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Journal of Siberian Federal University. Mathematics & Physics 2020, 13(2), 213—217
DOI: 10.17516/1997-1397-2020-13-2-213-217 УДК 539.374
Anisotropic Antiplane Elastoplastic Problem
Sergei I. Senashov* Irina L. Savostyanova^
Reshetnev Siberian State University of Science and Technology
Krasnoyarsk, Russian Federation
Olga N. Cherepanova*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 10.11.2019, received in revised form 11.01.2020, accepted 20.02.2020 Abstract. In this work we solve an anisotropic antiplane elastoplastic problem about stress state in a body weakened by a hole bounded by a piecewise-smooth contour. We give the conservation laws which allowed us to reduce calculations of stress components to a contour integral over the contour of the hole. The conservation laws allowed us to find the boundary between the elastic and plastic areas. Keywords: anisotropic elastoplastic problem, antiplane stress state, conservation laws. Citation: S.I.Senashov, I.L.Savostyanova, O.N.Cherepanova, Anisotropic Antiplane Elastoplastic Problem, J. Sib. Fed. Univ. Math. Phys., 2020, 13(2), 213-217. DOI: 10.17516/1997-1397-2020-13-2-213-217.
Introduction
Fields of shifts and stresses in the case under consideration are the following [1]
u = v =0, w= w (x,y) ax = g y = az = rxy = 0, rxz = t1 (x,y), Tyz = t2 (x,y) . (1)
Here u, v, w are shift vector components, ax, ay, gz, Txy, txz, Tyz are stress components, x, y, z the Cartesian coordinates, axis directed parallel to the element. In the elastic zone there are the relations
dT1 dT 2
—--+ —— = 0 (equilibrium equation), (2)
d x d y
t 1 = d—, t2 = G2 dw (Hooke's law) . (3)
dx dy
Here Gi are constants called elastic moduli [2].
From (2), (3) there arise relations in the elastic zone
d2w d2w
Gi dxxw + g2 W = 0, (4)
dT1 dT2
g2 -W = G TZ- (5)
* sen@mail.sibsau.ru https://orcid.org/0000-0001-5542-4781 truppa@inbox.ru https://orcid.org/0000-0002-9675-7109 tcheronik@mail.ru © Siberian Federal University. All rights reserved
From (2) and (5) it follows that t ,t2 satisfy the system of linear equations
„ dT1 dr2 „ dr 1 dT2 . .
Fi = ^r + ^r=0> F2 = - nIT" = (6)
dx dy dy dx
where n = G1/G2.
In the plastic zone there holds the relation (2), and also
ai3(t 1)2 + a23(r2)2 = 1 (yield condition), (7)
2 dw 1 dw . .
t -j— = t -j— (Hencky s equation) . (8)
d X d y
Here a13,a23 are constants called anisotropy coefficients.
On the boundary of the elastic and plastic areas the stresses and shifts are supposed to be continuous.
1. Conservation laws
By a conservation law for the system of equations (6) we shall call the relation of the form of dA(x,y,r 1,t 2) dB(x,y,T 1,t 2) 0
- dx +- dy = F1 +" F2, (9)
where w% = wi(x,y,r 1,t2) are some functions not identically zero simultaneously.
Note. A more general definition of conservation laws and their use in mechanics of a solid body being deformed can be studied for example in [3-5].
For the purposes that are set in this article a simplified formulation in the form of (9) will suit fine.
In (9) the values A, B are called conserved current components. Let us assume that the components A, B appear as follows
A = a1T1 + /31t 2 + y1, B = a2T1 + /3 2t 2 + y2, (10)
where a1 = a1 (x, y), / = 3 (x, y), y% = Y% (x, y) are some smooth functions to be determined. Let us substitute (10) into (9), as a result we obtain
aXT1 + a1T1 + ßX t 2 + ß1r2x + y2 + ay t 1 + a2rl + ß2yr2 + ß2r2y + y2 =
u1 T + t2) + u2 T - nrX) = 0,
(11)
x 2 J ' \ 2 x)
where the index below stands for a derivative with respect to the corresponding variable. From (11) we obtain
a1 = w1, 31 = -nw2, a2 = w2, 32 = w1, a1 + a2y = 0, /1 + /32y = 0, y1 + Y2 = ° (12) From (12) excluding wl we obtain
a1 = 32, 31 = -na2, a1 - n,3ly =0, /1 + aly = 0, y1 + y2 = °. (13)
By virtue of relations (12) the conserved current components are written as
A = a1T1 + 31t 2 + y 1, B = -331 t 1 + a1T2 + y2 ■ (14)
Since the right-hand part (9) is equal to zero, according to Green's formula we obtain
(Ax + By ) dxdy = é Ady — Bdx = Jas
..It 1 , q\t 2 , Y,l)jy I T 1 + alT 2 + dX = 0,
dS
(a1T1 + ß1T2 + Y1)dy —
(15)
where S is the area, dS is its piecewise-smooth boundary. All the functions in (15) are supposed to be smooth.
2. Elastoplastic problem for an arbitrary hole in case when the plastic area surrounds the entire hole
Assume C is a piecewise-smooth contour, there is a load applied to it
, 1 . , 2 I I ^ /12«23 + l2a13
¿1T + l2T = Tn, \Tn\ -,
V a13a23
(16)
where (li,l2) are normal's vector components to contour C. The plastic area's contour L surrounds entirely the hole C. See Fig. 1.
Fig. 1. Elastic-plastic border near the hole C
In this case on contour C, apart from the condition (16), also fulfilled is the yield condition (7). Thus on C there are two conditions:
lit1 + l2r2 = rn = rn, ais (t 1)2 + a23(t2)2 = 1. From the conditions (17) we find the stress components on contour c:
l2 2 1 Tn 2 — h T + h , T
a13l2Tn T h\Jl2a23 + l2a13 — 013023^
l2a 23 + l2ai3
From this point on, to be definite, in formulas (18) we will be selecting the upper sign.
(17)
(18)
3. The use of conservation laws to find stress components in the area
Assume the point M (xm,ym) lies beyond the contour C. Let us draw a circumference with radius e with the centre at the point M. We have e : (x — xm)2 + (y — ym)2 = e2. Assume D is
1
a line connecting the point M with the contour C. We obtain a closed contour consisting of the circumference e, the segmant P and the contour C. See Fig. 2.
Fig. 2. Calculating the contour integral around the singular point M
From (15) we obtain
Ady - Bdx + / Ady - Bdx + I Ady - Bdx + f Ady - Bdx = 0. (19)
C J P+ J P- Je
The sum of the second and the third summands in (19) is equal to zero, because the integrals are calculated in different directions. Finally from (19) we have
I Ady — Bdx = — j) Ady — Bdx.
(20)
Let us convert the right-hand part of equation (20) introducing parametrisation x = e cos t, y = e sin t, 0 ^ t ^ 2n. As a result we have
j) Ady — Bdx = e j (A cos t + B sin t) dt.
Assume in (15)
Then from (21) we obtain
x2 + ny
y
x2 + ny2
(21) (22)
f i'2n i'2n
® A1 dy - B1dx = e (A1 cos t + B1 sint) dt = / t 1dt = 2nr1 (xm,ym). (23) Je Jo Jo
The last equality in (23) is obtained with the use of the mean-value theorem with e tending to zero.
Assume in (15)
1 _ Vny al 1 x
x2 + ny2
ß1 =
Then from (21) we obtain
® A2dy - B2dx = e (A2 cos t + B2 sint) dt = / t2dt = 2nr2 (xm,ym). (25)
e o o
The last equality in (25) is obtained with the use of the mean-value theorem with e tending to zero.
y/n x2 + ny2
2n
(24)
x
1
a
From formula (20), and also from (23) and (25) we obtain
/ Aidy - Bidx = -2пт1 (xm, ym), / A2dy - B2dx = -2пт2 (xm, ym). (26) jc JC
Conclusion
Formulas (26) offer the opportunity to find stress components in any point xm, ym beyond the contour C. This allows us to determine the boundary between the elastic and plastic areas. If the plasticity condition is met а13(т1)2 + а23(т2)2 = 1 at the point xm,ym then this point belongs to the plastic area, if in the point the condition а13(т1 )2 + а23(т2)2 < 1 is met, then to the elastic area.
Note. The formulas found above allow us to solve elastoplastic problems even if the plastic contour does not entirely surrounds the contour C, provided that on the contour C the plasticity condition (7) is fulfilled.
References
[1] B.D.Annin, G.P.Cherepanov, Elastic-plastic problem, Novosibirsk, Nauka, 1983 (in Russian).
[2] S.G.Lehnitsky, Theory of elasticity of an anisotropic body, Moscow, Nauka, 1977 (in Russian).
[3] S.I.Senashov, A.M.Vinogradov, Proc. Edinburg Math. Soc., 31(1988), no. 3, 415-439. DOI: 10.1017/S0013091500006817
[4] P.P.Kiryakov, S.I.Senashov, A.N.Yakhno, Application of symmetries and conservation laws to solving differential equations, Novosibirsk, Ros. Acad. nauk. Sib. otd., 2001 (in Russian).
[5] S.I.Senashov, O.V.Gomonova, A.N.Yakhno, Mathematical problems of two-dimensional equations of ideal plasticity, Krasnoyarsk, Izd. SibGAU, 2012 (in Russian).
Анизотропная антиплоская упругопластическая задача
Сергей И. Сенашов Ирина Л. Савостьянова
Сибирский государственный университет науки и технологий им. М. Ф. Решетнева
Красноярск, Российская Федерация
Ольга Н. Черепанова
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. В работе решена анизотропная антиплоская упругопластическая задача о напряженном состоянии в теле, ослабленном отверстием, ограниченном кусочно-гладким контуром. В статье приведены законы сохранения, которые позволили свести вычисления компонент тензора напряжений к криволинейному интегралу по контуру отверстия. Законы сохранения дали возможность найти границу между упругой и пластической областями.
Ключевые слова: анизотропная упругопластическая задача, антиплоское напряженное состояние, законы сохранения.
