historia de la plasticidad y formado

8/20/2019 Historia de La Plasticidad y Formado

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Journal of Materials Processing Technology 210 (2010) 1436–1454

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Journal of Materials Processing Technology

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j m a t p r o t e c

History of plasticity and metal forming analysis

Kozo Osakada∗

Osaka University, School of Engineering Science, Machikaneyama 3, Toyonaka, Osaka, Japan

a r t i c l e i n f o

Article history:

Received 6 November 2009Received in revised form 1 April 2010Accepted 1 April 2010

Keywords:

Metal FormingPlasticityHistory

a b s t r a c t

The research history of mechanics, physics and metallurgy of plastic deformation, and the developmentof metal forming analysis are reviewed. The experimental observations of plastic deformation and metalforming started in France by Coulomb and Tresca. In the early 20th century, fundamental investigation

into plasticity flourished in Germany under the leadership of Prandtl, but many researchers moved outto the USA and UK when Hitler came in power. In the second half of the 20th century, some analyzingmethods of metal formingprocesses were developedand installed ontocomputers as software, and theyare effectively used all over the world.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The phenomenon of plasticity has been studied from the viewpointsofmechanics,physicsandmetallurgy,andmanymathemati-cians contributed to refine themechanics of plasticity. Theresearchresults are applied to geophysics and strength of materials, and

of course, are used as indispensable tools for analyzing the metalforming processes.

Although the theories and the experimental results areexplainedin many text books,the building up process is notknownwell.In this article,the history ofplasticityin relationto theanalysisof metal forming is reviewed by putting emphasis on the personalprofile of researchers.

2. Strength of materials and plasticity before the 20th

century

2.1. Early days of strength of materials (Timoshenko, 1953)

Leonardo da Vinci (1452–1519) left many texts and sketchesrelated to science and technology although he did not write books.One of the examples he studied is the strength of iron wire, on

This paper was originally presented as a keynote at the 9th International Con-ference on Technology of Plasticity, hosted by the Korean Society for Technologyof Plasticity, under the Chairmanship of Professor D.Y. Yang. The paper is a uniquerecord of the history of a key area of interest for the Journal of Materials ProcessingTechnology, so we invited Professor Osakada to expand the paper and submit it totheJournal. Weare extremelygratefulfor thepermissionof ProfessorYang to allowus to present the paper here. J.M. Allwood and A.E. Tekkaya, April 2010.∗ Tel.: +81 78 841 2594; fax: +81 78 841 2594.

E-mail address: [email protected].

which hangs a basket being filled with sand. The strength of thewire can be determined by measuring the weight of sand whenthe wire is broken. Unfortunately, the idea and the advancementmade by da Vinci were buried in his note, and were not noticed byscientists and engineers of the time.

It is generally accepted that Galileo Galilei (1564–1642) is the

originator of modern mechanics. In his famous book “Two NewSci-ences”, he treated various problemsrelated to mechanics,includingan example of the strength of a stone beam. He organised hismethods applicable to stress analysis into a logical sequence. Hislecture delivered in the University of Padua attracted many schol-ars gathered from all over Europe, and disseminated the methodsof modern science.

Robert Hooke (1635–1704) published the book “Of Spring” in1678 showing that the degree of elongation of the spring is in pro-portion to the applied load for various cases. It is generallybelievedthat Hooke came up with the idea of elastic deformation when hecarried out experiments on the compressibility of air at OxfordUniversity as an assistant of Robert Boyle (1627–1691), who putforward Boyles’s law.

2.2. Torsion test of iron wire by Coulomb (Bell, 1984)

In the paper submitted to the French Academy of Sciences in1784, C.A. de Coulomb showed the results of torsion tests of ironwire carried outwiththe simpledevice given inFig.1. He estimatedthe elastic shearing modulus of the material from the frequency of torsionalvibration, and measuredthe recovery angleafter twisting.For a wire of length 243.6mm and diameter 0.51 mm, the shearingelastic modulus was estimated to be about 8200 kgf/mm2.

Fig. 2 shows the relation between the number of rotations intwisting and the angle of spring back. When the number of rota-

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K. Osakada / Journal of Materials Processing Technology 210 (2010) 1436–1454 1437

Fig. 1. Torsion test by Coulomb (Bell, 1984).

tions exceeds about 0.5, the angle of recovery becomes smallerthan the angle of twisting, and the recovery angle increases onlyslightly when the number of rotation exceeds two times. This phe-nomenon suggests that plastic deformation starts on the surfaceof the wire when the rotation is about 0.5, and then the plastic

zone expands towards the centre of the wire up to two rotations,and work-hardening proceeds gradually as the number of rotationincreases further.

Let us estimate the yield stress and the flow stress of this wirefrom its dimensions and the elastic modulus. The shear strain andstress after 0.5rotations are calculated respectively to be 0.003 and24kgf/mm2,whichisalittlelargerthanthepresentlyknownshear-ingyieldstress of iron, butmay be reasonable if plastic deformationhas already proceeded for 0.5 rotations. For the spring back angleof 450◦, the shear stress is calculated to be k =50kgf/mm2 if it isdistributed uniformly across the cross-section. This shearing flowstress seems to be reasonable, too.

Fig. 2. Number of rotation and recovery angle in twisting of iron wire carried out

by Coulomb.

Fig. 3. C.A. Coulomb.

Charles A. de Coulomb (1736–1808)(Fig. 3) entered the militarycorps of engineers after receiving preliminary education in Paris.He was sent to the island of Martinique in the West Indies for 9years. There he studied the mechanical properties of materials. In1773, he submitted his first paper on the fracture of sandstone tothe Academy. He concluded that fracture of sandstone occurredwhen the shear stress reached a certain value, similarlyto the yieldcondition due to maximum shear stress.

After returning to France, he worked as an engineer, and con-

tinued to carry out research. In 1781, he won an Academy prize forhis paper on friction, now known as Coulomb friction, and in thesame year he was elected to membership of Academy.

2.3. Elasticity and stress–strain curve

In the early 19th century, the mathematical theory of elastic-ity began to flourish due to efforts of the scholars related withthe École Polytechnique such as S.D. Poisson (1781–1840), Navier(1785–1836), A. Cauchy (1789–1857) and G. Láme (1795–1870), inparallel withthose at the Universityof CambridgeUniversity, beingT. Young (1773–1829) and G. Green (1793–1841) (as reported byTimoshenko, 1953).

To determine the elastic constants, measurements of the

stress–strain relations of metals were begun, and after extensionin the elastic range, stress–strain curves in the plastic range weremeasured. Fig. 4 is the stress–strain curve of piano wire measuredby F.J. Gerstner (1756–1832) andpublished in 1831 (Bell, 1984). Heapplied the load to a piano wire of 0.63mm in diameter and 1.47min lengthwith a seriesof weights. It is obvious that theplastic strainis measured after unloading.

2.4. H. Tresca (Bell, 1984)

H.E. Tresca carried out experiments on metal forming such aspunching, extrusion and compression using various metals, andmeasured the relation between the forming load and ramdisplace-ment. He presented a series of papers to the French Academy of

Sciences, starting in1864. In Fig. 5, the cross-section of an extruded


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1438 K. Osakada / Journal of Materials Processing Technology 210 (2010) 1436–1454

Fig. 4. Stress–strain curve of piano wire measured by Gerstner (Bell, 1984).

billed made of 20 lead sheets is given. Tresca was interested in themetal flow as suggested by the title of his first paper, ‘Mémoire surl’éncoulement des corps solides á de fortes pression (On the flowof a solid body subjected to high pressure)’, rather than yielding inmaterial testing.

Tresca assumed that the extrusion force P could be expressed interms of the shear stress k, and estimated the value of k from themeasured forming load of various processes. Because the valuesof shear stress k estimated from the forming loads occurred in acertain range, he concluded that the metal flow occurred under aconstant maximum shear stress. The values of shearing flow stressmeasured by Tresca are given in Table 1. It seems that the flowstress values are reasonable even from the present view point.

Henri E. Tresca (1814–1885) (Fig. 6) graduated from the ÉcolePolytechnique at the age of 19 in 1833, and sought a career in thedesign of civil structures. But his ambition was deterred by seriousillness, and he spent many years teaching, building and performingtests on hydraulics. In 1852 he began to work at Conservatoire desArts et Metie in Paris as an engineer. He suddenly started researchwork when he was promoted to a major experimental physicistat the age of 50, and soon published many papers. After 8 years’

Fig. 5. Extruded rod by Tresca (Bell, 1984).

Table 1

Shearing flow stress measured by Tresca.

Material Shearing flow stress (kgf/mm2)

Lead 1.82Pure titanium 2.09Lead–titanium alloy 3.39Zinc 9.00Copper 18.93Iron 37.53

concentrated research activity, he was elected as a member of theFrench Academy of Sciences.

2.5. Saint-Venant and Lévy (Timoshenko, 1953)

When Tresca presented his paper to the French Academy of Sci-ences, Barré de Saint-Venant (1797–1886) was the authority of mechanics in France, elected a member of the Academy in 1868.After reading the experimental results of plastic flow by Tresca, hisattention was drawn to the area of plasticity. In 1871, he wrote apaper on elastic–plastic analysis of partly plastic problems, such asthe twisting of rods,bending of rectangularbeams and pressurizing

of hollow cylinders.Saint-Venant assumed that (1) the volume of material does notchange during plastic deformation, (2) the directions of principalstrains coincide with those of the principal stresses (now knownas total strain theory), and (3) the maximum shear stress at eachpoint is equal to a specific constant in the plastic region.

The last assumption is now known as the Tresca yield crite-rion which is expressed with the maximum principal stress 1,theminimum principal stresses 3 and flow stress Y as:

1 − 3 = 2k = Y (1)

Although his analyses are not complete from our current pointof view, it can be said that plastic analysis started from this paper.

Fig. 6. H.E. Tresca.


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Fig. 7. Bauschinger effect.

2.6. Bauschinger and Mohr (Timoshenko, 1953)

In the second half of the19th century, the Technical Universi-ties in German speaking areas became important research centresin plasticity and metal forming. They were established as PS:Polytechnische Schule and then changed to university level TH:Technische Hochschule.

Johann Bauschinger (1833–1893) graduated from Munich Uni-versity and became a professor of Munich PS in 1868. He installeda 100tons tension–compression universal testing machine with anextensometer of his own invention, and carried out a vast num-ber of measurements of stress–strain relations. He found that theyield stress in compression after plastic tensile deformation wassignificantly lower than the initial yield stress in tension. Fig. 7 isthe experimental result from tests carried out in 1885, in whichthe compression test is performed after a tension test up to a strainof 0.6%. It is seen that the initial yield stress was 20.91kgf/mm2

and the yield stress in compression after tensile deformation was9.84kgf/mm2.

In 1882, Otto Mohr (1835–1913) presented a graphical repre-sentationof stressat a point.On a graphwithaxesindicatingnormaland shear stress components, the stress state of a point on a plane

is expressed by a circle. Mohr used his representation of stress todevise a strength theory.

Fig. 8 shows the stress circles for cast iron tested in tension,compression, and in torsion. Mohr suggested that the envelope of the circles was a fracture limit. This idea was extended to a yieldconditionin which shearing yieldstress wasaffectedby hydrostaticpressure. This condition is often called “Mohr’s yield condition”.

Mohr graduated from Hannover PS and worked as a structuralengineer. When he was 32 years old, he was already a well-knownengineer and was invited by Stuttgart TH. After teaching engi-neering mechanics there until 1873, he moved to Dresden TH andcontinued teaching.

Fig. 8. Mohr’s stress circle.

Fig. 9. Experimental result plotted on principal stress plane by Guest (1900).

2.7. J. Guest (Guest, 1900)

In 1900, James Guest (University College London) published apaper from the Royal Society on the strength of ductile materialsunder combined stress states. By carrying out tension and torsiontestsofinternallypressurizedtubes,heexaminedtheoccurrenceof

yielding. Guest is the first person to differentiate yielding of ductilemetalfrombrittlefracture,wherepreviously‘failure’hadbeenusedto express the strength limit of material both due to yielding andbrittle fracture.

He came to the conclusion that yielding occurs when the max-imum shear stress reaches a certain value. In Fig. 9, the yieldingpoints are plotted on a graph of principal stresses in plane stress.Although his conclusion was the same as that of H. Tresca, he natu-rally thought that he had found a new criterion of yielding becausehe did not recognize that the large plastic flow observed by Trescaand the initial yielding he observed were essentially the same phe-nomenon.

3. Yield criteria and constitutive equations

3.1. Progress of research in yield condition

During the 19th century, the maximum shear stress criterionwas established by Tresca, Saint-Venant, Mohr andGuest. The yieldcriterion of elastic shear-strain energy, mostly called Mises yieldcriterion, was putforward in the early 20th century. It is written bythe following equation with the maximum,medium andminimumprincipal stresses 1 ≥ 2 ≥ 3 as:

1√ 2

( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2 = Y (2)

In 1904, M.T. Huber proposed this criterion (Engel, 1994)although limited to compressive hydrostatic stressconditions. This

paper was not known for 20 years by the researchers of plasticity


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because it was written in the Polish language. von Mises (1913)wrote his paper from the view point of mathematics without dis-cussing the physical background.Hencky(1924)introduced Hube’spaper and derived the yield criterion of elastic shear-strain energy.Nádai (1937) showed that the criterion could be interpreted as aprediction that yielding occurs when the shear stress on the octa-hedral plane in the space of principal stresses reaches a criticalvalue.This idea is nowoftenused in text books of plasticity becauseof its simple graphical representation, although it has no physicalmeaning.

The difference between the Tresca yield criterion Eq. (1) andthe Mises criterion Eq. (2) when expressed with principal stressesis that the medium principal stress ( 2) has an effect (Mises) ordoes not (Tresca). This difference was experimentally examined byLode (1926) f ollowing a suggestion by Nádai, and then Taylor andQuinney (1932). Their conclusions were that the medium principalstressdidinfluencetheyieldingconditionformildsteel,aluminiumand copper and thus the Mises criterion offered a better approxi-mation for the yield condition.

Hill (1948) proposed a yield criterion for anisotropic materi-als, and since then many researchers have also tried to expressthe yielding behaviour of anisotropic materials. Yield criteria forother materials which are not incompressible and isotropic have

also been put forward: one example is the criterion for porous orcompressible metals proposed by Shima and Oyane (1976).

3.2. Constitutive equation

In his paper published in 1872, M.Lévy (according toTimoshenko,1953) used an incrementalconstitutive equation. vonMises proposed the same constitutive equation because Lévy’spaper was not known outside of France. Mises considered that theincrements of plastic strain components dε p1, dε

p2, dε

p3 were in pro-

portion to the deviatoric stress components 1, 2,

3, where for

example 1 = 1 − ( 1 + 2 + 3) /3. Thus,

dε p1

1 =

dε p2

2 =

dε p3

3(3)

In the plastic strain range of an elastic–plastic material, theincrements of elastic strain componentsdεe1, dε

e2, dε

e3, and the plas-

tic strain increments dε p1, dε p2, dε

p3, should be handled separately.

Prandtl (1924) treated this problem for plane-strain, and Reuss(1930) (Budapest Technical University) showed the expression forall of the strain components. For example,

dε1 = dεe1 + dε p1 =

1E

d 1 − (d 2 + d 3)

+ dε̄Y

1 −

12

( 2 + 3)

(4)

where dε̄ is an equivalent strain increment which is expressed

in terms of the plastic strain increments, and Y is the flow stress.In the 1960s, when the finite element analysis of elastic–plasticmaterial was under development, a key topic was the inversion of the above Prandtl–Reuss equation to express the stress incrementsin terms of strain increments. However it was eventually foundthat R. Hill had already done this work in his book published in1950 (Hill, 1950).

3.3. Letter of J. Maxwell (Timoshenko, 1953)

The letters from James Clerk Maxwell (1831–1879: famous forMaxwell’s equation) to his friend William Thomson (Lord Kelvin:1824–1907) were published in 1937 in a book, and it was thenfound that Maxwell had written about the occurrence of yielding

as early as 1856.

Maxwell showed that the total strain energy per unit volumecould be resolved into two parts (1) the strain energy of uniformtension or compression and (2) the strain energy of distortion. Thetotal elastic energy per unit volume is expressed as:

F = 12

E

3 (1 − 2) ( 1 + 2 + 3)2

+13

E

2 (1+

2) ( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2 (5)

where E is Young’s modulus, is Poisson’s ration and, 1, 2, 3 are principal stresses. The first term on the right side of theequation is the energy for volume change due to uniform tensionor compression, and the second term is the energy of distortion.

Maxwell made the statement: “I have strong reasons for believ-ing that when thestrain energyof distortion reaches a certain limit,then the element will begin to give way.” Further on the states:“This is the first time that I have put pen to paper on this sub-

ject. I have never seen any investigation of the question, given themechanical strain in three directions on an element, when will itgive way?” Unfortunately he did not return to this subject again.

3.4. M.T. Huber (Engel, 1994; Olesiak, 2000)

In 1904, M.T. Huber proposed that yielding was determined byelastic shear-strain energy distortion when the hydrostatic stresswas compressive, and by the total elastic energy when the hydro-static stress was tensile. Huber’s paper in the Polish language didnot attract general attention until H. Hencky introduced it in 1924.

Maksymilian TytusHuber (1872–1950) (Fig.10) graduatedfromLwów (now Lviv, Ukraine) Technical University in 1895, and stud-

Fig. 10. M.T. Huber.


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ied mathematics at Berlin University. In 1899 he began to workat a technical school in Kraków and wrote his paper on the yieldcondition. In 1909 he was invited to head the Chair of TechnicalMechanics at Lwów Technical University.

When World WarI began,he was calledto theAustria–Hungaryarmy but captured by Russian troops. After World War I, he wentback to Lwów TechnicalUniversity andbecame thepresident of theuniversity. In 1928 he moved to Warsaw Technical University, andactively took part in various advisory andexpert bodies. He becamea member of the Polish Academy of Learning.

During World War II, he was not able to work in the universityand lost all his belongings in a fire ignited by Germans troops dur-ing the Warsaw uprising of 1944, but after the war he was able tocontinue research work in Gdansk Technical University.

3.5. R. von Mises

von Mises (1913) manipulated the maximum shear stresses onthe principal stress planes:

1 = 3 − 2

2 , 2 =

1 − 32

, 3 = 2 − 1

2 (6)

It is apparent that simple summation of the maximum shear

stresses is always zero:

1 + 2 + 3 = 0 (7)

In the space of the maximum shear stresses, he expressed thecriterion of maximum shear stress:

| 1| ≤ k, | 2| ≤ k, | 3| ≤ k (8)

which Mises called Mohr’s yield criterion, and is represented inthe cube shown in Fig. 11. The yield condition of maximum shearstressisexpressedastheintersectionofthecubeandEq.(7) isgivenby the hexagon in the figure.

Then he considered a sum of squares of the shear stresses as thesphere in the figure.

k2 = 21 + 22 + 23 = 14

( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2

(9)

The circle resulted as the intersection of the sphere and Eq. (7)is an approximation of the Mohr (Tresca) criterion. While the Mohrcriterion cannot be expressed by a simple mathematical equation,the newcriterion is easy to handle mathematically, as is often donein mathematical plasticity.

Richard von Mises (1883–1953) (Fig. 12) was born in Lemberg(now Lviv, Ukraine) andgraduated in mathematicsfrom theViennaUniversity of Technology. In 1908 Mises wasawardedhis doctoratefrom Vienna. In 1909, at the age of 26, he was appointed professorin Straßburg (now Strasbourg, France) and received Prussian cit-izenship. There he wrote his paper on the yield criterion. DuringWorld War I, he joined the Austro–Hungarian army and flew as a

Fig. 11. Handling of maximum shear stresses by Mises.

Fig. 12. R. von Mises.

test pilot, and then supervised the construction of a 600HP aircraftfor the Austrian army.

After the war Mises held the new chair at Dresden TH. In 1919,

he was appointed director of the new Institute of Applied Mathe-matics created in the University of Berlin. In 1921 he became theeditor of the newly founded journal “Zeitschrift für AngewandteMathematik und Mechanik” and stayed until 1933.

With the rise of the Nazi party to power in 1933, Mises felt hisposition threatened. He moved to Turkey, where he held the newlycreated chair of Pure and Applied Mathematics at the University of Istanbul. In 1939, amid political uncertainty in Turkey, he went tothe USA, where he was appointed in 1944 as the Gordon-McKayProfessor of Aerodynamics and Applied Mathematics at HarvardUniversity.

3.6. A.L. Nádai

Arpad L. Nádai (1883–1963) (Fig. 13) was born in Hungary andgraduated from Budapest University of Technology, and then stud-ied in Berlin TH getting his doctorate in 1911. In 1918 he movedto L. Prandtl’s Institute of Applied Mechanics in Göttingen and waspromoted to professor in 1923. In 1927, he moved to the Westing-house Laboratory in the USA as the successor of P.E. Timoshenko.Thus his paper in 1937 on the yield criterion was based on workperformed in the USA.

In 1927, Nádai published a book of plasticity in German andthis was translated into English as “Plasticity – A Mechanics of thePlastic State of Matter” (Nádai, 1931) as the first English book of plasticity. The characteristic feature of this book is that it consistsof two parts, (1) plasticity of metals and (2) application of plastic-ity in geophysics problems. In 1950 the first part of this book wasrewritten and published as “Theory of Flow and Fracture of Solids”.


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Fig. 13. A.L. Nádai.

4. Physics and metallurgy of plastic deformation

4.1. Plastic deformation of a single crystal

In 1923, P.W. Bridgman invented a method to make a singlecrystal of metal bypulling it out of molten metal. Since M. von Laue(1879–1960) had already established the method for determiningthe direction of the crystal lattice by X-ray diffraction, the studyof plastic deformation of single crystals could immediately begin.Taylor and Elam (1923) carried out a tension test of an Al singlecrystal (Fig. 14), and found that plastic deformation occurred by

sliding on a certain crystallographic (sliding) plane in a definite(sliding) direction, and the critical shear stress cr on the plane wascalculated.They continuedexperiments withsingle crystals of iron,gold, copper and brass. In Germany, the groups of Schmid (1926)and Göler and Sachs (1927) presented their results from tensiontests of single crystals.

Sachs (1928) calculated the yield stress of polycrystalline metalas an average of those of single crystals with random orienta-tions. The calculated average yield stresses in tension and shearingwere y = 2.24 cr and y = 1.29 cr , respectively, and the ratio was y/ cr = 0.577. This ratio was the same as that derived from Misesyield condition, but the calculated constant 2.24 was too smallcompared with the experimental value. Taylor (1938) proposed a

Fig. 14. Stress–strain curve of single crystal by Taylor and Elam (Bell, 1984).

Fig. 15. Rotation of crystal by plastic sliding.

methodto relatethe yield stress of polycrystalline metalswith thatof single crystals by taking account of the constraints provided byneighbouring grains, giving y = 3.96 cr , which was quite near tothe value obtained by experiments.

When a single crystal is plastically stretched, the direction of the crystal rotates as demonstrated in Fig. 15 due to sliding overthe specific planes, and the sliding planes tend to become parallelto the stretching direction irrespective of the initial orientation.

This means that anisotropy is developed by plastic deformation of polycrystalline metals. Boas and Schmid (1930) were the first tostudy the development of anisotropy.

4.2. Dislocation theory

When the initially polished surfaces of single crystals wereobserved after plastic deformation, slip bands (Fig. 16) wereobserved suggesting that sliding occurred on a limited number of sliding planes. Since an extremely large shear stress, 1000–10,000times as large as the measured critical shear stress, would beneeded to overcome the atomic bonding stress, many researchersworked to understand the mechanism of plastic deformation.Taylor (1934), Polanyi (1934), and Orowan (1934) independently

proposed the sliding mechanismby crystal defects, i.e.dislocations.Fig.17 shows theexplanationby Taylorabout dislocationsin a crys-tal lattice during plastic deformation. The existence of dislocationswas proved in the 1950s after electronic microscopy was invented.

WhenthegeneralassemblyoftheInternationalUnionofPhysicswas held in Tokyo in 1953, N.F. Mott (1905–1999), the presidentof the Union, told the delegates that the first person to recognizethe existence of dislocations was K. Yamaguchi. Yamaguchi (1929)showed a representation of dislocations as shown in Fig. 18 toexplain the cause of the warping of a single crystal after plasticdeformation. Yamaguchi carried out the research in the Institute of Physics and Chemistry at the laboratory of M. Mashima, who had aclose relation with the laboratory of G. Sachs in Germany. In 1937,Yamaguchiwasappointeda professorof OsakaImperialUniversity,

when it was established.


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Fig. 16. Slip bands on polished single crystal.

4.3. Response of metals to high deforming speed

The measurement of stress–strain curves began to attractresearchers in the second half of the 19th century but appropriatemeasuring methods for high speed phenomena did not exist.Dunn(1897) carried out compression tests by using a drop hammer, andmeasured the displacement of the hammer optically and recordedit on a film attached to a rotating drum. By differentiating the dis-

Fig. 17. Dislocation model proposed by Taylor and Elam (1923).

Fig. 18. Dislocation model suggested by Yamaguchi (1929).

placement, he calculated the hammer velocity, and then obtainedthe acceleration or force applied to the hammer by differentiatingthe velocity. From the measured displacement and the calculatedforce, he was able to determine the stress–strain curves. In theearly 20th century, high speed stress–strain curves were obtainedby some groups in Europe with similar measuring methods.

Itihara (1933) (Tohoku Imperial University) measured theshearing stress–shearing strain curve at high speed and high tem-perature up to 1000 ◦C. Fig. 19 shows the equipment of the torsiontest in which the torque was determined by the twisting angle of the measuring bar.

Manjoine and Nádai (1940)measured the stress–stain curves ina high speed tension test at up to 1000◦C as shown in Fig. 20 byusing a load cell with strain gauges.

Kolsky (1949) used the split Hopkinson bar, which was devel-oped by B. Hopkinson at the University of Cambridge in 1914 (Bell,1984). To measure a high strain rate stress–strain curve, a spec-imen was sandwiched between two long bars, one end of whichwas struck and the transmitted elastic wave was measured at theother.

4.4. P.W. Bridgman (Bridgman, 1964)

Although von Kármán carried out compression tests on mar-ble under high pressure and published the results in 1911, the

mechanical behaviour of metals under high hydrostatic pressurewas mainly studied by P.W. Bridgman during the first half of the20th century. Although he found that the ductility of metal was

Fig. 19. High speed torsion test used by Itihara (1933).


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Fig. 20. Stress–strain curves at high speed tension test by Manjoine (1940).

Fig. 21. Fracture strain and pressure measured by Bridgman (1964).

remarkably enhanced by pressure as Fig. 21, he was more inter-ested in the effect of pressure on the stress, which is only a littleaffected by pressure as shown in Fig. 22.

Percy Williams Bridgman (1882–1961) (Fig. 23) studied physicsin Harvard University and received his Ph.D. in 1908. He was

appointed Instructor (1910), Assistant Professor (1919), beforebecoming Hollis Professor of Mathematics and Natural Philoso-phy in 1926. He was appointed Higgins University Professor in1950. From 1905, Bridgman continued the experiments under highpressure for about 50 years, and published the results on plasticdeformation of metals in the book “Studies in Large Plastic FlowandFracture”.He invented a methodfor growing singlecrystalsandproposed a calculation method for the stress state in the neck of atensile test specimen. A machinery malfunction led him to modifyhis pressure apparatus; the result was a new device enabling himto create pressures eventuallyexceeding 100,000kgf/cm2 (10GPa).This newapparatus brought about a plentyof newfindings, includ-ing the effect of pressure on electrical resistance, and on the liquidand solid states. In 1946, he received the Nobel Prize in physics for

his work on high pressure physics.

Fig. 22. Average stress in tension test as a function of pressure measured by Bridg-

man.

Fig. 23. P.W. Bridgman.

4.5. G.I. Taylor

Geoffrey Ingram Taylor (1886–1975) (Fig. 24) was born in Lon-don, and studied mathematics at Trinity College, Cambridge. At theoutbreak of World War I he was sent to the Royal Aircraft Factory

at Farnborough to apply his knowledge to aircraft design.

Fig. 24. G.I. Taylor.


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After the war, Taylor returned to Trinity working on an applica-tion of turbulent flow to oceanography. In 1923, he was appointedto a Royal Society research professorship as a YarrowResearchPro-fessor. This enabled him to stop teaching which he had been doingfortheprevious4yearsandwhichhebothdislikedandforwhichhehadnogreataptitude.Itwasinthisperiodthathedidhismostwideranging work on thedeformation of crystalline materials which ledon from his war work at Farnborough.

During World War II Taylor again worked on applications of his expertise to military problems. Taylor was sent to the UnitedStates as part of the British delegation to the Manhattan project.Taylor continued his research after the end of the War serving onthe Aeronautical Research Committee and working on the devel-opment of supersonic aircraft. Though technically retiring in 1952,he continued researching for the next 20 years.

4.6. M. Polanyi

Michael Polanyi (1891–1976) was born into a Jewish family inBudapest, Hungary and graduatedfrom medical schoolof BudapestUniversity. Hisscientificinterests ledhim to further study in chem-istry at the Karlsruhe TH in Germany and he was awarded his

doctorate in 1917. In 1920, he was appointed a member of theKaiser Wilhelm Institute for Fibre Chemistry, Berlin, where hedeveloped new methods of X-ray analysis and he made contribu-tions to crystallography including dislocation theory.

In 1933, he resigned his position in Germany when Hitler cameto power. Within a few months he was invited to take the chair of physical chemistry at the University of Manchester in England. Hebelieved from his experience in science that there was a necessaryconnectionbetweenthepremisesofafreesocietyandthediscoveryof scientific truths. In 1938 he formed the Society for the Freedomof Science.

4.7. E. Orowan

Egon Orowan (1902–1989) (Fig. 25) was born in Budapest andreceived his doctorate from Berlin TH on the fracture of mica in1932. He had difficulty in finding employment and spent the nextfew years ruminating on his doctoral research, and completed hispaper on dislocations.

After working for a short while on the extraction of kryptonfrom the air for the manufacture of light bulbs in Hungary, Orowanmoved in 1937 to the University of Birmingham where he workedon the theory of fatigue collaborating with R. Peierls (1907–1955).

In 1939, he moved to the University of Cambridge where W.L.Bragg (1890–1971: X-ray analysis) inspired his interest in X-ray diffraction. During World War II, he worked on problems inmunitions production, particularly developing an understandingof plastic flow during rolling. In 1950, he moved to MIT where,in addition to continuing his metallurgical work, he developed hisinterests in geological and glaciological deformation and fracture.

5. Slip-line field method

5.1. Progress of the slip-line field method

Prandtl (1920) presented an analysis of the plane-strain inden-tation of a flat punch into a rigidplastic solid body as shown inFig.26. He assumed a rigid-perfectlyplasticmaterialwithout work-hardening but with a pressure sensitive flow stress (Mohr yieldcriterion). By solving the equilibrium equation, he constructed aseries of lines having directions parallel to the maximum shearstress as shown inFig.26. He correctlyobtained the indenting pres-

sure for a material with shearing flow stress k without pressure

Fig. 25. E. Orowan.

sensitivity as:

p = 2k

1 + 2

(10)

Hencky (1923) derived a general theorem of stress states forslip-line fields which now carries his name. A statically admissiblestressfieldwhichsatisfiestheequilibriumequation,yieldconditionand boundary force is not always correct, because the velocity fieldassociatedwiththestressstatemaynotsatisfyvolumeconstancyormayleadtonegativeenergyconsumption. Geiringer(1930) derivedan equation in relation to the velocity field by considering theincompressibility condition in plastic deformation and the relationbetween strain rate and velocity.

In 1933, when the fundamentals of slip-line field theory wereestablished, the Nazis came to power in Germany and forced the

Jewish researchers to leave from university positions. The remark-able progress attained in the field of plasticity was thus halted inGermany. The researchers expelled from Germany tried to find asafehaveninTurkey,theUnitedStatesandEngland,whereresearchwork on plasticity was replanted.

During World War II, R. Hill used the slip-line field method topredict the plastic deformation of a thick plate being penetrated bya bullet. He proposed slip-line fields for various problems such aswedge indentation, compression of thin plates with friction, platedrawing (Fig. 27) andtensiontestsforanotchedplate.Heusedslip-

Fig. 26. Slip-line field by Prandtl (1920).


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Fig. 27. Slip-line fields for drawing and extrusion proposed by Hill (1950).

line theory which had been developed as a mathematical methodfor the purpose of practical engineering purposes.

Sokolovskii (1948) reported that active research works wereperformedin thearea of slip-linefieldsin theSoviet Union too. Thisis possibly due to the influence of H. Hencky, who established theslip-line theoryand stayedin theSovietUnionto carry outresearchuntil 1938.

Hill (1950), and Prager and Hodge (1951), first presented a sys-tematic account of slip-line theory and displayed the engineeringworth of the approach. Prager’s introduction of the hodograph, orvelocity plane diagram, in 1953 introduced a vast simplificationinto thehandlingof slip-linesolutions andremoved thedifficulties.

During the 1950s and 60s, many new slip-line fields were pro-posed ( Johnson et al., 1970) for extrusion, rolling, drawing andmetal cutting. Since the slip-line method was the only mean toallow prediction of the stress state in the deforming material atthat time, it was used widely eventhough the required assumptionof plane-strain behaviour was not realistic for most metal formingoperations. When finite element methods enabled precise stresscalculation in axi-symmetric and later 3D problems, the use of theslip-line field method decreased from around 1980, although itsacademic value was not lost.

5.2. L. Prandtl

Ludwig Prandtl (1870–1953) (Fig. 28) received his engineer-ing education at the Munich TH. After graduating, he remainedat the school as an assistant of A. Föppl (1854–1924: successorof J. Bauschinger), and carried out doctoral work on the bendingof circular plates. After working in industry for a while, he wasappointedas a professor of industrial mechanics at Hannover TH in1900. There he proposed the membrane analogy of torsion and theboundary layer of fluid flow. In 1904 he was invited to the Insti-tute of Mechanics in Göttingen University. Soon he began to study

plasticity such as plastic buckling and bending. He was appointedthe leader of the laboratory of aerodynamics, and studied wingtheorems and other important works of fluid dynamics.

In 1922, Prandtl established the society of applied mathemat-ics and mechanics, “Gesellschaft für Angewandte Mathematik undMechanik”, andled theareaof applied mechanics. He is also famousas the teacher of many leaders in mechanics in the 20th centurysuch as Th. von Kármán (California Institute of Technology), S.P.Timoshenko (Stanford University), A. Nádai (Westinghouse Labo-ratory), W. Prager (Brown University) and others.

5.3. H. Hencky

Heinrich Hencky (1885–1951) (Fig. 29) graduated from Darm-

stadt TH and began to work in Ukraine as an engineer of a railway

company in 1913at the age of 28. Soon World War I began and thearea was occupied by Russian troops, and he was kept in a camp inUral, where he married a Russian woman.

Although he could not find a permanent job after the war inGermany, he was awarded his Habilitation (qualification for pro-fessorship) from Dresden TH and found a job in Delft TechnicalUniversityin1922.Hecarriedoutresearchintoslip-linefieldtheoryin Delft, and stayed until 1929.

In 1930, he moved to MIT in the USA, but his scientific approachto engineering was not accepted there because practical tech-nologies were overwhelming, and he resigned from MIT afteronly 2 years. In 1936, Hencky was invited to the Soviet Unionby B.G. Galerkin (1871–1945: variational method) and carried outresearch in Moscow University. But in 1938, as relations betweenthe Soviet Union and Germany worsened, he returned to Germany,and worked in a bus manufacturing company in Mainz.

Fig. 28. L. Prandtl.


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Fig. 29. H. Hencky.

5.4. H. Geiringer

Hilda Geiringer (1893–1973) (Fig. 30) was born in Vienna andreceived her doctorate in 1917 from the University of Vienna witha thesis about Fourier series. From 1921 to 1927 she worked at

the Institute of Applied Mathematics in the University of Berlin

Fig. 30. H. Geiringer.

as an assistant of von Mises. Her mathematical interests hadswitchedfrompuremathematics to probability and the mathemat-ical development of plasticity theory. In 1927, Geiringer becamePrivatdozent (lecturer). During this period she had a brief marriageand had one daughter.

In 1930, her work in plasticity theory led to the development of the fundamental Geiringer equations for plane-strain plastic defor-mations. Geiringer remained at the University of Berlin until forcedto leave when Hitler came to power. After a brief stay as a researchassociate at the Institute of Mechanics in Belgium, she became aprofessor of mathematics at Istanbul University in Turkey whereshe stayed for 5 years.

In 1939, she emigrated to the United States with the help of A. Einstein, and became a lecturer at Bryn Mawr College. Whileat Bryn Mawr she married R. von Mises who was then teachingat Harvard. In 1944, Geiringer became professor and chair of themathematics department at Wheaton College in Massachusetts.Attempts to find a position at some of the larger universities nearBostonrepeatedly failed, often because of hergender. From 1955 to1959, she worked as a research fellow in mathematics at Harvardin addition to her position at Wheaton to complete her husband’sunpublishedmanuscripts “Mathematical Theory of Probability andStatistics” after his death in 1953. Geiringer was elected a fellow of

the American Academy of Arts and Science.

5.5. W. Prager (Hopkins, 1980)

William Prager (1903–1980) (Fig. 31) was born in Karlsruhe,and studied at Darmstadt TH receiving his doctorate in 1926 atthe age of 23. From 1929 to 1933 he worked as the acting directorof Prandtl’s Applied Mechanics Institute at Göttingen. At the age of 29, he was appointed at Karlsruhe TH as the youngest professor inGermany, but soon he was dismissed when Hitler came to power.He was invited to Istanbul University, Turkey and acted as a specialadviser in education to the government. Prager remained in Istan-bul until 1941. The expansion of World War II made the position

Fig. 31. W. Prager.


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Fig. 32. R. Hill.

of the German refugees insecure, and he accepted the invitation of Brown University in the USA made on the recommendation of A.Einstein.

TheGraduateDivisionofAppliedMathematicsatBrownUniver-sity was created in 1946 with Prageras itsfirst Chairman,a position

he held until 1953. By his effort, Brown University became the cen-tre of applied mechanics, especially in the area of plasticity in the1950s and 60s.

5.6. R. Hill (Hopkins and Sewell, 1982)

Rodney Hill (1921–) (Fig. 32) was born in Yorkshire, Englandand read mathematics at Gonville and Caius College, Cambridge,where E. Orowan was teaching. In 1943, amid World War II, he

joined a theoretical group on armaments led by N. Mott and he wasassigned the problem of deep penetration of thick armour by high-velocity shells. This aroused Hill’s interest in the field of plasticity.From 1946, he began to work with the group of metal physicistsunder E. Orowan at the Cavendish Laboratory in Cambridge. He

solved various metal forming problems using plasticity theory, andobtained his Ph.D. in 1948. In1949, he was invited to bethe head of a new Section of the Metal Flow Research Laboratory of the BritishIron and Steel Association (BISRA).

Hill expanded his Ph.D. thesis and published the book “TheMathematical Theory of Plasticity” in 1950 when he was 29 yearsold.Thisbookwassoonacceptedasastandardofmechanicsofplas-ticity. In 1952, he became the Editor in Chief of a new journal, the“Journal of Mechanics and Physics of Solids”, which was eventuallyknown as thehighest level journal in mechanics. In 1953 he appliedfor and was offered the post of a new Chair of Applied Mathematicsin Nottingham University, and undertook administrative work ontop of the research works of plasticity until his retirement from theuniversity in 1962. From 1963, Hill moved back to Cambridge and

continued research work in solid mechanics.

Fig. 33. Model of slab method used by Siebel (1923).

6. Slab method

6.1. Analysis of forging by E. Siebel

Siebel (1923) wrote a paper on the analysis of forging. Heassumed a thin area (slab) on which to define an equilibrium equa-tion. In the case of the compression of a cylinder of diameter d andheight 2h shown in Fig. 33, he considered a thin layer of a thicknessdx andaheightthesameasthecylinder,anddefinedanequilibriumof forces acting on the layer.

For the case of the yield stress Y and friction coefficient , hederived the average contacting pressure P̄ as:

P̄ =

Y 1 +1

3d

h (11)

Using more recent applications of the slab method, the averagepressure is calculated to be

P̄ = 2Y

h

d

2exp

d

2h

− d

2h − 1

∼= Y

1 + 1

3

d

h

(12)

Thelast equation,identical to Siebel’s solution,is an approxima-tion of the second equation when d/h is sufficiently smaller than1.0. Siebel numerically calculated the average pressures for sometypical cases of forging, and discussed ways to apply the result tobackward extrusion.

Soon after Siebel’s paper, similar methods were used by von

Kármán (1925) f or analyzing rolling of sheet metal and by Sachs(1927) f or solving wire drawing. Using this method, Siebel contin-ued to analyze various processes, and many researchers extendedthe method. Since the results are mathematically analytical, theyare widely used in industry (Lippmann and Mahrenholts, 1967;Lange, 1985).

Erich Siebel (1891–1961) (Fig. 34) received his doctorate fromBerlin TH in 1923 at the age of 32, on the topic of the calculationof load and energy in forging and rolling. After working in the steelindustry for a short time, he became a leader of the metal formingdivision at Kaiser Institute in Dusseldorf (steel), and carried outanalysis of rolling and forging. In 1931, he became a professor of Stuttgart TH, and treated almost all areas of metal forming such asdeep drawing and wire drawing. He continued theoretical research

into metal forming until retirement in 1957.


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Fig. 34. E. Siebel.

6.2. Th. von Kármán (von Kármán, 1967 )

In 1925, Th. von Kármán presented a short paper on rolling toa meeting of applied mechanics. In three pages, the fundamentaldifferential equation, a result of pressure distribution and energyefficiency were given. Although this paper gave a great influence

to the subsequent researches in rolling technology, Kármán neverreturned to this subject again.

Theodore von Kármán (1881–1963) (Fig. 35) was born intoa Jewish family in Budapest, Austria – Hungary and he studiedengineering at the Royal Technical University in Budapest. Aftergraduating in 1902, he joined Prandtl’s Institute at Göttingen Uni-versity, and received his doctorate in 1908. Then he taught atGöttingen for4 years. During this periodhe measured stress–straincurves for marble under high pressures. He was also interested invibrationinducedby fluid flowandfoundthe Kármánvortex,whichmade him famous in the area of fluid dynamics.

In 1912, he accepted a position as director of the Aeronau-tical Institute at RWTH Aachen. His time at RWTH Aachen wasinterrupted by service in the Austro–Hungarian Army 1915–1918,

where he designed an early helicopter. In his own biography,he does not mention his work on rolling analysis in 1925, pos-sibly because his main interest was building a wind tunnel inAachen.

In 1927, Kármán stayed in Japan for a while as an advisor toan airplane company to build a wind tunnel. In 1930, he acceptedthe directorship of the Guggenheim Aeronautical Laboratory at theCalifornia Institute of Technology (Caltech) and emigrated to theUnited States. He is one of the founders of the Jet Propulsion Labo-ratory, which is now managed and operated by Caltech. In 1946 hebecame the first chairman of the Scientific Advisory Group whichstudied aeronautical technologies for the United States Army AirForces. At age 81 von Kármán received the first National Medal of Science, bestowed in a White House ceremony by President John F.

Kennedy.

Fig. 35. Th. Von Kármán.

6.3. Development of rolling analysis (Lippmann and Mahrenholts,

1967 )

For the case of flat rolling shown in Fig. 36, von Kármán (1925)derived the equilibrium equation for the position x, plate thickness

h and roll angle , as:

d

h · q

2

= p

tan ∓ tan f

dx (13)

where q is the horizontal pressure acting within the plate, p is thepressure acting on the roll surface, and the friction between theplate and the roll is =tan f . The minus (−) sign in the equation isfor the entrance side and the (+) sign is for the exit side.

This equation is valid when the plate thickness is small and thestress in the thickness direction is almost constant, and further,friction is described by Coulomb’s law, as is the case for cold rollingof thin plate.

Tresca’s yield condition Eq. (1) is written as:

p1 − q = Y (14)where p1 is the vertical pressure (perpendicular to q), which can

be calculated from the roll pressure p and the frictional stress pas:

p1 = p ± p tan (+ for entrance side) (15)

Since the parameters x, h and are related to each other whenthe roll radius is given, it is necessary to use only one of them insolving the differential equation. The stresses p, q and p1 are notindependent, and one of them should be chosen as the variable.Thus the equation to be solved may be in the form of p/q/p1

= f

x/h/

(16)

but the result of this problem cannot be presented explicitly.


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Fig. 36. Model of flat rolling by von Kármán (1925).

The distribution of roll pressure in Fig. 37 by Kármán wasobtained numerically, but for designing the rolling mills it is nec-essary to present the rolling torque, roll force and the energyexplicitly. By introducing various approximations, Trinks (1937),Nádai (1939) and Hill (1950) made approximate mathematicalexpressions leading to nomograms to be used in industry.

In applying theoretical results to cold rolling, roll flattening can-not be neglected because the roll is elastically deformed and thelocal radius is changed. The method of treating roll flattening by

Fig. 37. Roll pressure calculated by von Kármán (1925).

Trinks and Hitchcock (1935) was often used in the subsequentanalyses of rolling.

In hot rolling, the plate thickness is generally large, and thusthe stress state cannot be assumed uniform in the thickness direc-tion. Further, the friction in hot rolling is generally high, so stickingor constant friction stress is considered to be suitable. To includethe stress distribution in the thickness direction and a friction lawother than Coulomb friction,Orowan (1943) proposed a more gen-eralized differential equation than Kármán’s equation. Since thesolution of this differential equation cannot be expressed explic-itly, Orowan and Pascoe (1946), Bland and Ford (1948) and manyothers worked to establish approximate mathematical expressionsfor rolling force, torque, load and necessary power.

In the 1950s in the USA and UK, and in the 1960s in Japan,automation of strip rolling in the steel industry began, and manyengineers studied and improved these theories in industry.

7. Upper bound method

7.1. Progress of the upper bound method

Theupperboundmethodprovidesanapproximateformingload

whichisneverlowerthanthecorrectvalue.Becauseofthis,theloadcalculated by this methodis safe in selecting forming machines anddesigning tools, and thus this method has been used practically.

From the relation between velocity and strain rate, the associ-ated strain rate distribution can be determined in the deformingregion. With a kinematically admissible velocity field, which satis-fies the condition of volume constancy and the velocity boundarycondition, together with the flow stress value of the material, anenergy dissipation rate and a forming load greater than or equalto the correct values are obtained. This is guaranteed by the limittheorem for rigid-plastic material.

The upper bound theorem came tobe known when it was intro-duced in the book by Hill (1950) together with other boundingtheorems. Hill states that Markov wrote a paper about the case of rigid-perfectly plastic material in 1947 in the Russian language.

Let us consider a simple plane-strain case in which a rigid-perfectly plastic body with a shearing flow stress k is deformingby external force T due to a tool moving with velocity v . A kine-maticallyadmissible velocity fieldonly withvelocitydiscontinuouslines S∗

d with sliding v * is assumed, where (*) means kinematically

admissible field. The upper bound theorem states:

T v ≤

S∗kv ∗dS∗d (17)

where the left side is the correct working rate and the right sideis the energy dissipation rate for the plastic deformation along thevelocity discontinuities. This inequality means that the energy dis-sipation rate of the right side is greater or equal to the correct value

of the left side. The upper bound value of the forming load T isobtained by dividing the calculated value of the right side by v .Green (1951) applied this theorem to plane-strain compression

between smooth plates, and compared the result with that of theslip-line field method as shown in Fig. 38. While a slip-line fieldrequires a long time to draw, a kinematically admissible field withonly velocity discontinuity lines can be constructed easily with thehelp of a hodograph, and gives a reasonably good result.

With the upper bound method, Green solved the problem of sheetdrawingandbendingofnotchedbarsintheearly1950s.Fromthelate 1950s, W. Johnson (1922, working atthe University of Cam-bridge and at UMIST in Manchester) carried out extensive researchwork by using the upper bound method for plane-strain forging,extrusion, rolling and other forming problems. He optimized the

velocity field by assuming proper variables. In the case of extrusion


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Fig. 38. Velocity field for upper bound method used by Green (1951) and comparison of obtained pressure with slip-line field solution.

shown in Fig. 39 ( Johnson and Mellor, 1973), the angle is taken asthe parameter for optimization.

Kudo (1960) proposed a method for applying the upper boundmethod to axi-symmetric forging and extrusion. He divided theaxi-symmetric billet into several hypothetical units, and derivedmathematical expressions for the velocity of each unit by satisfy-ing the condition of volume constancy and the requirement thatsurface velocities be consistent with those of neighbouring unitsand the tool surfaces as demonstrated in Fig. 40.

Fig. 39. Velocity field used for optimization by Johnson and Mellor (1973).

Fig. 40. Velocity field for axi-symmetric extrusion by Kudo (1960).

In the 1960s, when cold forging of steel was increasingly usedin the automotive industry,prediction of forming pressure becamea very important subject to avoid fracture of the expensive tools.The upperbound method for axi-symmetric deformation appeared

just in time and was used extensively in the cold forging industry.Kudo showed various examples of the use of his method in axi-

symmetric forging and extrusion in the book written with Johnson( Johnson and Kudo, 1962). In the 1960s and 70s, his method led tomuch research activity aimed at finding new types of velocity field,for instance by Kobayashi (1964), Avitzur (1968) and many others.

In the 1970s, the upper bound method was expanded to threedimensional problems byYang andLee (1978)andothers,andthenit was combined with the finite element method, and grew up asthe rigid-plastic finite element method as will be explained in thelater chapter.

7.2. H. Kudo

Hideaki Kudo (1924–2001) (Fig. 41) graduated from TokyoImperial University in 1945 just after World War II, and started

Fig. 41. H. Kudo.


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his career at the Institute of Science and Technology in Tokyo Uni-versity under theguidanceof S. Fukui.He developedaxi-symmetricanalysis as an approximate energy method without knowing thathis method was related to the upper bound theorem. He wasawarded his doctoral degree for a thesis of the analysis of forging.

From 1959 to 1960, Kudo stayedin Hannover TH with O. Kienzleand in Manchester University with W. Johnson and wrote papersas well as a book on the upper bound method.

In 1960, he joined the GovernmentMechanical EngineeringLab-oratory in Tokyo as the team leader for cold forging. He workedhard for the promotion of the industrialization of cold forging andpublished many papers and solved various practical problems.

In 1966, Kudo was appointed to a professorship at YokohamaNational University where he held the chair for metal forming until1989. He carried out fundamental studies of slip-line fields, lubri-cation, material properties and so on as well as unique formingmethods such as tension-aided can extrusion.

Kudo is one of the originators of the Japanese Society for theTechnology of Plasticity and was President of JSTP for 1985/6. Healsooriginatedthe InternationalConferenceforTechnologyof Plas-ticity (ICTP) and served as the chairman of the first meeting held inTokyo in 1984.

8. Finite element method

8.1. Elastic–plastic finite element method

Thefiniteelementmethod (FEM) wasdeveloped forelastic anal-ysis of airplane structures in the 1950s. In this method, a plate wasdivided into many hypothetical elements, and equations of equi-librium at nodal points were developed. Since the equations werelinear in terms of the nodal displacements, they could be solvedwith the matrix method using digital computers, which had justbecame useful in some limited research facilities in the USA.

O.C. Zienkiwitcz and Y.K. Cheung published a book entitled“The finite element method in structural and continuum mechan-ics” (Zienkeiwicz and Cheung, 1967), in which the method wasexplained in detail with the software written in the FORTRAN lan-guage. Referring to this book, many groups in the world began todevelopsoftwareusingdigitalcomputerswhichhadbecomeusablein many countries.

The elastic–plastic FEM was developed as an extension of elastic FEM. Marcal and King (1967) published a paper with anelastic–plastic analysis of a plate specimen with a hole in whichthe development of a plastic zone was given as shown in Fig. 42.Theyemployed a stepwise computation to followthe deformation.The nodal coordinates and the components of stress and strain inthe element were renewed after each step calculation by addingthe increments to the values before the step. Next year, Yamadaet al. (1968) presented a paper on the stress–strain matrix forelastic–plastic analysis.

This method had a significant impact on researchers in plastic-ity and soon, many papers on the elastic–plastic analysis of metalforming problems began to appear. Fig. 43 is the result of analysisof the initial state of hydrostatic extrusion by Iwata et al. (1972).In spite of the great potential of the method, it was found that thecalculation error accumulated as plastic deformation proceeds.

In this method, the small deformation formulation, the stressvalue is renewed at the end of each step of the calculation as:

A x = B x + x (18)

where A x and B

x are the stresses after and before the step com-putation, and x is the incremental stress during the step. Whenan element rotates during the step, the stress state B x fixed to the

element also rotates as shown in Fig. 44. This means that Eq. (18)

Fig.42. Extensionof plasticzoneina platespecimenwitha hole analyzedby Marcal

and King (1967).

Fig. 43. Elastic–plastic analysis of hydrostatic extrusion by Iwata et al. (1972).

Fig. 44. Rotation and deformation of an element during a step computation.


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cannot hold without modifying B x , because A x and

B x are defined

in different coordinate systems.To solve this problem, a formulation for large elastic–plastic

deformation was put forward by McMeeking and Rice (1975).It is not known well that Kitagawa and Tomita (1974) ana-lyzed a large elastic–plastic deformation problem in a paperpublished earlier in 1974 due to the paper written in the

Japanese language. In the 1980s, commercial software forelastic–plastic deformation appeared, and with the splen-did increase of computing speed, the method began tobe used in the industry from around 1990. It should benoted that very small time steps are needed even withthis formulation to avoid error accumulation, and thuselastic–plastic analysis requires very long computing timeseven now.

8.2. Rigid-plastic finite element method

Hayes and Marcal (1967) presented a paper on the usage of theFEMfor optimizing the upper bound methodof a plane stress prob-lem. In the case of the plane stress problem, the stress state couldbe calculated from the optimized velocity field, but this was nottrue for other cases.

A normalstress componentcan be decomposed into a deviatoricstresscomponent x = x − m andahydrostaticstresscomponent m =

x + y + z

/3. Although the deviatoric stress is related to

the strain rate by Eq. (3), the hydrostatic component is not. Forexample, x may be written as:

x = x + m =23

Y

˙̄εε̇ x + m (19)

where m is left indeterminate with the strain rate associatedwith the optimized velocity.

If the hydrostatic stress could be determined, this method wasexpected to have a great potential. Because the stress could becalculated at each step without error accumulation, a drasticallyshorter computing time than the elastic–plastic FEM was possible

although a non-linear problem must be solved by an optimiza-tion.

Lee and Kobayashi (1973) and Lung and Mahrenholtz (1973)published papers that enabled stresscalculation in the rigid-plasticanalysis. Their theoretical basis was the variational principle witha Lagrange multiplier, which had been presented in the bookby Washizu (1968). This principle states that when the rigid-plastic problemis optimizedwith the Lagrange multiplier to handlevolume constancy, the multiplier coincides with the hydrostaticstress.

In FEM with the above principle, one or more multipliers areneeded for each element to obtain the velocity field which satis-fies the incompressibility condition. Since the Lagrange multipliersincreases the number of variables, the computation time becomes

very long for large scale problems. K. Mori and K. Osakada devel-oped a finite element method allowing for slight volume changewithout increasing the number of variables. In this method thehydrostatic stress was calculated directly from the slight volumechange which didnot give significant influence to the deformation.Fig. 45 shows a result of rolling simulated with this method (Moriand Osakada, 1982).

In the 1980s, the rigid-plastic finite element method began tobe used by many researchers in metal forming to solve practicalproblems (Osakada, 1980; Kobayashi et al., 1989). Similarly to theelastic–plastic finite element method, the rigid-plastic finite ele-ment method was expanded to three dimensional problems andwas installed in commercial software, especially for forging in thelate 1980s, andused in industry.Recentdevelopments aresumma-

rized in Mori (2002).

Fig. 45. Rolling analysis with rigid-plastic FEM by Mori and Osakada (1982).

9. Concluding remarks

With the fast development of information technology in the last20 years, the finite element method has become the main toolof metal forming analysis. To realize more accurate simulation,detailed research is still needed into various areas related to metalforming such as anisotropy development and changes of metal-lurgical and mechanical properties during deformation, inelasticbehaviour in unloading and reloading, lubrication, friction, seizureand fracture.

It seems to be inevitable that the finite element method willbe used more directly in industry. Simulation software may beintegrated into CAD based systems, in which forming simulationis carried out directly from CAD of forming tools. In order to enablesmall scale metal forming enterprises to use simulation, low costsoftware with simplified operations is required.

Although it is impossible for the author to predict far into thefuture, on-line control of metal forming processes with simultane-ous simulation maybecome possible by the increased computationspeed and new computing algorithms.

Once the simulation method is well advanced, metal formingengineers will be able to concentrate on more creative and inno-vative works, e.g., developments of forming processes for productswith very high dimensional accuracy, forming machines for silentenvironments, tool coatings for dry metal forming and thermo-mechanical processes with low tool pressure and high productstrength, etc.

Acknowledgements

The author would like to express sincere thanks to: Dr. R.Matsumoto (Osaka University), Dr. M. Otsu (Kumamoto Univer-sity), Prof. K. Mori (Toyohashi University of Technology), Dr. H.Utsunomiya (Osaka University), Prof. F. Fujita (Tohoku University),Dr.H. Furumoto(Mitsubishi HeavyIndustries) and Prof. T. Ishikawa(NagoyaUniversity), Z.S.Olesiak (University of Warsaw) for provid-ing the materials, and Dr. J. Allwood and Mark Carruth (CambridgeUniversity) and Dr. B. Dodd (University of Reading), for providingthe information and for helping to improve the manuscript.

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historia de la plasticidad y formado

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