Ondas de extensión

En el apartado anterior se han tratado las deformaciones elásticas estáticas o cuasi-estáticas en una barra o placa. Si, por el contrario, las deformaciones tienen lugar de forma dinámica a una velocidad finita, entonces no sólo la elasticidad del material juega un papel fundamental a la hora de determinar lo que ocurre, sino que también se debe considerar su inercia. Para ello, se establece un equilibrio de fuerzas similar al de la ecuación (2.6). El resultado puede obtenerse de forma inmediata sustituyendo la presión sonora p por el esfuerzo de tracción (negativo) σxx. Además, llevando a cabo el mismo proceso de linealización que en el Apartado 2.2, esto es, reemplazando la aceleración total por la aceleración local y la densidad total ρt por su valor medio ρ0, obtenemos:

(3.20)

Sound waves in isotropic solids 199 When the degree of bending varies with x the same is true of the moment D. Hence the moment D(x + dx) at the right cross section in Figure 10.7b may differ from that at the left side right, D(x), the difference

D(x + dx) − D(x) = ∂D_∂xdx

must be kept in equilibrium by a pair of forces consisting of two lateral forces ±Fyat distance dx:

−Fydx = ∂D_∂xdx (10.11)

When the lateral force Fy is also a function of x, each length element is

associated with a difference force dFy= Fy(x + dx) − Fy(x) = ∂Fy

∂x dx

Combining this equation with eqs. (10.11) and (10.9) gives the result: dFy= B∂

4_η

∂x4dx (10.12)

which must be somehow balanced, for instance, by exterior forces (which we excluded) or by inertial forces as will be detailed in Subsection 10.3.3.

The elasticity constants Y and ν are related to the Lamé constants which were introduced already in Section 3.3. These relations are:

µ= Y

2(1 + ν) and λ =

νY

(1 + ν)(1 − 2ν) (10.13) The constant µ is identical with the shear modulus or torsion modulus G often used in technical elasticity. Inserting these relations into eqs. (10.2) and (10.3) shows that the ratio of cLund cTdepends only on Poisson’s ratio:

!_c

L

c_T "2

= 2_{1 − 2ν}1 − ν (10.14)

Table 10.2 lists Young’s modulus and Poisson’s ratio of a few materials. 10.3.2 Extensional waves

The preceding subsection dealt with static or quasistatic elastic deformations of a straight bar or a plate. If, on the contrary, deformations take place at finite speed then not only the elasticity of the material determines what happens but its inertia becomes noticeable as well. To account for it we set

200 Sound waves in isotropic solids

Table 10.2 Young’s modulus and Poisson’s ratio of solids

Material Density

(kg/m3₎ Young’s modulus₍₁₀10_N/m2₎ Poisson’s ratio

Aluminium 2700 6.765 0.36 Brass (70% Cu, 30% Zn) 8600 10.520 0.37 Steel 7900 19.725 0.30 Glass (Flint) 3600 5.739 0.22 Glass (Crown) 2500 7.060 0.22 Plexiglas 1180 0.3994 0.40 Polyethylene 900 0.0764 0.45

up a force balance similar to that of eq. (3.5). The result can be immediately taken on by replacing the sound pressure p with the (negative) tensile stress σxx. Furthermore; we carry out the same linearisations as in Section 3.2; in

particular, we replace the total acceleration by the local one and the total density ρtwith its average value ρ0. Then we arrive at:

∂σxx ∂x =ρ0 ∂vx ∂t =ρ0 ∂2ξ ∂t2 (10.15)

Combining this relation with eq. (10.8) leads to the following wave equation: ∂2ξ ∂x2 = ρ0 Y ∂2ξ ∂t2 (10.16)

By comparing this equation with earlier wave equations, for instance, with eq. (3.21), we see that the wave velocity of extensional waves on a bar is

cE1=! Y

ρ0 (10.17)

The general solution corresponds to eq. (4.2). In a similar way the propaga- tion of extensional waves in plates with parallel boundaries is derived. Their wave velocity is found to be

cE2=

! Y

ρ0(1 − ν2) (10.18)

It is slightly higher than cE1 due to the fact that the elastic constraint in a

bar is lesser than that in a plate where stress relief due to lateral contraction can only occur in one direction, namely, perpendicular to the plate surfaces. By employing eq. (10.13) it is easily verified that

ONDAS MECÁNICAS EN SÓLIDOS

52 Combinando esta relación con la ecuación (3.10) se obtiene la siguiente ecuación de onda:

(3.21)

Al comparar esta ecuación con las ecuaciones de onda anteriores, por ejemplo, con la ecuación (2.24), llegamos a la conclusión de que la velocidad de onda de las ondas de extensión en una barra es

(3.22)

La solución general de dicha ecuación de onda es de la forma:

p(x,t)=f(x-ct)+g(x+ct) (3.23)

De forma similar puede deducirse la propagación de ondas de extensión en placas ilimitadas con límites paralelos. Su velocidad de onda resulta ser

(3.24)

Esta es ligeramente superior a cE1 debido a que la fuerza elástica en una barra es menor que en una placa donde el alivio del esfuerzo debido a la contracción lateral sólo puede tener lugar en una dirección, la perpendicular a las superficies de la placa. A partir de las ecuaciones (3.17) y (3.18), se puede verifica fácilmente que

(3.25)

La Figura 3.5a muestra las deformaciones asociadas a una onda de extensión viajando horizontalmente. Debido a la contracción lateral el movimiento de las partículas del material no es puramente longitudinal, sino que también existen componentes del desplazamiento perpendiculares a la superficie. La barra o placa es más gruesa donde la compresión longitudinal del material es máxima. Por tanto, las ondas de extensión no son puramente ondas longitudinales aunque el desplazamiento longitudinal prevalece. Así, habitualmente se denominan ondas cuasi-longitudinales.

200 Sound waves in isotropic solids

Table 10.2 Young’s modulus and Poisson’s ratio of solids

Material Density

(kg/m3₎ Young’s modulus₍₁₀10_N/m2₎ Poisson’s ratio

Aluminium 2700 6.765 0.36 Brass (70% Cu, 30% Zn) 8600 10.520 0.37 Steel 7900 19.725 0.30 Glass (Flint) 3600 5.739 0.22 Glass (Crown) 2500 7.060 0.22 Plexiglas 1180 0.3994 0.40 Polyethylene 900 0.0764 0.45

up a force balance similar to that of eq. (3.5). The result can be immediately taken on by replacing the sound pressure p with the (negative) tensile stress σxx. Furthermore; we carry out the same linearisations as in Section 3.2; in

particular, we replace the total acceleration by the local one and the total density ρtwith its average value ρ0. Then we arrive at:

∂σxx ∂x =ρ0 ∂vx ∂t =ρ0 ∂2ξ ∂t2 (10.15)

Combining this relation with eq. (10.8) leads to the following wave equation: ∂2ξ ∂x2 = ρ0 Y ∂2ξ ∂t2 (10.16)

By comparing this equation with earlier wave equations, for instance, with eq. (3.21), we see that the wave velocity of extensional waves on a bar is

cE1=! Y

ρ0 (10.17)

The general solution corresponds to eq. (4.2). In a similar way the propaga- tion of extensional waves in plates with parallel boundaries is derived. Their wave velocity is found to be

cE2=

! Y

ρ0(1 − ν2) (10.18)

It is slightly higher than cE1 due to the fact that the elastic constraint in a

bar is lesser than that in a plate where stress relief due to lateral contraction can only occur in one direction, namely, perpendicular to the plate surfaces. By employing eq. (10.13) it is easily verified that

cL>cE2>cE1>cT

200 Sound waves in isotropic solids

Table 10.2 Young’s modulus and Poisson’s ratio of solids

Material Density

(kg/m3₎ Young’s modulus₍₁₀10_N/m2₎ Poisson’s ratio

Aluminium 2700 6.765 0.36 Brass (70% Cu, 30% Zn) 8600 10.520 0.37 Steel 7900 19.725 0.30 Glass (Flint) 3600 5.739 0.22 Glass (Crown) 2500 7.060 0.22 Plexiglas 1180 0.3994 0.40 Polyethylene 900 0.0764 0.45

up a force balance similar to that of eq. (3.5). The result can be immediately taken on by replacing the sound pressure p with the (negative) tensile stress σxx. Furthermore; we carry out the same linearisations as in Section 3.2; in

particular, we replace the total acceleration by the local one and the total density ρtwith its average value ρ0. Then we arrive at:

∂σxx ∂x =ρ0 ∂vx ∂t =ρ0 ∂2ξ ∂t2 (10.15)

Combining this relation with eq. (10.8) leads to the following wave equation: ∂2ξ ∂x2 = ρ0 Y ∂2ξ ∂t2 (10.16)

By comparing this equation with earlier wave equations, for instance, with eq. (3.21), we see that the wave velocity of extensional waves on a bar is

cE1=! Y

ρ0 (10.17)

The general solution corresponds to eq. (4.2). In a similar way the propaga- tion of extensional waves in plates with parallel boundaries is derived. Their wave velocity is found to be

cE2=

! Y

ρ₀(1 − ν2) (10.18)

It is slightly higher than cE1 due to the fact that the elastic constraint in a

bar is lesser than that in a plate where stress relief due to lateral contraction can only occur in one direction, namely, perpendicular to the plate surfaces. By employing eq. (10.13) it is easily verified that

cL>cE2>cE1>cT

200 Sound waves in isotropic solids

Table 10.2 Young’s modulus and Poisson’s ratio of solids

Material Density

(kg/m3₎ Young’s modulus₍₁₀10_N/m2₎ Poisson’s ratio

Aluminium 2700 6.765 0.36 Brass (70% Cu, 30% Zn) 8600 10.520 0.37 Steel 7900 19.725 0.30 Glass (Flint) 3600 5.739 0.22 Glass (Crown) 2500 7.060 0.22 Plexiglas 1180 0.3994 0.40 Polyethylene 900 0.0764 0.45

up a force balance similar to that of eq. (3.5). The result can be immediately taken on by replacing the sound pressure p with the (negative) tensile stress σxx. Furthermore; we carry out the same linearisations as in Section 3.2; in particular, we replace the total acceleration by the local one and the total density ρtwith its average value ρ0. Then we arrive at:

∂σxx ∂x =ρ0 ∂vx ∂t =ρ0 ∂2ξ ∂t2 (10.15)

Combining this relation with eq. (10.8) leads to the following wave equation: ∂2ξ ∂x2 = ρ0 Y ∂2ξ ∂t2 (10.16)

By comparing this equation with earlier wave equations, for instance, with eq. (3.21), we see that the wave velocity of extensional waves on a bar is

cE1₌! Y

ρ0 (10.17)

The general solution corresponds to eq. (4.2). In a similar way the propaga- tion of extensional waves in plates with parallel boundaries is derived. Their wave velocity is found to be

cE2= !

Y

ρ0(1 − ν2) (10.18)

It is slightly higher than cE1 due to the fact that the elastic constraint in a bar is lesser than that in a plate where stress relief due to lateral contraction can only occur in one direction, namely, perpendicular to the plate surfaces. By employing eq. (10.13) it is easily verified that

cL>cE2>cE1 >cT

200 Sound waves in isotropic solids

Table 10.2 Young’s modulus and Poisson’s ratio of solids

Material Density

(kg/m3₎ Young’s modulus₍₁₀10_N/m2₎ Poisson’s ratio

Aluminium 2700 6.765 0.36 Brass (70% Cu, 30% Zn) 8600 10.520 0.37 Steel 7900 19.725 0.30 Glass (Flint) 3600 5.739 0.22 Glass (Crown) 2500 7.060 0.22 Plexiglas 1180 0.3994 0.40 Polyethylene 900 0.0764 0.45

up a force balance similar to that of eq. (3.5). The result can be immediately taken on by replacing the sound pressure p with the (negative) tensile stress σxx. Furthermore; we carry out the same linearisations as in Section 3.2; in particular, we replace the total acceleration by the local one and the total density ρtwith its average value ρ0. Then we arrive at:

∂σxx ∂x =ρ0 ∂vx ∂t =ρ0 ∂2ξ ∂t2 (10.15)

Combining this relation with eq. (10.8) leads to the following wave equation: ∂2ξ ∂x2 = ρ0 Y ∂2ξ ∂t2 (10.16)

By comparing this equation with earlier wave equations, for instance, with eq. (3.21), we see that the wave velocity of extensional waves on a bar is

cE1=! Y

ρ0 (10.17)

The general solution corresponds to eq. (4.2). In a similar way the propaga- tion of extensional waves in plates with parallel boundaries is derived. Their wave velocity is found to be

cE2= !

Y

ρ0(1 − ν2) (10.18)

It is slightly higher than cE1 due to the fact that the elastic constraint in a bar is lesser than that in a plate where stress relief due to lateral contraction can only occur in one direction, namely, perpendicular to the plate surfaces. By employing eq. (10.13) it is easily verified that

ONDAS MECÁNICAS EN SÓLIDOS

53 Figura 3.5. a) Onda de extensión (onda cuasi-longitudinal);

b) onda de flexión

Las ecuaciones (3.22) y (3.24) son válidas siempre que el espesor de la barra o de la placa sea pequeña en comparación con la longitud de onda de la onda de extensión. Si no es el caso, la velocidad de las ondas de extensión depende del espesor de la barra o de la placa y también de la frecuencia, es decir, la onda estará sujeta a dispersión.