On some mathematical models in nonlinear elasticity

Nonlinear

Analysis, Theory,

Pergamon

Methods&Applications, Vol. 30, NO. 8. pp. 5101-5108, 1997 Proc. 2nd World Congress of Nonlinear Analysts 0 1997 Elsevis science Ltd

printedin Great Britain. AU rights reserved 0362-546X/97

PII: so362-546x(!%yJo146-0

$17.00 + 0.00

ON SOME MATHEMATICAL MODELS IN NONLINEAR ELASTICITY* GIOVANNI PROUSE Dipartimento Piazza Key words

di Matematica

Leonardo

de1 Polite&co di Milano da Vinci 32, 20133 Milano, Italy

Mathematical models, Nonlinear elasticity, Hamilton’s principle, Consistency significant solutions, Approximable solutions.

and phrases:

ditions, Physically

con-

1. INTRODUCTION

The aim of the present paper is to introduce and study some nonlinear models related to the motion of one dimensional elastic systems, i.e. string and rod. We shall always assume that the motion is transversal and that the material is homogeneous, while no assumptions will be made on the “smallness” of the motion and the linearity of the elastic behaviour of the material. It will also be shown that, linearizing the equations obtained, these reduce to well known mathematical models for string and rod, i.e. to: u) The equation of D’Alembert 77ttfor the string; b) The Euler-Bernouilli

vzrlzz= f

(1.1)

equation

(1.2) for the rod without rotatory inertia and shear stress; c) The Rayleigh-Love equation vtt +

(1.3)

EIqz... - Irlzztt = f

for the rod without shear stress; d) The model of Timoshenko I ptt

+

EIllZZZl -

(I+ g)rlrrrr +qtt = $ftt - gfzz +f

(1.4)

for the rod, taking into account bending moment, rotatory inertia and shear stress. The general procedure followed in deducing the equations governing the nonlinear models will consist in substituting to the string or rod a finite-dimensional “approximating” *

The research

has been supported

by MURST

40% and 60% Research

5101

Contracts

5102

Second

WorldCongressof Nonlinear Analysts

system, depending on a real parameter it :b 0, determining, corresponding equations of motion, and then letting h --) 0.

by Hamilton’s

principle,

the

2. THE VIBRATINGSTRING (l) Consider a homogeneous string of proper length lo and mass M, fixed on the cc-axis at the points L = 0, z = 2 > lo and subject to an external force f(z, t) normal to the z axis; let moreover T = PI(T)

(2.1)

be the stress-strain law relative to the material. We associate to this string an “approximating” elastic system (h-string) R springs of length h = f linked by frictionless hinges of mass m = &;

constituted by in their rest

position, the hinges are placed on the z-axis at the points 2, = ih (i = 0,. . ,n). Denoting by Pi(t) = (ih, vi(t)} the position of the i-th hinge at the time t and setting 67; = Q+~ -Q. the kinetic and potential energies of the h-string are given respectively by

Ez?(t)

Lh = 2(2 + h)

2 qc2(t), i=. E

(2.2)

E;:)(t)= h n-1 c Ib(l+w*)ii

pl(zp

i=O

_

zo)dp >

(2.3)

I

which correspond to the hamiltonian

7-I(t) = Calculating obtain

J0

i(E::‘ -(t) E;;)(t) +

h2

fi(t)qi(t))dt.

(2.4)

i=O

from (2.4) the equations of motion of the h-string and letting then h -+ 0, we M Cqtt

-

&Z(l +11:>+ -

~0)11,(1+ rl:)-f)

= f,

(2.5)

which is the equation associated to our nonlinear model. It is obvious that, if the motion is “small” (1 + 71: N 1) and the relationship (2.1) is linear (7’ = k~, Hooke’s law), (2.5) reduces to (1.1). Equation (2.5) has been studied in [2], where a global existence and uniqueness theorem for the Cauchy-Dirichlet problem has been proved; a numerical analysis of (2.5) has been carried out in [2], [3], al so in relation to other nonlinear models of the string.

Cl1

For a detailed analysis of this model, see [l], [z], [s].

Second

3.

5103

WorldCongressof Nonlinear Analysts

THE VIBRATINGROD: A SPECIAL CASE (2)

We now consider a rod, in the same framework of the Euler-Bernouilli model, i.e. taking into account the bending moment, but not the rotatory inertia and the shear stress. We shall again assume that the rod, of proper length Es, is stretched it its rest position on the z-axis between the points z = 0 and z = 1 2 ls and that (2.1) holds; morover, we assume that bending moment and curvature are linked by the relationship

M = /Q(c).

(3.1)

The corresponding discrete model is obtained in a natural way by adding, to the model of the string, helicoidal springs acting on the hinges; the potential energy generated by the bending moment is then given by

(3.2) Proceeding

as for the string, we then obtain the equation

(3.3) Here again, if we assume that 1 + 7: - 1, that lo N 1 and that (2.1), (3.1) reduce to

T = kr,

M =

Elc,

(3.4)

(3.3) becomes the Euler-Bernouilli equation (1.2). Equation (3.3) h as b een studied in [4], where an existence and uniqueness theorem has been proved for so-called physically significant solutions, i.e. solutions which satisfy, in some neighbourhood (0, t’) of t = 0, the consistency conditions

T
WI

< N2,

1~1 <

(3.5)

N3.

Ni, N2, N3 are constants that correspond respectively to the maximum tension, bending moment, shear stress that can be borne by the material without breaking. Hence, if the solutions of (3.3) are such that (3.5) are not satisfied, the model necessarily breaks down. 4.

THE VIBRATINGROD: PRELIMINARIES

ON THE GENERAL

CASE

(3)

In order to take into account also rotatory inertia and shear stress, it is necessary modify the discrete model of the rod introduced in section 3. (2)

For L detailed analysis of this model, see [4].

(3)

For a more detailed description

of the model considered here, see [5].

to

5104

Second World Congress of Nonlinear Analysts

We shall, first of all, assume that the hinges have a moment of inertia given by Ih; moreover, the hinges are divided into two parts, each of which can move freely orthogonally to the z-axis, linked by a vertical spring with unstretched length=O. We shall then denote rsp1acement, at the time t, of the left (right) hand half of the hinge by q;(t) (G(t)> the d’ which, in its rest position, is placed at t,he point 2, = ih. The 2(n + I)-dimensional system thus obtained will be called the “h-rod”. Assuming that the relationship between the slip and the shear stress is given by r =

(4.1)

P3(C),

setting [i = (Ci - qi)/h, zi = (Sr);/h) -
dh)(t) = Cl

&“f)012(t)+(77X4+ WW) t=O

l3$‘(t) = ;lh &z:(t)(l +z;(t))-1)2 i=O

/~+z~~t)i”

E;:)(t) = h g n-1

E;;)(t) = h c i=l

E;;)(t)

(4.2)

pl(Zp - Zo)dp

t=o

I

(1+&t))-

Iti I

= h 2 J““) i=o 0

arctan%,-I(l)

*6 T;

dP)dp

,u3(p)dp.

Hence, by Hamilton’s principle, the equations of motion of the h-rod clamped at r = 0 and 5 = 1 are, for 0 < t 5 t,

where we have set Nt)

(i+$)h 1 fCh’(z t)dx = h s (i_$)h ’

(lim fCh) = f) h-+0

>+

$ arctan

+p2

(

(lfy$

yi

g(( >[

1 + yf)-l (1 +y$

Xi) _ t(arctan (1+y$

yi)Yi xi I ’

SecondWorld Congressof Nonlinear Analysts

5105

with no(t) = q,(t) = to(t) = vn(t) = qn-r(t) = m-r(t) = 0; equations (4.3) must hold Vlcp;(t),$;(t) satisfying the same boundary conditions as q;(t),
1

2

M -j-w

0

+

1

(

(I

/Q(Z(1+ Z”)i - lo)

+ztt2)2

rt -

2c1

;z2,3

Y

>

-

(4.6)

(l+ZZ2)1y-

Or, in “strong” form,

M -?

a2 - I&at (1 5)’

+ :

((1+1,2,1

3

t” - g (1 ; f2)3

- 21z

&2((1

,“:z,,

(P1(1(1$#

- lo)

>+

(1 +:Z)t

-f=O

>>

(4.7) +

1

a

- (1 +22)% azP2 ( (1 +Z:g)f >

/11(1(1+ Z2)b - lo) -

p3([)

=

(1 +:z)t

-

0.

is obvious that, if we do not take into account the energies due to shear stress and rotation of the sections (6 = I = /13= 0), equations (4.7) re d uce to (3.3) considered in section 3. Let us now linearize equations (4.7) and assume that the functions pi are also linear, so that (2.1), (3.1), (4.1) are replaced by

It

T = k(Z - Zo)z, Equations

M = EIz,,,,

T = G[.

(4.8)

(4.7) then become +t

- Izztt -

k(Z- zo)zz t EIz,,z -f=O

Iztt + k(Z - Zo)z - Elz,,

(4.9)

- G[ = 0.

Differentiating the second of (4.9) with respect to x and adding the first of (4.9), we obtain yvtt - f = Gt,, and, consequently, eliminating & and setting p = y-, &t

+

- k(l(4)

Having assumed

that

(I+ gqrrlt+(p+ !$I _ zo)rltt_

lO)rlZZ- iftt the rod is clamped

can therefore

extend

homogeneous

initial conditions.

modifications.

EIllrzzz -

the summations

+ $fZZ

- ( $(I

- lo) + 1)f

at both ends, we can suppose

from -CO to +co,

General boundary

denoting

these by xi.

and initial conditions

that

(4.10)

= 0.

Ili,si=O

when ill,

i_>n-I.

We shall also, for simplicity,

could however by considered

We

assume

with only minor

Second WorldCongressof NonlinearAnalysts

5106

Setting

p = 1 and assuming 1: While

REMARK models

the equations

are parabolic,

coincides

1 N lo? (4.10)

corresponding

system

(4.7) is hyperbolic;

(4.3)

ob viously represent

with the Timoshenko

to the EulerBernouilli the relative

equation

(1.4).

and Rayleigh-Love

characteristics

are given by the

equations

2: Equations

REMARK

spect to the 5 variable) since they sponding

of (4.6).

are the equations

difference

governing

the motion

approximation

have a concrete

(with re-

interpretation,

of the finite-dimensional

system

corre-

to the h-rod. 5. THE

In order to study general

case,

solution.

(5)

First the initial

if:

a finite

H ence, such approximations

GENERAL VIBRATING ROD: EXISTENCE

the equations

it is necessary

of all, we shall say that and boundary

(4.6)

{7(x,

g overning

(or (4.7))

to introduce

the notion

t), ((2, t)}

the motion

of physically

1s . a solution

of the rod in the

aignijcant in (O,t)

approximable

of (4.6)

satisfying

conditions

7(x,0) = %(X,0) = 0

(0 5

r/(0, t) = T/(Z,t) = z(0, t) = z(l, t) = 0

(0 I t I Q

32 5

1)

(5.1)

n W(0,t;L2), t(t) E L2(0,t;L2), 2(t) = q,(t) - E(t) E llrl+l) i) q(t)E Lm(lO,t;Ho L”(0,

t; Ho p7z+‘) nfP~“(O,t; L2) n @“(0,X;

ii) q(t),<(t)

satisfy

(4.6)

Vp(t),$(t)

such that

H-l);

~(0) = 0;

y(t) E L2(0,?;H~'(y'+1)"') n H1(0,?;L2),

+(t) E P(0,t;L2), y(t) = pz(t) -t(t) E L2(0,1;H:~(Yz+1)'~2)n H1(0,t;L2), &> = This solution

will be called physically

significant

if there exists an interval

0.

(0, t’) (t’ 5 i)

such that, in Qtl = (0,l)x (0,t')

where Nj are positive

constants;

to the rod and correspond (5)

relations

(5.2) constitute

to limits imposed

the consistency

on the velocity,

tension,

conditions

bending

moment,

relative shear

In what follows we shall always assume that the functions pi(p) satisfy the following conditions, of obvious

physical interpretation: a) ~r.(p)EC’[O,oo), fu(O)=O, O<&sM<+m (k=1,2,3); b) 3 positive constants c;k, yk, with yb
Second World Congressof Nonlinear Analysts

5107

stress of the rod. As already mentioned in section 3, if (5.2) are not satisfied, the model obviously breaks down. Finally, we shall say that a physically significant solution is upproximable if it is the limit, in an appropriate topology, when hk -+ 0, of a sequence of physically significant solutions {~ih~(t),
max Em0 t; = 0

p)

max FI,

t6 = t’ > 0

In the%rst case, the interval in which the solutions relative to the h-rod are physically significant -+ 0 with h; there is then no physically significant approximable solution of (4.6). In case ,8), on the other hand, such a solution may exist; its actual existence is proved by the following theorem. THEOREM 1. Assume that j”(t) E L’(O,?; H-‘) and that pi satisfy the assumptions made in section 5 (footnote (‘)). There exists then in (0,t’) at least a physically significant, approximable solution of (4.6), (5.1). The proof of this theorem is given in [6].

6.

THE GENERALVIBRATING ROD: UNIQUENESS

It has not, so far, been possible to prove a general uniqueness result under the same assumptions made in Theorem 1. The following uniqueness theorem holds, however, in the special case in which the terms in (4.7) which derive from the rotation of the sections are linearized (‘1. THEOREM 2. Under the assumption made above, there cannot exist more than one physically significant solution of (4.6), (5.1). For the proof, see [6]. Another uniqueness result, based on a different approach to the problem, can be stated as follows. Let ‘H be the “space of solutions” of (4.6), (5.1), corresponding to any known term f belonging to an appropriate functional class A and let H’, N** be the subsets of X constituted respectively by the physically significant solutions and by the non unique physically significant solutions. If we introduce in ‘H a measure (for instance a Gauss measure) such that m(B) = 1,the following theorem holds. THEOREM 3. Assume that pi are analytic functions satisfying the conditions of theorem 1; then m(E*) > 0, while m(‘H**) = 0. In other words, the physically significant solutions are “nearly all” unique. (6)

For physically

significant

solutions

of (4.3)

we mean those solutions

for which the discrete

analogues

of (5.2)

hold

in (O,ti) (7)

This assumptions

due to the rotation

seems reasonable,

of the sections

since, due to the fact that the rod approximates

is “small”.

a “slender”

beam,

the energy

5108

Second World Congress of Nonlinear Analysts

For a precise formulation

and proof of this theorem, see [6].

REFERENCES

1. PROUSE G., ROLAND1 F. k ZARETTI 2.

IANNELLI 3, 149-177

3.

GOTUSSO

A., PROUSE

IANNELLI

5.

PROUSE Milano,

L., PROUSE

Quad.

of a nonlinear

Quad.

IAC, s&e III, 111, l-27 (1977).

model of the vibrating

string,

NoDEA,

G. & VENEZIANI

A. & PROUSE

184/p,

A., On two nonlinear

models of the vibrating

string,

Mem. di Mat.,

(1994). G., Analysis

G., On some nonlinear

6. GRASSINI

A., Analysis

(1996).

112, XVIII, 201-225 4.

A., Sulla corda e verga vibranti,

G. & VENEZIANI

of a nonlinear

models associated

model of the vibrating to the motion

rod, to appear

of a rod, Quad.

Dip.

on NoDEA. Mat.

Politecnico

l-8 (1995).

E., IANNELLI

Dip. Mat.

Politecnico

A. & PROUSE

G., On a nonlinear

di Milano, 224/p.

l-23

(1996).

generalization

of the Timoshenko

beam equation,

di