Geometrically nonlinear gravito-elasticity: Hyperelastostatics coupled to Newtonian gravitation

International Journal of Engineering Science 49 (2011) 1452–1460

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Geometrically nonlinear gravito-elasticity: Hyperelastostatics coupled to Newtonian gravitation q Paul Steinmann Applied Mechanics, University of Erlangen-Nuremberg, Germany

a r t i c l e

i n f o

Article history: Available online 27 April 2011 Keywords: Newtonian gravitation Hyperelasticity Coupled gravito-elasticity

a b s t r a c t A theory of nonlinear elasticity is presented that incorporates a spatially non-constant Newtonian gravitational field as is appropriate if deformable heavy masses of finite volume are considered. For Newtonian gravitation the mass density is a sink for the (scaled) gravitational field. This Gauss-type law for gravitation is incorporated into the mechanical balance equations of linear and angular momentum as well as into the thermomechanical balance equations of energy and entropy. To this end, a total energy density as well as total Piola and Cauchy stresses are introduced that directly capture the contribution of Newtonian gravitation. For the nonlinear elastic case the pertinent relations for the gravito-elastically coupled problem are recovered from a variational setting in terms of the nonlinear deformation map and a gravitational potential. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Newtonian gravitation (Newton, 1686) is the paradigm for far-field forces (action at a distance). It is an approximative theory for the effects of gravitation that can for example successfully model observations of our planetary system as described by the laws of Kepler. Since the work of Einstein on general relativity it is commonly accepted, however, that Newtonian gravitation has some shortcomings, for example it fails to describe with high accuracy the precession of the perihelion of Mercury’s orbit and that it incorrectly predicts the angular deflection of light rays due to gravitation. Furthermore, Newtonian gravitational interaction occurs instantaneously and thus with infinite velocity (thus ignoring the speed of light as the limiting velocity). Nevertheless, as long as no extremely massive and dense objects are considered, i.e. u/c2  1 whereby u is the gravitational potential (to be introduced subsequently) and c is the speed of light, and as long the objects under consideration move much slower than the speed of light, i.e. v2/c2  1, Newtonian gravitation may be considered the low-gravity limit of general relativity and thus as a good approximation. If we restrict ourselves to those situations where Newtonian gravitation is applicable then its striking similarity to Coulomb’s law of electro-statics motivates our attempt to formulate a theory of nonlinear elasticity that incorporates the effects of gravitation in a similar manner as the recent theory of nonlinear electro-elasticity incorporates the effects of electro-statics (see Dorfmann & Ogden, 2005; Eringen & Maugin, 1990; Maugin, 1988; Vu & Steinmann, 2010; Vu, Steinmann, & Possart, 2007). Thus, by analogy we are tempted to baptize this new theory as gravito-elasticity. It shall be noted carefully, however, that historically it was Coulomb and Ampère that followed Newton by adopting his law of action at a distance to the cases of electro-magnetism. Possible applications of gravitoelasticity are heavy bodies of finite extent that deform elastically (and possibly geometrically nonlinearly) under the action of their own gravitation, whereby the nonlinear deformation and thus spatial redistribution of mass leads to a perturbation of the initial gravitational field that penetrates both the body and the whole of the surrounding free space. This problem has q

Submitted to the Eringen Special Issue of International Journal of Engineering Science. E-mail address: [email protected]

0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.03.017

P. Steinmann / International Journal of Engineering Science 49 (2011) 1452–1460

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long since been treated in the analysis of the mechanical behavior of the earth under the action of gravity (see Hoskins, 1909, 1921). Due to the precision of contemporary micro-gravity measurement instrumentation this is still relevant in modern geophysics, see for instance the discussion in Charo, Fernandez, Luzon, and Rundle (2006, 2007). In these works however a geometrically linear approximation as proposed already in Love (1911) is usually adopted. Along similar lines Barber (2004) investigates the stresses induced in the earth’s crust by the gravitational forces exerted by a massive asteroid passing close to the earth. An existence theorem for Newtonian gravitation coupled to geometrically nonlinear elasticity, based on a simplified Hooke-type elasticity law, is proven in Beig and Schmidt (2003), and time-independent configurations of two gravitating elastic bodies are constructed in Beig and Schmidt (2008). The present manuscript aims to combine the pertinent equations of geometrically nonlinear continuum mechanics, specialized to the case of elasticity for the sake of concreteness, with the continuum version of Newton’s law of universal gravitation. To this end we first reiterate the relevant relations of Newtonian gravitation. We then combine Newtonian gravitation and nonlinear continuum mechanics into what we call continuum gravito-mechanics. Thereby, the presentation rests heavily on the introduction of so-called gravitational stresses and in particular on the introduction of total stresses, a concept that we borrow again from nonlinear electro-elasticity. Likewise we investigate the contribution of Newtonian gravitation on the formulation of nonlinear continuum thermomechanics. To this end we endeavor to reformulate the balance of energy and the balance of entropy (including a positive entropy production term) so as to incorporate the previously introduced total stresses. Finally, for the conservative case and based on the notion of a total potential energy we set up a variational formulation that renders as the Euler–Lagrange equations exactly the field equations and boundary conditions as considered in the previous sections. The variational setting will serve as a prerequisite for a computational treatment in a later contribution. 2. Recapitulation of Newtonian gravitation In Newtonian gravitation two heavy point masses attract each other with a force, oriented along the direction vector connecting the masses, that is proportional to the masses and inversely proportional to the square of their separation distance (see Newton, 1686) for the origin of this concept. Thus, the force fij exerted on mass mi due to the presence of mass mj is given by

fij ¼ c

mi mj rij : jr ij j2 jrij j

ð1Þ

Here rij = rj  ri denotes the direction vector pointing from the mass mi located at ri to the mass mj located at rj, see Fig. 1. The proportionality constant c is the gravitational constant and takes the value

c ¼ 6:67428  1011

m3 : kg s2

ð2Þ

Obviously, e.g. Coulomb’s law of electro-statics formally follows the paradigm of Newtonian gravitation as an action at a distance with the difference that mass can only take positive values as opposed to charge. Moreover charges of equal sign repel each other whereas masses always exert an attractive force on each other. We introduce the normalized force gij exerted on mass i due to the presence of mass j as the gravitational vector

gij ¼ c

mj r ij : jrij j2 jr ij j

ð3Þ

For the case of many discrete masses the resulting gravitational vector gi at location i due to the presence of masses j = 1, . . . , nma, with nma the number of masses, is simply computed by superposition

Fig. 1. System of discrete masses at locations ri, rj, rk with direction vector rik = rk  ri pointing from the mass mi located at ri to the mass mk located at rk. In (a) the force fij exerted on mass mi due to the presence of mass mj is depicted, in (b) the corresponding gravitational vector gij , defined as the normalized force exerted on mass mi due to the presence of mass mj, is shown.

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gi ¼ c

nma X mj

r ij : jr ij j2 jr ij j

j¼1

ð4Þ

Accordingly, for a continuous distribution of mass with q(x0 ) denoting the mass density at the field point x0 , the corresponding gravitational field gðxÞ at the source point x follows from

gðxÞ ¼ c

Z S

qðx0 Þ jx0

x0  x dv :  xj jx0  xj

ð5Þ

2

Here S denotes the whole space so that all field points x0 2 S. Note again that, e.g., the electric field in electro-statics compares formally to the gravitational field. Next, considering the divergence of the integrand with respect to the source point x, i.e.

!

x0  x

divx

¼ 4pdðx  x0 Þ;

jx0  xj3

ð6Þ

renders, based on the well-known properties of the Dirac distribution d(x), as a fundamental result the Gauss-type law of Newtonian gravitation, i.e. sinks of the gravitational field are due to (positive) mass density

divx gðxÞ ¼ 4pcqðxÞ:

ð7Þ 0

Furthermore, the gradient with respect to the source point x of the inverse distance to a field point x , i.e.

rx



 1 x0  x ¼ ; jx0  xj jx0  xj3

ð8Þ

introduces a potential u(x), the so-called gravitational potential, for the gravitational field, i.e.

uðxÞ ¼ u0  c

Z S

qðx0 Þ jx0  xj

dv ;

ð9Þ

where u0 is a constant of integration. Thus the gravitational field follows from

gðxÞ ¼ rx uðxÞ

ð10Þ

with the consequence that g has a vanishing curl, i.e. curlg ¼ 0. Furthermore, the Poisson equation

Dx uðxÞ ¼ 4pcqðxÞ

ð11Þ

determines the gravitational potential from the spatial distribution of mass density. The gravitational field g has the same units as acceleration, i.e. m/s2, thus the gravitational potential u has the same units as velocity squared, i.e. [m/s]2. In summary the force exerted on a small test mass m in the gravitational field g is given by

f ¼ mg:

ð12Þ

Thus the power expended by moving a test mass an infinitesimal distance dx against a gravitational field is computed as

mg  dx ¼ mrx u  dx ¼ mdu;

ð13Þ

so that the gravitational potential u(x) takes the interpretation of the work performed to move a test mass from infinity to the point x normalized by the test mass

uðxÞ ¼ 

Z

x

gðx0 Þ  dx0 :

ð14Þ

1

3. Nonlinear continuum gravito-mechanics Here we consider a material body consisting of matter occupying its material configuration B0 surrounded by free space that occupies the material configuration S 0 , see Fig. 2. Due to the deformation of the body the matter and the free space will occupy spatial configurations Bt and S t , however with B0 [ S 0 ¼ Bt [ S t . The boundary of the body @B0 in the material configuration defines the interface between free space and matter. The deformation is locally characterized by the deformation gradient F = rX u, i.e. the material gradient of the nonlinear deformation map x = u(X) with the material coordinate X 2 B0 . Note that even if there is no matter outside the body and thus there exists no physical deformation map u in free space that justifies a material setting, we may nevertheless imagine a fictitious deformation map u that extends u = u(X) from the body to its exterior. In the context of a geometrically nonlinear formulation we shall distinguish tensor fields in the tangent space to the spatial configuration by those in the tangent space to the material configuration by using lower and upper case letters, respectively. Moreover, two-point tensor fields will also be denoted by capital letters. Finally densities per unit volume in the material configuration will be discriminated from densities per unit volume in the spatial configuration by sub-indexes 0 or t, respectively.

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Fig. 2. Material body consisting of matter occupying its material configuration B0 surrounded by free space that occupies the material configuration S 0 . The boundary of the body @B0 in the material configuration defines the interface between free space and matter.

3.1. Balance of mass The Gauss-type law of Newtonian gravitation may be expressed with respect to the material or the spatial configuration, i.e. in terms of the divergence operator with respect to the material or the spatial coordinates, whereby q0 and qt denote the corresponding mass densities, respectively, as

DivM ¼ q0

and divm ¼ qt :

ð15Þ

Here we introduced besides the spatial gravitational field g the scaled spatial gravitational field m together with the corresponding referential quantities

G¼gF

and M ¼ m  cofF:

ð16Þ

Thereby the referential quantities follow from appropriate pull-back operations (i.e. either by the chain rule or the Piola transformation in terms of the co-factor cofF of F) from their spatial counterparts. Note the different flavors of the vector fields g and m arise from being transformed either co- or contravariantly. The scaled gravity field reads explicitly

M¼

1 1 JB  G and m ¼ g: 4pc 4pc

ð17Þ

Here J denotes the Jacobian determinant J = detF and B is the inverse Cauchy–Green (or rather the Piola) strain B = [Ft  F]1. Since we do not consider separate mass densities distributed at the boundary of the body, the corresponding jump conditions follow from the usual pill-box argument applied to the Gauss-type law of Newtonian gravitation as

sMt  N ¼ 0 and smt  n ¼ 0:

ð18Þ

Here N and n denote the outward pointing normal to the material and spatial boundary of the body, respectively, i.e. pointing from matter to free space, and the jump of a quantity () at @B0 is defined as s()t = ()free space  ()matter. From the definition of the gravitational field as the gradient of the (continuous, i.e. sut = 0) gravitational potential and again from a pill-box argument we obtain tangential continuity of the gravity field. Moreover, the additional normal continuity of the scaled gravitational field renders the overall continuity of m across surfaces and thus the continuity of the gradient rxu of the gravitational potential

n  sgt ¼ 0 ! smt ¼ 0 and srx ut ¼ 0:

ð19Þ

The resultant mass M of the body follows with the help of the Gauss-type law of Newtonian gravitation as

M¼

Z Bt

qt dv ¼ 

Z

divmdv ¼ 

Z

Bt

m  nda:

ð20Þ

@Bt

Likewise, expressed in referential quantities we obtain

M¼

Z B0

q0 dV ¼ 

Z

DivMdV ¼ 

B0

Z

M  NdA:

ð21Þ

@B0

Thus we may conclude the following useful relations between the gravitational field at the boundary of the body and the total mass of the body

Z @Bt

g  nda ¼ 4pcM

and

Z @B0

JG  B  NdA ¼ 4pcM:

ð22Þ

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The conservation of mass renders eventually q0 = q0(X) and thus M ¼ MðXÞ which, however, does not contradict G ¼ GðX; tÞ due to the dependence of G on F = F(X, t). 3.2. Balance of linear momentum Next we introduce a (free space) gravitational energy density G0 = JGt per referential unit volume or Gt per spatial unit volume, respectively, (compare to the similar case of electro-mechanics)

1 1 G  MðF; GÞ and Gt ¼ g  m: 2 2

G0 ¼ G0 ðF; GÞ ¼

ð23Þ

The corresponding Piola-type gravitational stress Pgrv = rgrv  cofF is then

P grv ¼

@G0 ¼ Gt cofF  g  M: @F

ð24Þ

For the above derivation we used the helpful intermediate result

@B ¼ ½F 1  B þ B  F 1 : @F

ð25Þ

The special dyadic products  and  order indices so that ½F 1  BIJkL ¼ ½F 1 Ik ½BJL and [B  F1]IJkL = [B]IL [F1]Jk, respectively. Moreover, in our index notation lower case and upper case indices refer to the spatial and material base vectors, respectively. Then the obviously symmetric Cauchy-type gravitational stress rgrv is expressed as

rgrv ¼ Gt i  g  m ¼

 1 2 jgj i  g  g : 4pc 2 1



ð26Þ

In the above i denotes the second-order spatial unit tensor with coefficients dij. It is now straightforward to prove that the divergence of the Cauchy-type gravitational stress renders

divrgrv ¼ gdivm þ m  curlg ¼ qt g:

ð27Þ

To derive the previous result we used the following identity

m  curlg ¼ m  rx g  rx g  m:

ð28Þ

Likewise for the jump of the Cauchy-type gravitational stress one obtains

srgrv t ¼ fmg  sgti  fgg  smt  sgt  fmg:

ð29Þ

Here {()} denotes the average of a quantity () across the boundary of the body. Accordingly, based on the previous results for the jumps in g and m we obtain for the jump in the gravitational traction the obvious result

srgrv t  n ¼ fggsmt  n þ fmg  ½n  sgt ¼ 0:

ð30Þ

To prove the previous result the following useful identity is used

fmg  ½n  sgt ¼ ½fmg  sgtn  ½fmg  nsgt:

ð31Þ

It is interesting to note that the divergence statement for the gravitational stress is also obtained directly (after some straightforward manipulations) by multiplying the Gauss-type law of Newtonian gravitation by the gradient of the gravitational potential. This is in line with Noether’s theorem that associates additional conservation laws (here the divergence-free gravitational stress in vacuum) with the field equations (here the Gauss-type law of Newtonian gravitation). Next we compute the resultant gravitational force acting on the body expressed in terms of a distributed volume force

Z Bt

qt gdv ¼

Z Bt

grv

bt dv ¼

Z

divrgrv dv ¼

Bt

Z

rgrv  nda:

ð32Þ

@Bt grv

Thus the resultant gravitational force exerted on the body may be expressed by a distributed gravitational volume force bt (and vanishing gravitational traction densities t grv t ) grv

bt

¼ divrgrv ¼ qt g ðand t grv ¼ 0Þ: t

ð33Þ

With these preliminary results at hand the global format for the (quasi-static) balance of linear momentum is stated as

0¼

Z

t t da þ @Bt

Z

grv

Bt

½bt þ bt dv :

ð34Þ

Here tt and bt, respectively, denote the mechanical tractions at the surface of the body and all other distributed body forces grv except gravity, which is already contained in bt . With the Cauchy theorem tt = r  n the corresponding localized form of the balance of momentum then reads grv

0 ¼ divr þ bt þ bt

¼ divrtot þ bt

with r  n ¼ t t :

ð35Þ

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In the above we introduced the total Cauchy stress that comprises both mechanical and gravitational parts as

rtot ¼ r þ rgrv :

ð36Þ

The total Cauchy stress has to satisfy the following boundary condition in terms of the (prescribed) mechanical traction and an additional gravitational traction grv rtot matter  n ¼ t t þ rfree space  n:

ð37Þ

In referential quantities the (quasi-static) balance of linear momentum reads grv

0 ¼ DivP þ b0 þ b0 ¼ DivP tot þ b0

with P  N ¼ t 0

ð38Þ

with the total Piola stress defined as follows

P tot ¼ P þ P grv :

ð39Þ

Obviously the total Piola stress has to satisfy the following boundary condition grv P tot matter  N ¼ t 0 þ P free space  N:

ð40Þ

3.3. Balance of angular momentum The resultant gravitational moment acting on the body expressed in terms of a distributed volume couple is

Z Bt

Z

x  ½qt gdv ¼

grv

x  bt dv ¼

Bt

Z

c grv t dv :

ð41Þ

Bt

Thus, the gravitational volume couple c grv may be expressed in terms of the symmetric gravitational Cauchy stress as t grv

c grv ¼ x  bt t

¼ divðx  rgrv Þ:

ð42Þ

With these preliminaries at hand the global form of the (quasi-static) balance of angular momentum is

0¼

Z

x  t t da þ

Z

½x  bt þ c grv t dv :

ð43Þ

Bt

@Bt

Thus, incorporating the (quasi-static) balance of momentum the corresponding localized format then reads straightforwardly as

0 ¼ divðx  rtot Þ þ x  bt ¼ i  rtot :

ð44Þ t

Here the cross product of two second order tensors a and b is defined as a  b = e:[a  b ] with e denoting the third order permutation tensor. Of course this is nothing but the symmetry condition for the total Cauchy stress and thus also for the mechanical Cauchy stress. In terms of axial vectors these conditions read

axlrtot ¼ 0 and axlr ¼ 0:

ð45Þ

We remark that this is a much simpler situation as compared to the case of nonlinear electro-elasticity that displays a nonsymmetric mechanical Cauchy stress due to the presence of polarization. 4. Nonlinear continuum gravito-thermo-mechanics 4.1. Balance of energy Before stating the balance of energy we express the resultant working of the distributed gravitational volume forces in referential quantities as

Z B0

grv

b0  v dV ¼

Z

v  P grv  NdA 

@B0

Z

P grv : rX v dV:

ð46Þ

B0

_ denotes the usual velocity field. Then the global form of the balance of the standard internal energy (recall that Here, v ¼ u we do restrict ourselves to the quasi-static case), expressed in terms of its density e0 per unit volume in the reference configuration, reads as

Z B0

ext e_ 0 dV ¼ Eext sur þ Evol :

ð47Þ

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ext Here the external power input Esur across the surface of the body is due to the working of the mechanical and gravitational forces (mechanical and gravitational power input) together with the thermal power input due to a Piola-type heat flux H

Eext sur ¼

Z

½v  ½t 0 þ P grv  N  H  N dA:

ð48Þ

@B0

Likewise the external power input Eext vol within the body is due to the working of the distributed mechanical and gravitational forces (mechanical and gravitational power input) together with the thermal power input due to referential heat sources h0

Eext vol ¼

Z h

v  b0  P grv : F_ þ h0

i

dV:

ð49Þ

B0

Localizing the above statement renders merely the usual format for the balance of (internal) energy that, however, does not involve the total Piola stress but only the common mechanical Piola stress

e_ 0  P : F_ þ DivH  h0 ¼ 0:

ð50Þ tot

Since we want to express the localized balance of energy in terms of the working of the total Piola stress P express the global form of the balance of energy as

Z B0

ext ext e_ tot 0 dV ¼ Esur þ Evol :

we prefer to

ð51Þ

Here etot 0 ¼ e0 þ G0 denotes the total (internal) energy that includes explicitly the previously introduced gravitational energy G0. Thereby, based on the above definition of G0 ¼ G0 ðF; GÞ, we observe that

_ G_ 0 ¼ P grv : F_ þ M  G:

ð52Þ Eext sur

It is simple to verify that the external power input across the surface of the body is indeed unmodified, however it now proves more convenient to express it in terms of the working of the total Piola traction as

Eext sur ¼

Z

½v  P tot  H  NdA:

ð53Þ

@B0

The modified external power input Eext⁄ vol within the body, however, now takes a slightly different expression

Eext vol ¼

Z



v  b0 þ M  G_ þ h0



dV:

ð54Þ

B0

Consequently, the localized format for the balance of the total (internal) energy tot _ þ DivH  h0 ¼ 0: e_ tot : F_  M  G 0 P

etot 0 follows as ð55Þ

Observe that the above statement now includes the working of the total Piola stress Ptot, as desired. In addition there is a power density term related to the referential scaled gravitational field M. 4.2. Balance of entropy The global format of the balance of entropy, expressed in terms of its density per unit volume r0 in the reference configuration, reads as

Z B0

r_ 0 dV ¼ 

Z

h1 H  NdA þ

Z

@B0

h1 ½h0 þ d0 dV:

ð56Þ

B0

Here we have directly incorporated the common assumption for the entropy flux and the entropy source as being expressed in terms of the corresponding heat flux and heat source divided by the absolute temperature h > 0 (as the integrating denominator). Likewise, the entropy production is expressed in terms of the dissipation power density d0 and is restricted to be positive

Z

h1 d0 dV P 0 and d0 P 0:

ð57Þ

B0

Localizing the above statement eventually renders

d0 ¼ ½DivH  h0  þ r_ 0 h  rX ln h  H P 0:

ð58Þ

Next we introduce the Helmholtz energy density as the Legendre transformation of the internal energy in order to exchange the entropy density for the absolute temperature in the parametrization of the energy as

w0 ðF; h;   Þ ¼ e0 ðF; r0 ;   Þ  hr0 :

ð59Þ

P. Steinmann / International Journal of Engineering Science 49 (2011) 1452–1460

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Furthermore, incorporating the standard version of the balance of (internal) energy renders the common version of the dissipation power density inequality that, however, does not incorporate the power expended by the total Piola stress, as

d0 ¼ P : F_  w_ 0  r0 h_  rX ln h  H P 0:

ð60Þ

Alternatively, incorporating the balance of energy in terms of the power expended by the total Piola stress renders finally the desired result

_  w_ tot  r0 h_  rX ln h  H P 0: d0 ¼ P tot : F_ þ M  G 0

ð61Þ

wtot 0

Here ¼ w0 þ G0 abbreviates the total free energy. The above dissipation power density inequality is well-suited for determining the constitutive expressions for the total Piola stress Ptot and the referential scaled gravitational field M. For example, with the restriction to thermo-hyperelasticity, i.e. w0 = w0(F, h) and based on a standard Coleman–Noll exploitation argument we obtain for the constitutive relations incorporating the gravitational energy density

P tot ¼

@wtot 0 ; @F

r0 ¼ 

@wtot 0 @h

and M ¼

@wtot 0 : @G

ð62Þ

Finally, the reduced or in this case rather the conductive dissipation power density inequality remains as usual as

dcon ¼ rX ln h  H P 0: 0

ð63Þ

The conductive dissipation power density inequality may be satisfied by the common Fourier law of heat conduction. 5. Variational setting of nonlinear gravito-elasticity For the case of isothermal gravito-elasticity we define the total internal potential energy density W0 at the fixed reference temperature h = href by

W 0 ðF; GÞ ¼ G0 ðF; GÞ þ w0 ðF; hÞjh¼href :

ð64Þ

Then the total potential energy densities U0 and u0 in the bulk and at the surface of the body, respectively, are written as

U 0 ðu; F; u; GÞ ¼ W 0 ðF; GÞ þ V 0 ðu; uÞ and u0 ¼ u0 ðuÞ:

ð65Þ

Here V0(u, u) denotes the external potential energy density in the bulk parameterized in the deformation map u and the gravitational potential u, the external potential energy density at the surface of the body only depends on the deformation map u (since there are no separate surface mass densities). Consequently the constitutive relations for gravito-elasticity expressed in terms of the total potential energy density U0 in the bulk are

P tot ¼

@U 0 @F

and M ¼

@U 0 : @G

ð66Þ

Moreover, the mechanical body forces b0 and the mechanical tractions t0 together with the mass density q0 are defined via the external potential energy densities as

b0 ¼ 

@U 0 ; @u

t0 ¼ 

@u0 @u

and q0 ¼

@U 0 : @u

ð67Þ

With these preliminary results at hand we define a total energy functional as

Iðu; uÞ ¼

Z

G0 ðF; GÞdV þ

S0

Z

U 0 ðu; F; u; GÞdV þ

Z

B0

u0 ðuÞdA:

ð68Þ

@B0

For the sake of simplicity of exposition we shall ignore any energy contributions from the boundary of the free space @S 1 at infinity in this contribution. Then we seek a stationary point of the above defined functional

Iðu; uÞ ! stat:

ð69Þ

Using the previous definitions the variation of I(u, u) with respect to u renders

Z

rX du : P grv dV þ

Z

S0

½rX du : P tot  du  b0 dV  B0

Z

du  t 0 dA ¼ 0:

ð70Þ

@B0

Requiring the above variational statement to hold for all admissible du results in the following Euler–Lagrange equations

: DivP tot ¼  b0

in B0

: and DivP grv ¼ 0 in S 0

ð71Þ

together with the jump or boundary condition at @B0

: sP tot t  N ¼  t 0

)

: grv P tot matter  N ¼ t 0 þ P free space  N:

Likewise the variation of I(u, u) with respect to u produces

ð72Þ

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Z

rX du  MdV þ

S0

Z

½rX du  M  duq0 dV ¼ 0:

ð73Þ

B0

Requiring the above variational statement to hold for all admissible du results in the following Euler–Lagrange equations

: DivM ¼  q0

in B0

: and DivM ¼ 0 in S 0

ð74Þ

together with the jump or boundary condition at @B0

: sMt  N ¼ 0

)

Mmatter  N ¼ Mfree space  N:

ð75Þ

Thus, in conclusion, the variational problem formulated as above renders the relevant field equations and boundary conditions of coupled nonlinear gravito-elasticity. Clearly a variational setting is a necessary prerequisite for certain computational treatments such as the finite element method. 6. Conclusion Inspired by the formal similarity of Newton’s universal law of gravitation as the paradigm of an action at a distance to the (more recent) Coulomb’s law of electro-statics we formulated a straightforward theory of nonlinear gravito-elasticity. The resultant formulation resembles the by now well-established theory of electro-elasticity. Geometrically nonlinear gravitoelasticity is restricted to situations where neither the gravitational potential nor the velocity (squared) of the objects considered take values in the order of the speed of light (squared). Under these restrictions the theory applies to heavy bodies with finite extent that are capable of nonlinear elastic deformations. Paradigmatic examples are from the area of geophysics, where changes in the gravitational field measured by current micro-gravity instrumentation and the associated deformations in the earth’s crust are attributed to geodetic precursors to volcanic eruptions. In particular the variational setting produces the pertinent coupled equations and boundary conditions and will serve as a starting point for a computational treatment in a later contribution. References Barber, J. R. (2004). Stresses in a half space due to Newtonian gravitation. Journal of Elasticity, 75, 187–192. Beig, R., & Schmidt, B. G. (2003). Static self-gravitation elasticity bodies. Proceedings of Royal Society of London A, 459, 109–115. Beig, R., & Schmidt, B. G. (2008). Celestial mechanics of elastic bodies. Mathematische Zeitschrift, 258, 381–394. Charo, M., Fernandez, J., Luzon, F., & Rundle, J. B. (2006). On the relative importance of self-gravitation and elasticity in modeling volcanic ground deformation and gravity changes. Journal of Geophysical Research, 111, 1–12. B03404. Charo, M., Luzon, F., Fernandez, J., & Tiampo, K. F. (2007). Topography and self-gravitation interaction in elastic-gravitational modeling. Geochemistry, Geophysics, Geosystems, 8, 1–11. Q01001. Dorfmann, A., & Ogden, R. W. (2005). Nonlinear electroelasticity. Acta Mechanica, 174(12), 167–183. Eringen, AC., & Maugin, G. A. (1990). Electrodynamics of continua. Springer. Hoskins, L. M. (1909). The strain of a gravitating compressible elastic sphere. Transactions on American Mathematical Society, 11, 203–248. Hoskins, L. M. (1921). The strain of a gravitating sphere of variable density and elasticity. Transactions on American Mathmatical Society, 21, 1–43. Love, A. E. H. (1911). Some problems in geodynamics. Cambridge University Press. Maugin, G. A. (1988). Continuum mechanics of electromagnetic solids. North-Holland. I. Newton, Philosophiae Naturalis Principia Mathematica, 1686. Vu, D. K., Steinmann, P., & Possart, G. (2007). Numerical modelling of nonlinear electroelasticity. International Journal of Numerical Methods in Engineering, 70, 685–704. Vu, D. K., & Steinmann, P. (2010). A 2D coupled BEM-FEM simulation of electro-elastostatics at large strain. Computer Methods in Applied Mechanics and Engineering, 199, 1123–1124.