1951] P. G. HODGE, JR. 381
5. Conclusion. The methods presented above have been applied to more complicated
frames and readily give the required solutions for collapse design under constant or
varying loads. For shakedown design, the iterative numerical method converges fairly
rapidly. It may be found easier for highly redundant frames to obtain a new elastic
solution at each stage using the numerical values obtained from the previous analysis,
since an analysis with numerically unspecified flexural rigidities is extremely tedious.
For all three types of design, the introduction at an early stage of the numerical values
of the loads simplifies the work greatly, since it will be found that a large proportion
of the inequalities generated become redundant and can be ignored.
Only examples of concentrated loads on straight members of uniform cross-section
between joints have been examined, making it possible to pick by inspection the critical
cross-sections. However, the basic ideas are not altered by the introduction of other
variables; the analysis will be more complicated, but aids to calculation may be in-
troduced which leave the basic problem unchanged.
THE METHOD OF CHARACTERISTICS APPLIED TO PROBLEMS OF
STEADY MOTION IN PLANE PLASTIC STRESS*
By P. G. HODGE, JR. (University of California at Los Angeles)
A method is outlined for obtaining the stress, strain, and thickness distribution in a
thin sheet which is strained plastically in its plane. For the particular case of steady
motion, a method is given for obtaining directly the final solution to certain types of
boundary value problems. A step by step procedure is indicated for the general case of
non-steady motion.
1. Introduction. This paper is concerned with the stress and strain distribution in
a thin sheet which is strained plastically in its plane, under conditions of plane stress.
It will be shown that three types of problems may be distinguished. In certain special
cases the stress distribution may be found independently of the velocity or thickness
by solving three equations in as many unknowns. For general steady motion problems
it will be necessary to solve six equations simultaneously for three stress components,
two velocity components, and the thickness. These equations will be stated in Sec. 2,
and reduced to a system of five first order, quasi-linear differential equations under the
assumption of initial isotropy. For suitable boundary conditions, it will be possible to
find the final stress, strain, and thickness distribution of the material directly, using
the method of characteristics. The details of this method of solution will be described
in Sees. 3 and 4. Finally, in Sec. 5, a step-by-step procedure for solving problems of
non-steady motion will be briefly indicated.
2. Basic equations. Let the sheet be referred to a set of Cartesian axes such that
the x,y plane coincides with the middle surface, and let z = ±%h{x,y) be the equations
of the bounding surfaces. Under the assumptions of generalized plane stress, the only
non-vanishing averaged stress components are ax , ay , and rxy . These components
must satisfy the equations of equilibrium
*Received March 29, 1950.
382 NOTES [Vol. VIII, No. 4
|~(.h*x) + = X, (la)
|'(hrxt) + -~(hav) = F. (lb)
Here X and F are the components of body force in the x and y directions, respectively.
They may be functions of x, y, and h.
In addition, if the material is to behave plastically, the stress components must
satisfy a yield condition which may be written in the form
F(<rx , <jv , txv) — 0. (lc)
Strictly speaking, Eq. 1(c) must be satisfied by the actual stress components. However,
if the variation of the stress components across the thickness is not too great, it will be
satisfied by the averaged stress components to a high degree of accuracy.
In the particular case where h is known, and where the boundary conditions in
stresses are sufficient to set a fully plastic problem, Eqs. l(a-c) may be solved directly
for the stresses with no reference to the velocities.1 However, in general neither of these
conditions will be satisfied, and it is necessary to consider the stress-strain relations.
According to the "plastic potential" stress-strain law,2
du/dx dv/dy _ du/dy + dv/dx
dF/dax dF/day dF/drxv
Since F is a known function, Eqs. 1(d) provide two additional equations, but introduce
two additional functions u and v, the averaged velocity components in the x and y
directions respectively. The final necessary equation is obtained from the assumption
that the plastic material is incompressible. For steady flow, this condition may be
written in the form3
|(*») + - 0. (le)
Equations l(a-e) are a set of six equations for the six unknown functions <rx , <rv , txv ,
h, u, and v.
It will prove convenient to make the substitutions
ax = 2k[u + X sin 20], txv = —2kx cos 29, )
> (2a)
<jy = 2&[<u> — xsin 20], H — log h. )
The function 6 may be interpreted as the angle between a fixed direction and a principal
direction (see, for instance, Ref. footnote 1), so that if the yield condition is isotropic,
it may be written
F(<*x , <ry , txu) = x — /(w) = 0. (2b)
'This has been done by the author in another paper. P. G. Hodge, Jr., Yield conditions in plane plastic
stress, to be published in the J. Math, and Physics.
2R. v. Mises, Mechanik der plastischen Formaenderung von Kristallen, ZS. angew. Math. Mech., 8,
161-185 (1928).
3R. Hill, Plastic distortion of non-uniform, sheets, Phil. Mag. (7) 40, 971-983 (1949).
1951] P. G. HODGE, JR. 383
The substitution of Eqs. 2 into Eqs. 1 leads to the five equations
(1 + /' sin 26) ux —/' cos 26 uv + 2/(cos 26 6X + sin 26 6U)
+ (to + / sin 26)HX — / cos 26 Hy = X,
— f cos 26 ux + (1 — /' sin 26)uu + 2/(sin 26 dx — cos 26 0„)
— / cos 26 Hx + (co — / sin 26)HV = Y,
ux + vy + u Hx + v H„ — 0, (3c)
— 2 cos 26 ux + (/' — sin 20)(w„ + vx) = 0, (3d)
(/' + sin 20)(w„ + vx) — 2 cos 20 vv = 0, (3e)
where subscripts now indicate differentiation. Equations 3(a-e) are a set of 5 quasi-
linear first order equations for the 5 unknown functions w, 6, u, v, H. Once these quantities
have been determined, the stress components are easily found from Eqs. 2.
3. Characteristic equations. The characteristics of Eqs. 3 may be defined as those
curves across which it is possible for the derivatives of the unknown functions to exhibit
finite jump discontinuities, the functions themselves being continuous. Let x = x(s),
V — y(s) be the parametric equations of a characteristic curve, and let «, 6, u, v and H
be given along the curve as functions of the parameter s. Then, along the curve,
ux dx + Uy dy = du, vx dx + vv dy = dv, (4a)
ux dx + uv dy = du, 6X dx + 6y dy — dd, 1
> (4b)
Hz dx + Hy dy = dH. )
Equations 3 and 4 may be regarded as a set of 10 linear algebraic equations for the 10
unknowns ux , co„ , 6X , • • • , Hv . In general they will possess a unique solution, so that
a discontinuity in these unknown derivatives will not be permitted. An exception can
occur only if the slope of the curve, dy/dx, is such that the determinant of the coefficients
D0 vanishes. By means of Laplace's expansion,4 the 10 X 10 determinant of D0 may be
reduced to the product of three determinants:
where
D'0 =
D0 = Da-D" • Do",
1 + /' sin 26 —/' cos 26 2/ cos 26 2} sin 26
—/' cos 26 1 — /' sin 26 2/ sin 26 —2/ cos 26
dx dy 0 0
0 0 dx dy
D'o' =
dx dy
4See, for instance, A. C. Aitken, Determinants and matrices, Oliver and Boyd, London, 1939.
384 NOTES [Vol. VIII, No. 4
and
D'0" = -2 cos 26 f - sin 26 f - sin 26 0
0 /' + sin 26 f + sin 26 —2 cos 26
dx dy 0 0
0 0 dx dy
Do and D'0" each vanish along either of the curves
dy/dx — —cot a, dy/dx = tan /3, (5a)
where the substitutions
/' = sin 2\f/(\ ip ] < ir/ifi), a = 6 + x, = 6 — X
have been made. Obviously these characteristics are real only if | /' | < 1; n the re-
mainder of this paper it will be assumed that | /' | is actually less than one.5
Each of the curves 5(a) represent a multiple characteristic. The final family of
characteristics is obtained by setting D" = 0:
dy/dx = v/u. (5b)
The curves 5(a) are called the first and second characteristics, respectively, and the
curves 5(b) are called the streamlines.
4. Conditions along the characteristics. The problem of finding conditions which
must be satisfied along the characteristics is complicated by the existence of multiple
characteristics. It will prove convenient to use one method for Eqs. 3 (d, e) and find the
velocity conditions along the characteristics. The velocity derivatives will then be
regarded as known, and a different method applied to yield the stress relations along
the characteristics and the relations along the streamlines.
Consider the characteristic curves (Eqs. 5a) as curvilinear coordinates £ = const.,
rj — const., where £ and ij are functions of x and y. It is easily shown that the relations
1]x sin a + 57,, cos a = 0, cos /3 + £„ sin /3 = 0, (6)
must be valid along a first and second characteristic, respectively.
The substitution of Eqs. 6, together with the relations ux = u£x + , etc. into
Eqs. 3(d, e) leads to the equations
sin /3[m£ sin a — cos a] — cos a[w, cos /3 + y, sin /?] = 0, i
\ (7)
cos a cos2 /3[u( sin a — v( cos a] — ijx sin /3 sin2 a[u„ cos /3 + y, sin /3] = 0 J
If Eqs. 7 are regarded as simultaneous linear equations for the expressions in square
brackets, it is seen that the only solution is the trivial one. In view of the definitions of
£ and T], this may be written as
du sin a — dv cos a = 0 (8a)
6The meaning of this restriction has been discussed by the author (see footnote 1) and by R. Hill
(see footnote 3).
1951] P. G. HODGE, JR. 385
along a first characteristic, and
du cos p + dv sin /3 = 0 (8b)
along a second characteristic.
To find the relations between co, d, and H along the characteristics and streamlines,
let us assume that u and v and their derivatives are known from Eqs. 8 (a, b) and con-
sider Eqs. 3(a-c) as a set of three equations for the three unknowns co, 6, and H. The
characteristics of these three equations will be those curves for which the determinant
of the coefficients of cox , • • • , Hy vanishes. Let this determinant be denoted by A0 ,
and let Ak be the determinant formed by replacing the fcth column of A0 by the inhomo-
geneous terms (i.e., the terms not containing cox , • • ■ , //„) of Eqs. 3(a-c) and 4(b).
In order for the derivatives cox , • • • , //„ to be able to exhibit finite discontinuities across
the characteristics, it is necessary for all of the determinants A0 , , • • • , A6 to vanish.
The first condition leads to the three curves given by Eqs. 5, of course, and the re-
maining conditions lead to the relations
(u cos a + v sin a)[2/ dd — cos 2\p du — (X sin /? — Y cos 0)(dy/cos a)]
= [(u cos /3 + v sin 0)co + (—w sin a + v cos a)/] dH (8c)
+ («, + f„)(/ — co sin 2if)(dy/cos a)
along a first characteristic,
(u sin /3 — v cos /3)[2/ dd + cos 2ip dw — (X cos a + Y sin a)(dy/sin /3)]
= [(—u sin a + v cos oi)u + (u cos /3 + v sin /?)/] dH (8d)
+ (ux + vy)(f — co sin 2^)(%/sin 0)
along a second characteristic, and
(ux + vu) dy + v dH = 0 (8e)
along a streamline.
5. Conclusion. Equations 5 and 8 may be replaced by finite difference equations and
used to obtain numerical solutions of boundary value problems. There are a great num-
ber of possible such problems, corresponding to various combinations of the Cauchy
and Riemann problems for a system of two hyperbolic equations. The details of the
method may be readily adopted from corresponding treatments in compressible flow.6
For the general case of non-steady motion, it is necessary to first solve the problem
at the instant t0 when plastic flow begins. Since II is an initially known function, this
involves solving Eqs. 3(a, b, d, e) for u, 9, u, and v. This problem is similar to the one
discussed in the preceding sections, and, in fact, (5a) and 8(a, b, c, d) may be used,
since they are not dependent upon the motion being steady. The right hand side of
6See, for instance, W. Tollmien, Stationaere ebene und rotationssymmetrische Uberschallstroemungen,
Technische Hochschule Dresden, 1940. This work has been translated by the Air Materiel Command,
Wright Field, Dayton, Ohio as translation A9-T-1. Contract W33-038-ac-15004 (16351), Brown Uni-
versity, Providence, Rhode Island.