ictp08 history

7/29/2019 ICTP08 History

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History of Plasticity and Metal Forming Analysis

K. OsakadaOsaka University, Japan

Summary

The research history of mechanics, physics and metallurgy of plastic deformation, and thedevelopment of metal forming analysis are reviewed. The experimental observations of plasticdeformation and metal forming 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 out to the USA and UK when Hitler came in power. In the secondhalf of the 20th century, some analyzing methods of metal forming processes were developed andinstalled onto computers as software, and they are effectively used all over the world.

1. IntroductionThe phenomenon of plasticity has been studied from the view points of mechanics, physics and

metallurgy, and many mathematicians contributed to refine the mechanics of plasticity. The researchresults are applied to geophysics and strength of materials, and of course, are used as indispensabletools for analyzing the metal forming processes.

Although the theories and the experimental results are explained in many text books, the buildingup process is not known well. In this article, the history of plasticity in relation to the analysis of metalforming is reviewed by putting emphasis on the personal profile of researcher.

2. Strength of materials and plasticity before the 20th century

2.1. Early days of strength of materials [1]

Leonardo da Vinci (1452 - 1519) left many texts and sketches related with science and technologyalthough he did not write books. One of the examples he studied is the strength of iron wire, whichhangs a basket being pored with sand. The strength of the wire can be determined by measuring theweight of sand when the wire is broken. Unfortunately, the idea and the advancement made by daVinci were buried in his note, and were not succeeded by the scientists and engineers.

It is generally accepted that Galileo Galirlei (1564 - 1642) is the originator of the modern mechanics.In the famous book Two New Sciences, he treated various problems related with mechanics,including an example of the strength of stone beam. He put his methods applicable in stress analysisinto a logical sequence. His lecture delivered in the University of Padua attracted many scholarsgathered from all over Europe, and it disseminated the method of modern science.

Robert Hooke (1635 1704) published the book Of Spring in 1678 showing that the degree ofelongation of spring is in proportion to the applied load for various cases. It is generally believed thatHooke came up with the idea of elastic deformation when he carried out experiments ofcompressibility of air in Oxford University as an assistant of Robert Boyle (1627 1691), who putforward the Boyless law.

2.2. Torsion test of iron wire by Coulomb [2]

In the paper submitted to the French Academy of Sciences in 1784, C. A. de Coulomb showed theresults of tortion test of iron wire carried out with the simple device given in Fig.1. He estimated theelastic shearing modulus from the cycle of torsional vibration, and measured the recovery angle aftertwisting. For the wire of the length of 243.6mm and diameter of 0.51mm, the shearing elastic modulus

was estimated to be about 8200kgf/mm2


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Fig. 2 shows the relation between the number of rotation in twisting and the angle of spring back.When the number of rotation exceeds about 0.5, the angle of recovery becomes smaller than the angleof twisting, and the recovery angle increases only slightly when the number of rotation exceeds twotimes. This phenomenon may be interpreted as that plastic deformation starts on the surface of thewire when the rotation is about 0.5, and then the plastic zone expands towards the centre of the wireup to 2 rotations, and work-hardening proceeds gradually as the number of rotation increases further.Let us estimate the yield stress and the flow stress of this wire from its dimensions and the elastic

modulus. The shear strain and stress at the time of 0.5 rotation are calculated respectivelyto be 0.003and 24kgf/mm2 , which is a little larger than the presently known shearing yield stress of iron, but maybe reasonable if plastic deformation has already proceeded at 0.5 rotation. For the spring back angle of450, the shear stress is calculated to be =k 50kgf/mm2if it distributes uniformly across the cross-section. This shearing flow stress seems to be reasonable, too.

Charles A. de Coulomb (1736 -1808) entered the military corpsof engineers after receiving preliminary education in Paris. He wassent to the island of Martinique in the West Indies for nine years.

There he studied mechanical properties of materials. In 1773, he

submitted his first paper on fracture of sandstone to the Academy.He concluded that fracture of sandstone occurred when the shearstress reached a certain value, similarly to the yield condition dueto maximum shear stress.

After returning to France, he worked as an engineer, andcontinued to carry out research. In 1781, he won an Academy prizeon his paper of friction, presently known as Coulomb friction, andin the same year he was elected to membership of Academy.

2.3. Elasticity and stress-strain curve

In the early 19th century, the mathematical theory of elasticity began to flourish by the efforts of thescholars related with cole Polytechnique such asS.D.Poisson (1781 - 1840), Navier (1785 1836),A.Cauchy (1789 1857) and G.Lme (1795 1870), andof Cambridge University as T. Young (1773-1829) andG.Green (1793-1841) [1].

To determine the elastic constants, measurement ofstress-strain relations of metals started, and as anextension of elastic range, stress-strain curves in theplastic range were measured. Fig.3 is the stress-straincurve of piano wire measured by F.J .Gerstner (1756 1832) and published in 1831 [2]. He applied the load to a

piano wire of 0.63mm in diameter and 1.47min lengthwith a series of weights. It is obvious that plastic strain ismeasured after unloading.

Fig.1 Torsion test by Coulomb [2]

C.A. Coulomb

1 32 54 76 98

1 turn =360

Number of Rotation

200100

400

300

500

Rec

overyangle

/

0 1 32 54 76 98

1 turn =360

Number of Rotation

200100

400

300

500

Rec

overyangle

/

0

Fig.2 Number of rotation and recovery angle intwisting of iron wire carried out by Coulomb

Fig.3 Stress-strain curve of piano wiremeasured by Gerstner [2]


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Maurice Lvy (1838 1910), a student of Saint-Venant, changed the assumption (2) of Saint-Venantto as: (2) the directions of increments of principal strains coincide with those of the principal stresses,and published a paper in 1872. This is the first attempt of using incremental flow rule. But exactlyspeaking, his method is a little different from the present theory because he did not differentiate theplastic strain from the total strain which includes elastic strain even in plastic region.

2.6. Bauschinger and Mohr [1]

In the second half of the19th century, the TechnicalUniversities in German speaking area became importantresearch centres of plasticity and metal forming. They wereestablished as PS: Polytechnische Schule and then changed touniversity level TH: Technische Hochschule.

Johann Bauschinger (1833 1893) graduated from MunichUniversity and became a professor of Munich PS in 1868. Heinstalled a 100 tons tension-compression universal testingmachine with the extensometer of his invention, and carried out

vast amount of measurements of stress-strain relations. Hefound that the yield stress in compression after plastic tensile deformation was significantly lower thanthe initial yield stress in tension. Fig.5 is the experimental result carried out in 1885, in whichcompression test is done after tension test up to a strain of 0.6%. It is seen that the initial yield stresswas 20.91kgf/mm2and the yield stress in compression after tensile deformation was 9.84kgf/mm2.

In 1882, Otto Mohr (1835 1913) presented agraphical representation of stress at a point. On thecoordinates of normal and shear stress components, thestress state of a point on a plane is expressed by a circle.Mohr used his representation of stress to devise a strengththeory.

Fig. 6 shows the stress circles for cast iron tested in

tension, compression, and in torsion. Mohr suggested thatthe envelope of the circles was fracture limit. This ideawas extended to the yield condition in which shearingyield stress was affected by hydrostatic pressure. Thiscondition was often called Mohrs yield condition.

Mohr graduated from Hannover PS and worked as a structural engineer. When he was 32 years old,he was already a well-known engineer and was invited by Stuttgart TH. After teaching engineeringmechanics there until 1873, he moved to Dresden TH and continued teaching.

2.7. J . Guest [3]

In 1900, James Guest (University College London) published apaper from the Royal Society on the strength of ductile materials undercombined stress states. By carrying out tension and torsion tests ofinternally pressurized tubes, he examined the occurrence of yielding.Guest is the first person to differentiate yielding of ductile metal frombrittle fracture, where previously failure had been used to express thestrength limit of material both due to yielding and brittle fracture.

He came to a conclusion that yielding occurs when maximum shearstress reaches a certain value. In Fig.7 the yielding points are plotted onthe graph of principal stresses in plane stress. Although the conclusionwas the same as that of H.Tresca, he naturally thought that he found anew criterion of yielding because he did not recognize the large plastic

flow observed by Tresca and the initial yielding he observed wereessentially the same phenomenon.

Compression

Shear

Tension

c t

Fracturecriterion

Compression

Shear

Tension

c t

Fracturecriterion

Fig.6 Mohrs stress circle

20.91

9.84

9.84

/kgfmm-2

0.2 0.4 0.6Strain /%

Stress

-0.2-0.4-0.6-0.8 0

20.91

9.84

9.84

/kgfmm-2

0.2 0.4 0.6Strain /%

Stress

-0.2-0.4-0.6-0.8 0

Fig.5 Bauschinger effect

MaximumshearstressMaximumshearstress

Fig.7 Experimental resultplotted on principal stressplane by of Guest [3]

tensioncompression

tension


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3. Y ield criteria and constitutive equations

3.1. Progress of research in yield condition

During the 19th century, the maximum shear stress criterion was established by Tresca, Saint-

Venant, Mohr and Guest. The yield criterion of elastic shear-strain energy, mostly called as Misesyield criterion, was put forward in the early 20th century. It is written by the following equation withthe maximum, medium and minimum principal stresses 321 as:

(2)

In 1904, M.T.Huber proposed this criterion [4] although limiting to compressive hydrostatic stresscondition. This paper was not known for 20 years by the researchers of plasticity because it waswritten in Polish language. In 1913, R.von Mises [5] wrote his paper from the view point ofmathematics without discussing physical background. In 1924, H.Hencky [6] introduced the Huberspaper and derived the yield criterion of elastic shear-strain energy. In 1937, A.L.Ndai [7] showed thatthe criterion could be interpreted as that yielding occurs when the shear stress on the octahedral plane

in the space of principal stresses reaches a critical value. This idea is now often used in the text booksof plasticity because of its simple graphical handling although it has no physical meaning.

The difference between the Tresca yield criterion eq.(1) and the Mises criterion eq.(2) whenexpressed with principal stresses is that the effect of the medium principal stress ( 2 ) exists (Mises) ornot (Tresca). This difference was experimentally examined by W.Lode [8] in 1926 with the suggestionby Ndai, and then G.I.Taylor and H.Quinny [9] in 1932. Their conclusions were that the mediumprincipal stress did give influence to the yielding condition for mild steel, aluminium and copper andMises criterion offered a better approximation of yield condition.

In 1948 R.Hill [10] proposed a yield criterion of anisotropic material, and since then manyresearchers tried to express the yielding behaviour of anisotropic materials. Y ield criteria for thematerials other than incompressible isotropic materials were also put forward; one example is the

criterion for porous or compressible metals proposed by S.Shima and M.Oyane [11] in 1976.

3.2. Constitutive equation

In the paper published in 1872, M.Lvy [1] used an incremental constitutive equation. Von Misesproposed the same constitutive equation because Lvys paper was not known outside of France.Mises considered that the increments of plastic strain components ppp ddd 321 ,, were in proportion

to the deviatoric stress components 321 ,, , where for example 3/)(' 32111 ++= .

(3)

In the plastic strain range of a elastic-plastic material, the increments of elastic strain componentseeeddd 321 ,, and the plastic strain increments

pppddd 321 ,, should be handled separately. L.Prandl

[12] treated this problem for the plane strain problem in 1924, and A.Reuss (Budapest TechnicalUniversity) showed the expression for all of the strain components in 1930 [13].For example,

(4)

where d is 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-plastic material was under development,

expression of the stress increments in terms of strain increments by inverting the above Prandtl-Reussequation was an important subject. But it was eventually found that R.Hill had already done this workin the book published in 1950 [14].

3

3

2

2

1

1

=

= ppp ddd

( ) ( ) ( ) Y=++ 2132

322

212

1

( ){ } ( )

+

++=+= 311311111 2

11Y

dddd

Eddd

pe


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3.3. Letter of J . Maxwell [1]

The letters from James Clerk Maxwell (1831 1879: famous for Maxwells equation) to his friendWilliam Thomson (Lord Kelvin: 1824 1907) were published in 1937 in a book form, and then it wasfound that Maxwell had written about occurrence of yielding as early as in 1856.

Maxwell showed that the total strain energy per unit volume could be resolved into two parts (1) thestrain energy of uniform tension or compression and (2) the strain energy of distortion. The totalelastic energy per unit volume is expressed as:

(5)

where E is Youngs modulus, is Poissons ration and 321 ,, are principal stresses. The

first term in the right side of the equation is energy for volume change due to uniform tension orcompression, and the second term is the energy of distortion.

Maxwell made the statement: I have strong reason for believing that when [the strain energy ofdistortion] reaches a certain limit, then the element will begin to give way. Further on he states:This is the first time that I have put pen to paper on this subject. I have never seen any investigation

of the question, given the mechanical strain in three directions on an element, when it will give way?Unfortunately he did not return to this subject again.

3.4. M. T. Huber [3]In 1904, M.T. Huber proposed that the yielding was determined by

elastic shear-strain energy distortion when the hydrostatic stress wascompressive, and by the total elastic energy when the hydrostaticstress was tensile. Hubers paper in Polish language did not attractgeneral attention until H.Hencky introduced it in 1924.

Maksymilian Tytus Huber (1872 1950) graduated from Lww

(now Lviv, Ukraine) Technical University in 1985, and studiedmathematics in Berlin University. In 1899 he began to work at atechnical school in Krakw and wrote the paper on the yieldcondition. In 1909 he was invited to head of the Chair of TechnicalMechanics at Lww Technical University.

When the World War I began, he was called to the Austria-Hungary army but captured by the Russian troop. After the World War I, he went back to Lww

Technical University and became the president of the university. In 1928 he moved to WarsawTechnical University, and actively took part in various advisory and expert bodies. He became amember of the Polish Academy of Learning.

During the World War II, he was kept again in a concentration camp, but after the war he was ableto continue research work in Gdansk Technical University.

3.5. R. von Mises

In the paper published in 1913, R.von Mises [5] manipulated the maximum shear stresses on theprincipal stress planes:

. (6)

It is apparent that simple summation of the maximum shear stresses is always zero:0321 =++ (7)

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

k1 k2 k3 , (8)which Mises called Mohrs yield criterion, as the cube shown in Fig. 8.

,2

231

= ,

231

2

=

212

3

=

M.T.Hube

( ) ( ) ( ) ( ){ }2132322212321 )21(231

)21(321

+++

+++

=EE

F


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The yield condition of maximum shear stress is expressed as the intersection of the cube and eq. (7) asgiven by the hexagon in the figure.

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

The circle resulted as the intersection of the sphere and eq. (7) is an approximation of the Mohr(Tresca) criterion. While the Mohr criterion is hardly expressed by a simple mathematical equation,the new criterion is easy to handle mathematically, as is often done in mathematical plasticity.

Richard von Mises (1883 1953) was born in Lemberg (now Lviv,Ukraine) and graduated in mathematics from the Vienna Universityof Technology. In 1908 Mises was awarded doctorate from Vienna.In 1909, at the age of 26, he was appointed as professor in Straburg(now Strasbourg, France) and received Prussian citizenship. There hewrote the paper on the yield criterion.

During the World War I, he joined the Austro-Hungarian armyand flew as a test pilot, and then he supervised the construction of a600HP aircraft for the Austrian army.

After the war Mises held the new chair at the Dresden TH. In 1919,he was appointed as director of the new Institute of AppliedMathematics created in the University of Berlin. In 1921 he becamethe editor of the newly founded journal Zeitschrift fr AngewandteMathematik und Mechanik and stayed until 1933.

With the rise of the Nazi party to power in 1933, Mises felt his position threatened. He moved toTurkey, where he held the newly created chair of Pure and Applied Mathematics at the University ofIstanbul. In 1939, amid political uncertainty in Turkey, he went to the USA, where he was appointedin 1944 Gordon-McKay Professor of Aerodynamics and Applied Mathematics at Harvard University.

3.6. A.L. Ndai

Arpad L. Ndai (1883 1963) was born in Hungary and graduatedfrom Budapest University of Technology, and then studied in Berlin

TH getting doctorate in 1911. In 1918 he moved to L.PrandtlsInstitute of Applied Mechanics in Gttingen and was promoted toprofessor in 1923. In 1927, he moved to the WestinghouseLaboratory in the USA as the successor of P.E.Timoshenko. Thus hispaper in 1937 on yield criterion was the work in the USA.

In 1927, Ndai published a book of plasticity in German and thiswas translated into English as Plasticity A Mechanics of thePlastic State of Matter [15] in 1931 as the first English book ofplasticity. The characteristic feature of this book is that it consists of

two parts, (1) plasticity of metals and 2) application of plasticity ingeophysics problems. In1950 the first part of this book was rewrittenand published as Theory of Flow and Fracture of Solids.

( ) ( ) ( ){ }2132322212322212 41

++=++=k

R. von Mises

A. L. Ndai

Fig.8 Handling of maximumshear stresses by Mises

(9)


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4. Physics and metallurgy of plastic deformation

4.1. Plastic deformation of single crystal

In 1923, P.W.Bridgman invented a method to make single crystal of metal by pulling out of molten

metal. Since M. von Laue (1879 1960) had already established the method of determining thedirection of crystal lattice by X-ray diffraction, the study of plastic deformation of single crystalstarted abruptly. In 1923, G.I.Taylor and C.F.Elam [16] carried out tension test of Al single crystal(Fig.9), 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 was calculated.

They continued experiments with single crystals of iron, gold, cupper and brass. In Germany, thegroups of E.Schmid (1926) [17] and V.Gler and G.Sachs (1927) [18] presented their results oftension test of single crystals.

G.Sachs(1928) [19] calculated the yield stress of polycrystalline metal as an average of those of

single crystals with random orientations. The calculated average yield stresses in tension and shearingwerecry = 24.2 and cry = 29.1 , respectively, and the ratio was

577.0/ = yy . This ratio was the same as that derived from Mises yield

condition, but the calculated constant 2.24 was too small compared with theexperimental value. G.I.Taylor(1938) [20] proposed a method to relate theyield stress of polycrystalline with that of single crystals by taking theconstraint of the neighboring grains as

cry . 963= , which was quite near to

the value obtained by experiments.When a single crystal is plastically stretched, the direction of the crystal

rotates as demonstrated in Fig.10 due to the sliding over the specific planes,and the sliding planes tend to become in parallel with the stretching direction

irrespective of the initial orientation. This means that anisotropy is developedby plastic deformation of polycrystalline metals. W. Boas and E. Schmid(1930) [21] studied the development of anisotropy first.

4.2. Dislocation theory

When the initially polished surfaces of single crystals were observed after plastic deformation, slipbands (Fig.11) were observed suggesting that sliding occurred over limited number of sliding planes.Since an extremely large shear stress, 1000 -10000 times as large as the measured critical shear stress,should be needed to overcome the atomic bonding stress, many researchers tackled to find themechanism of plastic deformation. G.I.Taylor [22], M.Polanyi [23], and E.Orowan [24] independentlyproposed the sliding mechanism by crystal defect, i.e. dislocation, in 1934. Fig.12 shows theexplanation by Taylor about dislocation in crystal lattice during plastic deformation. The existence ofdislocation was proved in 1950s after the electronic microscopy was invented.

Fig.9 Stress-strain curve of single crystal by Taylor and Elam [16]

Fig.10 Rotation ofcrystal by plasticsliding


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When the general assembly of International Union of Physics was held in Tokyo in 1953, N. F.Mott (1905 1999), the president of the Union, told the attendants that the first person whorecognized the existence of dislocation was K. Yamaguchi. In the paper published in 1929 [25],

Yamaguchi showed a figure of dislocation as Fig.13 to explain the cause of warping of single crystalafter plastic deformation. Yamaguchi carried out the research in the Institute of Physics and Chemistryat the laboratory of M.Mashima, who had a close relation with the laboratory of G. Sachs in Germany.In 1937, Yamaguchi was appointed a professor of Osaka Imperial University, when it was established.

4.3. Response of metals to high deforming speed

Measurement of stress-strain curve began to attract researchers in the second half of the 19th

century but appropriate measuring method of high speed phenomenon did not exist. In 1897, B.Dunn[26] carried out compression test by using drop hammer, and measured the displacement of thehammer optically and recorded it on a film attached to a rotating drum. By differentiating thedisplacement, he calculated the hammer velocity, and then obtained the acceleration or force appliedto the hammer by differentiating the velocity. From the measured displacement and the calculatedforce, he was able to determine the stress-strain curves. In the early 20th century, high speed stress-strain curves were obtained by some groups in Europe with the similar measuring method.

In 1933, M. Itihara (Tohoku Imperial University) measured the shearing stress-shearing strain curveat high speed and high temperature up to 1000C[27]. Fig.14 shows the equipment of the torsion testin which the torque was determined by the twisting angle of the measuring bar.

Fig.12 Dislocation model proposed by Taylor [22]

Fig.13 Dislocation modelsuggested by Yamaguchi [25]

Fig.14 High speed torsion test usedby Itihara [27]Fig.15 Stress-strain curves at high speedtension test by Manjoine [28]

Fig.11 Slip bands on polished single crystal


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In 1940, M.J .Manjoine and A.L.Ndai measured the stress-stain curves in high speed tension test upto 1000C as shown in Fig.15 [28] by using a load cell with strain gauges.

In 1949 H.Kolsky [29] used the split Hopkinson bar, which was developed by B.Hopkinson in 1914[2]. To measure a high strain rate stress-strain curve, a specimen was sandwiched between two longbars, one of which end was struck and the transmitted elastic wave was measured by the other.

4.4. P.W.Bridgman [30]

Although von Krmn carried out compression test of marble under high pressure and published theresult in 1911, the mechanical behaviour of metals under high hydrostatic pressure was mainly studiedby P.W. Bridgman during the first half of the 20th century. Although he found that the ductility ofmetal was remarkably enhanced by pressure as Fig.16, he was more interested in the effect of pressureon the stress, which is a little affected by pressure as shown inFig.17.

Percy Williams Bridgman (1882-1961) studied physics in Harvard University and received Ph.D. in1908. He was appointed Instructor (1910), Assistant Professor (1919), before becoming Hollis

Professor of Mathematics and Natural Philosophy in 1926. He wasappointed Higgins University Professor in 1950.

From 1905, Bridgman continued the experiments under high pressurefor about 50 years, and published the results on plastic deformation ofmetals in the book Studies in Large Plastic Flow and Fracture. Heinvented a method of growing single crystal and proposed thecalculation method of stress state in the neck of tensile test specimen.

A machinery malfunction led him to modify his pressure apparatus;the result was a new device enabling him to create pressures eventuallyexceeding 100,000 kgf/cm (10 GPa). This new apparatus brought abouta plenty of new findings, including the effect of pressure on electricalresistance, and on the liquid and solid states. In 1946, he received NobelPrize in physics for his work on high pressure physics.

4.5. G.I. Taylor

Geoffrey Ingram Taylor (1886 - 1975) was born in London, and he studied mathematics at TrinityCollege, Cambridge. At the outbreak of World War I he was sent to the Royal Aircraft Factory atFarnborough to apply his knowledge to aircraft design.

After the war, Taylor returned to Trinity working on an application of turbulent flow tooceanography. In 1923 he was appointed to a Royal Society research professorship as a YarrowResearch Professor. This enabled him to stop teaching which he had been doing for the previous fouryears and which he both disliked and had no great aptitude for. It was in this period that he did his

most wide ranging work on the deformation of crystalline materials which led on from his war work atFarnborough.

P.W. Bridgman

Fig.16 Fracture strain and pressure measuredby Bridgman [30]

Fig.17 Average stress in tension test andpressure measured by Bridgman [30]


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During the World War II Taylor again worked on applications ofhis 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 on theAeronautical Research Committee and working on the development ofsupersonic aircraft. Though technically retiring in 1952 he continuedresearching for the next twenty years.

4.6. M. Polanyi

Michael Polanyi(1891 1976) was born into a Jewish family inBudapest, Hungary and graduated from medical school of BudapestUniversity. His scientific interests led him to further study inchemistry at the Karlsruhe TH in Germany and got doctorate in 1917.

In 1920 he was appointed a member of the Kaiser Wilhelm Institutefor Fibre Chemistry, Berlin, where he developed new methods of X-ray analysis and he made contributions to crystallography including the

dislocation theory.In 1933, he resigned his position in Germany when Hitler came inpower. Within a few months he was invited to take the chair ofphysical chemistry at the University of Manchester in England.He believed from his experience in science that there was a necessaryconnection between the premises of a free society and the discovery ofscientific truths. In 1938 he formed the Society for the Freedom ofScience.

4.7. E. Orowan

Egon Orowan (1902 1989) was born in Budapest and receiveddoctorate from Berlin TH on the fracture of mica in 1932. He haddifficulty in finding employment and spent the next few years ruminatingon his doctoral research, and completed the paper on dislocation.

After working for a short while on the extraction of krypton from theair for the manufacture of light bulbs in Hungary, Orowan moved in1937 to the University of Birmingham where he worked on the theory offatigue collaborating with R. Peierls (1907-1955:metal physics).

In 1939, he moved to Cambridge University where W.L.Bragg (1890-1971:X-ray analysis) inspired his interest in X-ray diffraction. DuringWorld War II, he worked on problems of munitions production,particularly that of 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 slip-line field method

In 1920, L.Prandtl [31] presented an analysis ofplane strain indentation of flat punch into a rigid-plastic solid body as Fig.18. He assumed a rigid-perfectly plastic material without work hardening butwith a pressure sensitive flow stress (Mohr yieldcriterion). By solving the equilibrium equation, heconstructed a series of lines having directions to the

M. Polanyi

E. Orowan

Fig.18 Slip-line field by Prandtl [31]

G.I. Taylor


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maximum shear stress as Fig.18. He correctly obtained the indenting pressure for the material of ashearing flow stress kwithout pressure sensitivity as;

(10)

In 1923, H. Hencky [32] derived a general theorem of stress state for slip-line field which nowcarries his name. A statically admissible stress field which satisfies the equilibrium equation, yieldcondition and boundary force is not always correct, because the velocity field associated with thestress state may not satisfy the volume constancy or may lead to negative energy consumption. In1930, H. Geiringer [33] derived an equations in relation to the velocity field by considering theincompressibility condition in plastic deformation and the relation between strain rate and velocity.

In 1933, when the fundamentals of slip-line field theory were established, the Nazi came to powerin Germany and forced the Jewish researchers to leave from the university positions. The remarkableprogress attained in the field of plasticity was halted in Germany. The researchers expelled fromGermany tried to find their safe haven in Turkey, United States and England, where the researches onplasticity were replanted.

During the World War II, R. Hill used the slip-line field method to predict the plastic deformation

of thick plate being penetrated by bullet. He proposed slip-line fields for various problems such aswedge indentation, compression of thin plate with friction, plate drawing (Fig.19) and tension test ofnotched plate. He utilized the slip-line theory which had been developed as a mathematical method forthe purpose of practical engineering purposes.

V.V.Sokolovskii [34] reported that active research works were done in the area of slip-line field inthe Soviet Union too. This is possibly due to the influence of H.Hencky, who established the slip-linetheory and stayed in the Soviet Union to carry out research works until 1938.

The book by R.Hill in 1950[35], and that by W.Prager and P.G.Hodge in1951 [36], first presentedsystematic account of slip-line theory and displayed the engineering worth of the approach. Pragersintroduction of the hodograph, or velocity plane diagram, in 1953 introduced a vast simplification intothe handling of slip-line solution and removed the difficulties.

During the 1950s and 60s, many new slip-line fields were proposed [37] for extrusion, rolling,drawing and metal cutting. Since only the slip-line method providedthe stress state in the deforming material at the time, it was effectively

used although plane strain metal forming was not realistic. But whenthe finite element methods enabled precise stress calculation in axi-symmetric and later 3D problems, the use of slip-line field methoddecreased from around 1980, although its academic value was not lost.

5.2. L . Prandtl

Ludwig Prandtl1870 1953received engineering education atthe Munich TH. After graduating, he remained at the school as anassistant of A.Fppl (1854 1924: successor of J .Bauschinger), andcarried out doctoral work on bending of circular plates. After working

in the industry for a while, he was appointed as a professor ofindustrial mechanics of Hannover TH in 1900. There he proposed

+=

212

kp

L.Prandtl

Fig.19 Slip-line fields fordrawing and extrusion proposedby Hill [14]


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membrane analogy of torsion and boundary layer of fluid flow. In 1904 he was invited to the Instituteof Mechanics in Gttingen University. Soon he began to study plasticity such as plastic buckling andbending. He was appointed the leader of the laboratory of aerodynamics, and studied wing theoremand other important works of fluid dynamics.

In 1922 Prandtl established the society of applied mathematics and mechanics, Gesellschaft frAngewandte Mathematik und Mechanik, and led the area of applied mechanics. He is also famous asthe teacher of many leaders in mechanics in the 20th century such as Th. von Krmn (CaliforniaInstitute of Technology), S.P. Timoshenko (Stanford University), A.Ndai (Westinghouse Laboratory),W. Prager (Brown University) and others.

5.3. H.Hencky

Heinrich Hencky (1885 1951) graduated from Darmstadt TH and began to work in Ukraine as anengineer of a rail-way company in 1913 at the age of 28. Soon the World War I began and the areawas occupied by Russian troops, and he was kept in the camp inUral, where he married a Russian woman.

Although he could not find permanent job after the war in

Germany, he got Habilitation (qualification for professorship) fromDresden TH and found a job in Delft Technical University in 1922.He carried out the research of slip-line field theory in Delft, andstayed until 1929.

In 1930 he moved to MIT in the USA, but his scientific approachto engineering was not accepted there because practical technologieswere overwhelming, and he resigned MIT only after 2 years. In 1936,Hencky was invited to the Soviet Union by B.G..Galerkin (1871 -1945: variational method) and carried out research in MoscowUniversity. But in 1938, the relation between the Soviet Union andGermany was worsened and he returned to Germany, and worked ina bus manufacturing company in Mainz.

5.4. H. Geiringer

Hilda Geiringer (1893 - 1973) was born in Vienna and receiveddoctorate in 1917 from the University of Vienna with a thesis aboutFourier series. From 1921 to 1927 she worked at the Institute ofApplied Mathematics in the University of Berlin as an assistant ofvon Mises. Her mathematical interests had switched from puremathematics to probability and the mathematical development ofplasticity theory. In 1927 Geiringer became Privatdozent (lecturer).During this period she had a brief marriage and had one daughter.

In 1930 her work in plasticity theory led to the development ofthe fundamental Geiringer equations for plane-strain plasticdeformations. Geiringer remained at the University of Berlin untilforced to leave when Hitler came to power. After a brief stay as a research associate at the Institute ofMechanics in Belgium, she became a professor 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 atBryn Mawr College. While at Bryn Mawr she married R. von Mises who was then teaching at Harvard.In 1944 Geiringer became professor and chair of the mathematics department at Wheaton College inMassachusetts. Attempts to find a position at some of the larger universities near Boston repeatedlyfailed, often because of her gender. From 1955 to 1959 she worked as a research fellow inmathematics at Harvard in addition to her position at Wheaton to complete her husband's unpublished

manuscripts Mathematical Theory of Probability and Statistics after his death in 1953. Geiringerwas elected a fellow of the American Academy of Arts and Science.

H.Henck

H.Geiringer


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5..5. W. Prager [38]

William Prager (1903 - 1980) was born in Karlsruhe, and studied in Darmstadt TH receivingdoctorate in 1926 at the age of 23. From 1929 to 1933 he worked as the acting director of PramdtlsApplied Mechanics Institute at Gttingen. At the age of 29, he was appointed professor in Karlsruhe

TH as youngest professor in Germany, but soon he was dismissed when Hitler came in power. He wasinvited to Istanbul University, Turkey and acted as a special adviser in education to the government.Prager remained in Istanbul until 1941. The expansion of the World War II made the position of theGerman refugees insecure, and he accepted the invitation of Brown University in the USA made onthe recommendation of A. Einstein.

The Graduate Division of Applied Mathematics of Brown University was created in 1946 withPrager as its first Chairman, a position he held until 1953. By his effort, Brown University became thecentre of applied mechanics, especially in the area of plasticity in the 1950s and 60s.

5.6. R. Hill [39]

Rodney Hill (1921 - ) was born in Yorkshire, England and read mathematics in Gonville and Caius

College, Cambridge, where E.Orowan was teaching. In 1943, amid the World War II, he joinedtheoretical group of armament led by N.Mott and he was assigned a problem of deep penetration ofthick armour by high-velocity shell. This aroused Hills interest in the field of plasticity. From 1946,he began to work with the group of metal physicist under E.Orowan at the Cavendish Laboratory inCambridge. He solved various metal forming problems using plasticity theory, and obtained PhD in1948. In 1949 he was invited to head of new Section of Metal Flow Research Laboratory of BritishIron and Steel Association (BISRA).

Hill expanded his Ph.D. thesis and published the book The Mathematical Theory of Plasticity in1950 when he was 29 years old. This book was accepted as a standard of mechanics of plasticity. In1952, He became the Editor in Chief of a new journal Journal of Mechanics and Physics of Solids,which was eventually known as the highest level journal in mechanics. In 1953 he applied and wasoffered the post of a new Chair of Applied Mathematics in Nottingham University, and undertook

administrative work on top of the research works of plasticity until his retirement from the universityin 1962. Since 1963, Hill moved back to Cambridge and continued research work in solid mechanics.

6. Slab method

6.1. Analysis of forging by E. Siebel

In 1923 E.Siebel wrote a paper on the analysis of forging[40]. He assumed a thin area (slab) to make equilibriumequation. In the case of compression of cylinder of diameterd and height h2 shown in Fig.20, he considered a thin

layer of a thickness xd and a height same as the cylinder,and made an equilibrium of forces acting to the layer.

For the case of the yield stress Y and friction coefficient, he derived the average contacting pressure p as:

(11)

In the case of present slab method, the average pressure is calculated to be

(12)

The last equation, identical to the Siebels solution, is an approximation of the second equationwhen h/d is enough smaller than 1.0. Siebel numerically calculated the average pressures forsome typical cases of forging, and discussed the way to apply the result to backward extrusion.

+=h

dYp

31

1

+

=

h

dY

h

d

h

d

d

hYp

31

1122

exp22

Fig.20 Model of slab method usedby Siebel [40]


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Soon after the Siebels paper, similar methods were used by Th. vonKrmn for analyzing rolling of sheet metal in 1925[41], and byG.Sachs for solving wire drawing in 1927[42]. Using this method,Siebel continued to analyze various processes, and many researchersextended the method. Since the analyzed results are mathematicallyexpressed, they are effectively used in the industry [43,44].

Erich Siebel (1891 1961) received the doctorate from Berlin TH in1923 at age of 32, on the topic of calculation of load and energy inforging and rolling. After working in a steel industry for a short time, hebecame a leader of the metal forming division at Kaiser Institute inDusseldorf (steel), and carried out analysis of rolling and forging. In1931 he became a professor of Stuttgart TH, and treated almost all areasof metal forming such as deep drawing and wire drawing. He made thebasis of theoretical research of metal forming until retirement in 1957.

6.2. Th. von Krmn [45]

In 1925, Th. von Krmn presented a short paper of rolling to ameeting 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 thesubsequent researches in rolling technology, Krmn never returned tothis subject again.

Theodore von Krmn (1881 1963) was born into a Jewish family atBudapest, Austria-Hungary and he studied engineering at the Royal

Technical University in Budapest. After graduating in 1902 he joinedPrandtls Institute at Gttingen University, and received his doctorate in1908. Then he taught at Gttingen for four years. During this period hemeasured stress-strain curves of marble under high pressures. He was

also interested in vibration induced by fluid flow and found Krmnvortex, which made him famous in the area of fluid dynamics.

In 1912 he accepted a position as director of the Aeronautical Institute at RWTH Aachen. His timeat RWTH Aachen was interrupted by service in the Austro-Hungarian Army 19151918, where hedesigned an early helicopter. In his own biography, he does not mention about his work on rollinganalysis in 1925, possibly because his main interest was building a wind tunnel in Aachen.

In 1927, Krmn stayed in Japan for a while as an advisor to an airplane company to build a windtunnel. In 1930 he accepted the directorship of the GuggenheimAeronautical Laboratory at the California Institute of Technology(Caltech) and emigrated to the United States. He is one of thefounders of the Jet Propulsion Laboratory, which is now managedand operated by Caltech. In 1946 he became the first chairman of theScientific Advisory Group which studied aeronautical technologiesfor the United States Army Air Forces. At age 81 von Krmnreceived the first National Medal of Science, bestowed in a WhiteHouse ceremony by President John F. Kennedy.

6.3. Development of Rolling Analysis [43]

For the case of flat rolling shown in Fig.21, Krmn derived theequilibrium equation for the position , plate thickness h and rollangel , as: [41]:

(13)

Th. von Krmn

E.Siebel

Fig.21 Model of flatrolling by Krmn [41]

( )dxfpqhd tantan2

m=


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where q is the horizontal pressure acting within the plate, p is the pressure acting on the roll surface,

and the friction between the plate and the roll is ftan= . Minus (-) sign in the equation is for theentrance side and (+) sign is for the exit side.

This equation is valid when the plate thickness is small and the stress in the thickness direction is

almost constant, and further, friction is described by Coulombs law, as the case of cold rolling of thinplate.Trescas yield condition eq.(1) is written as,

Yqp =1 (14)

where 1p is the vertical pressure (perpendicular to q), which can becalculated from the roll pressure pand the frictional stress p as;

= tan1 ppp (+for entrance side) (15)

Since the parameters , h and are related to each otherwhen the roll radius is given, it is necessary to use only one of themin solving the differential equation. The stresses p, q and 1p are

not independent, and one of them should be chosen as the variable.Thus the equation to be solved may be in the form of( )= //)//( 1 hxfpqp (16)

but the result of this problem cannot be presented explicitly.The distribution of roll pressure in Fig.22 by Krmn was obtained

numerically, but for designing the rolling mills it is necessary to present the rolling torque, roll forceand the energy explicitly. By introducing various approximations, W.Trinks (1937)[46], A.L.Nadai(1939)[47], R.Hill (1950) [35 ] made approximate mathematical expressions and monograms to beused in the industry.

In applying the theoretical results to cold rolling, roll flattening cannot be neglected because the rollis elastically deformed and the local radius is changed. The method of treating roll flattening byW.Trinks and J.H.Hitchcock in 1935 [48] was often used in the subsequent analyses of rolling.

In hot rolling, the plate thickness is generally large, and thus the stress state cannot be assumeduniform in the thickness direction. Further, the friction in hot rolling is generally high and sticking orconstant frictional friction stress is considered to be suitable. To include the stress distribution in thethickness direction and a friction law other than Coulomb friction, E.Orowan (1943) [49] proposed amore generalized differential equation than the Krmns equation. Since the solution of thisdifferential equation cannot be expressed explicitly, E.Orowan and K.P.Pasco (1946)[50], D.R.Blandand H.Ford (1948)[51] and many others worked out to establish approximate mathematicalexpressions and monograms of 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 steelindustry began, and many engineers studied and improved the theories in the industry.

7. Upper Bound Method

7.1. Progress of Upper bound method

The upper bound method provides an approximate forming load which is never lower than thecorrect value. By this character, the load calculated by this method is safe in selecting the formingmachines and designing the tools, and thus this method has been used practically.

From the relation between velocity and strain rate (definition of strain rate), the associated strainrate distribution can be determined in deforming region. With a kinematically admissible velocity field,which satisfies the condition of volume constancy and the velocity boundary condition, together withthe flow stress value of the material, energy dissipation rate and forming load higher or equal to thecorrect values are obtained. This is guaranteed by a limit theorem for rigid-plastic material.

The upper bound theorem became to be known when it was introduced in the books by Hill [35] in1950 together with other bounding theorems. Hill states that Markov wrote a paper about the case of

Fig.22 Roll pressurecalculated by Krmn [41]


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rigid-perfectly material in 1947 in Russian language.Let us consider a simple plane-strain case that a rigid-perfectly plastic body with a shearing flow

stress k is deforming by external force T through a tool moving with a velocity v. A kinematically

admissible velocity field only with velocity discontinuous lines d*S with sliding velocity*v is

assumed, where (*) means kinematically admissible field. The upper bound theorem states:

d*S**ddSvkTv (17)

where the left side is the correct working rate and the right side is the energy dissipation rate for theplastic deformation along the velocity discontinuities. This inequality means that the energydissipation rate of the right side is greater or equal to the correct value of the left side. The upperbound value of the forming load T is obtained by dividing the calculated value of right side with v.

In 1951, A.P.Green [52] applied the theorem to the plane-strain compression between smooth plates,and compared the result with that of slip-line field method as shown in Fig.23. While a slip-line fieldrequires a long time to draw, a kinematically admissible field with only velocity discontinuity linescan be constructed easily with the help of hodograph, and gives a reasonably good result.

With the upper bound method, Green solved the problem ofsheet drawing and bending of notched bar in the early 1950s.

From the late 1950s, W.Johnson (1922- , UMIST) carried outextensive research work by using the upper bound method forplane-strain forging, extrusion, rolling and other forming problems.He optimized the velocity field by assuming proper variables. Inthe case of extrusion shown in Fig.24 [53], the angle is takenas the parameter for optimization.

In 1960, H. Kudo[54] proposed a method for applying the upperbound method to axi-symmetric forging and extrusion. He dividedthe axi-symmetric billet into several hypothetical units, andderived mathematical expressions of velocity in the units bysatisfying the condition of volume constancy and the surface

velocities to be consistent with those of the neighbouring units andthe tool surfaces as demonstrated in Fig.25.In the 1960s, when cold forging of steel was increasingly used in

the automotive industry, prediction of forming pressure became avery important subject to avoid fracture of the expensive tools. Theupper bound method for axi-symmetric deformation appeared justin 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[55]. In the 1960s and 70s, his method arose active research worksof finding new type of velocity field by S.Kobayashi [56], B,Avitur [57]and many others.

In the 1970s, the upper bound method was expanded to three dimensional problems by D.Y .Yang

[58] and others, and then it was combined with the finite element method, and grew up as the rigid-plastic finite element method as will be explained in the later chapter.

Fig.25 Velocity field foraxi-symmetric extrusion byKudo [54]

Fig.23 Velocity field for upper bound method used by Green and comparison of obtainedpressure with slip-line field solution [50]

Fig.24 Velocity field used foroptimization by W.Johnson[53]


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7.2. H. Kudo

Hideaki Kudo (1924- 2001) graduated from Tokyo Imperial University in 1945 just after the WorldWar II, and started his career at the Institute of Science and Technology in Tokyo University under theguidance of S. Fukui. He developed axi-symmetric analysis as an approximate energy method withoutknowing that his method was on the line of the upper bound theorem. He was awarded doctoral degreeon the thesis of the analysis of forging.

From 1959 to 1960 Kudo stayed in Hannover TH with O. Kienzleand Manchester University with W. Johnson and wrote the papers onthe upper bound method and the book.

In 1960 he joined the Government Mechanical EngineeringLaboratory in Tokyo as the team leader of 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,

lubrication, 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 the

Technology of Plasticity and was President of JSTP for 1985/6. Healso originated the International Conference for Technology of Plasticity (ICTP) and served as thechairman of the first meeting held in Tokyo in 1984.

8. Finite Element Method

8.1. Elastic-plastic finite element method

The finite element method (FEM) was developed for elasticanalysis of airplane structure in the 1950s. In this method aplate was divided into many hypothetical elements, andequations of equilibrium at nodal points were made. Since theequations were linear in terms of the nodal displacements,they were suitable to be solved with the matrix method usingdigital computers, which just became to be usable in somelimited research facilities in the USA.

In 1967 O.C.Zienkiwitcz and Y.K.Cheung published abook entitled The finite element method in structural andcontinuum mechanics[59], in which the method wasexplained in detail with the software written in FORTRAN

language. Referring to this book, many groups in the worldbegan to develop software because digital computers hadbecome usable in many countries.

The elastic-plastic FEM was developed as an extension ofelastic FEM. In 1967, P.V.Marcal (Brown University) andI.P.King [60] published a paper of elastic-plastic analysis ofplate specimen with a hole in which the development ofplastic zone was given as shown in Fig.26. They employedthe stepwise computation to follow the deformation. Thenodal coordinates and the components of stress and strain inthe element were renewed after each step calculation byadding the increments to the values before the step. Next

year Y.Yamada , N.Yoshimura and T.Sakurai [61]presenteda paper on the stress-strain matrix for elastic-plastic analysis.

H.Kudo

Fig.26 Extension of plastic zone inplate specimen with a holeanalyzed by Marcal and King [59]

Extrusion Pressure

Enddisplacemente

Extrusion Pressure

Enddisplacemente

Fig.27 Elastic-plastic analysis ofhydrostatic extrusion by Iwata,

Osakada, Fujino [62]


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This method gave an impact to the researchers of plasticity, and papers of elastic-plastic analysisincluding metal forming problems began to be published soon. Fig.27 is the analyzed result of initialstate of hydrostatic extrusion by K.Iwata, K.Osakada and S.Fujino [62] in 1972. In spite of the greatpotential of the method, it was found that the calculationerror accumulated as plastic deformation proceeds.

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

x

B

x

A

x += (18)

where Ax and

B

x are the stresses after and before the

step computation, andx is the incremental stress during

the step. When an element rotates during the step, thestress state B

x fixed to the element also rotates as shown

in Fig.28. This means that eq.(17) cannot hold withoutmodifying B

x , becauseA

x andB

x are defined in different coordinate systems.

To solve this problem, a formulation for large elastic-plastic deformation was put forward by

O.M.McMeeking and J.R. Rice in 1975[63]. It is not known well that H.Kitagawa and Y.Tomita [64]analyzed a large elastic-plastic deformation problem in a paper published earlier in 1974 but inJapanese language. In the 1980s, commercial software for elastic-plastic deformation appeared, andwith the splendid increase of computing speed, the method began to be used in the industry fromaround 1990. It should be noted that very small time step is needed even with this formulation to avoiderror accumulation, and thus elastic-plastic analysis requires very long computing time even now.

8.2. Rigid-plastic finite element method

In 1967, D.J .Hayes and P.V.Marcal [65] presented a paper of the usage of the FEM for optimizingthe upper bound method of a plane stress problem. In the case of the plane stress problem, the stressstate could be calculated from the optimized velocity field, but this was not true for other cases.

A normal stress component can be decomposable into deviatoric stress component mxx = andhydrostatic stress component 3/zyxm ++= . Although the deviatoric stress is related with the

strain rate by eq.(3), the hydrostatic component is not. For example, x may be written as,

mxmxx

Y+

=+= &

&32

' (19)

where m is left indeterminate with the strain rate associated with the optimized velocity.If the hydrostatic stress could be determined, this method was expected to have a great potential.

Because the stress could be calculated at each step without error accumulation, a drastically shortercomputing time than the elastic-plastic FEM was possible although a non-linear problem should beoptimized.

In 1973, C.H.Lee and S.Kobayashi [66] and M.Lungand O.Mahrenholtz [67] published papers that enabledstress calculation in the rigid-plastic analysis. Theirtheoretical base was the variational principle withLagrange multiplier, which had been presented in the bookby K.Washizu in 1968 [68]. This principle states that whenrigid-plastic problem is optimized with the Lagrangemultiplier to handle volume constancy, the multipliercoincides with the hydrostatic stress.

In FEM on the above principle, one or more multipliersare needed for each element to obtain the velocity fieldwhich satisfies the incompressibility condition. Since theLagrange multipliers increases the number of variables, thecomputation time becomes very long for large scale

Strain

Stress

StepComputation

Strain

Stress

StepComputation

Fig.28 Rotation and deformation ofan element during a step computation

Fig.29 Rolling analysis with rigid-plastic FEM by Mori and Osakada [69]


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problems. K.Mori and K.Osakada developed a finite element method allowing for slight volumechange without increasing the number of variables. In this method the hydrostatic stress wascalculated directly from the slight volume change which did not give significant influence to thedeformation. Fig.29 shows a result of rolling simulated with this method [69].

In the 1980s, the rigid-plastic finite element method became to be used by many researchers ofmetal forming to solve practical problems [70, 71]. Similarly to the elastic-plastic finite elementmethod, the rigid-plastic finite element method was expanded to three dimensional problems and wasinstalled in commercial software, especially for forging in the late 1980s, and used in the industry. Therecent developments are summarized in [72] by K.Mori.

9. Concluding Remarks

With the fast development of the information technology in the last 20 years, the finite elementmethod has become to be the main tool of metal forming analysis. To realize more accurate simulation,detailed researches are needed into various related areas to metal forming such as anisotropydevelopment and changes of metallurgical and mechanical properties during deformation, inelasticbehaviour in unloading and reloading, lubrication, friction, seizure, and fracture.

It seems to be inevitable that the finite element method will be used more directly in the industry.The simulation software may be integrated into CAD based systems, in which forming simulation iscarried out directly from CAD of forming tools. In order to enable small scale metal formingenterprises to use simulation, it is necessary to supply low cost software with simplified operation.

Although it is impossible for the author to predict long future, on-line control of metal formingprocesses with simultaneous simulation may become possible by the increased computation speed andnew computing algorithms.

Once the simulation method is well advanced, metal forming engineers will be able to concentrateinto more creative and innovative works, e.g., developments of forming processes for product withvery high dimensional accuracy, forming machines for silent environment, tool coating for dry metalforming, thermo-mechanical processes with low tool pressure and high product strength, etc.

Acknowledgements

The author would like to express sincere thanks to: Dr. R. Matsumoto (Osaka University), Dr. M.Otsu (Kumamoto University), Prof. K. Mori (Toyohashi University of Technology), Dr. H.Utsunomiya (Osaka University), Prof. F. Fujita(Tohoku University), Dr.H. Furumoto (MitsubishiHeavy Industries) and Prof. T. Ishikawa (Nagoya University), Dr. J . Allwood (Cambridge University)and Dr. B. Dodd, for providing the information and for helping to improve the manuscript.

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