The Complementing and Agmon’s Conditions in Finite Elasticity

Abstract

We consider the complementing condition and Agmon’s condition for linearized elasticity in two-dimensions. With an elasticity tensor $\mathsf{C}$ derived from a compressible, isotropic stored energy $W$, linearized about a homogeneous deformation $\mathbf{f}_{0}$, we apply the complementing and Agmon’s conditions to a traction portion of the surface of a body with unit normal $\mathbf{n}$. We show these conditions are independent of $\mathbf{n}$ for arbitrary $W$ and $\mathbf{f}_{0}$. We also consider the case of failure of the complementing condition for the pure traction problem.

Appendix

We recall the linearized 4-tensor in (4.5) in two dimensions

$$ \mathsf{C}[\mathbf{H}] = \left [ \textstyle\begin{array}{c@{\quad }c} K H_{11}+N H_{22} & P H_{12}+L H_{21} \\ L H_{12}+P H_{21} & N H_{11}+T H_{22} \end{array}\displaystyle \right ], $$

(A.1)

for $\mathbf{H} \in \mathrm{M}_{2}$. In this appendix we consider some positivity properties of $\mathsf{C}$ as well as some consequences if $(\mathsf{C},\mathbf{n})$ fails the complementing condition for the boundary value problem (A.4) below. Note $\mathsf{C}$ is symmetric.

From (A.1) we have

$$ \mathbf{H} \cdot \mathsf{C}[\mathbf{H}] = [H_{11} \ \ H_{22}] \mathbf{D}_{1} \left [ \textstyle\begin{array}{c} H_{11} \\ H_{22} \end{array}\displaystyle \right ] + [H_{12} \ \ H_{21}] \mathbf{D}_{2} \left [ \textstyle\begin{array}{c} H_{12} \\ H_{21} \end{array}\displaystyle \right ] $$

(A.2)

where

$$ \mathbf{D}_{1} = \left [ \textstyle\begin{array}{c@{\quad }c} K & N \\ N & T \end{array}\displaystyle \right ],\qquad \mathbf{D}_{2} = \left [ \textstyle\begin{array}{c@{\quad }c} P & L \\ L & P \end{array}\displaystyle \right ],\qquad \mathbf{H} = \left [ \textstyle\begin{array}{c@{\quad }c} H_{11} & H_{12} \\ H_{21} & H_{22} \end{array}\displaystyle \right ]. $$

We say $\mathsf{C}$ is positive, $\mathsf{C} > 0$, iff $\mathbf{H} \cdot \mathsf{C}[\mathbf{H}] > 0$ for all $\mathbf{H} \in \mathrm{M}_{2} \setminus \{0\}$; and $\mathsf{C}$ is nonnegative, $\mathsf{C} \ge 0$, iff $\mathbf{H} \cdot \mathsf{C}[\mathbf{H}] \ge 0$ for all $\mathbf{H} \in \mathrm{M}_{2}$. From (A.2) we clearly have that $\mathsf{C} > 0$ iff $\mathbf{D} _{1} > 0$ and $\mathbf{D}_{2} > 0$; and additionally $\mathsf{C} \ge 0$ iff $\mathbf{D}_{1} \ge 0$ and $\mathbf{D}_{2} \ge 0$. (Here $\mathbf{D}_{i} >0$ means $\mathbf{D}_{i}$ is a positive definite matrix, and $\mathbf{D}_{i} \ge 0$ means $\mathbf{D}_{i}$ is a positive semidefinite matrix.)

Proposition A.1

Suppose$\mathsf{C} \ge 0$and$\mathbf{H} \in \mathrm{M}_{2}$. Then$\mathbf{H} \cdot \mathsf{C}[\mathbf{H}] = 0$iff$\mathsf{C}[ \mathbf{H}] = \mathbf{0}$.

Proof

Clearly $\mathsf{C}[\mathbf{H}] = \mathbf{0}$ implies $\mathbf{H} \cdot \mathsf{C}[\mathbf{H}] = 0$. Conversely suppose $\mathbf{H} \cdot \mathsf{C}[\mathbf{H}] = 0$. Then by semidefiniteness of $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ in (A.2) we have ${\mathbf{D}}_1\left[\begin{array}{c}{H}_{11}\\ H_{22}\end{array}\right]={\mathbf{D}}_2\left[\begin{array}{c}{H}_{12}\\ H_{21}\end{array}\right]=\mathbf{0}$. This yields $\mathsf{C}[\mathbf{H}] = \mathbf{0}$ from (A.1). □

In the case $\mathsf{C}$ is strongly elliptic (Theorem 4.1), we can characterize the set of $\mathbf{H}$ that satisfy $\mathsf{C}[ \mathbf{H}] = \mathbf{0}$ as follows. We denote the kernel of $\mathsf{C}$,

$$ \operatorname{ker} \mathsf{C} = \bigl\{ \mathbf{H} \in \mathrm{M}_{2} : \mathsf{C}[\mathbf{H}] = \mathbf{0} \bigr\} . $$

Proposition A.2

Suppose$P > 0$and either$K > 0$or$T > 0$. Then

• (i) $\mathsf{C}[\mathbf{H}] = \mathbf{0}$iff${\mathbf{D}}_1\left[\begin{array}{c}{H}_{11}\\ H_{22}\end{array}\right]={\mathbf{D}}_2\left[\begin{array}{c}{H}_{12}\\ H_{21}\end{array}\right]=\mathbf{0}$.

• (ii) If$KT - N^{2} \ne 0$, $P^{2} - L^{2} \ne 0$then$\ker \mathsf{C} = \{\mathbf{0}\}$.

• (iii) If$KT - N^{2} = 0$, $P^{2} - L^{2} \ne 0$then$ker\mathsf{C}=span\left\{\left[\begin{array}{cc}-N& 0\\ 0& K\end{array}\right]\right\}$.

• (iv) If$KT - N^{2} \ne 0$, $P^{2} - L^{2} = 0$then$ker\mathsf{C}=span\left\{\left[\begin{array}{cc}0& -L\\ P& 0\end{array}\right]\right\}$.

• (v) If$KT - N^{2} = 0$, $P^{2} - L^{2} = 0$then$ker\mathsf{C}=span\{\left[\begin{array}{cc}-N& 0\\ 0& K\end{array}\right],\left[\begin{array}{cc}0& -L\\ P& 0\end{array}\right]\}$.

Proof

(i). This follows directly from the fact that $\mathsf{C}[\mathbf{H}] = \mathbf{0}$ iff each entry of $\mathsf{C}[\mathbf{H}]$ in (A.1) is zero.

(ii)–(v). These follow from (i) and examining the cases when $\det \mathbf{D}_{i}$ is zero or not; note rank $\mathbf{D}_{i} \ge 1$ by the positivity assumptions of $K,T,P$. □

In the case of the Lame tensor in (2.24),

$$ \mathsf{C}[\mathbf{H}] = \mu \bigl(\mathbf{H} + \mathbf{H}^{T} \bigr) + \lambda (\mathbf{H} \cdot \mathbf{I}) \mathbf{I}, $$

we have $K = T = 2 \mu + \lambda$, $N = \lambda$, $P = L= \mu$. It is strongly elliptic iff $\mu > 0$ and $2\mu + \lambda > 0$ (see (4.6)). It is nonnegative iff $\mu \ge 0$ and $\mu + \lambda \ge 0$. It is not positive, as defined above, since $\mathsf{C}[\mathbf{W}] = \mathbf{0}$ for any skew $\mathbf{W} \in \mathrm{M}_{2}$.

Next, fixing the tensor $\mathsf{C}$ in (A.1), we assume in the following, that it is strongly elliptic (cf. (4.6)); in particular, $K > 0$, $T > 0$, $P > 0$. Then for any unit vector $\mathbf{n} \in {\mathbb{R}}^{2}$, by Theorem 4.4, $(\mathsf{C},\mathbf{n})$ fails the complementing condition iff

$$ A = P\bigl(KT - N^{2}\bigr) + \sqrt{KT} \bigl(P^{2} - L^{2}\bigr) = 0. $$

(A.3)

We note, that, if $A = 0$, then, $\det \mathbf{D}_{1} = KT - N^{2} = 0$ holds iff $\det \mathbf{D}_{2} = P^{2} - L^{2} = 0$ holds.

Suppose $\varOmega \subset {\mathbb{R}}^{2}$ is a bounded, connected, open set and its boundary, $\partial \varOmega$, is $C^{2}$-smooth. Consider the linearized traction boundary value problem with constant tensor $\mathsf{C}$

$$\begin{aligned} \begin{aligned} \operatorname {div}\mathsf{C} [\nabla \mathbf{u}] &= \mathbf{0} \quad\mbox{on } \varOmega , \\ \mathsf{C} [\nabla \mathbf{u}] \mathbf{n} &= 0 \quad\mbox{on } \partial \varOmega , \end{aligned} \end{aligned}$$

(A.4)

where $\mathbf{n}$ is the outward unit normal to $\partial \varOmega$; here the solution $\mathbf{u}:\varOmega \to {\mathbb{R}}^{2}$ is in the Sobolev space $W^{2,2}(\varOmega )$. Let

$$ S = \mbox{set of solutions } \mathbf{u} \in W^{2,2}(\varOmega ) \mbox{ of (A.4)}. $$

Note $S$ contains the constant functions on $\varOmega$. By Agmon, Douglis, Nirenberg [3], since $\mathsf{C}$ is strongly elliptic, if $(\mathsf{C},\mathbf{n})$ satisfies the complementing condition on all of $\partial \varOmega$, then the elliptic estimates hold for the operators in (A.4). From this, since $\varOmega$ is bounded, $S$ is a finite dimensional subspace of $W^{2,2}(\varOmega )$ (cf. Peetre [29]). If, on the other hand, the complementing condition for $(\mathsf{C}, \mathbf{n})$ fails as in (A.3) then we show that $\dim S = \infty$ is possible.

In part (iv) of the following Theorem A.3, we will denote, for an open set $\tilde{\varOmega } \subset {\mathbb{R}}^{2}$,

$$\begin{aligned} \mathcal{U}(\tilde{\varOmega }) ={}& \bigl\{ \mathbf{v} = (v_{1},v_{2}) \in C ^{\infty }\bigl(\tilde{ \varOmega },{\mathbb{R}}^{2}\bigr) : v_{1} + i v_{2} \mbox{ is an analytic function of } x_{1} + ix_{2} \mbox{ on } \tilde{\varOmega }, \\ &\ \,(x_{1},x_{2}) \in \tilde{\varOmega } \bigr\} \end{aligned}$$

(A.5)

viewing $\tilde{\varOmega }$ also as an open subset of ℂ.

Theorem A.3

Suppose$\mathsf{C}$is strongly elliptic and$\mathsf{C} \ge 0$.

• (i) If$KT - N^{2} \ne 0, P^{2} - L^{2} \ne 0$then$S = \{ \mathbf{a} \in {\mathbb{R}}^{2} \}$.

• (ii) If$KT - N^{2} = 0, P^{2} - L^{2} \ne 0$then$S=\{c\left[\begin{array}{cc}-N& 0\\ 0& K\end{array}\right]\mathbf{x}+\mathbf{a}:c\in \mathbb{R},\mathbf{a}\in {\mathbb{R}}^2\}$.

• (iii) If$KT - N^{2} \ne 0, P^{2} - L^{2} = 0$then$S=\{c\left[\begin{array}{cc}0& -L\\ P& 0\end{array}\right]\mathbf{x}+\mathbf{a}:c\in \mathbb{R},\mathbf{a}\in {\mathbb{R}}^2\}$.

• (iv) If$KT - N^{2} = 0, P^{2} - L^{2} = 0$, let$\mathbf{D} = \operatorname{diag} [ \sqrt{PK}, -(\operatorname{sgn} N) \sqrt{|N||L|} ]$,

  $\mathbf{B} = \operatorname{diag} [ \sqrt{K|L|}, \sqrt{P|N|} ]$, $\tilde{\varOmega } = \mathbf{B}^{-1} \varOmega = \{\mathbf{B}^{-1} \mathbf{x} : \mathbf{x} \in \varOmega \}$, $T\mathbf{u} ( \tilde{\mathbf{x}}) = \mathbf{D}\mathbf{u}(\mathbf{B} \tilde{\mathbf{x}})$for$\tilde{\mathbf{x}} \in \tilde{\varOmega }$. Then

  $$ S = \bigl\{ \mathbf{u} \in C^{\infty }(\varOmega ): T \mathbf{u} \in \mathcal{U}(\tilde{\varOmega }) \bigr\} . $$

  (A.6)

Proof

Suppose $\mathbf{u} \in S$. By the divergence theorem $\int _{\varOmega } \nabla \mathbf{u} \cdot \mathsf{C}[\nabla \mathbf{u}] \, dx = - \int _{\varOmega } \mathbf{u} \cdot \operatorname {div}\mathsf{C} [\nabla \mathbf{u}] \, dx + \int _{\partial \varOmega } \mathbf{u} \cdot \mathsf{C}[\nabla \mathbf{u}]\mathbf{n} \, da = 0$. Then nonnegativity of $\mathsf{C}$ implies $\nabla \mathbf{u} \cdot \mathsf{C} [\nabla \mathbf{u}] = 0$ a.e. $\varOmega$ and Proposition A.1 yields $\mathsf{C} [\nabla \mathbf{u}] = \mathbf{0}$ a.e. $\varOmega$. Conversely, if $\mathbf{u} \in W^{2,2}(\varOmega )$ satisfies the last equation then clearly $\mathbf{u} \in S$. Thus $\mathbf{u} \in S$ iff $\mathsf{C} [\nabla \mathbf{u}] = \mathbf{0}$ a.e. $\varOmega$. Next we appeal to Proposition A.2.

(i) If $KT - N^{2} \ne 0, P^{2} - L^{2} \ne 0$ then $\mathbf{u} \in S$ iff $\nabla \mathbf{u} = \mathbf{0}$ a.e. $\varOmega$ iff $\mathbf{u}= \mbox{const. on }\varOmega$.

(ii) If $KT - N^{2} = 0, P^{2} - L^{2} \ne 0$ and $\mathbf{u} \in S$ then $\mathrm{\nabla }\mathbf{u}\in span\left\{\left[\begin{array}{cc}-N& 0\\ 0& K\end{array}\right]\right\}$, so $K \frac{\partial u_{1}}{\partial x_{1}} + N \frac{ \partial u_{2}}{\partial x_{2}} = 0, \frac{\partial u_{1}}{\partial x _{2}} = \frac{\partial u_{2}}{\partial x_{1}} = 0$ a.e. $\varOmega$. Thus $u_{1}$ is a function of $x_{1}$ only and $u_{2}$ is a function of $x_{2}$ only. Then $\nabla \mathbf{u}$ is constant (note $N \ne 0$) with $\mathrm{\nabla }\mathbf{u}=c\left[\begin{array}{cc}-N& 0\\ 0& K\end{array}\right]$ for some constant $c \in {\mathbb{R}}$.

(iii) If $KT - N^{2} \ne 0, P^{2} - L^{2} = 0$ and $\mathbf{u} \in S$, by similar reasoning in (ii), $u_{1}$ is a function of $x_{2}$ only and $u_{2}$ is a function of $x_{1}$ only with $\mathrm{\nabla }\mathbf{u}=c\left[\begin{array}{cc}0& -L\\ P& 0\end{array}\right]$ for $c \in {\mathbb{R}}$.

(iv) Let $KT - N^{2} = 0, P^{2} - L^{2} = 0$; we show below $NL < 0$. If $\mathbf{u} \in S$ then $K\frac{\partial u_{1}}{\partial x_{1}} + N \frac{\partial u_{2}}{\partial x_{2}} = 0$, $P \frac{ \partial u_{1}}{\partial x_{2}} + L \frac{\partial u_{2}}{\partial x _{1}} = 0$ a.e. $\varOmega$. Rescaling variables via $\tilde{u}_{1} = a _{1} u_{1}, \tilde{u}_{2} = a_{2} u_{2}, \tilde{x}_{1} = b_{1}^{-1} x _{1}$, $\tilde{x}_{2} = b_{2}^{-1} x_{2}$ with $a_{1} = \sqrt{PK}$, $a _{2} = - \operatorname{sgn} N \sqrt{|N||L|}$, $b_{1} = \sqrt{K|L|}$, $b_{2} = \sqrt{P|N|}$ yields the equations $\frac{\partial \tilde{u}_{1}}{ \partial \tilde{x}_{1}} - \frac{\partial \tilde{u}_{2}}{\partial \tilde{x}_{2}} = 0, \frac{\partial \tilde{u}_{1}}{\partial \tilde{x} _{2}} + \frac{\partial \tilde{u}_{2}}{\partial \tilde{x}_{1}} = 0$, for a.e. $\tilde{\mathbf{x}} = (\tilde{x}_{1},\tilde{x}_{2}) \in \tilde{\varOmega }$ with $\tilde{\varOmega }$ defined above (A.6), $\tilde{x} = \mathbf{B}^{-1} \mathbf{x}$. Thus $(\tilde{u}_{1}, \tilde{u}_{2})$ satisfies the Cauchy-Riemann equations on $\tilde{\varOmega }$ and therefore $\tilde{u}_{1} + i \tilde{u}_{2}$ is an analytic function of $\tilde{x}_{1} + i \tilde{x}_{2}$ on $\tilde{\varOmega }$. Also $(\tilde{u}_{1}(\tilde{\mathbf{x}}), \tilde{u} _{2}(\tilde{\mathbf{x}})) = T \mathbf{u}(\tilde{\mathbf{x}})$ and this yields (A.6).

Lemma

$NL < 0$.

Proof of lemma

$KT = N^{2}$, $P^{2} = L^{2}$, $M = L + N$ and strong ellipticity (4.6) $P + \sqrt{KT} > |M|$ yield $|L| + |N| > |L + N|$ which implies $NL < 0$. □

Theorem A.4

Suppose$\mathsf{C}$is strongly elliptic and$KT - N^{2} = 0, P^{2} - L^{2} = 0$. Then the complementing condition for$(\mathsf{C}, \mathbf{n})$fails on all$\partial \varOmega$and the solution set$S$is given by (A.6). In particular$\dim S = \infty$.

Proof

Since $\det \mathbf{D}_{1} = \det \mathbf{D}_{2} = 0$, $K,T,P > 0$ then $\mathbf{D}_{1} \ge 0, \mathbf{D}_{2} \ge 0$. This implies $\mathsf{C} \ge 0$ and Theorem A.3 yields the rest. □

This generalizes Simpson and Spector [40], Mikhlin [25] for the strongly elliptic Lame operator (cf. (2.24)) since $KT - N^{2} = 4 \mu (\mu + \lambda )$, $P^{2} - L^{2} = 0$, $A = 4 \mu ^{2} (\mu + \lambda )$, $\mu > 0$, $2\mu + \lambda > 0$. The hypotheses of Theorem A.4 are satisfied iff $\mu + \lambda = 0$. In fact $(\mathsf{C}, \mathbf{n})$ satisfies the complementing condition iff $\mu + \lambda \ne 0$ by Theorem 4.4 (also Simpson and Spector [41, 43]). Then $\mathbf{D} = \mathbf{B} = \mu \mathbf{I}$ in Theorem A.3(iv) and, by scaling out $\mu$, we can take $\tilde{\varOmega } = \varOmega$, $T\mathbf{u} = \mathbf{u}$ so $u_{1} + i u_{2}$ is an analytic function of $x_{1} + i x_{2}$ on $\varOmega$.