On Weingarten-Volterra Defects

Abstract

The kinematic theory of Weingarten-Volterra line defects is revisited, both at small and finite deformations. Existing results are clarified and corrected as needed, and new results are obtained. The primary focus is to understand the relationship between the disclination strength and Burgers vector of deformations containing a Weingarten-Volterra defect corresponding to different cut-surfaces.

Appendix: Strain Compatibility on a simply connected region

For the sake of completeness we collect some classical results related to questions of strain compatibility in the following sections. All regions considered in these appendices are simply connected, unless mentioned otherwise. We repeatedly use the argument that if $f_{,k} = 0$ on the domain then $f$ is a constant, which uses the fact that the region in question is path connected.

1.1 A.1 Small deformation

Theorem A.1

Given a$C^{2}$second-order tensor field$\boldsymbol{\varepsilon }$on a simply connected region, it is necessary and sufficient for the existence of a$C^{3}$displacement field${\boldsymbol{u}}$on it satisfying$(\operatorname{grad} {\boldsymbol{u}})_{sym} = \boldsymbol{\varepsilon }$that the St.-Venant compatibility condition (1)₃be satisfied.

Proof

Necessity—$u_{i}$ exists satisfying

$$ \begin{aligned} & u_{i,k} = \varepsilon _{ik} + \omega _{ik}; \qquad \varepsilon _{ik} = \frac{1}{2} ( u_{i,k} + u_{k,i} ) ; \qquad \omega _{ik} := \frac{1}{2} ( u_{i,k} - u_{k,i} ) . \\ & \omega _{ik,l} = \frac{1}{2} ( u_{i,kl} + u_{l,ki} - u_{l,ki} - u_{k,li} ) = e_{il,k} - e_{kl,i}. \end{aligned} $$

(25)

The infinitesimal rotation $\boldsymbol{\omega }$ is twice continuously differentiable and therefore, $\omega _{ik,lm} - \omega _{ik,ml} = 0$ implies (1)₃.

Sufficiency—We assume (1)₃ is satisfied and define

$$ \begin{aligned} & E_{ikl} := \varepsilon _{il,k} - \varepsilon _{kl,i}, \\ & \omega _{ik} ({\boldsymbol{x}}) = W_{ik} + \int _{{\boldsymbol{c}}{({\boldsymbol{x}}_{0},{\boldsymbol{x}})}} E _{ikl} ({\boldsymbol{c}}) dc_{l}, \end{aligned} $$

(26)

where ${\boldsymbol{c}}_{({\boldsymbol{x}}_{0},{\boldsymbol{x}})}$ is some path in the region from ${\boldsymbol{x}}_{0}$ to ${\boldsymbol{x}}$ and ${\boldsymbol{W}}$ is an arbitrary skew-symmetric tensor. Since $E_{ikl,m} = E_{ikm,l}$ due to (1)₃, $\boldsymbol{\omega }$ as defined in (26)₂ is independent of path, and defines a smooth function that satisfies

$$ \omega _{ik,l} = E_{ikl} $$

(27)

on the region which is unique up to the choice of $\boldsymbol{\omega }({\boldsymbol{x}}_{0}) = {\boldsymbol{W}}$. Since (27) and (26)₁ imply

$$ \varepsilon _{ik,l} + \omega _{ik,l} - ( \varepsilon _{il,k} + \omega _{il,k} ) = 0, $$

the definition

$$ u_{i}({\boldsymbol{x}}) = u^{0}_{i} + \int _{{\boldsymbol{c}}_{({\boldsymbol{x}}_{0},{\boldsymbol{x}})}} \bigl( \varepsilon _{ik} ({ \boldsymbol{c}}) + \omega _{ik}({\boldsymbol{c}}) \bigr) dc_{k}, $$

for any ${\boldsymbol{u}}^{0}$ an arbitrary constant vector, is independent of the path ${\boldsymbol{c}}_{({\boldsymbol{x}}_{0}, {\boldsymbol{x}})}$ chosen to connect ${\boldsymbol{x}}_{0}$ and ${\boldsymbol{x}}$, and hence defines a smooth displacement field whose symmetrized gradient equals the given $\boldsymbol{\varepsilon }$ field. □

1.2 A.2 Rigidity for smooth infinitesimal deformations

Theorem A.2

If two$C^{1}$displacement fields on a region have identical strain fields, then they differ at most by an infinitesimally rigid deformation. The region here need not be simply-connected.

Proof

Let ${\boldsymbol{u}}^{1}$ and ${\boldsymbol{u}}^{2}$ be the two displacement fields with strain fields $\boldsymbol{\varepsilon }^{1}$ and $\boldsymbol{\varepsilon }^{2}$ and rotation field $\boldsymbol{\omega }^{1}$ and $\boldsymbol{\omega } ^{2}$, respectively, defined from the corresponding displacement fields by relations (25) and let $\boldsymbol{\varepsilon }^{1} = \boldsymbol{\varepsilon }^{2}$. Then

$$ \omega ^{1}_{ij,k} - \omega ^{2}_{ij,k} = \varepsilon ^{1}_{ik,j} - \varepsilon ^{2}_{jk,i} = 0, $$

using a similar computation as in (25), and therefore ${\boldsymbol{W}}: = \boldsymbol{\omega }^{1} - \boldsymbol{ \omega }^{2}$ is a constant skew-symmetric tensor on the region (which is path-connected). We then have

$$ u^{1}_{i,j} - u^{2}_{i,j} = W_{ij}, $$

and integrating along paths from an arbitrarily fixed ${\boldsymbol{z}}$ to all points ${\boldsymbol{x}}$ in the region and rearranging terms we obtain

$$ {\boldsymbol{u}}^{1}({\boldsymbol{x}}) - {\boldsymbol{u}}^{1}({ \boldsymbol{z}}) = {\boldsymbol{u}}^{2}({\boldsymbol{x}}) - { \boldsymbol{u}}^{2}({\boldsymbol{z}}) + {\boldsymbol{W}} [ { \boldsymbol{x}}-{\boldsymbol{z}} ] . $$

Since ${\boldsymbol{z}}$ was arbitrarily fixed, this implies that ${\boldsymbol{u}}^{1}$ and ${\boldsymbol{u}}^{2}$ are related by an infinitesimally rigid deformation by Definition 6. □

1.3 A.3 Finite deformation

The treatment here is from Sokolnikoff [12]. While the considerations below relate to fundamental relations in Riemannian geometry, we intentionally emphasize the purely algebraic fact that if the functions $h,g,x,y$ (defined below) satisfy (28), then they necessarily satisfy (31) and the definition (29) implies (32).

Let $\varOmega _{x}$ and $\varOmega _{y}$ be two 3-d coordinate patches, i.e., open bounded regions of $\mathbb{R}^{3}$, and let $y:\varOmega _{x} \rightarrow \varOmega _{y}$ be a $C^{2}$ diffeomorphism with a $C^{2}$ inverse that we denote by $x:\varOmega _{y} \rightarrow \varOmega _{x}$. Let $h:\varOmega _{y} \rightarrow \mathbb{R}^{3\times 3}$ and $g:\varOmega _{x} \rightarrow \mathbb{R}^{3\times 3}$ be two prescribed matrix fields with range in the set of symmetric, positive definite $3\times 3$ matrices. In the following all indices range over the set $\{ 1,2,3\}$. Assume

$$ h_{ij} = x^{\alpha }_{,i} ( g_{\alpha \beta } \circ x ) x ^{\beta }_{,j} \Longleftrightarrow g_{\alpha \beta } = y^{i}_{,\alpha } ( h_{ij} \circ y ) y^{j}_{,\beta } $$

(28)

holds. Then, using the symmetry of $g_{\alpha \beta }$ (and switching some dummy indices),

$$ \begin{aligned} h_{ij,k} & = \bigl( x^{\alpha }_{,ik} x^{\beta }_{,j} + x^{\beta } _{,i} x^{\alpha }_{,jk} \bigr) ( g_{\alpha \beta } \circ x ) + x^{\alpha }_{,i} x^{\beta }_{,j} x^{\gamma }_{,k} ( g_{\alpha \beta ,\gamma } \circ x ) , \\ h_{ik,j} & = \bigl( x^{\alpha }_{,ij} x^{\beta }_{,k} + x^{\beta } _{,i} x^{\alpha }_{,kj} \bigr) ( g_{\alpha \beta } \circ x ) + x^{\alpha }_{,i} x^{\gamma }_{,k} x^{\beta }_{,j} ( g_{\alpha \gamma ,\beta } \circ x ) , \\ h_{jk,i} & = \bigl( x^{\alpha }_{,ji} x^{\beta }_{,k} + x^{\beta } _{,j} x^{\alpha }_{,ki} \bigr) ( g_{\alpha \beta } \circ x ) + x^{\beta }_{,j} x^{\gamma }_{,k} x^{\alpha }_{,i} ( g_{\beta \gamma ,\alpha } \circ x ) . \end{aligned} $$

Now define $\hat{\varGamma }:\varOmega _{x} \rightarrow \mathbb{R}^{3 \times 3 \times 3}$ and $\hat{H}:\varOmega _{y} \rightarrow \mathbb{R} ^{3 \times 3 \times 3}$ as

$$ \begin{aligned} \hat{H}_{ijk} & := \frac{1}{2} ( h_{ik,j} + h_{jk,i} - h_{ij,k} ) , \\ \hat{\varGamma }_{\alpha \beta \gamma } & := \frac{1}{2} ( g_{ \alpha \gamma ,\beta } + g_{\beta \gamma ,\alpha } - g_{\alpha \beta ,\gamma } ) . \end{aligned} $$

(29)

Then,

$$ \hat{H}_{ijk} = x^{\alpha }_{,i} x^{\beta }_{,j} x^{\gamma }_{,k} ( \hat{\varGamma }_{\alpha \beta \gamma } \circ x ) + x^{ \alpha }_{,ji} x^{\beta }_{,k} ( g_{\alpha \beta } \circ x ) . $$

Next we define $H: \varOmega _{y} \rightarrow \mathbb{R}^{3 \times 3 \times 3}$ and $\varGamma : \varOmega _{x} \rightarrow \mathbb{R}^{3 \times 3 \times 3}$ by

$$ H^{k}_{ij} = h^{km} \hat{H}_{ijm} \quad \mbox{and} \quad \varGamma^{\rho }_{\alpha \beta } = g^{\rho \nu } \hat{ \varGamma }_{\alpha \beta \nu }, $$

where $h^{km}$ and $g^{\alpha \beta }$ are the components of the matrices $h^{-1}$ and $g^{-1}$, respectively. We note that $\hat{H}, H, \hat{\varGamma }, \varGamma$ are all symmetric in their lower first two indices. Noting from (28) that

$$ h^{km} = \bigl( g^{\alpha \beta } \circ x \bigr) \bigl( y^{k}_{, \alpha } \circ x \bigr) \bigl( y^{m}_{,\beta } \circ x \bigr) , $$

we obtain

$$ H^{k}_{ij} = \bigl( y^{k}_{,\alpha } \circ x \bigr) x^{\alpha } _{,ji} + \bigl( y^{k}_{,\rho } \circ x \bigr) x^{\alpha }_{,i} x ^{\beta }_{,j} \bigl( \varGamma^{\rho }_{\alpha \beta } \circ x \bigr) , $$

which, after rearrangement of terms, yields

$$ x^{\mu }_{,ji} = x^{\mu }_{,k} H^{k}_{ij} - x^{\alpha }_{,i} x^{ \beta }_{,j} \bigl( \varGamma^{\mu }_{\alpha \beta } \circ x \bigr) . $$

(30)

Of course, the computations above indicate that interchanging the list $( x,y, g,g^{-1}, h, h^{-1}, \hat{\varGamma }, \hat{H}, \varGamma , H )$ by $( y,x, h, h^{-1}, g,g^{-1}, \hat{H}, \hat{\varGamma }, H, \varGamma )$ in the above formulae is admissible. We thus have

$$ y^{i}_{,\alpha \beta } = y^{i}_{,\gamma } \varGamma^{\gamma }_{\alpha \beta } - y^{j}_{,\alpha } y^{k}_{,\beta } \bigl( H^{i}_{jk} \circ y \bigr) . $$

(31)

Another result we will need for our compatibility argument to follow is as follows:

$$ \begin{aligned} \hat{\varGamma }_{\alpha \gamma \beta } + \hat{\varGamma }_{\beta \gamma \alpha } & = g_{\alpha \beta ,\gamma }, \\ \bigl( g_{\alpha \beta } g^{\beta \mu } \bigr) _{,\gamma } & = 0 \end{aligned} $$

so that

$$ g^{\rho \mu }_{,\gamma } = - g^{\rho \alpha } ( \hat{\varGamma } _{\alpha \gamma \beta } + \hat{\varGamma }_{\beta \gamma \alpha } ) g^{\beta \mu } = - \bigl( g^{\rho \alpha } \varGamma^{\mu }_{\alpha \gamma } + g^{\beta \mu } \varGamma^{\rho }_{\beta \gamma } \bigr) , $$

which implies

$$ g^{\alpha \beta }_{,\mu } = - \bigl( g^{\alpha \gamma } \varGamma^{ \beta }_{\gamma \mu } + g^{\gamma \beta } \varGamma^{\alpha }_{\gamma \mu } \bigr) . $$

(32)

Let $\varOmega$ be a simply connected region and let ${\boldsymbol{C}}: \varOmega \rightarrow {\mathcal{{P}}}_{sym}$ be a prescribed $C^{2}$ field on $\varOmega$, where $\mathcal{{P}}_{sym}$ is the set of all positive-definite, symmetric tensors on $V_{3}$. Let $\varOmega _{x}$ be a Rectangular Cartesian coordinate patch parameterizing $\varOmega$ and $C:\varOmega \rightarrow \mathbb{R}^{3 \times 3}$ be the component map of ${\boldsymbol{C}}$ with respect to the rectangular Cartesian basis of the parametrization of $\varOmega$ by $\varOmega _{x}$.

Theorem A.3

If there exists a deformation${\boldsymbol{y}}: \varOmega \rightarrow {\mathcal{{E}}}_{3}$satisfying

$$ ( \operatorname{grad} {\boldsymbol{y}} ) ^{T} \operatorname{grad} { \boldsymbol{y}}= {\boldsymbol{C}}, $$

(33)

then (1)₂is satisfied in$\varOmega$. Conversely, if the matrix field$C$satisfies (1)₂, then there exists a deformation${\boldsymbol{y}}: \varOmega \rightarrow {\mathcal{ {E}}}_{3}$that satisfies (33).

Proof

Necessity of (1)₂ for (33)—Let $y: \varOmega _{x} \rightarrow \varOmega _{y} \subset \mathbb{R}^{3}$ be the coordinate map representing the parametrization of ${\boldsymbol{y}}(\varOmega )$ by the same Rectangular Cartesian system for $\mathcal{{E}}_{3}$ used to parametrize $\varOmega$. Then

$$ y^{i}_{,\alpha } \delta _{ij} y^{j}_{,\beta } = C_{\alpha \beta } $$

holds. Making the identification of $h = I$ and $g = C$ in (28), we have $H = 0$ and (31) implies

$$ \begin{aligned} y^{i}_{,\alpha } & = F^{i}_{\alpha }, \\ F^{i}_{\alpha ,\beta } & = \varGamma^{\gamma }_{\alpha \beta } F^{i} _{\gamma }. \end{aligned} $$

(34)

Due to the smoothness of ${\boldsymbol{C}}$ and (34) we obtain

$$ \begin{aligned} & y^{i}_{\alpha ,\beta \rho } = y^{i}_{\alpha , \rho \beta } \\ & \quad \Longrightarrow F^{i}_{\gamma ,\rho } \varGamma^{\gamma }_{ \alpha \beta } + F^{i}_{\gamma } \varGamma^{\gamma }_{\alpha \beta , \rho } - F^{i}_{\gamma ,\beta } \varGamma^{\gamma }_{\alpha \rho } - F ^{i}_{\gamma } \varGamma^{\gamma }_{\alpha \rho , \beta } = 0 \\ &\quad \Longrightarrow F^{i}_{\mu } \bigl( \varGamma^{\mu }_{\alpha \beta ,\rho } - \varGamma^{\mu }_{\alpha \rho ,\beta } + \varGamma^{\mu } _{\gamma \rho } \varGamma^{\gamma }_{\alpha \beta } - \varGamma^{\mu }_{ \gamma \beta } \varGamma^{\gamma }_{\alpha \rho } \bigr) = 0 \end{aligned} $$

and since $F$ is an invertible matrix (due to ${\boldsymbol{C}}\in {\mathcal{{P}}}_{sym}$), (1)₂ holds.

Sufficiency of (1)₂ for (33)—By a theorem of Thomas [13] (also the Froebenius theorem in the differential geometry literature), we have that a solution to (34) exists, with freely specifiable value of $F$ and $y$ at arbitrarily chosen points of $\varOmega _{x}$, if $\varGamma^{\gamma }_{\alpha \beta } = \varGamma^{\gamma }_{\beta \alpha }$ (which holds by definition of $\varGamma$), and (1) ₂ hold.

Specify the value of $F$ at an arbitrarily chosen point $x_{0} \in \varOmega _{x}$ such that $F^{i}_{\alpha }(x_{0}) F^{i}_{\beta }(x_{0}) = C_{\alpha \beta } (x_{0})$, written alternatively as $F_{0}^{T} F_{0} = C_{0} \Longrightarrow F_{0} C^{-1}_{0} F_{0}^{T} = I$. For an $F$ field satisfying (34) with the identification $C = g$ in (29) and (32), consider now

$$ \begin{aligned} \bigl( C^{\alpha \beta } F^{i}_{\alpha }F^{j}_{\beta } \bigr) _{, \mu } = \bigl( C^{\alpha \beta }_{,\mu } + C^{\gamma \beta } \varGamma^{\alpha }_{\gamma \mu } + C^{\alpha \gamma } \varGamma^{\beta } _{\gamma \mu } \bigr) F^{i}_{\alpha }F^{j}_{\beta }= 0, \end{aligned} $$

by (32). Thus we have that

$$ FC^{-1}F^{T} = I \Longrightarrow C = F^{T}F \quad \mbox{on } \varOmega _{x}, $$

and defining ${\boldsymbol{y}}= y^{i} {\boldsymbol{e}}_{i}$ where ${\boldsymbol{e}}_{i}, i = 1,2,3$ is the (spatially constant) natural basis of the Rectangular Cartesian coordinate system used to parametrize $\varOmega$, we find that ${\boldsymbol{y}}$ satisfies (33). □

1.4 A.4 Rigidity for smooth finite deformations

Let ${\boldsymbol{y}}^{*}:\varOmega \rightarrow {\mathcal{{E}}}_{3}$ and ${\boldsymbol{z}}:\varOmega \rightarrow {\mathcal{{E}}}_{3}$ be two $C^{2}$ deformations of $\varOmega$. Consider a rectangular Cartesian parametrization of $\mathcal{{E}}_{3}$ under which $\varOmega$ maps to the set of coordinates $\varOmega _{x} \subset \mathbb{R}^{3}$, ${\boldsymbol{y}} ^{*}(\varOmega )$ maps to $\varOmega _{y}$, and ${\boldsymbol{z}}(\varOmega )$ maps to $\varOmega _{z}$. Let $y^{*}:\varOmega _{x} \rightarrow \varOmega _{y}$, $z:\varOmega _{x} \rightarrow \varOmega _{z}$, $y:\varOmega _{z} \rightarrow \varOmega _{y}$, and $x: \varOmega _{z} \rightarrow \varOmega _{x}$ be the corresponding deformations, represented in coordinates. $\varOmega$ need not be simply connected.

Theorem A.4

If${\boldsymbol{y}}^{*}$and${\boldsymbol{z}}$have the same right Cauchy-Green tensor fields then they are rigidly related to each other.

Proof

We have, following [11],

$$ \begin{aligned} & y^{*i}_{,j} y^{*i}_{,k} = z^{m}_{,j}z^{m}_{,k} = C_{jk} \\ &\quad \Rightarrow \bigl( y^{i}_{,m} \circ z \bigr) z^{m}_{,j} \bigl( y^{i}_{,n} \circ z \bigr) z^{n}_{,k} = z^{m}_{,j} z^{m}_{,k} \\ &\quad \Rightarrow y^{i}_{,p} y^{i}_{,q} = y^{\alpha }_{,p} \delta _{\alpha \beta } y^{\beta }_{,q} = x^{j}_{,p} \bigl( z^{m} _{,j} \circ x \bigr) \bigl( z^{m}_{,k} \circ x \bigr) x^{k}_{,q} = \delta ^{m}_{p} \delta ^{m}_{q} = \delta _{pq}, \end{aligned} $$

(35)

and identifying $x$, $h$, and $g \circ x$ in (28) with $y$, $I$, and $I$, respectively, here, we have from (30) that

$$ y^{\alpha }_{,pq} = 0 \quad \mbox{on } \varOmega _{z}. $$

Due to the path connectedness of $\varOmega _{z}$ (induced from $\varOmega _{x}$), this implies that there exists a constant matrix $R$ satisfying $y^{\alpha }_{,p} = R^{\alpha }_{p}$ on $\varOmega _{z}$ with $R^{T}R = I$ from (35)₃. Integrating along an arbitrarily chosen path from $z(w_{0}) \in \varOmega _{z}$ to $z(w) \in \varOmega _{z}$ for $w_{0}, w \in \varOmega _{x}$, we obtain

$$ y^{\alpha }\circ z (w) = y^{\alpha }\circ z (w_{0}) + R^{\alpha }_{p} \bigl[ z^{p}(w) - z^{p}(w_{0}) \bigr] \Longrightarrow y^{*}(w) = y ^{*}(w_{0}) + R \bigl[ z(w) - z(w_{0}) \bigr] $$

for any $w \in \varOmega _{x}$ and $w_{0} \in \varOmega _{x}$ and therefore ${\boldsymbol{y}}^{*}$ and ${\boldsymbol{z}}$ are rigidly related to each other by Definition 5. □