Property Of Inertia

THE PROPERTY OF INERTIA IN A TRUE SPATIAL CONTINUUM BJØRN URSIN KARLSEN To Elizabeth Abstract. A traditional elastic medium like steel consists of material particles bound together by electromagnetic forces. The medium gets its inertial properties from the masses of these particles, and if it is compressed, its mass density increases because more particles are squeezed into a smaller volume. On the other hand mass is proportional with energy according to Einstein’s relation, E = M c2 . In this paper I will discuss a couple of ways a true spatial continuum can acquire inertia and how mass density changes by being compressed. Finally I will discuss how the wave speed is affected.

1. Spatial continuum mechanics 1.1. Strain. In this section I will introduce some necessary tensors in order to describe the deformation of an initially homogeneous spatial continuum. In a true spatial continuum it will be possible to displace spatial points away from their original positions. We consider the configuration of spatial points B0 at time t0 in a 3D Euclidian space E3. In B0 , space is undeformed and unstressed. The position vector of a point P0 of B0 relative to the origin O of an orthogonal Cartesian coordinate system is denoted by X = X i ii ,

where X i : X 1 , X 2 , X 3

are Lagrangian or material coordinates and ii = ii are unit vectors along the X i -axes. We now suppose the spatial continuum to take at a certain time t a new configuration B in E3. Thus, the point P0 is moved into the position P which will be determined with respect to the same origin by the position vector x = xi ii ,

where xi : x1 , x2 , x3

are called Euler or spatial coordinates. We now assume the mapping of B0 into B such that the correspondence of the point P0 and P is one to one and may be described by the transformation xi = xi (X 1 , X 2 , X 3 , t)

→

x = x(X, t),

X i = X i (x1 , x2 , x3 , t)

→

X = X(x, t).

which is reversible

Date: 20:08:08. Thanks to a friend who wants to be anonymous, because he has supplied me with important books, and given me the term ”The spatial continuum”. 1

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BJØRN URSIN KARLSEN

This condition is satisfied if the functions xi and X i are single valued and at least once continuously differentiable with respect to their arguments. In addition the Jacobian must be positive: ¯ ∂xi ¯ ¯ ¯ J =¯ ¯ > 0. ∂X j We now imbed a convective coordinate system into the spatial continuum, i.e. the coordinate system undergo the same deformations as the spatial continuum itself. In such a coordinate system any point will maintain the same coordinates, Θi : (Θ1 , Θ2 , Θ3 ), within the course of the deformation. It follows that (1.1)

X i = X i (Θ1 , Θ2 , Θ3 )

→

X = X(Θ1 , Θ2 , Θ3 ),

xi = xi (Θ1 , Θ2 , Θ3 )

→

x = x(Θ1 , Θ2 , Θ3 ).

and (1.2)

The coordinate system will in general be curvilinear, but it will always be possible to select it as Cartesian coordinates for example Θi = xi in B at a given time t, which would make the corresponding coordinate lines in B0 curvilinear. I will keep this possibility in mind for later use. For now we suppose the mapping of B0 into B to be described by curvilinear coordinates Θi . The line elements in the undeformed state can be derived from (1.1) ∂Θi ∂X i j j i dX , dX = dΘ , ∂X j ∂Θj and, similarly in the deformed state, from (1.2) at a fixed time t dΘi =

∂Θi j ∂xi dx , dxi = dΘj . j ∂x ∂Θj From (1.1) we also derive the base vectors related to P0 dΘi =

∂X ∂X k = ik , ∂Θi ∂Θi

Gi = X,i =

∂Θi k i , ∂X k and accordingly from (1.2) the base vectors related to P Gi =

gi = x,i =

∂x ∂xk = ik , ∂Θi ∂Θi

∂Θi k i , ∂xk The position of point P relative to P0 is called the displacement vector and denoted u. We introduce for u two sets of components Ui and ui gi =

u = u(Θi , t) = (x − X) = Ui Gi = ui gi . defined with respect to the undeformed basis Gi and the deformed basis gi , respectively. The vectorial line elements dX and dx related respectively to the material and spatial coordinates X and x are given by ∂X dX = dΘi = Gi dΘi , ∂Θi

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∂x dΘi = gi dΘi . ∂Θi The deformation gradient is defined by: dx =

F = gi ⊗ Gi ,

(1.3)

and accordingly the inverse tensors by F−1 = Gi ⊗ gi .

(1.4)

For later use we introduce the Lagrangian gradient of x referring to the undeformed state ∂x ∂Θi j ∂x j grad x = ⊗ i ⊗ i = gi ⊗ Gi = F, (1.5) = ∂X j ∂Θi ∂X j and similarly the spatial or Euler gradient grad X =

∂X ∂X ∂Θi j j ⊗ i = ⊗ i = Gi ⊗ gi = F−1 . ∂xj ∂Θi ∂xj

Hence (1.6)

F = grad (X + u) = G + grad u.

Note that G = I, the identity tensor in the undeformed base. By contracting (1.3) and (1.4) by Gi and gi respectively we acquire (1.7)

FGi = (gm ⊗ Gm )Gi = gm (Gm · Gi ) = gm δim = gi

(1.8)

F−1 gi = (Gm ⊗ gm )gi = Gm (gm · gi ) = Gm δim = Gi

from which it clearly follows that the deformation gradient transforms the convective curvilinear coordinate system from the undeformed to the deformed basis and vise versa. Furthermore by contracting (1.7) and (1.8) by dΘi we obtain gi dΘi = (FGi )dΘi = F(Gi dΘi ) Gi dΘi = (F−1 gi )dΘi = F−1 (gi dΘi )

→ →

dx = FdX, dX = F−1 dx,

from which it is clear that F can also be used to transform a line element in the undeformed basis to a line element in the deformed basis and conversely. These formulas clearly are independent of which coordinate system we choose to use. As with any second-order tensor the polar decomposition theorem states that the deformation gradient F can be multiplicatively decomposed in the form (1.9)

F = RU = vR

into a rotation tensor R, which is orthogonal, and a right stretch tensor U or a left stretch tensor v, which are supposed to be positive definite and symmetric. These translations can geometrically be interpreted as a stretch of a volume element by U, a rotation by R, and a translation by u, or a translation by u, a rotation by R, and a stretch by v. In general any symmetric second-order tensor, in this case U and v, possesses three eigenvalues λi and three orthogonal principal axes which can be determined by the eigenvectors Ni . If the tensor U refers to the orthonormal basis Ni ij

U = U Ni ⊗ Nj

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BJØRN URSIN KARLSEN

Figure 1. Deformation of volume element with edges showing in principal directions Ni or ni as a 2D illustration. it possesses only three non-vanishing components U

ii

= U ii = λi ,

U

ij

= U ij = 0

f or

i 6= j.

This permits to represent U by the spectral decomposition theorem as U=

3 X

λi Ni ⊗ Ni .

i=1

The eigenvalues λi are determined through the condition det(U − λI) = 0, and the unit vectors Ni satisfying the homogenous equation (1.10)

(U − λi I)Ni = 0

determines the mutually orthogonal principal axes of U. In accordance with (1.9) we now express U by U = RT F = RT vR. Next we insert the above expression for U into (1.10) and contract it from the left side by R. This delivers (RRT vR − λi RI)Ni = 0 (v − λi I)RNi = 0

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by taking the orthogonality condition RRT = I and the identity RI = IR into account. By inserting a new unit vector (1.11)

Ni = RT ni

ni = RNi ,

it becomes (v − λi I)ni = 0. Thus the spectral decomposition of v is given by v=

3 X

λi ni ⊗ ni

i=1

with the same eigenvalues λi as for the spectral decomposition of U and the principal direction given by (1.11). 1.2. Stress-strain relations. In this section I will postulate a stress-strain relationship that is valid for great deformations. I will also consider a general deformation on top of a uniform compression of space. In the preceding section it was pointed out that strain can be represented as a translation by u, a rotation by R, and a stretch by v. It was also pointed out that like any symmetric tensors, the stretch tensor v can be represented by the spectral decomposition theorem (1.12)

v=

3 X

λi ni ⊗ ni ,

i=1

where the eigenvalues λi are the principal values and the orthogonal base vectors ni are the associated principal directions of the stretch tensor v. The spatial continuum is per definition homogeneous, isotropic, and elastic. Hence a deformation as shown in Figure 1 will be closely related to real internal stress that generally can be expressed by the second order symmetric Cauchy stress tensor σ = σ(F), which has got to be some function of the deformation gradient F. Moreover, since it is a second order symmetric tensor, it can be decomposed by the same decomposition theorem in much of the same way as the stretch tensor v, and since the spatial continuum by definition is isotropic, it will have the same principal axes as v, so σ=

3 X

γi ni ⊗ ni ,

i=1

where the eigenvalues γi are the principal values of σ. Aside from the more or less obvious assumption that γi is some function of λi , it is for several reasons not a suitable way to go further along the classical route leading to the definition of the forth-order material tensor C, so here I have come to a point where I have got to delve into some calculated guesses and assume some more or less probable connections. Let two opposite directed forces, say outwards along the n1 -axes, act on two complementary surfaces of a small cube in the spatial continuum, see Figure 2. In order to keep the volume unchanged from the undeformed to the deformed state,

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BJØRN URSIN KARLSEN

Figure 2. Stress acting on a volume element with edges showing in principal directions Ni or ni as a 2D illustration.

we apply four mutually equal compressible forces on the four faces of the cube. Hence J = λ1 · λ3 · λ3 = 1. My first assumption is that the force in the n1 direction is proportional to ln(λ1 ) keeping J constant like 1. Next let the force in the n1 -direction be such that λ1 remains unchanged, i.e. λ1 = 1. The forces in the remaining two directions are such that the volume of the cube is changed to some arbitrary values, e.g. dV /dV0 = 1 · λ2 · λ3 = J. My next assumption is that the forces acting on the faces orthogonal to those in the n1 -dirction needed to keep λ1 unchanged is proportional to ln(J). Finally, if both the volume and the stretch along one or more of the principal axes are changed, I will combine these two assumptions and postulate that the stress-strain relation for the spatial continuum is given by (1.13)

γi = 2µ ln(λi ) + 2µβ ln(J) = 2µ ln(λi · J β ),

where 2µ and β are constants1. Hence the Euler stress tensor is given by the fairly simple relation (1.14)

σ=

3 X

¢ ¡ 2µ ln λi · J β ni ⊗ ni .

i=1

1I have chosen to include the number 2 because then µ turns up to be identical with the Lame ´ elastic constant.

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Figure 3. Stress caused by two successive deformations, one pure compression followed by an infinitesimal strain, as a 2D illustration.

Generally ln I = ln

3 X

ni ⊗ ni ,

i=1

and for any scalar a we have ln(aI) = ln[I + (aI − I)] = ln[I + I(a − 1)] = I(a − 1) − 12 I2 (a − 1)2 + 13 I3 (a − 1)3 · · · £ ¤ = I (a − 1) − 21 (a − 1)2 + 13 (a − 1)3 · · · (1.15)

= I ln a,

so by (1.12) Equation (1.14) can finally be write σ = 2µβ ln

3 X

(J ni ⊗ ni ) + 2µ ln

i=1

(1.16)

3 X

(λi ni ⊗ ni )

i=1

= 2µβ ln(J I) + 2µ ln v.

There is still another situation to be considered. Suppose that space initially is compressed and then gets an additional deformation. The goal would then be to consider what additional stress this deformation would cause. In Figure (3) I have illustrated the situation as a 2D representation. We first have a uniform compression of space bringing a volume element from B0 to B and then an arbitrary deformation bringing the volume element from B to B. The principal axes for B and B follow by the spectral decomposition theorem, and since the deformation between B0 and B is uniform without rotation, Ni can be chosen at will like Ni .

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BJØRN URSIN KARLSEN

The first step can be described by the the relation σ = 2µ

3 X

ln(J β λ N1 ⊗ N1 ),

i=1

and the next step would bring the stress tensor up to σ = 2µ

3 X

ln(J β · δJ β · λ · δλ1 n1 ⊗ n1 )

i=1

(1.17)

= 2µ

3 X

ln(J β λ n1 ⊗ n1 ) + 2µ

i=1

3 X

ln(δJ β · δλ1 n1 ⊗ n1 ).

i=1

The change of stress can be represented by the difference between these two stress components and we finally arrive at δσ = 2µ

3 X

ln(δJ β · δλ1 n1 ⊗ n1 ),

i=1

or by changing the notation δσ, δJ and δλi into σ, J and λi respectively we get (1.18)

σ = 2µ

3 X

ln(J β λ1 n1 ⊗ n1 ).

i=1

Hence the additional strain-stress relation for deformations in a uniformly compressed or expanded space is the same as it is in the undeformed space. 1.3. Velocity and acceleration. In order to find the acceleration of points in the spatial continuum special care has got to be taken if spatial description is considered. The following notation for the ’material’ time derivative in spatial notation is adopted from the mechanics of solids [1, page 115 ff.] Df = f˙ , Dt where f˙ is given by ∂ f˙ f˙ = + ( grad f )v. ∂t The velocities, v, of spatial points are given by the intrinsic function v = u˙ =

∂u + ( grad u)v, ∂t

a=u ¨=

∂ u˙ + ( grad u) ˙ u. ˙ ∂t

and the acceleration by

By infinitesimal deformations the last term is negligible and the acceleration is simply a=u ¨=

∂2u . ∂t2

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1.4. The strain-stress relation for small deformations. In this section I will show that a small variation on top of a great compression of space obeys the same law that we find in a classical continuum. The spatial continuum is per definition isotropic, and a uniform compression of space will not alter this property. It can be shown that in an isotropic continuum a stress function σ(v) can be obtained simply by setting v = (FFT )1/2 , see [1, page 147]. By this relation and (1.6) the strain-stress relation (1.16) can be developed into σ = 2µβ ln(J I) + 2µ ln(FFT )1/2 ¡ ¢ = 2µβ ln(λ1 λ2 λ3 I) + 2µ 12 ln F + ln FT = 2µβ(ln λ1 + ln λ2 + ln λ3 )I h ¡ ¢i + 2µ 12 ln(I + grad u) + ln I + ( grad u)T . By linearization of the tensor valued function of σ with respect on u we get (1.19)

Lσ(u, ∆u) = σ(u) + ∆σ(u, ∆u),

where the second term on the right hand side is the Gateaux-derivative defined by ∆σ(u, ∆u) =

d σ(u + ε∆u). dε

Equation (1.19) can be developed into ∆λ1 ∆λ2 ∆λ3 + + )I λ1 λ2 λ3 1 h ( grad ∆u) ( grad ∆u)T i¯¯ + 2µ + ¯ . 2 I + grad u I + ( grad u)T ε=0

Lσ(u, ∆u) = σ(u) + 2µβ(

By the identity (u = 0) ⇒ (λi = 1 ∧ grad u = 0) we have

(1.20)

Lσ(u = 0, ∆u) = 2µβ(∆λ1 + ∆λ2 + ∆λ3 )I i £ + 2µ 21 ( grad ∆u) + ( grad ∆u)T .

We now introduce the linear tensor valued function at the point where u = 0 (1.21)

²=

¢ 1¡ grad ∆u + ( grad ∆u)T 2

From (1.12) we have that tr v = vii = λ1 + λ2 + λ3 , so (1.22)

∆λ1 + ∆λ2 + ∆λ3 = tr (∆v).

By (1.9) and (1.5) we have v = FRT = (I + grad u)RT

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so

BJØRN URSIN KARLSEN

³

´ ∂v ∆( grad u) ∂( grad u) £ T ¤ £ ¤ = tr R grad (∆u) = tr grad (∆u) £ ¡ ¢¤ = tr sym grad (∆u) = tr ².

tr (∆v) = tr

(1.23)

By (1.21), (1.22), and (1.23) Equation (1.20) finally takes the form (1.24)

σ = λ tr ² I + 2µ²,

where λ = 2µβ. This is the same relation as we encounter in literature on the strain stress-relation for small deformations in a homogeneous and isotropic elastic continuum, and it follows that the elastic potential is given by W =

λ (tr ²)2 + µ tr ²2 , 2

see e.g. [1, page 167]. As shown in Equation (??), (??), and (??), the strain-stress relation (1.24) and the Cauchy equation of motion ρ¨ u = div σ + b [1, page 126], where b is some hypothetical force per unit volume and ρ is the the inertial density to be discussed in the next section, can be developed into the Navier-Cauchy equation (1.25)

(λ + 2µ) grad div u − µ curl curl u + b = ρ¨ u,

and further into two wave equations (1.26)

∇2 (div u) −

1 ∂ 2 (div u) = 0, c12 ∂t2

(1.27)

∇2 (curl u) −

1 ∂ 2 (curl u) = 0, c22 ∂t2

representing longitudinal waves moving with speed c1 and transversal waves moving with speed c2 respectively. The velocities are s c1 =

λ + 2µ , ρ r

(1.28)

c2 =

µ . ρ

The relation between c1 and c2 is s λ + 2µ p (1.29) = 2(β + 1), c1 /c = µ and we notice that longitudinal waves are travelling with the double of the speed of transversal waves if β should happen to be unity. Finally we notice that both µ and λ are true universal constants according to the definitions in Equation (1.13) and (1.24).

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1.5. Inertia and the speed of waves as a function of the compression of space. Elastic waves can only propagate through space if it has inertial properties. In this section I will consider two possible ways through which inertia can enter the stage. The classical approach. The simplest possibility is the classical assumption that inertia is an intrinsic property of the continuum such that the spatial density in uncompressed space is like a universal constant ρ0 and increases or decreases if space is being compressed or inflated according to the relation ρclassical = ρ0

dV0 = ρ0 D. dV

By the classical approach the propagating velocity of transversal waves as a function of compression are given by r µ . c2 (D) = ρ0 D Let us consider small variations of the waves speed c in an area of space that initially is compressed by a factor D. This can be achieved by the linearization of c2 with respect to D which is defined by (1.30)

L c2 (D, ∆D) = c(D) + ∆c(D, ∆D),

where the second term on the right-hand side as usual is the Gateaux-derivative defined by ¯ d ¯ ∆c2 (D, ∆D) = c(D + ε∆D)¯ , (1.31) dε ε=0 so ¯ ³ µ ´1/2 d ¯ ∆c2 (D, ∆D) = (D + ε∆D)−1/2 ¯ ρ0 dε ε=0 ¯ ³ µ ´1/2 ³ 1 ´¡ ¯ − D + ε∆D)−3/2 · ∆D¯ = ρ0 2 ε=0 r µ ∆D 1 (1.32) · . =− 2 ρ0 D D Generally we have the identity (1.33)

1 V →0 V

ZZ

div u = lim

∆V , V →0 V

u n dA = lim A

and the inverse of the Jacobian is given by D = J −1 = lim

V →0

V0 , V

so by (1.33) ∂D −V0 ∆V ∆V = lim ∆V = −D lim V →0 V 2 V →0 V ∂V = − D div (∆u),

∆D = (1.34)

where div (∆u) is the spatial divergence in Euler coordinates.

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BJØRN URSIN KARLSEN

By (1.34) and (1.32) we finally acquire from (1.30) the variations of c as a function of small deformations u in a space that initially is compressed by a factor D corresponding to a wave speed of c r µ (1.35) c2 = (1 + 12 div u) = c(1 + 12 div u). ρ0 D In later discussions it is the gradient of c2 that is of most interest so we write (1.36)

grad c2 = α c grad div u,

where α = 12 .

(1.37)

We notice that by this approach the wave speed of transversal waves varies as a function of small displacements u independent of how much space initially is compressed or inflated. The spatial approach. Another possibility is that space in its undeformed state has no inertia at all and only gets its inertia by the energy going into it by being compressed or inflated. This is more in line with how earthly matter gets its mass according to the famous equation e = mc2 where e is the energy dencity in a body and c is the speed of light. The property m then becomes the definition of mass. If this is how Nature works, the spatial mass density might be given by ρs = e/κ 2 , where e is the deformation energy per unit volume going into space by compression or inflation, and κ is a universal constant. This is a very interesting possibility because it would mean that the speed of any elastic waves will approach infinity if space is in a nearly undeformed state. In order to investigate this possibility, it will be necessary to find the energy going into space by compression. For simplicity I will consider a space where the constant β is set to unity. By uniform compression of space, a little sphere with radius r0 will shrink to a sphere with radius r. The force acting on a unit surface element of the sphere is given by the Cauchy stress vector t = σn where n is the unit vector normal to the surface. Since the pressure is uniform in all directions the Jacobian is given by J=

dV = λ3 dV0

where λ = λ1 = λ2 = λ3 .

The Euler stress vector is normal to any surface we choose, e.g. on a surface whose unit normal points in the n1 -direction t1 = σn1 = 2µ

3 X

ln λ4 (ni ⊗ ni )n1 = 2µ ln λ4 n1

i=1

Hence t is normal to the surface of the sphere and is pointing outwards by stretch. The energy going into the sphere as potential energy is given by the surface of

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the sphere times the Cauchy stress vector times the enlargement of the radius, so according to (1.18) we acquire dEr = 4πr2 (σn)ndr = 8πµr2 ln(λ4 )dr ³r´ = 32πµr2 ln dr r0 = 32πµr2 (ln r − ln r0 ).

(1.38)

The whole energy going into the sphere if it is inflated or deflated from an undeformed state (or previous uniformly compressed or expanded state) with radius r0 to its final radius r is given by the definite integral Z r Er =32πµ (r2 ln r − r2 ln r0 )dr r0

ir r3 r3 − ln r0 3 9 3 r0 h r3 3 3 r r r 3 =32πµ ln r − 0 ln r0 − + 0 3 3 9 9 i r3 r03 − ln r0 + ln r0 3 3 3h r 3i 32πµr 3 ln r − 3 ln r0 + 1 − 03 = 9 r ³ r ´3 ³ r ´3 i 8µ h 0 0 = Vr 1 − − ln 3 r r Er 8µ £ −1 −1 = J − 1 − ln J ], Vr 3 =32πµ

(1.39)

h r3

ln r −

where Vr is the volume of the deformed sphere. If we define the compression of space as D = J −1 = dV0 /dV we finally get for the energy density of the compressed space measured by Euler or spatial coordinates (1.40)

es =

¤ 8µ £ (D − 1) − ln D . 3

If we chose to postulate that ρ is a function of the energy density of space, then it is given by (1.41)

ρs =

¤ 8µ £ (D − 1) − ln D . 3κ 2

By (1.41) and (1.28) the speed of transversal waves in a space with varying compression is given by ³ ´1/2 £ ¤−1/2 (1.42) . c2 (D) = κ 38 (D − 1) − ln D We notice that the wave-speed only is depending on the constant κ and the compression of the spatial continuum. Next let us consider small variations of the waves speed c in an area of space that initially is compressed by a factor D0 . This can be achieved by the linearization of c with respect to D which is defined by

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BJØRN URSIN KARLSEN

¯ ¯ L c2 (D, ∆D) = c(D) + ∆c(D, ∆D)¯

c2 =c, D=D0

,

where the second term on the right-hand side is the Gateaux-derivative defined by (1.31). Thus we obtain from (1.42) by means of (1.31) ∆c2 (D, ∆D) ³ 3 ´ 12 d h i− 12 =κ (D + ε∆D − 1) − ln(D + ε∆D) 8 dε c D−1 ∆D =− 2 (D − 1) − ln D D c³ ln D ´−1 = (1.43) 1− div (∆u). 2 D−1 Finally we acquire for variations of c2 as a function of small deformations u = ∆u in a space that initially is compressed by a factor D corresponding to a wave speed of c c³ ln D ´−1 c2 = c + (1.44) 1− div u, 2 D−1 and the gradient of c is given by (1.45)

grad c2 = α c grad div u,

where (1.46)

α=

1³ ln D ´−1 1− , 2 D−1

or by (1.40) 4(D − 1)µ . 3es By the spatial approach the gradient of the wave speed is proportional to grad div u as it was by the classical approach only differing from that by a somewhat smaller constant depending on the initial compression of space.

(1.47)

α=

References 1. Yavuz Ba¸sar and Dieter Weichert, Nonlinear continuum mechanics of solids, Springer, 1999. E-mail address: [email protected]