Chapter V: Shape Analysis and Energy Stability of Conductive Liquids
5.2 Lagrangian Specification in Continuum Mechanics
the Rayleigh limit, the second shape variation to its total potential energy is a strictly positive quadratic functional and hence implies local stability in the energy landscape.
to be the vectoriol rate at which the position of a particle initially labeled with material coordinate X changes with respect to parameter . Again if we take χ(X,0) = X and to be time, then V(X, ) is the velocity of the particle which starts out at the reference position X inΩ. The spatial counterpart of material velocity V(X, ) is the spatial velocity
v(x, ) =V(χ−1(;x), ) (5.5) which is the velocity vector of the particular marker particle currently occupying the spatial position x. Here χ(;·) : Ω → ω is the invertible configuration map when is held fixed. In the same spirit, a spatial scalar field φ(x, ) in general can be viewed as a material fieldΦ(X, ) if we identify
Φ(X, ) =φ(x(X, ), ) (5.6) for all ≥0 in the valid range.
Convective curvilinear coordinate
Let the vectorial functionX(ξ1, ξ2, ξ3)denote the position vector of the material frame constructed out of coordinates(ξ1, ξ2, ξ3)drawn from a subsetOin the Euclidean space E3. Here we differentiate the ambient three-dimensional space R3 from the Euclidean parameter space E3 although both of them are flat spaces. The material coordinate curves, i.e. curves of constantξi, form a net in the material configuration. The covariant basisGi are the tangent vectors to these material coordinate curves ξi,
Gi= ∂X
∂ξi, Gi·Gj =δij, Gij =Gi·Gj, (5.7) where Gi are the contravariant basis vectors and Gij are the metric coefficients (δji is the Kronecker delta). Given knowledge of the configuration mapχ(·, ), we can express the spatial coordinates xat≥0 in terms of the same curvilinear coordinates as well,
x(ξ1, ξ2, ξ3) =χ(X(ξ1, ξ2, ξ3), ). (5.8) The covariant and contravariant basis vectors,gi andgi of the spatial curvilinear coor- dinates are given by
gi= ∂x
∂ξi, gi·gj =δji, gij =gi·gj, (5.9) where gij is the spatial metric coefficient. The curvilinear coordinates x are said to be convective if coordinate curves are attached to material particles and deform with the body so that each material particle has the same parameter coordinates (ξ1, ξ2, ξ3) in both the material configuration X and spatial configuration x. The mapping triad between parametrization (ξ1, ξ2, ξ3) ∈ O ⊂ E3, material configuration X ∈ Ω ⊂ R3 and spatial configurationx∈ω ⊂R3 is illustrated in figure 5.1. A useful fact to have
X(ξ1 , ξ2
, ξ3) x(ξ1, ξ2, ξ3, ) Parameter space
(ξ1,ξ2,ξ3) Configuration map
χ(X, ) Material frame Ω⊂R3
X
Spatial frame ω⊂R3 x(χ(X, ), ) Ψ = 0
Ψ = 1 Γ
ψ= 0
ψ= 1 γ
Figure 5.1: Triad of parameter, material and spatial frames: material coordinates X send(ξ1, ξ2, ξ3) from the parameter spaceE3 into the material frameΩin the ambient spaceR3with boundaryΓ; Spatial frameωand its boundaryγinR3are convected from the material coordinates X through the configuration map χ(·, ); the composition of χ(X, )andX(ξ1, ξ2, ξ3)can be viewed as a set of curvilinear coordinatesx(ξ1, ξ2, ξ3) drawn from the parameter space. The thin solid curves are the coordinate lines in each frame while the thick solid curves represent the boundariesΓ and γ of the same conductor (white region) before and after deformation. The material harmonic potential Ψ and its spatial counterpart ψremain equipotential on these boundaries. The dashed lines are immobile boundaries of the vacuum domain exterior to the conductor.
in mind is that, despite being curvilinear, both x(ξ1, ξ2, ξ3) and X(ξ1, ξ2, ξ3) are a re-parametrization of the ambient three-dimensional space R3 which is Euclidean, flat and torsion-free.
In what follows, we use operator D(·) to represent the (covariant) gradient vector with respect to material coordinates X and d(·) with respect to spatial coordinates x. For example, whenD (or d) acts on a scalar fieldΦ (orφ) we have
DΦ= ∂Φ
∂ξiGi, dφ= ∂φ
∂ξigi. (5.10)
When gradient D (or d) acts on a vector field, the resulting object becomes a rank-2 tensor,
DV = ∂V
∂ξi ⊗Gi, dv= ∂v
∂ξi ⊗gi. (5.11)
The deformation gradient tensor
F =Dx= ∂x
∂ξi ⊗Gi (5.12)
is defined as the gradient of spatial configurationxwith respect to material coordinates X. The deformation gradient F is a two-point tensor which maps the tangent space of the material configuration at position X ∈ Ω to the tangent space of the spatial configuration at positionx∈ω. With convective curvilinear coordinates(ξ1, ξ2, ξ3), it’s easy to show that F behaves like a linear operator that transforms the covariant basis vectors Gi at X of the material frame into the covariant basis vectors gi at x of the spatial frame,
F =gi⊗Gi, FGi=gi. (5.13)
In addition to the deformation gradient F, its transpose, inverse and inverse-transpose are two-point tensors as well which map a set of basis vectors from one frame onto the other (Kelly,2013),
F−1 =Gi⊗gi, F−1gi =Gi, F>=Gi⊗gi, F>gi =Gi, F−>=gi⊗Gi, F−>Gi =gi.
(5.14)
With various two-point tensors introduced in (5.14), the material gradient of a scalar field Φ(X) and the spatial gradient of its spatial counterpart φ(x) can be transformed back and forth through the relations,
dφ= (DΦ)F−1, dv = (DV)F−1. (5.15) It is convenient to introduce the right Cauchy-Green tensorC and its inverseC−1,
C =F>F, C−1 =F−1F−>. (5.16)
Physically, the Cauchy–Green tensor C measures the squared local change in distances due to body deformation asGiCGi =gi·gi. In the same fashion, we define the velocity gradient as a measure of the rate at which a material is deforming. It’s common to use the spatial velocity gradient tensor
l =dv. (5.17)
The differential volume element of the material configurationX is given by dΩ=√
Gdξ1dξ2dξ3, G= detGij, (5.18) whereGis the determinant of material metric tensorGij. Similarly, the volume element of the spatial configurationxis given by the determinantg of the spatial metric tensor gij,
dω =√
gdξ1dξ2dξ3. (5.19)
SinceF is the essentially the Jacobian matrix ofx(X, ), the material volume element dΩ and the spatial volume elementdω are related through the relation,
dω=JdΩ, J = detF, (5.20)
where J is the coordinate Jacobian. A similar identity exists for the differential area elements. Let Γ (orγ) be the boundary of the material ( or spatial) domain Ω (or ω) andN (orn) be the unit normal vector of the boundary. The material differential area elementdΓ is related to the spatial area elementdγ by the Nanson’s formula (Wriggers, 2008)
ndγ =JF−>NdΓ. (5.21)
Kinematics: rate of quantities
We next consider how fields transform when material body undergoes deformation. The total change of a scalar field at a spatial coordinate x following a material particle initially labeled with position X is given by the chain rule,
dφ(x, )
d = dφ(x(X, ), )
d = ∂φ(x, )
∂ +v(x, )·dφ·. (5.22) Following the convention in literature of shape analysis, we define
φ0(x, ) = ∂φ(x, )
∂ (5.23)
to be the Eulerian derivative ofφ and
φ(x, ) =˙ φ0(x, ) +v(x, )·dφ(x, ) (5.24) to be the material derivative ofφ. The name “material derivative” stems from the obser- vation that, alternatively the total derivative (5.22) of spatial fieldφ(x, )is equivalent to the partial derivative of its material counterpart Φ(X, ),
φ(x, ) =˙ dφ(x, )
d = dφ(x(X, ), )
d = ∂Φ(X, )
∂ . (5.25)
It can be shown that the time derivatives of various forms of deformation gradient tensor F can be expressed in terms of the spatial velocity gradient l,
∂F
∂ =l F, ∂F−1
∂ =−F−1l, ∂F>
∂ =F>l>, ∂F−>
∂ =−l>F−>. (5.26) The Eulerian derivative of spatial velocity gradient is frequently encountered in this work.
If we introduce the material and spatial accelerations, V˙ = ∂2χ(X, )
∂2 , v˙ = ˙V(χ−1(;x), ), (5.27) then ∂l/∂ can be computed as
∂l
∂ = ∂dv
∂ = ∂(DV)F−1
∂ =D∂V
∂
F−1−(DV)F−1l =dv˙ −l l, (5.28) where we have used (5.26) and (5.15) to simplify. Here dv, similar to spatial velocity˙ gradientl =dv, is the spatial acceleration gradient. A similar expression can be derived for the transposed quantity, i.e. ∂l>/∂= (∂l/∂)>.
The rate at which spatial volume element changes is given by the first partial derivative of Jacobian J with respect to,
∂J
∂ = (detF)tr∂F
∂F−1
=Jdivv, (5.29)
where the divergence operator div is understood to act on the spatial vector fields (Div acts on material vectors). Differentiating ∂J/∂ again yields the second partial derivative of Jacobian J,
∂2J
∂2 = ∂
∂
(detF)tr∂F
∂F−1
= ∂J
∂tr(l F F−1) +Jtr ∂2F
∂2 F−1
!
+Jtr ∂F
∂
∂F−1
∂
!
=J
(divv)tr(l) + trhD∂x
∂2
F−1i−tr(l F F−1l)
=Jh(divv)2+ div( ˙v)−tr(l l)i, (5.30) which we will use later.