Zeitschrift für Angewandte Mathematik und Mechanik: Band 61, Heft 8 August 1981 [Reprint 2021 ed.] 9783112549605, 9783112549599

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U N T E R M I T W I R K U N G V O N E. B E C K E R • H. B E C K E R T • L. BERG • L. B I T T N E R • W . F I S Z D O N • H. G A J E W S K I • H. G Ö R T L E R • J . H E I N H O L D • H . H E I N R I C H A.JU. I S C H L I N S K I • R. K L Ö T Z L E R • P. H.MÜLLER • H. N E U B E R • K. OSWATITSCH L. S C H M E T T E R E R • J . W . S C H M I D T • H . S C H U B E R T • G. G . T S C H O R N Y • F. W E I D E N H A M M E R U N D F. Z I E G L E R H E R A U S G E G E B E N V O N G . S C H M I D T , BERLIN C H E F R E D A K T E U R : G. SCHMIDT

L. C O L L A T Z • J.HULT • A. S A W C Z U K H.UNGER

REDAKTEURE: W. H E I N R I C H , H. WEINERT

BAND 61

1981

HEFT 8

A U S DEM I N H A L T H A U P T A U F S Ä T Z E K. Z.Markov: On the Dilatation Theory of Elasticity/ A. Chakrabarti / K. Manivachakan: On the Transform Method of Solution of an External Crack Problem / M. N. Farah: Laminare, instationäre Strömung im Ringraum zwischen zwei Rohren, von denen das innere harmonisch schwingt / M. Tasche: Eine einheitliche Herleitung verschiedener Interpolationsformeln mittels der Taylorschen Formel der Operatorenrechnung KLEINE

M I T T E I L U N G E N

B U C H B E S P R E C H U N G E N

A K A D E M I E - V E R L A G ZAMM EVP 18,— M

Bd. 61

Nr. 8

S. 345—408

•

B E R L I N Berlin, August 1981 34115

INHALT Hauptaufsätze K. Z. Markov: On the Dilatation Theory of Elasticity A. Chakrabart! / K.Manivachakan: On the Transform Method of Solution of an External Crack Problem . : M. N.Farah: Laminare, instationäre Strömung im Ringraum zwischen zwei Rohren, von denen das innere harmonisch schwingt M.Tasche: Eine einheitliche Herleitung verschiedener Interpolationsformeln mittels der Taylorschen Formel der Operatorenrechnung

Seite 349 359 365 379

Kleine Mitteilungen F. Beichelt: A Replacement Policy Based on Limits for the Repair Cost Rate L. Berg: Stabile Iterationsverfahren beliebiger Ordnung zur Berechnung von Wurzeln . . . . . P. H.M.Wolkenfelt: On the Numerical Stability of Reducible Quadrature Methods for Second Kind Volterra Integral Equations . . P. Rajagopalan / R. Purushothaman: Mean Fldw Induced by a Travelling Wave in a Rotating Fluid

395 396 395 401

Buchbesprechungen

404

Wir bitten, Manuskriptsendungen zweifach (Original,und eine Kopie, sprachlich einwandfrei, Formeln mit Maschine oder in Druckschrift geschrieben) an folgende Anschrift zu richten: Zeitschrift für Angewandte Mathematik und Mechanik, Institut für Mechanik der Akademie der Wissenschaften der DDR DDR-1199 B e r l i n , Rudower Chaussee 5, Zu den Arbeiten, die als Hauptaufsätze bestimmt sind, ist auf gesondertem Blatt eine Zusammenfassung von 5 bis 10 Zellen in englischer und (mögllchst)«deutscher und russischer Sprache beizufügen. Ausführliche Hinweise für die Autoren, um deren strikte Berücksichtigung gebeten wird, finden sich im Anschluß an das Inhaltsverzeichnis des Jahrganges 60(1980). Die Autoren erhalten von den Hauptaufsätzen 75, von den Kleinen Mitteilungen 25 Sonderdrucke ohne Berechnung, darüber hinaus weitere Sonderdrucke gegen Berechnung. Der Verlag behält sich für alle Beiträge das Recht der Vervielfältigung und Übersetzung vor. Bestellungen sind zu richten — in der DDR an den Postzeitungsvertrieb unter Angabe der Kundennummer des Bestellenden oder an den AKADEMIE-VERLAG, DDR-1080 Berlin, Leipziger Straße 3 - 4 — Im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Berlln(West) an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber OHG, D-7000 Stuttgart 1, Wilhelmstraße 4 - 6 — In den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Aüslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, Ch-8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen'Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen 'Demokratischen Republik, DDR-7010'Leipzig, Postfach 160; oder an den AKADEMIE-VERLAG, DDR-1080 Berlin, Leipziger Straße 3 - 4 ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND MECHANIK Herausgeber und Chefredakteur: Prof. Dr. Günter Schmidt. Redaktion: Dr. Winfried Heinrich, Dr. Horst Weinert, Dipl.-Math. Friedhild Dudel, Helga Rühl, Institut für Mechanik der Akademie der Wissenschaften der DDR. Verlag: Akademie-Verlag, DDR-1080 Berlin, Leipziger Straße3—4; Fernruf: 2236221 oder 2236229. Telex-Nr.: 114420; Bank: Staatsbank der DDR, Berlin, Kto.-Nr.: 6836-26-20712. Anschrift der Redaktion: Institut für Mechanik der Akademie der Wissenschaften, DDR-1199 Berlin, Rudower Chaussee 5, Fernruf: 6702841. . Veröffentlich! unter der Lizenznummer 12B2 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckerei ,.Thomas M ü n t z e r " , DDR-5820 Bad Langensalza. Erscheinungsweise: Die Z e i t s c h r i f t f ü r Angewandte Mathematik und Mechanik erscheint monatlich. Die 12 Hefte eines Jahres einschließlich Tagungshefte bilden einen Band. Bezugspreis je Band 360,— M zuzüglich Versandspesen (Preis für die DDR 216, — M). Bezugspreis je Heft 30, — M (Preis für die DDR 18, — M). Bestellnummer dieses Heftes: 1009/61/8. © 1 9 8 1 by Akademie-Verlag Berlin • Printed in the German Democratic Repubiic. A N (EDV) 35937

ISSN 0044-2267 K . Z. MABKOV: On the Dilatation Theory of Elasticity

349 ZAMM 61, 3 4 9 - 3 5 8 (1981)

K . Z . MARKOV

On the Dilatation Theory of Elasticity In dieser Arbeit wird die lineare Theorie des Festkörpers, für welche das Verschiebungsfeld und die Volumendilatation unabhängig sind, in Einzelheiten erforscht. Die Theorie ist für eine Beschreibung des Elastizitätskörpers, der eine große Zahl Mikroporen enthalt, angewandt. Für diesen Fall wurden die in der Theorie vorkommenden Materialkonstanten berechnet. In this paper a linear theory for a solid, for which the displacement and dilatation (volume change) fields are independent, is investigated in details. The theory is applied to describe the mechanical behaviour of an elastic solid containing a large number of microvoids and the material constants, entering the theory, are calculated for this case.

PaSoTa nocBHiueiia jjeTajiwiOMy H3yieHHK> jiHueitnoii Teoprni TBepnoro ne$opMHpyeMoro Tejia, JJJIH KOToporo nojiH nepeMememm n iiMJiJiTamiH (oG'teMnoro H3M6H6HHH) He3aBHCHMM. Teopan npHMeHeHa HJIH oimcaniiH MexaHH^ecKoro noBefleHHH ynpyroro Tejia cofl;ep>Kamero ßojitmoe KOJIHHCCTBO MHKponop. JJ,JIH aToro cjiyqaa noHCimaniii MaTepnajibULie nocTOimiiLie, yiacTByioiUHe B Teopnn. 1. Introduction In the classical continuum mechanics there is a fundamental assumption that the basic characteristic of the motion is the field u of the body points displacement. This field is basic in the sense that all the rest deformation characteristics can be found provided the field u is known. For example, under the supposition of small strains the simple relations hold true, viz. 0

=

V X tt ,

0 = V- M

(1.1)

which describe the rotation (P and dilatation 0 for a small volume of the body. If however the microstructure of the material is taken into account, we should reject the relations (1.1) and consider the fields

0, as it follows from (2.8). («+»?)% The equations (2.12) and (2.13) form a system of nine equations which specifies the stress characteristics Ta and A for an isotropic dilatation elastic solid. To bring this Section to a conclusion we note that the GREEN tensor and general solution for the system (2.9) could be constructed in a more or less standard manner, having decomposed the displacement field as a sum of two potentials: u = b + ^rp. Here, we give the final results only. The first one concerns the displacement Ue(x — as') and dilatation 0e(x — x') fields, generated by unit force acting along the direction e, |e| = 1, and concentrated at the point x'. They are of the form v

0e — D • e ,

his) = — (1 s

D = —^—SJh(Rlm); 4 7iqxm '

with the constants m2 =

e~s) ,

^ , qx = (A + 2/j, + rf) d — rf, q2 = (X

9i>

B = \x -

x'\ ,

(2.14)

ft + f]) d — rj2. The second result

refers to the general solution of the PAPKOVICH-NEUBER type for the system (2.9). It has the form ( 5 i

a = = 6 + V

e

_ J ^

\ 2i

r

.

6 + & 0 ),

¿h

0 = 0O + — V • & >

/

(2.15)

Hi

where»' =x, b and b0 are arbitrary harmonic functions and 0O is an arbitrary solution of the equation m2 A 0 — 0 = 0. For details, concerning (2.14) and (2.15), cf. [4], 3. Plane Problem for the Dilatation Elasticity I n the case of plane deformation the stress and strain fields are functions of two co-ordinates; for simplicity, we choose them as the Cartesian co-ordinates x and y. The first equilibrium equation (2.1) is the same as that in the classical elasticity. That is why it can be satisfied by introducing the AIRY function q> = tp(x, y) [7], so that „

_

-

9

>

92/2

'

,.

T*V

_

-

92(

p

~ QX dy>

„

_

9

V

dx2

•

/o l \

V-V

We shall use in what follows the AIRY function cp and the free dilatation 0 C as basic functions (potentials) for the plane problem in discussion. Body forces are supposed to vanish, / = 0. I t is to note first the relation (X + 2/x) Ae — rj A0C = 0

(3.2)

which is an obvious consequence of (2.9)!. Making use of (2.9)2 and (3.2), we get the following equation M2 A A 0 C -

A0C = 0

for the field of free dilatation. 25*

(3.3)

352

K . Z. MABKOV: On the Dilatation Theory of Elasticity

On the other hand, we have

as it follows from (2.7) and (3.1). Equations (3.2) to (3.4) yield now that the function

is biharmonic, i.e. AArp = 0. I t is worth mentioning another relation between the potentials

= 4

ox

1

+ if,2,

ay

g(z) = 1 / G(w) dw = gt + ig2 , R =

= 4 ^ = ox

(3.12)

4 ^ . oy

Having used these two relations in (3.9), we find eventually that 2¡m x = — ^

+ 4(1 — v)fx — 4(1 — 2v) ^

+ const, (3.13)

2¡iuy = — ^ + 4(1 — v)f2 — 4(1 — 2v) rjg2 + const. This is the desired generalization of the LOVE formulae [8] for the case of a dilatation elastic solid undergoing plane deformation. As a simple example of application of LOVE'S type relations (3.13), we shall consider the displacement field generated by an edge dislocation in an unbounded dilatation elastic body. Let Q, OC be the polar co-ordinates, so that a = 0 be the slip direction and o = 0 — the centre of the dislocation.

K. Z. MABKOV: On the Dilatation Theory of Elasticity

353

We first choose the biharmonic function tp from (3.5) to be the same as that in the case of elastic material [9], namely, ip = cp +

m2®e

5 sin « ,

=

(3.14)

and the solution of (3.3) for the free dilatation as follows &C =

Here

(ï

)

+ aaKl(Qlm)

sin

+ft4"

(3-15)

j = 1 to 4, are some constants and Kx stands for the MCDONALD function of the first order. The analytical functions (3.11), which correspond to the potentials (3.14) and (3.15), are very simple, viz.

AJ,

=

(3.16)

G{z) = - ^ .

Making use of (3.12) to (3.16), we find after simple calculations that 2(iux = — ^ — (2a t (l — v) + »7(1 — 2v) a2) x + const, 2uuy — — ^ — (2^(1 — v) + v(l — 2v) a2) In q + const. dy

(3-17)!

The relations (3.17) give the displacement field generated by an edge dislocation in a dilatation elastic solid. The constants a1 and A2 should be specified through the two facts we know: first, the BURGERS vector b is given and second, the functions

0 and \J© which are invariant with respect to inversion, do not enter (4.3). Having inserted (4.3) into (4.2), we receive the following stress-strain relations for the body under consideration

Ta = {le - rjO") I + 2/iTy + 2 « T J , Tft = pi tr Tx + 2yT% + 2eT% , A = %V0,

s -p

(4.4)

= 00" + (, so that (Q c ") = = 0O. Then the relation (6.9) ensures the "equivalence" of those two spheres in the following sense. We cannot tell which one is made of elastic and which one of dilatation elastic material, using merely measuring of their surface displacement under hydrostatic pressure — because this displacement would be one and the same for both spheres provided (6.9) holds true. As such an equivalence represents a natural requirement as far as we want to simulate the mechanical behaviour of a microporous elastic solid through the dilatation elastic model, we shall consider (6.9) as the desired calculation for the first of the two dilatation constants we are seeking for. I t is to note that the supposition of microporosity 0 O 1 for the elastic sphere was used above only to provide the bulk modulus k in (6.9) to coincide with that of the solid part of the sphere. I f the void ratio 0O is not small, the foregoing considerations remain valid, with the only change that the bulk modulus k = k(0a) for the porous solid should enter (6.9). Explicit expressions for function k(0o) could be taken from various theories, predicting the overall moduli for porous elastic solids (cf., e.g., [11], [12]). I t should be pointed out that, as a matter of fact, the same equality (6.9) has been obtained by E S H E L B Y , who has derived equation (5.2)x and also noted the analogy between body expansion generated by temperature increasing and that caused by damaging [18]. In order to specify the last unknown dilatation constant we should examine another problem, concerning a microporous elastic solid and its simulation by means of the dilatation elastic model. 6.2. S p h e r i c a l H o l e in U n b o u n d e d M i c r o p o r o u s S o l i d Consider an unbounded dilatation elastic material containing a spherical hole with a radius R. The material undergoes an internal pressure P acting on the surface of the hole, so that the same boundary condition (6.1) holds. We then look for the solution of the system (5.2). In the case in discussion, the free dilatation field 0C has the form

K . Z. Mabkov: On the Dilatation Theory of Elasticity

357

cf. (6.4)2. For the potential/, according to (6.5) and (6.10), we get f =

J

1 +

2p.

0

r

-

C

-

(

6

.

1

1

K

)

'

Keeping in mind (6.10), (6.11) and (2.7)!, we obtain the radial stress ay to be 4fir\ 1 +

D^x)

2/i

2fi

x

r>R,

T

®3 ~~

(6.12)

where

In order to satisfy the boundary condition (6.1), the unknown constants a 3 and a 4 should comply with the relation 4ah

DM)

2«

_

,

.

As our basic idea in this section consists in simulating of the elastic porous material by means of a dilatation elastic one, we are bound to examine now the same unbounded body containing the spherical hole, but made of elastic material whose void ratio changes r in accordance with (6.10). As it can be seen from (6.10), the magnitude of the void ratio @ c is not small in the vicinity of the hole. This fact makes us take into account the decrease of the elastic moduli due to porosity. Here, in order to simplify the calculations in what follows, we shall apply the relations p* = ^(1 _ 20c) ,

k* = £( 1 - 20")

(6.14)

for the shear and bulk moduli respectively for a porous solid with void ratio 0", having assumed for simplicity that the POISSON ratio of the solid part v — 0 . 2 [ 1 2 ] , [ 2 1 ] . However, the solution of the centrosymmetric problem considered for a body, whose elastic moduli vary along r in accordance with (6.14) and (6.10), cannot be explicitly done. That is the reason to make one more simplification supposing that the void ratio &c(r) is replaced by a step-constant function 0c(r) such that g.(r)= p s A ,

R < r < L ,

[0,

r > L ,

e~djd, with the value -^asD0 which is a half of the maximal one for the function Oe(r), given by (6.10). The parameter L is to be found from the obvious condition

D0 =

f&e(r) t>R

d F =

/ 6e(r) r>R

dF ,

which specifies L to be (Ljm')3

=

d3 +

6d(l +

d) .

(6.16)

Consider now a solid whose elastic constants fx* and k* are defined by means of (6.14) and (6.16). The centrosymmetric problem we deal with, namely, the expansion of a spherical hole undergoing internal pressure in an unbounded matrix, can be readily solved for such a partially homogeneous elastic solid. The solution looks as follows r > L ;

ur =

A1r

+

B < r < L ,

(6.17)

where P 0

~ 1 + 6