An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications
Stanislav Sysala, Jaroslav Haslinger, Daya Reddy, Sergey Repin
AAn abstract inf-sup problem inspired by limit analysisin perfect plasticity and related applications
S. Sysala ∗ , J. Haslinger , B. D. Reddy , S. Repin , Institute of Geonics of the Czech Academy of Sciences, Ostrava, Czech Republic University of Cape Town, South Africa V.A. Steklov Institute of Mathematics at St. Petersburg, Russia University of Jyv¨askyl¨a, Finland
September 9, 2020
Abstract
This work is concerned with an abstract inf-sup problem generated by a bilinearLagrangian and convex constraints. We study the conditions that guarantee no gapbetween the inf-sup and related sup-inf problems. The key assumption introducedin the paper generalizes the well-known Babuˇska-Brezzi condition. It is based onan inf-sup condition defined for convex cones in function spaces. We also applya regularization method convenient for solving the inf-sup problem and derive acomputable majorant of the critical (inf-sup) value, which can be used in a posteriorierror analysis of numerical results. Results obtained for the abstract problem areapplied to continuum mechanics. In particular, examples of limit load problems andsimilar ones arising in classical plasticity, gradient plasticity and delamination areintroduced.
Keywords: convex optimization, duality, inf-sup conditions on cones, regularization,computable majorants, plasticity, delamination, limit analysis ∗ corresponding author, email: [email protected] a r X i v : . [ m a t h . O C ] S e p Introduction
This paper is concerned with analysis of the abstract duality problem λ ∗ := sup x ∈ P inf y ∈ YL ( y )=1 a ( x, y ) ? = inf y ∈ YL ( y )=1 sup x ∈ P a ( x, y ) =: ζ ∗ , (1.1)where P ⊂ X is a closed, convex set with 0 X ∈ P , X, Y are Banach spaces, L is a non-trivial continuous linear functional in Y , and a : X × Y → R is a bilinear form continuouswith respect to both arguments. Henceforth the problem in the right hand side of (1.1)is called primal, while the one in the left hand side is called dual. It is easy to checkthat 0 ≤ λ ∗ ≤ ζ ∗ ≤ + ∞ . In general, necessary and sufficient conditions for λ ∗ = ζ ∗ areunknown (therefore, (1.1) uses the symbol ? =). One of our main goals is to identify caseswhere (1.1) holds as the equality.Problem (1.1) and similar problems appear in various applications, from mechanics toeconomics [12, 9, 3]. In finite dimensions, minimax and maximin variants of these problemsare known in game theory [19] and linear, cone or convex programming [11, 5, 21, 18].In classical elastic-perfect plasticity, (1.1) is known as the limit analysis problem . Inthis case, λ ∗ is the factor that determines the critical load λ ∗ L ( L is a linear functionalassociated with external loads), subject to the constraint set P of plastically admissiblestresses; see for example [17, 9, 32, 8, 10, 29, 30, 28, 16]. For the load λL with λ > ζ ∗ , nosolution of the primal and dual problems exists; the body is unable to sustain the loadingand collapses. Also, we note the similarity between (1.1) and the shakedown analysisproblem (see [33] and the references therein).Although the limit analysis problem has been studied for several decades, it is still un-solved in the general setting and presents a challenging problem from the theoretical andnumerical points of view. There are several reasons that stimulate further analysis of theproblem. First, we notice that the equality λ ∗ = ζ ∗ can be analyzed in a rather gen-eral framework introduced in [12] or by using particular results from [9, 32, 16]. However,these results do not cover any interesting cases. Second, additional and hidden constraintsappear in the primal and dual problems (that follow from their inf- and sup-definitions).They often make the numerical analysis difficult. The third reason is related to the choiceof the function spaces X and Y . This question becomes especially important if the primalproblem is related to minimization of a functional with linear growth at infinity and acertain problem relaxation must be done to find a minimizer (see e.g. [17, 32, 29, 10]).Then we arrive, for example, at a formulation in which the BD - or BV - spaces of func-2ions of bounded deformation and bounded variation, respectively, are appropriate for theproblem setting [32]. Nevertheless, standard Sobolev spaces seem to be sufficient or evenmore appropriate for analysis of numerical errors [27, 28, 16]. Finally, reliable estimatesof λ ∗ and ζ ∗ are often required because they define safety factors of structures. Lowerbounds of λ ∗ and upper bounds of ζ ∗ can be found by analytical approaches for specificgeometries [8] or, more generally, by finite element methods; see [30] and the referencestherein. Computable majorants of ζ ∗ can be found in recent papers [28, 16].In order to investigate the abstract problem (1.1), we use the ideas applied in [14, 15,28, 16] for analysis of limit load problems. This extension is not always straightforwardand requires innovative techniques. In particular, we derive conditions for the equality λ ∗ = ζ ∗ to hold, the existence of a solution to the dual problem in (1.1), a regularizationmethod for solving (1.1) with related convergence results, and a computable majorant of ζ ∗ , which can be used for a posteriori analysis of numerical results.One of the key assumptions in the results presented is the so-called inf-sup condition onconvex cones which was introduced in [16]. This condition generalizes the Babuˇska-Brezzicondition defined on function spaces [1, 4]. Conditions of this type are important for anal-ysis of saddle point problems generated by various mixed finite element approximations[3].Generalization and abstraction of results is a basic procedure that allows results andinsights in a particular application to be applied to broad classes of problems. In ourcase, we show that the results presented here are useful in problems of gradient-enhancedplasticity and in delamination problems. We choose the strain gradient model studied in[25, 24, 6, 26] and use (1.1) for the description of a global yield surface and for limit loadanalysis. In related work, limit analysis has been considered for a model in which size-dependence is through the gradient of a scalar function of plastic strain, see [13, Section 7]or [23]. One can expect further applications of the problem (1.1), at least within nonlinearmechanics.The rest of the paper is organized as follows. In Section 2, we introduce the primal anddual problems, discuss them in more detail, and present criteria ensuring their solvabilityand the principal duality relation λ ∗ = ζ ∗ . One of the criteria is based on the inf-supcondition on convex cones. The proof of this new result is carried out in Section 3 andits extensions are studied in Section 4. Section 5 is devoted to a regularization of theproblem (1.1). The regularized problem provides a lower and sufficiently sharp bound of λ ∗ , reduces the constraints in the dual problem, and thus it is convenient for numericalsolution. In Section 6, a computable majorant of the quantity ζ ∗ is derived. Section 73ontains particular examples of the abstract problem (1.1), including classical and strain-gradient plasticity and a delamination problem. First, we recapitulate the basic assumptions used in the problem (1.1):(A1)
X, Y are two Banach spaces equipped with the norms (cid:107) . (cid:107) X and (cid:107) . (cid:107) Y , respectively.The corresponding dual spaces are denoted by X ∗ and Y ∗ ;(A2) a : X × Y → R is a continuous bilinear form;(A3) L : Y → R is a non-trivial continuous linear functional (i.e., L (cid:54) = 0 in Y ∗ );(A4) P ⊂ X is a nonempty, closed and convex set with 0 X ∈ P .The primal problem in (1.1) reads ζ ∗ = inf y ∈ YL ( y )=1 sup x ∈ P a ( x, y ) = inf y ∈ YL ( y )=1 J ( y ) , (2.1)where J : Y → R ∪ { + ∞} , J ( y ) := sup x ∈ P a ( x, y ) , y ∈ Y. (2.2)The functional J is convex, proper and 1-positively homogeneous. In addition, the effec-tive domain dom J is a convex cone; see Section 6 for more details. We shall assume thatall cones considered in the text have a vertex at zero, so henceforth do not emphasize thisproperty. We say that the problem (2.1) has a solution if the functional J has a minimizerin the feasible set dom J ∩ { y ∈ Y | L ( y ) = 1 } . Using the positive homogeneity of J , weobtain the following useful and equivalent definition of ζ ∗ : ζ ∗ = sup { λ ∈ R + | J ( y ) − λL ( y ) ≥ ∀ y ∈ Y } . (2.3)To rewrite the dual problem in (1.1) we define the functional I ( x ) := inf y ∈ YL ( y )=1 a ( x, y ) = (cid:40) λ, ∃ λ ∈ R : a ( x, y ) = λL ( y ) ∀ y ∈ Y, −∞ , otherwise , x ∈ X, (2.4)4nd the related set Λ λ := { x ∈ X | a ( x, y ) = λL ( y ) ∀ y ∈ Y } . (2.5)Then, we have λ ∗ = sup x ∈ P inf y ∈ YL ( y )=1 a ( x, y ) = sup x ∈ P I ( x ) = sup { λ ∈ R + | P ∩ Λ λ (cid:54) = ∅} . (2.6)We shall say that the problem (2.6) has a solution if λ ∗ < + ∞ and there exists ¯ x ∈ P ∩ Λ λ ∗ .Now, we present three different results ensuring the equality λ ∗ = ζ ∗ and the existence ofprimal or dual solutions. The first result follows from [12, Proposition VI.2.3 and RemarkVI.2.3]. Theorem 2.1.
Let (A1)–(A4) be satisfied and assume in addition that(B) P is a bounded set in X .Then λ ∗ = ζ ∗ and the dual problem (2.6) has a solution. Unfortunately, the set P can be unbounded in plasticity and other applications. Therefore,we also need other criteria. The second result has been introduced in [9, Theorem 2.1]and also used in [10, Theorem 5.7]. It is convenient for use with non-reflexive spaces suchas L ∞ . Theorem 2.2.
Let (A1)–(A4) be satisfied together with the following: ( C P has a non-empty interior in X ; ( C There exists x ∈ X such that a ( x , y ) = L ( y ) for any y ∈ Y ; ( C For any M ∈ X ∗ such that inf x ∈ P M ( x ) > −∞ ,a ( x, y ) = 0 ∀ y ∈ Y = ⇒ M ( x ) = 0 , there exists y ∈ Y satisfying a ( x, y ) = M ( x ) for any x ∈ X .Then λ ∗ = ζ ∗ and the primal problem has a solution. Theorem 2.3.
Let (A1)–(A4) be satisfied and, in addition, assume the following: ( D X is a reflexive Banach space; ( D Y is a Hilbert space with a scalar product ( ., . ) Y and the induced norm (cid:107) . (cid:107) Y ; ( D For any x ∈ P there exist x A ∈ P A and x C ∈ P C such that x = x A + x C , where P A ⊂ X is closed, convex and bounded and P C ⊂ P is a closed convex cone; ( D
4) inf x C ∈ P C x C (cid:54) =0 X sup y ∈ Yy (cid:54) =0 Y a ( x C , y ) (cid:107) x C (cid:107) X (cid:107) y (cid:107) Y = c ∗ > . (2.7) Then λ ∗ = ζ ∗ . Moreover, if λ ∗ < + ∞ then the dual problem (2.6) has a solution. It is worth noting that for the validity of the theorem it suffices to assume that the set P C is only closed and convex in X and satisfies ( D a ( x C , y ) (cid:107) x C (cid:107) X (cid:107) y (cid:107) Y = a ( αx C , y ) (cid:107) αx C (cid:107) X (cid:107) y (cid:107) Y ∀ α > . This fact (independence of the scaling parameter) explains why we assume that P C is aconvex cone. In addition, we shall see in Section 6 that the cones P C and dom J areclosely related.We also note that any closed linear subspace of X is a special case of the cone P C .Then, we arrive at the standard inf-sup condition on function spaces. This case will beconsidered in Theorem 4.2 and in Section 7. Within this section we assume that the conditions (A1)–(A4), (D1)–(D4) are satisfiedand also λ ∗ < + ∞ (notice that Theorem 2.3 holds trivially for λ ∗ = + ∞ ). To prove thistheorem we define auxiliary functions ϕ : R → R + and Φ λ : X → R + : ϕ ( λ ) := inf x ∈ P Φ λ ( x ) , Φ λ ( x ) := sup y ∈ Yy (cid:54) =0 Y a ( x, y ) − λL ( y ) (cid:107) y (cid:107) Y . (3.1)6heir basic properties are introduced in the following lemma. Lemma 3.1.
The function Φ λ is nonnegative, convex and Lipschitz continuous in X forany λ ∈ R + . The function ϕ is nonnegative, nondecreasing, and Lipschitz continuous in R + .Proof. It is straightforward to verify that Φ λ and ϕ are nonnegative and convex. Let x , x ∈ X . Then, using continuity of the bilinear form a , we haveΦ λ ( x ) = sup y ∈ Yy (cid:54) =0 Y a ( x , y ) − λL ( y ) + a ( x − x , y ) (cid:107) y (cid:107) Y ≤ Φ λ ( x ) + (cid:107) a (cid:107)(cid:107) x − x (cid:107) X , where (cid:107) a (cid:107) is the norm of a . Similarly, Φ λ ( x ) ≤ Φ λ ( x ) + (cid:107) a (cid:107)(cid:107) x − x (cid:107) X and so | Φ λ ( x ) − Φ λ ( x ) | ≤ (cid:107) a (cid:107)(cid:107) x − x (cid:107) X proving the Lipschitz continuity of Φ λ in X .Since P is convex and 0 X ∈ P , we have x/α ∈ P for any x ∈ P and α ≥
1. Hence, ϕ ( αλ ) = α inf x ∈ P Φ λ ( x/α ) ≥ α inf x ∈ P Φ λ ( x ) = αϕ ( λ ) ≥ ϕ ( λ ) ∀ α ≥ , i.e, ϕ is nondecreasing in R + . Let λ, ¯ λ ∈ R + , λ < ¯ λ . Then ϕ ( λ ) ≤ ϕ (¯ λ ) and ϕ (¯ λ ) = inf x ∈ P sup y ∈ Yy (cid:54) =0 a ( x, y ) − λL ( y ) − (¯ λ − λ ) L ( y ) (cid:107) y (cid:107) Y ≤ ϕ ( λ ) + (¯ λ − λ ) (cid:107) L (cid:107) Y ∗ . Thus ϕ is Lipschitz continuous in R + with modulus (cid:107) L (cid:107) Y ∗ .The next lemma shows that the function ϕ is closely related to the problems (2.6) and(2.1). Lemma 3.2.
The function ϕ defined in (3.1) satisfies the following relations: ϕ ( λ ) = 0 if λ ≤ ζ ∗ , ϕ ( λ ) > if λ > ζ ∗ , (3.2) and λ ∗ = ζ ∗ if and only if ϕ ( λ ) > ∀ λ > λ ∗ . (3.3) Proof.
To prove (3.2) we use the Lagrangian L ( x, y ) := 12 (cid:107) y (cid:107) Y + a ( x, y ) − λL ( y ) , x ∈ P, y ∈ Y. (3.4)7he mapping y (cid:55)→ L ( x, y ) is coercive, convex, and continuous in Y for any x ∈ P while x (cid:55)→ L ( x, y ) is linear for any y ∈ Y and the set P is closed and convex in X . Therefore,by [12, Proposition VI 2.3], we know thatmin y ∈ Y sup x ∈ P L ( x, y ) = sup x ∈ P inf y ∈ Y L ( x, y ) . (3.5)For any given x ∈ P , there exists a unique element y x ∈ Y such that L ( x, y x ) ≤ L ( x, y ) ∀ y ∈ Y, or equivalently ( y x , y ) Y = λL ( y ) − a ( x, y ) ∀ y ∈ Y. (3.6)Consequently, (cid:107) y x (cid:107) Y = sup y ∈ Yy (cid:54) =0 Y ( y x , y ) Y (cid:107) y (cid:107) Y = sup y ∈ Yy (cid:54) =0 Y − ( y x , y ) Y (cid:107) y (cid:107) Y = sup y ∈ Yy (cid:54) =0 Y a ( x, y ) − λL ( y ) (cid:107) y (cid:107) Y = Φ λ ( x ) (3.7)andsup x ∈ P inf y ∈ Y L ( x, y ) = sup x ∈ P (cid:110) − (cid:107) y x (cid:107) Y (cid:111) (3 . = −
12 inf x ∈ P Φ λ ( x ) = − (cid:18) inf x ∈ P Φ λ ( x ) (cid:19) = − ϕ ( λ ) . (3.8)From (3.5) and (3.8), we have: − ϕ ( λ ) = min y ∈ Y sup x ∈ P L ( x, y ) = min y ∈ Y (cid:26) (cid:107) y (cid:107) Y + J ( y ) − λL ( y ) (cid:27) ∀ λ ∈ R + , where J is the primal functional defined by (2.2). Thus, ϕ ( λ ) = (cid:18) − y ∈ Y (cid:26) (cid:107) y (cid:107) Y + J ( y ) − λL ( y ) (cid:27)(cid:19) / . (3.9)From (2.3), one can see that J ( y ) − λL ( y ) ≥ λ < ζ ∗ and y ∈ Y . Hence, ϕ ( λ ) = 0for any λ < ζ ∗ and ϕ ( ζ ∗ ) = 0 using the continuity argument. On the other hand, if λ > ζ ∗ then there exists ¯ y ∈ Y such that J (¯ y ) − λL (¯ y ) <
0. Hence,12 (cid:107) α ¯ y (cid:107) Y + J ( α ¯ y ) − λL ( α ¯ y ) = α (cid:110) α (cid:107) ¯ y (cid:107) Y + J (¯ y ) − λL (¯ y ) (cid:111) < α > ϕ ( λ ) > λ > ζ ∗ .8herefore, (3.2) holds. It is easy to see that (3.3) follows from (3.2) and the inequality λ ∗ ≤ ζ ∗ .Next, consider the following problem: given λ ≥ find x λ ∈ P : Φ λ ( x λ ) ≤ Φ λ ( x ) ∀ x ∈ P. (3.10) Lemma 3.3.
Let (3.10) have a solution for any λ ≥ . Then λ ∗ = ζ ∗ . In addition, P ∩ Λ λ ∗ (cid:54) = ∅ ; that is, the dual problem (2.6) has a solution.Proof. Let λ > λ ∗ be fixed but arbitrary and x λ ∈ P be the solution to (3.10). From(2.6) and the choice of λ , it follows that x λ (cid:54)∈ Λ λ . Using the definition (2.5) of Λ λ , we seethat there exists ¯ y ∈ Y such that a ( x λ , ¯ y ) − λL (¯ y ) >
0. Hence, ϕ ( λ ) = Φ λ ( x λ ) >
0. ByLemma 3.2, we have λ ∗ = ζ ∗ . If λ ≤ λ ∗ then ϕ ( λ ) = Φ λ ( x λ ) = 0 and so x λ ∈ P ∩ Λ λ ,proving the existence of a solution to (2.6). Proof of Theorem 2.3.
The proof is based on Lemma 3.3. We show that the problem(3.10) has a solution for any λ ≥ λ is convex and Lipschitz continuous in X for any λ ∈ R + .Using the assumptions (D3) and (D4) we prove that Φ λ is also coercive in P for any λ ≥ x ∈ P and λ ∈ R + , we have:Φ λ ( x ) = sup y ∈ Yy (cid:54) =0 Y a ( x C , y ) + a ( x A , y ) − λL ( y ) (cid:107) y (cid:107) Y ≥ c ∗ (cid:107) x C (cid:107) X − (cid:107) a (cid:107)(cid:107) x A (cid:107) X − λ (cid:107) L (cid:107) Y ∗ ≥ c ∗ (cid:107) x (cid:107) X − ( c ∗ + (cid:107) a (cid:107) ) (cid:107) x A (cid:107) X − λ (cid:107) L (cid:107) Y ∗ ≥ c ∗ (cid:107) x (cid:107) X − ( c ∗ + (cid:107) a (cid:107) ) ρ A − λ (cid:107) L (cid:107) Y ∗ ∀ x ∈ P, (3.11)where c ∗ > ρ A is a positive constant characterizingthe boundedness of A . Since X is a reflexive Banach space, the properties of Φ λ guaranteethat (3.10) has a solution for any λ ≥ Now we present three different generalizations (or extensions) of Theorem 2.3. Since theirproofs are quite analogous, we only sketch them.9irst, the assumption (D4) cannot hold if the subspace H := { x ∈ X | a ( x , y ) = 0 ∀ y ∈ Y } (4.1)contains an element x ∈ P C such that x (cid:54) = 0. In Section 7, we will show that thiscase may arise in some applications. To weaken the assumption (D4) we introduce thequotient space X/H (with the norm (cid:107) · (cid:107)
X/H ) whose elements are the equivalence classesinduced by the equivalence relation: x ∼ = x if and only if x − x ∈ H, x , x ∈ X. Let
P/H denote the set of equivalent classes generated by the set P . It is easy to verifythat P/H is a closed, convex, and nonempty set in
X/H . Similarly, one can introduce thesets P A /H and P C /H , where P A and P C are defined in accordance with the assumption(D3). These sets have properties analogous to properties of P A and P C . In particular, forany x ∈ P/H there exists x A ∈ P A /H and x C ∈ P C /H such that x = x A + x C . We notethat if X is a Hilbert space then X/H can be identified with the orthogonal complement H ⊥ of H in X and P/H with the projection of P onto H ⊥ . Theorem 4.1.
Let the assumptions (A1)–(A4) and (D1)-(D3) of Theorem 2.3 be satisfied, H be defined by (4.1) , and inf x C ∈ P C /Hx C (cid:54) =0 X sup y ∈ Yy (cid:54) =0 Y a ( x C , y ) (cid:107) x C (cid:107) X/H (cid:107) y (cid:107) Y = c ∗ > . (4.2) Then λ ∗ = ζ ∗ . If, in addition λ ∗ < + ∞ then the dual problem (2.6) has a solution.Sketch of the proof: It suffices to show that (3.10) has a solution in P for any λ ≥ λ ≥ λ defined in (3.1) satisfiesΦ λ ( x + x ) = Φ λ ( x ) ∀ x ∈ X, ∀ x ∈ H. (4.3)Therefore, (3.10) has a solution in P if and only if Φ λ has a minimum in P/H . From(4.2), one can prove the coercivity of Φ λ in P/H analogously as in the proof of Theorem2.3. Therefore, Φ λ has a minimum in P/H and thus the result of Theorem 4.1 holds.Second, it turns out that the assumption (D2) of Theorem 2.3 can be extended to somereflexive Banach spaces associated with a bounded Lipschitz domain Ω ⊂ R d , d = 2 , heorem 4.2. Let the assumptions (A1)–(A4) and (D1), (D3)-(D4) be satisfied and(D2’) Y = W ,p (Ω , R m ) , equipped with the standard Sobolev norm (cid:107) y (cid:107) Y = (cid:18)(cid:90) Ω |∇ y | p + | y | p dx (cid:19) /p . Then λ ∗ = ζ ∗ . In addition, if λ ∗ < + ∞ then the dual problem (2.6) has a solution.Sketch of the proof: It suffices to modify formulae (3.4)–(3.9). To this end, we set L ( x, y ) := 1 p (cid:107) y (cid:107) pY + a ( x, y ) − λL ( y ) , x ∈ P, y ∈ Y. (4.4)Then (3.5) holds and there exists a unique element y x ∈ Y such that L ( x, y x ) ≤ L ( x, y ) ∀ y ∈ Y, or equivalently (cid:90) Ω |∇ y x | p − ∇ y x : ∇ y + | y x | p − y x · y dx = λL ( y ) − a ( x, y ) ∀ y ∈ Y. (4.5)Consequently, (cid:107) y x (cid:107) p − Y = sup y ∈ Yy (cid:54) =0 Y (cid:82) Ω |∇ y x | p − ∇ y x : ∇ y + | y x | p − y x · y dx (cid:107) y (cid:107) Y = sup y ∈ Yy (cid:54) =0 Y a ( x, y ) − λL ( y ) (cid:107) y (cid:107) Y = Φ λ ( x )(4.6)andsup x ∈ P inf y ∈ Y L ( x, y ) = sup x ∈ P (cid:110) − q (cid:107) y x (cid:107) pY (cid:111) (4 . = − q inf x ∈ P Φ qλ ( x ) = − q (cid:18) inf x ∈ P Φ λ ( x ) (cid:19) q = − q ϕ q ( λ ) , (4.7)where 1 /q = 1 − /p . The rest of the proof is analogous to that of Section 3.The third extension illustrates that Theorem 2.3 remains valid even if the space Y isreplaced by its conic subset. Theorem 4.3.
Let the assumptions (A1)–(A4) and (D1)–(D3) of Theorem 2.3 be satis- ed, Y C ⊂ Y be a closed convex cone, and inf x C ∈ P C x C (cid:54) =0 X sup y ∈ Y C y (cid:54) =0 Y − a ( x C , y ) (cid:107) x C (cid:107) X (cid:107) y (cid:107) Y = c ∗ > . (4.8) Then λ ∗ := sup x ∈ P inf y ∈ Y C L ( y )=1 a ( x, y ) = inf y ∈ Y C L ( y )=1 sup x ∈ P a ( x, y ) =: ζ ∗ . (4.9) In addition, λ ∗ = max { λ ∈ R + | P ∩ Λ λ (cid:54) = ∅} , where Λ λ = { x ∈ X | a ( x, y ) ≥ λL ( y ) ∀ y ∈ Y C } . Sketch of the proof:
The following two changes in the proof of Theorem 2.3 are necessary:1. We define Φ λ ( x ) := sup y ∈ Y C y (cid:54) =0 Y − a ( x, y ) + λL ( y ) (cid:107) y (cid:107) Y and consider the minimization problemfind y x ∈ Y C : L ( x, y x ) ≤ L ( x, y ) ∀ y ∈ Y C with L defined by (3.4). Since Y C is a closed convex cone, the corresponding neces-sary and sufficient condition characterising y x reads( y x , y ) Y ≥ λL ( y ) − a ( x, y ) ∀ y ∈ Y C , ( y x , y x ) Y = λL ( y x ) − a ( x, y x ) . We obtain (cid:107) y x (cid:107) Y = sup y ∈ Y C y (cid:54) =0 Y ( y x , y ) Y (cid:107) y (cid:107) Y ≥ sup y ∈ Y C y (cid:54) =0 Y − a ( x, y ) + λL ( y ) (cid:107) y (cid:107) Y ≥ − a ( x, y x ) + λL ( y x ) (cid:107) y x (cid:107) Y = (cid:107) y x (cid:107) Y , so that (cid:107) y x (cid:107) Y = Φ λ ( x ).2. To prove Lemma 3.3, we modify (2.6) as follows: λ ∗ = sup { λ ∈ R + | P ∩ Λ λ (cid:54) = ∅} , Λ λ := { x ∈ X | a ( x, y ) ≥ λL ( y ) ∀ y ∈ Y C } . Then the proof of Theorem 2.3 is applicable without any substantial changes.12
Regularization method
Regularization methods are often used for solving nonsmooth, constrained, or ill-posedproblems. As an example, we mention proximal point methods [22] which can be used forsolving the problems (2.6) and (2.1).Here we consider another regularization method which has been subsequently developedin [31, 7, 14, 15] and used also in [28, 16]. In these recent papers, this method has beencalled either the “indirect incremental method” or the “penalization method”. Below, wegeneralize, results of [14, 15] and show that some of these can be established in a simplerway. Within this section it is assumed that the conditions (A1)–(A4) from Section 2 hold.To regularize the functional J defined by (2.2) we introduce the functional J α : Y → R , J α ( y ) := max x ∈ P (cid:26) a ( x, y ) − α (cid:107) x (cid:107) X (cid:27) , (5.1)where α > J α is convex and finite-valued in Y (unlike the functional J ) and J α ≤ J α ≤ J for any α , α > α ≤ α . Lemma 5.1.
Let J and J α be defined by (2.2) and (5.1) . Then lim α → + ∞ J α ( y ) = J ( y ) ∀ y ∈ Y. (5.2) Proof.
Let y ∈ Y be fixed. As mentioned above, the sequence {J α ( y ) } α is nondecreasing.Therefore, it has a limit which is less than or equal to J ( y ). On the other hand,lim α → + ∞ J α ( y ) ≥ lim α → + ∞ (cid:26) a ( x, y ) − α (cid:107) x (cid:107) X (cid:27) = a ( x, y ) ∀ x ∈ P. Thus (5.2) holds.The regularization of the primal problem (2.1) with respect to the parameter α definesthe function ψ : R + → R + : ψ ( α ) := inf y ∈ YL ( y )=1 J α ( y ) , α > . (5.3)13n view of (5.1) and [12, Proposition VI 2.3], it holds ψ ( α ) = inf y ∈ YL ( y )=1 max x ∈ P (cid:26) a ( x, y ) − α (cid:107) x (cid:107) X (cid:27) = max x ∈ P inf y ∈ YL ( y )=1 (cid:26) a ( x, y ) − α (cid:107) x (cid:107) X (cid:27) . (5.4)Thus, the main duality relation holds without any gap, unlike the original primal-dualproblem (1.1). The properties of the function ψ are set out in the following theorem. Theorem 5.1.
The function ψ is continuous, nondecreasing and lim α → + ∞ ψ ( α ) = λ ∗ ≤ ζ ∗ , (5.5) where λ ∗ and ζ ∗ are defined by (2.6) and (2.1) , respectively.Proof. From the properties of {J α } α> , it is easy to see that ψ is nondecreasing and thusit has a limit as α → + ∞ . Comparing (2.6) and (5.4) we see that ψ ( α ) ≤ λ ∗ . In addition,for any x ∈ P we havelim α → + ∞ ψ ( α ) (5.4) ≥ lim α → + ∞ inf y ∈ YL ( y )=1 (cid:26) a ( x, y ) − α (cid:107) x (cid:107) X (cid:27) = inf y ∈ YL ( y )=1 a ( x, y ) . Making use of the definition of λ ∗ , we arrive at (5.5).Let β > α . Since 0 X ∈ P , we have ( α/β ) x ∈ P if x ∈ P . Hence, ψ ( α ) (5.4) ≥ inf y ∈ YL ( y )=1 max x ∈ P (cid:26) a (( α/β ) x, y ) − α (cid:107) ( α/β ) x (cid:107) X (cid:27) = αβ ψ ( β ) . This relation and the monotonicity of ψ imply αβ ψ ( β ) ≤ ψ ( α ) ≤ ψ ( β ) . Hence, lim sup β (cid:38) α ψ ( β ) = lim sup β (cid:38) α αβ ψ ( β ) ≤ ψ ( α ) ≤ lim inf β (cid:38) α ψ ( β ) . (5.6)Let β < α . By interchanging α and β in (5), we obtain βα ψ ( α ) ≤ ψ ( β ) ≤ ψ ( α ) or ψ ( β ) ≤ ψ ( α ) ≤ αβ ψ ( β ) . β (cid:37) α ψ ( β ) ≤ ψ ( α ) ≤ lim sup β (cid:37) α αβ ψ ( β ) = lim inf β (cid:37) α ψ ( β ) . (5.7)From (5.6) and (5.7), we havelim sup β → α ψ ( β ) ≤ ψ ( α ) ≤ lim inf β → α ψ ( β ) or lim β → α ψ ( β ) = ψ ( α ) , implying the continuity of ψ .It is worth noting that for any value of α >
0, the quantity ψ ( α ) is a lower bound of λ ∗ and ζ ∗ . Upper bounds of λ ∗ and ζ ∗ will be derived in the next section.From the numerical point of view, it is useful if the functional J α is differentiable in theGˆateaux sense. Below we establish this property of the regularized functional. Lemma 5.2.
Let X be a Hilbert space with the scalar product ( ., . ) X and define Π α : Y → P, Π α y := arg max x ∈ P (cid:26) a ( x, y ) − α (cid:107) x (cid:107) X (cid:27) . (5.8) Then Π α is Lipschitz continuous in Y and J (cid:48) α ( y ; z ) := lim t → t [ J α ( y + tz ) − J α ( y )] = a (Π α y, z ) ∀ α > , ∀ y, z ∈ Y. (5.9) Proof.
Since X is a Hilbert space, it is easy to see that there exists a unique Π α y solving(5.8) and satisfying the variational inequality1 α (Π α y, x − Π α y ) X ≥ a ( x − Π α y, y ) ∀ x ∈ P, ∀ y ∈ Y. (5.10)Hence, we derive the inequalities1 α (Π α y, Π α ( y + tz ) − Π α y ) X ≥ a (Π α ( y + tz ) − Π α y, y ) , α (Π α ( y + tz ) , Π α y − Π α ( y + tz )) X ≥ a (Π α y − Π α ( y + tz ) , y + tz ) , which hold for any y, z ∈ Y and any t ∈ R . By adding these inequalities, we obtain1 α (cid:107) Π α ( y + tz ) − Π α y (cid:107) X ≤ ta (Π α ( y + tz ) − Π α y, z ) ≤ t (cid:107) a (cid:107)(cid:107) Π α ( y + tz ) − Π α y (cid:107) X (cid:107) z (cid:107) Y . α is Lipschitz continuous in Y .From (5.1) and (5.8) we have, for any t ∈ R and any y, z ∈ Y , J α ( y ) = a (Π α y, y ) − α (cid:107) Π α y (cid:107) X ≥ a (Π α ( y + tz ) , y ) − α (cid:107) Π α ( y + tz ) (cid:107) X , J α ( y + tz ) = a (Π α ( y + tz ) , y + tz ) − α (cid:107) Π α ( y + tz ) (cid:107) X ≥ a (Π α y, y + tz ) − α (cid:107) Π α y (cid:107) X . Hence, a (Π α y, z ) ≤ t [ J α ( y + tz ) − J α ( y )] ≤ a (Π α ( y + tz ) , z ) , proving (5.9).Using the differentiability of J α , one can rewrite the problem (5.3) as a system of nonlinearvariational equations. Theorem 5.2.
Let X be a Hilbert space with the scalar product ( ., . ) X and let y α be aminimizer in (5.3) . Then there exists λ α ∈ R + such that the pair ( y α , λ α ) is a solution ofthe system: a (Π α y α , z ) = λ α L ( z ) ∀ z ∈ Y,L ( y α ) = 1 . (cid:41) (5.11) Conversely, if ( y α , λ α ) is a solution to (5.11) then y α solves (5.3) . Remark 5.1.
In [14], the function ˜ ψ : α (cid:55)→ λ α was introduced and analysed for the caseof Hencky plasticity. It is worth noticing that this function is well defined even if (5.3)does not have a minimizer in Y . In addition, ˜ ψ is continuous and nondecreasing, with ψ ( α ) ≤ ˜ ψ ( α ) ≤ λ ∗ for any α >
0, and ˜ ψ ( α ) → λ ∗ as α → + ∞ . One can expect that theseconsiderations from [14] can be extended to our abstract problem. ζ ∗ For classical limit analysis problems, computable majorants of ζ ∗ have been derived in[28, 16]. The aim of this section is to derive a more general majorant valid for the abstractproblem (2.1). In our analysis, we shall use the assumptions (A1)–(A4) and (D1)–(D4) ofTheorem 2.3. The following alternative to the assumption (D3) will also be considered:(D3 (cid:48) ) P = P A + P C = { x ∈ X | x = x A + x C , x A ∈ P A , x C ∈ P C } , where P , P A and P C have the same properties as in (D3). 16e note that (D3 (cid:48) ) is more restrictive than (D3); it has been used in [16].From the definition of ζ ∗ (see (2.6)), we have the following simple upper bound of ζ ∗ : ζ ∗ ≤ J ( y ) L ( y ) ∀ y ∈ Y, y ∈ dom J , L ( y ) > . (6.1)Unfortunately, if the set P is unbounded then it is difficult or even impossible to find y ∈ dom J in such a way that the bound (6.1) would be sufficiently sharp. The aim ofthis section is to derive an upper bound of ζ ∗ for a larger class of functions y ∈ Y , notnecessarily belonging to dom J .First, we need to characterize the set dom J . For this purpose, we define the closedconvex cone K := { y ∈ Y | a ( x, y ) ≤ ∀ x ∈ P C } , (6.2)and the convex, finite-valued functional J A : Y → R , J A ( y ) := max x ∈ P A a ( x, y ) , y ∈ Y. (6.3) Lemma 6.1.
Let the assumptions (A1)–(A4) and (D1)–(D4) be satisfied. Then dom J = K and J ( y ) ≤ J A ( y ) ∀ y ∈ K . (6.4) Moreover, if (D3 (cid:48) ) holds then J ( y ) = J A ( y ) for any y ∈ K .Proof. Assume that y (cid:54)∈ K . Then there exists x C ∈ P C such that a ( x C , y ) >
0. From(D3), it follows that αx C ∈ P for any α ≥
0. Hence, J ( y ) ≥ lim α → + ∞ a ( αx C , y ) = lim α → + ∞ αa ( x C , y ) = + ∞ . Let y ∈ K . Then J ( y ) ≤ sup x A ∈ P A a ( x A , y ) + sup x C ∈ P C a ( x C , y ) = J A ( y ) + 0 = J A ( y ) < + ∞ . (6.5)If (D3 (cid:48) ) holds then P A = P A + { X } ⊂ P . Hence, J ( y ) ≥ sup x ∈ P A a ( x, y ) = J A ( y ) . (6.6)From (6.5) and (6.6), it follows that J ( y ) = J A ( y ) for any y ∈ K .17rom the definition of J A and the boundedness of a and P A , we easily derive the usefulestimates |J A ( y ) − J A ( y ) | ≤ J A ( y − y ) , ∀ y , y ∈ Y, (6.7)and J A ( y ) ≤ (cid:37) A (cid:107) a (cid:107)(cid:107) y (cid:107) Y , ∀ y ∈ Y, (cid:37) A := max x ∈ P A (cid:107) x (cid:107) X . (6.8)In order to estimate ζ ∗ using y (cid:54)∈ K , it is important to measure the distance between y and K . Define the quantity (cid:107) Π C y (cid:107) X := (cid:18) max x ∈ P C {−(cid:107) x (cid:107) X + 2 a ( x, y ) } (cid:19) / , y ∈ Y. (6.9) Remark 6.1.
The notation (cid:107) Π C y (cid:107) X including the norm in X is justified if X is a Hilbertspace. Indeed, define the operatorΠ C : Y → P C , Π C y := arg max x ∈ P C {−(cid:107) x (cid:107) X + 2 a ( x, y ) } , y ∈ Y. (6.10)From the cone property of P C , (6.10) is equivalent to (cid:107) Π C y (cid:107) X = a (Π C y, y ) and (Π C y, x ) ≥ a ( x, y ) ∀ x ∈ P C . Hence, we obtain (6.9).It is also useful to note that if y ∈ K then (cid:107) Π C y (cid:107) X = 0. We have the following result. Lemma 6.2.
Let the assumptions (A1)–(A4) and (D1)–(D4) be satisfied and c ∗ > , K , (cid:107) Π C y (cid:107) X be defined by (2.7) , (6.2) , and (6.9) , respectively. Then min z ∈K (cid:107) y − z (cid:107) ≤ C ∗ (cid:107) Π C y (cid:107) X , ∀ y ∈ Y, C ∗ := c − ∗ > . (6.11) Proof.
Using (6.2), [12, Proposition VI 2.3] and the substitution z (cid:55)→ z + y , we conse-18uently derivemin z ∈K (cid:107) y − z (cid:107) = min z ∈ Y sup x ∈ P C (cid:8) (cid:107) y − z (cid:107) + 2 a ( x, z ) (cid:9) = sup x ∈ P C min z ∈ Y (cid:8) (cid:107) y − z (cid:107) + 2 a ( x, z ) (cid:9) = sup x ∈ P C min z ∈ Y (cid:8) (cid:107) z (cid:107) + 2 a ( x, z ) + 2 a ( x, y ) (cid:9) ∀ y ∈ Y. (6.12)For any x ∈ X , there exists a unique z x ∈ Y such that( z x , z ) Y = − a ( x, z ) ∀ z ∈ Y. Hence, (cid:107) z x (cid:107) X = sup z ∈ Yz (cid:54) =0 Y a ( x, z ) (cid:107) z (cid:107) Y and min z ∈ Y (cid:8) (cid:107) z (cid:107) + 2 a ( x, z ) (cid:9) = −(cid:107) z x (cid:107) . (6.13)Inserting (6.13) into (6.12), we find thatmin z ∈K (cid:107) y − z (cid:107) = sup x ∈ P C min z ∈ Y (cid:8) (cid:107) z (cid:107) + 2 a ( x, z ) + 2 a ( x, y ) (cid:9) = sup x ∈ P C − sup z ∈ Yz (cid:54) =0 Y a ( x, z ) (cid:107) z (cid:107) Y + 2 a ( x, y ) (2.7) ≤ sup x ∈ P C (cid:8) − c ∗ (cid:107) x (cid:107) X + 2 a ( x, y ) (cid:9) = max x ∈ P C (cid:8) − c ∗ (cid:107) x/c ∗ (cid:107) X + 2 a ( x/c ∗ , y ) (cid:9) = 1 c ∗ max x ∈ P C (cid:8) −(cid:107) x (cid:107) X + 2 a ( x, y ) (cid:9) (6.10) = C ∗ (cid:107) Π C y (cid:107) X ∀ y ∈ Y, (6.14)which gives the desired result.Using Lemma 6.1 and 6.2, we derive the following upper bound of ζ ∗ . Theorem 6.1.
Let the assumptions (A1)–(A4) and (D1)–(D4) be satisfied and y ∈ Y besuch that L ( y ) > C ∗ (cid:107) Π C y (cid:107) X (cid:107) L (cid:107) Y ∗ . (6.15) Then ζ ∗ ≤ J A ( y ) + (cid:37) A C ∗ (cid:107) a (cid:107)(cid:107) Π C y (cid:107) X L ( y ) − C ∗ (cid:107) Π C y (cid:107) X (cid:107) L (cid:107) Y ∗ . (6.16)19 roof. Let y ∈ Y satisfy (6.15). By Lemma 6.2 there exists z y ∈ K such that (cid:107) y − z y (cid:107) Y ≤ C ∗ (cid:107) Π C y (cid:107) X . (6.17)For any λ > J A ( y )+ (cid:37) A C ∗ (cid:107) a (cid:107)(cid:107) Π C y (cid:107) X L ( y ) − C ∗ (cid:107) Π C y (cid:107) X (cid:107) L (cid:107) Y ∗ , we have J ( z y ) − λL ( z y ) (6.4) ≤ J A ( z y ) − λL ( z y )= J A ( y ) − λL ( y ) + [ J A ( z y ) − J A ( y )] + λL ( y − z y ) (6.7) , (6.8) ≤ J A ( y ) − λL ( y ) + ( (cid:37) A (cid:107) a (cid:107) + λ (cid:107) L (cid:107) Y ∗ ) (cid:107) y − z y (cid:107) Y (6.17) ≤ J A ( y ) − λL ( y ) + C ∗ ( (cid:37) A (cid:107) a (cid:107) + λ (cid:107) L (cid:107) Y ∗ ) (cid:107) Π C y (cid:107) X = J A ( y ) + (cid:37) A C ∗ (cid:107) a (cid:107)(cid:107) Π C y (cid:107) X − λ [ L ( y ) − C ∗ (cid:107) L (cid:107) Y ∗ (cid:107) Π C y (cid:107) X ] < . Hence, L ( z y ) > J ( z y ) /λ ≥ ζ ∗ (6.1) ≤ J ( z y ) L ( z y ) < λ ∀ λ > J A ( y ) + (cid:37) A C ∗ (cid:107) a (cid:107)(cid:107) Π C y (cid:107) X L ( y ) − C ∗ (cid:107) Π C y (cid:107) X (cid:107) L (cid:107) Y ∗ . This implies (6.16).
Remark 6.2.
If the assumption (D3 (cid:48) ) holds and y ∈ K then J A ( y ) = J ( y ), (cid:107) Π C y (cid:107) X = 0,and thus the bounds (6.1) and (6.16) coincide. Remark 6.3. If y ∈ Y is sufficiently close to the cone K then the assumption (6.15) is sat-isfied. This can be achieved by a convenient numerical method, e.g., by the regularizationmethod presented in the previous section. Remark 6.4.
The bound (6.16) is computable if estimates of (cid:107) L (cid:107) Y ∗ , (cid:107) Π C y (cid:107) X and C ∗ are at our disposal. The computable bounds of (cid:107) L (cid:107) Y ∗ , (cid:107) Π C y (cid:107) X are available in theliterature on a posteriori error analysis. Computable bounds of the inf-sup constant C ∗ have appeared in the literature quite recently, see [16] and references therein. Remark 6.5.
In [28], a computable majorant of the limit load was used in the Henckyplasticity problem to prove convergence of the standard finite element method and todetect locking effects that may arise when the simplest P1 elements are used.20
Examples
In this section, we illustrate the abstract problem (1.1) on particular examples fromnonlinear mechanics and discuss the validity of the assumptions (A1)–(A4), (B) and (D1)–(D4) presented in Section 2. In all examples we consider a bounded domain Ω ⊂ R d , d = 2 ,
3, with Lipschitz continuous boundary ∂ Ω. The outward unit normal to ∂ Ω isdenoted by ν . The abstract spaces X and Y will be represented by L and H spaces,respectively, for the sake of simplicity. Details of the mathematical theory of limit analysis in classical perfect plasticity may befound in [32] or [10]. For its engineering applications we refer, for example, to [8, 30]. Theaim is to find the largest load factor at which plastic behaviour may be sustained, in thecontext of proportional loading. We briefly recapitulate results presented in [16, 28, 15,14].A body occupying the domain Ω is fixed on a part Γ ⊂ ∂ Ω and surface forces f : Γ f → R d act on the remaining part Γ f of ∂ Ω. We assume that Γ and Γ f have a positive surfacemeasure. Let F : Ω → R d denote the volume force. The external loads are parametrizedby a scalar factor λ ≥ R d × dsym . TheCauchy stress field σ : Ω → R d × dsym satisfies the equilibrium equation and traction boundarycondition div σ + λF = 0 in Ω , (7.1a) σν = λf on Γ f , (7.1b)and is plastically admissible in the sense that σ ∈ B in Ω , B := { τ ∈ R d × dsym | ϕ ( τ ) ≤ } . (7.2)Here, ϕ : R d × dsym → R , ϕ (0) <
0, is a convex function representing a yield criterion. For thesake of simplicity, we assume that ϕ and thus B are independent of the spatial variable.The infinitesimal strain rate ε : Ω → R d × dsym and the displacement rate v : Ω → R d satisfy21he relations ε := ε ( v ) = 12 [ ∇ v + ( ∇ v ) (cid:62) ] in Ω , v = 0 on Γ . (7.3)The last ingredient of the perfectly plastic model is a plastic flow rule that relates σ and ε ,and which is based on the set B . This relation is represented by the principle of maximumplastic dissipation in quasistatic models or by a generalized projection of R d × dsym onto B intotal strain models. We skip its definition, for the sake of brevity.Formally, the limit load factor λ ∗ is defined as the supremum over λ ≥ λ ∗ more precisely and in the form (2.6), it is necessaryto introduce a convenient function space X for stress fields. For this purpose define theHilbert space X := L (Ω; R d × dsym ) = { σ : Ω → R d × dsym | σ ij ∈ L (Ω) , i, j = 1 , , . . . d } equipped with the scalar product and norm( σ, ε ) X := ( σ, ε ) = (cid:90) Ω σ : ε dx, (cid:107) σ (cid:107) X := (cid:107) σ (cid:107) = (cid:112) ( σ, σ ) , where σ : ε = σ ij ε ij with the summation convention on repeated indices. The correspond-ing primal space Y is chosen as follows: Y := { v ∈ W , (Ω; R d ) | v = 0 a.e. in Γ } . It is also a Hilbert space representing rates of displacements with the following scalarproduct and norm: ( u, v ) Y := ( ∇ u, ∇ v ) , (cid:107) v (cid:107) Y := (cid:107)∇ v (cid:107) . Using the spaces
X, Y and Green’s theorem, a weak formulation of (7.1a) and (7.1b) forfixed σ reads as follows: a ( σ, v ) = λL ( v ) ∀ v ∈ Y, (7.4)where a ( σ, v ) := (cid:90) Ω σ : ε ( v ) dx, L ( v ) := (cid:90) Ω F · v dx + (cid:90) Γ f f · v ds, v ∈ Y, (7.5)with σ ∈ X , F ∈ L (Ω; R d ) and f ∈ L (Γ f ; R d ). It is easy to see that a is a continuous22ilinear form in X × Y and L ∈ Y ∗ . Using the notation from Section 1, one can write λ ∗ = sup { λ ∈ R + | P ∩ Λ λ (cid:54) = ∅} = sup σ ∈ P inf v ∈ YL ( v )=1 a ( σ, v ) , where P := { σ ∈ X | σ ∈ B a.e. in Ω } , Λ λ := { σ ∈ X | a ( σ, v ) = λL ( v ) ∀ v ∈ Y } . (7.6)The sets P and Λ λ are closed, convex and non-empty in X and represent plastically andstatically admissible stresses, respectively.We note that the set P is defined in a pointwise sense. Consequently, the sets P A , P C andthe functions J , J α , Π α and Π C introduced in the previous sections may be also definedin a pointwise sense. To illustrate, we choose the von Mises yield criterion defined by ϕ ( σ ) := | σ D | − γ, γ > , σ D = σ − d (tr σ ) I, | σ | := √ σ ij σ ij , (7.7)where I is the unit d × d matrix, tr σ denotes the trace of σ , σ D is the deviatoric part of σ and γ > P can be decomposed according to P = P A + P C , where P A = { τ ∈ X | | τ | ≤ γ a.e. in Ω } , P C = { τ ∈ X | ∃ q ∈ L (Ω) : τ = qI } . Clearly, P A is bounded in X and P C is a closed subspace of X , that is, a convex cone. Toprove (2.7), we use the well-known inf-sup condition for incompressible flow media with c Ω > x C ∈ P C x C (cid:54) =0 X sup y ∈ Yy (cid:54) =0 Y a ( x C , y ) (cid:107) x C (cid:107) X (cid:107) y (cid:107) Y = inf τ ∈ P C τ (cid:54) =0 X sup v ∈ Yv (cid:54) =0 Y (cid:82) Ω τ : ε ( v ) dx (cid:107) τ (cid:107) (cid:107)∇ v (cid:107) = 1 √ d inf q ∈ L (Ω) q (cid:54) =0 sup v ∈ Yv (cid:54) =0 Y (cid:82) Ω q div v dx (cid:107) q (cid:107) (cid:107)∇ v (cid:107) ≥ c Ω √ d . (7.8)Thus, the condition (2.7) holds with c ∗ = c Ω / √ d . Consequently, the assumptions (A1)–(A4), (D1)–(D4) from Section 2 are satisfied and from Theorem 2.3 it follows that λ ∗ = ζ ∗ = inf v ∈ YL ( v )=1 sup σ ∈ P a ( σ, v ) = inf v ∈ YL ( v )=1 J ( v ) . Notice that if Γ = ∂ Ω then it is necessary to use Theorem 4.1 with the weaker assumption(4.2) instead of (D4). In this case, we replace the space L (Ω) in (7.8) by L (Ω) = { q ∈ L (Ω) | (cid:82) Ω q dx = 0 } , see [28, 16]. 23he primal functional J for the von Mises yield criterion is given by J ( v ) = sup σ ∈ P a ( σ, v ) = (cid:90) Ω γ | ε ( v ) | dx, div v = 0 in Ω , + ∞ , otherwise , ∀ v ∈ Y. This functional may have no minimizers in Y . To guarantee that the primal problemis solvable, it is necessary to use another choice of X and Y , as was done, for example,in [9, 32, 10]. In particular, the assumptions (C1)–(C3) of Theorem 2.2 were verified in[9, 10].The functions J α , J A and Π C for the von Mises yield criterion can be found in thefollowing forms: J α ( v ) := (cid:90) Ω j α ( ε ( v )) dx, j α ( ε ) = (cid:40) α | ε | , α | ε D | ≤ γ d α (tr ε ) + γ | e D | − γ α , α | e D | ≥ γ, , J A ( v ) = (cid:90) Ω γ | ε ( v ) | dx, (cid:107) Π C v (cid:107) = d − / (cid:107) div v (cid:107) ∀ v ∈ Y. Let us recall that they are important for the regularization method and the computablemajorant presented in the previous sections. We refer to [14, 15, 28, 16] for more details.
Remark 7.1.
If we choose the Drucker-Prager or Mohr-Coulomb yield criteria in (7.2)instead of von Mises then it is also possible to find an appropriate split P = P A + P C suchthat the assumptions (D3) and even (D3 (cid:48) ) are satisfied. But for these criteria the cone P C is not a subspace of X . Therefore, it is necessary to work with the inf-sup conditionon convex cones, see [16]. In the next two subsections, we consider as further examples the models of strain-gradientplasticity presented in [25, 24, 6, 26]. First, following [26], we introduce a subproblem thatenables us to decide whether a given stress tensor is plastically admissible or not. We notethat this problem is simple in classical plasticity where the yield criterion can be verifiedpointwisely (see, for example the definition of P in (7.6)). However, plastic yield criteriain strain-gradient plasticity are non-local and the verification is strongly non-trivial.Beside the space R d × dsym defined in Section 7.1, we also use the following spaces of the second24nd third order tensors, respectively: R d × dsym, := { π ∈ R d × dsym | tr π = 0 } , R d × d × dsym, := { Π ∈ R d × d × d | Π ijk = Π jik , i, j, k = 1 , , . . . , d, Π ppk = 0 , k = 1 , , . . . , d } . Thus, the third order tensor Π belongs to R d × d × dsym, if it is symmetric and deviatoric withrespect to the first two indices.We assume that σ : Ω → R d × dsym is a given stress field and σ D : Ω → R d × dsym, denotes itsdeviatoric part. The theory of strain gradient plasticity makes use of second- and third-order tensors π : Ω → R d × dsym, and Π : Ω → R d × d × dsym, that represent microstresses. We saythat σ is plastically admissible if there exists a pair ( π, Π) such that σ D = π − div Π in Ω , Π ν = 0 on Γ F , (7.9) ϕ (cid:96) ( π, Π) := (cid:112) | π | + (cid:96) − | Π | − γ ≤ , (7.10)where γ > (cid:96) > | Π | = Π ◦ Π := Π ijk Π ijk and Γ F ⊂ ∂ Ω. The part of ∂ Ω complementary to Γ F in ∂ Ω is denoted by Γ H .We note that the yield criterion (7.10) can be viewed as an extension of the classicalcondition (7.7). Indeed, setting Π = 0 we derive the sufficient condition | σ D | ≤ γ for σ to be plastically admissible. Unlike the classical case, the stress σ can be plasticallyadmissible even if | σ D | > γ .If σ is plastically admissible then λσ is also plastically admissible for any λ ∈ [0 , find the maximal value λ ∗ of λ ≥ for which λσ is plastically admissible in the sense of (7.9) and (7.10). Clearly,if λ ∗ > σ is admissible.Let us define λ ∗ more precisely, using the abstract problem (2.6). We assume that allcomponents of σ , π and Π belong to L (Ω), that is, σ ∈ L (Ω; R d × dsym ), π ∈ L (Ω; R d × dsym, )and Π ∈ L (Ω; R d × d × dsym, ). The space X is defined as the space of pairs ( π, Π) endowed withthe scalar product (( π, Π) , (¯ π, ¯Π)) X := (cid:90) Ω ( π : ¯ π + Π ◦ ¯Π) dx. The primal space Y := { q ∈ L (Ω; R d × dsym, ) | ∇ q ∈ L (Ω; R d × d × dsym, ) , q = 0 on Γ H }
25s the Hilbert space of admissible plastic strain rates with the scalar product( q, ¯ q ) Y := (cid:90) Ω ( q : ¯ q + ∇ q ◦ ∇ ¯ q ) dx. Using the spaces X and Y , we introduce the following weak form of (7.9): (cid:90) Ω [ π : q + Π ◦ ∇ q ] dx = (cid:90) Ω σ D : q dx ∀ q ∈ Y, (7.11)and define the forms a : X × Y and L ∈ Y ∗ by a (( π, Π) , q ) := (cid:90) Ω [ π : q + Π ◦ ∇ q ] dx, L ( q ) := (cid:90) Ω σ D : q dx. Then the dual problem (2.6) reads λ ∗ = sup { λ ∈ R + | P ∩ Λ λ (cid:54) = ∅} = sup ( π, Π) ∈ P inf q ∈ YL ( q )=1 a (( π, Π) , q ) , where P := { ( π, Π) ∈ X | (cid:112) | π | + (cid:96) − | Π | ≤ γ a.e. in Ω } , Λ λ := { ( π, Π) ∈ X | a (( π, Π) , q ) = λL ( q ) ∀ q ∈ Y } . From (7.10), it follows that P is bounded in X , i.e. the assumption (B) of Theorem 2.1is satisfied. Thus we have λ ∗ = ζ ∗ = inf q ∈ YL ( q )=1 sup ( π, Π) ∈ P a (( π, Π) , q ) = inf q ∈ YL ( q )=1 J ( q ) . In this case, the functional J can be found in the form J ( q ) = (cid:90) Ω γ (cid:112) | q | + (cid:96) |∇ q | dx ∀ q ∈ Y. Although J is finite-valued everywhere, it is not coercive in Y . Therefore, a certainrelaxation of the problem is necessary if we wish to properly define a minimizer of J andguarantee its existence. Such an analysis has not been done for this problem and we leavethis as a topic for further investigation.The primal and dual problems have been solved by regularization (penalization) methods26n [26]. In particular, the regularized functional J α defined by (5.1) takes the form J α ( q ) := (cid:90) Ω D α ( q, ∇ q ) dx, D α ( q, ∇ q ) = α ( | q | + (cid:96) |∇ q | ) , (cid:112) | q | + (cid:96) |∇ q | ≤ α (cid:112) | q | + (cid:96) |∇ q | − α , (cid:112) | q | + (cid:96) |∇ q | ≥ α . Reliable lower and upper bounds of λ ∗ have also been estimated in [26] using the regular-ization methods. Remark 7.2.
Other choices of yield functions are possible in (7.10). For example, thefollowing more general function has been considered in [26, 24]: ϕ (cid:96),r ( π, Π) := (cid:40) [ | π | r + ( (cid:96) − | Π | ) r ] /r − γ, ≤ r < + ∞ , max {| π | , (cid:96) − | Π |} − γ, r = + ∞ . (7.12)The set P corresponding to this function remains bounded and thus the equality λ ∗ = ζ ∗ holds. Denoting r (cid:48) = (1 − /r ) − we find the functional J in the following form: J ( q ) = (cid:82) Ω γ [ | q | r (cid:48) + (cid:96) |∇ q | r (cid:48) ] /r (cid:48) dx, ≤ r (cid:48) < + ∞ , (cid:82) Ω γ max {| q | , (cid:96) |∇ q |} dx, r (cid:48) = + ∞ . (7.13) Limit analysis in gradient-enhanced plasticity has been studied in [13, 23] for a model inwhich size-dependence is through the gradient of a scalar function of the plastic strain.Here, we consider the model from [25, 24, 6, 26] where the gradient is applied to the entireplastic strain.We use the same tensors σ , π , Π and external forces F and f as in Sections 7.1 and 7.2.Let us note that the pair of boundaries (Γ F , Γ H ) defined in Section 7.2 may differ from(Γ , Γ f ) introduced in Section 7.1. The limit analysis problem for the strain gradientplasticity reads: find the supremum λ ∗ over all λ ≥ for which there exist σ , π , Π suchthat div σ + λF = 0 in Ω , σν = λf on Γ f , (7.14) σ D = π − div Π in Ω , Π ν = 0 on Γ F , (7.15) ϕ (cid:96) ( π, Π) = (cid:112) | π | + (cid:96) − | Π | ≤ γ in Ω , γ, (cid:96) > . (7.16)We see that (7.14) coincides with (7.1a) and (7.1b) from Section 7.1. However, we nowuse the definition of plastically admissible stresses from Section 7.2 (see (7.15) and (7.16))27nstead of (7.10).To rewrite this problem in the form (2.6) or (2.1), we split σ as follows: σ = pI + σ D = pI + π − div Π in Ω . (7.17)We denote by X the L -space of all admissible triples ( p, π, Π). The equations (7.14) and(7.15) can be rewritten using (7.17) to the following weak form: a (( p, π, Π) , v ) = λL ( v ) ∀ v ∈ Y, where a (( p, π, Π) , v ) := (cid:90) Ω [ p div v + π : ε ( v ) + Π ◦ ∇ ε ( v )] dx,L ( v ) := (cid:90) Ω F · v dx + (cid:90) Γ f f · v ds, and Y := { v ∈ W , (Ω; R d ) | v = 0 on Γ , ε ( v ) = 0 on Γ H } . The space Y is equipped with the standard norm denoted by (cid:107) . (cid:107) Y . The set Λ λ remainsthe same as in (2.5) and the set P of plastically admissible stresses reads P := { ( p, π, Π) ∈ X | (cid:112) | π | + (cid:96) − | Π | ≤ γ a.e. in Ω } . Thus, we can define the limit analysis problem as follows: λ ∗ = sup { λ ∈ R + | P ∩ Λ λ (cid:54) = ∅} = sup ( p,π, Π) ∈ P inf v ∈ YL ( v )=1 a (( p, π, Π) , v ) . For analysis of the primal problem (2.1), it is convenient to use the split P = P A + P C ,where P A := { ( p, π, Π) ∈ X | p = 0 , (cid:112) | π | + (cid:96) − | Π | ≤ γ a.e. in Ω } ,P C := { ( p, π, Π) ∈ X | π = 0 , Π = 0 } . It is easy to check that P A is bounded in X and P C is a closed linear subspace of X . Wehave ζ ∗ = inf v ∈ YL ( v )=1 sup ( p,π, Π) ∈ P a (( p, π, Π) , v ) = inf v ∈ YL ( v )=1 J ( v ) , J ( v ) = (cid:40) (cid:82) Ω γ (cid:112) | ε ( v ) | + (cid:96) |∇ ε ( v ) | dx, if div v = 0 in Ω , + ∞ , otherwise . The inf-sup term in (2.7) becomesinf ( p,π, Π) ∈ P C ( p,π, Π) (cid:54) =0 sup v ∈ Yv (cid:54) =0 a (( p, π, Π) , v ) (cid:107) ( p, π, Π) (cid:107) X (cid:107) v (cid:107) Y = inf p ∈ L (Ω) p (cid:54) =0 sup v ∈ Yv (cid:54) =0 (cid:82) Ω p (div v ) dx (cid:107) p (cid:107) (cid:107) v (cid:107) Y . (7.18)For the equality λ ∗ = ζ ∗ to be satisfied it suffices to show that the right-hand side of(7.18) is positive on an appropriate factor space of L (Ω). Such an analysis seems to bemore involved and we leave this as a topic for further investigation. Remark 7.3.
If we replace the yield functions ϕ (cid:96) in (7.16) with ϕ (cid:96),r defined by (7.12)then the set P C and the inf-sup expression (7.18) remain the same. The correspondingfunctional J ( v ) is the same as in (7.13) for div v = 0. The last example is devoted to a model for delamination, inspired by [2]. Let Ω ⊂ R denote the domain occupied by an elastic body, with boundary ∂ Ω. The body is alaminated composite, comprising two distinct materials. The geometry is idealized withone material, referred to as the bulk, comprising the entire domain with the exceptionof a thin layer of the second material. This thin layer is treated as a line Γ b ⊂ Ω, andseparation or delamination may occur along this line.We follow [2] and consider a problem with a symmetric geometry and loading, as shownin Figure 1(a). Zero displacements in the normal ( x ) direction are prescribed along theboundary Γ (cid:96) , while on Γ f a surface force λf is applied, where λ ≥ t is unconstrained and traction-free. The surface force as wellas a body force λF act symmetrically along the x axis so that F ( x , x ) = F ( x , − x ),the same applying to f . Given the symmetry of the problem we may confine attention tothe upper half Ω + of the domain, shown in Figure 1(b).The boundary conditions set out above have to be augmented with a condition alongΓ b . This takes the form of conditions on the traction vector t = σν : from symmetry thetangential component σν · τ := σ must be zero. Here and henceforth subscripts ν and τ refer respectively to normal and tangential components. The condition in the normal29
If we consider the case in which the body is fixed on Γ (cid:96) as in [2], then K = { Y } , which implies that λ ∗ = ζ ∗ = + ∞ . Thus the related delamination problemmay have a solution even if the composite is completely debonded.32 emark 7.5. The complete formulation of the delamination problem requires also acondition of non-interpenetration (that is, a Signorini condition) along Γ b . For the sym-metrized problem this amounts to defining the conic set Y C := { v ∈ Y | v ≥ b } ofadmissible displacement fields, replacing the last of equations (7.21) with σ = 0 , − σ ∈ [0 , γ ] on Γ b , and consequently, replacing P with P := { ( σ, Ξ) ∈ X | Ξ ∈ [0 , γ ] on Γ b } . According toTheorem 4.3, we have the duality problem λ ∗ = sup x ∈ P inf y ∈ Y C L ( y )=1 a ( x, y ) ? = inf y ∈ Y C L ( y )=1 sup x ∈ P a ( x, y ) = ζ ∗ . By combining Theorems 4.1 and 4.3 it is possible to show that λ ∗ = ζ ∗ . In particular, if (cid:90) Ω + F dx + (cid:90) Γ f f ds > This work has been concerned with an inf-sup problem posed on abstract Banach spaces.The main feature of this convex and constrained problem has been the presence of abilinear Lagrangian, which appears in applications leading to linear, cone or convex pro-gramming problems. Conditions for ensuring duality without any gap have been intro-duced. We have introduced and extended an innovative framework based on an inf-supcondition on convex cones generalizing the well-known Babuˇska-Brezzi conditions. Wehave also suggested a new regularization method and derived a computable majorant tothe problem.Applications of the abstract problem to various examples in mechanics have been pre-sented. First, the problem of limit analysis in classical plasticity has been revisited in thecontext of the duality framework of this work. Then, we have shown that the abstractframework may be used in the case of two different subproblems related to strain-gradientplasticity, viz. the determination of plastically admissible stresses and the determinationof limit loads, and for a delamination problem.33he techniques presented in this paper could be extended to more general duality problemswhere the Lagrangian contains, in addition to the bilinear form, linear forms with respectto primal or dual variables. Such an extension would be applicable to a wider range ofproblems in mechanics.
Acknowledgment:
SS and JH acknowledge support for their work from the CzechScience Foundation (GA ˇCR) through project No. 19-11441S. BDR acknowledges supportfor his work from the National Research Foundation, through the South African Chair inComputational Mechanics, SARChI Grant 47584.
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