AD/A-004 433

EQUATION OF STA-E OF STATIC AND DYNAMICPROBLEMS OF THERMOPLASTiCITY UNDERCOMPLEX LOADING

Yu. G. Korotkikh, et al

Foreign Technology DivisionWright-Patterson Air Force Base, Ohio

26 November 1974

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EQUATION OF STATE OF STATIC AND DYNAMIC PROBLEMS OF*THERMOPLASTICITY UNDER~ COMPLEX LOADING

acESCRIPTVE NOTKS (Ty'po efrowt* and inclusive data&)

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I U T0400115 (Fire? no.e. mniddle initife. elas ",Me*

Yu. G. Korotk ikh, S.M. Belevich

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FTD-MT. 24-2514-74

EDITED MACHINE TRANSLATIONFTD-MT-24-2514-74 26 November 1974

AR2041250

EQUATION OF STATE OF STATIC AND DYNAMIC PROBLEMSOF THERMOPLASTICITY UNDER COMPLEX LOADING

By: Yu. G. Korotkikh, S. 'M. Belevich

English pages: 14

Source: Uchenyye Zaplski Gor'kovskiyGosudarstvennyy Universit6t. SeriyaMekhanika, No. 108, 1970, pp. 80o-90

Country of Origin: USSR I :Requester: FTD/PDTNThis document is a SYSTRAN machine aidedtranslation, post-edited for technical accuracy,by: K. L. DionApproved for public release;distribution unlimited.

THI1 "RANSLATION IS A RENDITION OF THE ORIGI.NAL FOREIGN TEXT WITHOUT ANY ANALYTICAL OREDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED BY-ADVOCA1 EDOR IMPLIED ARE THOSE OF THE SOURCEAND DO NOT NECESSARILY REFLECT THE POSITION TRANSLATION DIVISIONOR OPINION OF THE FOREIGN TECHNOLOGY DI. FOREIGN TECHNOLOGY DIVISIONVISION. WP.AFB, OHIO.

FTm-MT- 24-2514-74 Date 26 Nov 19 74

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Block Italic Transliteration Block Italic Transliteration

* A a AD A, a P p p p R, r

-6 55 B, b C c Cc S, s

Be B. Vv TT Tm T, t

rr r r G,g Y y Y y Uu

A A X Ds d ¢o F3 f

E e E 8 Ye, ye; E, e* X x X x Kh, kh

W J* Zh, zh L LA L4 Z/ Ts, ts

a 3 Z, z 1 4 V Ch, ch

H H /4 W I, I W w LU w Sh, 3h

A R a Y, y 9 u. Ui e Shph, shch

H1,i K a X L, k b Y, y

*M M M M M, m b - h b IH H H n N, n 3 a a I E, e

0 o 0 o 0, o 0I1 10 / o Yu, yu

n n 17 n P, p F1 F1 X Ya, ya

*ye initially, after vowels, and after b, b; e elsewhere.When written as . in Russian, transliterate as ye or e.The use of diacritical marks is preferred, but such marksmay be omitted when expediency dictates.

GRAPHICS D!SCLAIMER

All figures, graphics, tabl•s, equations, etc.merged into this translation were extractedfrom the best quality copy available.

FTD-MT-24-2514-74 if

RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS

Russian Englishi

sin sin

Cos Cos

tg tanI

sec see

cosec CBC

sh sinh

ch cosh

th tanh

cth coth

sch sech

csch csch

arc sin sin 1

arc cos Cos'

are tg tan-l

arc ctg cot 1

arc sec sec 1l

arc cosec CBC1

aro. sh sinh-l

arc ,!h cosh-1

arc th tanh1

arc cth coth-1

arc sch sech-1

arc csch csch 1l

rot curl

ig log

FTD-MT-21I-2514-74

GREEK ALPHABET

Alpha A c a Nu N

Beta B 8 xiGamma r' y .Omicron 0 0

Delta A 6 Pi 1 7rEpsilon E c i Rho Pp P

Zeta Z • Sigma a

Eta H r Tau T tTheta 0 e $ Upsilon T

Iota I i Phi $

Kappa K X K z Chi X XLambda A X Psi '

Mu M v Omega Q

FTD-MT-2 1 -25 1 4-71 4,.

EQUATION OF STATE OF STATIC AND DYNAMICPROBLEMS OF THERMOPLASTICITY UNDERCOMPLEX LOADING

Yu. G. Korotkikh, S. M. Belevich

References [l)-[8] examine the coupling equations between istresses and strains in differential form, i.e., the connectionbetween the differentials of the stress and strain tensors. In

this case it is assumed that reutral loading (loading tangentiallyto the surface of the loading) does not cause plastic deformation, -I

but the hyperspherical surface of loading is replaced at the

point of loading by a hyperplane. The latter is exhibited inthe fact that the ccndition that the new surface of loading

passes through the terminus of vector of the final load is replacedby the condition of its passage through the terminus of the vector

which is the projection of the vector of final load on the direction

of the gradient to the surface of loading at the point of loading.

Experimental determination of the yield surface established

the specific tolerance to plastic strain (~0.002). A decrease in

this tolerance means that the fundamental positions of the

phenomenological theory of plasticity cease to be valid. This A

fact leads to this position that, on thf one hand the experinmentally

obtained increases in the stres- and strain tensors cannot bo

FTD-MT-24-2514.7- 1

considered as differentials, but on the other hand, in practical

calculations there is no sense to determine plastic deformations

to within less than tolerance, i.e., also it is not possible to

consider them as infinitesimal. Furthermore, in practical cal-culations, especially during the solution of dynamic problems,in many instances it is difficult to regulate the size of the

tensors of stress and strain increments, and at isolated points Ithey can reach a significant magnitude.

This work attempts to derivie the relationships between

increases in the stress and strain tensors without relying on

the asbumption of their smallness, and gives the "physical"

algorithm of practical realization of these relationships when

solving problems of thermoplasticity.

DERIVATION OF FUNDAMENTAL PRINCIPLES

During derivation of fundamental principles, as usual, the

stress-strain tensors and their increase are broken down into

spherical and deviator parts, and the connection is established

separately tfor each part.

The complete stress tensor eij is broken down into elastic e_

and plastic ep parts:

At a certain point in time let the stress and strain state of

the material be characterized by the fields of the stress-stain

rs acm and and at the following poP" t in time by tensorsaensand e

ij ii

Then it is possible to write the relationships:

TD-90-24-2514-4(2)

FTD-MT-2i4-2514-74 2

where 0-G (T, d) is the shear modulus;

AG - change in the modulus due to change in the temperature

T of the medium and the hydrostatic pressure a= .

By deducting from equality (2) equality (3) and carrying out

some transformations, we will obtain

A- y Ao;; -= 20PAe A O+P. cH'r Y

where (see Fig, 1)

AzPY2OA e;1 + A '

Ao, =2G A e'+ C 3l;B-I G + AG ... '

.1.1,1 v y ~I. *+- A :;'x,IICI,'(b,)'ir

Aeij=AeY +AePj - strain increment tensor;

ij/\' iiti

S.... A;"-- * ( for case a

if if ik)tX.tw- ~'V --. '*

AU*i A for caseb b ,

FTO-MT-24-2514-74 3

., -. • ;• ,• • • .. .• . i , ,vr' 'q , • . . .

Lot us assume that the tensor of increase in the plastic strainr

Is coaxial to a certain tensor S*P whose sense will be explainedbelow:

By utilizing (I) and (5), we will obtain:

Coittracting in ) -

we will obtain "or X:

2(G+ GSs;, , (7)

Let us explain the sense of tensor Si.

. - - -U "- .. "- " -

'I I- I,;I I /

I3

10

.00C?. bFigure l

If we reject the previous surface, i.e., consider that the

tensor of increase in plastic strain is normal to the previoussurface of loading at the point of its intersection with vector

A0,then by S it is necessary to understand S (see Fig. 1).

In this case the terminus of vector AaI will lie at the point ofintersection of the new surface with the line passing through

point c (terminus of vector Aaij) parallel to vector SO point D).This assumption can give considerable error with large values of

increase in the strain tensor Aei(Aa)eij

In this case it is expedient to accept as S the vector:

""ii 1•PI;,• + 0 • . - ' I, Iwhere 0 and 01 are certain constants:

IV 0when 0=01=0 the vector of increase in the plastic strain will be

normal to the new surface of loading at point D. With 0=01=;

we will obtain the first case.

The connection between the tensor of increase in permanentmicrostrain and the tensor .,f increase in the plastic strain is

given by the expression:

A1 - -C Ail (9)

where

,,.,, r,,,, I,.u , I e,, I ), =c~ l u • C t;

i.s determined as follows.

\ , ....... . . .

If for diffezent materials we use experimental data to graph

ias a fuiation of temperature *.f T, where pIj are coordinates

cr the center of the yield surface with simple elongation at

different values of ep and temperature T-0 0 C.

i 3 are coordinetes of the cen'ker of the yield surface at

certain temperature T (for ezample, see Fig. 2), then accurate to

within experimental values it is possible to consider the ratio

as independent of the degree of plastic strain. This graph

has the following features:

up to a certain value T=T LE•LL4 const.-l, then it sharply

decreases to 0 in a narrow temperature i,•nge. By having the

experimental curve. 1-l1as a function of T, it is possible to

determine c from the formula

T- when ATT > (9*)

where c=O when AT<O.

The first and second terms in the right side of formula (8)

reflect the two really existing coexisting coexisting processes:

the process of formation of microheterogeneities with plastic

strain and the process of recrystallization. Temperature T* is

the temperature of recrystallization. Here we do not consider

the fact that the process of recrystallization occurs with holdcing

at constant temperature (annealing).

•,|.6

-O 0 -0-2-

0,

0,4---

100, XO "j00 40Q 50 -Soo

Z f6

-4C

:0 J0 400 JVQ 1600Figure2

I

"PHYSICAL" ALGORITHM

Cases a and b Fig. 1 are not distinguished below. Case b

differu from case a in that in case a: I

-AC = Ac"l [Acr;yj p ;• I A '" A

and in case b: AC = 7t, .... A0.

The iterative process of determination of the true value of

increase in the tensor of the stress deviator (Aa, )=BD can be

represented by the following algorithm:

1) In the beginning we take 6=el--l. Through the terminus of

vector "Acid (point c) we draw a straight line parallel to

{S~j up to the intersection with the previous yield surface, Ishifted and changed because of AT, ..a., e (point D'). If there isl

no intersection, in any manner select point D1 (deflection along

normal or tangentially). The vector of stress tensor increment A(Ao 3 )P in the lst approximation is defined as the half-sum of i

vectors AD' and AC.

Knowing (Aa•)in the let approximation, we compute X

according to formula (7), and then according to formulas (5) we

compute (AePj)and (Ap*)l=g (AePj)

We compute (S) according to formula (8) at some values

0 and e1. We compute the new value of the radius of the loadingsurfacei as a function of the new value of the strengtheningparameter, calculated in termns of value (Ae~j) 1 according to

the formula:

-2GAe 1 -- (Ae,. 1] + (•'); [A(10)- ~ -i li (10)--.

8

Ii

we find the value of the vector of the stress increment. which

corresponds to this value (Ae~ 3 ) 1 . We test to see whether the

terminus of vector (HO )2 CaT +(A ) 2 lies on the new loading

surface with center

S),- + -(A + C(l)

radius equal to: 17C + (A%) 1 "7 This condition must be

fulfilled accurate to a certain e>O. If this condition is not

satisfied, we find the intersection of the loading surface obtained

in this approximation with the straight line passing through

point c parallel to (Si) (point D ). For vector (Aa~l) 2 in

the second approximation we select the vector equal to the half-sum

of vectors (.AM ) and vector BYDHOBI etc.

.The procedure for the correction of the stress tensor proposed

in work [9] is a particular case of the given method (p=E, aT~const.,

e=-e-0) and does not require the iterative process. If there is

a yield delay effect, the given algorithm is put into use only

after fulfillment of the conditions:

\"1, ...t• ; (ll*)

where Atk i the k-th time step, and tk is the yield delay time§

[lU], which corresponds to an excess ofu T at this step, and

ak is the value of stress intensity on this step. In this case,u

after fulfillment of the conditions (11), it is possible to use

different laws governing change in the dynamic yield point.

The above relationships between the strain increment tensor'

and the stress increment tensor can be presented in ani.otropic

9 5

form. By transforming (5), (7) and (4) , it is possible to

obtain:

ItG+ -, (12)(2G+AG) S.S.. 2 0 ., Is' S

" ' . It .• .

k-i:

By utilizing then the equality Ae =Aey +Ae j, it is possible to

obtain the unknown connection between the increase in the stress

tensor and strain tensor in anisotropic form. By converting

dependences (12) and (13), it is possible to obtain the expressions

of stress tensor increment through the strain tensor increment.

The connection between the spherical stress-strain tensor KIcomponents in static problems can be presented in the form:

Se =-: a +• A )(14)3k 3k-

where k=k(T) - the bulk modulus, a - the coefficient of linearexpansion, Aeij

In the case of dynamic problems with the large hydrodynamic

pressures which occur during shock wave propagation, it is

necessary to utilize the equations of' state of solid bodies at

high pressures and temperatures in order to know the connection

between spherical components.

As shown in work [II], a large role in these equations

belongs to the dependence of pressure on internal energy.

Subsequently, the connection between the spherical stress-strain

10

I1

1... .. .... 1 iii!tensor components is established on the basis of equations of'

state of Mie-Gruene1-en [12-13]

AV) + J[E E,-.

where P n(v) - the potential part of the pressure, caused by

compression of crystal lattice at 00 Kelvin. This dependencecan be obtained from the experimental data [14] in the form: !-

( i13P ai (16)

I _4where ai - certain experimentally determined coefficients, and

p - density of element (p/p 0 )=(vov), (E-Ev)=PT - thermLl part

of pressure.

y=y(v) - Grueneisen's coefficient. It is given in the form:

7 7.+ P 1 + -? 1 ' : )-2 (17)

where A, B, C are experimentally determined'coefficients.

If the shock adiabat is known for just one material with

normal and low density (porous), then y(v) can be determined by

the formula:2v(P 0 -- (18)-- V(J; P-

or calculated from certain approximation formulas given in [16).

E - specific internal energy.

ii-

1.1

~~(v ?'~~ E T p (v)d + CX'4- ,19)T-L

where -ZA energy of deviator strain, and I

4

Cv. specific heat of lattice at constant pressure.

To equation (15) it is necessary to add a certain thermodynamic

F- identity, whIch characterizes the strain condition. This can beeither condition on the discontinuity -the Rankine-Hugoniot

equation

E-E~=-(jI+"P*)(lve`-V)'.z 1 (20)

or the equation of the adiabatic curve

dS .,Q (21)

where dS is the increase in the entropy, or any other thermodynamic

relationship, which expresses the condition of conservation ofI ~energy. The use of the indicated relation~ships makes it possibleto consider the fact that the shock loading of a material passes

along the shock adiabat, and unloading passes along the isentrope.

Utilizing equations (15), (20) or (21), it is possible tocalculate the change in temperature at a particular point of the

body [12); temperature is determined from the integral relationship.

-- T

- EO+ c~dTj (22)

FTD-MT-24-2514p.7J4 12

where F.0 is initial thermal energy: Et) v and c s z

determinek from formula [i2]: .

I!

cc= Th r4D(.L i (23)

R - the universal gas constant,

- molecular weight,

P - Debye function

0 - Debye temperature.

T" 15 0 "

A-ikG (214 )

6 ~ whenj T<i2 , 0¥e <

Since at T=O, cv reaches 96% of its maximum value, when Týe c v

is considered constant, and Debye function D will take the form

I I(25)

Energy sources of neutron irradiation type are considered through

the increase in internal energy of' the element. After determining

the connection between the spherical and deviator stress-strain

tensor components, we obtain the complete relationships between

stresses and strains, which determine the behavior of the medium

with nonisothermal elasto-plastic deformations.

FTD-MT-24-2514-74 13

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1. B a K y a~ e u K o 'A. A. 0 CBR3UX Nte)y iiaIIpRI~eKt1qM1 H Ae~opmaw.,IMIYR~ 1'OTPOihIWX Ireynayri.x cpe,~ax. B c6.: Hcc41ejoaamiRi no nyo[T.13YMTP r It. .

If r~lc~nIfcTH ~ , M2,113. JIy, 1963.2. B Hp r ep H4. A. PacqeT KO0HC'rPy1(LLfif C jneTOMt nldaCTIMffCTH it no13!

qe~i. 13A. AH- CCCP, MlexaHIuKH, Xg 2, 1965. --

3. K a A a we a H 10. H4., H-10 80 W11710oai B. B. TeopHS1 ruacTKIo0CTI1;,.yq)fTh~laigaiota ocTaToqlibe m~ixpoiianpR)KeHHSI. [1VmM, 22, we~ I ..(1958)

4. Ho0 Bo )K fao10B B. B. 0 cnotio)KiO IfaFp~?KBH1i H nepCi~eKTH138X,-4~meH~j1HqecKoro no~lxoAa K, HMeCAe2oaHH10 31111pO1taripH)KeHIh.- U1MM, 28,.

5. [1 a 11 er inH 3 pitroweH'le :TeopHtt ocTaTO41Hb1 MH 0aipsazeuniK.He~f3oTepMtiqecKomy- JAecjPMPHpo~aalin0, 142. l CC ,mealHia,

JN'& 2, 1965. --

-3. KOPOIKIIX 10. r, Beaeniaii C. M. Ocfioamie ypaBHeH"HR-?*'.epmo-rWJ~cTIIqHOCTII IIPH CAO)KHOM Harpy)KeHilfn. Yq. _.3an. rry,"c, Cpý'a Mei-ua-T.,.

, \ 0 pUT KIf X 10. r., 5 .-I e B ~i it C. Ml. TIbOCKag 3aaatila Tep~mO aTI4flOCTH flpK MA0HOiM Harpy)KeHPIX-. B c6.: Tenaioswe Hanp~weinitr B wie-ý

sfewiax )COMCPYKLIHI, KHeB, 1968.BbiT89167.' . 13. 5 en~ e- n it q C. Al. -Pacqer. nAocxoft 3a~aa4i Tep.Mo0J1a'r8f4OcT111 np) t

caxo)KtoM HarpywewflH. B HaacoxfrneM 010pIfIIIIe.9. Y X A K If Hi C. Pac~qe'r ynpyro-OaacTiitteCKHx Tetteffilf. BmtiIuciTeb)1Ue Ue--.

TOlbI a rifAP0,a1HHimHKe, 1967.>). 3o..loen~o.KAIIOBIILU~r, 10.51 . ji~ttla.WCUInee LK-

'tICTHI, 1965.ý,I ( in Rfjlfll ., A n cien D., Maxwell D. Computed shock respouse ofporolls ijuminum ,J. Appi Physe, 1968, 39, X! 10.

* 2.A a o it B . 11I., K a a~ it it it Ii B. A. YpaBijetirg COCT0OHJ4H TBePAWbX 7e.1!9; I34C'<~X an~eiii~x,1968.

1:'. .'1 11. I ~ CA(,;11 IQ TROI)aUbX W.1 C1II.Jh11AIM1i yaIapHbINMH BO1;AHWI.ii; :~~r~: .!.I-1-'01tcCIh1:C rL~eP,.1.tX Te.i np~i rt!-?ioowIx ;ai~-

1.1. AnlbTWuV-lp.1 . 13 ., Kop.Nep C. B., BaKaHoBa A. A., T py-11111 P. (1). YpaBeffffl COCTOMMIHBS1a.i-0IM1H&H, %tal H CBHHUtxa As 06-.IUCT11IINtbCOKIIX aaaem1ih~r, W3 it TOl, 38, BunT. 3 (1968).

i.. )r 6ait03.B11 [. M., K iifiK 01I1. A. 11oae~ieHHe Beu.kecTa nojg AaBqerni-fQM, 113.1 AW1Y, 1962.

I C). 3" e J1 bj ., B 11 qI B. R. (DH31iKa y~xpHUX BOJIH if rI1ApAHHa.MH4CiCKI1X Bbl-* v~~OKOTIeMflepar'!HUX qwietuiaf. 1966. -

FTD-MT--24-251 11-75 14l