023 WAVES IH FIUITS STRAIN
We consider a moving surface H ("t) t propa
gated into an isotropic elastic medium at rest in its unstrained position, over which there may occur a discontinuity In the density, the stress, or the relo
ci ty of the material particles. The dynamical rela
tions that must be satisfied along ^ , ** derived by Thomas / 3^ -J7» are
(
12.
1.
1)
1 2 .1 . Introduction.
(
12.
1.
2)
I ■ ■ *
where 2/ denotes the components of the unit nonal vector to 2 (t) , directed into the unstrained portion of the medium, and (> °j is the coordinate velocity of propagation of X In this direction. The barred
quantities f* and Vj- denote density and velocity eoaponents o!l the rear side of 21 snd the quantity
is the normal component of this velocity.
We use the double bracket (L 3) to denote the differ
ence in the values on the two sides of Z ( t ) of the quantity enclosed by the bracket. ?or definiteness, we take that the bracket is defined as the value of the quantity in question on the rear of 21 minus its value on front of this surface, this latter value will usually be zero due to the fact that the medium on the front
side of £; is unstrained and at rest. In such a ease the bracket expression will merely give an evaluation of the quantity enclosed on the rear of I . thus the above relations (1 2 .1 .2 ) are equivalent to the relations
<77- =, - f £ v . . (1 2 .1 .3 ) d
We shall assume that {£ u.( j| - o or, In other words, that there is no separation or sliding of the medium over the moving surface Z(t;! • The deformation i * t therefore, continuous over the surface X it) * It will*
however, be assumed explicitly that there is a dlscooti”
nuity in the spatial derivatives across "k. (jb) j
’ i . e . , not all of the quantities vanish. Such • surface [t] Is said to be singular of order one relative
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to the deformation, or to be a wave surface of order one, In the elastic medium. The object, here, ia to
find out Gj , the coordinate velocity of propagation of £ in case when the medium is subject to finite deformation and also to discuss some other aspects of the propagation of 2
1 2 .2 . Solution.
Since (t U-il) =o * i® follows from the compati
bility conditions of the first order derived by Thooas t 2)\ J that wa must have relations of the form
(1 2 .3.1)
where are functions defined over the surface X ( t ) Hot all the functions U>; can vanish from the aasump- tion that Z (tr) is a wave surface of order one. It i s , therefore, natural to define the strength VJ of this wave surface by the equation
\a/ — \J C*£ ~^L
(1 2 .2 .8 )We use the stress-strain relation
(1 2 .2 .3 )
where, unlike Xhonas L 39 -7, the strain tensor is given by
* e (12.2.4)
and X and jx are elastic constants. Using (12.1.1) and ( 1 2 .2 .1 ) , it can be seen that
«i - - ri o , . (1 2 .8 .9 ) f
With the help of ( 1 2 .1 .3 ) , ( 1 2 .2 .1 ) , ( 1 2 .2 .3 ) , (1 2 .2 .4) and ( 1 2 .2 .5 ) , we can write
■j
[* Sy0
V n ) - jK
^ ( uv i!k) + /«~ jL ( o k i>£ W K 1) ^ , f V u f . I f ;
(1 2 .8 .6)
where we have ased the identity
il^jj = faj]I ^ +1^1 v ^lUil
(1 2 .2 .7 )- ;'V!17S
M Cw* 3v )’ ’
X
+ /* ^ +'ua30
= |f • (12.2.8)
Case (i)
We assume that GJ^ 0 on * Multi
plying both sides of (1 2 .2 .8 ) by a)- and summing up os the repeated index i , we obtain
Simplifying ( 1 2 .2 .6 ), we have
Si = [k+ijk) ( I __ ^
r *- ^ pi / \ 'ij. a- (1 2 .2 .9 )
r “t t f
Case (i i )
We assume that
2
^ = o on Z".(j^ • Multiplying both sides of (1 2 .2 .8 ) by &v and summing on til*
L'
repested index i , we get
1 2 .3 . Discussion.
How substituting the value of 6, given by (1 2 .2 .9 ) into (1 2 .2 .3 ), we get
= caa>t. , c ^ = ^ £ 6 _
(12.3.1)
Combining (1 2 .3 .1 ) with ( 1 2 .2 .1 ) , we writs
£
(12.3.2)and
t at-L = l^ic as. - G, ,
(1 2 .3 .3 )
where C& 0 (case i ) . From ( 1 2 .3 .2 ) , it follows that
U •
v (1 2 .3 .4 )
Hence the rotation vanishes Immediately behind the wav®
front in the case of a wave whose velocity is given by ( 1 2 .2 .9 ) . Corresponding to waves whose velocity is given by ( 1 2 .2 .1 0 ) , the dialatation U.c ^ = 0 iaraedlstsly
177
behind the wave front* We call the wave In Case (1) as irrotational wave and the wave in Case (li) as equi*
voluminal wave. Accordingly we have
Q = ( a + i u ) F. ^ a> _r_ __ j k\a f
f *• f <*l»l r
(for irrotational waves)
(12.3.5) and
b* ~ A i r (for equivoluminal waves) (12*3*6) r
From ( 1 2 .2 .5 ) , we have
V-ft- for irrotational waves, (12.3*7)
= o ) for equivoluminal waves. (12.3*8)
Combining (1 2 .3 .7 ) with (1 2 .1 .1 ) and (1 2 .3 .8 ) with ( 1 2 .1 .1 ) we conclude that
’ j° - f £(.— ^ Irrotational waves, (12*3*9) P — P for equivoluminal waves, (12.3.10)
Jsing (1 2 .3 .1 ) and ( 1 2 .3 .9 ), we may write (1 2 .3 .5 ) in the form
Q / 0 ££ 0 —cc^ —
f f
( I 2 .3 .II )
from (1 2 .3 .9 ) and (1 2 .3 .1 0 ), It follows that density is continuous across the equivolumlnal wave but dlscooti~
nuous across the irrotational wave. Further from
(12.3.11), since W varies during the propagation of the wave, we conclude that in case of finite strain, the velocity of propagation of irrotational wave is no longer constant and consequently the successive positions of the irrotational wave surface do not form a family of parallel surfaces in space. Tor equivolumlnal waves, however, the velocity of propagation is still constant i . e . , sT/Mf* and the successive positions of the
equivolumlnal wave surface^ ^ [t) constitute a family of parallel surfaces. I f Co is small and CO<~0(.W) then from (1 2 .3 .1 1 ) it follows that the value of G*
reduces to (a *- ^ ) [(' 8S calculated in equation (4 .4 ) of Thomas' analysis.
179
jl-if. Summary.
The propagation of a wave surface of order one into an elastic medium, at rest in its unstrained posi
tion, is considered by following Thomas’ s / jty _ / method*.
When unlike Thomas, we retain the quadratic terms in the strain tensor, it is found that for irrotatiooal waves the velocity of propagation is no longer constant and consequently positions of the irrotational wave surface
T it) do not form a family of parallel surface* in space. ?or equivoluminal waves, however, the velocity or propagation is still constant.
Some of the assumptions that are made in order to derive the constitutive equations of anisotropic fluids may be possible to remove. It may take us a
little nearer to the real fluids which show both viscous and elastic effects. One may also include temperature and find out the corresponding system of governing equa
tions. No doubt, this creates difficulties in solving the problems. Infact, the proper formulation of the theory itself will become a Job. But such difficulties are always there when we try to approach real materials.
Another point worth investigation is to see whether the equation
is indispensable for describing the behaviour of ani
sotropic fluids or not. Sreen / (3 _/ has suggested that it is not so on the ground that it is an extra equation not based on well established principles of continuum mechanics. It is ofcourse admitted that th®
above equation i3 quite reasonable to take with the other constitutive equations. Following Noll's|/)idea Green
° proposes to use, in the constitutive equations of aniso
tropic fluids, the rotation vector ^ instead of vector
•n . Which of the two points of view is more useful to
181 -
describe anisotropic fluids can be decided only by the tractability of the problems and the new effects that one gets from them. Erickson's non-linear theory of anisotropic fluids, which we have used in solving the various problems in this thesis, has many advantages.
It is not merely simple and properly Invariant but is also capable of describing the behaviour of Bingham type materials with the help of a single set of constitutive equations unlike the theories of Oldroyd and others.
This theory may also be applied to the ideal plastic flow problems.
The problem part of hypoelasticity is not ye^lt fully developed, perhaps, due to the difficult algebra involved in finding a consistent solution to the ten non-linear differential equations. There are
i
^J
a few solutions available for grade zero but they are approximate. In the present thesis, we have been able
to present solutions only to simple problems. We have not proceeded beyond grade one. Even in grades zero and one, one faces a lot of trouble. In the case of siaple extension problem, we make an interesting observation viz, the shear stresses, i f once non-zero, will develop with time. It has however not been possible to obtain *
* solution with non zero shear stresses and at the same time consistent with the basic equations.
The question still remains open.
It will be worth examining the classical prob
lems of torsion and flexure with the help of hypoelastic equations without negleotlng the inertia terms. Due to
the presence of rate of strain in hypoelasticity appli
cations to plasticity can also be obtained on the lines of Green / _ /• A couple of them have been included in this thesis. The experiments £ 5j _ / are yet to prove whether hypoelastic aquations can really determine the yield point in a certain material. Another direction in which one can think of working is to analyse the thermal stresses sat up in hypoelastic medium with the help of governing equations. We have attempted to for
mulate such a set of equations in the present thesis.
Statical and dynamical problems need be solved on this topic.
It has also been possible to find the deformation energy for hypoelastic bodies of different grades
Bricksen £ >~l _ / derived the conditions under which hypoelastic potential can exist. Stipulation of exis
tence of a strain energy function in general i s , however, excluded on the grounds that we do not assume in hypo
elasticity that there is no dissipation through any fion- mechanical phenomena. Natural state is not the suppo
sition in hypoelasticity. One measures the strain with
183
reference to immediately preceding state. All these questions require a ftirther probe.
Wave propagation in an hypoelastic medium may be studied on the lines of Thomas £ ^ _/• Some work in this direction has been done by Hariboli J_ ^5 _ / quite recently. In a few special cases, it may be possible with the help of this technique to predict fracture.
Nariboli / 6) _ / has shown it for a hypoelastic bar.
Linear fluent materials include viscous fluids and non-Newtonian fluids as special cases. I f it is possible to get solutions to problems without assuming certain restrictions on the coefficients, it may be possible to Infer some general results. Again the difficulty lies in the complicated and intractable
algebra involved. But the resilient and fluent bodies, as presented her®, give a good link between solid and fluid states.
Various non-linear theories like non-linear ther
moviscoelasticity or electroela3tostates or thermo- magnetoelasticity or magnetohydrodynantics have come up recently. In all these theories one is concerned with the interaction of different fields. Here there is good
* scope for future research. In this thesis, we have also included a few problems on this type of interdiscipline
work. For instance we have seen the effect of polari
sation on the symmetrical expansion of a hollow spheri
cal dielectric under internal and external pressures and subject to large deformation. In another problem, we examine the effect of a uniform axial magnetic field on the torsional vibrations of a right circular cylinder of infinite conductivity. Since a body possesses to some extent all rheological properties, it is expected that a
study of interaction fields will be more suitable than a single theory for describing its behaviour under different types of forces.
There is yet another field open for research. In elasticity theory, it is generally assumed that the forces across a plane elemental area placed at a point inside a solid reduce to a single force Instead of a single force and a couple. Now i f this couple is taken as non zero,
the stress tensor becomes asymmetric and we get what is called coupled stress theory. Very few problems have so
far been attempted in this branch of solid mechanics.
Here again one faces non-linearity.
What is most needed to-day is a new calculus that can render the non-linear differential equations amenable to solution. For this purpose, we may have to define our derivatives afresh or an Integral approach may be given.
One may suggest the use of computers as Is being done
v>
now-a“days but then the range In which the solution is valid is highly restricted.
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