THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Technical Report THE TILTING PUNCH R. M. Haythornthwaite ORA Project 05894 under contract with: DEPARTMENT OF THE ARMY ORDNANCE TANK-AUTOMOTIVE COMMAND DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-AMC-0980T DETROIT, MICHIGAN administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1965

TABLE OF CONTENTS Page LIST OF FIGURES v SUMMARY vii 1. NOTATION 1 2. INTRODUCTION 2 3. CONCENTRIC LOADING 3 4. ECCENTRIC LOADING 6 Collapse loads 9 Soils 10 5. TRUNCATED WEDGE 13 REFERENCES 16 iii

LIST OF FIGURES Figure Page 1 A possible deformation mode for the tilting punch with concentric load. 4 2 Deformation mode for the punch with an eccentric load. One possible statically admissible extension of the stress field is shown in (a). 7 3 Yield curve for the tilting punch drawn in terms of the non-dimensional variables m = 8M/ps2; w = W/ps. 11 4 Deformation mode for eccentric loading of a punch bearing on the end of a truncated wedge. One possible statically admissible extension of the stress field is shown in (a). 14 v

SUMMARY 1 The solution of Prandtl for plastic indentation of a half-space by a rigid punch in plane strain is extended to cases where the load is applied eccentrically. Complete solutions are obtained for all possible combinations of applied load. The solution is also given for the obtuse angle truncated wedge loaded at the end, of which the former problem is a special case. vii

1. NOTATION M applied moment W applied load c,cp material constants for a Coulomb body k shear stress at yield m,w dimensionless variables p indentation pressure s width of punch a width ratio P wedge semi-angle b displacement e rotation f- = sO/8 1

2. INTRODUCTION The extensive literature on the theory of punch indentation seems devoid of any reference to tilting modes. The latter are to be expected when the load 2-4 is applied eccentrically, and they have been observed in tests on sands and clays 3 An estimate of the failure load can be found easily3 by introducing 1 an eccentrically placed stress field of the type used by Prandtl, with the zone of surface pressure extending over part of the width of the punch, the remainder being unloaded. The solution is incomplete in the sense this term is used in plasticity theory, unless the load can be shown to be statically admissible and unless at least one kinematically admissible mode of deformation can be associated with the stress field. It is the purpose of this note to supply these ingredients for the plane strain indentation of a level surface and also for the more general case of an obtuse angle truncated wedge loaded at the end. It will also be shown that a series of tilting mechanisms are possible even when the Loading is concentric. The latter point will be dealt with first. 2

3. CONCENTRIC LOADING Three velocity fields have been proposed for plane strain indentation of a half-space by a smooth rigid punch, l6 7 but none accommodates rotation of the punch. A broad class of velocity fields accommodating rotation can be defined by noting the restriction placed on the velocities by the stress field, which requires only that the sign of the plastic power be everywhere positive. An example of a possible velocity field is shown in Figure 1, where the extent of the deforming zone is shown in (a), and the corresponding hodograph in (b). The material has been assumed to be one in which the yield point stress is independent of the hydrostatic component of stress, so that the stress deformation characteristics form an orthogonal net and there is no dilation during plastic flow. The instantaneous center for motion of the punch has been placed in an arbitrary position outside a zone defined by vertical lines drawn through the corners of the punch B and B1, so that the indenting surface of the punch does not lift at any point. The slip between punch and surface, bb' at B, is defined at once from the requirement that the velocity in ABC near AB must be normal to AB, in view of the presence of rigid material below AC. The slip at other points on the punch surface is not uniquely determined but must meet certain conditions. There is a one-to-one correspondence between points on BB1 in (a) and points on bbl in (b). Progression from B to B1 is accompanied by progression from b to bl. In a progression from B to B1, the change in the velocity component in direction EF cannot be positive and the change in direction EF1 cannot be negative, in order that no element of material should act as an 5

qftm. qound Bu-iFqTq- aq JOJ apom uo~q.BttGIoja p a-TqTssod V 1- 9Ani ldDJo60pOq ( ) Iq q (q —-q qD) S6I/,O/9A 0 \p 3 D \aopns M( II (') \\ s$/l/a Ole O asojns M

engine during the motion. It follows that the change in velocity for the indented material on line BB1 for a progression from B to B1 must have a negative projection on ob' and a positive projection on obl'. In terms of the hodograph, Figure lb, vectors for increments of velocity associated with progression from B to B1 must lie within an arc between the directions b'o and b'b. In the figure, the path b'e"e'j'bl'bi' has been chosen arbitrarily to meet this condition and to illustrate various types of discontinuity that can occur. It is interesting to note that a velocity discontinuity oa' on ACD must be accompanied by a discontinuity b.i" b'" in surface velocity at B1. The point b' could be chosen on ob' and in that case, zone AC1DlB1 is rigid and motion is confined to zone ACDBB1. On the other hand, no choice of surface slip velocities can eliminate deformation from any part of the latter zone. The surface velocities for points on BD cannot be the same, but must decrease from B to D, with possible discontinuities, as at H and D in the example shown. On B1D1, the velocity can be the same at various positions, as between M1 and H1, and in particular zero, as between H1 and D1 in the example. The non-tilting punch can be considered as a limiting case of the tilting punch when the instantaneous center is removed an indefinite distance either to the right or to the left of the punch. The velocity field (Figure 1) then coincides with that given by Spencer7 as the limiting (quasi-static) field for dynamic indentation without tilting. On 1 6 the other hand, the velocity fields proposed by Prandtl and Hill are inadmissible in the presence of even the smallest amount of tilting, because it becomes impossible to ensure that every element of material in the deforming zone dissipates work. 5

4. ECCENTRIC LOADING A case in which the instantaneous center of motion of the punch lies between the vertical edges of the punch is shown in Figure 2. Clockwise motion about the center 0 is assumed, so that points on the indenting surface to the left of B1 (Figure 2a), will rise and points to the right will fall. It appears reasonable to introduce a Prandtl stress field ACDD1C1 which is associated with contact pressure over BB1 and which leaves the rest of the punch surface unloaded. To prove this choice of stress field correct, it is necessary to establish an associated velocity field and also to show the existence of an extension of the stress field which is statically admissible throughout the half-space. A velocity field can be found quite simply by adapting the field discussed for the case of concentric loading (Figure 1). Due to the presence of the punch surface to the left of B1 (Figure 2a), the vertical component of velocity just to the left of B1 must tend to zero, and this condition is met by the field shown (Figures 2a and 2c). A statically admissible extension of the Prandtl stress field was first given by J. F. W. Bishop. An alternative suggested by Shield9 will be repeated here because it is simpler and deserves more recognition. The field (Figure 2a) is obtained by continuing the stresses of the radial zone ABC outwards and introducing a discontinuity AE chosen so that the principal stresses just to the left of AE are horizontal and vertical. The path of the discontinuity is constructed.starting at A (Figure 2a) using a succession of constructions of the type illustrated in the stress plane, (Figure 2b) which shows the discontinuity 6

W M O -- — r -- *" - surface velocities X// - 1 B D (a).ei k b/ b b (b) stress plane C ) hodograph Figure 2. Deformation mode for the punch with an eccentric load. One possible statically admissible extension of the stress field is shown in (a). 7

for point E (Figure 2a) To the left of AE, the horizontal and vertical stresses on AE a.re assumed to be maintained in columns of material downwards and to the left. The pressure P2 decreases monotonically as the depth increases, so the material in the zone EAE1 is below yield. The cycloidal trace of poles for stresses in zone ABC is shown in Figure 2b and it is confirmed that the velocities shown in Figure 2c are associated with absorbtion of work by every element of material. The deformable zone can be restricted to ACDBB1 in two steps. Zone EAE1 is below yield and hence rigid, so it will be rigid in all solutions.5 Inextensibility of the slip lines ACD and AC1D1 then ensures that the entire zone below DCACiDi is necessarily rigid in this (and hence in all) solutions. In establishing the velocity field, it has been noted that the velocity at B1 just to the left of AB1, must be zero to avoid interference with the punch at B1; however, in view of the sense of the shear stress in zone B1C1Dl, the velocity in the direction ABi cannot increase across any element along the line BC1. As a monotonic function zero at both B1 and C1, the velocity must be zero everwhere on BaC1 and so the zone AB1CiD1 must be rigid. The deformable zone is no) confined to ACDBBE Contact with the punch must be maintained along BB1 in the assumed stress solution A.lso( there is no velocity across AB1 in view of the rigidity of ABiC1; hence velocities in ABB1 and thus in the rest of the deforming zone ACDB1 are completely determined by the motion of the punch. The deforming reg:ion must reach out to D and there is no alternative possible. 8

In the case of concentric loading discussed in Section 2, an extended stress field of the type shown in Figure 2 can be constructed below ACDD1C1 of Figure 1 and hence the material below that zone can be shown to be necessarily rigid. The deformable zone comprises the whole of ACDD1C1. COLLAPSE LOADS 1 The Prandtl indentation pressure is P1 = (2 + r)k (1) where k is the yield stress in simple shear. This pressure acts across the interface BB1, and equilibrating it with an applied force W and a moment M applied to the center of the punch, we obtain W = aspi, ^1> (2) M = (l-a)sW/2 J where s is the width of the punch and as is the length of BB1. The yield curve in the M, W plane for the range M > 0 is found by eliminating a between equations (2). The analysis is repeated for M < 0. It is convenient to introduce the dimensionless variables m = 8M/pis 2 (3) w = W/pis and the results can then be summarized by the single expression Iml = 4w(l -w) (4) 9

which defines the yield curve shown in Figure 3. The rate of energy dissipation is Ws + MG, where 6 is the displacement rate of W and 0 the rotation rate of M. It can be rewritten D = pis(wb + m4) (5) where s = se/8. Hencell w and m can be regarded as generalized stresses and 6 and' as corresponding generalized strains for the purposes of plasticity theory. It follows that the yield surface shown in Figure 5 is necessarily convex, and if a corresponding generalized strain vector is drawn for any point on the surface, it will lie in the direction of the outwards drawn normal to the yield surface. With this interpretation, the various features of Figure 3 can be readily identified. Point A corresponds to concentric loading, and the presence of a corner in the yield surface at that point permits a fan of normals to various supporting planes; each corresponding to a different position for the instantaneous center 0 (Figure 1) as it ranges over all possible locations outside the punch. These many modes of deformation, with various ratios of sinking and tilting are all associated with the same indentation load. Point B corresponds to lifting of the punch with no indentation, and hence zero load. The arrows shown at that point in Figure 3 correspond to tilting about one or other of the corners. The curves between A and B represent the loads associated with the tilting mode described in this section. SOILS The analysis can be repeated for the indentation of material, the yieldpoint stress of which is sensitive to the level of mean stress. Such materials 10

w A +1 /\B +1 m Figure 3. Yield curve for the tilting punch drawn in terms of the nondimensional variables m = 8M/ps2; w = W/ps. 11

are commonly used as models for granular media and other soils. In order to illustrate the differences in procedure, a material of this type will be used in the following discussion of the truncated wedge. 12

5. TRUNCATED WEDGE The analysis for indentation of a half space can be adapted to the general case of the truncated wedge without introducing new difficulties. The material 12 of the wedge will be assumed to obey the Coulomb yield criterion in which it is postulated that deformation can occur when, on any plane in the material, the shear stress reaches the value c - a tan cp, where a is the normal stress across the plane and c,cp are material constants. This model is often used to represent granular media. In terms of the stress plane (Figure 4b), yield will occur when the largest Mohr circle associated with the stress state touches the sloping line defined by the dimension c and the angle Cp. If the body is ideally plastic, 13 yield will be accompanied by dilation such that relative motions will not be parallel to the directions of slip, but will subtend an angle cp with them. Details of the construction for a wedge with semi-angle 45 are shown in Figure 4, where the same notations have been used as in Figure 2 to facilitate 14 comparisons. The pressure across BB1 is P2 = c.cotcp [e tanp 2(T/4 + p/2) - 1] (6) where P is the wedge semi-angle. The non-dimensional yield curve shown in Figure 3 applies to this case if the pressure defined in equation (6) is substituted in equation (3) for the Prandtl indentation pressure pi = (2 + r)k. For material insensitive to mean stress, the pressure across BB1 is p3 = (2 + 23)k (7) 13

b I', W Ml,/ss p^^ne" 1 —-J /t [. ^velodtie h( ograph (a) _b-_i c P'velocities tb' (a) Figure 4. Deformation mode for eccentric loading of a punch bearing on the end of a truncated wedge. One possible statically admissible extension of the stress field is shown in (a).

Again, the yield curve shown in Figure 3 applies if p3 is substituted for pi in equation (3). The solutions are valid for jr/2 > P > O. If P < 0, the wedge is reduced in size at points remote from the punch and a local failure would occur there. If P > t/2, the extension of the stress field AC1D1B1 cannot be constructed due to the presence of the punch interface within the required zone. 15

REFERENCES 1. L. Prandtl, Nach. Koeniglichen Ges. Wiss. Goettingen, Math.-Phys. K1. 74 (1920). 2. C. Ramelot and L. Vandeperre, Compt. Rend. Rech., I.R.S.I.A., Brussels, No. 2 (1950). 3. G. G. Meyerhof, Proc. Third Int. Conf. Soil Mech., Foundation Engng., Zurich 1, 440 (1953). 4. A. R. Jumikis, Proc. Am. Soc. Civil Engnrs. 82 (SM1), 1(1956). 5. J. F. W. Bishop, A. P. Green, and R. Hill, J. Mech. Phys. Solids 4, 256 (1956). 6. R. Hill, Quart. J. Mech. 2, 40 (1949). 7. A.J. M. Spencer, J. Mech. Phys. Solids 8, 262 (1960). 8. J. F. W. Bishop, J. Mech. Phys. Solids 2, 43 (1953). 9. R. T. Shield, J. Appl. Mech. 21, 193 (1954). 10. W. Prager, Trans. Roy. Inst. Tech., Sweden, No. 65 (1953). 11. W. Prager, Proc. 8th Internat. Congr. Appl. Mech. (Istanbul, 1952) 2, 65 (l956). 12. R. T. Shield, J. Mech. Phys. Solids 4, 10 (1955). 13. D. C. Drucker and W. Prager, Q. Appl. Math. 10, 157 (1952). 14. R. T. Shield, J. Matho Phys. Z3, 144 (1954). 16