Mechanical analysis of coin striking: Application to the study of byzantine gold solidi minted in constantinople and carthage

Journal o f Mechanical Working Technology, 10 (1984) 253--271

253

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

A M E C H A N I C A L A N A L Y S I S OF COIN S T R I K I N G : A P P L I C A T I O N TO T H E S T U D Y O F B Y Z A N T I N E G O L D S O L I D I M I N T E D IN CONSTANTINOPLE AND CARTHAGE

F. DELAMARE and P. MONTMITONNET Ecole des Mines de Paris, Centre de Mise en Forme des Mat$riaux, Sophia Antipolis, 06565 Valbonne (France)

(Received February 20, 1984; accepted March 7, 1984)

Industrial S u m m a r y A mechanical analysis of the forging of a coin, using an upper-bound method, is briefly presented: the geometry of the coin, simplified, consists of an axisymmetric disk with a central circular design and an outer annular legend. The energy and strain necessary to form the design are calculated and the shape of the blank then computed from the shape of the coin. This analysis is applied to numismatic problems, in the present instance considering whether mechanics is able to explain the evolution of the geometry of Byzantine gold solidi at the Mints of Carthage and Constantinople between the VIth and XIIth centuries. It is shown that, in Carthage, this evolution was aimed at saving energy, and in Constantinople, the evolution of the geometry of solidi was due to variation in the fineness of the gold.

Introduction F o r t h o u s a n d s o f years, c o i n i n g has been o n e o f t h e m e t a l w o r k i n g technologies t h a t have received t h e m o s t a t t e n t i o n f r o m the a u t h o r i t i e s : this is p r o b a b l y w h y the s y m b o l o f this j o u r n a l is a h a m m e r e r striking a coin. Surprisingly, t h e r e is a drastic lack o f k n o w l e d g e a b o u t this t e c h n o l o g y . V e r y f e w papers have been w r i t t e n , and m o s t o f these b y n u m i s m a t i s t s [ 1 - - 3 , for e x a m p l e ] . Most technical papers c o m e f r o m metallurgists [e.g. 4 , 5 ] , w h o t r y t o use s t r u c t u r a l observations t o m a k e d e d u c t i o n s a b o u t t h e m i n t i n g process. I n so doing, h o w e v e r , t h e y neglect t h e m e c h a n i c a l depend e n c e o f metallurgical s t r u c t u r e : t h e latter d e p e n d s n o t o n l y o n the n a t u r e o f t h e alloy and its t e m p e r a t u r e , b u t also o n its strain and strain rate. A m o n g s t the literature studied, o n l y o n e p a p e r deals w i t h m e c h a n i c s [6]. This p a p e r aims at a b e t t e r u n d e r s t a n d i n g o f t h e flow o f m e t a l d u r i n g the plastic d e f o r m a t i o n o f t h e blank. T h e u p p e r - b o u n d m e t h o d has been applied t o s o m e simple a x i s y m m e t r i c a l geometries in an earlier p a p e r [ 7 ] , b u t a m o r e c o m p l i c a t e d g e o m e t r y is c o n s i d e r e d herein. I t will also be

0378-3804/84/$03.00

© 1984 Elsevier Science Publishers B.V.

254

indicated why it is difficult to obtain accurate numerical values in this field. The model presented will help in the study of Byzantine gold coinage and in the explanation of i m p o r t a n t features o f its evolution. An upper b o u n d model of

forging

Upper-bound m et hods [8] have proven their efficiency in the computing of forging loads. Automatic processes for dividing the metal into standard blocks -- whatever the shape of the die may be -- have been described [9] and have achieved remarkable success [10].

Peculiarities of minting Three peculiarities make the analysis of minting difficult: the complex coin geometry; the u n k n o w n geometry o f the blank; and the u n k n o w n yield stress of the m o n e t a r y alloy. The surface to pogr a phy of a coin consists of an inner device with, generally, a portrait on the obverse: this head may usually be approximated by a circumscribing cylindrical element, the height of which must be the mean height of the design, so that the volume is the same. The reverse design is generally harder to model: in the present study, it will be taken as a cylindrical element of the same radius as t h at on the obverse side, although its height may be different. This lack of s y m m e t r y calls for a particular procedure described at the end of the kinematical analysis. A coin generally bears an outer legend, consisting o f a n u m b e r of letters surrounding the design. Again its geometry will be simplified: it becomes annular, the height of this annulus being chosen so that the volume is approximately the same. The simplified geometry is summarized in Fig. 1. A cylindrical shape will be assumed for the blank, though there is evidence that ancient blanks may sometimes have been lens-shaped or even globular. Further, as only the volume is known, a radius and height must be chosen. A selection procedure will be advanced: the chosen blank geometry is the one leading to the final known coin with minimum strain.

R3

R4

o

~I

I

Fig. 1. Initial and c u r r e n t g e o m e t r y o f the dies and t h e coin.

255 Finally, the rheology of the alloy m u s t be considered. Owing to the large plastic strains involved, elasticity will be ignored, the metal being assumed to be rigid--plastic. Strain hardening is accounted for only by taking a yield stress corresponding to half the final strain. It remains to choose the yield stress which, for most m o n e t a r y alloys, is unknown. Also, the striking conditions impose very high strain rates, ~, with a possibility o f high temperatures, T: in the present work, data at low strain rates and temperatures are taken from reference [15].

Kinematical analysis The workpiece of Fig. 1 is divided into six blocks, as shown in Fig. 2. From volume constancy,

R~ V, + (R]-R~) V + V: (R]- R~) + (R 2-R~) V + 2RhU = 0 Z

?Vl

(1)

J'Vl-LI 2"

a

3 r

Fig. 2. K i n e m a t i c basis o f t h e

r

analysis.

(See Figs. 1 and 2 for meaning of the symbols.) The velocity fields satisfying incompressibility and the boundary conditions are: block 1:

lul = -Vlr/2h wl = Vlz/h

(2)

~ = I Vll/h

(3)

block 2: I w2U2= -Vr/2hvz/h + ( V - V~)R~/2hr

(4)

~2 = I Yl v/l+4A~/3r4/h

A2 = ( V - VI)R~/2V.

(5)

u 2 = 0 if r = r o ~ = ~ , u 2 > 0 if r > r 0 2 , u 2 < 0 i f r < r 0 2 . IfRl
R~ <~R~ x/'I + Va/IV] <<-R 2

(6)

256

A neutral radius thus exists in block 2 only when V1 > 0, which means that material from block 5 'climbs' into the groove of the die. In this case, this 'climbing' obviously has to be fed with metal coming from block 1. block 3:

ua = - V 2 r / 2 h + [V2R~ - V I R 2,- V(R~ -R2,)]/2hr

(7)

w3 = V2 z / h

~3 = I V21v/l+4A~/3r4/h

(8)

A3 = [ V 2 R ~ - V I R ~ - V ( R ~ - R 2 , ) ] / 2 V 2

There is a neutral radius r03 = ~

in block 3 if

R: ~< r0a ~< Ra block 4:

(9)

! u, = - Y r / 2 h + [ ( V - V:)R~ - V ( R ~ - R 2 , ) + (V2R~ - V , R ~ ) ] / 2 h r)l"0 w4 = V z / h

~4 = I V I v / l + 4 A ~ / 3 r 4 / h

(11)

A , = [ ( V - V2)R~ - V ( R ~ - R ~ ) + (V2R~ - V, R 2 , ) I / 2 V if R3 ~< ~ to4 = ~

~< R, there is a neutral radius in block 4. = R ~I+2hU/VI~

(12)

In the rigid blocks 5 and 6, us = u6 = 0, es = e6 = 0, ws = V~, w6 = 172. The possibility of the existence of a neutral point has been demonstrated [6] (Fig. 3). The directions of flow were determined and are indicated by little arrows. The direction of flow changes from outwards to inwards under the legend; this is the definition of a neutral point. Dead metal zones also appear, corresponding to blocks 5 and 6. These data qualitatively agree with the present model. There are three u n k n o w n velocities, U, V~, 172, and one relation (1): the problem has two degrees of freedom, for instance V1 and V2. Figure 4 shows the possible flow patterns as a function of V1 and V2, accounting for the possible existence of neutral points (eqns. (6), (9), (12)). The V~--V2 plane has three restrictions: V~ ~> V, V2~ > V (lines 1 and 2) and U~> 0 (line 3). The vertical straight lines 4 and 5 limit the part of the triangle where a neutral point exists in block 2; lines 3 and 6, a neutral point in block 4; lines 5 and 6, a neutral point in block 3. Thus, for instance, if V~ and V2 are negative, the metal will flow outwards and there is no neutral point ( V I < 0 , V 2 < 0 or slightly positive). If V1 becomes positive, some metal must come from block 3 to fill the hollow part: there is a neutral point in block 2. If V~ is still greater, the o u t p u t from block 2 is no longer sufficient. The neutral point is then farther, in block 4: all the metal escaping from block 2 plus some metal from block 4, 'feeds' block 1.

257

_L._.__',<

t~

\

~ \

FIGURE DESIGN

~

._~

~

, \

COMPRESS'ON

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DESIGN

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REVERSE S/DE

CONPRESSION/ NINOR BACK EXTRUSION DEAD METAL ZONE \ F,GURE DESIGN ~

OEAO METAL

/

~, ,~

_i.

"~

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~

~

_.

EXTRUSION

HEAD

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O~VERSE SlOE ,

f~

0 . 1 1 6 6 INCH G A U G E

COIN

DEAD METAL

\

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1 ,,/

,

HEAD DESIGN

\

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_

,~',-..1~

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~:.~:~.--i ~-..~!-~.!"..-:.~ RE'VERSE S~E

|

v~,c,~/I

BACK ~XToM~SIj:Ij HORIZONTAL EXTRUSION'

(b) COIN 0.0292

\ APEx \ ~ OBVERSES/DE" \

EXTRUSI~

,I

INCH G A U G E

Fig. 3. F l o w pattern observed after forging o f a m o d e l coin: note the changes in the direction o f the velocity, i.e. neutral points (after reference [ 6 ] ) (1 inch =- 25.4 A m ) .

258

__ V2 ~-

%1

__,~

/ ,6 1

1]11]1

I

I]111111 Illl lJll]

Fig. 4. Possible flow patterns in a coin during striking as a function of V1 and V~. V1 and V2 are calculated b y minimizing the dissipated power, the equation of which is n o t displayed here because of its length. Once V~ and V2 are known, U is calculated from (1) and the incremental change of geometry computed: the calculation is then iterated until the final shape is reached, or until the whole of the input energy is dissipated (as in the case of hammer striking, for instance). Influence of striking conditions on strain and energy dissipation

Difference between carved and flat dies The model detailed above gives, as a function of the die stroke (or of time): the speeds U, V1 and V2; the dissipated power V¢; the integral of the latter, energy 'W; and geometrical parameters (external radius, current height, height of design and legend hgl and hg2).

259

Figure 5 shows the energy consumption during striking versus Ah -- h0 - h, in three cases. The first case is t h a t of the compression of a cylinder (G1 = G: = 0}; the second is that of a cylindrical coin with just a figure (G~ ~ 0, G2 =0); whilst the last is t h a t of a complete coin with figure and legend (G~ ¢ 0 , G2 ¢ 0). For a given value of Ah, the energy is almost the same in the three different cases. The presence of the design slightly reduces the energy consumption, since part of the matter remains u n d e f o r m e d (blocks 5 and 6): the difference never exceeds 10% in the geometry chosen.

iVVo~Jo,: - . . . .

0

' ''

0.5

''

h o - h f (rnm I h~

Fig. 5. Energy consumption W as a function of the stroke (Ah).

Thus, if the initial (blank) and final (coin) geometries are k n o w n the dissipated energy m a y be simply estimated by neglecting design and inscription. In this case (compression of a cylinder), the energy may be evaluated from w

9 - h04 R0([h013'2_ 1)IL _j

Moreover, Fig. 5 gives the instant when design and legend are completed h~*, which corresponds to an energy W* and an approximate mean strain ~* = ln(h0/h~).

Influence of blank thickness Figure 6 illustrates the influence of h0 on hf* and W*, when R0 is constant. The case of a constant volume of material, as in coin striking, will be discussed in section 3. The question arises as to whether the shape of the curves can be explained. In most cases, V I ~ 0 , so t h a t hf~ho-Max(G~,G2), or ln(ho/hf)~

V/(ho-V). Thus, for a given value o f G, if ho decreases down to G, ~* increases tremendously. The branch for high h0 may also be explained. This geometry disfavours the forming of the figures (VI~.V, V2~V
260

As for W*, ~* consists of three terms: (i) dissipation by friction, which depends on ~ and on the area of the friction surfaces; (ii) dissipation by shear on surfaces of velocity discontinuity; and (iii) dissipation by plastic deformation Wp, which is proportional to ~ = ln(ho/hf) and to the volume of material V. i

w~Jl

i

I

v

T

i

100

--a

c

I I I I

0 ho

0.5

-----1

o/I

,fi~

I

h f*

05 /

'\ 0 O5 1' b 0

I

I

I

f

1

2

3

4

d ho

l 1

2

holmm)

Fig. 6. Dissipated energy (a,c) and initial/final heights ratio h o/hf (b,d) at the end of the formation of the design and of the legend, a0 = 2 0 0 MPa. a , b : R 1 = 2 r a m , R2=4 m m , R3 = 6 r a m , R 0 = 8 r a m . c , d : RI = 4 ram, R2 = 6 r a m , R3 = 7 m m , R 0 = 8 ram. Friction coefficients on all contacting parts are e q u a l .

In Fig. 6, R0 is constant, so that Wp is proportional to holn(ho/hf). As this term is the dominating one, it imposes the shape of the W* (h0) curves. Note that there is an "optimum initial shape" which leads to minimum

261 energy dissipation. This calculation can be used to compare the energy dissipated in striking a coin and in striking its pieford, i.e. the same figure struck on a thick blank, and destined to be used as a pattern or proof.

Design and legend geometry Comparison of Figs. 6(a,c) and 6 ( b , d ) confirms the importance of the geometry of the design and of the legend. In the second case, the area of the inscriptions is slightly greater: consequently e* is greater, and so is W*. Note that a very thick design (0.5 mm) has been assumed. More classical values (~0.1 or 0.2 mm) would lead to much lower strains and energies. Friction and yield stress Unexpectedly, the influence of the friction coefficient ~ is weak. In fact, as the value of ~ increases, radial flow becomes more difficult and, as V1 and V2 increase, ~* decreases. The higher friction energy partly compensates for this effect in the cases chosen. Finally, W* is proportional to the yield stress o0. Case of imperfect symmetry: different design thickness on obverse and reverse The analysis presented above supposes a symmetry between obverse and reverse, b u t it may be generalized. Suppose that the obverse design has the same radius, b u t a greater thickness (Go) than the reverse one (Gr). As all computations consider only half the coin (owing to symmetry considerations), the following procedure becomes possible: (i) formation of symmetrical designs on the obverse and the reverse, with the thickness being that of the reverse one; (ii) the reverse half of the coin then becomes rigid, and it only remains to form a cylindrical design of height (Go-Gr) on one half of the coin, which is precisely the scope of the model.

Numismatic applications Such a modelisation begs for application to numismatics problems. On the advice of the numismatist C. Morrisson, the authors have attempted to complete a recent study on the evolution of the fineness of Byzantine gold solidi from the VIth to the XIIth century [11--13].

Blank geometry computation The geometry of the blank is generally unknown, which is a major difficulty, since it makes it impossible to calculate the strain due to the striking. Presented here is a c o m p u t a t i o n of the blank geometry derived from the foregoing analysis. As seen in Fig. 6, if the blank geometry is known, the instant when design and legend are formed may be calculated: the height is then hf*, and corresponds to the minimal strain ~* and energy W*. Conversely, if only h~ is known, there exists a blank thickness h0* which will

O

1

20

mm

,

2

r

e

3 O

0 5

Fig. 7. Evolution if the geometry of gold solidi minted in Constantinople : ( 1 ) AnasLase I, emission date 492-507, BN 07; (2) Constant II, 662--667 BN 6 6 ; ( 3 ) M i c h e l [I, 821--829, BN 02; (4) Basile II, 1005--1025, BN 15; (5) Constantin IX, 1042--1055, BN 05; (6) 2onstantin X, 1059--1067, ~N 04; (7) Nicephore III, [078--1081; (8) Alexis Ist ~omnene, 1092--1118, BN )4 ; (9) Manuel Ist Comnene, flectrum aspron trachy (BN ~L 13, type 4).

T~

263

give a completely formed design just when h = h~ : if h0 < h0*, the design will be incomplete when h = h~; if h0 > h0*, a greater strain will be necessary to reach h~, and energy will be wasted. This thickness h0* may be calculated in a few iterations. Thus, h0* may be considered as the minimal and ideal thickness of the cylindrical blank in the striking of a given coin. Unlike the coins in Fig. 7, some Byzantine gold solidi are struck offcentre: on these coins, part of the blank remains untouched after minting. The observation of such "fleur de coin" coins (i.e. uncirculated and therefore free of wear) shows t h a t the blanks were flattened by hammering, and a mean thickness could therefore be measured and compared with the "ideal thickness" (h0*) c o m p u t e d from the present model. Consider the solidus of Justin Ist (No.3 in Table 1). The mean blank thickness is 2h0 = 0.95 mm. From the mass of this coin and its composition [11], its volume can be estimated (V= 0.23 cm3), giving R0 = 8.8 mm. On the area that has been struck, the dot border radius is seen to be 10.0 mm, and the design mean thickness is approximately 0.25 mm. It is assumed that the friction coefficient is ~ = 0.5. Under these conditions, computation yields a completely formed design and legend when 2h~* = 0.60 m m and the TABLE 1 Solidi m i n t e d in C o n s t a n t i n o p l e . (ahave b e e n analysed in r e f e r e n c e s [ 1 1 ] and [ 1 3 ] ; b has b e e n analysed e l s e w h e r e . ) U n d e r l i n e d n u m b e r : coin illustrated in Fig. 7 No

Emperor

Ia 2a 3 4a 5 6 7 8 9a 10 11 12 13 14 15 16 b 17 18 19 20 a 21 22 a 23

Anastase I Justin I Justin I Maurice Maurice Phocas Phocas Heraclius Heraclius Heraclius C o n s t a n t II C o n s t a n t i n IV C o n s t a n t i n VI Basile I C o n s t a n t i n VII Basile II Constantin IX Theodora Constantin X Michel VII Alexis I C o m n e n e J e a n II C o m n e n e Manuel I C o m n e n e

Production d a t e (A.D.) 492--507 522--527 522--527 584--602 584--602 603--607 609--610 610--613 616--625 629--632 662--667 669--674 790 868--870 945--969 1005--1025 1042--1055 1055--1056 1059--1067 1071--1078 1092--1118 1118--1143 1143--1180

Reference B.N. BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN BN

07 05 11 03 09 03 15 01 15 30 66 02 05 02 16 15 05 01 04 02 03 04 02

264 m a x i m u m thickness is 1.10 m m . These values are close to the e x p e r i m e n t a l l y m e a s u r e d values {0.65 and 1.15 m m ) which establish t h e validity o f the model. N o t e t h a t the e n e r g y dissipated, W = 30 J, m a y be p r o v i d e d easily b y o n e h a m m e r b l o w at r o o m t e m p e r a t u r e .

Evolution of the geometry of Byzantine gold solidi minted in Constantinople B a r r a n d o n and Morrisson [ 1 1 , 1 2 ] have s h o w n h o w close the c o n n e c t i o n was b e t w e e n the e v o l u t i o n o f the c o m p o s i t i o n o f m o n e t a r y alloys and political events in t h e B y z a n t i n e Empire. S i m u l t a n e o u s l y , o n e can see a dramatic e v o l u t i o n o f the shape o f the gold solidi m i n t e d at C o n s t a n t i n o p l e (Fig. 7). Such an evolution, at the present time, c a n n o t be a c c o u n t e d for: it is perhaps a p r o p e r topic for a m e c h a n i c a l study. All the coins studied (see Table 1) belong t o the c o l l e c t i o n o f the Cabinet des M~dailles, Biblioth~que Nationale (B.N.), Paris, and are described in its subject catalogue [ 14]. In Fig. 8 ( a f t e r r e f e r e n c e [13] ), the e v o l u t i o n o f the fineness o f gold in the solidi m i n t e d at C o n s t a n t i n o p l e , f r o m the Vth to the X I V t h c e n t u r y , can be observed. In the same figure can be seen the s i m u l t a n e o u s changes o f the flow stress o f the c o r r e s p o n d i n g alloys -- calculated f r o m hardness m e a s u r e m e n t s [15] -- bearing in mind the reservations m a d e previously [ 1 6 ] . It will be n o t e d t h a t the decreasing fineness leads t o a significant increase in the flow stress o f the m o n e t a r y alloy.

%gold

I

!

,I

'

~/ogold

90

:%

I'

~'~

(MPa) 300

/

80

j/

200

/

70

/

100

60 50

I

500

I

700

I

!

900

1100 date Fig. 8. Gold fineness of the solidi minted at Constantinople versus period (after references [ 11,13 ] ) and the corresponding flow stress evolution (~ = 0.1 ). The following figures deal with the evolution in the g e o m e t r y o f the solidi. As it was impossible to d o statistics o n every t y p e o f solidus with the coins available, t h e choice o f the coins was p l a c e d in a n u m i s m a t i s t ' s care (C. Morrison). T h e y m u s t be representative o f the t y p e and very little w o r n (uncirculated if at all possible). This sampling is t h e r e f o r e n o t e x a c t l y the same as t h a t for previous studies [ 1 1 , 1 3 ] , b u t it is as representative as it is possible f o r it to be. M e a s u r e m e n t s were m a d e o n the coins with calliper, square and m i c r o m eter.

265

Figure 9 gives the evolution of two global parameters: the mass and the volume of the solidi (after [13] ). As noted [13], the decrease in fineness is compensated for b y an increase in the volume rather than b y mass variation. Simultaneously, a slight increase in the d o t border diameter, and a striking one for the module or external diameter {Fig. 10), can be observed. Thus, the plastically deformed volume decreases {Fig. 11), as do the mean thickness of the coins and the relief of the device (Fig. 12). Gradually, the coins become concave; Fig. 13 shows that the dip of the coin depends upon the difference between the module and the d o t border. g

M

j cm 3 /~"---'"

4..~

/ t

4.2

/ V __ .,- _ I n 700 900

mm 3C

.0.26

25

0/~)

I

0.24

20 ~.-v,_,q~-oe l l C l L ~ ~e----~.~--.9-_ ~ _ . : J =-

i ~I 1100

0.22

15

i 700

O0

~'~v: o / fv J v ' w

~ 900

--

n 1100

Fig. 9. Mass and volume versus period for solidi minted at Constantinople (after references [11,13] }. Fig. 10. Module (external diameter, ~e) and dot border diameter (~g) versus period. Constantinople.

((~}g/(~1)2t

!

I

I

|

!

!

500

700

900

ii

/

1100

Fig. 11. Deformed volume versus period. Constantinople. i

2h mlT 0,£

u

p I

hg mm

f mlT ~ 3

0.2

2

I

0.1

1

/

0

0

i

0.-~ 0.5

h'~]A

0.: 500

! 700

| 900

1100

!

• 0

I

I

• I 0,2

I

I

I

I 0.4

1 -{gl~Pe

Fig. 12. Minimum thickness (2 h) and device relief (hg) versus period. Constantinople. Fig. 13. Dip of concave coins (f) versus normalised width of the unstruck part of the coin. Co nstantinople.

266

The evolution in the thickness and diameter of the ideal blank, computed as described above, is shown in Fig. 14: the diameter increases after 950 A.D., whereas the height falls. The knowledge of the blank geometry makes it possible to c o m p u t e the overall strain due to striking (Fig. 15). It can be observed that it increases after 600 A.D., remains high, and then decreases around 1000 A.D. Figure 16 shows the evolution of the mean flow stress during striking. After 900 A.D., it increases dramatically and is increased by a factor of 2.7 in 1100 A.D. Figure 17 represents the evolution of the energy W* dissipated in striking the solidi. Some dispersion can be noted in the results, which is due to the variety of solidi struck. Coin No. 2, for example, has a very marked relief, and needs a high value of W*. In spite of this scatter, some progress in the saving of energy during the VIIth century can be observed, followed b y stabilization at a value corresponding to a mean hammer blow [16], and then growth in the XIth century. At about 1050 A.D., the increase in o0 results in a doubling of W*, despite the decreasing of the deformed volume and of the device height. Moreover, when the device height remains high, W* exceeds 80 Joules (three hammer blows).

2h~l

I

I

I

I

I

I

~m

15

1,3 12

1.1 2

.

4.

1.1 7. 19__ 2h~ 6.- 9 5. 8

0.9

34 • • r'--I I I

1.4 _

13

30

1-

26

19./

0.7

22

lb__l.,..~llJ]8"lm;~1:~ 0.5 i I-

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lip

•

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- i l r i~ . . . . . .

•

. . . .

•

. . . .

.J

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18

21 22 •"23 II h°

I 14 I I I I 1000 600 800 Fig. 14. C o m p u t e d height (ho*) and d i a m e t e r (90*) of the blank. Constantinople. The numbers refer to Table 1. I 0 I rO I UII (h*-h) UgO,..) u u u • 14 15 C 0.6 13 • 0 O 18 12

0.3

0.8

• ~3

6o81.° 11 ,

5o~7

O O

"

21

0.7 0.6 ~1

5., 8

I].

16 0.5

i 6OO

z

800

1000

•

2O o !

2_3 0.4

22 • i0.2

Fig. 15. C o m p u t e d strain Q ((ho*-hf)/hg) and generalized strain • (~). Constantinople. The numbers refer to Table 1.

267 The XIIth century then restores the situation, the primary reason being the economic reformation of Alexis, and the improved fineness of the gold alloy. Secondly, the design height remains very small. Finally, coins are made still flatter to further decrease the deformed volume; but the difference between module and d o t border radius is such t h a t coins become concave. Thus, the evolution in the solidus geometry is due to the technical solutions intended to mediate the increase in the alloy flow stress, i.e. the decrease in the fineness of gold. ao( 12j MPa

i

I

I

i

/

30C 20C

I

.I

i

i

i 100

i

o

500 700 900 1100 Fig. 16. Computed mean flow stress of the different alloys for each type of coin versus period. Constantinople. |

!

!

|

!

| !

W ~

I

(J)

80-

18 "19/i

2

/!

oo

71

4o

.

,~

4

13

^

1~

•

L-"

/1~

10 • =12 nmnu 8 11

20

i¥

a~-"~

I i I

0

I

I

I

I

a

il

600 800 1000 Fig. 17. Computed energy dissipated during the striking of the coins (W*)versus period. Constantinople. The solutions chosen consist o f decreasing the device height (hence the strain) and the deformed volume. The former, though it m a y hinder legibility, is particularly efficient. As for the latter, two possibilities existed: increasing the module, or decreasing the d o t border diameter. The first solu-

268 t i o n was c h o s e n - - p r o b a b l y f o r aesthetic reasons - - but, u n f o r t u n a t e l y , increasing t h e m o d u l e leads t o a decrease in t h e t h i c k n e s s a n d this f l a t t e r g e o m e t r y n e e d s m o r e energy. What is saved b y decreasing t h e d e f o r m e d v o l u m e is lost b y t h e c h o i c e o f t o o flat a g e o m e t r y . As a whole, the t e c h n i c a l failure is o b v i o u s , as W~ r e m a i n s high.

Comparison between solidi minted in Constantinople and Carthage D u r i n g t h e V I I t h c e n t u r y , solidi were m i n t e d in d i f f e r e n t w o r k s h o p s w i t h t h e s a m e alloy, t h e s a m e mass, b u t w i t h r a t h e r d i f f e r e n t g e o m e t r i e s . T h e r e a s o n b e h i n d this d i f f e r e n c e is u n k n o w n . C o m p a r e d h e r e are t h e solidi s t r u c k at C o n s t a n t i n o p l e (Fig. 7) w i t h t h o s e m i n t e d at Carthage, w h i c h b e c o m e m o r e and m o r e g l o b u l a r (the d i a m e t e r decreases a n d t h e t h i c k n e s s increases). Five coins w e r e c h o s e n t o illustrate the e v o l u t i o n o f t h e solidi t o w a r d s the g l o b u l a r f o r m in Carthage (Table 2). A t the beginning, the g e o m e t r y o f t h e solidi m i n t e d at C a r t h a g e is t h e s a m e as t h a t at C o n s t a n t i n o p l e ( c o m p a r e C a r t h a g e No. 2 a n d C o n s t a n t i n o p l e Nos. 5, 6, 8, 9, f r o m t h e s a m e t i m e period). T h e b l a n k s are also identical. Later, Fig. 18 s h o w s t h a t the d o t b o r d e r d i a m e t e r a n d e t h e m o d u l e decrease strikingly. S i m u l t a n e o u s l y , t h e t h i c k n e s s o f t h e c o i n increases, b u t t h e device height decreases (Fig. 19). TABLE 2 Solidi minted in Carthage studied (ahave been analysed in references [12] and [13]) No.

Emperor

Production date (A.D.)

Reference B.N.

1a 2 3a 4a 5

Maurice Phocas Heraclius Heraclius Constant II

600--601 606--607 610--611 631--632 667--668

B.N. 14 1971/87 h B.N. 01 B.N. 11 B.N. 09

f ] ' l FI"

16

~ .

12

8

I 500 600 700 Fig. 18. Module (~e) and dot border diameter (9¢) versus period. Carthage. The numbers refer to Table 2.

269

Figure 20 gives the c o m p u t e d diameters and thicknesses of the corresponding ideal blanks, whose slenderness increases continuously. From Fig. 20 it is noted that the strain due to striking decreases with time, being reduced by a factor of a b o u t 3. It follows that the mean flow stress decreases (from 150 to 100 MPa), the composition of the alloy remaining constant (Fig. 21). Finally, it can be seen that the energy dissipated during striking decreases from 18 to 2 joules (Fig. 22). 2h mrr

!

hg mm

_

'

•

/

~~U ~ h~o

•

16

•,/V

0.2

2

12

• \

.

0,1 0

iO0

I 600

2h~ mm

•

2h

1

*~ mrr

1

m

0 700

8

500

6;30

7000

Fig, 19, Minimum thickness (2 h) and device relief (hg) versus period, Carthage, Fig. 20. Computed thickness (h0*) and diameter (¢*) of the blanks versus period. Carthage. Oo

I

MPa

,\

ram2

(J

200

0.2

\

1C

\

100 0

!

500

600

0

7()0

0

I

500

600

700

Fig. 21. Computed generalized strain (e-) and mean flow stress of the alloy (a0(~/2)). Carthage. Fig. 22. Computed energy dissipated during the striking of the coin (W*). Carthage. The numbers refer to Table 2.

In an easier historical background, since the alloy fineness remains constant, this is an example of optimization o f the process. Increasing the slenderness and decreasing the relief o f the device leads to the saving of a lot of energy: around 668 A.D., the mint of Carthage consumed ten times less energy to strike a solidus than the mint of Constantinople. In this energy range, two aims are possible: either increasing the production, or reducing die wear. The Carthaginian method, because it substantially decreases the normal stresses at the die/coin interfaces, must be very efficient

270 from this point of view, as a greater n u m b e r of coins can be minted with the same dies. Thus, Carthage f ound a remarkable technical improvement, b u t the coins obtained are mu c h less legible due t o their small surface. Conclusions The aim o f this work was to demonstrate that mechanics can bring new data to numismatics. In spite of the rather complicated shape of coins, it has been shown that it is possible to com put e, approximately, the blank g e o metr y and the energy dissipated during striking with an upper bound model o f forging. F r o m its application to a series o f gold solidi struck at the mint of Constantinople, it may be concluded t h s t their geometric evolution is an a t t e m p t to answer the technical problems caused by the lowering o f the fineness of the gold alloy. On the other hand, at the mint of Carthage, the effects of technical research leading to substantial savings of energy are observed. Whether this was justified by the need of increasing production, or of lengthening the die life, is still a m a t t e r for conjecture. It should be born in mind that the precision of the present model depends less on the c o m p u t a t i o n itself than on the simplified geometry to be used and the lack of data on the mechanical properties of gold and m o n e t a r y alloys. R ef in eme nt of the model requires that these two problems be solved first. Acknowledgements We are greatly indebted to C. Morrisson (Laboratoire d'Histoire et Civilisation de Byzance, Coll~ge de France, Paris} who proposed this subject to us, chose the most representative coins, and helped us in the numismatic part of this study. Grateful acknowledgement is made to Mr. M. Rey (Centre de Recherches Arch~ologiques du C.N.R.S., Sophia Antipolis, France) for taking the photographs of the coins. References 1 T. I4ackens, Terminologie et technique de fabrication. In: Numismatique antique, probl~mes et m~thodes. Annales de l'Est, 44 (1975) 3. 2 P. Grierson, Numismatics. Oxford University Press (1975). 3 D. Sellwood, Minting. In: D. Strong and D. Brown (Eds.), Roman Crafts, Duckworth, 1976. 4 C.F. Elam, J. of the Inst. of Metals, 45 (1931) 57. 5 M. Barral, K. Gruel, J. Barralis and F. Widemann, Quelques dl~ments de m~taUurgie mon~taire gauloise. Journ~es de Pal~om~tallurgie, Compi~gne, France 1983. 6 F.T. Barwell, Tribology in metal working, developments in perspective. In: Inst. of Mech. Engineers, Conference Publications, 1980, p. 51.

271 7 8 9

10 11 12 13 14 15 16

F. Delamare and P. Montmitonnet, Introduction ~ une ~tude m~canique de la frappe des monnaies. Journ~es de Pal~om~tallurgie, Compi~gne, France (1983). P. Baque, E. Felder, J. Hyafil and Y. D'Escatha, Mise en forme des m~taux, calculs par la plasticitY. Dunod, Paris (1973). R.P. Mc Dermott and A.N. Bramley, An elemental upper bound technique for general use in forging analysis. Proc. 15th Int. MTDR Conf., 1974, MacMillan, London, 1975. F.H. Osman, A.N. Bramley, Metal flow prediction in forging and extrusion using UBET. Proc. 20th Int. MTDR Conf., 1979, MacMillan, London (1980). J.N. Barrandon and C. Morrisson, De Rome ~ Byzance: l'or monnay~. Cahiers E. Babelon no. 2, ed. C.N.R.S., 1984. C. Morrisson, J.N. Barrandon and J.L. Poirier. Jahrbuch des ~)sterreichischen Byzantinistik, 33 (1983) 267. J. Poirier, Contribution ~ l'analyse de l'or antique. Application aux monnayages du monde m~diterran~en du II~me au XIV~me si~cle. Thesis, Orleans, 1983. C. Morrisson, Catalogue des rnonnaies Byzantines de la Biblioth~que Nationale. B.N., Paris, 1970. L. Sterner-Rainer, Z. Metallkunde, 1 8 ( 5 ) ( 1 9 2 6 ) 143. F. Delamare and P. Montmitonnet, Evolution of coin striking processes: a mechanical survey. I. Hammer striking. Submitted to J. Mech. Work Tech.