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If | ε 1 | > 1 , take σ 11 ' = K ε 1 and σ 22 ' = 0 for some K 0 so that the expression now becomes K | ε 1 | - K | ε 1 | 2 < 0 , and let K so that the minimum goes to - . Similarly, the minimum is - if | ε 2 | > 1 . Otherwise, if | ε 1 | , | ε 2 | 1 , the minimum is zero. To see this, let σ i i = 0 if | ε | < 1 and σ i i ' = K ε i if | ε i | = 1 .

Substituting in this minimum, the dual problem is now

Maximize Γ ( u 1 f 1 + u 2 f 2 ) d s subject to | ε i ( u ) | 1

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 | ε i | = 1 . When ε 1 = 1 and ε 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 2 θ α β = 0 on the angle θ between the horizontal and the direction of the strain ε 1 = 1 , and the secondary property ε 1 + ε 2 = u 1 x + u 2 y = 0 . Here, the geometrical problem is identical to that of slip lines in plane strain, where the eigenvalues of stress equal ± σ 0 . In the cases when ε 1 = ε 2 = 1 or ε 1 = ε 2 = - 1 there is a similar correspondence with pure hydrostatic pressure in the stress case.

Because [link] is equal to 0, for every admissible σ and u ,

min div σ = 0 σ : n = f Ω | λ 1 | + | λ 2 | = max u Γ u · f d s

so that

Ω ( | λ 1 ( σ ) | + | λ 2 ( σ ) | d x d y Γ u · f d s

with equality holding if the optimality condition is satisfied.

If σ and ε ( u ) are simultaneously diagonal, the possibilities for σ can be restated as

if λ 1 ( σ ) = 0 , then | ε i ( u ) | 1 if λ i ( σ ) 0 , then ε i ( u ) = sgn λ i

because in a diagonal matrix, λ i ( σ ) = σ i i ' .

The dual of the problem [link] becomes

min | λ i | Λ | λ 1 ( σ ) | + | λ 2 ( σ ) | - ε , σ

and the principal axis transformation leads to

min | λ i | Λ | λ 1 ( σ ' ) | + | λ 2 ( σ ' ) | - ε 1 σ 11 ' - ε 2 σ 22 '

Because λ i cannot be arbitrarily large, - is no longer reached when | ε i | > 1 . Hence, there are now three cases, depending on the value of | ε i | :

if | ε i | > 1 , then λ i = Λ sgn ε i if | ε i | = 1 , then λ i = | λ i | sgn ε i , 0 | λ i | Λ if | ε i | < 1 , then λ i = 0

In the second and third cases, the minimum value in [link] and [link] is still zero. However, in the first case, when | ε i | > 1 , the minimum is no longer - . When σ ' is diagonal with σ i i ' = λ i = Λ sgn ε i , | λ i | - ε i σ i i ' = Λ ( 1 - | ε i | ) . Thus, the minimum in [link] and [link] can be expressed as

M Λ = Λ min ( 0 , 1 - | ε 1 | ) + Λ min ( 0 , 1 - | ε 2 | )

Here, λ i is being chosen through the minimum expression depending on the fixed ε i . Hence the dual problem can now be stated as

Maximize Ω M Λ d x d y + Γ ( u 1 f 1 + u 2 f 2 ) d s

The cases of interest arise when one family of bars is in tension and the other compression, meaning λ 1 λ 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 ε 1 > 1 , ε 2 < - 1 , λ 1 = Λ , λ 2 = - Λ . In this case, ψ represents the difference 1 2 ( λ 1 - λ 2 ) , which has the constant value Λ , which is the yield surface in plane strain. The Michell situation occurs when ε 1 = 1 , ε 2 = - 1 , λ 1 = 0 , λ 2 = 0 .

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

Maximize Ω M k d x d y + Λ ( u 1 f 1 + u 2 f 2 ) d s subject to div u = 0

The kinematic constraint div u = 0 arises because the trace of σ is unconstrained in [link] . The optimality conditions relate the principal strains ε i to the eigenvalues λ i D = λ i ( σ D ) : ε and σ D are simultaneously diagonal, and

if | ε | > 1 , then λ i D = k sgn ε i , if | ε | = 1 , then λ i D = | λ i D | sgn ε i , 0 | λ i D | k if | ε | < 1 , then λ i D = 0 .

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.

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Source:  OpenStax, Michell trusses study, rice u. nsf vigre group, summer 2013. OpenStax CNX. Sep 02, 2013 Download for free at http://cnx.org/content/col11567/1.2
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