MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 2 pdf

The first equation governs thelocal equilibrium at time t of a material particle located atr and the second equationstands for elastic boundary conditions.. Finally, by specifying the st

Though vectors may be considered as being merely tensors of first rank, it ispreferred to mark the gradient of a scalar quantity by an upper arrow instead of adouble bar in order to stress that the result is a vector The first equation governs thelocal equilibrium at time t of a material particle located at
r and the second equationstands for elastic boundary conditions No prescribed motion has been assumed,

as it would bring nothing new to the formalism, at least at this step Finally, thesystem [1.39], taken as a whole, is said to be homogeneous if no external loading

of any kind is applied either to (V), or to (S), even as non-zero initial conditions.Otherwise, it is said to be inhomogeneous

1.3 Hamilton’s principle

Hamilton’s principle has already been introduced and extensively used in[AXI 04] for deriving the Lagrange equations of discrete systems It is recalledthat this variational principle is expressed analytically as:

The dimension of all the vectors just mentioned is equal to the number ND of

the degrees of freedom (DOF) of the system Here, Hamilton’s principle will beextended to continuous media, providing us with a very efficient analytical tool for

dealing with:

1 the kinematical constraints,

2 the boundary conditions,

3 various numerical methods for obtaining approximate solutions of the ential equations of static and dynamic equilibrium

differ-1.3.1 General presentation of the formalism

The spatial domain occupied by the body and its boundary are still denoted by(V)and (S) respectively, though, depending on the dimension of the Euclideanspace considered, (V) may be either a volume, or a surface, or a line; accordingly(S)may be either a surface, a line, or a finite set of points To formulate Hamilton’sprinciple in a continuous medium, one proceeds according to the following steps:

1 A continuous displacement field and its associated strain tensor is suitablydefined in (V) The components of the displacement field are functions ofspace and time They can be defined by using the coordinate system which isthe most suitable in relation to the geometry of (V) The displacement field

X(
r; t), which is a continuous vector function of space, extends the independentgeneralized displacements used in the discrete systems to the continuous case

2 The Lagrangian L is defined again as the difference between the kinetic energy

Eκand the internal potential energy Ep, plus the work W of extra external orinternal forces which are eventually applied within (V) and/or on (S) Calcu-lation involves a spatial integration over (V) and/or (S) of the correspondingenergy and work densities, denoted eκ, ep and w respectively Fortunately,the actual calculation of W can be avoided Indeed, because of the variationalnature of Hamilton’s principle, only the virtual variation δ[W] is needed δ[W]

is far more easily expressed than W itself, when one has to deal with internalforces which are neither inertial nor potential in nature

3 Hamilton’s principle is applied, according to which the action integral of theLagrangian between two arbitrary times t1 and t2is stationary with respect

to any admissible variation δ[
X] To be admissible δ[
X] must comply withthe boundary conditions of the problem and must vanish at t1and t2 In mostcases, integrations by parts are needed to formulate such variations explicitly

in terms of the components of the displacement field Boundary terms arisingfrom the spatial integrals contribute to define the boundary conditions of theproblem At this final step, the equilibrium equations are obtained in terms ofgeneralized forces Obviously, they are necessarily identical to the equilibriumequations which would result from the Newtonian approach

4 As in the case of discrete systems, the kinematical constraints which can beeventually prescribed on the body may be conveniently introduced by usingLagrange’s multipliers (cf [AXI 04], Chapter 4)

5 Finally, by specifying the stress-strain relationships, the equilibrium equationscan be expressed in terms of displacement variables only.

Summarizing briefly the above procedure, it can be said that the major ences between the mechanics of discrete and continuous systems originate from thereplacement of a countable set of independent displacement variables by a continu-ous displacement field, which is a function of the position vector
r in (V), and fromthe emergence of boundary conditions, which specify the contact forces and/or thekinematical conditions prescribed on the boundary (S)

differ-1.3.2 Application to a three-dimensional solid

For the sake of simplicity, we consider here a 3D solid with either free or fixedboundary conditions, though extension to more general elastic conditions wouldnot lead to particular difficulties

1.3.2.2 Hilbert functional vector space

The displacement field is a vector of the three-dimensional Euclidean space.Using a Cartesian coordinate system, it is written in symbolic notation as:

X(
r; t) = X(x, y, z; t)
i + Y (x, y, z; t)
j+ Z(x, y, z; t)
k [1.42]

As X, Y , Z are real functions of the Cartesian components x, y, z of
r,
Xbelongs

also to a functional vector space provided with the functional scalar product:

where
U·
V is the usual notation for the scalar product in the Euclidean space and

U, V (V)is the notation for the scalar product in the functional space

In contrast to the case of discrete systems, the dimension of the functional vectorspace is infinite, and even uncountable Here, it will be asserted, without performingthe mathematical proof, that it is complete, which means that any Cauchy sequence

of functional vectors is convergent to a functional vector within the same space

This space is thus an Hilbert space The definition holds independently from the

dimension of the Euclidean space in which (V) is embedded The reader interested

in a more formal and detailed presentation of the functional vector spaces is referred

to the specialized literature, for instance [STA 70]

1.3.2.3 Variation of the kinetic energy

The density of kinetic energy is defined as the kinetic energy per unit volume

of a fictitious material particle of infinitesimal mass ρ dV:

eκ(
r; t) = 12ρ



˙X (
r; t) ·
˙X(
r; t) [1.44]The total kinetic energy is:

1.3.2.4 Variation of the strain energy

To obtain an explicit formulation of the strain energy, a material law describing

the mechanical behaviour of the material must be specified first Nevertheless, ifthe problem is limited to infinitesimal strain variations, it is always possible to write

Figure 1.8 Virtual work of the stresses

the variation of the strain energy as the contracted tensor product:

if esis a differentiable function of ε, as in the case of elasticity, σ can be calculated

by using the following formula:

1.3.2.5 Variation of the external load work

Writing the variation of the external work is immediate, leading to the followingvolume and surface integrals:

1.3.2.6 Equilibrium equations and boundary conditions

By substituting the relations [1.46] to [1.49] into [1.41], Hamilton’s principlecan be written in indicial notation as:

of δXi exclusively Gathering together the volume terms in one integral and thesurface terms in another one, the equation [1.50] is thus transformed into:

of the volume integral produces the equations of dynamical equilibrium of thebody, whereas the kernel of the surface integral provides the boundary conditions

Of course, it is immediately apparent that such equations are identical to thosealready established in section 1.2 (cf system [1.32]) Moreover, if the boundary,

or a part (Sk)of it is free (admissible δXi = 0) the disappearance of the kernel

of the surface integral leads to the disappearance of the stresses on (Sk) On thecontrary, if the displacement is constrained by the condition Xi = 0 on (Sk), aLagrange multiplier i is associated with the locking condition and the surfaceintegral becomes:



(S k )(i − σijnj)δXidS and δXi = 0 [1.52]Letting the integral [1.52] vanish produces the reaction force at the fixed boundary:

i = σij(
r)nj ∀
r ∈ (Sk) [1.53]

1.3.2.7 Stress tensor and Lagrange’s multipliers

A comment is in order here concerning the relation between stresses andLagrange’s multipliers in a constrained medium Indeed, rigidity of a solid may

be understood as a particular material law expressed analytically by the vanishing

of the strain tensor Turning now to the problem of determining the stress-strainrelationship associated with the law ε ≡ 0, the relation εij = 0 is interpreted as

a holonomic constraint with which a Lagrange multiplier ij is associated Thus,the stress tensor describes the internal reactions of the rigid body to an externalloading The simplest way to prove this important result is to consider the staticequilibrium of a rigid body loaded by contact forces only (body forces could beincluded but are not necessary) The constrained Lagrangian is:

L′=



(V)(−es+ ijεij) dV+

δ[L′] =



(V)(ij)δ εijdV+

as the reactions of a rigid body to external loading This point of view is useful, atleast conceptually, to define stress components under rigidifying assumptions, forinstance the pressure in an incompressible fluid (constraint condition div(
X)= 0),

as further detailed in the following example

example –Water column enclosed in a rigid tube

As shown in Figure 1.9, a rigid tube at rest contains a column of liquid The fluid

is subjected to a normal load T(e)which is applied through a rigid waterproof piston

We are interested in determining the pressure field in the fluid Obviously, thecondition of local and/or global static equilibrium leads immediately to a uniformpressure P = −T(e)/S where S is the tube cross-sectional area (section normal

to the piston axis) This result is clearly independent of the material law of the

Figure 1.9 Column of liquid compressed in a rigid tube

fluid However, we want to define the pressure in a logical manner starting fromthe material behaviour of the fluid, which is supposed here to be incompressible.Let us assume that the problem is one-dimensional, as reasonably expected Thelaw of incompressibility reduces to ∂X/∂x = 0 and the pressure is given by theLagrange multiplier associated with this condition The variation of the constrainedLagrangian is:

δL′= S

 L 0

∂(δX)

∂x dx+ T(e)δ X(L)= 0After integrating by parts,

δL′= −S

 L 0

1.3.2.8 Variation of the elastic strain energy

In the preceding subsections the material law has not yet been specified, except

in the limit case of rigidity It is now particularized to the case of linear elasticity.The virtual variation of the elastic energy density per unit volume is expressed as:

δ ee= σijδεij = hij kℓεkℓδ εij = hij kℓεijδ εkℓ= hij kℓδ12εijεkℓ



or in symbolic notation:

δ ee =12δε: h : εThus, the elastic energy density is found to be:

Then, using the infinitesimal strain tensor [1.25], eecan be further written in terms

of the displacement field as the quadratic form:

i

∂xj

2+∂X∂xij

T

which is symmetric and positive, or eventually null if displacements of rigid bodyare included

1.3.2.9 Equation of elastic vibrations

The material is supposed to be isotropic and linear elastic The external loadsare either contact forces
t(e)or/and body forces
f(e) It is recalled that in order toavoid redundancy in the boundary loading by contact and body forces, the latterare assumed to vanish at the boundary As the contribution of the external loading

to the equilibrium equations gives rise to no difficulty, we concentrate here on thevariation of internal terms Retaining the inertial and elastic terms solely, Hamilton’sprinciple is written in indicial notation as:



niδXijdS= 0

Again, as the variation of action must vanish whatever the admissible δXimay

be, the equation of motion is obtained by equating to zero the kernel within thebrackets of the volume integral whereas the boundary conditions are given byequating to zero the kernel of the surface integral In the absence of any externalloading, or any elastic support, this reduces to a condition of either a free boundarysuch that δXi = 0 and the kernel within the brackets equal to zero, or that of afixed boundary δXi = 0 and the kernel within the brackets not equal to zero

Finally, it is possible to shift from the indicial to the symbolic notation, by usingthe following identities:

where = div grad() is the Laplace operator (see Appendix A.2)

The vibration equations are thus found to agree with the intrinsic form [1.39],

as suitable

1.3.2.10 Conservation of mechanical energy

As above, the solid occupies the finite volume (V) closed by the surface (S)

At time t = 0 the stable and unstressed configuration of equilibrium is chosen asthe state of reference; then the external loads characterized by a volume density

By using the equations of equilibrium [1.32], the result [1.62] is expressed in terms

of internal forces as:

T





= σ : ˙ε [1.65]

To establish this relation, the symmetry of the stress tensor and the hypothesis

of small displacements are used The work rate supplied to the solid may thus beexpressed in terms of kinetic and elastic energies (cf Clapeyron formula [1.57]) Ifthe material is elastic σ :˙ε = dee/dt, and [1.64] leads to:

 t 0P(τ )dτ = Eκ(t )+ Ee(t )= Em(t ) [1.67]

1.3.2.11 Uniqueness of solution of motion equations

Starting from the conservation law of energy, it is possible to prove the theorem

of uniqueness of the solution of any linear elastodynamic problem, which is stated

as follows:

The equations of motion of a linear elastic solid, subjected to suitably prescribed loading and/or displacement fields,(including the boundary conditions) have a solution which is unique.

The proof is due to Neumann [NEU 85] First it is noted that provided theproblem is linear, the principle of superposition can be applied Accordingly, let

X1,
X2be the respective solutions of the two following problems:

ρ
¨X1− div σ1+
f1(i)=
f1(e)(
r; t); ∀
r ∈ (V)

Then
X= α
X1+ β
X2will be solution of:

ρ
¨X− divσ +
f(i)=
f(e)(
r; t); ∀
r ∈ (V)

σ ·
n(
r) =
t(e)(
r; t); ∀
r ∈ (S1)

σ·
n(
r) − KS[
X] = 0; ∀
r ∈ (S2)

X(
r; 0) =
D(
r);
˙X(
r; 0) =
˙D(
r)

[1.72]

Because in [1.72], the initial conditions and the external loading are nil, nomechanical energy is provided to the system Therefore the kinetic and the elasticenergy are zero at any time, so the system remains at rest:

X(
r; t) =
X2(
r; t) −
X1(
r; t) ≡ 0 ∀
r, t

To conclude this subsection, it is worth mentioning that a similar theorem wasobtained by Kirchhoff in the case of statics Kirchhoff’s theorem of uniquenessdiffers from that of Neumann, since in statics, kinetic energy is discarded Hence,

if the body is free (i.e not provided with any support) it is always possible to add

a uniform, and otherwise arbitrary, displacement field to a given static solution

1.4 Elastic waves in three-dimensional media

1.4.1 Material oscillations in a continuous medium interpreted as waves

When the particles contained in an elastic medium are removed from their ition of static equilibrium – assumed here to be stable – the stresses related to thelocal change of configuration have the tendency to take them back to the position

pos-of static equilibrium, but the inertia forces are acting to the opposite, having thetendency to make the particles overshoot it As a result, they start to oscillate Due

to the principle of action and reaction, the particles lying in the immediate ity are also excited and start to oscillate too In this way, the motion is found topropagate throughout the whole solid In the absence of inertia, the propagationwould have the instantaneous character of the elastic forces The inertia introduceshowever a delay in the propagation, in such a way that the speed is finite Such

vicin-progressive oscillations are termed travelling waves It is important to point out

first that in this “chain reaction” what propagates is not matter but mechanicalenergy A discrete version of material waves was already discussed in [AXI 04],Chapters 7 and 8 As schematically illustrated in Figure 1.10, two kinds of waves

can be distinguished They are termed transverse waves if the particles oscillate

in a direction perpendicular to that of wave propagation and longitudinal waves if

Figure 1.10 Discrete model of the oscillations of material points in transverse and

longitudinal waves

the particles oscillate in the direction of wave propagation On the other hand, in

an infinite conservative medium, the amount of mechanical energy conveyed bythe waves is constant during the propagation However, in reality nonconservativeforces are always present, so the waves are damped out, or alternatively amplified,depending on the sign of the energy transfer to the wave

1.4.2 Harmonic solutions of Navier’s equations

In a solid, the material waves are governed by Navier’s equations [1.39] It isappropriate to study first their general properties independently of external loading.Furthermore, it is also suitable to start by assuming a medium extending to infinity

in all directions, in such a way that boundary effects can be discarded Navier’sequations are thus reduced to the homogeneous vector equation:

dif-A well known mathematical technique used to solve this kind of equation is the

method of variables separation If the problem is further particularized to

har-monic oscillations of pulsation ω, solutions sought can be written as the complexfield:

X(x, y, z; t)= eiωt

{X
i + Y
j+ Z
k}

X(x, y, z)= fx(x)gx(y)hx(z)

Y (x, y, z)= fy(x)gy(y)hy(z)Z(x, y, z)= fz(x)gz(y)hz(z)

[1.74]

Substitution of [1.74] into [1.73] allows one to reduce the problem of solvingthe partial differential vector equation [1.73] to one of solving nine ordinary differ-ential equations However, determination of the appropriate nine space functionsinvolved in [1.74] is not a simple task and it is advisable to particularize the problemfurther, in order to obtain comparatively simple analytical solutions which can beeasily discussed from a physical point of view This is the object of the followingsubsections

1.4.3 Dilatation and shear elastic waves

The relation −−−−−−−→

grad(div
X) =
X + (curl curl(
X)) (see formula [A.2.17] inAppendix A.2) allows a meaningful simplification of [1.73] to be made, by

separating the motion into two physically distinct types, namely:

1.4.3.1 Irrotational, or potential motion

which satisfies automatically the condition curl
X= 0

1.4.3.2 Equivoluminal, or shear motion

which satisfies automatically the condition div
X= 0

1.4.3.3 Irrotational harmonic waves (dilatation or pressure waves)

The system [1.75] describes waves in which the volume of the medium fluctuatessince div
X= 0 (otherwise the
Xterm would vanish identically); for this reasonsuch waves are often referred to as ‘volume’ or dilatation waves To point out theirmajor features, the easiest way is to study the plane harmonic waves which travelalong the Ox axis, of unit vector
i The displacement field reduces thus to thecomplex amplitude:

The condition curl
X= 0 is obviously satisfied by [1.79] If this form is tuted into the first equation [1.75], the following ordinary differential equation is

cL=! κρ =

"

1− ν(1+ ν)(1 − 2ν)

E

The delay τ (x) = −|x|/cLis the time spent by the harmonic waves X±eiωt

to cover a distance ±|x| The negative sign agrees with the principle of ality, according to which the response of the medium cannot anticipate theexcitation

caus-In terms of phase angle, τ (x) is replaced by the phase shift between the lations located at the source and at a distance x from the source It is given by

oscil-ψ (x)= −ω|x|/cL Accordingly, cLis interpreted as the phase speed of the

dilata-tion waves The wave which travels from left to right (x > 0) has the magnitude

X+and phase shift−ωx/cLand the wave which travels from right to left (x < 0)

Figure 1.11 Propagation of plane waves

has the magnitude X−and phase shift−ω|x|/cL As a general definition, the phasespeed of a wave, denoted cψ, is such that:

ψ (x)= −ωc|x|

ψ = −2πλ|x| = −k|x| ⇒ cψ = ωk [1.83]

where the wavelength λ is the distance travelled by the wave during one period

T = 2π/ω of oscillation and k = 2π/λ is the wave number.

If the phase speed is independent of the pulsation, the wave is said nondispersive,

as is the case of dilatation waves, cf [1.82], if not it is said to be dispersive To

understand the meaning of this terminology, it is appropriate to consider first acompound wave defined as the superposition of two distinct harmonic waves offrequency f1and f2respectively Its complex amplitude is written as:

X(x; t)= ei2π f1 (t −x/c 1 )

+ ei2π f2 (t −x/c 2 )

If c1= c2= c, each component travels at the same speed, so the time profile ofthe wave is the same from one position to another, and the same holds for the spaceprofile from one time to another If c2differs from c1, each component travels at isown speed, so the time profile of the wave changes from one position to another, asthe space profile does from one time to another This is illustrated in Figure 1.12,where the real part of X(x; t) is plotted versus time at two distinct positions x1= 0and x2 = 1.75λ1where λ1 = c1f1 The spectral components are at f1 = 10 Hzand f2= 20 Hz, the period of the compound wave is T = 1/f1= 0.1s

Figure 1.12a refers to the nondispersive case c1= c2= c The shape of the wave

at x2is the same as that at x1= 0, the time profile being simply translated to theright by the propagation delay τ = x2/c Figure 1.12b refers to the dispersive case

c1= c2 The time profile of the wave at x2differs from that at x1= 0 Thus, in thedispersive case, propagation cannot be described simply in terms of propagationdelay

Such elementary considerations can be extended to more complicated waves,such as transients by using the Fourier transformation Transients are described

in the time domain by the displacement field X(x; t) which usually vanish side a finite time interval 0 ≤ t ≤ t1 It is necessary to stress that here X(x; t)denotes a real valued function, in contrast with the former case where it denotedthe complex amplitude of superposed harmonics waves Shifting to the spec-tral domain, the transients are described by the Fourier transform of X(x; t),denotedX(x; ω).⌢

out-Figure 1.12a Time profile of the compound wave, nondispersive case:

c1= c2= 5000 m/s

Figure 1.12b Time profile of the compound wave, dispersive case:

c1= 5000 m/s, c2= 3000 m/s

It is recalled that by definition of the Fourier transformX(x; ω) and X(x; t ) are⌢related to each other by:

[1.84]Substituting t− τ for t, the shift theorem follows:

[1.85]

where τ is the time delay

Let X(t) stand for the displacement field of a transient plane wave emitted at

x = 0, starting from t = 0 In the nondispersive case, the wave observed at x > 0

is X+(x; t) = 0.5X(x; t − τ ) where τ = x/cψ as sketched in Figure 1.13 and

in the spectral domainX⌢+(ω, x)= 0.5X(ω)e⌢ −iωx/cψ Of course, the same resultholds for a wave X−(t, x) travelling in the domain x < 0, with τ = |x|/cψ This

is precisely the reason why the multiplying factor 0.5 appears in the travellingwaves, in such a way thatX⌢+(ω, 0)+X⌢−(ω, 0) =⌢X(ω) Contrasting with suchsimple results, if cψ is frequency dependent, the Fourier transform of X+(x; t)

Figure 1.13 Propagation of nondispersive waves: c= cψis constant