Metal Machining - Theory and Applications Episode 2 Part 8 pptx

However, in plane strain conditions, and if the work hardening of the material is negligible, the integration of the equilibrium and compatibility equations, under the constraint of the

∂s xx ∂sxy ∂s xz ∂s yx ∂s yy ∂s yz ∂s zx ∂s zy ∂s zz

—— + —— + —— = —— + —— + —— = —— + —— + —— = 0

}

—— + —— + —— = 0

(A1.15)

where u˘, v˘ and w˘ are the x, y and z components of the material’s velocity The general

three-dimensional situation is complicated However, in plane strain conditions, and if the work hardening of the material is negligible, the integration of the equilibrium and compatibility equations, under the constraint of the constitutive equations, is simplified by describing the stresses and velocities not in a Cartesian coordinate system but in a curvi-linear system that is everywhere tangential to the maximum shear stress directions The net

of curvilinear maximum shear stress lines is known as the slip-line field Determining the shape of the net for any application and then the stresses and velocities in the field is achieved through slip-line field theory This theory is now outlined

A1.2.1 Constitutive laws for a non-hardening material in plane strain

When the strain in one direction, say the z-direction, is zero, from the flow rules (equation (A1.13)) the deviatoric stresses in that direction are also zero Then s zz = sm= (1/2)(s xx+

s yy) The yield criterion, equation (A1.12), and flow rules, equation (A1.13), become

(s xx – s yy)2 + 4s2

xy = 4k2

1/2(s xx – s yy) 1/2(s xx – s yy) s xy When the material is non-hardening, the shear yield stress k is independent of strain If,

in a plastic region, the x, y directions are chosen locally to coincide with the maximum shear stress directions, s xx becomes equal to s yy (and equal to sm), so (s xx – s yy) = 0 Equation (A1.16) becomes a statement that (i) the maximum shear stress is constant throughout the plastic region and (ii) there is no extension along maximum shear stress directions The consequences of these statements for stress and velocity variations throughout a plastic region are developed in the next two subsections

A1.2.2 Stress relations in a slip-line field

Figure A1.4(a) shows a network of slip-lines in a plastic field The pressure p (= –sm) and

the shear stress k is shown at a general point O in the field The variation of pressure

throughout the field may be found by integrating the equilibrium equations along the slip-lines How this is done, and some consequences for the shape of the field, are now described

First, the two families of lines, orthogonal to each other, are labelled a and b Which is labelled a and which is b is chosen, by convention, so that, if a and b are regarded as a

right-handed coordinate system, the direction of the largest principal stress lies in the first

quadrant (This means that the shear stresses k are positive as shown in the figure.) The

Perfectly plastic material in plane strain 333

direction of an a line at any point is described by its anticlockwise rotation from a datum direction, for example f from the +X direction.

By stress analysis (Figure A1.4(b)), the stresses p and k at O have (x,y) components (s xx,

s yy , s xy)

s xx = –p – k sin2f; s yy = –p + k sin2f; s xy = k cos2f (A1.17) Substituting these into the equilibrium equations

∂s xx ∂s xy ∂s yy ∂s xy

gives, after noting that k is a constant,

– —— – 2k cos 2f —— – 2k sin2f —— = 0

– —— + 2k cos 2f —— – 2k sin2f —— = 0

If the direction of X is chosen so that f = 0, that is so that the a slip-line is tangential to X, sin 2f = 0 and cos 2f = 1 and

—— (p + 2kf) = 0; —— (p – 2kf) = 0 (A1.18c)

or

p + 2kf = constant along an a-line; p – 2kf = constant along a b-line.

(A1.18d)

If the geometry of the slip-line field and the pressure at any one point is known, the pressure at any other point can be calculated Equation (A1.19) relates, for the example of

Figure A1.4, pressures along the a-lines AB and DC, and along the b-lines AD and BC

Fig A1.4 (a) A slip-line net and (b) free body equilibrium diagrams around O

pA+ 2kfA= pB+ 2kfB; pC+ 2kfC= pD+ 2kfD

(A1.19)

pA+ 2kfA= pD– 2kfD; pB – 2kfB= pC+ 2kfC }

Geometry of the field

The inclinations fA, fB, fCand fDare not independent The pressure pCat C may be calcu-lated from that at A in two ways from equations (A1.19), either along the path ABC or

ADC For pCto be single valued

fB– fA = fC– fD; fD– fA = fC– fB (A1.20) Figure A1.5(a) gives some common examples of curvilinear nets that satisfy this condi-tion: a grid of straight lines in which the pressure is constant, a centred fan and a net constructed on two circular arcs Systematic methods for constructing more complicated

fields are described by Johnson et al (1982).

Force boundary conditions

Figure A1.5(b) shows a and b slip-lines meeting a tool surface on which there is a friction stress tf Equilibrium of forces on the triangle ABC, in the direction of tf, gives

Thus, the magnitude of the friction stress relative to k determines the angle z at which the a-line intersects the tool face Similarly, a and b slip-lines meet a free surface at 45˚ (tf/k

= 0) Because there is no normal stress on a free surface, p = ± k there, depending on the direction of k.

Perfectly plastic material in plane strain 335

Fig A1.5 (a) Nets satisfying internal force equilibrium and (b) slip-lines meeting a friction boundary

A1.2.3 Velocity relations in a slip-line field

Analogous equations to equations (A1.18d) exist for the variation of velocity along the slip lines However, the statement that there is no extension along a slip line (Section A1.2.1) may directly be used to develop velocity relations and further rules for the geometry of a slip-line field Figure A1.6(a) repeats the net of Figure A1.4(a) Figure A1.6(b) represents,

in a velocity diagram, possible variations of velocity in the field Because there is no exten-sion along a slip-line, every element of the velocity net is perpendicular to its correspond-ing element in the physical plane of Figure A1.6(a) Thus, equations (A1.20) also apply in the velocity diagram

Velocity boundary conditions

Other constraints on slip-line fields may be derived from velocity diagrams (in addition to the obvious boundary condition that the velocity of work material at an interface with a tool must be parallel to the tool surface) Figure A1.7(a) shows proposed boundaries AB and CDE between a plastic region and a rigid region in a metal forming process Because this is a book on metal machining, the example is of continuous chip formation, but any example could have been chosen in which part of the work is plastically deformed and part

is not

First, the boundary between a plastic and a rigid region must be a slip-line Secondly, the boundary between a plastic region and a rotating rigid region (for example CDE in Figure A1.7(a) must have the same shape in the physical plane as in the velocity diagram Both these can be shown by considering the second case

Suppose that any boundary such as CD is not a slip-line Then any point such as H inside the plastic region can be joined to the boundary in two places by two slip-lines, for example to F and G by HF and HG Figure A1.7(b) is the velocity diagram The velocities

vFand vGof points F and G are determined from the rigid body rotation of the chip to be

wOF and wOG, where w is the angular velocity of the chip The velocity vHrelative to vF

is perpendicular to HF and that of vHrelative to vGis perpendicular to HG By comparing

the positions of vF, vGand vH relative to vO, the origin of the velocity diagram, with the positions of F, G and H relative to the centre of rotation O in the physical diagram, it is

Fig A1.6 (a) The physical net of Figure A1.4(a) and (b) a possible associated velocity diagram

seen that the velocity of H is part of the rigid-body rotation: if the boundary CD is not a slip-line, it cannot accommodate velocity changes that must occur in a plastic field

If the boundary is a slip-line, a point H can only be joined to the boundary in two places

by three slip-lines: thus, the argument above can no longer be made For continuity of flow between a plastic and a rigid region, the boundary between the two must be a slip-line

Figure A1.7(b) also shows the whole boundary vCvDvE It is visually obvious that only if

it has the same shape relative to the origin of velocity that CDE has relative to O, can it be consistent with a rigid body rotation

Velocity discontinuities

The usual procedure in slip-line field analysis is to construct fields that satisfy the geom-etry and force requirements of a problem and then to check that the velocity requirements are met In this last part, one more feature of the theory must be introduced: the possibil-ity of velocpossibil-ity jumps (discontinuities) occurring Figure A1.7(c) returns to the considera-tion of the velocity of a point H in the plastic field H is connected to the boundary by slip-lines, both directly to G and indirectly to F through H′ It is possible for there to be a finite velocity difference between H and G, however short is the length HG, i.e a discon-tinuity If there is a discontinuity, then the rules for constructing the velocity net require that there be a discontinuity of equal size between H′ and F A velocity discontinuity can exist across a slip line, but only if it is of constant size along the line It is not implied that there is a discontinuity in the condition of the example described here: examples of actual machining slip-line fields are given in Section A1.2.5

A1.2.4 Further considerations

Slip-line fields must satisfy more than the force and velocity conditions considered in Sections A1.2.2 and A1.2.3 First, they must (as must every plastic flow) satisfy a work criterion, that everywhere the work rate on the flow is positive This means that the direc-tion of the shear stresses in the physical diagram must be the same as the direcdirec-tion of the shear strain rates deduced from the velocity diagram

Perfectly plastic material in plane strain 337

Fig A1.7 (a) A possible machining process with (b) a partial velocity diagram and (c) an illustration of a velocity

discontinuity across a slip-line

It must also be checked that it is possible that regions in the work material that are assumed to be rigid can in fact be rigid For example, in Figure A1.7(a), in the rigid regions KBA and LCD, the loads change from values on BA and CD determined by the plastic flow to zero on the free surfaces KB and LC It must be checked that such load changes can be accommodated without the yield stress being exceeded in the rigid regions in the neighbourhood of the vertices B and C Checking for overstressing is introduced in another context in Appendix A5 The overstressing limits developed in Appendix 5 (Hill, 1954) apply here too

A1.2.5 Machining slip-line fields

Figure A1.8 collects a range of slip-line fields, and their velocity diagrams (due to Lee and Shaffer, 1951, Kudo, 1965, and Dewhurst, 1978), which describe steady state chip forma-tion by a flat-faced cutting tool

The first is Lee and Shaffer’s field It describes formation of a straight chip The work

velocity Uwork is transformed to the chip velocity Uchip by a discontinuous change UOA

tangential to the slip-line OA The angle at which OA meets the free surface is not set by a free surface boundary condition A is a singularity where the surface direction is not defined Instead, the direction of OA is determined by its being perpendicular to BD The inclination

of BD to the rake face is given by equation (A1.21) Because all the slip-lines are straight, the hydrostatic pressure is constant along them (equation (A1.19)) The chip region above

ADB is free, i.e there are no forces acting on it This determines that p = k and AD = DB.

The second is due to Kudo It may be thought of as a modification of Lee and Shaffer’s field in which the primary shear plane OA is replaced by a fan-shaped zone of angular

extent d, still with a singularity at the free surface A It still describes a straight chip The slip-lines intersecting the rake face do so at a constant angle z: the field therefore

contin-ues to describe a condition of constant friction stress along the rake face The free-chip

boundary condition still requires p = k on AD and DB and AD = DB However d can take

a range of values, from zero up to a maximum at which the region below AE becomes

overstressed For the same friction condition, tool rake angle and feed, f, as in the Lee and

Shaffer field, the Kudo field describes chip formation with a larger shear plane angle and

a shorter contact length

Two further fields suggested by Kudo are the third and fourth examples in Figure A1.8 These describe rotating chips The boundaries ADB in the physical plane between the fields and the chips can be seen to transform into their own shapes in their velocity diagrams The third field may be thought of as a distortion of the Lee and Shaffer field and the fourth as a distortion of Kudo’s first field The slip-lines in the secondary shear zone

intersect the rake face at angles z which vary from O to B: these fields describe conditions

of friction stress reducing from O to B Because the slip-lines are curved, the hydrostatic stress now varies throughout the field Again the allowable fields are limited by the requirement that material assumed rigid outside the flow zone around A must be able to be rigid However, the possibility arises that it is the chip material downstream of A that becomes overstressed

The last example shows another way in which a rotating chip may be formed A fan region OED is centred on the cutting edge O and the remainder DA of the primary shear region is a single plane With this field, the slip-lines intersect the rake face at a constant

angle, so that it describes constant friction stress conditions The fan angle y can take a

Perfectly plastic material in plane strain 339

Fig A1.8 Metal machining slip-line fields (left) and their velocity diagrams (right), due to (1) Lee and Shaffer (1951),

(2–4) Kudo (1965) and (5) Dewhurst (1978)

range of values, limited only by its effect on overstressing material around A For the same

friction condition, tool rake angle and feed, f, as in the Lee and Shaffer field, this last field

describes chip formation with a lower shear plane angle and a longer contact length

A1.3 Yielding and flow in a triaxial stress state:

advanced analysis

A1.3.1 Yielding and flow rules referred to non-principal axes – analysis

of stress

The yield criterion is stated in equation (A1.7) in principal stress terms It is extended to non-principal stresses in equation (A1.12): this has been justified in the two special cases when it represents principal stress and maximum shear stress descriptions of stress It is now justified more generally, by showing that the function of stress which is the left-hand side of equation (A1.12) has a magnitude that is independent of the orientation of the

(x,y,z) coordinate system If it is valid in one case (as it is when the axes are the principal

axes), it is valid for all cases Tensor analysis is chosen as the tool for proving this, in part

to introduce it for later use

Tensor notation and the summation convention

Figure A1.9 shows two Cartesian coordinate systems (x,y,z) and (x*,y*,z*) rotated arbitrar-ily with respect to each other In the (x,y,z) system the stresses are s ij with i and j denoting

Fig A1.9 (x,y,z) and (x*,y*,z*) co-ordinate systems

x, y or z as appropriate In the (x*,y*,z*) system the stresses are s* kl , with k,l denoting x*, y*

or z* The figure also shows a tetrahedron OABC, the faces of which are normal to the x, y, z and x* directions Writing the direction cosines of x* with x, y and z as a x*x , a x*y and a x*z, with

similar quantities a y*x , a y*y , a y*z and a z*x , a z*y , a z*z for the direction cosines of y* and z* with

x, y and z, first, by geometry, the ratios of the areas OAC, OAB and OBC to ABC are respec-tively a x*x , a x*y and a x*z Then, from force equilibrium on the tetrahedron, for example

s* x*x* = a x*x a x*x s xx + a x*x a x*y s yx + a x*x a x*z s zx + a x*y a x*y s yy + a x*y a x*x s xy + a x*y a x*z s zy (A1.22a)

+ a x*z a x*z s zz + a x*z a x*y s yz + a x*z a x*x s xz

In general and more compactly, any of the stresses s* klmay be written

3 3

s* kl= ∑ ∑a ki a lj s ij (A1.22b)

j=1 i=1

Quantities which transform like this are called tensors, and the study of the properties of the transformation is tensor analysis

By the summation convention, the summation signs are omitted, but are implied by the

repetition of the suffixes i and j among the coefficients a ki and a lj Thus equation (A1.22b) becomes

Furthermore, the repetition of k and l, between the left and right-hand sides of the equa-tion, implies that it represents all nine equations for the components of s* The meaning

of the equation is unchanged by substituting another pair of letter suffixes, say m and n, for i and j: suffixes such as i and j, repeated on the same side of an equation, are called dummy suffixes and are said to be interchangeable Suffixes such as k and l are called free suffixes In the special case when k = l, the summation convention extends to include

s* kk = s * x*x* + s * y*y* + s * z*z* (A1.22d)

Properties of the direction cosines

Because the angle between a direction i and another direction k is the same as the angle between the direction k and the direction i, a ik = a ki

Because the scalar product of two unit vectors is unity if they are parallel and zero if they are perpendicular to each other, the same is true of the sum of the scalar products of

their components in any other coordinate system In repeated suffix notation, a ik a jk= 1 if

i = j and 0 if i ≠ j This can be written

where d ij is defined as 1 or 0 depending respectively on whether or not i = j.

Transformations of stress

Now consider the summation of the direct stresses

s* kk = a ki a kj s ij = d ij s ij = s ii (A1.24)

This demonstrates that the sum of direct stresses s* kk in the (x*,y*,z*) system equals the sum s in the (x,y,z) system One of the systems could be the principal stress

Yielding and flow in a triaxial stress state 341

system: thus, the hydrostatic stress smis a stress invariant (it is known as the first stress invariant)

Consider now the product of stresses s* kl s* lk, with the transformations of equation (A1.22c), the interchangeability of dummy suffixes and equation (A1.23):

s* kl s* lk = (a ki a lj s ij )(a lm a kn s mn ) = (a ki a kn s ij )(a lm a lj s mn)

(A1.25)

= (d in s ij )(d mj s mn ) = s nj s jn = s ij s ji

In principal stress space, s ij s ji = s2r So sris also a stress invariant (it is known as the second stress invariant) From equation (A1.25)

s2 r= s2

xx + s2

yy + s2

zz + 2(s2

xy + s2

yz + s2

As smand srare stress invariants, so is sr′ From Figure A1.2(a) sr′2= s2r – 3sm2 From this, equation (A1.4) and similar manipulations as in equations (A1.6) to (A1.7), the yield criterion becomes

2s–2 ≡ 3(sr2– 3sm2) ≡ 3(s xx′2+ s yy′2+ s zz′2) + 6(s2

xy + s2

yz + s2

zy) ≡

(s xx – s yy)2+ (s yy – s zz)2+ (s zz – s xx)2+ 6(s2

xy + s2

yz + s2

zy ) = 6k2or 2Y2

(A1.26b) which is the same as equation (A1.12) of Section A1.1

Strain transformations

The strain increments also transform as a tensor:

It follows, as for stress, that the resultant strain increment and the equivalent strain incre-ment are invariants of the strain The extension of the definition of resultant strain to a general strain state is

der2= de2

xx + de2

yy + de2

zz + 2(de2

xy + de2

yz + de2

where, as in equation (A1.13), de xy = de yx= (1/2)(∂u/∂y + ∂v/∂x) and similarly for de yzand

de zx Equivalent strain increments are √(2/3) times resultant strain increments

1.3.2 Further developments

The repeated suffix notation may be used to write the plastic flow rules (equation (A1.13)) more compactly and to express various relations between changes in equivalent stress and the deviatoric stress components that will be of use in Section A1.4 First, from equation (A1.13),

The dependence of ds– on its components s′klis

∂s–

∂s′