on the plastic flow beneath a blunt axisymmetric … · on the plastic flow beneath a blunt ......

M. C. S H A W Professor,

Mem. ASME

G. J. DeSALVO Assistant Professor.

Mechanical Engineering Dept., Carnegie-Mellon University,

Pittsburgh, Pa.

Introduction

On the Plastic Flow Beneath a Blunt Axisymmetric lnden ter A new approaclt to large strain plasticity problems i n which the material i s considered to behave i n a plastic-elastic fasltion, instead of as a plastic-rigid body, i s applied to the axisymmetric blunt indenter. The ratio of the mean stress on the pt~nclj face to tlte uniaxial flow stress of the material (constraint factor C ) i s found to be 2.82 for a n extensive specimen. However, i t i s slzown that a small part of the punclt face i s elastically loaded, and i f the loaded punch area i s assumed equal to the size of the plastic impression, the% the constraint factor to be used i s 3.00 instead of 2.82. Th i s i s the value to be used in interpreting the ordinary brine11 test. Hardness values are shown to be independent of the degree of friction on the face of a blunt indenter and of the elasticity of the indenter. The amount of material required beneath a n axisymmetric indenter i n order that there be no upward flow i s found to be about 2.6 times the diameter of the impression for steel. However, tlte exact value depends on Young's modulus of elasticity, Poisson's ratio, material hardness, and tlte depth of the impression relative to the diameter of the indenting sphere. W h e n two opposing axisymmetric indenters are employed, the influence of one on the other will be less t l u n one percent when their spacing i s approximately 11.5 times the indefttation diameter. .I blunt indenter m a y be defined a s one wlzick gives 710 npward flow i n a hardness test. Upward flow should be avoided i n kardness tesfing since i t causes the mean stress on tlte punch face to produce a given impression to be sensitive to friction and the tendency of the metal to strain harden. Upward flow m a y be prevented by use of a n extensive specimen relative to the depth of the impression, large indenter angle (160- 180 deg), and high indenter friction (roziglt surface and no lubricant). The$ow stress measured by a blunt indenter i s that correspouding to the onset of plastic flow. W h e n upward flow i s permitted, the flow stress measwed by a n indentation hardness test will correspond to a n appreciable plastic strain which increases a s the included a n g b of the indenter decreases. The quantity measured by a n indenter that performs wi th upward fEow i s , therefore, quite ambigzbous when the material tested strain hardens.

I N A ~ n ~ v r o u s paper (Shaw and DeSalvo [1]) ,1 a new approach to large strain plasticity was presented and applied to the two-dimensional blunt indenter. The elastic solutions for two distributions of load (Hertzian and concentrated) on a blunt rigid punch acting on a semi-infinite specimen were considered together to determine the value of constraint pertaining (C =

mean stress on punch face/mean uniaxial flow stress), which was

found to be 2.62. This value, which is higher than the upper bound value obtained from slip line field theory (2.57), is of en- tirely different origin.

The new analysis considers the specimen to be a plastic-elastic body instead of plastic-rigid one assumed in the slip line field approach. The new constraint is associated with the force required to establish the elastic field beneath the indenter, while t,he slip line field constraint is concerned with the force required to cause upward flow of the material beneath the punch.

The new approach to the blunt indenter is useful in estimat,ing Numbers in brackets designate References at end of paper.

Contributed by the Production Engineering Division for presenta- residual st,resses following indentation and predicts asecond plastic

tion at the Winter Annual Meeting, Los Angeles, Calif., November flow on unloading, as well as a primary one on loading. I t also 16-20, 1969, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, explains a number of observations concerning the indentation Manuscript received at ASME Headquarters, July 1, 1969. Paper process that have hitherto gone unexplained. However, most No. 68-WA/Prod-12. practical problems are concerned with axisvmmet,rical indenters.

Copies will be available until September 21, 1970. which have significantsly different characteristics from those of i

8, deg.

2

C

a

Fig. 2(a) Variation of constraint factor (C) with semicone angle (8) of conical indenter [offer Locken [lo] and Haddow and Danyluk [ I l l ) , (b) Slip line flelds obtained by Lockett [lo] for semicone angles of 80 and 52.5 deg

Fig. I(@) Slip line fleld for circular flat smooth punch, (b) Extension of slip line field into rigid region to show that flow criterion is not violated (afler Shield [9]) -

0 3 0 60 9 0

3- - 3

two-dimensional indenters. The present paper, therefore, ex- tends the new treatment to the axisymmetric indenter.

-

------- LOCKETT I - - HADDOW AND DANYLUK - I

I I

A number of authors have applied the two-dimensional slip line field solutions of the flat punch of Prandtl [ 2 ] and Hill 131 to axisymmetric indenters. Hencky [4] assumed slip line fields similar to those used in plane strain (Prandtl) to obtain an ap- proximate solution for the circular flat punch. In this treatment., Hencky employed the Haar-von Karman [.?I hypothesis that the circumferential stress is equal to one of 6he principal st,resses on a meridional plane during plastic flow. Hill [ 3 ] has crit,icized this hypothesis, which is int,roduced so that the number of equations

C

.o

may be made to equal the number of unknown quantities. Ishlinsky [ 6 ] calculated the actual slip line field around a cir-

cular punch using a graphical method and obtained a value for the constraint factor of 2.84. The Haar-von Karman hypothesis was also used in this analysis.

Shield and Drucker [ 7 ] employed limit analysis to the solution of a square punch. Using the Tresca yield criterion, they obtained a value of 2.855 for the constraint factor ( C ) for this case. Levin 181 determined the constraint factor (C = 2.92) for a circular smooth punch loaded uniformly. This upper bound analysis was based on a slip line field that resembled Hill's for a two- dimensional indenter.

Shield [ 9 ] applied the Haar-von Karman hypothesis and Tresca yield criterion to the circular smooth punch. His method essentially follows that of Ishlinsky but employs a numerical approach in place of the graphical method of Ishlinsky. Despite this fact, the value he obtained for the constraint factor (C =

2.845) was remarkably close to that of Ishlinsky (2.84). Shield obtained an admissible velocity field compatible with the stress field, and hence his solution is both an upper and lower bound one

and thus represents a cotl~plete solut~iot~. His slip line field for the circular punch is shown in Fig. l ( a ) . I t should be noted t,hat stress is not uniformly dhributed over the punch face as in Levin's case. Shield also extended the stress field into the rigid region [Fig. l ( b ) ] and showed that the yield criterion was' not violated in so doing.

Lockett [ lo ] has extended the work of Shield to rigid cones of varying cone angle using the same assumptions and techniques. This problem is a more complex one than that of the flat punch, since the shape of the lip that is formed by upward flow is not straight and must be determined by iteration. Values of constraint factor C obtained by Lockett are given by the dotted curve in Fig. 2(a) . This curve stops a t a semicone angle of 52.5 deg since Lockett's solution is degenerate for smaller angles. The slip line fields det,ermined by Lockett for semicone angles ( 8 ) of 80 and 52.5 deg are shown in Fig. 2(b) . Lockett's slip line field cannot be extended below 0 = 52.5 deg, since angle $in Fig. 2(b) would then be negative. A different form of slip line solut,ion is, therefore, needed for semicone angles below 52.5 deg. ' Haddow and Danyluk [ l l ] have also extended the work of

Shield [9] to the case of rigid cones inserted into prepared match- ing cavities. In this case, there is no lip. The values of wn- straint factor ( C ) obtained by these authors are given by the solid curve in Fig. 2 ( a ) for different values of 0. The value for 0 = 90 deg is from Shield [ 9 ] who used the same assumptions and approach in his analysis of the flat circular punch as did both Lockett and Haddow and Danyluk. The constraint factors of Shield, Lockett and Haddow, and Danyluk are summarized in Table 1.

The foregoing literature review indicates that a number of solutions for the flat circular punch are available all based on the Haar-von Karman hypothesis, and all assuming the material to be plastic-rigid. The constraint value obtained by Shield (2.845) is an accurate exact solution for these assumptions, and the slip line field on which it is based is shown in Fig. l ( a ) .

Transactions of the ASME

Table 1

1Jocket.t [I01 2.85 (Shield [9] ) 2.60

Hnddow arid L)anylak /I11

2.85 (Shield (91 )

Experiment Fig. 3 shows deformed grid patterns on the meridional plane of

specimens indented by spheres. The deformed region shown in Fig. 3(c) clearly does not resemble the slip line field of the Shield analysis shown in Fig. l(a). The difficulty lies ill the assumption that the material beyond the deformed region is rigid. I n renlity, it is elastic and, as has been shown in a previous paper (Shaw and DeSalvo [I]), the elasticity of the specimen plays a major role in determining the shape of the plastic-elastic boundary and, hence, relative to the origin and magnitude of constraint factor ( C ) .

The large difference between the flow patterns that are observed and predicted by slip line theory has been noted by previous authors, but the inadmissible assumption has not been identified, nor has an alternative theory been presented. San~uels and i\iIulhearn [12] and Mulhearn [13], using etch patterns in- volving the Frey reagent, showed that the deformation beneath a blunt indenter corresponds to a "radial" compression instead of the cutting type of pattern predicted by the slip line approach. Atkins and Tabor [14, 151 have presented additional evidence for compressive type flow with a blunt indenter.

Williams and O'Neill (161 have shown lines of constant hardness (isosklars) obtained by making microhardness indentations on the meridional plane for a large indentation. These patterns strongly resemble the etch patterns obtained by Samuels and Mulhearn and have little resemblance to the assumed slip line solutions. However, interpretation of such results are difficult due to the large effects associated with residual stresses induced by the initial indentation. For example, a decrease in microhardness near the center of the impression at the smlace is re- ferred to as strain softening, while it is really due to the presence of the important macroresidual stresses to be discussed later in this paper.

With the appearance of the digital computer, i t has become possible to solve the large number of linear simultaneous equations associated with a finite element approach to elastic-plastic problems. Examples are to be found in the work of Akyux [17] and Akyuz and Merwin [18]. However, the emphasis of such studies appears to b,e upon computational techniques, rather than upon the physics of the problems considered.

I n the analysis which follows, the material being indented is considered to be plastic-elastic, and the rationale employed is aa follows. Initially, the applied load is considered to be elastically supported. The volume displaced by the indenter must then be equal to the total decrease in volume in the elastic body due to the elastic compressive stresses developed, and there will be no upward flow. Since the Tresca flow criterion will be exceeded by the elastic stresses in the vicinity of the surface, the material will flow plastically until the stresses so developed cause the flow criterion to be satisfied. However, it is assumed that this plastic flow takes place in such a way that the external load is not increased beyond that required by the elastic solution. The additional stresses due to plastic flow will be residual elastic ones. After plastic flow has taken place and with the load still present, the same elastic stress field as initially present will be there plus the additional residual stresses associated with plastic flow.

I I PLASTIC /

\ I /

/

'--A/

BOUNDARY

(c)

Fig. 3 Deformed grid patterns on the meridional plane of specimens indented by a sphere: (a) plasticine, (b) mild steel, (c) interpretation of (0) and (b)

When the load is released, the elastic stress field will collapse, and the residual stress system will be all that remains. If the flow criterion is now again exceeded, a second plastic flow must occur. The specimen will finally be left in a stmate of residual stress that is compat,ible with the flow criterion.

Elastic Analysis Fig. 4(a)'shows lines of maximum elastic shear st,ress beneath

a smooth spherical indenter obtained from a Hertz analysis (see for example Timoshenko and Goodier [19] or Davies [20] ). The load distribution is hemispherical, and the ratio of mean ( p ) to maximum ( p o ) pressure is 2/3. The numbers on the lines of constant maximum shear st,ress shown are equal to the maximum shear st'ress divided by the mean stress on the punch face ( M I =

r/p). Fig. 4(b) is a plot of ~lf' versus the nondimensional distance beneath the surface (z/a) for the vertical center!ine of

Journal of Engineering for Industry

Fig. 5 Boussinesq solution for concentrated load P on semi-infinite elastic body

the edges of the punch, it is said to be fully developed, and this is the situat.ion involved in an indentation hardness test.

By the maximum shear t,heory, one of the lines of maximum shear stress in Fig. 4(a) should correspond to the plastic-elastic boundary for fully developed plastic flow. I n determining which of these lines to select, and in subsequent calculations, it is con- venient to consider a second type of loading on the punch-that of a concentrated load (P) that is equal in rnagnit,ude to the distributed load.

Fig. 5 shows the Boussinesq solution for a concentrated load (P) actring on a semi-infinite elastic body. The sLresses acting at a point A having coordinates r, 8, and z will be as follows:

Fig. 4(a) Hertz liner of constant maximum shear stress M' = 716 for a smooth spherical indenter. The radius of the indentation in the plane of the surface is a. (b) Variation in M' with z/a for vertical centerline of Fig. 4(a)

Fig. 4(a). The base of the punch in Fig. 4(a) is shown as a flat surface.

I n reality, this is a spherical surface of large radius. As a matter of convenience, the punch will be drawn wit,h a flat base in all subsequent figures, even though the base of the punch is really a sphere of large radius.

Plastic flow will occur first aL the point beneath the surface corresponding to Ab' = 0.468 [Fig. 4(a)]. By t,he maximum shear theory, this will occur when the stress a t this point, is equal to Y/2 . Or, when the mean stress on the punch p is

As the load on the punch is increased, the plastic flow zone will grow and extend to the surface of t,he punch. When it approaches

where v = Poisson's rat,io. The resnltunt. stress acting on a hori- zontal plane (mn in Fig. Ti) will be

where d is the diameter of the circle shown in Fig. 5. Stress (S) will have a constant value for points on a single circle and will be directed toward (0) a t all points. The circles of Fig. 5 are not lines of constant shear stress as they were for the two-dimensional case (Shaw and DeSalvo [I]).

Fig. 6 shows the concentrated load situation equivalent to


PLASTIC-ELASTIC

As an example, assume $o = 71 deg (b/a = a = 0.344, and d la = 3.249), then the maximum shear stress a t point8 E and D will be as follows:

The maximum shear stress a t point D on circle OED is obviously less than that a t point E. The point on the vertical cent,erline (Dl) having the same maximum shear stress as point E may be found as follows: Fig. 6 Equivalent concenfrafed load arrangement

that of a flat punch on a semi-infinite elastic body. The point of application of the load (0) will lie some distance (b) within the punch, and stress will be distributed from 0 to the punch face as though the punch were a cone ( C ) passing through the edges of the punch (E). I t may be readily shown that the stress at any point within the cone is given by equations (1) to (5) with P re- placed by P I 1 - c ~ s ~ $ ~ , where $0 is the semicone angle which, in turn, is equal to cot-' (b/a).

If a circle is drawn through points 0 and E in Fig. 6, this will intersect t,he vertical centerline at point D. From equations (1) to (4)) t,he maximum shear stress a t points E and D may be found in terms of P, dl and b/a, where d is the diameter of the circle passing through points 0, El and D.

As a convenience, let

where d, is the value of d corresponding to point D' or

and

From Fig. 4(b) the value of M' corresponding to this value of e,/a is 0.177, and the maximum shear stress a t point D' from the Hertz analysis will be

then, If the elastic-plastic boundary is assumed to pass through points E and D' for the concentrated load case, then from the Tresca flow criterion and equation (18)

Y M'P' - 2 7ra2 and

tl b e - - - - + - = 1 a a a

ff +- a

$0 = cot-' a (9)

Taking Poisson's ratio to be 0.3, the maximum shear stress ( T ~ ) a t point E(r = a, z = b), of Fig. 6 will be

For the assumed value of qo = 71 deg, M' = 0.177, and hence the constraint factor C = 2.82.

This is the answer we are seeking, provided the angle $0 h~ been correctly assumed.

One check on the assumed value of $0 (and hence on the distance concentrated load P is above the plane of loading) may be had by comparing the values of shear stress a t point D' form the Hertz and concentrated load solutions. By the principle of Saint-Venant, these two values of stress should be essentially the same when the concentrated load (P) and the distributed load (P') are equal.

From equations (17) and (18)

where

where f(a) is the function of ( a ) obt,ained by substituting equa- t,ions (11) t,o (13) into equation (10).

Similarly, t,he maximum shear stress (rD) at point D(r = 0, z = d) of Fig. 6 will be

For the assumed value of $o = 71 deg

Journal of Engineering for Industry

Fig. 7(a) Construction for checking correct assumption for $0, (b) Mohr's circle diagram for stresses at edge of punch (point E )

which is obviously close to unity. However, other values of $0

in the vicinity of $o = 71 deg give equally satisfactory values for T ' ~ ~ / T ~ ' , and hence this check on $o is not sufficient.

A second more discriminating check is obtained as follows: Fig. 7(a) shows lines of constant shear stress (Hertz) together

with the assumed value of $0 = 71 deg and the corresponding concentrated load P. Point D' on the elastic-plastic boundary is located a t dJa = 2.16, for this value of $0, and the corresponding interpolated line of constant shear stress for i1.I' = 0.177 is shown by the dashed line. The elastic-plastic boundary will follow this Hertz line of constant shear stress to the point (T) which is closest to the edge of the punch at E. At this point, the elastic- plastic boundary will, in turn, follow the direction of maximum shear stress to the face of the punch a t E'.

Line T S is the locus of points of closest approach (to point E) of successive Hertz lines of maximum shear stress. Point 1' is the intersection of TS and the particular Hertz line of constant shear stress corresponding to the elastic-plastic boundary for the assumed value $0 = 71 deg. The direction of maximum shear stress E 'T is obtained by constructing the Mohr's circle diagram [Fig. 7(b)] a t point E for the concentrated load solution and assuming the direction of maximum shear stress to be essentially the same at points T and E.

The shear stress at E ' should correspond to Y/2, and the check on the assumed value of $0 consists of showing that this is consistent with the value of constraint (C) obtained from equation (20). The value of C corresponding to $0 = 71 deg was previously found to be 2.82, and since the maximum Hertz stress (po) on the punch face is 3/2 the mean value (p), it follows that po = 3/2 (2.82)Y = 4.23Y. The vertical scale of normal stress on the punch face (p) in Fig. 7(a) has been laid off such that po =

4.23Y. If now, a vertical line is erected from El i t will intersect the semicircle of punch face pressure distribution at point IT which corresponds to p = Y. Thus, the pressure on the punch face a t E ' is seen to be Y, and since there is no friction for the Hertz solution, this will give a value of maximum shear stress at

this point of Y/2, which corresponds to t,he value indicated from t,he pladc-elast,ic boundary.

I t is thus seen that a value of $o = 71 deg is consistent with the ,

st.ress dist,ribution across the punch face and wit,hin the deformed material. This value of $0 which provides the check indicated was, of course, arrived at by trial.

The value 2.82 is seen to be very close t,o that obtained by Shield (2.845). However, the closeness of these two values is apparently fortuitous since the physical origin of the t,wo constraint,~ are so very different. The value obtained here is an elastic constraint, while that obtained by Shield is a flow const,raint.

In the concentrated load analysis, the material was assumed to go plastic all the way to the edge of the punch. Hence, the size of the dent left behind in the surface would correspond to dimen- sion (2a) in such a case. However, in the Hertz analysis the material a t the very edge of the punch [E in Fig. 7(a)] will be elastically loaded, and i t will not flow plastically until point E' is reached. Thus, the radius of the dent left behind in the Hertz case will be O'E' instead of O'E.

The distance O'E' may be computed, since the Hertz load tiis- tribution is known to be hemispherical with a peak stress equal to 4.23Y. Thus,

OE' d ( 4 . 2 3 ~ ) ~ - Y2 - - - a 4.23Y

OE' = 0.97a (23)

The diameter of the impression left behind in the surface will be 3 percent less than (2a) for t,he Hert,z case. If the size of dent left. in the surface is used as a measure of the total loaded area, t,he~i this area will be too low by 6 percent for a Hertz-type load dis- t,ribution. The true hardness would then be 0.94 times the apparent value. Or, the effective constraint factor would be


Table 2 Valuer of normal (P) and tongential (7) stresses across punch face for concentrated load

psi c cos $ SA/Y p l y 7/Y f

0 1.0000 37.8 37.8 0 0 0 . 2 5 ~ 0.8097 15.8 12.8 9 . 3 0.73 0 . 5 0 ~ 0.5679 3.87 2.20 3.19 1.45 0 . 7 5 ~ 0.4179 1.15 0.48 1.05 2.44 1 . 0 0 ~ 0.3256 0.42 0.14 0.39 2.91

b d = - cos"

Therefore,

cos J. = s~ (*)

Resolving into normal (p) and tangential (7) components

cos $ , = s, (--) cos lc cos $0

cos $ rA = sE (-) sin +

cos $0

The local coefficient of friction (f) will be

Values of p, 7, and f are given in Table 2 and Fig. 8(b) for different points across the punch face in terms of the three-dimensional uniaxial %ow stress ( Y ) .

The peak value of normal stress for the concentrated load case (37.8Y) is seen to be much higher than that for the Hertz case (4.231').

In the con~ent~rated load case, the cone-shaped punch is elastic and is considered to have the same Young's modulus as the material indented.

I t is evident that t,he two loading conditions considered here differ widely relative to both friction and elasticity of punch. The fact that the same constraint factor (C) is obtained for these two cases is consistent with t,he observations that lubrication and stiffness of indenter are relatively unimportant in hardness testing wit,h a blunt indenter.

Fig. 8 Distribution of stress on punch face: (a) Hetiz case, (b) concentrated load case

(2.82/0.94) = 3.0. Since, in practice, the size of impression left behind in the sur-

face is used as a measure of the loaded area without correct,ion, the constraint value to be used in such cases is 3.0 instead of 2.82.

Load Distribution on Punch The distribution of elastic stress on the punch face will be as

shown in Fig. 8(a) for the Hertz case. The normal stress will follow a circular arc. The mean stress will be 2.82Y, and the maximum stress will be 3/2 times this value (4.23Y). There will be no shear stresses on the punch face in this case, and hence the Hertz distribution of load corresponds to a frictionless sit,uation.

The distribution of elastic stress on the punch face with the concentrated load may be obtained from equation (5) adjusted for the case of a wedge of angle $0

Plastic Action The left side of Fig. 9(a) shows the elastic-plastic boundary

for the concentrated load case shown solid and for the distributed load case (Hertz) shown dotted. , The right-hand side of this diagram shows the variation of maximum shear stress along the vertical centerline for the concentrated load case in units of Y, the uniaxial flow stress of the material.

Fig. 9(b) is a Mohr's circle diagram for the elastic stresses for any point on the vertical centerline. The largest principal stress (al) is directed vertically in the z direction and is 15 times as large as the other two principal stresses, which are equal to each other and tensile in character.

Fig. 9(c) shows the elastic stresses on a particle on the vertical centerline that is within the plastic-elastic boundary, but before plastic flow has occurred. When plastic flow occurs, stresses HI will develop in the two transverse directions. These stresses will be equal in accordance with the Haar-von Karman assumption (1909). Flow will continue until the Tresca flow criterion is

If the value of d for the particular circle passing through 0 and E [Fig. 8(b)] (the plast,ic-elastic boundary) is do then for point E

Or, for point A

satisfied; and hence

CT~ - HI = Y (33

but,

The vertical component of force a t each point will not change as

Journal of Engineering for Industry 7

SECONDARY FLOW ZONE

PLASTIC-ELASTIC BOUNDARY

L - J 1 (dl I a ( ~ l - 1 -

I I LOAD REMOVED

Fig. 9 Stress within the plastic zone

PLASTIC-ELASTIC BOUNDARY

(cl Fig. 10 Grid pattern of spherical indentation an meridional plane, ball, 6/16 diameter tungsten carbide; material, annealed a brass, 3 in. dia and 2 in. high. (a) Surface with etched grid, 100 lines per inch, (b) mating surface with grid transferred by coining, (e) interpretation of (a) and (b) (Osias [22]).

plastic flow occurs, because otherwise the external load would change and, hence, the size of the plastic-elastic boundary. The plastic-elastic boundary would then no longer pass through the edge of the punch (E) in Fig. 7.

When the external load (P) is removed, stress UI will disappear. The state of stress will be as shown in Fig. 9(e) unless HI exceeds uniaxial flow stress (Y). If HI > Y, then a second plastic flow will occur on unloading and stress Hz [Fig. 9(j)] will develop until

in accordance with the Tresca flow criterion. The extent of the zone of secondary plastic flow will corre-

spond to points where the maximum shear stress is equal to Y. Thus, the limit line for secondary shear will intersect the vertical centerline a t A in Fig. 9(a) and the remainder of the limit line will be as shown.

The plastic material beneath an axisymmetric indenter will contain residual compressive stresses that are several times the uniaxial flow stress (Y). For example, the maximum lateral residual stress developed a t the center of an indentat,ion near the surface will equal 3.23Y for a Hertz-type load distribution.

The extent of the primary zone of plastic flow (ez/a) will be 1.82 beneath the original surface where 2a is the diameter of the indentation. This value is to be compared with 2.99 for the case of a two-dimensional indenter (Shaw and DeSalvo [I]). The extent of secondary plastic flow (ella) for the axisymmetric indenter is 1.18, which was found to be 1.35 for the two-dimensional case. Thus, both the primary and secondary plastic flow zones penetrate deeper into the material in the case of a two- dimensional indentation.

Fig. 10(a) shows an annealed cr brass specimen (3 in. dia by 2 in. high) that was coated with a photosensitive plastoic and ex-


where:

v = Poisson's ratio E = Young's modulus of elaqt.ir!it,y

Fig. 11 Sphere indenting plane surface

posed to ultraviolet radiation through a negative containing 100 grid lines per inch. After development (removal of exposed plastic), the surface was etched in acid to provide a grid etched into the surface of the specimen. An unetched mating member was securely clamped to the etched one using a massive steel split block, and a 5/16-in. dia tungsten carbide ball was forced into the interface. The unetched mating snrface is shown in Fig. 10(b) after deformation. The grid from the etched surface was coined into the second one over a very definite area. This is the area over which the transverse stress HI in Fig. 9(d) is equal to Y or greater. The limit of the transferred region is, therefore, the locus of the secondary flow zone. The elastic-plastic boundary may be obtained from Fig. 10(a), and this is shown by the solid line in Fig. lO(c) together with a tracing of the line for which HI = Y from Fig. lO(b), shown dotted. The shapes of these two curves are seen to be in excellent agreement with the equivalent analytically derived curves shown in Fig. 9(a).

Required Sppimen Size I n order that the elastic constraint fully develop, i t is necessary

that sufficient material be located beneath the punch so that the volume displaced may be accommodated elastically. Otherwise there will be some upward flow, and not all of the resistance to penetration will be associated with an elastic ~onst~raint. When a thin layer of soft material is located on an essentially rigid substrate, upward flow must occur, and a slip line solution giving rise to a velocity constraint will pertain.

When a brine11 ball of diameter D is forced into a surface to a given depth (Fig. 11), the diameter of the impression in the plane of the surface will be (2a), and to a good approximation, the volume of material displaced will be (if 2n < 0.40)

For a purely elastic constraint to develop, sufficient material must surround this impression so that the change in volume due t,o the elastic stresses equals this volume.

The change in volume per unit volume in an elastic body is equal to the sum of the principal strains (el + €2 + €3). This, in turn, may be expressed in terms of the principal stresses as follows.

The total change in volume due to the elast,ir, stxesses may then he obtained by integrat,ion

where PI and pg are the radii shown in Fig. 11. I t may be shown that the sum of the principal stresses at any

point having coordinates (p, $) will be as follows for the case of a concentrated load P acting on a cone of half angle $0

X [3 sin2 $ cos $(1 + cot2 $) - cos3 $ sin2 $1 (38)

Substituting into equation (37) and integrating

The quantity in brackets may be evaluated by let,ting $0 = 71 deg and hence

E q u a h g TG and V2 and solving for p2/2a

where

As an example consider an indentation in steel where

E = 30 X 106 psi

v = 0.3

p = 3Y = 300,000 psi

The value 2a/U will normally lie between 0.2 and 0.4 (the latter more c o ~ i s e ~ ~ a t i v e value will be used), and a good est,imate for p1/2a = 0.5 (Fig. 11).

Upon substituting these values

Thus, to insure no upward flow, the specimen should extend about 2.6 times the extent of the impression for a three-dimensional indenter. The corresponding value was previously found to be much larger (approximately 15) for a two-dimensional indenter (Shaw and DeSalvo [I]).

From equation (41), i t is evident that the material propert,ies that influence the amount of material that must surround an indenter for the full elastic const,raint to develop are

In Table 3, values of this quantity are given and compared with those for steel.


Table 3

Rela- - - tive to

Material Cold-rolled steel

E, psi Y 3Y (1 - 2v)(31') Steel 30 X 106 0.30 300,000 250 1 . O

Cold-rolled aluminum alloy 9 .9 Cold-rolled pure copper 15.6 Annealed pure copper 15.6

It is evident that the amount of material required to develop a fully elastic constraint increases as Young's modulus increases but as the hardness of the material decreases. For annealed copper, about 20 times the indentation diameter (2a) will be required if upward flow is to be avoided. Since annealed copper strain hardens extensively and since strain hardening has a strong influence on observed hardness when upward flow is present but has very little influence when there is no upward flow (since the quantity measured with full elastic constraint is the initial yield stress a t the plastic-elastic boundary), i t is very important to avoid upward flow with such materials if consistent and meaning- ful hardness values are to be obt,ained. I n addition to having sufficient material beneath the indenter, upward flow may be avoided by use of a very blunt indenter (ball of very large diameter) that has a relatively rough surface and is not lubricated. The experimental result,s presented in the Discussion support this view.

Two Opposing Indentations When two opposing indenters act upon a specimen (Fig. 12))

there will be no interaction, and the full constraint will be developed when the spacing of the two punches is sufficiently great. I t is of interest to find this limiting value of punch spacing for an axisymmetric indenter.

It has previously been shown that for a single punch the distance between the effective point of load application and the lowest point on the elastic-plastic boundary (D' in Fig. 6) is equal to 2.16~. The maximum shear stress at this point (D' in Fig. 6) will equal Y/2 and

By definition the constraint factor (C) is

and sttbstituting into equation (43)

When two punches are present (Fig. 12), the stress at D' in Fig. 12 will receive a contribution from each, and when the material a t this point just goes plastic

where dzl is the distance from the second load to D' in Fig. 12. Equating (45) and (46), we obtain

The punch spacing corresponding to any degree of interaction between the loads may be found from this equation. For example, if the interaction reduces C by 1 percent, then C'/C is 0.99 and d,/d,' will be 0.1. From Fig. 12, it is evident that

h a, d,' b lldz b - - - - + - - 2 - = - - - - 2a 2a 2a 2a 2a a

(48)

Fig, 12 Two opposing spheres indenting a specimen

As previously found for a single punch d,/a = 2.16 and b/a =

cot $o = 0.344 and upon substituting these into (48), we find.

Thus, when the two punches are spaced such that h/2a =

11.54, the load required will be 1 percent less than when the indenters act independently (h/2a = a).

Similarly, for a 5 percent reduction in the constraint factor (C1/C = 0.95), the value of h/2a is found to be 5.56.

This calculation should not be extended beyond this point, because the values of b/a and d,/a that have been substituted are t,hose obtained from t,he single load analysis. As the degree of interaction increases, a more complex analysis is required in which b/a changes with the degree of interaction. I n fact, the analysis would become very complex since this then represents a case where both elastic and velocity constraints are operative.

While slip line field solutions exist for plane strain indenters which show that the constraint factor C' approaches one as h/2a approaches unity, a parallel treatment is apparently not available for pairs of axisymmetric indenters.

Discussion Dugdale [23] has studied the indentation characteristics of

well-lubricated conical indenters in work-hardened specimens of mild steel, pure copper, and pure aluminum. The flow stress of the material was determined from torsion tests. Fig. 13 shows the variation of the observed constraint factor with cone angle. 111 accordance with the Tresca flow criterion, the uniaxial flow stress has been taken equal to 2k. In these experiments, the specimens were 1 in, square and 0.6 in. deep, while indentations were up to 0.2 in. in diameter. The ratio of depth of specimen to impression diameter was, therefore, as low as three in some in-


r

f STEEL OCOPPER AALUMINUM

I EXPERIM~NTAL ----- HADDOW AND DANYLUK (1964) LOCKETT,I1963)

THEORY -D

I I 0

I 60 120 180 CONE ANGLE, 29, deg.

Fig. 13 Variation of constraint factor C = j/2k with cone angle (20) for cold-worked metals and smooth cones well lubricated with calcium stearate grease (Dugdale 1231). The nonsolid curves are from the theories of Haddow and Danyluk [I 11 and Locken [lo]

stances. While this will provide sufficient material beneath t,he indenter when 28 is small and there is substantial upward flow, it will be insufficient to allow a fully developed elastic constraint to be established for blunt indenters. The low values of constraint for steel a t cone angles of 120 and 140 deg (Fig. 13) is believed to be due to the spcimens being too shallow.

The experimental curve (shown solid) has, therefore, been drawn through the upper points in Fig. 13. This curve is seen to intersect C = 3.0 a t an angle 280 170. According to the analysis presented here, the con~t~raint factor (C) should be 3.0 until upward flow commences, which for a low friction indenter operating in a nonst.rain hardening material will be for values of 28 from about 170 to 180 deg. As the cone angle drops below 170 deg the value of constraint should fall, but not so rapidly as a theory based on upward flow would indicate, since there would be an elastic contribution to the total constraint (Sham and DeSalvo, [I]).

Also shown in Fig. 13 are the values predicted by Shield [9], Lockett [lo], and Haddow and Danyluk [I l l . These curves also plotted in Fig. 2 are seen to be in poor agreement with the experimental data. While the Haddow and Danyluk curve fits the data better, i t is for the case where no lip is present due to upward flow. The lip that is formed increases as 28 decreases which accounts for the divergence of the two theoretical curves as 28 decreases.

In order to check the Haddow and Danyluk curve in Fig. 13, specimens should be prepared by machining a conical pattern in the surface that precisely fits the indenter. If the mean punch pressure (p) required to initiate plastic flow is then measured, the values should be in better agreement with theory. When Haddow and Danyluk [Ill performed such tests the data given in Table 4 were obtained. The experimental values given have been corrected for friction by dividing by 1 + p cot 8, where p = 0.06.

Table 4

Cone semiangle

(@I deg 45 60 75 90 ,

Mean Stress P -c= -- on Cone (p), psi Y

Experi- Experi- mental Theory mental Theory 76,600 67,822 2.52 2.23 78,100 74,420 2.57 2.45 79,500 79,100 2.62 2.60 55,306 86,490 2.80 2.85

The experimental values are now seen to be greater than the theoretical ones (except for 8 = 90 deg where the theory presented here is t,o be preferred over t,he slip line solution used by Shield, since for 8 = 90 deg there will be no upward flow). This is probably due to the fact that, for all values of 8 other than 8 =

90 deg, the origin of the constraint will be mixed, being partly

- NEW THEORY 3- 1

/ r---*

,J& 2

P, I _.pP

p=8'

Fig. 14 Variation of consfraint factor C = ;/2k with square pyramid semiangle (0) for: (a) well-lubricated indenters, (b) roughened indenters (Dugdale [24])

I

0

due to upward flow and part,ly of elastic origin. Ignoring the elastic component of const.raint will give a predicted value that is too low which corresponds to what is observed.

Dugdale (241 has presented similar data for square pyramidal indenters operat,ing in cold-worked specimens of mild steel, pure

*STEEL OCOPPER bALUMINUM

copper, and silicon aluminum. The steel and copper specimens had the same dimensions as before (1 in, square by 0.6 in. deep), while the aluminum specimens were l l / z in, cubes. Flow stress values in shear (k) were determined from torsion tests; and by t,he Tresca flow criterion, the uniaxial flow st,ress wm taken to be 2k.

Fig. 14 shows observed constraint factors versus the pyramid semiangle 8 for: (a) carefully lubricated smooth indenters, and (b) for indenters t,hat had been roughened by etching in nitric

0 30 60 90

acid. The aluminum points are seen to lie distinctly above those for steel or copper in Fig. 14(a) due to the greater depth of the aluminum specimens. The dotted curve drawn through the aluminum points is believed to better represent the constraint to be expected with specimens of sufficient depth to allow the elastic constraint to fully develop.

Fig. 15 shows the size of lip formed due to upward flow for each of the experiments of Fig. 14(a). Upward flow is seen to cease at an angle (0) of about 75 deg for the aluminum specimens, but not until 8 E 85 deg for the copper specimens.

The dott.ed curve for the low friction indenters in Fig. 14(a) is seen to intersect the C = 3.0 line a t an angle B0 75 deg. There will be no upward flow for values of 8 > 75 deg in this case. Thus, a smooth indenter operating on cold-worked materials may be considered blunt (no upward flow) when 20 is 150 to 180 degs.

The dotted curve for the high friction situation in Fig. 14(b) intersects the C = 3.0 line a t 8 E 50 deg. Thus, a high friction indenter operating on a cold-worked material may be considered blunt when 28 is as low as 100 degs.

The constraint factor is seen to become more sensitive to the angle of the indenter and the degree of friction as the angle of the indenter decreases. For a pyramid having 8 = 30 deg, the constraint factor C is seen to vary from about 1.8 when the friction is low to 3.6 when the friction is high.

Dugdale [25] has given square pyramid data for annealed copper. These result,s are shown plotted in Fig. 16(a) for indenters of low and high friction. In this case, constraint factors are now shown due to the large change in torsional flow stress (k) with


8, deg. I 80

Fig. 15 Height of lip produced by square pyramid indenters of Fig. 14(a) (Dugdale 124))

shear strain [Fig. 16(b)], which makes i t uncertain as to the value to be used when finding C = p/2k when there is upward flow. Therefore, observed values of p are shown plotted against 8 in Fig. 16. The value'of p is seen to rise as 0 decreases, par- ticularly when friction is high.

Atkins and Tabor [15] have also studied the indentation characteristics of conical indenters. Specimens of pure copper and mild steel were tested a t different degrees of strain hardening ranging from the fully annealed state to natural strains as high as 1.39. All copper specimens initially measured 3/, D by 1 in. high. Cold work was introduced by reducing the height of the specimens in uniaxial compression. The fully work-hardened copper specimens had a height of but in. The steel specimens had initial dimensions of D by 3/1 in., and the most highly worked specimens had a final height of but 3/16 in.

The constraint factor (C) for the most heavily worked specimens (both steel and copper) is shown plotted in Fig. 17 against cone angle (20). In determining these values of C, Atkins and Tabor took the flow stress to be that corresponding to the cold-worked material before indentation. No additional work hardening was considered to occur during indentation of the heavily worked material. Fig. 17 corresponds to results of Dug- dale shown in Fig. 13. Beth curves have a minimum, and anomalously low values of C are observed for large values of 28. The low values of C for large values of 28 are believed due to the specimen height being too small to allow'a fully developed elastic constraint to be produced. The difference in the magnitude and location of the minimum values of C in Figs. 13 and 17 is believed due to differences in lubrication (friction) specimen height relative to indentation diameter, degree of work hardening, and the met,hod employed in determining the flow stress (Y) used in finding the constraint factor (C = p l y ) . The basic flow curves were obtained from toision tests in the case of Fig. 13, but from compression tests in Fig. 17. Also, a correction for friction was made in the case of Fig. 13, but not in the case of Fig. 17.

However,.both Figs. 13 and 17 appear consistent with the new theory, when t.he fact that the specimens were too shallow for blunt indenters is taken into account.

Atkins and Tabor [15] also found an increase in C with decrease in degree of previous work hardening, part~icularly for low values of 20.

From the foregoing discussion, it is apparent t,hat the constraint factor for an axisymnietric indenter will vary with the

8, deg. (a)

6 0

4 0

2 0

0

SHEAR STRAIN, y (b)

Fig. 16(a) Variation of mean pressure (I;) on square pyramid punch with pyramid semiangle (8) for annealed copper. (6) Shear flow stress (k ) for annealed copper against shear strain (7) in torsion test (Dugdale [251).

I J 60 120 ' 180

CONE ANGLE, 28, deg.

'.J

Fig. 17 Variation of constraint factor C = (5/Y with cone angle (20) for fully worked steel and copper specimens lubricated by a water base graphite grease (Atkins and Tabor 1151)

indenter angle (28), friction, and degree of strain hardening as shown diagrammatically in Fig. 18. The advantage of using a blunt indenter (no upward flow) is that it is essentially invariant relative to indenter friction and the tendency for the metal to work harden. Upward flow may be prevented by use of a rough indenter, no lubricant, a large indenter angle (28), and a n extensive specimen. Another advantage of an indenter that gives no upward flow is that the flow stress being measured is obvious. This corresponds to the strain in the elastic-plastic boundary. When there is upward flow, t,he flow stress involved in the constraint factor is that corresponding to the mean strain in the zone of flow, which increases significantly as the wedge angle decreases (Atkins and Tabor [I51 ).

i


0,"

-----------

0 LUBRICATED INDENTER 0 ROUGH INDENTER

0 3 0 6 0 90

Fig. 18 Schematic diagram showing variation of constraint factor C for axisymmetriz indenters of different semiangle 8. Dotted curve is for blunt indenter.

Acknowlegment The authors wish t'o acknowledge a grant from the National

Science Foundat,iorl in support of this work.

References 1 Shaw, M. C., and DeSalvo, G., "A New Approach to Plasticity

and Its Application to Blunt Two Dimensional Indenters," to be published.

2 Prandtl, L., " ~ b e r die Haerte Plastischer Koerper," Nach- richten der Akademie TT~issenschaffen in Gottinoen, Mathematisch- Physikalische Klasse, 1920, p. 74.

3 Hill, R., The Mathematical Theory of Plasticity, Oxford, 1950. 4 Hencky, H., "Uber einige statisoh bestimmte Faelle des

Gleichgewichts in Plastischen Koerpern," Zeitschrift fuer Angaw andte iWathematik nnd Mechanik, Vol, 3, 1923, p. 241,

5 Haar, A., and van Karman, T., "Zur Theorie der Spannungs- zustaende in Plastischen und Stanartigen Medien," ivachrkhten der Akademie IT7issenschaften i n Gtittingen, Afathematisch-Physikalische Rlasse, 1909, p. 204.

6 Ishlinsky, A. I., "The Problem of Plasticity with Axial Sym- metry and Brinell's Test," Prikladnaia Mathemalika i Melzhanika, Vol. 8, 1944, p. 201.

7 Shield, R. T., and Dmcker, D. C., "The Application of Limit Analysis to Punch-Indentation Problems," JonrnaL of Applied Mechanics, Vol. 20, 1953, p. 483.

8 ,Levin, E., "Indentation Pressure of a Smooth Circular Punch," Quarterly Applied Mechanics, Vol. 13, 1954, p. 133.

9 Shield, R. T., "On the Plastic Flow of Metals under Conditions of Axial Symmetry," Proceedings of the Row1 ij'ocietu, Vol. A23X 1955, p. 267.

10 Lockett, F. J., "Indentation of a Rigid/Plastic Material by a Conical Indenter," Journal of thr Merhanics and Physics of Solids, Vol. 11, 1963, p. 345.

11 Haddow, J. B., and Danyluk, H. T., "Indentation of a Rigid- Plastic Semi-Infinite Medium bv a Smooth Rigid Cone Fitted in a Prepared Cuvity," ~nternat ional-~ournal of M e ~ h a n i c d Science, Vol. 6, 1964, p. 1.

12 Samuels, L. E., and Mulhearn, T. O., "An Experimental In- vestigation of the Deformed Zone Associated with Indentation Hard- ness Impressions," Jottrnal of the Mechanics and Physics of Solids, Vol. 5, 1957, p. 125.

13 Mulhearn. T . 0.. "The Deformation of Metals bv Vickers-Ts~e Pyramidal ~ndenters," Journal of the Mechanics &d ~ h y s i c s o f Solids, Vol. 7, 1959, p. 85.

14 Atkins, A. G. and Tabor, D., "On Indenting with Pyramids," Znternalional Jo~irnal of Mechanical Science, Vol. 7, 1965, u, 647.

15 Atkins, A. G., and Tabor, D., "Plastic ~ndentatiod in Metals with Cones," Jonrnal of the Mechanics and Physics of Solids, Vol. 13, 1905, p. 149.

16 Williams, G. H., and O'Neill, H., "The Straining of Metals by Indentation Including Work-Softening Effects," Journal of the Iron and Steel Institute, Vol. 182, 1956, p. 266.

17 Akyuz, F. A,, "Indentation of an Elastoplastic Halfspace by Rigid Punch," Ph.D. Dissertation, Rice University, Houston, Texas, 1966.

18 Akyuz, F. A., and Merwin, J. E., "Solution of Nonlinear Prob- lems of Electroplasticity by Finite Element Method," A Z A A Jour- nal, Vol. 6, 1968, p. 1825.

19 Timoshenko, S., and Goodier, J. H., "Theory of Elasticity," McGraw-Hill, New York, 1951,

20 Davies, R. IM., "The Determination of Static and Dynamic Yield Stresses Using a Steel Ball," Proceedings of the Royal Society, Val. A197, 1949, p. 416.

21 Boussinesq, J., Comptes Rendus, 1892, pp. 114, 1465, 1510. 22 Osias, J., "A Study of the Indentation of Metals," unpublished

report, Mechanical Engineering Department, CarnegieMellon Uni- versity, 1968.

23 Dugdale, D. S., "Cone Indentation Experiments," Jolcrnal of the Mechanics and Physics of Solids, Vol. 2, 1954, p. 265.

24 Dugdale, D. S., "Experiments nith Pyramidal Indenters- Part I," Journal of the Mechanics and Physics of Solids, Vol. 3, 1955, p. 197.

25 Dugdale, D. S., "Experiments with Pyramidal Indenters- Part 11," Jottrnal of the Mechanics and Physics of Solids, Vol. 3, 1955, p. 206.

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13 Journal of Engineering for Industry

on the plastic flow beneath a blunt axisymmetric … · on the plastic flow beneath a blunt ......

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