Lie Point Symmetries, Conservation and Balance Laws ... - Springer Link

J Elasticity (2007) 88:5–25 DOI 10.1007/s10659-007-9105-5

Lie Point Symmetries, Conservation and Balance Laws in Linear Gradient Elastodynamics Markus Lazar · Charalampos Anastassiadis

Received: 24 November 2006 / Accepted: 15 February 2007 / Published online: 27 June 2007 © Springer Science + Business Media B.V. 2007

Abstract The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether’s theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity. Keywords Conservation laws · Symmetries · Gradient elasticity · Path-independent integrals Mathematics Subject Classifications (2000) 73A05 · 73R05 · 73B10 · 73V25

1 Introduction Conservation laws are of great interest for problems in elastostatics and elastodynamics [15, 22]. The translational, rotational and scaling symmetries and the corresponding conservation laws have been investigated in elastostatics by Günther [11], Knowles and Sternberg [16], Olver [30] and in elastodynamics by Fletcher [6], Morse and Feshbach [29]. The translational, rotational and scaling conservation laws correspond to the J, L and M integrals [2]. They are of importance for problems concerning defects (cracks, dislocations) and phase transitions. More from the mathematical point of view, Lie group analysis and the Noether theorem have been applied by Knowles and Sternberg [16], Olver [30] to obtain conservation laws in elasticity.

M. Lazar (B) · C. Anastassiadis Emmy Noether Research Group, Department of Physics, Darmstadt University of Technology, Hochschulstr. 6, D 64289 Darmstadt, Germany e-mail: [email protected]

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M. Lazar, C. Anastassiadis

One possible generalization of elasticity and elastodynamics are the so-called gradient elasticity and gradient elastodynamics (Mindlin [26], Mindlin and Eshel [27], Mindlin [28]). In a gradient theory, the Lagrangian possesses higher gradients of the physical variables. In general, gradient elastodynamics can be obtained from a socalled restricted Mindlin continuum or pseudo micromorphic elastodynamics. Such a gradient elastodynamics is the case II in Mindlin’s theory [26]. For the first time, in gradient elasticity of grade-2 Eshelby [4, 5] derived the static energy-momentum tensor, which is called in continuum mechanics the Eshelby stress tensor. The J, L and M integrals were derived by Lazar and Kirchner [18] for anisotropic, linear, gradient elasticity with inhomogeneities and dislocations. Maugin and Trimarco [21], Maugin [22] gave the Eshelby stress tensor for a simple gradient elastodynamics of grade-2. Gradient elasticity and elastodynamics coupled with electric fields were investigated by Huang and Batra [12] and Kalpakides and Agiasofitou [14], respectively. But these authors have assumed the usual form of the kinetic energy neglecting terms which are obtained in the limit from a restricted Mindlin continuum to gradient elastodynamics. The aim of the present paper is the investigation of the Lie point symmetries and conservation laws via the Noether theorem for linear gradient elastodynamics of grade-2. We take into consideration second gradients of the displacement and first gradients of the velocity. Both the Lie point symmetries of the Euler-Lagrange equations and the variational and divergence symmetries will be determined. Finally, we generalize the obtained conservation laws to material balance laws in order to introduce configurational forces, configurational vector moments and scalar moments in gradient elastodynamics.

2 Gradient Elastodynamics We consider a theory of gradient elasticity which was originally derived from micromorphic elasticity. It is obtained as a so-called restricted micromorphic elasticity or a restricted Mindlin continuum (pseudo micromorphic elasticity) where the micro-distortion is equal to the displacement gradient. In a restricted micromorphic continuum, like in a restricted Cosserat theory, the relative distortion is zero [7, 26]. On the other hand, the micro-deformation gradient is changed by second gradients of the displacement vector of the macro-medium. Thus, we have to consider gradients of the velocity and second gradients of the displacement. Their reason are microstructural effects. Let the Lagrangian (Lagrange density) of gradient elastodynamics be of the form

L = L(uα , u˙ α , uα,i , u˙ α,i , uα,ij).

(2.1)

It corresponds to the case II in Mindlin’s theory [26], the long frequency, very longwave approximation of micromorphic elastodynamics. The Euler-Lagrange equation of such a gradient theory is calculated as Eα (L) =

∂L ∂L ∂L ∂L ∂L − Dt − Di + Dt Di + Di Dj = 0. ∂uα ∂ u˙ α ∂uα,i ∂ u˙ α,i ∂uα,ij

(2.2)

Lie point symmetries, conservation and balance laws in linear gradient elastodynamics

7

Here Di and Di are the total derivatives: Dt =

∂ ∂ ∂ ∂ ∂ + u¨ α + u˙ α, j + u¨ α, j + ... , + u˙ α ∂t ∂uα ∂ u˙ α ∂uα, j ∂ u˙ α, j

Di =

∂ ∂ ∂ ∂ ∂ ∂ + uα,i + uα,ij + u˙ α,i + uα,ijk + u˙ α,ij + . . . . (2.4) ∂ xi ∂uα ∂uα, j ∂ u˙ α ∂uα, jk ∂ u˙ α, j

(2.3)

The Lagrangian is given in terms of kinetic energy and strain energy according to

L = T − W,

(2.5)

where the ‘kinetic’ energy density 1 1 pα u˙ α + pα ju˙ α, j , 2 2

T=

(2.6)

and the strain energy density reads W=

1 1 τα juα, j + τα jk uα, jk . 2 2

(2.7)

It can be seen that in (2.6) the ‘classical’ kinetic energy is changed due to the gradient of the velocity. Its origin are the micro-inertia term and micro-velocity in micromorphic elasticity. On the other hand, in (2.7) the ‘classical’ strain energy is modified due to second gradient terms of the displacement. The canonical conjugate quantities are defined as pα =

∂L , ∂ u˙ α

pα j =

∂L , ∂ u˙ α, j

τα j = −

∂L , ∂uα, j

τα jk = −

∂L , ∂uα, jk

(2.8)

where pα is the physical momentum vector, τα j is the stress tensor and τα jk is the double stress tensor. pα j is canonical conjugate to the velocity gradient tensor or deformation rate tensor. Using (2.2), the Euler-Lagrange equation is of the form Dt pα − Dj pα j − Dj τα j − Dk τα jk = 0, Form A. (2.9) In (2.9), the term peff α = pα − Dj pα j ,

(2.10)

has the meaning of an effective momentum vector and ταeffj = τα j − Dk τα jk ,

(2.11)

is an effective stress tensor of form A, which is an objective tensor. The additional term in pα j is coming from inertia. With these definitions, (2.9) takes the form eff Dt peff α − Dj τα j = 0.

(2.12)

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Because Dt and Dj commute trivially, we can rewrite (2.9) in the following form Dt pα − Dj τα j − Dk τα jk + Dt pα j = 0, Form B, (2.13) where one may consider the term, σα j = τα j − Dk τα jk + Dt pα j,

(2.14)

as the so-called effective stress tensor of form B. Since the last term, it is not an objective tensor. Here pα j has the form of a kind viscosity. With this definition, (2.13) takes the form Dt pα − Djσα j = 0.

(2.15)

We want to note that this form B has been used, e.g., by Georgiadis and Vardoulakis [7], Georgiadis [8], Georgiadis et al. [9]. For an isotropic theory, the linear constitutive equations become pα = ρ u˙ α ,

(2.16)

pα j = ηI (u˙ α, j + u˙ j,α ) + ηII δα ju˙ k,k ,

(2.17)

τα j = μ(uα, j + u j,α ) + λ δα juk,k ,

(2.18)

1 τα jk = a1 δα jul,lk + δαk ul,lj + a2 δα juk,ll + δαk u j,ll + 2δ jk ul,lα 2 + 2a3 δ jk uα,ll + 2a4 uα, jk + a5 uj,kα + uk, jα ,

(2.19)

where ρ is the mass density, ηI and ηII are the coefficients for the velocity gradients, μ and λ are the Lamé constants, and a1 , · · · a5 are Mindlin’s gradient coefficients. The field equation of isotropic gradient elastodynamics (form A and B) expressed in terms of the displacement vector is given by ρ u¨ α − h21 u¨ α − h22 u¨ j, jα − μ 1 − 21 uα − (λ + μ) 1 − 22 u j, jα = 0, (2.20) with 21 = 2 h21 =

a3 + a4 , μ

ηI , ρ

22 = 2 h22 =

a1 + a2 + a5 , λ+μ

ηI + ηII . ρ

(2.21) (2.22)

Equation (2.20) is a system of three linear partial differential equations of fourth order: ≡ (1 , . . . , 3 ) = 0. Mixed fourth-order space-time derivatives appear in (2.20) like in the regularized long-wave Boussinesq equation (see, e.g., [23, 24]). If we neglect velocity gradients and thus pα j = 0, we obtain from (2.9) and (2.13) the Euler-Lagrange equation Dt pα − Dj τα j − Dk τα jk = 0, Form C (2.23) and in terms of the displacement vector it reads ρ u¨ α − μ 1 − 21 uα − (λ + μ) 1 − 22 u j, jα = 0,

(2.24)

where denotes the Laplacian. In such a gradient theory the kinetic energy is left unchanged. It only contains higher gradients of the displacement vector. Here such


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gradient theory is called gradient elasticity of form C. Gradient elasticity of form C has been investigated by Huang and Batra [12], Maugin and Trimarco [21], Maugin [22], Mindlin and Eshel [27], Mindlin [28] and Kalpakides and Agiasofitou [14].

3 Mathematical Preliminaries The present goal is to find the Lie group of invariance of (2.20). Here x ∈ Rn and t ∈ R are the independent and u are the dependent variables. They build the space (x, t, u) ∈ Rn × R × R p (in our case we have n = 3 and p = 3). The variables x and t denote the independent variables which are the material space variables and u is the dependent variable. Thus, u is the physical space variable. Let G be the Lie group of invariance of equation (2.20). The infinitesimal group action has the form xi = xi + ε Xi (x, t, u) + · · · ,

(3.1)

t = t + ετ (x, t, u) + · · · ,

(3.2)

uα = uα + εU α (x, t, u) + · · · , and the infinitesimal generators of G are defined by ∂ x ∂t Xi (x, t, u) := i , τ (x, t, u) := , ∂ε ε=0 ∂ε ε=0

U α (x, t, u) :=

(3.3) ∂uα , ∂ε ε=0

(3.4)

where ε is a group parameter, i = 1, . . . , n and α = 1, . . . , p. The vector field v is called the infinitesimal generator of G [1, 13, 31]: v = Xi (x, t, u)

∂ ∂ ∂ + τ (x, t, u) + U α (x, t, u) . ∂ xi ∂t ∂uα

(3.5)

The second order prolongation of the vector field v, defined on the corresponding second order jet space, is given by pr(2) v = v + U¯ αi

∂ ∂ ∂ ∂ ∂ + U¯ αt + U¯ αij + U¯ αit + U¯ αtt , ∂uα,i ∂ u˙ α ∂uα,ij ∂ u˙ α,i ∂ u¨ α

(3.6)

where U¯ αi = Di (U α − Xk uα,k − τ u˙ α ) + Xk uα,ki + τ u˙ α,i ,

(3.7)

U¯ αt = Dt (U α − Xk uα,k − τ u˙ α ) + Xk u˙ α,k + τ u¨ α ,

(3.8)

U¯ αij = Di Dj(U α − Xk uα,k − τ u˙ α ) + Xk uα,kij + τ u˙ α,ij,

(3.9)

U¯ αit = Di Dt (U α − Xk uα,k − τ u˙ α ) + Xk u˙ α,ki + τ u¨ α,i , ... U¯ αtt = Dt Dt (U α − Xk uα,k − τ u˙ α ) + Xk u¨ α,k + τ u α .

(3.10) (3.11)

Using the characteristic Qα of the vector field v Qα = U α − X juα, j − τ u˙ α ,

(3.12)

the second order prolongation (3.6) is now given by pr(2) v = pr(2) v Q + Xi Di + τ Dt ,

(3.13)

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where ∂ ∂ ∂ + Di Qα + Dt Qα ∂uα ∂uα,i ∂ u˙ α

pr(2) v Q = Qα

+ Di Dj Qα

∂ ∂ ∂ + Di Dt Qα + Dt Dt Qα . ∂uα,ij ∂ u˙ α,i ∂ u¨ α

(3.14)

The group G is a group of invariance of a system of partial differential equation of fourth order if and only if [31] pr(4) v() = 0,

= 0,

whenever

(3.15)

(4)

for every infinitesimal generator v of G. Here pr v denotes the prolongation of fourth order. Using (2.20), one finds the infinitesimal invariance criterion (3.15) to be ¨ ρ U¯ α − h21 U¨¯ α − h22 U¨¯ j, jα − μ 1 − 21 U¯ α − (λ + μ) 1 − 22 U¯ j, jα = 0 =0 , (3.16) which must be fulfilled. By the help of existing computer algebra systems (e.g. Desolv), one can easily find the symmetries. It turns out that the Lie algebra of infinitesimal symmetries of the field equation (2.20) is spanned by the following Lie symmetries: vi1 =

∂ ∂ xi

v2 =

∂ ∂t

vi3 = ijk x j v 4 = ui

∂ ∂ + uj ∂ xk ∂uk

(translations in space),

(3.17)

(translation in time),

(3.18)

(rotations),

(3.19)

(scaling),

(3.20)

(addition of solutions),

(3.21)

∂ ∂ui

and the linear symmetry v 5 = fi (x)

∂ ∂ui

where fi (x) is an arbitrary solution of (2.20). The first and the second pieces of (3.19) are the generators for orbital (space) and intrinsic (spin) rotations (rotation transformation of the independent and dependent variables). v 4 is the generator for intrinsic scaling (scaling of the dependent variables). Unlike classical elasticity, no scaling of the independent variables v = xi ∂xi appears as symmetry of the Euler Lagrange equation.

4 Conservation Laws A conservation law for = 0 is of the form Di Ai + Dt A4 = 0,

(4.1)


11

where Ai is a vector flux and A4 is a conserved density. Using the divergence theorem, one finds the conservation law in integral form

Ai ni dS + S

Dt A4 dV = 0,

(4.2)

V

where ni is the normal to the surface S and V is the volume element. A Lie group G is a variational or divergence symmetry of L if and only if [31] pr(2) v(L) + L (Di Xi + Dt τ ) = Di Bi + Dt B4 ,

(4.3)

where Bi and B4 are opportune analytic functions. If Bi = 0 and B4 = 0, then v is the generator of a divergence symmetry of the Lagrangian L. If Bi = 0 and B4 = 0, v generates a variational symmetry of L. Every variational or divergence symmetry of the Lagrangian L is also a symmetry of the associated Euler-Lagrange equations. But not vice versa. Now we substitute the prolongation formula (3.13) into the infinitesimal invariance criterion and find Di Bi + Dt B4 = pr(2) v Q (L) + Di (L Xi ) + Dt (Lτ ),

(4.4)

with pr(2) v Q (L) = Qα

∂L ∂L ∂L ∂L ∂L + Di Qα + Dt Qα + Di Dj Qα + Dt Di Qα . ∂uα ∂uα,i ∂ u˙ α ∂uα,ij ∂ u˙ α,i (4.5)

Using (4.5), the first term on the right hand side of (4.4) can by integrated by parts. Due to the mixed space-time term Dt Di in (4.5) we have two possibilities for the integration by parts. The first possibility which corresponds to the Form A of the Euler-Lagrange equation is

∂L ∂L ∂L ∂L pr v Q (L) = Qα Eα (L) + Di Qα + Dj Qα − Qα Dj + Dt Qα ∂uα,i ∂uα,ij ∂uα,ij ∂ u˙ α,i ∂L ∂L . (4.6) − Qα Dj + Dt Qα ∂ u˙ α ∂ u˙ α, j (2)

Combining now (4.4) and (4.6) and using the Euler-Lagrange equation Eα (L) = 0, we finally find the following fluxes:

∂L ∂L ∂L ∂L + Dj Qα − Dj + Dt Qα + L Xi − Bi , Ai = Qα ∂uα,i ∂uα,ij ∂uα,ij ∂ u˙ α,i

∂L ∂L − Dj + Lτ − B4 . A4 = Qα ∂ u˙ α ∂ u˙ α, j

(4.7) (4.8)

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Using the characteristic (3.12), we obtain ∂L ∂U α ∂U α ∂L − Dj + uβ, j + ∂uα,i ∂uα,ij ∂xj ∂uβ ∂ Xk ∂τ ∂τ ∂L − Xk uα,kj − u˙ α − u˙ α uβ, j − τ u˙ α, j ∂uβ ∂xj ∂uβ ∂uα,ij

Ai = L Xi + U α − Xk uα,k − τ u˙ α − uα,k

∂ Xk − uα,k uβ, j ∂xj

∂U α ∂ Xk ∂ Xk ∂τ ∂U α − uα,k − Xk u˙ α,k − u˙ α + u˙ β − uα,k u˙ β ∂t ∂uβ ∂t ∂uβ ∂t ∂L ∂τ − τ u¨ α − Bi , −u˙ α u˙ β ∂uβ ∂ u˙ α,i

∂L ∂L − Dj − B4 . A4 = Lτ + U α − Xk uα,k − τ u˙ α ∂ u˙ α ∂ u˙ α, j +

(4.9) (4.10)

The other possibility corresponding to Form B of the Euler-Lagrange equation is given by ∂L ∂L ∂L ∂L pr(2) v Q (L) = Qα Eα (L) + Di Qα + Dj Qα − Qα Dj − Qα Dt ∂uα,i ∂uα,ij ∂uα,ij ∂ u˙ α,i ∂L ∂L + Dj Qα . (4.11) + Dt Qα ∂ u˙ α ∂ u˙ α, j Combining now (4.4) and (4.6) and using the Euler-Lagrange equation Eα (L) = 0, we obtain the following fluxes:

Ai = Qα A4 = Qα

∂L ∂L ∂L ∂L − Dj − Dt + Dj Qα + L Xi − Bi , ∂uα,i ∂uα,ij ∂ u˙ α,i ∂uα,ij

∂L ∂L + Dj Qα + Lτ − B4 . ∂ u˙ α ∂ u˙ α, j

(4.12) (4.13)

Using the characteristics Qα given in Eq. (3.12), the fluxes can be expressed as

∂L ∂L ∂L Ai = L Xi + U α − Xk uα,k − τ u˙ α − Dj − Dt ∂uα,i ∂uα,ij ∂ u˙ α,i ∂U α ∂ Xk ∂ Xk ∂U α + uβ, j − uα,k − uα,k uβ, j − Xk uα,kj + ∂xj ∂uβ ∂xj ∂uβ ∂τ ∂τ ∂L − u˙ α − u˙ α uβ, j − τ u˙ α, j − Bi , (4.14) ∂xj ∂uβ ∂uα,ij ∂L ∂U α ∂ Xk ∂U α + + uβ, j − uα,k A4 = Lτ + U α − Xk uα,k − τ u˙ α ∂ u˙ α ∂xj ∂uβ ∂xj ∂L ∂ Xk ∂τ ∂τ − uα,k uβ, j − Xk uα,kj − u˙ α − u˙ α uβ, j − τ u˙ α, j − B4 . (4.15) ∂uβ ∂xj ∂uβ ∂ u˙ α, j


13

4.1 Translations in Space and Time Because the translation is a variational symmetry, Bi = 0 and B4 = 0. The translation group acts on the independent variables, only. The translation in space and time is given by the formulae: xi = xi + εk δki ,

(4.16)

t = t + ε4 δ44 , uα

(4.17)

= uα .

(4.18)

The corresponding generators of infinitesimal transformations are Xki = δki ,

τ = δ44 ,

U α = 0.

(4.19)

4.1.1 Form A Using (4.9), (4.10) and (4.19), the flux quantities are given by ∂L ∂L ∂L ∂L − Dj − u˙ α,k , − uα,kj Aki = L δki − uα,k ∂uα,i ∂uα,ij ∂uα,ij ∂ u˙ α,i ∂L ∂L − Dj , Ak4 = −uα,k ∂ u˙ α ∂ u˙ α, j ∂L ∂L ∂L ∂L − Dj − u˙ α, j − u¨ α , A4i = −u˙ α ∂uα,i ∂uα,ij ∂uα,ij ∂ u˙ α,i ∂L ∂L − Dj . A44 = L − u˙ α ∂ u˙ α ∂ u˙ α, j

(4.20) (4.21) (4.22) (4.23)

In terms of stresses they read Pki := −Aki = −L δki − uα,k ταi − Djταij − uα,kj ταij + u˙ α,k pαi , Pk := Ak4 = −uα,k pα − Dj pα j , Si := A4i = u˙ α ταi − Djταij + u˙ α, j ταij − u¨ α pαi , H := −A44 = u˙ α pα − Dj pα j − L.

(4.24) (4.25) (4.26) (4.27)

The tensor Pki is the total Eshelby stress tensor generalized to gradient elasticity of form A. The vector Pk is the pseudomomentum or field momentum density and the vector Si is called the field intensity or Poynting vector of form A. The scalar H is the total energy density called the Hamiltonian. We note that (4.27) is close to the result obtained by Podolsky and Kikuchi [32] in generalized electrodynamics. On the other hand, the Hamiltonian (4.27) differs from the usual structure: H = T + W. It can be rewritten in the form: H = T + W − Dj(u˙ α pα j). Thus, it differs by the term Dj(u˙ α pα j). It is zero, since the integral of a divergence over all space equals the net outflow integral at infinity. An expression like the pseudomomentum (4.25) can be found in [24] (see (4.6.24)). Also the pseudomomentum derived in one-dimensional elastic models by [25] (see (13)) has a similar form.

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4.1.2 Form B With (4.14), (4.15) and (4.19), we obtain for the flux quantities ∂L ∂L ∂L ∂L Aki = L δki − uα,k − Dj − Dt − uα,kj , ∂uα,i ∂uα,ij ∂ u˙ α,i ∂uα,ij Ak4 = −uα,k A4i = −u˙ α

∂L ∂L − uα,kj , ∂ u˙ α ∂ u˙ α, j

(4.29)

∂L ∂L ∂L − Dj − Dt ∂uα,i ∂uα,ij ∂ u˙ α,i

A44 = L − u˙ α

(4.28)

− u˙ α, j

∂L , ∂uα,ij

∂L ∂L − u˙ α, j . ∂ u˙ α ∂ u˙ α, j

(4.30) (4.31)

In terms of stresses they have the form Pki := −Aki = −L δki − uα,k ταi − Djταij + Dt pαi − uα,kj ταij,

(4.32)

Pk := Ak4 = −uα,k pα − uα,kj pα j, Si := A4i = u˙ α ταi − Djταij + Dt pαi + u˙ α, j ταij,

(4.34)

H := −A44 = u˙ α pα + u˙ α, j pα j − L.

(4.35)

(4.33)

The tensor Pki is the total Eshelby stress tensor in gradient elasticity of form B. The vector Pk is the pseudomomentum or field momentum density and the vector Si is the field intensity or Poynting vector of form B. The scalar H = T + W is the total energy density called the Hamiltonian of form B. 4.1.3 Form C If we set pα j = 0, the flux quantities are now given by ∂L ∂L ∂L − Dj − uα,kj , Aki = L δki − uα,k ∂uα,i ∂uα,ij ∂uα,ij Ak4 = −uα,k A4i = −u˙ α

∂L , ∂ u˙ α ∂L ∂L − Dj ∂uα,i ∂uα,ij

A44 = L − u˙ α

(4.36) (4.37)

− u˙ α, j

∂L , ∂uα,ij

∂L . ∂ u˙ α

(4.38) (4.39)

In terms of stresses they are

Pki := −Aki = −L δki − uα,k ταi − Djταij − uα,kj ταij,

(4.40)

Pk := Ak4 = −uα,k pα , Si := A4i = u˙ α ταi − Djταij + u˙ α, j ταij,

(4.42)

H := −A44 = u˙ α pα − L.

(4.43)

(4.41)


15

The fluxes of form A, B, and C fulfill the following conservation laws Dt Pk − Di Pki = 0,

(4.44)

Dt H − Di Si = 0.

(4.45)

Equation (4.44) is the conservation law of the Eshelby stress tensor and the pseudomomentum tensor in gradient elasticity. Equation (4.45) represents the equation of energy conservation for gradient elasticity. By the help of the Gauss theorem, we get the momentum and energy conservation laws in integral form Dt Pk dV − Pki ni dS = 0, (4.46) V

S

Dt H dV − V

Si ni dS = 0.

(4.47)

S

Using partial differentiation, we rewrite (4.40) in the following expression (4.48) Pki = − L δki + uα,k ταi + 2uα,kj ταij + Dj uα,k ταij . Equation (4.48) has the form of the ‘effective’ Eshelby stress tensor of gradient elasticity given by Maugin and Trimarco [21], Maugin [22] and Kalpakides and Agiasofitou [14]. Ignoring all the gradient terms in (4.40)–(4.43), we recover the formulae for elastodynamics derived by Morse and Feshbach [29] and Fletcher [6]. If we neglect the kinetic energy term in the Lagrangian L, (4.40) is in agreement with the expression originally given by Eshelby [4, 5]. So we recover the static Eshelby stress tensor [4, 5] in gradient elasticity: Pki = Wδki − uα,k ταi − Djταij − uα,kj ταij. (4.49) The J-integral is given by means of (4.48) and reads Jk :=

Pki ni dS = 0.

(4.50)

S

It is the J-integral generalized from elasticity to gradient elasticity. J1 is the so-called Rice’s integral of gradient elasticity earlier derived by Chen et al. [3], Georgiadis [8] and Georgiadis and Grentzelou [10]. 4.2 Rotations in Space Since the rotation symmetry is a variational symmetry, Bi = 0 and B4 = 0. The three-dimensional group of rotations, SO(3), acts in the space of independent and dependent variables (see (3.19)). Its infinitesimal action is given by: xi = xi + kji x jεk ,

(4.51)

t = t,

(4.52)

uα

= uα + kβα uβ εk .

(4.53)

16


Eventually, the infinitesimal generators are easily obtained Xik = ikj x j,

τ = 0,

U αk = αkβ uβ .

(4.54)

4.2.1 Form A Substituting the generators (4.54) into (4.9) and (4.10), we obtain the following rotational fluxes

∂L ∂L ∂L ∂L − Dl − u˙ α, j − uα, jl Aki = kmj xm L δij − uα, j ∂uα,i ∂uα,il ∂uα,il ∂ u˙ α,i ∂L ∂L ∂L − Dl + kmα um, j + kmα um ∂uα,i ∂uα,il ∂uα,ij ∂L ∂L + kmα u˙ m , ∂uα,ij ∂ u˙ α,i

(4.55)

∂L ∂L ∂L ∂L − Dl − x juα,i − Dl . Ak4 = ijk u j ∂ u˙ i ∂ u˙ i,l ∂ u˙ α ∂ u˙ α,l

(4.56)

+ kmjuα,m

In terms of stress and momentum we obtain Mki := −Aki = kmj xm Pji + um [τ ji − Dl τ jil ] + um,l τ jil + ul,m τlij − u˙ m pji , Mk := Ak4 = kmj xm Pj + um pj − Dl pjl .

(4.57) (4.58)

From the field theoretical point of view [34], one may decompose the total angular momentum tensor of gradient elasticity according to (o)

(i)

(a)

Mki = Mki + Mki + Mki ,

(4.59)

Mki = kmj xm Pji , (i) Mki = kmj um [τ ji − Dl τ jil ] + um,l τ jil − u˙ m pji ,

(4.60)

where (o)

(a)

Mki = kmj ul,m τlij. (o)

(4.61) (4.62)

(i)

Here Mki is the orbital angular momentum tensor, Mki is the intrinsic (or spin) (a) angular momentum tensor and Mki is an additional angular momentum tensor, which does not appear in the conventual case (classical elasticity). From the rotation (o) (a) transformation of the independent variables we only get Mki and Mki . The piece (i) Mki is obtained from the rotation of the dependent variables.1 On the other hand,

1 We

would like to note that Kalpakides and Agiasofitou [14] used the three-dimensional rotations in the space of independent variables (material space) only. Due to this reason they just obtained the (a) orbital angular momentum tensor and the part Mki . The spin angular momentum part is missing in their approach.


17

the vector Mk is called the material inertia vector. From (4.1), (4.55) and (4.56) we obtain the so-called isotropy condition for gradient elasticity of form A

kmj ui, j τim + u j,i τmi + ui, jl τiml + u j,li τmli + ul,ij τlim − u˙ i, j pim − u˙ j,i pmi = 0. (4.63) It is the generalization of Eshelby’s isotropy condition [4] to gradient elasticity.

4.2.2 Form B If we substitute the generators (4.54) into (4.14) and (4.15), we find the rotational fluxes

∂L ∂L ∂L ∂L − uα, jl − Dl − Dt Aki = kmj xm Lδij − uα, j ∂uα,i ∂uα,il ∂ u˙ α,i ∂uα,il ∂L ∂L ∂L − Dl − Dt + kmα um ∂uα,i ∂uα,il ∂ u˙ α,i ∂L ∂L + kmjuα,m , (4.64) ∂uα,ij ∂uα,ij ∂L ∂L ∂L ∂L ∂L = ijk u j − x j uα,i + uα,il + ul, j + u j,l . (4.65) ∂ u˙ i ∂ u˙ α ∂ u˙ α,l ∂ u˙ i,l ∂ u˙ l,i + kmα um, j

Ak4

In terms of stress tensors and momentum tensors they read Mki := −Aki = kmj xm P ji + um [τ ji − Dl τ jil + Dt p ji ] + um,l τ jil + ul,m τlij , Mk := Ak4 = kmj xm P j + um p j + um,l p jl + ul,m plj .

(4.66) (4.67)

We may decompose (4.66) according

(o)

Mki = kmj xm P ji , (i)

(4.68)

Mki = kmj um [τ ji − Dl τ jil + Dt p ji ] + um,l τ jil , (a)

Mki = kmj ul,m τlij.

(4.69) (4.70)

With (4.64) and (4.65) the so-called isotropy condition of gradient elasticity of form B is obtained as

kmj ui, j τim + u j,i τmi + ui, jl τiml + u j,li τmli + ul,ij τlim − u˙ i, j pim − u˙ j,i pmi = 0. (4.71) It has the same form as (4.63) in gradient elasticity of form A.

18


4.2.3 Form C With pα j = 0 we obtain the following rotational fluxes of form C

∂L ∂L ∂L − uα, jl − Dl Aki = kmj xm Lδij − uα, j ∂uα,i ∂uα,il ∂uα,il ∂L ∂L ∂L ∂L + kmα um, j − Dl + kmjuα,m , + kmα um ∂uα,i ∂uα,il ∂uα,ij ∂uα,ij ∂L . Ak4 = ijk u jδil − x jul,i ∂ u˙ l

(4.72) (4.73)

In terms of stress tensors and momentum tensors we obtain Mki := −Aki = kmj xm P ji + um [τ ji − Dl τ jil ] + um,l τ jil + ul,m τlij , Mk := Ak4 = kmj xm P j + um p j .

(4.74) (4.75)

The tensor Mki is the total angular momentum tensor of gradient elasticity of form C. We note that the second part of (4.75) is missing by Kalpakides and Agiasofitou [14] due to the reason discussed above. Ignoring all the gradient terms in (4.74) and (4.75), we recover the formulae for elastodynamics derived by Fletcher [6]. From (4.1), (4.74) and (4.75) we obtain the so-called isotropy condition of gradient elasticity

kmj um,i τ ji + ui,m τij + ul,mi τlij + um,li τ jil + ui,ml τijl = 0. (4.76) It is obvious that the isotropy condition always follows from rotational invariance of the material. Thus, (4.63), (4.71) and (4.76) are only fulfilled when the material is isotropic. The conservation law of the total angular momentum tensor and the material inertia vector of form A,B and C reads Dt Mk − Di Mki = 0. The rotational conservation law may be given in integral form Mki ni dS + Dt Mk dV = 0. S

(4.77)

(4.78)

V

It is a conservation integral for an isotropic medium. On the other hand, for anisotropic materials the rotational symmetry is broken. In the static limit, we obtain for the angular momentum tensor (4.79) Mki = kmj xm P ji + um [τ ji − Dl τ jil ] + um,l τ jil + ul,m τlij , where P ji is given by (4.48). The corresponding L-integral is defined as: Mki ni dS, Lk :=

(4.80)

S

which is zero for an isotropic gradient theory. It is the L-integral extended to gradient elasticity.


19

4.3 Addition of Solutions This symmetry is based on the superposition of solutions of the Euler- Lagrange equation. The vector field v 5 is a generator of a divergence symmetry. The addition of solutions is given by xi = xi ,

(4.81)

t = t,

(4.82)

uα

= uα + ε fα ,

(4.83)

where fα is an arbitrary solution of the field equation (2.20). The infinitesimal generators of addition of solutions read Xi = 0,

τ = 0,

U α = fα .

(4.84)

4.3.1 Form A Using Betti’s reciprocal theorem, the vector field Bi and the scalar field A4 are given by Bi = −u j τ ji ( f ) − Dk τ jik ( f ) − u j,k τ jik ( f ) + u˙ j p ji ( f ), (4.85) (4.86) B4 = u j p j( f ) − Dk p jk ( f ) . The notations τij( f ), τijk ( f ) mean that u is replaced by f . Finally, the corresponding conserved vector fluxes are Ai = − f j τ ji (u) − Dk τ jik (u) − f j,k τ jik (u) + f˙j pji (u) + u j τ ji ( f ) − Dk τ jik ( f ) + u j,k τ jik ( f ) − u˙ j pji ( f ), A4 = f j p j(u) − Dk pjk (u) − u j p j( f ) − Dk pjk ( f ) .

(4.87) (4.88)

4.3.2 Form B For gradient elasticity of from B the fields Bi and B4 read Bi = −u j τ ji ( f ) − Dk τ jik ( f ) + Dt pji ( f ) − u j,k τ jik ( f ),

(4.89)

B4 = u j p j( f ) + u j,k pjk ( f ).

(4.90)

The corresponding conserved vector fluxes are Ai = − f j τ ji (u) − Dk τ jik (u) + Dt pji (u) − f j,k τ jik (u) + u j τ ji ( f ) − Dk τ jik ( f ) + Dt pji ( f ) + u j,k τ jik ( f ), A4 = f j p j(u) + f j,k pjk (u) − u j p j( f ) − u j,k pjk ( f ).

(4.91) (4.92)

20


4.3.3 Form C With pij = 0 the fields Bi and B4 have now the form Bi = −u j τ ji ( f ) − Dk τ jik ( f ) − u j,k τ jik ( f ),

(4.93)

B4 = u j pj( f ).

(4.94)

The corresponding conserved vector fluxes are given by Ai = − f j τ ji (u) − Dk τ jik (u) − f j,k τ jik (u) + u j τ ji ( f ) − Dk τ jik ( f ) + u j,k τ jik ( f ), (4.95) A4 = f j p j(u) − u j p j( f ).

(4.96)

Like in classical elasticity it is the result of the standard Betti-reciprocal theorem [31]. It is a consequence of the linearity of the field equations (2.20) and (2.24). For static gradient elasticity the field A4 is zero. 4.4 Scaling Unlike classical elasticity the scaling group is neither a variational nor a divergence symmetry. The reason is that field theories like micropolar, microstretch, micromorphic elasticity and gradient elasticity are theories with internal length scales. Such constants with the dimension of length appearing in the Lagrangian (strain energy density) violate the dilatational (scaling) invariance. This fact was found e.g. in micropolar elasticity by Lubarda and Markenscoff [20], Pucci and Saccomandi [33], in microstretch and micromorphic elasticity by Lazar and Anastassiadis [17] and in micromorphic elastodynamics by Lazar [19]. Therefore, in gradient elasticity the scaling symmetry must be a broken variational symmetry. The scaling group acts in infinitesimal form on the independent and dependent variables (in material and physical space) xi = (1 + ε)xi ,

(4.97)

t = (1 + ε)t, uα

(4.98)

= (1 + ε du )uα ,

(4.99)

where du denotes the (scaling) dimension of the vector field u. The dimension of the vector field u is in d-“space-time” dimensions: du = −

d−2 , 2

d = n + 1.

(4.100)

In the present dynamic case we obtain du = −1, since n = 3 in the static case we have du = −1/2, since d = n = 3. The infinitesimal generators are given by Xi = xi ,

τ = t,

U α = du uα .

(4.101)


21

4.4.1 Form A If we substitute the relations (4.101) into (4.9) and (4.10), we obtain for the scaling fluxes

∂L ∂L Ai = xi L + du uα − xk uα,k − tu˙ α − Dj ∂uα,i ∂uα,ij ∂L + (du − 1)uα, j − xk uα,k, j − tu˙ α, j ∂uα,ij ∂L + (du − 1)u˙ α − xk u˙ α,k − tu¨ α , ∂ u˙ α,i

∂L ∂L . − Dj A4 = tL + du uα − xk uα,k − tu˙ α ∂ u˙ α ∂ u˙ α, j

(4.102) (4.103)

In terms of stress and momentum the scaling fluxes of form A read Yi := −Ai = x j P ji − tSi + du u j [τ ji − Dl τ jil ] + (du − 1)u j,k τ jik − (du − 1)u˙ j p ji , (4.104)

Y := A4 = x jP j − tH + du u j [ p j − Dk p jk ].

(4.105)

These scaling fluxes fulfill the relation Dt Y − Di Yi = −(u˙ i, j pij − ui, jk τijk ),

(4.106)

which is now a balance law or not a strict conservation law. The terms on the RHS break the scaling symmetry in gradient elasticity. The scaling symmetry is broken by dynamical terms, pij, and static terms, τijk . Because gradient elasticity of form A is a restricted micromorphic elasticity, the symmetry breaking terms have a similar form as the terms breaking the scaling in micromorphic elasticity (see Lazar [19]). 4.4.2 Form B Now we substitute the relations (4.101) into (4.14) and (4.15) and get the following scaling fluxes of form B

∂L ∂L ∂L Ai = xi L + du uα − xk uα,k − tu˙ α − Dj − Dt ∂uα,i ∂uα,ij ∂ u˙ α,i ∂L , + (du − 1)uα, j − xk uα,k, j − tu˙ α, j ∂uα,ij

(4.107)

∂L A4 = tL + du uα − xk uα,k − tu˙ α ∂ u˙ α ∂L + (du − 1)uα, j − xk uα,kj − tu˙ α, j . ∂ u˙ α, j

(4.108)

22


In terms of stress and momentum they are of the form Yi := −Ai = x j P ji − tSi + du u j [τ ji − Dl τ jil + Dt pji ] + (du − 1)u j,k τ jik ,

(4.109)

Y := A4 = x jP j − tH + du u j pj + (du − 1)ui, j pij.

(4.110)

Equations (4.109) and (4.110) satisfy the broken conservation law Dt Y − Di Yi = −(u˙ i, j pij − ui, jk τijk ),

(4.111)

which has the same form as (4.106). Equations (4.106) and (4.111) read in integral form Dt Y dV − Yi ni dS = − (u˙ i, j pij − ui, jk τijk ) dV. (4.112) V

S

V

4.4.3 Form C With pij = 0 we obtain from the scaling fluxes of form A and B:

∂L ∂L Ai = xi L + du uα − xk uα,k − tu˙ α − Dj ∂uα,i ∂uα,ij ∂L , + (du − 1)uα, j − xk uα,k, j − tu˙ α, j ∂uα,ij ∂L . A4 = tL + du uα − xi uα,i − tu˙ α ∂ u˙ α

(4.113) (4.114)

In terms of stress tensors and momentum vector the scaling fluxes read Yi := −Ai = x j P ji − tSi + du u j [τ ji − Dl τ jil ] + (du − 1)u j,k τ jik ,

(4.115)

Y := A4 = x jP j − tH + du u j p j.

(4.116)

The scaling fluxes satisfy the following relation Dt Y − Di Yi = ui, jk τijk .

(4.117)

Thus, (4.117) is a balance law instead a conservation law. Using the divergence theorem, we obtain the integral form of the balance law of scaling transformation: Dt Y dV − Yi ni dS = ui, jk τijk dV. (4.118) V

S

V

For static gradient elasticity we get the scaling vector flux from (4.74) Yi = x j P ji −

n−2 n u j (τ ji − Dl τ jil ) − u j,k τ jik . 2 2

The corresponding balance law is called the M-integral and is given by M := Yi ni dS = − ui, jk τijk dV. S

(4.119)

(4.120)

V

This is the M-integral generalized from elasticity to gradient elasticity. In gradient elasticity and in gradient elastodynamics the scaling symmetry is broken due to the presence of higher order stress. Thus, for scaling symmetry the conservation law is replaced by a balance law.


23

5 Balance Laws So far, we have examined conservation laws of homogeneous gradient elastodynamics without external sources. Now we want to investigate balance laws for nonhomogeneous gradient elastodynamics with external forces. We postulate the Lagrangian to be of the form

L = T − W − V,

(5.1)

where V is the potential of external forces. Let us assume that the Lagrangian depends explicitely on xi :

L = L(xi , uα , u˙ α , uα,i , u˙ α,i , uα,ij).

(5.2)

In this case, the material force (or inhomogeneity force) is defined by f iinh :=

∂L , ∂ xi

(5.3)

which is caused by material inhomogeneities. External body forces are defined by Fα := −

∂V . ∂uα

(5.4)

It turns out that the invariance with respect to space translation, rotation and scaling are broken. We obtain that the translational balance law has the form Dt Pk − Di Pki = f kinh .

(5.5)

This balance law is valid for linear, inhomogeneous, anisotropic, gradient elastodynamics of form A,B and C which are subjected by external body forces. The energy conservation law (4.45) is still valid. Equation (5.5) follows essentially from the lack of translational invariance and it can be called the canonical momentum balance law. Using the Euler-Lagrange equations, we obtain for the rotational balance law of form A and B Dt Mk − Di Mki = kjn ui, j τin + u j,i τni + ui, jl τinl + u j,li τnli + ul,ij τlin − u˙ i, j pin − u˙ j,i pni + kjn x j f ninh +u j Fn , (5.6) and for form C Dt Mk − Di Mki = kjn ui, j τin + u j,i τni + ui, jl τinl + u j,li τnli + ul,ij τlin + kjn x j f ninh + u j Fn .

(5.7)

The terms in the first parentheses on the right hand side vanish when the material is isotropic. The other terms on the right hand side are resulting vector moments caused by inhomogeneities and external body forces. Equations (5.6) and (5.7) may be called the canonical angular momentum balance law. The scaling balance laws are for the form A and B Dt Y − Di Yi = −(u˙ α, j pα j − uα, jk τα jk ) + xi f iinh +

d+2 u α Fα , 2

(5.8)

24


and for form C Dt Y − Di Yi = uα, jk τα jk + xi f iinh +

d+2 u α Fα . 2

(5.9)

The source terms are ‘scalar moments’ breaking the scaling symmetry. The additional source terms appear which account for material inhomogeneities and external forces. Equations (5.8) and (5.9) can be called the scalar moment of momentum balance law.

6 Conclusion In this paper, we investigated the Euler-Lagrange and variational symmetries in linear gradient elastodynamics of grade-2. We have investigated three forms of gradient elastodynamics obtained from pseudo micromorphic elasticity. They are called form A, B and C of gradient elastodynamics. We derived and discussed the Euler-Lagrange equations for these three forms of gradient elastodynamics. The Lie point symmetries are investigated and the conservation laws are derived for form A, B, and C. We found conservation laws corresponding to variational or divergence symmetries of the Lagrange density. Symmetries of the Euler-Lagrange equations and variational symmetries are the translation, the rotation and the addition of solutions. Addition of solutions provides a conservation law which is the result of Betti’s reciprocal theorem. The scaling symmetry is neither a variational nor a divergence symmetry of the corresponding Lagrangian. For nonhomogeneous gradient elastodynamics with external forces the conservation laws become balance laws. Acknowledgement The authors have been supported by an Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (Grant No. La1974/1-2).

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