PART II PART II
CHAPTER 71 CHAPTER 71
STEADY FLOW FORMATION OF LINEAfl FLUENT B0DI2S WITH SXTIOK
6 . 1. Statement of the problem.
We consider a steady flow of an Infinite mass of incompressible linear fluent material due to the motion of an infinite plate parallel to itself with uniform
velocity, \/ , in the case when a constant normal velo
city U of suction is imposed on the moving boundary.
We use the system of rectangular Cartesian coordinates fixed in space such that y-axis lies along the plate and the x-axis is along the normal to i t . Thus all the
physical quantities may be taken as Independent of ^ . The velocity field can be written in the form
U ,
^ V (X ) , U ) ,0 , ( 6 _U 1 )where U is a given constant. The boundary conditions O-(o) a V , xr C00) = o will be found to agree i f we choose 1)- (x) as follows
= \J e , . (6. 1. 2 )
where 4 is a constant to be determined. Thus we have
u.= U , ^ = \ / e ^ x , |-or= o -
, (6 .1 .3 )
We shall determine the stresses in the interior of the fluid, consistent with the basic equations and their vslues at the plate.
6 . 2. Reduced form of the basic equations.
The equation (5 .3 .3 ) is identically satisfied.
From (6. 1. 1 ) , we calculate
i- V
0 O’
li Vr o
( 6 . 2 . 1 )
where dash denotes differentiation with respect to X . We assume the extra stress tensor, f , in the fora .
T -
(
6.
2.
2)
76
where ^ ^ , V and S *re all ftinctiona of X.
only. The constitutive equations (5 .3 .1 ) reduce toS
(6 .2 .3 )
U v + ( ^ l4 ^ - i ) i
U % = X11/M +• ^ i / + - » 's ,
3 3 y ( 6 .2 .4 )
U <|) = \ °i/. ^ i. 'V + * * ( < , ) ,
(6. 2.**)
(
6.
2.
6)
where ^ ^ . The equations of motion ( 5 .3 .2 ) reduce to
) (6 .2 .7 )
i l
provided that there are no external forces and there le no gradient of ^ with respect to ^
S =. f U 1)- t !>;
(6 .3 .1 )
where is to be determined. After some easy calcul*' tlons, from equations (6 .2 .3 ) to ( 6 .2 .p) , (6 .3 .1 ) and (6 .1 .3 ) we get as follows:
H
^ = 1 1 VI* + l M n t) U ' lJ _le + r &V*
^{ili + (x,v j XO If 1] + (2 u l*
-J
(6 .3 .2 ) + A , l f ) -Xv ( i ^ + 3 ^ ) x
c 'e X|U^
+. | c I - ', ^ f
X & + ^A(+-3At.) U. ' j j
v y ~ * v i ir' [tffci-ji, u f y V 4 * *
t , M C ^ t ^ ^ T e ^ J + c ' e ^ '
( 6 .3 .3 )
(6 .3 *4 )
I W t X ^ x V v u , C y ^ , ^ -
}
0>c 'i^AK.
- f U H e * ^ K, ( f l l l l t K l . ) _ X V t e l , x | M , + ^ [ * U * « ' lu' l'‘
x (1.^+ U + itCi ) K
* a-1
(M^oflc
* { ( V <>) (^ + u_l > - ) - x ,. ( u J t w l) U 'l (;(t).x,u‘ ' ) * [ l +
+ )U .'] *j+if* v1*!
- ^(>-<Jt w , ) U . ' , ( » . U >, y . - ' j l U W ^ j + t ' t ^
^ i c f * 3i"->‘ lu j + |--J ft | /t e U a-|( V \ > r l)H -
- 2f U-l)_l+ C, ek,uHl]|
( 6 .3 * 0 )
On comparing the two sides of ( 6 .3 .P ) , we obtain
f 0 4 ^ -f U h, + (6.3.6)
X l / ^ t v * > ) - t v ^ ( t ^ u j I]
^ (6 .3 .8 )
Using these results, we get the extra stresses subject to ( 6 .3 .6 ) and ( 6 .3 .7 ) ,
-l|x , « .-I V - x (j 4u+-x,)
- U* V 1**)], (g .3. 9 ) j = f U V li i1 ' x ( * ( , u a , ) 1 f e w j t l ) -
X(jtt U-f A,+ J
-\
(6 ,3 .1 0 )
S ' *
j
9u v £.
- {(6 .3 .1 1 )
( 6 . 3 . 1 2 )
Considering equation ( 6 . 2 . 7 ) , w® write i) —
\
, where constant is taken to be zero. (6 .3 .1 3 ) Thus the stresses are- 80
(6 .3 .1 4 )
(6 .3 .1 « )
(6 .3 .1 6 )
(6 .3 .1 7 )
6 .4 . Remarks.
As x approaches infinity, all the stresses tend to zero. The normal stress In the direction perpendicu
lar to the plate is zero throughout the medium. I?
= I } will disappear. The values of $~~L"
and S*'} over the plate are respectively.
\ f |
i f U V L^ (a til A , ) f U . A ( W 3)
HYPOSLASTIC BODIES
7 .1 . Introduction.
In continuum mechanics, we are Interested in
knowing the response of a material body to forces which , tend to deform it. This response depends, through the equations of motion, upon the stress tensor. The geo
metry and the physical state of the body are character
ized by other kinematic and thermodynamic variables. A relation between the stress tensor and these variables is known as the constitutive equation of the medluB*
Various authors like Hooked 1678), CauchyW(19£3.), Hsvier&O (1 8 2 7 ), Green(*l(l939), Maxwell[«)(l867), Zaremba[^(1903),
Seth^(193S), R e i n e d 1948), H lvU n^(1948), Old royd0(1950), TruesdelljjijC 1955) and K o ll^l955) have made attempts to
formulate properly invariant constitutive equations to explain, (at least qualitatively), the different physi
cal phenomena Involved. It has not been^however, possi
ble, as pointed out earlier to arrive at any one set of constitutive equations which could explain all the
82 -
physical phenomena exhibited by the material in different states between solid and fluid. Noll gave the genera*
Used approach of fluent and resilient bodies explained in chapter V. Here, we shall 4eal with resilient bodies of which hypoelastic bodies are particular cases. A material is hypoelastic i f and only i f , for a given ini
tial stress, the stress at a final state depends only on the paths by which the material points reach the final state and not upon the rate at which they trawverse these paths. The constitutive equation for a resilient body ° L /8 J is
S + S W - W S s . J (l), S > p ) ^ (
7
.1
.1
)where s , W , D , p are stress, vorticlty, rate of deformation and density respectively and the dot denote*
the material derivative. In ( 7 .1 .1 ) ^ is such that 3- (o ,
S
, (°) = 0 for a certain range ofS
and fWe consider a resilient material for which ’jr doe* not depend on p so that ( 7 .1 .1 ) becomes
S + S W - W S = 9 ( 3), S ) . ( 7. 1. 2 )
$ + ■ S W - W S = t J i t K S f ' 1), (7.1.3)
where ( m) = M T LT* , f t ) s. “ f . If the equation (7 .1 .3 ) does not contain a material constant of the dimension of time, the material is called hypoelastic.
This is possible only i f X cancels out on the right hand side of ( 7 .1 .3 ) . This implies that 'J- is homoge
neous of degree one in 3) . Since 3 is assumed to be continuously differentiable in a neighbourhood of
]> = o » it follows that 3- (.D, S ) must be linear in 1) . Using a result of Rivlin and Sricksen / 4 | the equation ( 7 .1 .3 ) can be written as
S + S\
a) - W S - 3) -j- ( S ]> +-D s^) + fc> 3 ( S\j> + 3) s1-)
+ ( k i) f 3>sl) } l
* I
^ ^ S•)] s)
( 7 .1 .4 )
84 -
where f)-£ are dlmensionles3 analytic functions of the principle invariants of £ alone. In the case of an incompressible body', $ has to be replaced by '
T - S + |> X • I f we put ]) - o in 1} , then
^ S) = 0 . S o hypoelastic materials are automa
tically resilient bodies. Each choice of define#
a special hypoelastic body which corresponds to a possi
ble physical model, no idea of approximation being necessarily implied. ?or instance, for grade zero, we have
b, = I , = ^ # ; k r M v t )
and for grade one
b,= 4 , I l , 1) = ( X U / ) + O , ks-= ^ > 1*3 = ^ - b “ - = b
(7 .1 .8 )
(7 .1 .6 )
where <1* A are constants and j" is the first prin
cipal invariant of
Infact, in case of grade one, we take linearity in S as well. We can call the bodies of grade on*
as linear hypoelastic bodies. We can similarly define
higher grades* In addition to the constitutive equa
tions, one must use the equations o f motion and the equation of continuity* Thus we have ten equations for
• i
ten unknowns , v and jp ,- six constitutive equations, three equations of motion and one equation of continuity. These ten non-linear partial differential equations are solved in a particular problem, in a . manner conslstant with the initial and boundary condi
tions.
7 .2 . Main features of hypoela%ticity.
Some of the salient features of hypo-elastleity are as follows:
i ) In any given problem, there will be a relation between stress and strain but here its form is not
explicitly known in the beginning. It is obtained after solving the system of differential equations and hence depends entirely on the Individual problem we are going to solve. These differential equations are non-linear.
Most of the solutions sought are, however, exact. Ins£i*jxd, of neglecting inertia terms, we are interested in know
ing what part these terms play In hypoelastlc solutions.
i i ) ftn analytic difference from the theory of elas
ticity comes in the appearance of I in addition s
86
to <S in the constitutive equations. Thus in addition to boundary conditions such as are usual in elasticity, we may expect to be able to satisfy the condition of arbitrary initial stress \ S = S when i s o , where
S is any solution of the elastic problem.
i i i ) Exploration of th® phenomena indicative of yield is possible as shown in some examples by Green
and Truesdell / £| _ / .
iv) For some hypoelastic materials it is possible to introduce a scalar potential analogous to the strain
energy function in elasticity. Conditions necessary and sufficient for the existence of such a potential have been derived / 5~l __/• In fact we gave / £~J _ / the expre
ssion for deformation energy of hypoelastic materials of grade zero for the first time. Later on, Sinha deduced the expressions for higher grades.
v) All isotropic Cauchy-elastic materials for which the stress strain relations are invertible are hypoelas
tic but the converse is not true.
vi) Since the theory connects only rates, it gives
* the actual motion experienced by the £ody. There is no natural state assumed for measuring strain. It is mea
sured from the immediately preceding state.
7*3* Hypoelasticity and elasticity.
Elasticity is characterized by a stress-strain relation of the form
T ^ “ _ f A m , * h „ \
' r (£) . (7-3-i>
where 7" the ordinary Cauchy stress tensor, f and P are the densities in the actual and the reference
I . 4 i / j. L < , i s
configurations, x ^ - dx I d K , x = x L K ;x ) being the deformation, and t is a symmetrised function of the components of the symmetric tensor C , where
= p, . In order that the elastic
material be isotropic, it is necessary and sufficient that ( 7 .3 .1 ) reduces to a relation of the form
T = -f C % ) j ( 7 . 3 . 2 )
where £ is an isotropic function and where B h*a trie components jj>^ >v' = ^ * Dif cerentia~
ting ( 7 .3 .1 ) materially w.r.to time, we easily show that
t v\ t kn dJ - t - T M - ^ " K
2> ,
°\! >1/ 1 • ' V n
( 7 . 3 . 3 )
88
where X is the velocity field, 3) is the stretching
tensor ^ „ d
fxy ^ w i1 A
* yj * , x , X r * * V ;y • <7 C7.3.4)^
k ‘ i C rl
ItVK bfly
^or a given elastic material, the components C
are given functions of the finite strain and rotation.
Hypoelasticity is defined by an equation of the fOTO
-r . . W| *-p . . ft in
T - T
1- V I ^ - T > WV - H ^ (
7.
3.
5)
where l/\j is the spin Ha) ^ - *nd w^®r®
the components n r are given functions of the stress. Moreover, in order that (7 .3 .5 ) satisfies the principle of material indifference, it is necessary and
sufficient that be an isotropic func
tion of ~f and 2)
Sow both ( 7 .3 .3 ) and ( 7 .3 .5 ) may be written in the fora
" ^ J ( 7 . 3 . 6 )
but the functional dependence of 'J) is different in two cases.
In hypoelasticity, 1) is a given function of stressi in elasticity, 3) depends, in general, also on
tho strain and rotation from a reference configuration.
Elasticity will be Included in hypoelasticity only when the dependence of J on strain and rotation turns out to be merely apparent. For an isotropic elastic mate
rial, by substituting (7 .3 .2 ) into ( 7 . 3 .4 ) , we find
^ p s ' ( 7 . 3 . 7 )
hence C is a function of b and T " only. There
fore i f ( 7 .3 .2 ) is invertible, C is equal to a func
tion of T only and so is ]) . Thus Boll's theorem follows. However the common domain is not exhausted by Noll's theorem, since an easy calculation shows that elastic fluids with invertible pressure functions [(f)
* are always hypoelastic, although for them the stress strain relations ( 7 .3 .2 ) are never invertible.
• 90
However, no material that has a natural state and is anlsotropically elastic in small deformation froa i t , is hypoelastic. Svery hypoelastic material responds to infinitesimal strain from an unstressed state like an isotropic elastic material. The domain common to elas
ticity and hypoelasticity has thus been nearly delimited.
?or materials having a natural state, it is no larger than that of isotropic elasticity, at least for small strains. It is no smaller than that of Isotropic elas
ticity with Invertible stress-strain relation. Bfcrpo- elasticity includes elastic fluids I f their pressure functions are invertible (these have neither a natural state nor an invertible stress-strain relation. o|
-flu. itvOa
M aso ovfcc.
1 •
7 .4 . Hypoelasticity and plasticity.
Theories of plasticity for perfectly plastic and work hardening materials have received considerable attention in recent years L and «*ny applications have been made to practical problems. Remaining within
mm
bounds of incremental laws, Green / 5« _J has indicated the relation between hypoelasticity and plasticity. He has shown that several of the theories usually proposed for plastic bodies are included as special eases of limit cases within hypoelasticity. Using a special esse
of hypoelasticity, he has developed a general theory of work-hardening plastic materials, both compressible and incompressible. As in most theories of strain hardening he has established different constitutive equations for loading and unloading, imposing conditions on the hypo
elastic coefficisnt functions in order that the two sets be compatiabl®. Green / 5"^_/ draws special attention to the materials for which the stress-logarithmic strain curve for unloading in simple extension is linear. In simple extension problem, he finds that the stress
system is not a state of simple tension and is not homo
geneous. It depends on the strain and the rate of strain.
He obtains approximately a stress which tends ■onotoni- cally to a finite limit as strain increases, implying a gradual transition from the elastic to plastic region.
For the same special equations, the time derivative of the stress intensity reaches a certain value, suggesting a yield of the Maxwell Mises type. .4 similar observa
tion was made independently by Thomas / Sty _ /• Similar phenomena were found by Green for simple shear. Wa also discuss in this thesis a couple of problems on the line*
of Green. They are (i ) solid rotating shaft and (i i ) Pure shear. Our conclusions regarding the transition from
elastic to plastic range are also found to be similar to those of Green. There is a good scope for using
- 92
hypoelasticity in the treatment of problems of fracture given by Thomas / 3j _ / . The constitutive equations
for loading, as established by Green J_ S\ _/» for incom
pressible hypoelastic work hardening materials are
■*" " " t (j (7 .4 .1 )
where M d . >
t 1 J
tion tensor,
being the rate of deforma-
{[ - s- - 1 c L <;*-
r ) j (j s
*■ ’end
?r
& (7 .4 .8 )The equations of motion and the equation of continuity together with appropriate initial and boundary condi
tions should be the other basic equations besides (7 .4 .1 )
7 .5 * Bemarks.
There have been a number of solutions Jmv*>x& Q sr,s%
for hypoelastic materials of grade zero. Some of these
are presented here as well. There i s , however, a little improvement that can be made by taking resort to prob
lems of grade one. The equations of hypoelasticity of grade zero become the special equations of hypoelasti
city of TBaaaam Qte -ttpagfta* auatehomi «P tofftp- e&m M eb tjp of higher grades under different definitions of rate of stress. That is to say that the equations of hypoelasticity of grade zero are not independent of the choice of convected time flux. So we present some solu
tions for grades zero and one, in the next chapter.
While doing the problem of simple extension, we make an important observation that has a bearing on the theory as such. The question is still open. In other problems, we find that the departure from ordinary theory of elas
ticity lies in 3ome additional stresses and also in finding out th® effects of inertia terms. Method used to solve the various problems is the semi-inverse method.
The equations are non-linear and whatever solutions are of some significance have been given. Other solutions may also be existing but we are not interested in them.
In chapter IX, we give the derivation of deformation energy of grade zero and point out that this can further be found for higher grades.