0.4 A look at a paper by strang and kohn  (Page 3/3)

If $|{\epsilon }_1|>1$ , take ${\sigma }_{11}^{\text{'}}=K{\epsilon }_1$ and ${\sigma }_{22}^{\text{'}}=0$ for some $K\ge 0$ so that the expression now becomes $K|{\epsilon }_1|-K|{\epsilon }_1{|}^2<0$ , and let $K\to \infty$ so that the minimum goes to $-\infty$ . Similarly, the minimum is $-\infty$ if $|{\epsilon }_2|>1$ . Otherwise, if $|{\epsilon }_1|,|{\epsilon }_2|\le 1$ , the minimum is zero. To see this, let ${\sigma }_{ii}=0$ if $\left|\epsilon \right|<1$ and ${\sigma }_{ii}^{\text{'}}=K{\epsilon }_i$ if $|{\epsilon }_i|=1$ .

Substituting in this minimum, the dual problem is now

It's expected that this maximum value is equal to the minimum of [link] , corresponding to the minimum volume of the truss.

Consider the case when $|{\epsilon }_i|=1$ . When ${\epsilon }_1=1$ and ${\epsilon }_2=-1$ the principal strains have equal magnitude and opposite sign, which can happen only for a special class of displacement fields. There is a condition $\frac{{\partial }^2\theta }{\partial \alpha \partial \beta }=0$ on the angle $\theta$ between the horizontal and the direction of the strain ${\epsilon }_1=1$ , and the secondary property ${\epsilon }_1+{\epsilon }_2=\frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}=0$ . Here, the geometrical problem is identical to that of slip lines in plane strain, where the eigenvalues of stress equal $\pm {\sigma }_0$ . In the cases when ${\epsilon }_1={\epsilon }_2=1$ or ${\epsilon }_1={\epsilon }_2=-1$ there is a similar correspondence with pure hydrostatic pressure in the stress case.

Because [link] is equal to 0, for every admissible $\sigma$ and $u$ ,

so that

with equality holding if the optimality condition is satisfied.

If $\sigma$ and $\epsilon \left(u\right)$ are simultaneously diagonal, the possibilities for $\sigma$ can be restated as

because in a diagonal matrix, ${\lambda }_i\left(\sigma \right)={\sigma }_{ii}^{\text{'}}$ .

The dual of the problem [link] becomes

and the principal axis transformation leads to

Because ${\lambda }_i$ cannot be arbitrarily large, $-\infty$ is no longer reached when $|{\epsilon }_i|>1$ . Hence, there are now three cases, depending on the value of $|{\epsilon }_i|$ :

In the second and third cases, the minimum value in [link] and [link] is still zero. However, in the first case, when $|{\epsilon }_i|>1$ , the minimum is no longer $-\infty$ . When ${\sigma }^{\text{'}}$ is diagonal with ${\sigma }_{ii}^{\text{'}}={\lambda }_i=\Lambda \text{sgn}{\epsilon }_i$ , $|{\lambda }_i|-{\epsilon }_i{\sigma }_{ii}^{\text{'}}=\Lambda (1-|{\epsilon }_i\left|\right)$ . Thus, the minimum in [link] and [link] can be expressed as

Here, ${\lambda }_i$ is being chosen through the minimum expression depending on the fixed ${\epsilon }_i$ . Hence the dual problem can now be stated as

The cases of interest arise when one family of bars is in tension and the other compression, meaning ${\lambda }_1{\lambda }_2<0$ . Then there is a combination of Hencky-Prandtl nets, one coming from slip lines, and the other from Michell trusses.

The slip line net will occur when ${\epsilon }_1>1,{\epsilon }_2<-1,{\lambda }_1=\Lambda ,{\lambda }_2=-\Lambda$ . In this case, $\parallel \psi \parallel$ represents the difference $\frac{1}{2}({\lambda }_1-{\lambda }_2)$ , which has the constant value $\Lambda$ , which is the yield surface in plane strain. The Michell situation occurs when ${\epsilon }_1=1,{\epsilon }_2=-1,{\lambda }_1=0,{\lambda }_2=0$ .

The other constrained problem [link] differs from the dual only in the condition of incompressibility:

The kinematic constraint $\text{div}u=0$ arises because the trace of $\sigma$ is unconstrained in [link] . The optimality conditions relate the principal strains ${\epsilon }_i$ to the eigenvalues ${\lambda }_i^D={\lambda }_i\left({\sigma }^D\right):\epsilon$ and ${\sigma }^D$ are simultaneously diagonal, and

Open problems

• Find the most economical truss given a balanced point force field.
• Does the stress field in an optimal structure vary smoothly, so that the lines of principal stress are $C^1$ curves? The problem of regularity runs deeper than that of existence.