PAMM · Proc. Appl. Math. Mech. 11, 483 – 484 (2011) / DOI 10.1002/pamm.201110233

Geometrically Nonlinear Continuum Thermomechanics with SurfaceEnergies Coupled to Diffusion

Andrew McBride1,∗, Paul Steinmann1, Ali Javili1, and Swantje Bargmann2

1 Lehrstuhl für Technische Mechanik, Universität Erlangen-Nürnberg, Egerland-Str. 5, 91058 Erlangen, Germany2 Institute of Mechanics, Technische Universität Dortmund, Leonhard-Euler-Str. 5, 44227 Dortmund, Germany

Surfaces can have a significant influence on the overall response of a continuum body but are often neglected or accounted forin an ad-hoc manner. This work briefly describes a thermodynamically consistent, nonlinear, continuum thermomechanicsformulation which accounts for surface structures and includes the effects of diffusion.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The response of the surface can differ from that of the bulk due to factors such as oxidation and coating, which result in asurface composition distinct from the bulk, and the termination of interatomic bonds at the surface. A phenomenologicalapproach to capturing surface effects and describing scale-dependent phenomena at the nanoscale is to endow the surfaceand the bulk with their own Helmholtz energies. The surface is assumed to have negligible thickness and acts in a similarmanner to a membrane “wrapped” around the bulk. Heat flux and species diffusion occur on the surface as well as in the bulk.The assumption of infinitesimal surface thickness restricts the flux vectors associated with the various surface quantities tobe tangent to the surface. The objective of this work is to briefly describe the approach to derive the equations governing theresponse of a thermomechanical solid undergoing finite, inelastic deformations with diffusion and extend these results to thesurface. For extensive details and references the reader is referred to [1].

2 Summary of the basic concepts

The equations governing the coupled response of the bulk and the surface are obtained from the balances of solid and diffusingspecies mass, linear and angular momentum, and energy and entropy over a control region (see Fig. 1). These integral balanceexpressions are then localised at an arbitrary point, labelled X , on the surface S0 or in the bulk B0. The canonical controlregion considered here is one which possibly has, as part of its boundary, the surface S0 of the domain B0 and is denotedB0 ⊂ B0 with boundary ∂B0.

S0 := ∂B0

S •0 := ∂B0/S0

B0

∂B0

B0

C0 := ∂S0

S0 := S0 ∩ ∂B0

S0 M

M

M

N

N

�N

�N

if X ∈ B0 → S0 = ∅, S •0 = ∂B0 and

�

S0

{•} dA = 0

if X ∈ S0 → S0 �= ∅, ∂B0 = S0 ∪ S •0 and

�

B0

{•} dV = 0

localisation properties:

B0 Bt

Bt

St

m

∂Bt

m

S •tStn

�n

Ct

ϕ

initial configuration

current configuration

canonical control region

control regionin the bulk

control regionon the surface

Fig. 1 The domains B0 and Bt, and the canonical control region and its representation on the surface and in the bulk.

The region B0 is mapped via the motion ϕ to the current configuration Bt as Bt = ϕ(B0, t). The surface of Bt is denoted∂Bt = St ∪S •t with outward unit normal m. The spatial velocity is denoted v while the spatial velocity of the diffusingspecies is denoted v?.

The preceding definition of the control region is reconciled with the more commonly adopted view of a control region asa domain completely within the bulk by specifying the surface S0 to be an empty set when localising an integral balance

∗ Corresponding author: email andrew.mcbride@ltm.uni-erlangen.de, phone +09 131 852 8521, fax +09 131 852 8503

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

484 Section 7: Coupled problems

expression to an arbitrary point in the bulk. Critically, the equations governing the response of the bulk obtained by localisingthe integral balance equations over the canonical control region to a point in the bulk are identical to those obtained bylocalising the integral balance equations obtained from a conventional control region.

Conceptually, the process of localising an integral expression over a control region to a point on the surface is as follows.Firstly, the volume of the control region is decreased to zero in a manner such that the surfaces S0 and S •0 coincide andN ·M = −1 on S0. Hence all terms involving integrals over the volume B0 in the integral expression vanish. Theremaining terms are then localised to a point on the surface in the standard manner and the relationship N ·N = 0 holds. Theprocess of localising an expression over a control region to a point on the surface is in agreement with the assumption that thesurface has zero thickness and can be viewed as a membrane.

When localising an integral expression over a control region to a point in the bulk, the surface S0 is, by definition, emptyand all integrals over S0 vanish.

The integral form of the balance equations typically include terms due to convective or physical fluxes, or both, of a quantityover the internal surface S •0 ⊆ ∂B0. We denote by {•}0 the material flux vector associated with the aforementioned quantity.The contribution of the flux vector to the integral balance expression will generally be of the form

∫S •

0{•}0 ·M dA. This

integral expression when localised to a point on the surface S0 becomes−{•}0 ·N . When the integral expression is localisedto a point in the bulk, the divergence theorem will be used to convert the area integral over ∂B0 to a volume integral over B0.Neumann boundary conditions on the flux act on the exterior of the surface S0. The prescribed flux is denoted {•}p

0. Thecontribution of the prescribed term to the integral expression of the balance equations is of the form

∫S0{•}p

0 dA which whenlocalised to a point on the surface S0 becomes {•}p

0. There is thus generally a term of the form [{•}p0 − {•}0 ·N ] present

when the balance equations are localised to a point on the surface. This term represents the net flux due to a prescribed fluxacting from the exterior and the flux from the bulk in the interior.

We define by v the surface velocity; a surface tangent vector. The notation v does not denote the (bulk) velocity evaluatedat the surface St, i.e. v 6= v|St , rather it is the projection of the bulk velocity v onto the surface with normal n. Spatial scalarquantities in the bulk {•} (x, t) and on the surface {•} (x|St , t) are denoted {•}t and {•}t, respectively.

The material time derivative of an integral over a spatial control region Bt ⊂ Bt is given by a modified form of Reynolds’stransport theorem as

Dt

∫

Bt

{•}t dv+Dt

∫

St

{•}t da =

∫

Bt

dt {•}t dv+

∫

S •t

{•}t v ·m da+

∫

St

dt {•}t da+

∫

Ct

{•}t v ·n dl . (1)

Here Dt {•} and dt {•} denote the material and spatial time derivatives of a scalar {•}, respectively. The modified form ofReynolds’s transport theorem given in Eq. (1) simplifies to the standard form if the control region is completely within thebulk as St = ∅ and hence S •t = ∂Bt. With this assumption the second term on the left-hand side and the third and fourthterms on the right-hand side of Eq. (1) disappear. The resulting (bulk) Reynolds’s transport theorem equates the material rateof change of the integral of a spatial quantity {•}t in the bulk to the integral of the instantaneous rate of change of the quantityin the bulk and changes due to the convective flux of the quantity over the surface ∂Bt.

Consider the case now where S0 6= ∅. The second term on the left-hand side of Eq. (1) represents the contribution dueto the material rate of change of an integral of a surface quantity {•}t. The third and fourth terms on the right-hand sideaccount for the instantaneous rate of change of the integral of the spatial quantity {•}t over the surface, and changes due tothe convective flux of the quantity over the curve Ct, respectively.

The localised statement for the conservation of diffusing species concentration on the surface c0 is given to illustrate theresulting structure of the governing equations:

Dtc0 = −DivW + [W p + W ·N ] = −DivW + W on S0 . (2)

The quantity W is the net species flux on the surface and W the species flux vector within the bulk. It is clear that theprescribed species flux W p = W −W ·N contains contributions from both the surface W and the bulk W ·N and arises asa consequence of the balance equations. In the absence of surface effects, i.e. W = 0, one recovers the standard definition ofthe diffusional flux as W p = −W ·N .

3 Conclusion

The procedure outlined here is used to derive the key balance equations in the bulk and on the surface. Constitutive relationsare then obtained from a standard Colemann–Noll procedure wherein the bulk and the surface are both endowed with aHelmholtz energy. The numerical solution of the governing equations will form the focus of a future work.

References[1] A. T. McBride, A. Javili, P. Steinmann, and S. Bargmann, Geometrically nonlinear continuum thermomechanics with surface energies

coupled to diffusion, In review.

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