limit strains in the processes of stretch forming sheet metal

Int. J. Mech. Sci. Pergamon Press Ltd. 1967. Vol. 9, pp. 609-620. Printed in Great Britain

LIMIT STRAINS IN THE PROCESSES OF STRETCH-FORMING SHEET METAL

ZDZISLAW MARCINIAK and KAZIMIERZ KUCZYI~SKI Warsaw, Poland

(Received 10 February 1967)

Summary--The process of the loss of stability is analysed for sheet metal subjected to biaxial tension when the ratio of the principal stresses 0.5 ~< a~/al ~ 1. The loss of stability manifests itself by a groove running in a direction perpendicular to the larger principal stress. In this groove local strains begin to concentrate gradually. In the initial stage of the process the deepening of the groove is associated with a gradually fading strain in the regions adjacent to the groove. This fading strain attains a certain limiting value e*.

This paper contains both experimental results and a theoretical analysis of the process of the generation of the groove based on anisotropic plasticity theory.

The system of equations derived was solved numerically with the aid of a computer, which enabled the limiting strain of the sheet metal to be determined as a function of the following properties of the material: (i) Initial inhomogeneity of the sheet metal, (ii) exponent of the strain-hardening function, (iii) coefficient of normal anisotropy, (iv) initial plastic strain, (v) strain at which the fracture occurs.

The results are discussed and the properties are described that influence the drawability of sheet metal used in the stretch-forming process.

1. INTRODUCTION

AMONG various processes of sheet metal forming there are ones in which the required shape of a drawpiece is obtained by means of uniaxial or biaxial

(o)

(c)

N_N

FIo. 1. Examples of the biaxial stretch-forming processes.

stretching of the sheet of metal. These processes lead to an increase in area of the sheet metal surface at the cost of a reduction in its thickness. In this class belong: stretch drawing, hydroforming, bulging, tube bulging and the like. In the process of deep drawing the bottom of a cup undergoes biaxial stretching. Some examples of the stretch-forming processes are shown in Fig. 1.

6o9

610 ZDZlSLAW MARCI~AK and KAZIMIERZ KUCZY~SKI

One of the advantages of the processes belonging to this group is that the sheet metal is prevented from wrinkling. However, the defect of this method consists in a relatively small magnitude of strain at which fracture of the sheet metal takes place. To prevent this then, at no point of the drawpiece should the strains in the sheet metal exceed a certain limiting value e*. This magnitude depends upon both the properties of the deformed material and the mode of its loading, i.e. the ratio of the principal stresses and the loading history. The object of this paper is to analyse the factors which influence this limiting value of the strain in the sheet metal under stretching.

A starting point to further considerations is the observed fact that during the deep drawing of mild steel and plastic materials (those most frequently used in industrial practice), fracture of the sheet metal does not occur abruptly but is, as a rule, preceded by the loss of stability of the sheet metal. As a result of this loss of stability the strain concentrations in certain regions of the drawpiece begin to take place while its remaining parts lying outside the local thinning either undergo unloading or the plastic strains decrease. Some local necking of the sheet metal begins to take place and a loss of cohesion takes place at a certain instant. Thus, the phenomenon of deeohesion should in this case be looked upon as a secondary one following the former loss of stability. To determine the limit value of the strain in the sheet metal under stretching the loss of stability of a shell should first be analysed.

The loss of stability in the process of bulging a circular metal diaphragm clamped at the circumference and subjected to lateral fluid pressure was analysed by Hill 1, Swift s, Mellor a and others. It was assumed that the loss of stability of such a sheet takes place when the increment of its strains occurs with no simultaneous increase in the pressure. Such a definition of the point of loss of stability, which is sometimes identified with the point of fracture of the sheet metal, corresponds to the maximum load which the deformed shell can suffer under some special loading conditions, i.e. when the deformation of the shell has no influence upon the magnitude of the pressure acting on its surface-- as for example happens when the pressure is exerted by a gas whose volume is considerably greater than that of the forming cup.

The above-mentioned definition appears unsuitable for the analysis of the conditions that lead to the fracture of the sheet metal in a technological process. The reasons for this are:

(i) Under the actual conditions of press forming which use an almost incompressible fluid, the extreme pressure reached does not lead to the collapse of the drawpiece. It can continue to deform in a controlled manner at a gradually falling pressure. The same applies when forming with a rigid punch whose drawing force can, in some cases, decrease, but the drawpieee is then found neither to collapse nor to lose its stability.

(ii) The fracture of the sheet metal together with the preceding concentration of strains can occur either before or after the pressure has attained its extreme value. This phenomenon depends not upon the loss of stability of the shell as a whole, but only upon the possibility of occurrence of either a local discontinuity or a local concentration of strains.

Limit strains in stretch-forming sheet metal 611

This discontinuity can take the shape of a groove with a reduced thickness. The groove is inclined to the directions of the principal stresses at such an angle that its length remains constant during the straining process. However, it should be emphasized that discontinuities of this type follow from the existence of the characteristics of the system of equations describing the state of stress and can occur only in a certain stress field, namely if a S ~< al, where a 1 and a S are the principal stresses in the shell under plane stress; a a = 0.

In many bulging processes the stress state is found to be one of almost equal, biaxial tension and hence a 1 ~< a~ ~< 0-1. In this case the above-discussed local loss of stabil ity cannot take place.

In this paper it is shown that in the range 0-1 ~ O"2 ~ 0-1 a completely new mechanism of the local loss of stabil ity may appear. However, in order to describe this phenomenon we must depart from the treatment of instability as characterized by a certain end point, and, instead, analyse it as a certain process which takes root from an initial inhomogeneity of the sheet metal. In consequence, this process leads to fracture of the sheet metal in the most strained place. This approach is a development of the former author's papers. 5

2. EXPERIMENTAL RESULTS

To investigate experimentally the phenomenon of the loss of stability of the sheet metal under equal, biaxial tension a special method of sheet metal straining has been devised. The apparatus employed is shown in Fig. 2. A circular specimen 1, of diameter 215 nun and thickness 1.7 mm is clamped around its circumference by means of an annular

I 4

J

~ 5

FIG. 2.

blank holder, 2. A punch, 3, with a flat bottom exerts a pressure on the testpieee, not directly, but through a sheet metal ring, 4, which deforms together with the test- piece. Since, during the deformation of the ring, its point elements move in a radial direction more rapidly than those which would belong to the diaphragm without a hole frictional, radial forces appear in the region of contact between the specimen with the ring, 4. This friction prevents the test-piece from fracturing near the rounded edge of the punch and results in the largest strains taking place in the fiat part of the bottom of

612 ZDZISLAW M~CIZ~IAX and KAZIMZ]~RZ KUCZ~SEI

I-0

w

t -

c r -

u o

9~

9) "o

o

- o~

o

o"

e..

u

t -

09 ~. . . . _~ ~

~ . ~---"+

\',/,( 0'5

0"4

0.3 A 5 The x 6 consecutive

0.2 o 7 stoges of n 8 deformation

O.t +10

FIG. 5. Variation in the thickness of the sheet metal in the plane perpendicular to the forming groove.

1.2

I'1

1.0

0"9 I

O-B - -

0 .7

0 .6

0'5

0.4

0 '3

0.2" - -

C

-'~--~ Expq

0 '98

j. c" ' 0"97

%o_ ~(~

=0.995.

:o.~ I

cui've

0 0'1 O'Z 0-3 0"4 0"5 0-6 0"7 0"8 0,9 1"0 I'1 1.2 1"3 Fro. 6. The course of the stra in concentrat ion in the groove.

1.4

peal necking

Fracture

FIG. 3.

l A= 0.31 *= o-49 l A= 0.70 E,= 0.85

FIG. 4. Surface of the specimen at the consecutive stages of deformation.

f. p. 61:


the cup where the cracks are apt to occur. A grid of straight lines was drawn on the surface of the specimen and the whole process of sheet metal cracking was filmed.

The observations led to the conclusion that the process of the local loss of stability of the sheet metal under equal biaxial stretching is somewhat different from that which takes place during the uniaxial tension of cylindrical specimens. A grid of grooves appears on the surface of the sheet metal and at these the strains gradually begin to concentrate (Fig. 3). Differently then from the case of uniaxial tension, the process of the forming and deepening of the grooves is accomapnied by the gradual disappearance of strains in the sheet metal in the regions lying outside the grooves. The photograph in Fig. 4 shows the consecutive stages of the straining process. The dimensions of the squares drawn on the surface of the sheet metal that lie outside the groove continually increase at consecutive stages in spite of the neighbourhood of the groove. This process is also illustrated in Fig. 5, which represents an experimentally found variation in the thickness of lead sheet metal under tension along a line perpendicular to the forming groove. The particular lines correspond to the consecutive stages of the straining process.

Let eiA denote the strain outside the necking zone and ets the strain in the groove. The course of the process of strain concentration can now be shown in the diagram e~A against eis, see Fig. 6. The data derived from the experiments on the lead sheet metal are shown by dots. They form a certain curve (the dashed line) which gradually approaches a straight line parallel to the axis etB.

3. THEORETICAL ANALYS IS

The theoretical analysis of the groove-forming process in the sheet metal subjected to biaxial stretching will be based on the assumption that the sheet metal exhibits the same properties in all directions in its plane, but its properties in the perpendicular direction, i.e. through the thickness, are different. This type of anisotropy, termed "normal anisotropy", can be characterized by the coefficient of anisotropy

R = e2/e3 (1)

where e2 denotes the strain across the width of the specimen and e8 the strain through its thickness. The coefficient R will be assumed to remain constant during the straining process. The plastic properties of such a material axe based on the model of anisotropy put forward by Hill s and for the plane state of stress are as follows:

(i) The yield condition is

~4(2R + ] ) a'~ = (R + ]) a~' - 2R~x + (R + ]1 o| {T B (2)

In this equation a~ denotes the equivalent yield point for an isotropie material. The definition of this yield stress is based on the assumption that the area of the yield ellipse for the anisotropic material given by the equation (2) is equal to the area of the equivalent yield ellipse for the isotropic material described by the equation a~ = a~ - al as + a~.

(ii) The flow law is

de1 de, des de~ (R+ l )a l -Ra2 (R+ l )a I -Ra l -ax -a l 2~][(2R+ l)/3]a~> (3)

where the increment of representative strain de~ is expressed in terms of the strain components de 1, de2, des by

4 3

I t can be readily proved that the increment of the work done by the plastic deformation is expressed identically as in the case of the isotropie material by

dL --- a~ de~

(iii) The strain hardening function will be assumed to be of the form

a~ = c(eo+e~)" (4)

614 ZDZISLAW MAlZClNIAK and KAZlMIERZ KUCZYI~SKI

Consider an e lement of the stressed sheet ABDC, see Fig. 7, w i th the above-descr ibed propert ies conta in ing a groove that runs normal ly to the direct ion of the greater pr incipal stress 01 .

We shall confine ourselves to the case in which the rat io of the pr incipal stresses in the region A, outside the groove, remains constant dur ing the process, that is

d(~lA _ d(~2A _ daaA _ daTA (5) (~IA O'2A O'aA O'TA

Hence in th is region proport ional s t ra in ing takes place and the rat io of the stra in components is kept constant , i.e.

eaA H~aA - - - - - - ~x (6 )

~2 d82

On the other hand, the stress rat io alB/a2B in the groove, the region B, does gradual ly change as the non-homogenei ty of the sheet develops. At every ins tant the condit ion is

I I t to 2

A

h

c l I

A

- I

I l l FIG. 7.

satisfied of the constancy of the un i t force t ransmi t ted across the groove and the ad jacent mater ia l in the direct ion perpendicular to the groove. Suppose at a certa in ins tant the th ickness of the sheet meta l outside the groove is t A while that in the groove is t B, tB < tA. The equi l ibr ium of the forces perpendicular to the groove requires that

fflA tA = (71B tB (7) In t redue ing

~/ [3 (2R+ 1)] a~, u - 412(R+1) ] aT~

equat ion (7) can be wr i t ten in the form

~/[2(R + 1)] (8) alA tA -- ~/[3(2R+ 1)] aTBtBu

On di f ferent iat ing we obta in a differential equat ion descr ibing the process of groove formation,

daub de du dO'lA + deaA = + aB + (T1A (T~B U

where dt A dtB

deaA = ~A deaB = t~

Bear ing in mind the re lat ion (5), th is equat ion assumes the form

du daTA -- daub + deaA -- deaB (9)


The terms appearing on the right-hand side of the equation can be most conveniently expressed by a function of the common strain of both regions of the sheet in the direction (2).

de~ = dg2A = deaB (10)

The strain 82 will be treated as an independent parameter of the straining process. On differentiating the strain-hardening function (4) we obtain the relationships

and

da~A = n deiA ( 11 ) (Irma Eo + EiA

d(T~B -- n

(T~B 8 0 + EiB deiB (12)

Employing the flow law equation (3), and the assumption equation (4), the magnitudes 8iA, etB, deiA and de~B occurring in equations (11) and (12) can be expressed as the following functions of the component 82 :

- -=- -=de iA 8tA +24[(R+l)a2+a+l]_ (13) d82 82 ~/[3(2R + 1)]

deiB = _+ 4(1_u2 ) (14)

and

- 4 (1 -u~)

the strain increments deaA and desB through the thickness can be made dependent upon the increment of the parameter ev According to equation (6), we have

desA = ade2 (16) and

u 4(2R+ 1)+4(1 -u ~) deaB = ( ~ i ~]~)u2 ) de2 (17)

Substituting from equations (11-17) into equation (9) and rearranging, we finally arrive at the differential-integral equation

1 1 E 1 __ ~___L_+ Cu- f [d~l .4~) I d~ du [A + Be, -- D + B (18) 4(1 -u~)

which gives a relation between the strain component 82 and the coefficient u characterizing the stress state in the groove. The constants A, B, C, D and E, appearing in this equation depend merely upon the properties of the material under deformation R, n, e0 and the loading program for the sheet a. They are found to be

A = eo~/ [3(2R+ 1) 2n4[(R+ 1) a*+a+ 1 1

B=- n

c = 4(2R + 1) R+I

D -~ E n 412/(R + 1)] ~[(2R + 1)/3]

616 ZDZISLAW MARCINIAK and KAZIMIERZ KUCZY~SKI

4. RESULTS AND CONCLUSIONS

The solution of equation (18) enables the unknown functions u = u(s~) and e~ = ~is(e2) to be found provided the initial conditions regarding the initial inhomogeneity t~/t,4 of the sheet are given. An approximate determination of those functions obtained with the help of a computer are shown by the solid lines in the diagram in Fig. 6. I t refers to the case of the equal biaxial tension a S = al for the isotropic material R -- 1 with the strain- hardening exponent n = 0.2 and the initial hardening e0 -- 0.00136. These data correspond to the properties of the lead sheet used in the tests. The particular lines refer to the different values of the coefficient f = tB/t A of the initial inhomogeneity of the sheet material. Since the experimental ly obtained points lie between the lines corresponding to the initial inhomogeneity f = 0.98 and f = 0.99 it can be ascertained that the initial inhomogeneity of the sheet metal used was contained between the two limits. The diagram shows that the theoretical predictions are well confirmed by the experimental observations. In particular, at the initial stage of the loss of stabi l ity of the sheet the formation of the groove, or the net of grooves, is associated with the gradually fading strains of the sheet in the region adjacent to the groove. At the point C the strain eA attains a l imiting value which will be henceforth designated by e*. I t will be treated as the value of the limit strain sought which the sheet metal under tension is able to suffer before the local instabil ity develops.

After the point C is exceeded the strains concentrate in the grooves, while the remaining parts of the sheet undergo unloading and revert to the elastic state. This implies complete loss of stabil ity.

From the presented considerations it follows that the solution of equation (18) yields the value of the strain which the sheet metal can undergo outside the groove up to complete loss of stabil ity. The magnitude e* depends both on the loading programme of the sheet, the ratio of the principal stresses (YlA/[YlA and the following properties of the material under deformation: (i) the initial inhomogeneity of the sheet f = tB/t A, (ii) the coefficient of normal anisotropy R, (iii) the exponent n of the strain-hardening curve, and (iv) the initial strain-hardening e0.

Varying, in turn, the magnitudes of the above-l isted material constants as well as the ratio 01A/(Y2A, with the other parameters kept constant, the influence of each of the factors on the l imiting value of the strain can be investigated.

I t is, however, the initial inhomogeneity of the material f -- tB/t A which exerts the greatest influence upon the l imiting strain ~*. In the analysis presented above only the geometrical inhomogeneity of the sheet metal was taken into account, caused by the changes in its thickness and disregarding the changes in the plastic properties. The same result will be arrived at if the inhomogeneity of the sheet metal results from a non-uniform distr ibution of impurities, varying texture, different size and orientation of grains and so on. These types of physical inhomogeneity can always be reduced to the equivalent geometrical inhomogeneity and thus expressed by the coefficient f .

Fig. 8 shows that the value of the l imiting strain increases very rapidly as the inhomogeneity of the material diminishes, i.e. as the coefficient f increases. For the fully homogeneous material, f = 1, this strain attains an infinitely large value.

Thus, for a relatively homogeneous sheet metal even slight changes in the inhomogeneity coefficient f exert a very strong influence on the process of the loss of stabi l ity and cause a substantial change in the magnitude of the l imiting strain e*. The enormous sensitivity of the sheet metal under stretching to the degree of its inbomogeneity is the reason for the considerable scatter in experimental results and introduces a persistent degree of uncertainty. The degree of the inhomogeneity of a given specimen or a sheet metal blank can never be foreseen.

The influence of the coefficient of normal anisotropy R is shown in Fig. 9. Its increase makes for the loss of stabil ity and causes a decrease in the l imiting strain e*. Contrariwise, in the drawing processes an increase in R enhances the drawabil ity of the sheet metal.

Fig. 10 shows that the increase in the exponent n of the strain-hardening function diminishes the tendency of the sheet metal to lose its stabil ity and causes an increase in the l imiting strain. However, it is significant that even in the case of a non-hardening material, n = 0, the sheet metal retains a certain abi l i ty to deform under tension. In the case of the uniaxial tension of cylindrical, non-hardening specimens no uniform elongations occur.


The influence of the initial strain-hardening so of the material on the value s* is shown in Fig. 11. The l imiting value of the strain in the sheet decreases sl ightly as the initial strain s o increases.

0'( N~

% 0.5

6 0-4 "6

0.3 ._~ -J

0"2

O.I

o:0.05 ~:o" 2

0.98 0'96 0.94 0"92 0.90

F IG. 8. In f luence o f in i t ia ] non-homogene i ty o f the sheet meta] on the l imit strain.

%

c

m

E .J

0.5

0.4

0'3

0.2

0.1

0.8 I-0 1'2

6o:0.05 - -

1.4

FIG. 9. Influence of anisotropy R on the l imit strain.

J

/ o. /

"w 0.5 / -

/ toe5 4o

.--_ 0-3

J 0 .2

0.1

o o.t o .z o-~ o.4 0.5

Fxo. 10. Influence of strain-hardening n on the l imit strain.

The above considerations have been concerned merely with the phenomenon of the loss of stabil ity of sheet metal under stretching disregarding the possibil ity of its fracture

41

618 ZDZISLA~,V MARCI~IAK and KAZIMIERZ KUCZYI~SKI

dur ing the process. However, the process of the deepening of the groove can be at a certa in ins tant stopped due to the f racture of the sheet meta l at the most weakened place, that is a long the groove. Wi thout looking deeper into the cr iter ion of the loss of cohesion of the mater ia l i t can be roughly assumed that , for a given mater ia l , a given mode of the deformation, i.e. the rat io al /a 2 and the loading history, decohesion takes place when the largest s t ra in at f racture at ta ins a certa in l imit ing value @. This value may be looked upon as one more magn i tude that character izes the propert ies of the mater ia l trader deformation.

. .

(#)

E --I

0.2

0"1

0'5

0"4

0"3

0 FIG. 11.

J n=025

R=I

)

0.1 0.2 0"3 0 .4 0.5

Inf luence of init ial s t ra in ~0 on the lhYdt strain.

;= ,

.F.

(f l Of 2

FIG. 12.

0

m ~f3 ~f

Inf luence of decohesion stra in @ on the l imit strain.

The influence of the value ~/on the l imit ing stra in e* can be analysed in Fig. 12. For hard and sl ightly plast ic mater ials, when @ < e*, the phenomenon of decohesion is a deciding factor as to the value of the al lowable s t ra in in the sheet metal . F racture occurs at the point P1 (Fig. 12) which lies at the very beginning of the process of groove formation. The l imit ing st ra in is then a lmost equal to the f racture strain, e* ~@. For more plast ic mater ia ls f racture takes place at the po int P8 and the influence of the value ey on the l imit ing stra in is considerably less pronounced. F racture appears when the st ra in in the sheet meta l outside the groove approaches the l imit ing st ra in associated wi th the loss of stabi l i ty of the sheet. For a mater ia l with such dist inct plast ic propert ies as have a luminium, lead and the like f racture occurs at the point Pa lying far to the r ight of the point C which corresponds to the complete loss of stabi l i ty of the sheet. In this case the value of the st ra in at f racture E 1 exerts no influence upon the l imit ing st ra in e*, which depends solely upon the process of the loss of stabi l i ty and is equal to the ordinate of the point C, e~ = ec-

When account is taken of the decohesion of the mater ia l the l imit ing stra in e*, which the sheet meta l can suffer prior to its f racture while the rat io a l /a s is kept constant , can be expressed as a funct ion of the propert ies of the mater ia l


To investigate the influence of the part icular variables in the neighbourhood of a certain point, this relationship can be expressed in the form of the total differential

c~ * . . c~e* . c~e* . c~e * . ~e* de* = -xwdf -b~- dgq- -x - -dnT-z - - de0q--~r-dep

Making use of the data obtained from the computer and shown in Figs. 8-12, the values of the partial derivatives in the neighbourhood of a point can be determined. For example, for a material whose plastic properties are described by f = 0.95, R = 1, n = 0.25, s0 = 0.05, st t> 6, the total differential of the function under discussion in the vicinity of the above fixed point (depicted by a circle in Figs. 8-12) assumes the form

ds* = 7 .4dr - 0.294dR q- 1.25dn - 0.76de0. (19)

There can then be predicted the influence of using material with properties differing by dr, dR, dn, de0 and dsf, upon the value of the l imiting strain e*.

F rom the above considerations it follows that those kinds of sheet metal which exhibit the largest value of the l imiting strain, and hence the best drawabil ity, have the following properties :

(a) A high homogeneity, i.e. a uniform thickness, a homogeneous, fine-grained structure and the absence of impurit ies; and

(b) An abil ity to strain harden extensively, i.e. the hardening exponent n should be high and the initial hardening s o low.

The value of the coefficient of anisotropy R has a secondary effect on the behaviour of the sheet metal in this type of forming.

For the sake of comparison let us quote an analogous equation referred to the process of deep drawing and taken from another paper by the authors,

ds* = 0.3 dr+ 0.13 dR - 0.27 dn+ 0.15 de0. (20)

I t is seen that the influence of the part icular material properties on the l imiting strain depends upon the technological process involved. For example, a decrease in the l imiting strain due to the inhomogeneity of the material appears to be many times greater in the case of stretching than in the case of deep-drawing. On the other hand, an increase in the coefficient of anisotropy R leads to better drawabil i ty of the sheet metal and, at the same time, causes a deterioration in its behaviour under tension. Similar differences occur as with other properties of the material.

E -,I

o: ,! n=0.25 %=0.05

0.4 \ f =0.95 _

0-3 ~

0-2 ~

O.i ~ '~ "~-~ _

0'9 0"8 0'7 0"6 0-5 0"~" FIG: 13. Influence of stress ratio (Y=/O" 1 on the l imit strain.

The results presented in Figs. 8-12 refer to equal biaxial tension, al = as. I t should, however, be remembered that the process of losing stabil ity also depends upon the ratio of the principal stresses. As seen in Fig. 13 the state of equal biaxial tension proves most advantageous because it leads to the largest value of the l imiting strain e*. As the ratio ~2/al decreases the strain e* also decreases rapidly and attains its min imum when a2/al = 0.5 which means that a plane state of strain prevails, i.e. e2 = 0.

620 ZDZISLA'~V MARCINIAK and KAZIMIERZ KUCZY~SKI

A conclusion can therefore be drawn that when bulging an elliptical or rectangular shape the l imiting strain of the sheet metal will be considerably smaller than in the case of a circular one. I t can also be pointed out that the most probable direction of the groove will be perpendicular to the direction of the greater principal stress.

REFERENCES

1. R. HILL, Phil. Mag. 41, ser. 7, 1133 (1950). 2. H. W. SWIFT, J. mech. Phys. Solids 1, 1 (1952). 3. P. B. MELLOR, Engineer, Lond. March 25 (1960). 4. Z. MARCINIAK, Archwm. ~Iech. stosow. 4, 17 (1965). 5. R. HILL, The Mathematical Theory of Plasticity. Oxford University Press (1950).

limit strains in the processes of stretch forming sheet metal

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