, they are said to becold worked. Cold working increases the strength and hardness. The termswork hardeningandstrain 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.