acceleration waves in thermoelastic beams

Meccanica 35: 519–546, 2000.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Acceleration Waves in Thermoelastic Beams

LUCA SABATINI and GIULIANO AUGUSTIUniversità di Roma ‘La Sapienza’, Dipartimento di Ingegneria Strutturale e Geotecnica, Via Eudossiana 18;00184 Roma, Italy

(Received: 15 November 2000; accepted in revised form: 2 April 2001)

Abstract. The laws governing the propagation, growth and decay of acceleration waves in directed models ofelastic beams are deduced for materials with zero and non-zero heat flux, that is non-conductors and conductorsof heat. The equations depend explicitly on the geometric and inertial characteristic of the beam section and onthe mechanical properties of the material. Solutions are derived for the velocities of propagation and the evolution(amplitude variation) of each type of wave (extension, bending, shear, twist). Results are discussed and somenumerical examples presented.

Sommario. Si deducono le leggi che governano la propagazione, la crescita e l’ attenuazione delle onde diaccelerazione nei modelli diretti di travi elastiche, per materiali sia conduttori sia non conduttori di calore, cioè cone senza flusso termico. Tali leggi sono espresse da equazioni che dipendono esplicitamente dalle caratteristichegeometriche e inerziali della sezione della trave e dalle proprietà meccaniche del materiale. Si derivano quindisoluzioni in forma chiusa per le velocità di propagazione e l’ evoluzione (variazione di ampiezza) di ciascun tipodi onda (estensionale, flessionale, tagliante, torsionale). Infine, si applicano le formule ad alcuni semplici esempie si discutono i risultati numerici ottenuti, nonché i possibili sviluppi delle ricerche.

Key words: Discontinuity waves (solids), thermoelasticity, beams (direct models), mechanics of solids.

1. Introduction and Summary

The mathematical model of continuum with microstructures can describe well a wide rangeof mechanical problems that are strongly influenced by the material texture. In these models,two variable quantities are assigned to each point P of the continuum: the first one is a vectorand usually represents the placement of the point in the Euclidean space, the second one takesvalue in an n-dimensional connected paracompact manifold M and expresses all possibleconfiguration of the microstructures. In this way the interaction between the elements ofthe microstructure can be expressed by microstresses and self-forces that satisfy appropriatebalance equations [1–3].

Typical examples of continua with microstructure are the direct (or directed) models of oneor two-dimensional structural elements like beams, plates and shells: the three-dimensionalbody is reduced to a representative line or surface of material points; the motion and deforma-tion of the transversal section are described by attaching to each point two systems of vectors(kinematical descriptors) [4–11]. Section 2 illustrates how this model is applied to the straightuniform beam studied in this paper; then, accounting for the inertial terms pertaning to eachkinematical descriptor, the relevant balance and constitutive equations are obtained.

Direct models can be useful for many problems of beams, plates and shells, but in particularsimplify the treatment of acceleration waves, that is moving acceleration discontinuities, aresearch subject that has received recently great attention [12–17] and is very cumbersome if

520 L. Sabatini and G. Augusti

tackled with classical three-dimensional models. The basic concepts of discontinuity waves,as applied to direct models of beams, are set in Section 3.

The propagation equation of acceleration waves is obtained and discussed in Section 4for elastic beams of generic section and either heat-conductor or non-conductor material(respectively corresponding to homothermal and homoentropic waves). In the case of doubly-symmetrical cross-section, the equation is much simplified, and explicit formulae for thevelocities of propagation of each type of wave (extensional, bending, shear, twist) can beobtained, at least for homothermal waves.

It has also been possible to derive (§ 5) the equation governing the evolution of the amp-litudes of the acceleration discontinuity (jumps), again in beams of either conductor or non-conductor material. Explicit expressions for the coefficients, which allow solving the equa-tions can be found in [18].

Section 6 presents examples of application of the proposed equations to two simple cases.Finally, Section 7 summarizes the main results obtained and suggests some further research.

2. Position of the Problem: Field Equations

By definition, a moving surface intersecting a body is an acceleration wave if the kinematicaldescriptors of the continuum and their first derivatives are continuous across the surface buttheir second time derivatives may undergo a finite jump.

This paper deals with acceleration waves in infinitely long rectilinear beams of constantcross-section and linear isotropic elastic materials. In a Cartesian frame Oxyz, such a beamoccupies in the reference configuration (Figure 1) a semi-infinite straight prism obtained bythe translation along the z-axis of a limited plane figure S (section); the x and y axes areassumed parallel to axes of inertia of the section. The origin O is usually taken in the shearcentre (or centre of torsion) of the initial section of the semi-infinite cylinder, with the positivedirection of the z-axis ‘entering’ into the beam.

Figure 1. Semi-infinite beam.

In the direct model of the beam, the three-dimensional body is substituted by the semi-infinite straight line Oz of the shear centres: every point P = P(z) of Oz is a structuredpoint, that is an appropriate system of vectors di is attached to it (Figure 2). Each point P =P(zs) divides the beam into two disjointed parts B+ and B−, with common boundary, definedrespectively by z ∈ [0, zs) and z ∈ (zs,∞).Indicating the time by t , the motion of the directed model is fully described by:

1. The displacement vector s(z, t) of every point P = P(z) (whose components along x, yand z will be indicated by u(z, t), v(z, t), w(z, t) respectively).

Acceleration Waves in Thermoelastic Beams 521

Figure 2. Direct model of beam.

2. An orthogonal second-order tensor Q(z, t) that describes the rotations between the dir-ectors, initially assumed parallel to the coordinates axes, and the coordinates axes (In thefollowing, the analysis is restricted to linear theory only; it is possible thus to replacethe tensor Q(z, t) with its linearization, the skew-symmetric tensor of small rotations�(z, t) = QQQQT , where, � = eφ, e being in turn Ricci’s alternating tensor, and φ(z, t)the vector of small rotations. The components of φ(z, t) with respect to x, y, z axes willbe indicated by α(z, t),−β(z, t) and τ(z, t) respectively).

3. A warping function λ(z, t)ϕ(x, y), where the term ϕ(x, y) defines the shape of warpingand λ(z, t) furnishes the amplitude of this kind of deformation, that varies along the z-axis and in time. In the following, the kinematical descriptors will be collected into avector d = {τ, α, β, λ}T , denoted rotation-warping vector.

Indicating by a prime ()′ the derivation with respect to z, the deformations of the beamelement are defined by:

e = w′, elongation, (1)

{γx = u′ − αγy = v′ − β, shear deformation, (2)

{χx = α′

χy = β ′, bending deformations (curvature), (3)

χT = τ ′, twist deformation, (4)


δ = λ′, warping elongation. (5)

Indicating by a superimposed dot (.) the derivation with respect to time, the dynamicbalance (or ‘equilibrium’) equations are [5, 6, 8–10]:

F ′ + f = ρAw, (6)

Q′x + px = ρAu, (7)

Q′y + py = ρAv, (8)

T ′ + t = ρJpτ , (9)

M ′x −Qx +mx = ρJxα, (10)

M ′y −Qy +my = ρJyβ, (11)

L′ − S + r = ρ%ϕλ, (12)

where

• ρ is the mass density,• A the section area,• Jx and Jy the moments of inertia of the section S with respect to the x and y axis

respectively,• Jp the polar moment of inertia, and

• %ϕ is defined by %ϕ =∫A

ϕ2 dA, with ϕ = ϕ(x, y) the warping shape function. (13)

The components of internal actions are:

• Qx and Qy : shear forces,• F : normal force,• Mx andMy : bending moments,• T : torque,• L: bi-moment,• S and r: bi-shear and distributed bi-moment that arise from non-homogeneous warping of

sections [8–10].

The external forces distributed along the axis are defined by the terms:

• px , py and f : distributed forces, parallel to x, y, z respectively,• mx,my , t: distributed bending moments and torque.


The boundary conditions are immediate if the internal actions are known in a section:for instance, in a case of a semi-infinite beam, in the section z = 0 they coincide with theexternally applied forces and moments.

In this paper, besides mechanical actions, the beam is assumed subject to thermal actions,that is an internal energy ε exists together with mechanical potential and kinetic energies.Thus, if the material is a heat conductor, a heat flux q parallel to the axis of the beam arises.

Conservation of energy yields

ε = qqq′ + F e +Qxγx +Qyγy +Mxχx +Myχy + T χT + Rδ + Sλ, (14)

while the second law of thermodynamic is expressed by the Clausius–Duhem inequality

ρη − h′ � 0, (15)

where η is the entropy density and h the entropy flux.Assuming neither entropy nor heat source in the beam

h = qqq/θ, (15′)

where θ is the absolute temperature. Introducing (15′) and Helmoltz’s free-energy

, = ε − ηθ, (15′′)

equation (15) becomes

, − ηθ − F e −Qxγx −Qyγy −Mxχx −Myχy − T χT − Rδ − Sλ− 1

θqqq · θ � 0. (16)

If the material of the beam is a definite heat conductor [19], the relation between heat fluxq and temperature gradient θ ′ is given by Fourier’s law

qqq = −Kθ ′, (17)

withK is a positive function of the deformations (1)–(5), amplitude of warping λ and absolutetemperature θ . Similarly, also, Helmoltz’s free-energy is a function of deformations, amplitudeof warping and temperature

, = ,(e, γx, γy, χx, χy, χT , δ, λ, θ). (18)

From now on, let us assume that the beam material is linear elastic and isotropic. Develop-ments based on Clausius–Duhem inequality of thermodynamic equilibrium and application ofSaint-Venant’s and Vlasov’s theories [8–10, 20] lead to the following constitutive equations

F = ∂e, = EAe + ESxχx + ESyχy + E%δ, (19)

Qx = ∂γ x, = GA∗xγx +GA∗

yxγy −GSyγT +GSyλ, (20)

Qy = ∂γy, = GA∗xyγx +GA∗

yγy −GSxγT +GSxλ, (21)

Mx = ∂χx, = ESxe + EJxχx + EJxyχT + E%xδ, (22)

My = ∂χy, = ESye + EJyχy + EJxyχT + E%yδ, (23)


T = ∂XT, = −GSyγx +GSxγy +GJzχT −G2λ, (24)

L = ∂δ, = E%e + E%xχx + E%yχy + E%ϕδ, (25)

S = ∂λ, = GSyγy −GSxγx −G2γT +G2λ, (26)

η = −∂θ,. (27)

From equation (15), the following inequality is also derived

1

θq · θ � 0. (28)

In equations (19)–(26)

• Sx and Sy are static moments of section,• Jxy is the mixed inertial moment,• Jz is the torsional constant Jz = Jp/µz, while µz� 1 a scalar depending on the shape of

the section (twist coefficient),•

% =∫A

ϕ dA, %x =∫A

xϕ dA, %y =∫A

yϕ dA, (29)

2 =∫A

(ϕ2,x + ϕ2

,y) dA,

with ϕ the warping shape function ϕ(x, y) that can be obtained solving the torsion problemaccording to Saint-Venant’s theory.

• %ϕ has been already defined (13),• A∗

x, A∗y , are shear areas for actions parallel to the x and y axis respectively, and A∗

xy thereciprocal shear area, obtained dividing the transversal area section by the respective shearcoefficient µx,µy and µxy,

• E and G are respectively the Young modulus and the shear modulus, that in the examples(§ 6) will be assumed linear functions of absolute temperature,

E = E0 + Eθ,G = G0 +Gθ. (30)

3. Discontinuities and Discontinuity Surfaces

The kinematic of singular (or discontinuity) surface in three-dimensional continua have beenthoroughly and rigorously described, for instance by [15, 16] and others. As already under-lined, in direct model of beams, the three-dimensional continuum is substituted by a semi-infinite straight line Oz of structured points P = P(z), each dividing the beam in two disjointedparts B− and B+. The discontinuity surface is thus substituted by a discontinuity point S(that, for simplicity of reference, will be – somewhat improperly – indicated in the followingas discontinuity section) fully individuated by its abscissa zS that varies with time.

Let f (z, t) be a scalar function defined in z ∈ [0,∞) and t ∈ [0,∞), continuous with allits derivatives at all points (abscissae) z except zS. Let

4 = z − zs, (31)


and define the right and left limit of f in zS respectively by

f + = lim4→0+ f (zs +4, t), (32)

f − = lim4→0− f (zs +4, t).

Assume that both right and left limits exist and are finite.The jump of function f across S is defined by

[f ] = f + − f −. (33)

The jump of a vector function v = {f1, f2, f3, ...}T , indicated by [v], can be defined as thevector whose components are the jumps of the components of v, that is

[v] = {[f1], [f2], [f3], ...}T . (34)

An essential tool for the study of discontinuity sections is Hadamard’s theorem [15, 16] thatin the present case of a one-dimensional continuum (beam) can be expressed by the followingexpression

[f ′] = [f ]′ − 1

U[f ], (35)

where

U = dzsdt, (36)

is the velocity with which the discontinuity moves along the beam. Equation (35) shows that,if the time derivative of a function f is discontinuous across S, that is [f ] �= 0, also the spatialderivative of f is discontinuous, and vice versa.

Recalling from Section 2 the definitions of displacement vector s = {w, u, v}T and rotation-warping vector d = {τ, α, β, λ}T , an acceleration wave is a moving section S characterisedby

[s] = 0, [d] = 0, [θ] = 0,

[d′] = 0, [s] = 0, [d′] = 0, [d] = 0,

[s] �= 0, [d] �= 0. (37)

All second spatial and mixed derivatives of kinematical descriptors of the beam may presenta jump.

4. Wave Propagation Equations

Let us now derive the equations that govern the propagation of the acceleration waves in anelastic beam. Let us start with the first balance equation (6) that, applying the constitutiverelations (19)–(27), becomes

(EAe + ESxχx + ESyχy + E%δ)′ + f = ρAw. (38)

The deformations components and the temperature θ are functions of the abscissa z; theYoung modulus E also depends on z through the temperature θ (30). Developing spatialderivatives and substituting definitions (1)–(5) of the deformation components, one obtains

E(Aw′ + Sxα′ + Syβ ′ + %λ′)θ ′ + EAw′′ + ESxα′′ + ESyβ ′′ + E%λ′′ + f = ρAw. (39)


Evaluating jumps across S, applying Hadamard’s theorem and assuming that external forcesand moments are continuous across S, one obtains

−EU(Aw′ + Sxα′ + Syβ ′ + %λ′) [θ] ++EA[w] + ESx[α] + ESy[β] + E%[λ] = ρAU 2[w]. (40)

Applying the same procedure to equations (7)–(12), the following equations are derived

−GU(A∗xγx + A∗

xyγy − SyγT + Syλ) [θ] +GA∗x[u] +GA∗

xy[v] −GSy[τ ]= ρAU 2[u], (41)

−GU(A∗xyγx + A∗

yγy − SxγT + Sxλ) [θ] +GA∗xy[u] +GA∗

y[v] −GSx[τ ]= ρAU 2[v], (42)

−GU(−Syγx − Sxγy + JzγT +2λ) [θ ] +GSyγx[u] −GSx[v] +GJz[τ ]= ρJpU 2[τ ], (43)

−EU(Sxe + Jxχx + Jxyχy + %xδ) [θ] ++ESx[w] + EJx[α] + EJxy[β] + E%x[λ] = ρJxU 2[α], (44)

−EU(Sye + Jxyχx + Jyχy + %yδ) [θ] ++ESy[w] + EJxy[α] + EJy[β] + E%y[λ] = ρJyU 2[β], (45)

−EU(%e + %xχx + %yχy + %ϕδ) [θ] ++E%[w] + E%x[α] + E%y[β] + E%ϕ[λ] = ρ%ϕU 2[λ]. (46)

4.1. HOMOTHERMAL WAVES

In a definite heat conductor the heat flux q is continuous across S (homothermal wave); thenequation (17) yields

[θ ′] = 0, (47)

hence Hadamard’s theorem (35) also gives

[θ] = 0. (47′)

In this case the wave propagation is governed by the system of equations (40)–(46) thatbecomes

(KKKθ − U 2MMM)ξ = 0, (48)

where

ξ = {[w], [u], [v], [τ ], [α], [β], [λ]} (49)


is the seven components vector of unknowns (jump of the acceleration vector),

MMM = ρ diag (A,A,A, Jp, Jx, Jy, %ϕ) (50)

is the inertia matrix, and

KKKθ =

EA 0 0 0 ESx ESy E%

0 GA∗x GA∗

xy −GSy 0 0 0

0 GA∗xy GA∗

y −GSx 0 0 0

0 −GSy −GSx GJz 0 0 0

ESx 0 0 0 EJx EJxy E%x

ESy 0 0 0 EJxy EJy E%y

E% 0 0 0 E%x E%y E%ϕ

(51)

is the stiffness matrix in homothermal conditions. The system (48) admits non-zero solutionif and only if

det (KKKθ − U 2MMM) = 0. (52)

Thus, the determination of the velocities U of propagation of homothermal accelerationwaves coincides with the eigenvalue problem for the pair of matrices Kθ and M. Both matricesare symmetric and positive definite, hence the seven eigenvalues are real and positive, to eacheigenvalue two velocities of propagation correspond, equal in modulus but opposite in sign.Any acceleration wave may propagate in either sense of the beam axis.

4.2. HOMOENTROPIC WAVES

In a non-conductor of heat any acceleration wave is homoentropic, that is there is no discon-tinuity across S of the time derivative of the entropy

[η] = 0 (53)

and, again from Hadamard’s theorem (35),

[η′] = 0. (53′)

In this case [θ ] �= 0 and the seven equations (40)–(46) are not sufficient for the determina-tion of the conditions of propagation. However it is possible to deduce another equation fromthe homoentropy condition (53) and the constitutive relation (27)

−EU(Ae + Sxχx + Syχy + %δ) [w] −GU(A∗xγx + A∗

xyγy − SyγT + Syλ) [u] +−GU(A∗

xyγx + A∗yγy − SxγT + Sxλ) [v] −GU(−Syγx − Sxγy + JzγT +2λ) [τ ] +

−EU(Sxe + Jxχx + Jxyχy + %xδ) [α] − EU(Sye + Jxyχx + Jyχy + %yδ) [β] +−EU(%e + %xχx + %yχy + %ϕδ) [λ] + C[θ] = 0, (54)

where

C = −∂2θθ ,. (55)


Equations (40)–(46) and (54) give a homogeneous linear algebraic system that can be writtenin the following form:

(KKKη − U 2MMMη)ζ = 0, (56)

where

ζ = {[w], [u], [v], [τ ], [α], [β], [λ], [θ ]} (57)

is the eight component vector of unknown jumps,

MMMη = ρ diag (A,A,A, Jp, Jx, Jy, %ϕ, 0) (58)

and

KKKij

η = KKK ij

θ , i, j = 1...7, (59)

KKK18

η = KKK81

η = −EU(Ae + Sxχx + Syχy + %δ), (60)

KKK28

η = KKK82

η = −EU(A∗xγx + A∗

xyγy − SyγT + Syλ), (61)

KKK38

η = KKK83

η = −EU(A∗xyγx + A∗

yγy − SxγT + Sxλ), (62)

KKK48

η = KKK84

η = −GU(−Syγx − Sxγy + JpγT +2λ), (63)

KKK58

η = KKK85

η = −GU(Sxe + Jxχx + Jxyχy + %xδ), (64)

KKK68

η = KKK86

η = −GU(Sye + Jxyχx + Jyχy + %yδ), (65)

KKK78

η = KKK87

η = −EU(%e + %xχx + %yχy + %ϕδ), (66)

KKK88

η = C. (67)

4.3. DOUBLY SYMMETRIC SECTION: HOMOTHERMAL WAVES

If the cross-section of the beam has two axes of symmetry, equations (40)–(46) became muchsimpler; in fact in this case centroid and shear centre coincides, and not only MMM but also thematrix KKKθ is diagonal

KKKθ = diag (EA,GA∗x ,GA

∗y ,GJz,EJx,EJy,E%ϕ). (68)

The system (48) reduces to seven uncoupled equations and the determination of velocitiesof propagation of homothermal acceleration waves is immediate. Four different speeds ofpropagation are obtained:


• Four roots coincide in the first velocity

UE =√E(θ)

ρ(69)

associated with deformations of extensional type; it is the velocity of waves of extension,bending and warping (respectively [w], [α] and [β], [λ]). These are extensional wavesbecause the jumps of accelerations are parallel to the direction of propagation. UE isindependent of the shape of the transversal section and is function only of the beammaterial.

• The second root is the velocity of twist waves [τ ]

UT =√G(θ)

ρ

Jz

Jp=√G(θ)

ρ

1

µz. (70)

• Two further velocities of propagation exist, namely the velocities of shear waves [u] and[v]

USx =√G(θ)

ρ

A∗x

A=√G(θ)

ρ

1

µx, USy =

√G(θ)

ρ

A∗y

A=√G(θ)

ρ

1

µy. (71)

Twist and shear waves are transversal waves because the jump of acceleration is orthogonalto the direction of propagation. Their velocities depend on the shape of transversal sectionthrough the twist coefficient µz and the shear coefficients µx , µy . All four velocities areconstant if and only if the beam is homogeneous and at uniform temperature.

4.4. DOUBLY SYMMETRIC SECTION: HOMOENTROPIC WAVES

The diagonalization of the stiffness matrix noted for homothermal waves in a beam witha doubly-symmetric cross-section does not apply if the wave is homoentropic because theterms of the last row and column of the matrix KKKη are all different from zero. For thisreason an explicit formulation of the speeds of propagation of homoentropic accelerationwaves is not possible. For this kind of waves it is convenient to eliminate the terms includingjumps of time derivative of temperature and write a system which will be used in successivedevelopments (§ 5).

Still considering beams with a doubly-symmetric cross-section, the jump of the time de-rivative of temperature is given by the expression, linear in the unknown acceleration jumps,

[θ] = − 1

CU{EAw′[w] +GA∗

x(u′ − α) [u] +GA∗

y(v′ − β) [v] +GJzτ ′[τ ] +

+ EJxα′[α] + EJyβ ′[β] + E%ϕλ′[λ]}, (72)

that can be written in compact form as

[θ] = − 1

CUI · ξ, (73)

where I is the vector

I = {EAw′,GA∗x(u

′ − α),GA∗y(v

′ − β),GJzτ ′, EJxα′, EJyβ ′, E%ϕλ′}T . (74)


Introduction of (73) into system (56) and some algebra lead to a system containing onlymechanical terms

(KKKη − U 2MMM)ξ = 0, (75)

where

KKKη = KKKθ − 1

CI ⊗ I. (76)

The two matrices KKKθ (51) and KKKη (76) are the correspondents for direct models of beamsof the homoentropic and homothermal generalised acoustical tensors of three-dimensionalmultifield theories of continua [17]. KKKη is positive definite like KKKθ .

As usual, system (75) admits non-zero solutions if and only if

det (KKKη − U 2MMM) = 0. (77)

Equation (77) yields seven positive and distinct eigenvalues for the pair KKKη and MMM; thatis in the case of homoentropic waves seven different velocities of propagation exist. How-ever, no simple closed formulae like (69)–(71) exist. For each eigenvalue, the correspondingeigenvector furnishes a seven-component vector, that is the propagating wave.

5. The Evolution Equations of the Amplitude of the Acceleration Jumps

During the propagation of the acceleration waves, the amplitudes of the jumps may change;the equations that describe these variations will be denoted amplitude evolution equations(or, evolution equations). For simplicity, the following analyses have been developed only forsections with two axes of symmetry.

To obtain such an evolution equation, one follows the usual procedures for the solution ofhyperbolic systems of partial differential equations, presented for linear systems in [21] andextended in [22] to quasi-linear systems.

Each solution ξ or ζ of the propagation system (48) or (56) can be written respectively inthe form

ξ =n∑i=1

riσi(z) and ζ =n∑i=1

riσi(z), (78)

where ri is a constant vector that defines the directions of the propagating discontinuity vector,σi = σi(z) is a scalar that defines the amplitude of the acceleration discontinuity, and the sumis extended to the algebraic multiplicity of the eigenvalue, that is to the multiplicity of thecorresponding root of the characteristic equation (n = 1 for isolated roots).

The starting point for the determination of the evolution equations is the time derivation ofthe balance equations (6)–(11). From equations (6) and (19) one obtains immediately

d

dt

((Ee)′ + f ) = ρw. (79)

Let us assume also that distributed external forces and moments with their first-time deriv-atives are continuous across the section S; then, developing derivatives, applying Hadamard’stheorem (35), and introducing the identity

[=g] = [=]g+ + =+[g] − [=][g], (80)


valid for any pair of functions g and =, defined in [0,∞)× [t0, t1], one obtains the followingexpression

Ew′[θ ]′ − Ew′

U[θ] + E

{(w′′)+ − 1

U(w′)+

}[θ ] − 2

E

U 2[θ][w] +

+{

1

U 2E((θ)+ − U(θ ′)+

)− E(

1

U

)′}[w] + 2

E

U[w]′ =

=(ρ − E

U 2

)[...w]. (81)

Derivation of equations (7)–(12) with respect to time, taking into account equations (20)–(26),yields

1

µxGγx[θ]′ − 1

UµxGγx[θ] + G

µx

{− 1

U(u′)+ + (u′′)+

}[θ] + 2

G

µxU 2[θ ][u] +

+ 1

µx

{G

U 2

((θ)+ − U(θ ′)+

)−G(

1

U

)′}[u] + G

µxU[α] − 2

G

µxU[u]′ =

=(ρ − G

µxU2

)[...u ], (82)

1

µyGγy[θ]′ − 1

µyUGγy[θ] + G

µy

{− 1

U(v′)+ + (v′′)+

}[θ ] + 2

G

µyU 2[θ][v] +

+ 1

µy

{G

U 2

((θ)+ − U(θ ′)+

)−G(

1

U

)′}[v] + G

µyU[β] − 2

G

µyU[v]′ =

=(ρ − G

µyU 2

)[...v ], (83)

Gµzτ′[θ ]′ − Gµzτ

′

U[θ] +Gµz

{(τ )+ − 1

U(τ ′)+

}[θ] − 2

Gµz

U 2[θ ][τ ] +

+µz{

1

U 2G((θ)+ − U(θ ′)+

)−G(

1

U

)′}[τ ] + 2

Gµz

U[τ ]′ =

=(ρ − Gµz

U 2

)[τ ], (84)

EJxχx[θ]′ +− EJxχx

U[θ] +

{EJx

((α′′)+ − 1

U(α′)+

)−GA∗

xγx

}[θ] − 2

U 2EJx[θ ][α] +

+{EJx

U 2

((θ)+ − U(θ ′)+

)− EJx(

1

U

)′}[α] +GA∗

x

1

U[u] − 2

EJx

U[α]′ =

= Jx(ρ − E

U 2

)[...α ], (85)


EJyχy[θ]′ +− EJyχy

U[θ] +

{EJy

((β ′′)+ − 1

U(β ′)+

)−GA∗

yγy

}[θ] − 2

U 2EJy[θ][β] +

+{EJy

U 2

((θ)+ − U(θ ′)+

)− EJy(

1

U

)′}[β] +GA∗

y

1

U[v] − 2

EJy

U[β]′ =

= Jy(ρ − E

U 2

)[...β ], (86)

E%ϕδ[θ]′ − E%ϕδU

[θ ] +

+{E%ϕ

((λ′′)+ − 1

U(λ′)+

)+2(G(λ′)−Gλ)

}[θ ] − 2

U 2E%ϕ[θ][λ] +

+{E%ϕ

U 2

((θ)+ − U(θ ′)+

)− E%ϕ(

1

U

)′}[λ] + G2

U[τ ] − 2

E%ϕ

U[λ]′ =

= %ϕ(ρ − E

U 2

)[...λ ]. (87)

However, equations (81)–(87) and position (78) are not sufficient for the determinationof the evolution equation, because of the presence of time derivatives of temperature andaccelerations. The derivation of the eighth condition, relating time derivatives of temperatureand acceleration discontinuities and, of the evolution equations is illustrated in the followingSection 5.1 and 5.2, respectively for homothermal and homoentropic waves.

5.1. HOMOTHERMAL WAVES

If the wave is homothermal, the first derivative of the temperature is continuous across S,equation (47′), but its second time derivative may undergo a jump. Thus, in equations (81)–(87) all terms including [θ] disappear but those including [θ] remain. It is possible to deducea further relation between the jump of second-time derivative of temperature and the jumps ofaccelerations by introducing Fourier’s law (17) and the relation that expresses the balance ofentropy

[qqq′] = ρθ[η]. (88)

Applying Hadamard’s theorem (35), equation (17) yields

[qqq′] = [−Kθ ′′] = − 1

U 2K[θ ]. (88′)

Introducing also the constitutive equations (19)–(26), some algebra leads to

[θ] = ρUθK

{EAw′[w] +GA∗x(u

′ − α) [u] +GA∗y(v

′ − β) [v] +GJzτ ′[τ ] ++EJxα′[α] + EJyβ ′[β] + E%ϕλ′[λ]}. (89)


Substituting (89) into (81)–(87) one obtains

−EρθKw′{EAw′[w] +GA∗

x(u′ − α) [u] +GA∗

y(v′ − β) [v] +

+EJxα′[α] + EJyβ ′[β] +GJzτ ′[τ ] + E%ϕλ′[λ]} ++{

1

U 2E((θ)+ − U(θ ′)+

)− E(

1

U

)′}[w] + 2

E

U[w]′ =

=(ρ − E

U 2

)[...w], (90)

− GθµxK

(u′ − α){EAw′[w] +GA∗x(u

′ − α) [u] +GA∗y(v

′ − β) [v] ++EJxα′[α] + EJyβ ′[β] +GJzτ ′[τ ] + E%ϕλ′[λ]} +

+ 1

µx

{G

U 2

((θ)+ − U(θ ′)+

)−G(

1

U

)′}[u] + G

µxU[α] − 2

G

µxU[u]′ =

=(ρ − G

µxU 2

)[...u ], (91)

− GθµyK

(v′ − β){EAw′[w] +GA∗x(u

′ − α) [u] +GA∗y(v

′ − β) [v] ++EJxα′[α] + EJyβ ′[β] +GJzτ ′[τ ] + E%ϕλ′[λ]} +

+ 1

µy

{G

U 2

((θ)+ − U(θ ′)+

)−G(

1

U

)′}[v] + G

µyU[β] − 2

G

µyU[v]′ =

=(ρ − G

µyU 2

)[...v ], (92)

−Gθτ′

µzK{EAw′[w] +GA∗

x(u′ − α) [u] +GA∗

y(v′ − β) [v] +

+EJxα′[α] + EJyβ ′[β] +GJzτ ′[τ ] + E%ϕλ′[λ]} ++ 1

µz

{1

U 2G((θ)+ − U(θ ′)+

)−G(

1

U

)′}[τ ] + 2

G

µzU[τ ]′ =

=(ρ − G

µzU 2

)[...τ ], (93)

−EJxθKα′{EAw′[w] +GA∗

x(u′ − α) [u] +GA∗

y(v′ − β) [v] +

+EJxα′[α] + EJyβ ′[β] +GJzτ ′[τ ] + E%ϕλ′[λ]} +

+{EJx

U 2

((θ)+ − U(θ ′)+

)− EJx(

1

U

)′}[α] +GA∗

x

1

U[u] − 2

EJx

U[α]′ =

= Jx(ρ − E

U 2

)[...α ], (94)


−EJyθKβ ′{EAw′[w] +GA∗

x(u′ − α) [u] +GA∗

y(v′ − β) [v] +


+{EJy

U 2

((θ)+ − U(θ ′)+

)− EJy(

1

U

)′}[β] +GA∗

y

1

U[v] − 2

EJy

U[β]′ =

= Jy(ρ − E

U 2

)[...β ], (95)

−E%ϕθKλ′{EAw′[w] +GA∗

x(u′ − α) [u] +GA∗

y(v′ − β) [v] +


+{E%ϕ

U 2

((θ)+ − U(θ ′)+

)− E%ϕ(

1

U

)′}[λ] + G2

U[τ ] − 2

E%ϕ

U[λ]′ =

= %ϕ(ρ − E

U 2

)[...λ ]. (96)

Equations (90)–(96) form a linear system of partial differential equations (evolution equations)that can be written in the following form

−2KKKθ ξ′ + LLLθ ξ =

(KKKθ − 1

U 2MMM

)ξ , (97)

where KKKθ and MMM have already been defined, while the expressions of LLLθ the elements of canbe found in [18].

5.1.1. Waves corresponding to isolated roots UAs seen in Section 4.3 for twist and shear waves (provided µx �= µy), the velocity of propaga-tion of the wave U is an isolated root of the characteristic equation (52) and the jump of theacceleration vector (eq. 78) reduces to a single term

ξ = σ r. (78′)

Let us assume that

ξ =7∑L=1

γLrL, (98)

where γL are independent unknown constants.Introducing (98), system (97) takes the form

−2KKKθ (σ )′r + LLLθσ r =

(KKKθ − 1

U 2MMM

) 7∑L=1

γLrL, (99)

where the unknowns σ and γL appear.


Let us indicate by U a specific solution of the eigenvalue problem (49) and by r thecorresponding eigenvector. Left multiplication by rT and some algebra lead easily to

rT(KKKθ − 1

U 2MMM

) 7∑L=1

γLrL = KKKθ

(1 − U

2

U 2

)7∑L=1

γLrL = 0. (100)

Taking equation (100) into account, equation (99) becomes for each eigenvalue U

−2kkkσ ′ + lσ = 0, (101)

where

kkk = rT · KKKθrrr, (102)

l = rT · LLLθr. (103)

If σ0 is the initial amplitude of the acceleration jump, introducing the hypotheses of regularityof the functions kkk and l, equation (101) yields finally the laws of evolution of the homothermalacceleration waves

σ (z) = σ0 exp

{∫ z

z0

l2kkk

ds

}. (104)

kkk and l take a different value for each type of wave.If the beam material is a simple elastic material (i.e. its mechanical properties do not

depend on temperature) or if the beam temperature is uniform and constant in time, the term lin equation (104) vanishes: hence, both the velocity of wave propagation and the value of thejump are constant in time. If on the contrary the temperature varies in space and/or in time,also the velocity of propagation varies, and the value of the acceleration jump increases wherel/kkk > 0, is stationary where l = 0, and decreases where l/kkk < 0.

The just presented procedure can be applied immediately to twist and shear waves in beamswith doubly-symmetrical cross-section and µx �= µy , that – as already repeatedly stated –correspond to isolated roots of the characteristic equation, hence propagate independentlyfrom each other and can be analysed one by one.

For twist acceleration waves, equation (93) can be written in the form

2

√ρµz

G(σT )

′ − GθKτ ′{GJzτ

′ + µz(Gρ

µzG(θ)++

−√G

µzρ(θ ′)+

)−G

(√ρµz

G

)′}σT = 0. (105)

Recalling position (70), equation (101) becomes

2

√ρµz

Gσ ′T +

− Gθτ′

K

{GJzτ

′ + GµzρG

((θ)+ −

√G

µzρ(θ ′)

)− G

µz

(√µzρ

G

)′}σT = 0, (106)


where only properties of the beam material and of the thermodynamic state of the beam aheadof the wave appear. The solution of equation (101) is given by

σT (z) = σT 0 exp

{∫ z

z0

1

2

√G

µzρ

Gθτ ′

K

{GJzτ

′+

+ GµzρG

((θ)+ −

√G

µzρ(θ ′)

)−G

(√µzρ

G

)′}ds

}, (107)

where σT 0 is the initial amplitude of the jump of twist acceleration (z-component of angulardisplacement).

For shear waves, two different cases must be distinguished: namely actions parallel toeither the x or the y axis. In the first case the velocity of propagation of acceleration wave isfurnished by equation (71), and the evolution equation is

2

√Gρ

µx(σSx)

′ − GθK(u′ − α)

{GA∗

x(u′ − α)+ 1

µx

(Gµxρ

G(θ)++

−√G

µxρ(θ ′)+

)−G

(√ρµx

G

)′}σSx = 0. (108)

If σSx0 is the initial amplitude of the jump of acceleration of u, the solution of equation (108)gives the jump of u

σSx(z) = σSx0 exp

{∫ z

z0

Gθ

2K

√1

µxGρ(u′ − α)

{GA∗

x(u′ − α)+

+µx(Gµxρ

G(θ)+ −

√G

µxρ(θ ′)+

)−G

(√ρµx

G

)′}ds

}. (109)

Likewise, the amplitude of the jump of acceleration of v (the y-axis component of displace-ment) is given by

σSy(z) = σSy0 exp

{∫ z

z0

Gθ

2K

√1

µyGρ(v′ − β)

{GA∗

y(v′ − β)+

+µy(Gµyρ

G(θ)+ −

√G

µyρ(θ ′)+

)−G

(√ρµy

G

)′}ds

}. (110)

5.1.2. Waves corresponding to multiple roots UFor extension-bending-warping waves, and also for shear waves if µx = µy , the velocity ofpropagation is not an isolated root of equation (52): in equation (78) the sum is extended to thealgebraic multiplicity n of the correspondent eigenvalue, n= 4 for extension-bending-warping,and n = 2 for shear waves for cross-sections in which µx = µy . Equation (97) thereforebecomes

−2KKKθ

n∑M=1

(σM)′rM + LLLθ

n∑M=1

(σM) rM =(KKKθ − 1

U 2MMM

) 7∑L=1

γLrL. (111)


Collecting in a matrix R the n eigenvectors that correspond to each multiple eigenvalue, leftmultiplication for RT and application of relation (100) lead to the following linear differentialsystem

KKKσ ′ − LLLσ = 0, (112)

where σ is the n-component vector of the jumps of relevant accelerations,

KKKσ ′ = 2RTKKKθ

n∑M=1

(σM)′rM (113)

and

LLLσ = RTLLLθn∑m=1

(σM) rM. (114)

Following developments usual in system theory (see e.g., [23]), in each space interval [z0, z1]in which detKKK �= 0, equation (112) can be written in the following form

σ ′ = AAA(z)σ (115)

with

AAA = KKK−1LLL. (115′)

If σ0 = σ (z0) is the vector of initial amplitudes of accelerations jumps, the solution of equation(112) is given by

σ (z) = FFF (z, z0) σ0. (116)

FFF is the state transition matrix and expresses the relation between the amplitudes of accel-eration jumps in two different abscissae. The following relations link the matrices AAAand FFF

AAA(z) = ∂FFF (z, z)∂z

∣∣∣∣z=z, (117)

FFF (z, z0) = I +∫ z

z0

AAA(s1) ds1 +∫ z

z0

AAA(s1)

∫ s1

z0

AAA(s2) ds2 ds1 + ..., (118)

where I is the n× n identity matrix.If n linearly independent solutions of (115) are known and collected into a matrix S(z), the

state transition matrix takes the simple form

FFF (z, z0) = S(z)S−1(z0). (119)

If the system (115) has constant coefficients, developing the integrals in the right-hand mem-ber of (118) the state transition matrix becomes

FFF (z) = I + AAAz

1+ AAA2 z

2

2! + · · · =∞∑k=0

AAAkzk

k! = exp {AAAz}, (120)

and the solution of (115) is simply

σ (z) = exp {AAAz} σ0. (121)

Inspection of equation (121) allows to state that, under the set hypotheses:


• acceleration waves propagate with increasing amplitudes if all eigenvalues of AAA havepositive real part; vice versa if all eigenvalues of AAA have negative real part, the amplitudesof acceleration jumps decay,

• if all eigenvalues of AAA have zero real part, an oscillation of the amplitudes around zerooccurs during the propagation,

• the previous behaviours combine if the eigenvalues have real parts with different signs.


In homoentropic acceleration waves both the first and the second derivative of temperatureundergo a jump across S. From the homoentropy condition [η] = 0, equation (72) is derived.Moreover, it can be shown that [19]

[η] = 0. (122)

Thus it is possible to deduce the relation between the jump of the second-time derivative oftemperature [θ] and the accelerations jumps. In fact, introducing the definition of deformation(1)–(5) into the constitutive relation (27), and applying Hadamard’s theorem, some algebrayields

[θ] = −1

C

{EAw′

([w]′ − 1

U[...w]

)+GA∗

x(u′ − α)

([u]′ − 1

U[...u ] − [α]

)+

+GA∗y(v

′ − β)(

[v]′ − 1

U[...v ] − [β]

)+ EJxα′

([α]′ − 1

U[...α ]

)+

+EJyβ ′(

[β]′ − 1

U[...β ]

)+GJzτ ′

([τ ]′ − 1

U[...τ ]

)+

+E%ϕλ′(

[λ] − 1

U[...λ ]

)+G2λ[λ]

}. (123)

Expressions (72) and (123) can be substituted into equations (81)–(87) to obtain a system ofpartial differential equations that can be written in compact form as

KKKθ ξ′ + LLLηξ − QQQ(ξ ⊗ ξ) =

(KKKη − MMM

U 2

)ξ . (124)

The components of the tensors LLLη and QQQ can be found in [18].If U is a speed of propagation of a homoentropic acceleration wave ξ (isolated root of

characteristic eq. (77)) it is possible, as usual, to set

ξ = σ r (125)

and, after left multiplication by rT , to deduce the evolution equation for homoentropic accel-eration waves

σ ′ = q σ 2 − l σ, (126)

where

l = rTLLLηrKKK

, (127)


q = rTQQQ(r ⊗ r)KKK

, (128)

KKK = rTKKKη r. (129)

If σ0 is the initial amplitude of the acceleration jump, assuming that the whole thermo-dynamic response of the beam ahead of the wave is known, the solution of equation (126)is

σ (z) = σ0

exp{− ∫ z

z0l(s) ds

}1 − σ0

∫ zz0q(ς) exp

{− ∫ ς

z0l(s)ds

}dς. (130)

Equation (126) is the general equation that governs the evolution of acceleration waves: a fulldescription of its local and global behaviour may be found in [12, 13].

6. Numerical Examples

To show the practical applicability of the proposed models and procedures, this section presentssome numerical examples of propagation of acceleration waves in two extreme cases, namely:

• pure extensional homothermal waves,• homoentropic waves.

A semi-infinite beam of constant cross-section, fixed in the initial section, is considered; itis assumed that, before the wave propagation phenomenon, the beam is in a state of uniformdeformation.

6.1. EXTENSIONAL HOMOTHERMAL WAVES

In this case, that – as already said – occurs in heat conductor materials, it is assumed thatthe beam is immersed in an environment at 300 K and the heat flux is stationary, while thetemperature θ0 = 600 K of the initial section remains constant in time.

The following material properties have been assumed:

Density: ρ = 8900 Kg/m3

Heat conducibility: λ = 400 W/K·mYoung modulus; equation (30), E = 8 × 108 − 4 × 103 θ Pa.

Under these conditions, the temperature distribution along the beam axis is also constantin time and can be obtained from elementary heat transmission equations: it depends on thethermodynamical properties of the material, and also on the ratio P/A between the perimeterP and the area A of the cross-section.

An example of the temperature distribution is shown in Figure 3, in which the abscissaz is normalized with respect to the diameter d of the section (diameter of the circumscribedcircumference). It can be noted that for approximately z/d� 5 the temperature of the beam isequal to the temperature of the environment.

The velocity of the wave propagation, that also depends on P/A, is obtained from equation(69) and is plotted in Figure 4 for the same beam of Figure 3: it becomes constant sinceapproximately z/d = 5.


Figure 3. Heat conductor material: temperature along the beam.

The distribution of the jump σ of the waves is immediately determined from equation(104). It is plotted in Figure 5 for beams of square section of four different sizes, namely sides10, 15, 20 and 50 cm; σ0 is the initial value of the jump. It can be noted that in all cases thejump tends to a constant value (residual jump), but the decay is larger and more rapid in largersections. Figure 6 shows that the analogous plots for four rectangular sections with the sameperimeter: the decay is larger, but slower, the larger is the area A of the section.

Finally, the distributions of the wave jump in a few examples of thin-walled sections areshown in Figure 7. Four sections have been considered, all inscribed in a 20 × 20 cm square:a simple T section, a double-T section, a square-box section and a full-square section. It canbe noted that for thin-walled section the jump decays only in the initial part of the beam, andthe residual is of the order 80% of initial value, while for the full 20 × 20 cm square sectionthe residual jump is of the order of 15% of initial jump but the decay is slower.


These waves have been studied assuming a uniform initial beam temperature θ0, and thefollowing thermo-dynamical characteristics:

Mass density: ρ = 900 Kg/m3

Young’s modulus: E = 1.2 × 108 PaShear modulus: G = 0.6 × 108 PaSpecific heat: C = 3000 KJ K−1.

As already noted, during a homoentropic wave, all accelerations of kinematical descriptorsof the beam have a jump in the discontinuity section. The following results refer to the fastestwave, that is to the wave with the largest speed, that moves first along the previously un-


Figure 4. Propagation speed UE of extensional homothermal waves.

Figure 5. Homothermal waves: acceleration jump along beam axis in function of the abscissa zs , where theacceleration jump is located at each instant (square section).


Figure 6. Homothermal waves: acceleration jump along beam axis in function of the abscissa zs , where theacceleration jump is located at each instant (rectangular section).

Figure 7. Homothermal waves: acceleration jump along beam axis in function of the abscissa zs , where theacceleration jump is located at each instant (full-square section 20 × 20 cm and thin-walled section inscribedin a 20 × 20 cm square, wall section 1 cm).


Figure 8. Homoentropic wave: temperature distribution along the beam with square cross-section, 40×40 cm andθ0 =300 K, when the discontinuity section has reached z = 9d .

perturbed beam. Introducing the above values and θ0 =300 K into equation (76), this fastestspeed results U =288.3 m/s.

The homoentropic wave produces a slight temperature increase behind the advancing dis-continuity section: Figure 8 shows the temperature distribution along the beam, calculatedfrom equations (72) for θ0 =300 K, when the discontinuity section has reached the abscissaz = 9d.

Figure 9 shows the distribution of the acceleration jump along the beam, obtained fromequation (130), for several values of the initial (uniform) beam temperature θ0. It can be notedthat this distribution is not much affected by θ0: differenly from homothermal waves, in the

Figure 9. Homoentropic waves: decay of acceleration jump along beam axis in function of the abscissa zs , wherethe acceleration jump is located at each instant for five different initial temperature values θ0. Square cross-section,40 × 40 cm.


Figure 10. Homoentropic waves: acceleration jump along beam axis in function of the abscissa zs , where theacceleration jump is located at each instant (square cross-sections).

present case the jump practically vanishes when the wave front reaches z = 5d. As in the caseof homothermal waves, the influence of dimension and shape of the cross-section on decay ofthe jump has been investigated. Figure 10 refers to beams of square section of different sizes,Figure 11 to rectangular sections with basis 40 cm and height B. In all examined cases, thedecay is more rapid the larger is the section.

Figure 11. Homoentropic waves: acceleration jump along beam axis in function of the abscissa zs , where theacceleration jump is located at each instant (rectangular sections 40× B cm).


7. Concluding Remarks: Present Results and Open Problems

The direct models, well known and much used to simplify the description of the mechanicalresponse of beams, have been shown in this paper to be very useful also in the treatment ofthe propagation of acceleration waves.

An appropriate choice of the directors (kinematical descriptors) allows to describe not onlybending and extensional phenomena, but also non-uniform torsion and connected warpingdeformation: the one-dimensional direct model is used in conjunction with classical theoriesof strength of materials, like Saint-Venant’s and Vlasov’s. The resulting equations are differ-ential equations in one spatial variable: the coefficients contain the geometrical properties ofthe beam cross-section and the mechanical properties of the material. This facilitates greatlyformulation and solutions of the relevant problems, as it has been confirmed by the simplenumerical examples presented in Section 6.

Future refinements of the direct models may allow to take into account the deformability ofthe sections in their own plane (and to describe the corresponding striction waves). Moreover,a wider class of constitutive material could be considered if the isotropy assumption is relaxedand non-linear models are introduced. Many other problems are still open: an important classregards wave reflection in beams of finite length and/or with internal restraints, and in corres-pondence of different materials. Of course, direct models are one-dimensional, and thereforecan only describe waves propagating along the beam axis, nor heat transmission ‘across’ thebeam section.

Acknowledgements

This research has been supported by a grant from the Università ‘La Sapienza’ out of researchfunds obtained from the Ministry of University and Research (MURST 60%, 1998). Mostresults have been included in the Thesis prepared by L.S. for the title of ‘Dottore di Ricerca’[18]; a short preliminary version of this paper had been presented at the 14th Italian Congressof Theoretical and Applied Mechanics AIMETA ’99 (Como, October 1999). We are alsograteful to Dr. Ing. Paolo M. Mariano for the many useful comments and suggestions duringthe preparation of this paper.

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