THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Technical Report RADIAL EXPANSION IN PLATES COMPOSED OF IDEAL COULOMB MATERIAL p4 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 September 1965

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TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. INITIAL MOTION 3 3. CONTINUED MOTION 11 REFERENCES 20 iii

LIST OF FIGURES Figure 1 Plane stress section of Coulomb yield criterion. 2 Yield curve for 3= 3 and N 3. 3 Annular plate a) Undeformed, b) Deformed without thickening: f < ncr' c) Deformed with thickening: B > Ocr. Limit load as a function of plate size.

SUMMARY Analysis for the plastic expansion of an annular sheet subject to inplane radial forces is extended to ideally plastic materials in which the yield criterion is sensitive to the mean stress. Upper and lower bounds to the yield pressures, are found for the general initial motion problem where differential pressures are applied to the inner and outer perimeters. The subsequent motion problem is discussed for the particular case where the outer edge remains unloaded. vii

I. INTRODUCTION The radial expansion of thin sheets in the plastic range has been the subject of intensive investigation. Taylor [1], Hill [2], Prager [3], Hodge and Sankaranarayanan [4], and Sokolovsky [5] have provided various elements of plane stress solutions based on the rigid, ideally plastic materials, and Ford and Alexander [6] have done extensive numerical analysis for the elasticplastic material with strain hardening. Nordgren and Naghdi [7,8] have extended the work to the case of combined twist and expansion. All the above work has been based on a'structural' approximation for plane stress in which no attempt is made to follow in detail the nature of the lateral displacements. Some of the flows implied are impossible of realization and in consequence the associated loads will be lower bounds rather than exact yield loads. In the present work, reference will be made in the first instance to the more general analysis for the three-dimensional axisymmetric body due to Shield [9]J Furthermore, the analysis will be carried out for a material which follows the yield criterion of Coulomb [10], where the strength is dependent on the intensity of the mean stress. As a consequence of the latter generalization, the results may have some relevance to the strength of sheets composed of soil-like materials and ice. COULOMB YIELD CRITERION This criterion has been discussed by Shield [11] in a manner relevent to the present work. Being based on the concept of internal friction, the criterion 1

rests on the postulate that flow can occur whenever the shear stress developed across any plane reaches a value c - a tan p, where a is the normal (tensile) stress on the plane and C, cp are intrisically positive material constantso The criterion can also be expressed in the form 2 a1 = N a5 2cN (a3 > 2 > (1) where a1, a2, v) are the principal stresses and N = tan (. + ~) The six planes obtained by interchanging a1l a2 and oa in Equation (1) define the Coulomb yield surface in principal stress space. In the theory of ideally plastic solids, a stress strain relation is introduced in a form which ensures uniqueness of applied loads in boundary value problems with force boundary conditions. It is sufficient to postulate that an element of the material cannot release work in any closed loading cycle [12]. The yield surface must be convex, and the vector formed by the increments of plastic strain associated with a particular stress state on the yield surface must lie in the direction of the outwards drawn normal. The vector formed by the principal strain increments is defined uniquely when the yield surface has a continuously turning tangent but at corners it can lie perpendicular to any supporting plane. The yield surface for the Coulomb criterion comprises a series of six planes which intersect to form corners, and the flow rule will vary according to the position of the stress state point on the surface. For example, on the side represented by Equation (1) 1 E2 =E -1 0 N2 ( 2)

On the adjacent side (C2 > o3 _> 1)'2 2(3) ~: = * X - - N1 N 0 (3) and so at the corner common to these sides ~ -O - N N (4) l:: 3= -c - c N where a, are positive numbers. The material constant y? (and hence N = tan ( + (P)) is found to be 4 2 positive in practice, and the flow rules for all the various sides and corners then imply dilation of the material during flow. II. INITIAL MOTION An annular disc occupying the region r2 > r >-; ho/2> z >- ho/2 initially is subject to the boundary tractions &rl _ Crl r r] P2= or - P=;h =] = 0. (6) r=ao r r z=+ho/2 Z z=-ho/2 Upper and lower bounds for the combinations of pressure which induce initial deformation will be sought. If ho is either very large when compared with r1 or very small when compared with r2, the direction of the principal stresses will approach the axial, circumferential and radial directions, by symmetry. The bounds to be obtained are based on stress distribution in which these directions are maintained throughout, and in particular the axial stress is 3

zero throughout, but it is to be noted that certain of them are nevertheless valid for plates of any thickness, including those of intermediate thickness where the actual stress states may be quite different. The cross-section of the yield surface, Equation (1), formed by the plane Cz = 0 is shown in Figure (1). Plastic regimes associated with the various sides and corners will be investigated. REGIME AB The ordering of the principal stresses is ar > a > _ and Equation (1) becomes ae = Nar ~- Po (7) where Po 2cN is the compressive yield strength. For radial equilibrium dar ar-ae dr + 0 (8) dr r and on.integration after substitution for ae from Equation (7): P0 N2r 2 1 N -l (To > Or > () (9) Po 2 N2_1 N2- + C1N r The flow rule associated with Equation (7) is *Due to the singularity9 the solution for the Tresca yield criterion cannot be obtained by substituting N = l;however it can be retrieved by setting N = 01+ wher eE + 0

rEr'Es: E = N -1 0 (10) which, when written in terms of the velocities u, w, becomes au + N2 u 0 dr r a = 0 a6 az dr where the last equation expresses the necessary and sufficient condition for coi.ncidence of the principal directions of stress and strain rates. Setting z=0 = O, w = 0 everywhere, by the second of (11) and by the third of (11), u is then a function of r only. The first of (11) is now an ordinary differential equation and the solution is -N2 u = C2r (12) REGIME BC The ordering is Ur > C > oz. Equation (1) becomes N2r - 2CN (13) so the flow rule is er ~ E ~ e = N2 O 0 -1 In terms of velocities au + N2aw o dr dz7 5

U o =04 (14) r and also -— +-= 0 dz dr The velocities are zero everywhere, so no solution is possible. REGIME CD The ordering is aC > ar > oz. Equation (1) becomes z = N2Oe - 2CN (15) ~ 2 so the flow rule is er: Ez = O: N ~ -1. In terms of velocities 0 dr U +N2 (16) r z= and also au + aw -+ = 0 aw dr The displacements are zero everywhere and no solution is possible. REGIME DE The analysis parallels that given above for regime AB., and it is found that -2 Po N -1 (17) = + C r N2-l! N2

-1/N2 u = c4r1/ (18) REGIMES EF and FA These regimes are analogous to regimes BC and CD respectively and they also can be discarded because no compatible motion is possible. CORNERS A,B,D and E In view of radial equilibrium, Equation (8), these state points can be reached only at discrete radii. CORNER C This corner lies at the intersection of the facets BC, Equation (13) and CD, Equation (15), where or = oe = To and the flow rule is 2 2 where c,, are positive numbersso r + + N2 = 0 which becomes, when written in terms of the velocities au + u + N2 aw = dr r az (19) lu aw and also dz dr dz dr Several solutions have been given previously [13]; one of the simplest is u - 2 N2 - z2/r2' -1, 7(20) w = -tan -1 N2r2/z2 P 1 it~~

where z > 0, The deforming zone is restricted to the sector N > z/r > -N. CORNER F This corner is analogous to corner C discussed above. The stresses are ar a= - Po and a compatible velocity field, [13], is 2 - u = - -//N2 - z1/' (21) w = 2 tan-1 r2/z2N2 1 The deforming zone is restricted to the sector 1/N > z/r > 1/N. It is immediately evident that exact solutions will be available for certain ratios of the inside and outside pressure by the simple expedient of consigning the entire plate to the regimes AB or DE, or to the corners C or F. The solutions are exact for all plate thicknesses. For other ratios of rl and r2 these regimes can be used as the basis for upper bounds. When regime AB applies to the entire plate, substitution of the boundary conditions, Equation (16), in Equation (9) leads to 2 2 N -1 1 -P P2 = P1 (22) N2_1 ( o > P1, P2 >- 1/N2) where Pi = P1/Po; P2 = P2/Po and p=bo/aoo When regime DE applies throughout, Equation (17) leads to N-2_1 N-2-1 2 1 l-~ (23) (1 > p1, P2 > 0) 8

The singular points C, F give rise to complete solutions for the case P1 = P2. The above cases appear to exhaust the exact solutions available when no restriction is placed on the thickness ho. Exact solutions for other ratios of PI and P2 presumably reqidre stress distributions where the principal stresses are no longer in the directions r, E, z. These solutions will lie within the range of the general equations for axi-symmetric plastic flow [9, 14]. THE THIN PLATE In the case of the thin plate, the solutions given above still apply, and they can also be used as a basis for quite close inner and outer bounds. which are correct bounds within the framework of three-dimensional theory (Bounds obtained by the'structural' theory of plates in which displacement conditions in the lateral direction are often ignored are of dubious validity). Figure (2) shows the results of computations for the particular case where N2 = 3 2 = 3. Equation (22) becomes line fgh. Point g represents a complete solution but at other points the inequalities in Equation (22) are not satisfied and the points lie outside the yield surface, by the upper bound theorem of limit analysis [151e Equation (23) becomes line labc. In this case the inequalities are satisfied along ab and the remainder of the line represents an upper bound. The velocity fields associated with the singular points C,F can be employed to give very useful upper bounds. Placing the thin plate at z = 0, and equating the work done by the forces of the known solution to the work done by a general set of pressures P!, P2, we obtain the outer bound P2 - P1/ 1/N2 + 1/3N2 (24) 9

from corner C and the outer bound P2 = Pl/p + 1 - (2) from corner F These are shown as lines jkl and def respectively in Figure (2). Placing the thin plate at a height such that the radial velocity is zero at the inner edge for velocity fields (20) and (21), we obtain the outer bounds P2 = - 1/N2 (26) from corner C and P2 = + 1 from corner F. These are shown as lines hj and dc in Figure (2). Some inner bound lines are available at once by noting the property of convexity of the yield surface in the space of generalized stresses P1, P2. As a consequence of this property, any lines joining known solution points are inner bounds. Examples are bd and gj. Another bound can be obtained by assuming all stress state points lie in regime BC, Figure (1), where a~ = To. On integrating Equation (8), and substituting the boundary conditions (5), we obtain Equation (24) once more, subject to the requirement that 0 > Pi, P2 > - 1/N2. (27) Coincident upper and lower bounds have been found, so it follows that jk, Figure (2) represents the true yield line, even though a complete solution 10

has not been found. (The velocity field, Equation (20) is not compatible with stress state points on side BC, Figure (1).) A similar procedure based on side FE, Figure (1) confirms that Equation (25) is a lower bound, subject to the requirement! P15 P2 >o 1 > Pl' p2 >0 (28) and so line de, Figure (2) is also a true yield line. Points e and k, Figure (2) are evidently on the yield curve, so further inner bound lines may be obtained by joining eg and ka. III. CONTINUED MOTION* The general problem of continued motion presents great difficulty because the particular velocity field must be selected which allows the deforming body to remain in equilibrium at every instant [16]. For the expanding sheet, it so happens that the continued motion solution is straightforward when there is expansion without thickening (side ab and point g of Figure(2)). This is the only case where an incipient velocity field is known and for other cases resource must be made to the approximation of conventional plane stress theory. For the purposes of illustration, attention will be confined to an annular plate subject to internal pressure P1 = P and zero external pressure (P2 = 0), as shown in Figure(3a> Equation (23) reduces to *This section formed part of the Ph.oDo dissertation of M.D. Coon at The University of Michigano 11

1i-N: - p - - (29) N2- 1 This expression is valid when 1 > p > 0 or, in terms of A, N2N2/(N ) > P > 1. (30) For larger values of P, the yield load is not known, but a lower bound, p = 1, is established at once by noting that the stress field for the plate where = 2N /(N -1)(1) = is statically admissible for any larger plate. The expression for p in Equation (29) is shown as the solid lines in Figure(4). The dotted lines indicate the lower bound, p = 1, obtained above. Equation (31) gives a Ocr such that for < Or the plate expands without thickening as shown in Figure (3b) but for ~ > crthe determination will involve thickening as shown in Figure (3c). FIRST SOLUTION: X I ~cr The exact initial motion solution given above will be the starting point for this analysis. However, in this problem the stresses, velocities, and strain rates will be functions of time. Because there is no viscosity, time can be replaced by any variable which increases monotonically as the applied load increases. The coordinates will represent the current geometry, therefore, it will be convenient to take a (the inner radius) as a measure of time. 12

The velocity for this case can be found from Equation (18) by noting that when a is taken as the measure of time, the new variable r/a replaces the variable r. Therefore, Equation (18) gives = C4(a) (32) To determine u completely the velocity of some point must be specified and for convenience u = 1 at r = a is chosen. This implies that C4 = 1. The displacement, u, can be found from the definition of velocity, which becomes 1/N2 dr = u = (a) da r after making use of Equation (32). The boundary condition is r = r at a = a (34) o o where the initial coordinate of a point is given by ro. Its final coordinate will be r. The initial and final coordinates of a point can be related by ro = r - i(r,a). (35) By separating Equation (33) and using the above boundary conditions, one obtains r a 1/2 1 / N2 r-u 0 13

which can be integrated to give 2 (N2+)/N2 (N2/(N2+)/N N2+ = r -LNl N + aoN3 (57) Equation (37) implies that b b o b< be for all a > a ( a ao (38) Therefore, as the motion continues the plate becomes smaller (i.e. P becomes smaller) which implies that the entire plate remains on side DE of Figure(l). The stresses can be found from Equations (17) by noting that the constant C3 is now a function of time. This function of time can be obtained by using the boundary condition r = 0 at r = b which leads to e = PO q) (N -1)/N] N -1 L_ = N2(N21) 2 (b) (391)/N The time variation in the stresses is expressed in terms of b which can be related to a through Equation (37)~ In this solution, all of the equations for the three-dimensional problem have been satisfied and this is an exact solution to the continued motion problemr~ SECOND SOLUTION~ f > 1cr For this solution, the mode of deformation must involve thickening of the plate and it appears that there are no exact three-dimensional solu:aions

for this case. A solution can be obtained by averaging the strain rate Ez through the thickness. If this is done ez, is given by 1 Dh c -- (40) z h Dt where D a + u — (41) Dt 6t 6r Equation (41) denotes the rate of change following an element, i e. the material derivative. When this is done, the third of Equations (11) will not be satisfied, which implies that the principal directions of stress and strain rate do not coincide. Also, for the plate of variable thickness, the equilibrium equation will be (har) + h (r S) = 0 (4 kc) hr r where h is a function of r. As in the solution for p < Pcr7 the stresses, velocities, and strain rates will be functions of time, Statically admissible stresses can be found by introducing a rigid region on the range r2 < r K b (Figure(3c)). For rl < r < r2 the stresses can be taken on regime DE of Figure(1) with ar = 0 at r2 and ar = -Po at r1. Similar to the case for D < Bcrg it is convenient to choose some radius as a measure of time and r1 will be chosen. On the range rl < r < r2 the actual stress distribution can be found, as for the case I_ <K crn from Equation (17) using the boundary condition ~r = 0 at r = rg2 this gives

and 2 = |2 - (' /N and - N2(N2_1) r The time variation of the stresses is now expressed through r2 which can be related to r1 by noting that or = -Po at r = rl, and this leads to 2N2/(N2-1) r2 = r1 N (44) On the range a < r < rl the stresses can be taken at point E of Figure(l) where ar = -Po and = 0. (45) For the stresses to be statically admissible, the plate must thicken in this range and the thickening can be found from Equation (42) to be h-or h = (46) where the boundary condition h = ho (ho is the initial thickness of the plate) at r = r1 has been used. The stress solution is now complete. For the range rl < r < r2 the velocity can be found, as in the case of I < cr, from Equation (18). However, in this case, u = 0 at r = r2 and Equation (18) gives u identically 0 on this range. On the range a < r < r the velocity is found from the equation for the strain rates at point E, which is c0 + Ez + N2 r = 0 (47) This equation can be written in terms of u and h to give 16

N2 mu + u ah 1 ah +u o (48) 6r h ri1 h 6rl r Combining Equations (46) and (48) and using the boundary condition u = 0 at r = rl one obtains 1, r i NE r- * (49) The displacement u is found from the definition of velocity to be eu = r suI+iN +l)r - r1 ( o o) + at (t50) The solution for B > ~cr is now complete except that the inner radius a must be related to r1. This can be accomplished by rewriting Equation (47) in the following form d + Ez + Er = (1-N )r * (51) The right side of Equation (51) is known because Er is known. The left side of Equation (51) is the time rate of change of the change of the volume per unit volume. Thus.2. AV EO + EZ + Er = (l-N )E =V (52) where V is the current volume of the deformed material, i.e., all of the material inside the radius ri and AV is the time rate of change of the volume inside this radius. Because a0 is the initial radius of the hole and a its present radius the change in volume is given by 17

rl AV = 2ihrdr- Q%[r2l-a]. (53) a The rate of volume change per unit volume is then a An alternative expression for AV can be found ty substituting the values of the strain rates found from Equation (49) into Equation (52): 2 rl N -1 AV = 2 21thrdr. (5) N rl a Hence r1 r Hned i~ 2hrdr - ho[r2 -ao2 N -1 2hrdr. (56) lLa ] N2rl a The thickness h is given by Equation (46), and the radius a is a function of r1 which is to be determined. Equation (56) is equivalent to da 1 a - =' - al - - (57) drl N2 rl and with the boundary condition a = ao when rl = ao, the solution is a 2 Llrl +N a (N2+l)/N21 a 2+1 1 + (N — ]. (58) The first solution (I _ A r ) is an exact solution to three-dimensional problems for both the limit load and for finite displacement. The second solution (, > ner) is of the structural type used by previous authors for 18

analogous problems. In terms of the three-dimensional point of view, it gives a lower bound to the limit load. This solution cannot be exact in terms of classical plasticity theory, because the principal directions of stress and strain rate do not coincide. Also, the thickness changes in the plate are accounted for in the equilibrium equation, but the stress boundary conditions have not been satisfied on the part of the plate in which thickening occurs. 19

1REFERENCES 1. Taylor, G. I., "The formation and enlargement of a circular hole in a thin plastic sheet," Q. J. Mech., Appl. Math., Vol. 1, 1948, pp. 103-124. 2. Hill, R., "Plastic distortion of non-uniform sheets," Phil, Mag., Vol. 40, 1949y pp. 971-983. 3. Prager,, W. "tOn the use of singular yield conditions and associated flow rules" t J. Appl. Mech., Vol. 20, 1953, pp. 317-320. 4. Hodge, P. G., Jr. and R. Sankaranarayanan, "On finite expansion of a hole in a thin infinite plate." Q. Appl. Math,, Vol. 16, 1958, pp. 753-80. 5. Sokolovsky, V. V., "Expansion of a circular hole in a rigid/plastic plate," Applo Math., Mech. (Transl. PMM), Vol. 25, 1961, pp. 809-815. 6. Alexander, J. M. and H. Ford, "On expanding a hole from zero radius in a thin infinite plate,," Proco Roy. Soc. London Ser. A 226, 1954, ppo 543-561. Nordgren, R. P. and P. M. Naghdiq "Finite twisting and expansion of a hole in a rigid/plastic plate," J. Applo Mech., Vol. 30, 1963, pp. 60o5-612. 8.:Nordgren, R. P. and P. M. Naghdi, "Loading and unloading solutions for an elastic/plastic annular plate in the state of plane stress under confined pressure and couple," Intern. J. Engin. Sci,, Vol. 1. 1963 5 pp. 33-70 9. Shield, R. T., "On the plastic flow of metals under conditions of axial symmetry," Proc. Roy. Soc. London Ser A 2335 1955, ppo 267-286. 10. Coulomb, C. A,, "Essai sur une application des regles de maximis et rinimis a quelqnes problemes de Statique, relatifs a l'Architecture," Mem, Acad. Sci. (Savants Estrangeis), Vol. 7, 1773, pp. 343-382. 11. Shield, R. T., "On Coulomb's law of failure in soil," J. Mech. Phys. Solid, Vol. 49 1955, pp. 10-14. 12o Drucker, Do C, o "A more fundamental approach to plastic stressstrain relationst Proco 1st UoS. Nat, Congr. Applo Mech., (ASME 1951), pp. 487-491. 20

13. Haythornthwaite, Ro M., "The mechanics of the triaxial test for soils," Proc. Am. Soc. Civil Engrs., Vol. 86 (SM5), 1960, pp. 35-62. 14. Cox, A. B., Eason, G. and Hopkins, H. G., "Axially symmetric plastic deformation of soils," Phil. Trans. Roy. Soc. Ser. A 254, 1961, pp. 1-45. 15. Drucker, D. C., Greenberg, H. J., and Prager, W., "Extended limit design theorems for continuous media," Q. Appl. Math., Vol. 9, 1952, pp. 381-389. 16. Hill, R., "On the problem of uniqueness in the theory of a rigidplastic solid-III," J. Mech. Phys. Solids, Vol. 4, 19579 pp. 153-161. 21

FIGURE CAPTIIONS Figure 1 Plane stress section of Coulomb yield criterion. FiFgure 2 Yield curve for P2= 3 and N2= 3. Figure 3 Annular plate a) Undeformed, b) Deformed without thickening: < n Dcr c) Deformed with thickening: f > 3cr Figure 4 Limit load as a function of plate size. 22

To =,2c ~'e = N (Tr A D _![ N F IE e Po=2cN TO= N fFigure 1

p2 d F C inner bound from side FE1 / outer bound from corner F, /' // inner bounds: —i / D /C DE outer bounds: —-- I/' /solutions: - inner bound from side BC hL- 1 outer bound from corner C C I Figure 2

h 0 t~~~~ o a b h0 h I2 Figr 5bo

P(N.1) r2CN 2 N =2 N =3 2 bo 1 2 3 4 5 6? ao Figure 4

h o P \\\\ (a) a b holbo ho h (C) a I 8 I bo Figure 3

UNIVERSITY OF MICHIGAN III3 9015 02845 3184IIlI llI l 3 9015 02845 3184