POSPLO[ENATEORIJAPLASTI^NOSTI AGENERALIZEDTHEORYOFPLASTICITY

V. CHYGYRYNS’KYY ET AL.: A GENERALIZED THEORY OF PLASTICITY

A GENERALIZED THEORY OF PLASTICITY

POSPLO[ENA TEORIJA PLASTI^NOSTI

Chygyryns’kyy Victorovich Valeryy¹, Kachan Aleksey Yakovlevich¹, Ilija Mamuzi}², Ben’ Anna Nikolaevna¹

1Zaporozhskyy National Technical University, Str. Zukovsky 64, 69063 Zaporozhskye, Ukraine 2University of Zagreb, Faculty for Metallurgy, Aleja narodnih heroja 3, 44000 Sisak, Croatia

valerij@zntu.edu.ua

Prejem rokopisa – received: 2008-11-21; sprejem za objavo – accepted for publication: 2009-11-23

A closed solution of the plane problem of the generalized theory of plasticity and a model of the complex plastic medium were theoretically developed. Solutions with the use of the deformation theory and the theory of plastic yielding were developed. The solution for a simple strengthening medium was deduced.

Key words: metal plasticity, analytical solution, mathematical model, plastic medium, process parameters

Razvita sta bila zaprta re{itev splo{ne teorije plasti~nosti in teoreti~en model kompleksnega plasti~nega medija. Opredeljene so bile re{itve z uporabo teorije deformacije in teorije plasti~nega te~enja. Razvita je bila re{itev za preprost utrditveni medij.

Klju~ne besede: plasti~nost kovin, analiti~na re{itev, matermati~ni model, plasti~ni medij, parametri procesa

1 INTRODUCTION

A characteristic of the new method based on a closed solution of the plane problem of theory of plasticity is a simplified analysis of the deformation mode of the medium and the theoretical connection to the medium mechanical characteristics through the process para- meters. The analytical solution of the plane problem of the theory of plasticity for a strengthening medium is known.¹ The developed complex model for the strengthening of the plastic medium is based on the shear resistance to the plastic deformation and is a function of the coordinates of the nucleus of deformation. This approach offers a new possibility to evolve a new solution for a problem, including the generalized theory of plasticity. The approach includes equations and criteria: an equilibrium equation, and the criteria of yielding, the equation of incompressibility, of the deformation rate and the deformation as well as equations of continuity of the deformation rate and the deformation:

–the equilibrium equations

∂

∂

∂

∂ s_x t_xy

x + y =0;∂

∂

∂

∂ t_xy s_y

x + y =0 –the criterion of yielding

(sx−sy)²+ ⋅4 txy² = ⋅4 k²

–the constraint equations for the rates of deformation and deformation

s s

t

x x g

x y

xy

x y

xy

− F

⋅ = −

2 ' = ¹; s s

t

e e g

x y

xy

x y

xy

− F

⋅ = −

2 = ² (1)

–the equations of incompressibility for the rates of deformation and the deformation

xx+xy =0; ex+ey =0

–the equation of continuity for the deformation rates and the deformation

∂

∂

∂

∂

∂

∂ ∂

2 2

2 2

x_x xy 2gxy

y + x = y x'

;

∂

∂

∂

∂

∂

∂ ∂

2 2

2 2

e_x ey 2gxy

y + x = y x –the equation of heat conductivity

a T

x T y

T t

2 2

2 2

2

∂

∂

∂

∂

∂ + ∂

⎛

⎝⎜ ⎞

⎠⎟ =

The model of the complex plastic medium is defined with

T_i = ⋅c (H_i)^m1 ⋅(G_i)^m2 ⋅( )T ^m3 (2) The system of equations (1) includes the equations of the deformation theory of plasticity and the theory of plastic yielding with the addition of the equation of heat conductivity.² The model (2) is a real strengthening medium with the boundary conditions for stresses³

[ ]

tn =Ti ⋅sin AF−2 ,a T_i= k

or t s s

a- t a

n = −

⋅ ⋅

x y

2 sin2 xy cos2 (3)

The additional conditions are given by the specific contact forces (3) of the change of friction according to the sinusoidal law of deformational and high-speed strain hardening. All the intensities and the temperature depend on the coordinates of the deformation nucleus.

UDK 539.374.001.8.621.7-111 ISSN 1580-2949

Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(3)141(2010)

2 THEORETICAL DEVELOPMENT

With the aim to obtain the model (2), let us consider three second-order equations in form of non-uniform hyperbolic partial derivations:

∂

∂

∂

∂

∂

∂ ∂

2 2

2 2

2

2 2

t t

t²

xy xy

x − y = ⋅ y x k − xy,

∂

∂

∂

∂

∂

∂ ∂

2 2

2 2

2

1

2 1

x x

x x x

y − x = ⋅ y x F ⋅ x (4)

∂

∂

∂

∂

∂

∂ ∂

2 2

2 2

2

2

2 1

e e

e

x x

y − x = ⋅ y x F ⋅ x

The boundary conditions (3) correspond to the substitution txy = ⋅k sinAF. A complex dependence of the coordinates is assumed with k=f(G_i,H T x y_i, , , ). In this case, k=C_s⋅exp ', withq q'=f(G_i,H T x y_i, , , ), with Gi,Hi,Tstanding for the intensity of the deformation, the rates of deformation and the temperature.

The derivatives are taken as for the complex function,⁴and after substitution in the first equation (3) we obtain:

{

^{(q'H⋅Hx+ ⋅q'V Gx+ ⋅q't Tx)x+}

[

^{(q'H⋅Hx+ ⋅q'V Gx+}

]

+ ⋅qt^' Tx)+AFy ²−(qH^' ⋅Hy+ ⋅q_V^' Gy+ ⋅qt^' Ty)y−

[ ] }

− (q^'_H⋅H_y+ ⋅q^'_V G_y+ ⋅q^'_t T_y)−AF_x ²+2AF_xy ⋅

[ ]

{

⋅sin(AF)+ 2⋅ (q^'_H⋅H_x+ ⋅q^'_V G_x+ ⋅q^'_t T_x)+AF_y ⋅

[ ]

⋅ AF_x−(q_H^' ⋅H_y+ ⋅q^'_V G_y+ ⋅q_t^' T_y) +AF_xx−

−AF_yy− ⋅2 (q_HH^' ⋅H_x⋅H_y+q_H^' ⋅H_xy+

}

−q_VV^' ⋅G_x⋅G_y+ ⋅q_V^' G_xy+q^'_tt⋅T_x⋅ + ⋅T_y q^'_t T_xy) ·

·cos (AF) = 0 (5)

Equation (5) is equal to zero if the parts in the square brackets are equal to zero. Then,

qH^' ⋅Hx+ ⋅q_V^' Gx+ ⋅qt^' Tx =AFy

q_H^' ⋅Hy+ ⋅q_V^' Gy+ ⋅qt^' Ty =AFx

(qH^' ⋅Hx+ ⋅q_V^' Gx+ ⋅qt^' Tx)x =−AFyx

(qH^' ⋅Hy+ ⋅q_V^' Gy+ ⋅qt^' Ty)y =−AFxy

AF_yy=-(q_HH^' ⋅H_x⋅H_y+q_H^' ⋅H_xy+q_VV^' ⋅G_x⋅G_y+ +q_V^'⋅G_xy+q^'_tt⋅T_x⋅T_y+ ⋅q^'_t T_xy)

AF_xx= (q_HH^' ⋅H_x⋅H_y+q_H^' ⋅H_xy+q_VV^' ⋅G_x⋅G_y+ +q_V^'⋅G_xy+q^'_tt⋅T_x⋅T_y+ ⋅q^'_t T_xy)

The operations with the complex function allow us to determine the exponent index as the sum of three functions accounting for the effect of the deformation degree and the rate, and of the temperature:

q^'=−Aq q= ₁^' + +q₂^' q₃^' =−(A₁^'q+ q+ q)A₂^' A₃^' The shear resistance and the components of the tensor of the stresses are:

k=Cs⋅exp(−A1^'q)⋅exp(−A2^'q)⋅exp(−A3^'q)⋅sin(AF) (6)

sx =C_s⋅exp(−A₁^'q)⋅exp(−A₂^'q)⋅exp(−A₃^'q)⋅cos(AF)+

+s₀+f y( )+C

sy =−C_s⋅exp(−A₁^'q)⋅exp(−A₂^'q)⋅exp(−A₃^'q)⋅cos(AF)+

+s0+f x( )+C

with qx^' =(q₁^')x+(q₂^')x+(q₃^')x =−AFy

qy^' =(q₁^')y+(q₂^')y+(q₃^')y =AFx

By substituting the stress values into the equation of constraint we obtain:

s s

t

x y

xy

− A

2 =ctg F; x x g

x y

xy

− B

' =ctg ₁F

ctgAF=ctgB₁F=F₁ e e

g

x y

xy

B

−' =ctg ₂F

ctgAF=ctgB2F=F2

It is possible to establish the relation between the shears and the linear figures of the deformation rates and the deformations. Taking into account the equations of non-compressibility we obtain:

gxy^' = ⋅2 F1 ⋅xx ⋅xx⋅ B

1

= 2 tg 1F gxy = ⋅2 F1 ⋅ex ⋅ ⋅ex B

2

= 2 tg 2F In order to simplify, we define:

xx =C_x⋅expq₁^''⋅cosB₁F ex =C_e⋅expq₂^''⋅cosB₂F

By substituting these relations into the equations of continuity of the deformation rate and the deformation (1) or (4), we obtain:

[

^{−q1''xx−(q1''x+B1Fy)2+q1''yy+(q1''y−B1Fx)}

]

^⋅sinB1F+

[ ]

+ 2(B₁F_x−q₁^''_y) (⋅ q₁^''_x+B₁F_y) (+ B₁F_xx−B₁F_yy) ⋅

⋅cosB₁F= ⋅2 B₁F_xy⋅sinB₁F+ ⋅2 q₁^''_xy⋅cosB₁F (7) as well as

[

^{−q2''xx−(q2''x+B2Fy)2+q2''yy+(q2''y−B2Fx)}

]

^⋅sinB2F+

[ ]

+ 2(B₂F_x−q^''_2y) (⋅ q^''_2x+B₂F_y) (+ B₂F_xx−B₂F_yy) ⋅

⋅cosB₂F= ⋅2 B₂F_xy⋅sinB₂F+ ⋅2 q^''_2xy⋅cosB₂F (8) Brackets identical to (5) appear in equations (7) and (8). For

(q₁^'')_x =−B₁F_y (q₁^'')_y =B₁F_x (q₂^'')_x =−B₂F_y (q₂^'')_y =B₂F_x

the equations are transformed into identities, with, q₁^''=−B₁q,q₂^''=−B₂qas the indices of the exponents of the functions determining the fields of the deformation rate and the deformation, B₁F and B₂F are the trigo- nometric functions determining the fields of the defor- mation rate and the deformation.

The expressions for the deformation rate and the deformation are:

xx =− =xy C_x⋅expq₁^''⋅cosB₁F=

=C_x⋅exp (−B₁q)⋅cosB₁F (9) gxy^' =C_x⋅expq₁^''⋅sinB₁F =C_x⋅exp(−B₁q)⋅sinB₁F

H_i =2C_x⋅expq₁^''=2C_x⋅exp (−B₁q) ex =− =ey C_e⋅expq₂^''⋅cosB₂F=

=C_e⋅exp(–B₂q)⋅cosB₂F (10) gxy =C_e⋅expq₂^''⋅sinB₂F =C_e⋅exp(−B₂q)⋅sinB₂F

G_i =2C_e⋅expq₂^''=2C_e⋅exp (−B₂q) with (q₁^'')_y =B₁F_x (q₁^'')_x =−B₁F_y

(q₂^'')_y =B2F_x (q₂^'')_x =−B2F_y

With a comparison of expressions (9), (10) and (7) we can confirm that all the expressions have functional dependencies on the coordinatesqandF(the indices of the exponents and the examples of the trigonometric functions).

It is of interest to obtain similar dependencies for the solution for the temperature fields that could allow us to solve this task theoretically. Let us consider a differential equation for the stationary temperature field

∂

∂

∂

∂

2 2

2

2 0

T x

T + y =

For this case we look for the solution in the form of T =C_T ⋅exp (q₃^'') (sin⋅ B₃F+cosB₃F) (11) with: (q₃^'')_x =−B₃F_y, (q₃^'')_y =−B₃F_x

We will demonstrate that expression (11) is a solution of the Laplace equation. By substituting the derivatives (11) into the equation of the heat conductivity and a simplification we obtain:

{

^{(q3'')xx + (}

[

^q3'')x+B3Fy

] [

^{⋅ (q3'')x−B3Fy}

]

^+(q3'')yy+

[ ] [ ]}

+ (q₃^'')_y+B3F_x ⋅ (q₃^'')_y−B3F_x ⋅(sinB3F+ [

+cosB₃F)+ ⋅2(q^''₃)_x⋅B₃F_x+B₃F_xx+ ⋅2 (q^''₃)_y⋅

]

⋅B₃F_y+B₃F_yy ⋅(cosB₃F−sinB₃F)=0 (12) In the case of equality to zero, the brackets,

[

^(q3'')x+B3Fy

]

^,

[

^{(q3'')y−B3Fx}

]

in equation (12) esta- blish a relation in the form of:

(q₃^'')_xx =−B₃F_yx, (q₃^'')_yy =−B₃F_xy B₃F_xx =(q₃^'') ,_yx B₃F_yy =(q_xy^'' )

The last correlations correspond to Cauchy-Rieman condition and are functions determined by the Laplace equation corresponding to relation (11).

From the comparison of the solutions (7) to (11) (conditions superimposed on functions) it was concluded that q₃^''=−B₃q for the stressed and deformed conditions and the temperature fields can be used to determine a common parametric function, which is included into the fields of the stresses, the deformations, the rates of deformation and the temperatures, allowing us to express them mathematically, one with another. Thus,

exp (− ⋅

⎛

⎝⎜ ⎞

⎠⎟ = ⋅

⎛

⎝⎜ ⎞

⎠⎟ = q)=

x e

H

C C

i B

i B

2 2

1 1

1 G 2

= T

C_T B B

B

⋅ +

⎛

⎝⎜ ⎞

⎠⎟ (sin ₃ cos ₃ )

1 3

F F

With substitution into an expression for resistance to deformation, we obtain

( ) ( ) ^{( )}

T_i H_i T

A B

i A B

A

=c ¹¹ ⋅ ²² ⋅ B³³

' ' '

G ' (13)

The form of expression (13) corresponds to a depen- dence of the yield stress from the rate, the degree of deformation and the temperature proposed in¹.

3 ANALYSIS OF THE RESULTS

In the analysis the expressions (6) are used to study the stressed condition of the plastic medium in the case of a flat upsetting of rough plates. If the problem is reduced to a more simple mathematical model (A₂^' =A₃^'= 0), expression (6) will correspond to the solutions in⁵.

k=C_s⋅exp (−A₁^'q) txy =Cs⋅exp (−A₁^'q) sin(⋅ AF) sx =C_s ⋅exp (−A₁^'q) cos(⋅ AF)+s₀+f y( )+C sy =−C_s⋅exp (−A₁^'q) cos(⋅ AF)+s₀+f x( )+C (14) Applying the condition for plasticity s0 =− ⋅2k cosAF, C=k₀, the functions AF and q become harmonic. Starting from the Laplace equation and the Cauchy-Rieman conditions we obtain the expressions for determining the functions in the form of the coordinate polynomial.

AF=AA₆⋅ ⋅ −x y AA₁₃⋅ ⋅ ⋅x y x( ²−y²)

{ [

q^' =− 0 5. ⋅AA₆⋅(x²−y²)−AA₁₃⋅ 0 25. ⋅(x⁴+

] }

+y⁴)− ⋅15. x²⋅y²

The constants in the expressions were determined as proceeding from the real boundary conditions

AA⁶ l h 4 0

= ⋅ ⋅

Y , AA l h

l h l h

13 16 1 3 2

= ⋅ ⋅ − ⋅

⋅ ⋅ +

Y ( )

[ ]

Y₀ =arctg 2⋅ ⋅ −f (1 f)

[ ]

Y₁ =arctg 1 7. ⋅ ⋅ −f (1 f) and the coefficient

C k

s = A⁰ ⋅ −q0

cos 0 ^')

F exp ( A AA l h

AA l h l h F_{0 6 13}

2 2

4 4 4 4

= ⋅ ⋅

− ⋅ ⋅

⋅⎛ −

⎝⎜ ⎞

⎠⎟ q₀^' =− q =− . ⋅ ⋅⎛ −

⎝⎜ ⎞

⎠⎟ −

⎧⎨

A ⎩ AA l h

0 6

2 2

0 5 4 4

− ⋅⎛ +

⎝⎜ ⎞

⎠⎟ − ⋅ ⋅

⎡

⎣⎢

⎤

⎦⎥

⎫⎬

AA l h l h ⎭

13

4 4 2 2

0 25 16 16 15

. . 16

By substituting the components of the tensor of stresses into (14), we obtain

s q - q

0

x k

A A k

=− ⋅exp ( ^{' '})⋅ +

cos ⁰ cos

0

F F 0

s q - q

0

y k

A A k

=−3 ⋅ ⁰ ⋅ +

0

0

exp ( ^{' '})

cos cos

F F

t q - q

0

xy k

A A

= ⋅exp ( ^{' '})⋅

cos ⁰ sin

F0 F (15) The results of the calculation according to equations (15) in Figures 1 to 4 show that the distribution of the contact stresses is related to the factor of the shape of the stress nucleus l/h and the friction coefficient f, the relative normal stresses sy/2k0 and the relative tangent stresses txy/k0. The results of the calculation correspond to the real distribution diagrams of the contact stresses.⁶ It should be emphasized that expressions (15) are uni- form for the entire nucleus of deformation and there is no need to break it into separate zones of contact fric- tion.⁷Figures 5 and 6 show the distribution of stresses

Figure 4:Distribution of the normal stresses along the strip height during the upsetting with rough strikersf= 0,3,l/h= 2...15

Slika 4:Porazdelitev normalnih napetosti po vi{ini plo{~e pri kr~enju s te`kimi kladivif= 0,3,l/h= 2...15

Figure 2: Distribution of the normal and tangent stresses at the contact during upsetting with rough strikersf= 0,3,l/h= 1...15 Slika 2:Porazdelitev kontaktnih normalnih in tangetnih napetosti pri kr~enju s te`kimi kladivif= 0,3,l/h= 1...15

Figure 3: Distribution of the normal stresses along the plate height during the upsetting with rough strikersl/h= 8,f= 0,1...0,5 Slika 3:Porazdelitev normalnih napetosti po vi{ini plo{~e pri kr~enju s te`kimi kladivi:l/h= 8,f= 0.1…0.5

Figure 1: Distribution of the normal and tangent stresses at the contact during upsetting with rough strikersl/h= 8,f= 0,1...0,5 Slika 1:Porazdelitev kontaktnih normalnih in tangetnih napetosti pri kr~enju s te`kimi kladivil/h= 8,f= 0,1...0,5

in the nucleus of deformation, which also include the shape factor and the friction coefficient.

The obtained results display qualitatively and quanti- tatively the general patterns of the distribution fields of the tensor of stresses over the entire nucleus of defor- mation. The results meet fully the requirements of the boundary conditions. In particular, the proposed proce- dure and expressions¹⁵ can be recommended for the calculation of various problems in applications.

The proposed complex model of a plastic medium based on the closed solution can be considered as a generalization of the theory of plasticity, uniting the theories of deformation and of plastic yielding.

4 REFERENCES

1V. V. Chygyryns’kyy, I. Mamuzi}, G. V. Bergeman: Analysis of the State of Stress of a Medium under Conditions of Inhomogeneous Plastic Yielding; Metalurgija. Zagreb. 43 (2004), 87–93

2A. N. Tikhonov, A. A. Samarsky: Uravnenia matematicheskoy fisiki (Equations of mathematical physics); Nauka, (1977), 735

3N. N. Malinin: Prikladnaya teoria plastichnosti i polsychesti (Applied theory of plasticity and yielding); Mashinostroyeniye, (1975), 399

4L. M. Kachanov: Osnovy teorii plastichnosti (Fundamentals of the theory of plasticity); Nauka, (1969), 419

5G. E. Arkulis, V. G. Dorogobid: Teoria plastichnosti (Theory of plasticity); Metalurgiya, (1987), 251

6A. P. Chekmarev, P. L. Klimenko: Experimentalnoe issledovanie udel’nyh davleniy na kontaktnoy poverhnosti pri prokatke v kalibrah (Experimental investigations of partial pressures on the contact surface in rolling in calibers); Obrabotka metallov davleniem (Pressure forming of metals); Sbornik trudov Dnepropetrovskogo metallurgicheskogo instituta; Khar’kov, M., 1960. – Vypusk 39

7M. V. Storozhev, E. A. Popov: Teoria obrabotki metallov davleniem (Theory of pressure working of metals); Mashinostroyeniye, (1977), 422

LIST OF SYMBOLS

s– normal components of the stress tensor;

t– tangential components of the stress tensor;

x– linear components of the strain-rate tensor;

g– shear components of the strain-rate tensor;

e– linear deformation along the axes x and y;

c – factor of correspondence between the tangent stress and the temperature-deformation parameter of the centre to deformation;

Gi– intensity of the shift of the deformation;

tⁿ– tangential contact stress on an arbitrary inclined area;

a– angle of inclination of the contact area;

k– shearing plastic deformation strength;

F– harmonic function depending on the coordinates of the deforma- tion zone and the argument of a trigonometric function;

F0– argument of trigonometric function forx= ±l/2 andy= ±h/2;

A– constant characterizing the trigonometric function for the state of stress of the plastic medium;

A6,A13– constant factor, characterizing the shearing tangent stress in the zone of reduction

Y0,Y1– values taking into account the influence of the factor of fric- tion;

B– constant value characterizing the trigonometric function for the state of strain of the plastic medium;

q⁰– factor exhibitors forx= ±l/2 andy= ±h/2;

q^'– harmonic function, exponential index, characterizing the shearing stress distribution in the zone of reduction;

q^{' '}– a harmonic function, exponential index, characterizing the distri- bution of the rate of shearing in the zone of reduction;

C_s– constant value determining the state of stress of the plastic me- dium;

C_x– constant value characterizing the state of strain of the plastic me- dium;

T– temperature in the k-th point;

Ti– intensity of tangential stress;

Hi– intensity of shearing rates;

CT– constant value characterizing the temperature field;

a– coefficient of temperature conductivity;

landh– length and height of the deformation nucleus during strip up- setting;

f– friction coefficient;

k0– contact shear resistance at the beginning of the deformation nu- cleus