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A perturbation theory of antiplane elastic-plastic deformations
A PERTURBATION THEORY OF ANTIPLANE ELASTIC-PLASTIC DEFORMATIONS I. S. TUBA Westinghouse Research Laboratories, Pittsburgh, Pa. m--A perturbation theory of antiplane elastic-plastic deformations is presented based on a total or deformation theory of plasticity. A complete formal solution is given. A circular hole in a body under uniform antiplane shear is considered for illustration purposes.
1. INTRODUCIION
IN this paper a perturbation theory of antiplane elastic-plastic deformations is presented based on a total or deformation theory of plasticity. In addition to the formulation, the complete formal solution is given. The method is illustrated for a circular hole in a body under uniform antiplane shear.
2. THEORJZTICALCONSIDERATIONS
2.1 Formulationfor Cartesian coordinates Consider an orthogonal Cartesian coordinate system x, y, z. For antiplane deformation all stress components are zero except r_ and r,,. Let us assume that there are no body forces, that the material is isotropic and is at uniform temperature. The three equations of equilibrium reduce to the forms
a%* _ -z-
()
(la)
k, az
o
(IV
-=
From the first two of these it is concluded that t,, and rs,. are not functions of z. The third equilibrium equation can be satisfied automatically by a suitably selected stress function F aF ? ‘“=ay
(W aF
T =),=
--
51
ax
(2b)
52
I. S. TUBA
There are two equations identically :
of compatibility
of the total strains which are not satisfied
a2Yzr a2Yzx
---=
ax2
o
(34
o
(3W
axay
a2Yzxa%_ axay
dy2
Let us integrate the first of these with respect to x and the second with respect to y.
ayl,.
---
ax
avzx =f ay
w
(y)
1
tiz, :; .=f2(x) --ay By comparison fr(y) = -f2(x) = c The arbitrary constant c is set equal to zero and a single compatibility obtained.
a/,
ah
-=
ax
o
ay
equation
is
(5)
The total strain can be expressed as the sum of the elastic and the plastic components : Yzx = [al
+ v) 4’ +
361 rsx
@a)
y*? = [2(1 + v) 4’ + 3#‘] ?zy
(6b)
where 4’ and 4” are the reciprocals of the elastic and plastic moduli, i.e. #’ = l/E
(W
The plastic modulus is a variable quantity which depends on the state of stress and on the shape of the stress-strain curve. For the Ramberg-Osgood [l] type of stress-strain curves (Fig. 1) it can be shown that (7b) where cr, is the stress at a secant modulus of 0.7 E and n is a shape factor. The effective stress, 6, is related to the stress components B, = J3 . [?E + rf).])
(84
In terms of the stress function F 6, = 43.
[E)’
+ ($)I’
W
+ (!Y)~‘“-l~‘z
(7c)
The reciprocal of the plastic modulus is then $1, = &
@2
A pertur&adon theory of ontipkme elosticplattic &fommtions
53
l-5
1-o
rr 3
04
0
I
,
1 l
2.0
l-0
h 9 Fffi. 1. Rambergspood
Let@=#“Eandm=(n-
stress-straincurves.
1)/2,then
[(gJ2+ @;1”
tl+ Yzx
=
d[2(1
+ v) + 3@]E
YZS= - f [2(1 + v) + 3@] s When the stress-strain relations expressed by equations (10) are substituted compatibility condition (5), the following governing equation is obtained : 3 V2F + 2(1 + v)
=
0
into the
(11)
Since d is a function of F only, equation (11) is a function of F only. Any function F, which satisfies equation (11) and some appropriate boundary conditions, will be an exact solution to the elastic-plastic antiplane problem. 2.2 Conversion to complex form It is more convenient to consider F as a function of the complex variables z = x + iy
54
I. S. TUEIA
and Z = x - iy instead of as a function of x and y. Let F(x, y) = FC(z,Z), then drop the subscript c. In that case the governing equation (11) can be written as
+0 --= 1 aF 2a2F
E
rz.x=1
. aF
BF
r,).
=
-
aF
( > (dz+z > x-z
aF
a22
0
(12) (134
(13b)
By definition
K
=
3m+2 .22m-1 7( 1 + v) ap
(14)
2.3 Orthogonal transformation of geometry Similar to plane problems of elasticity, an orthogonal transformation of the geometry can be carried out in order to simplify the process of satisfying boundary conditions. Let z = o(c), Z = m = E(r) then F[z,L] = FCC&), m)] = F&r). In what follows we shall drop the subscript t. The governing equation (12), transforms to the form a2F -+Ki$$
ar a7 2
+2Q
aF aF d2F
(154
Tqzfayae+“l
where a=
l w’(C)m’(T)
(15b) (15d (154
The stresses are related to the stress function as
Ifz=~theno’=G’==P=Q,=l,Q, = 0 and equation (15) reduces to equation (12) ; furthermore, equations (16) reduce to the forms of equations (13).
of antiplane elartic-j7lastic &formations
A perturbation theory
55
Led c = u + iv. The sole purpose of the transformation of the geometry is to have some constant values of u or v coincide with the boundaries, therefore, it is desirable to express the shear stresses along the u, v coordinate lines. The results are
i ‘52”‘=#
1
aF aF z - x [
(174 um
2.4 Introduction of a characteristic load For conveniences in obtaining a solution, only those cases are considered here where the stress function can be expressed as FK, 7) = 0X,
b
(18)
where 6, is defined as the characteristic load, which shall be specified in particular problems from the knowledge of the load distribution. Substitute equation (18) into equations (lSa), (16) and (17), then drop the subscript f. The results will be identical to those listed in the previous section, except that K is replaced by 3n+1pn
P
o
la
z-2
7U + v) 6> 1
(19)
and the various components of 7 by the corresponding component of r/a,
3. FORMAL SOLUTION BY PEzRTURBAT~ON
Equation (15a), is a second order non-linear partial differential equation. At the present there are no methods available for the exact solution of nonlinear problems of this type. Equation (Ua), as modified by the introduction of the characteristic load, involves a dimensionless parameter p. Setting p = 0, the equation (15a) reduces to the first term, which is the governing equation for the elastic case. For monotonically increasing loads, at first elastic conditions exist, then with continuously increasing load the solution gradually changes. For small values of aC/al, the parameter p is also small. It can be assumed that the stress function is an analytic function of the parameter p in the neighborhood of p = 0, and therefore it can be expanded in Taylor series form.
The coefficient of each power of p is an unknown function of C and <. First the series (20) is substituted into the modified equation (15), then the terms having equal powers of ~1are collected and finally the coeflicicnts of each power of p are set equal to zero. This process provides an infinite set of “linear” second order partial differential equations. The one corresponding to /JOprovides the governing equation for the elastic case, which can be used as a starting point. All other equations are nonhomogeneous and involve solutions corresponding to the lower powers of /.L Since the actual boundary
I. S. TUBA
56
conditions are satisfied by the elastic solution 0, = 0), all other F, for p > 1 must satisfy homogeneous boundary conditions. The above outlined solution is quite lengthy so the details are not presented here, but the results are summarized in such a manner that specific problems can be solved by following the outlined steps. For the sake of compactness let aF -_=c(= f a#; ap = -aFP al x P=o
Define *p = f
(224
cz4BP-Y
q=o
mp*,yo + ‘il (PY,=e!l;
I,
4)bb,-,Up, - tiqyp-ql
411
=
WW
PI0
(m - w,h
r, = I);--‘;
r, =
A,=ai;
A,=
‘$P- 4)Rm-
+
W,-,q
cl=1
(224
P#o %a40
+
‘fl
(P
-
4)
[zap-,A,
- IL,r,-,I
-
4=1
%A,-,1
(224 pa0
p-1
2PSpBo + c
B,=j3;;
@ - 4) cvp-,Bq
4x1
BP=
-
BqbJl
PBO CP
=
f-h f. 4?(~1tlp-q - %s,-,) q=o
We) WfJ
D, = al f ~qVh~p-q- Q2ap-q)
Pg)
HP = P2 f $q.8p_q
Wh)
q=o 900
M,= C,+ D,+ 2H, [Q”6,Y,_, i-(m/2)i2"'-2T,Mp_q] 4=0
Tp = f
(22i) (22j)
4 caption
theoryoften
ei~~~~t~
~f~t~
57
In terms of the above quantities the infinite set of linear second-order partial differ~ti~ equations are so = 0 (23a) 6, = -T,_:l;p > 1 1W
The
corres~nding stress components are
or
Consider a circular hole in an infmite media under unifo~ ~tip~~ne shear t, as shown on Fig. 2. For this problem it is convenient to use a polar coordinate system, where z = o(t) = t = r em.in this case the stresses are related to a stress function through the relations
FIG. 2. Iliustration of geometry and loading.
58
I.S.Tvs~
In this problem the characteristic load is defined as 0, = ,/3 . ‘c,. At first the elastic solution is obtained from
(234 It can be shown that F,, = +
(27)
is a solution of equation (23a) and furthermore, it satisfies the boundary conditions that at r = a, t,, = 0 and that r = xl, r,, = 0 and T,? = T,. The procedure will now be illustrated for m = 1 and p = 1. From equations (21), (22) and (26)
W-4 VW (27~)
A0
=
B, =
co = 6, =
$
(1+$+$)
a2 $J‘T+-
G
2a4 + 2%’
a6 pp
(27d) (27e)
>
Ho = 0
(27f)
(27h) Forp = 1, equations (23b) and (27h) provide the governing equation for the first perturbation which can be integrated to result a1 =
j+z$pg
a4 ;;_AL_
-~-3-
3z3z3
&
+ fb(r)
1 (28)
where f-,(z) and f;(Z) are arbitrary integration functions. They must be determined from the homogeneous boundary conditions, because the actual boundary conditions are already satisfied by F,. Homogeneous boundary conditions require that (29)
A perturbation theory of mttpk
elastic+ntic
59
&f&mudons
It is noted that once ai and /Ii are known the stress function F, is not needed, therefore, it will not be given here. At this stage a first estimate for the elastic-plastic solution is known. 7x, =
-2
ir 2
a2
z+ [
;;*I
E
s+gg+z+
+!+
a6
a4
a6
+=--a+-
5
a6
5
E_;$f+;.$L&
7 a2 3 a4 7 -_6z#g++Zm+6m-2ztP
a6
7
a2
3
a4
-_-
7
a2
a2
3 a*
3
--
1
(3Oa)
a4
The poIar form of these equations can be obtained easily using z = t ers and Z = r e-@ The shear stress 7ze at r = u, 6 = 0 is of special interest since it can be used to compute the maximum stress concentration factor. At this point z = f = a. From equation (30b) (31) Let us define the stress concentration
factor as K, =
7zemu
(32)
7
0 and a loading parameter as R =
7aJ71
(33)
71 =
Cl/J3
(34)
where
In this problem = 31~+222~-1 7a cc
7(1 + v)
2m
0<
(35)
Then K,=2-
99 a2 42(1 + v)
(36a)
For v = 0.3 K, = 2 - 1~81318681Iz2
(36b)
The above process can be repeated for higher values of p. Since the equations expand in size rather rapidly, only the stress concentration factors are presented here for the number of perturbations for which the calculations have been carried out. Form=l,n=3 K, = 2 - I.81318681 A2 + 18.0304311 L4
(37a)
1. s. -fURA
60
For M = 2, n = 5 K, = 2 - 10.18681318 L4 + 681.1586506 A8
(37b)
K t = 2 - 4590423861 A6 + 187WO207112 Li*
(37c)
K, = 2 - 194.230769241”
(37d)
K r = 2 - 802*71086061110
(37e)
For m = 3, n = 7
Form=4,n=9
Form=5,n=
11
It should be noted that these stress concentration yielding.
factors apply only to the initial stages of
5. FINAL COMMENT
A straight-forward perturbation theory is presented here for a deformation theory of antiplane elastic-plastic deformations. The method is convenient for the study of the initial stages of plastic deformation. For higher degrees of loading, a large amount of work would be required to carry out a detailed solution.
Acknowledgemenr-The
assistance provided by Miss C. I. Muchow is greatly appreciated. (Received 19 June 1967; revised 1 December 1967)
REFERENCES [1] W. RAMBERG and W. R. Oscoo~, Description of strewtrain
curves by three parameters. NACA tech. Nore
902 (1943).
R&n&--On presente une theorie des perturbations des deformations tlastic*plastiques perpendiculaires a un plan ; cette thCorie est bas& sur la th&xie totale ou deform& de laplasticiti. On donne formellementune solution complete. Pour l’illustrer on consid&e un trou circulaire dam un corps soumis a un cisaillement uniforme perpendiculaire au plan. Zusauunenfassaag-Auf der Basis einer totalen oder Dcformationstheorie der Plastizit%t wird eine Perturbationstheorie filr elastisch-plastische Verformungen einer Antif&& entwickelt Eine vollstgndige, formale Losung wird angegeben. Zur Erlauterung wird ein kreisformiges Loch in einem K&per untersucht, der gleichformiger Antillilchenscherung unterworfen ist. A6eTpalclL~peficTannrrRrcrr ~eopwt BoeKyrseKti ynpyro-nnacTwtecKKx aawwxuw~xx* ne+op~a@4 OCHOBBHH~H Ka TOTUMIO~~ Rw @oprawoluroP Teopsm uzacrw+xocrK. gaK0 noJutoe @p*anbKoe pemcaKe. AnlrKnnM:~p~apaccra’lpKBa~cI KpyroBoeoTBepcFrre~cperce KaxoWrWeticK B COCTOIWMH o~opo~Horo aaKTKllnOCKOrO* cmKra.
1. INTRODUCIION
IN this paper a perturbation theory of antiplane elastic-plastic deformations is presented based on a total or deformation theory of plasticity. In addition to the formulation, the complete formal solution is given. The method is illustrated for a circular hole in a body under uniform antiplane shear.
2. THEORJZTICALCONSIDERATIONS
2.1 Formulationfor Cartesian coordinates Consider an orthogonal Cartesian coordinate system x, y, z. For antiplane deformation all stress components are zero except r_ and r,,. Let us assume that there are no body forces, that the material is isotropic and is at uniform temperature. The three equations of equilibrium reduce to the forms
a%* _ -z-
()
(la)
k, az
o
(IV
-=
From the first two of these it is concluded that t,, and rs,. are not functions of z. The third equilibrium equation can be satisfied automatically by a suitably selected stress function F aF ? ‘“=ay
(W aF
T =),=
--
51
ax
(2b)
52
I. S. TUBA
There are two equations identically :
of compatibility
of the total strains which are not satisfied
a2Yzr a2Yzx
---=
ax2
o
(34
o
(3W
axay
a2Yzxa%_ axay
dy2
Let us integrate the first of these with respect to x and the second with respect to y.
ayl,.
---
ax
avzx =f ay
w
(y)
1
tiz, :; .=f2(x) --ay By comparison fr(y) = -f2(x) = c The arbitrary constant c is set equal to zero and a single compatibility obtained.
a/,
ah
-=
ax
o
ay
equation
is
(5)
The total strain can be expressed as the sum of the elastic and the plastic components : Yzx = [al
+ v) 4’ +
361 rsx
@a)
y*? = [2(1 + v) 4’ + 3#‘] ?zy
(6b)
where 4’ and 4” are the reciprocals of the elastic and plastic moduli, i.e. #’ = l/E
(W
The plastic modulus is a variable quantity which depends on the state of stress and on the shape of the stress-strain curve. For the Ramberg-Osgood [l] type of stress-strain curves (Fig. 1) it can be shown that (7b) where cr, is the stress at a secant modulus of 0.7 E and n is a shape factor. The effective stress, 6, is related to the stress components B, = J3 . [?E + rf).])
(84
In terms of the stress function F 6, = 43.
[E)’
+ ($)I’
W
+ (!Y)~‘“-l~‘z
(7c)
The reciprocal of the plastic modulus is then $1, = &
@2
A pertur&adon theory of ontipkme elosticplattic &fommtions
53
l-5
1-o
rr 3
04
0
I
,
1 l
2.0
l-0
h 9 Fffi. 1. Rambergspood
Let@=#“Eandm=(n-
stress-straincurves.
1)/2,then
[(gJ2+ @;1”
tl+ Yzx
=
d[2(1
+ v) + 3@]E
YZS= - f [2(1 + v) + 3@] s When the stress-strain relations expressed by equations (10) are substituted compatibility condition (5), the following governing equation is obtained : 3 V2F + 2(1 + v)
=
0
into the
(11)
Since d is a function of F only, equation (11) is a function of F only. Any function F, which satisfies equation (11) and some appropriate boundary conditions, will be an exact solution to the elastic-plastic antiplane problem. 2.2 Conversion to complex form It is more convenient to consider F as a function of the complex variables z = x + iy
54
I. S. TUEIA
and Z = x - iy instead of as a function of x and y. Let F(x, y) = FC(z,Z), then drop the subscript c. In that case the governing equation (11) can be written as
+0 --= 1 aF 2a2F
E
rz.x=1
. aF
BF
r,).
=
-
aF
( > (dz+z > x-z
aF
a22
0
(12) (134
(13b)
By definition
K
=
3m+2 .22m-1 7( 1 + v) ap
(14)
2.3 Orthogonal transformation of geometry Similar to plane problems of elasticity, an orthogonal transformation of the geometry can be carried out in order to simplify the process of satisfying boundary conditions. Let z = o(c), Z = m = E(r) then F[z,L] = FCC&), m)] = F&r). In what follows we shall drop the subscript t. The governing equation (12), transforms to the form a2F -+Ki$$
ar a7 2
+2Q
aF aF d2F
(154
Tqzfayae+“l
where a=
l w’(C)m’(T)
(15b) (15d (154
The stresses are related to the stress function as
Ifz=~theno’=G’==P=Q,=l,Q, = 0 and equation (15) reduces to equation (12) ; furthermore, equations (16) reduce to the forms of equations (13).
of antiplane elartic-j7lastic &formations
A perturbation theory
55
Led c = u + iv. The sole purpose of the transformation of the geometry is to have some constant values of u or v coincide with the boundaries, therefore, it is desirable to express the shear stresses along the u, v coordinate lines. The results are
i ‘52”‘=#
1
aF aF z - x [
(174 um
2.4 Introduction of a characteristic load For conveniences in obtaining a solution, only those cases are considered here where the stress function can be expressed as FK, 7) = 0X,
b
(18)
where 6, is defined as the characteristic load, which shall be specified in particular problems from the knowledge of the load distribution. Substitute equation (18) into equations (lSa), (16) and (17), then drop the subscript f. The results will be identical to those listed in the previous section, except that K is replaced by 3n+1pn
P
o
la
z-2
7U + v) 6> 1
(19)
and the various components of 7 by the corresponding component of r/a,
3. FORMAL SOLUTION BY PEzRTURBAT~ON
Equation (15a), is a second order non-linear partial differential equation. At the present there are no methods available for the exact solution of nonlinear problems of this type. Equation (Ua), as modified by the introduction of the characteristic load, involves a dimensionless parameter p. Setting p = 0, the equation (15a) reduces to the first term, which is the governing equation for the elastic case. For monotonically increasing loads, at first elastic conditions exist, then with continuously increasing load the solution gradually changes. For small values of aC/al, the parameter p is also small. It can be assumed that the stress function is an analytic function of the parameter p in the neighborhood of p = 0, and therefore it can be expanded in Taylor series form.
The coefficient of each power of p is an unknown function of C and <. First the series (20) is substituted into the modified equation (15), then the terms having equal powers of ~1are collected and finally the coeflicicnts of each power of p are set equal to zero. This process provides an infinite set of “linear” second order partial differential equations. The one corresponding to /JOprovides the governing equation for the elastic case, which can be used as a starting point. All other equations are nonhomogeneous and involve solutions corresponding to the lower powers of /.L Since the actual boundary
I. S. TUBA
56
conditions are satisfied by the elastic solution 0, = 0), all other F, for p > 1 must satisfy homogeneous boundary conditions. The above outlined solution is quite lengthy so the details are not presented here, but the results are summarized in such a manner that specific problems can be solved by following the outlined steps. For the sake of compactness let aF -_=c(= f a#; ap = -aFP al x P=o
Define *p = f
(224
cz4BP-Y
q=o
mp*,yo + ‘il (PY,=e!l;
I,
4)bb,-,Up, - tiqyp-ql
411
=
WW
PI0
(m - w,h
r, = I);--‘;
r, =
A,=ai;
A,=
‘$P- 4)Rm-
+
W,-,q
cl=1
(224
P#o %a40
+
‘fl
(P
-
4)
[zap-,A,
- IL,r,-,I
-
4=1
%A,-,1
(224 pa0
p-1
2PSpBo + c
B,=j3;;
@ - 4) cvp-,Bq
4x1
BP=
-
BqbJl
PBO CP
=
f-h f. 4?(~1tlp-q - %s,-,) q=o
We) WfJ
D, = al f ~qVh~p-q- Q2ap-q)
Pg)
HP = P2 f $q.8p_q
Wh)
q=o 900
M,= C,+ D,+ 2H, [Q”6,Y,_, i-(m/2)i2"'-2T,Mp_q] 4=0
Tp = f
(22i) (22j)
4 caption
theoryoften
ei~~~~t~
~f~t~
57
In terms of the above quantities the infinite set of linear second-order partial differ~ti~ equations are so = 0 (23a) 6, = -T,_:l;p > 1 1W
The
corres~nding stress components are
or
Consider a circular hole in an infmite media under unifo~ ~tip~~ne shear t, as shown on Fig. 2. For this problem it is convenient to use a polar coordinate system, where z = o(t) = t = r em.in this case the stresses are related to a stress function through the relations
FIG. 2. Iliustration of geometry and loading.
58
I.S.Tvs~
In this problem the characteristic load is defined as 0, = ,/3 . ‘c,. At first the elastic solution is obtained from
(234 It can be shown that F,, = +
(27)
is a solution of equation (23a) and furthermore, it satisfies the boundary conditions that at r = a, t,, = 0 and that r = xl, r,, = 0 and T,? = T,. The procedure will now be illustrated for m = 1 and p = 1. From equations (21), (22) and (26)
W-4 VW (27~)
A0
=
B, =
co = 6, =
$
(1+$+$)
a2 $J‘T+-
G
2a4 + 2%’
a6 pp
(27d) (27e)
>
Ho = 0
(27f)
(27h) Forp = 1, equations (23b) and (27h) provide the governing equation for the first perturbation which can be integrated to result a1 =
j+z$pg
a4 ;;_AL_
-~-3-
3z3z3
&
+ fb(r)
1 (28)
where f-,(z) and f;(Z) are arbitrary integration functions. They must be determined from the homogeneous boundary conditions, because the actual boundary conditions are already satisfied by F,. Homogeneous boundary conditions require that (29)
A perturbation theory of mttpk
elastic+ntic
59
&f&mudons
It is noted that once ai and /Ii are known the stress function F, is not needed, therefore, it will not be given here. At this stage a first estimate for the elastic-plastic solution is known. 7x, =
-2
ir 2
a2
z+ [
;;*I
E
s+gg+z+
+!+
a6
a4
a6
+=--a+-
5
a6
5
E_;$f+;.$L&
7 a2 3 a4 7 -_6z#g++Zm+6m-2ztP
a6
7
a2
3
a4
-_-
7
a2
a2
3 a*
3
--
1
(3Oa)
a4
The poIar form of these equations can be obtained easily using z = t ers and Z = r e-@ The shear stress 7ze at r = u, 6 = 0 is of special interest since it can be used to compute the maximum stress concentration factor. At this point z = f = a. From equation (30b) (31) Let us define the stress concentration
factor as K, =
7zemu
(32)
7
0 and a loading parameter as R =
7aJ71
(33)
71 =
Cl/J3
(34)
where
In this problem = 31~+222~-1 7a cc
7(1 + v)
2m
0<
(35)
Then K,=2-
99 a2 42(1 + v)
(36a)
For v = 0.3 K, = 2 - 1~81318681Iz2
(36b)
The above process can be repeated for higher values of p. Since the equations expand in size rather rapidly, only the stress concentration factors are presented here for the number of perturbations for which the calculations have been carried out. Form=l,n=3 K, = 2 - I.81318681 A2 + 18.0304311 L4
(37a)
1. s. -fURA
60
For M = 2, n = 5 K, = 2 - 10.18681318 L4 + 681.1586506 A8
(37b)
K t = 2 - 4590423861 A6 + 187WO207112 Li*
(37c)
K, = 2 - 194.230769241”
(37d)
K r = 2 - 802*71086061110
(37e)
For m = 3, n = 7
Form=4,n=9
Form=5,n=
11
It should be noted that these stress concentration yielding.
factors apply only to the initial stages of
5. FINAL COMMENT
A straight-forward perturbation theory is presented here for a deformation theory of antiplane elastic-plastic deformations. The method is convenient for the study of the initial stages of plastic deformation. For higher degrees of loading, a large amount of work would be required to carry out a detailed solution.
Acknowledgemenr-The
assistance provided by Miss C. I. Muchow is greatly appreciated. (Received 19 June 1967; revised 1 December 1967)
REFERENCES [1] W. RAMBERG and W. R. Oscoo~, Description of strewtrain
curves by three parameters. NACA tech. Nore
902 (1943).
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