Optimal radius of a rigid cylindrical inclusion in nonhomogeneous plates with a crack Текст научной статьи по специальности «Физика»

Математические заметки СВФУ Январь—март, 2019. Том 26, № 1

UDC 517.946

OPTIMAL RADIUS OF A RIGID CYLINDRICAL INCLUSION IN NONHOMOGENEOUS PLATES WITH A CRACK N. P. Lazarev, A. Tani, and P. Sivtsev

Abstract. We consider equilibrium problems for a cracked inhomogeneous plate with a rigid circular inclusion. Deformation of an elastic matrix is described by the Timo-shenko model. The plate is assumed to have a through crack that reaches the boundary of the rigid inclusion. The boundary condition on the crack curve is given in the form of inequality and describes the mutual nonpenetration of the crack faces. For a family of corresponding variational problems, we analyze the dependence of their solutions on the radius of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the radius of the cylindrical inclusion is chosen as the control parameter. Existence of a solution to the optimal control problem and continuous dependence of the solutions with respect to the radius of the rigid inclusion are proved.

DOI: 10.25587/SVFU.2019.101.27246

Keywords: variational inequality, optimal control problem, nonpenetration, non-linear boundary conditions, crack, rigid inclusion.

1. Introduction

Engineering applications of composites are growing vastly along with the development of models and methods related to analysis of the mechanical properties and behavior of composite materials [1—19]. It is well known that the presence of defects such as cracks, holes and inclusions in elastic bodies can cause a high stress concentration. In this regard, the issues of justification and examination of mathematical models describing the mechanical influence of defects on stress-strain state of composites are very important.

In particular, general representation of the analytical solution for a crack lying on the line of centers between two rigid inclusions are given in [5]. Note that derivation of these expressions was carried out under the assumption that the appropriate shear stresses on the crack faces are equal to zero. The plane problems for a cracked body with a rectilinear crack located midway between two circular elastic or rigid grains (inclusions) are investigated in [2]. Namely, expressions for critical loads in the most dangerous points are found. These results have been obtained on

The first author was supported by the Ministry of Education and Science of the Russian Federation within the framework of the base part of the state task (project 1.7218.2017/6.7).

© 2019 N. P. Lazarev, A. Tani, and P. Sivtsev

the basis of the classic boundary conditions in the form of equations imposed on the crack curve.

It is well known that imposing linear boundary conditions on the crack may lead to physical inconsistency of mathematical models since a mutual penetration of the crack faces may happen [2,20]. In recent years, a crack theory with nonpenetration conditions has been under active study [10-14, 20-25]. This approach to solving crack problems is characterized by inequality type boundary conditions at the crack faces, is indeed what we employ in the present paper. Within this approach, various problems for bodies with rigid inclusions have been successfully formulated and investigated using variational methods, see for example [12, 26-31].

The result concerning the optimal radius of a rigid inclusion for a two-dimensional nonlinear model describing equilibrium of a cracked composite solid was obtained in [31]. The optimal control problem analyzed in this paper consists in the best choice of the radius r* £ [ro, R] of the circular rigid inclusion. A cost functional is defined using an arbitrary continuous functional in the solution space. The existence of the solution to the optimal control problem is proved. In addition, for a family of variational problems describing equilibrium of cracked plates with inclusions of different radiuses r £ [ro, R], we prove the continuous dependence of the solutions with respect to the parameter r.

Let Q C R2 be a bounded domain with a smooth boundary T G C0'1. We consider the family of open disks |wr} of radius r G [ro, R] such that

(a) ur' C for all r', r'' G [r0, R]: r' < r'';

(b) cjr C Q;

(c) the circles dwr, r G [r0, R], enclosing the disks wr intersect at one point P with coordinate xp = (x1p, x2p) (see Fig. 1).

Suppose that a smooth curve 7 C O without self-intersections has the following properties: 7 C O, exactly one endpoint of 7 coincides with P, the inequality x1p < xi holds for all points of curve x = (xi, £ 7.

We assume that 7 can be extended in such a way that this extension crosses r at two points, and O is divided into two subdomains O1 and O2 with Lipschitz

2. Family of equilibrium problems

r

Fig. 1. Geometry of the problem

boundaries OHi, d02, meas(r n Oili) > 0, i = 1, 2. This condition is sufficient for Korn's inequality to hold in the non-Lipschitz domain =

We define a three-dimensional Cartesian space {x1,x2,z} such that the set } x {0} C R3 corresponds to the middle plane of the plate. The curve 7 defines a crack (a cut) in the plate. This means that the cylindrical surface of the through crack specified by the relations x = (x1, x2) G 7, —1 < z < 1 where |z| is the distance to the middle plane. For a fixed parameter r G [ro,R] we suppose that the rigid cylindrical inclusion is specified by the set ur x [—1; 1], i.e. the boundary of the rigid inclusion is defined by the cylindrical surface 3ur x [—1,1]. Elastic part of the plate corresponds to the domain il\Ujr. Depending on the direction of the normal v = (v1;v2) to 7 we will speak about a positive face 7+ or a negative face 7- of the curve 7. The jump [q] of the function q on the curve 7 is found by the formula [q] = q|Y+ — q1 y- .

Denote by (W, w) the vector of mid-plane displacements (x G ), where W = (w1, w2) are the displacements in the plane and {x1; x2} and w are the displacements along the axis z. We denote the angles of rotation of a normal fiber by — = — (x) =

(—1,-2) (x G ).

Introduce the tensors describing the deformation of the transversely isotropic plate

11 dv

£ijW = £ij(W) = -{Wij + Wj,i), i, j = 1,2, v,i= —.

The tensors of moments m(^) = {mij(-0)} and stresses a(W) = {aij(W)} are expressed by the formulas (summation is performed over repeated indices) [32]

mij(-) = <iijri£ki(-), (W) = 3a,ijki£ki(W), i,j,k,l = 1, 2,

where the nonzero components of elasticity tensor A = {aijki} are as follows:

ana = D, anjj = Dk, a^j = aijji = D(1 — k)/2, i = j, i, j = 1, 2,

where D and k are the constants: D is a cylindrical rigidity of the plate, k is the Poisson ratio, 0 < k < 1/2. The transverse forces in the Timoshenko-type model are specified by the expressions

qi(w, —) = A(w,i +-i), i = 1, 2, (1)

where A = 2k'gh, k' is the shear coefficient, g is the shear modulus in areas perpendicular to the middle plane of the plate, and A is the constant [32]. Let B(G, ■, ■) be a bilinear form defined by the equality

B(G,X,X) = J{TOyWOeyWO +A(w,l+t(jl)(v,l + <Ttj{W)etj{W)} dx,

G

with some Lipschitzian subdomain G C f27, \ = X = The

potential energy functional of the plate occupying the region 07 has the form

1 2'

n{x) = \b{ÇI^x,x)~ j Fxdx, x = (W,w,i[>),

where F = (fi,/2,/3,^1,^2) G L2(Q7)5 is the vector specifying the external loads [32].

Introduce the Sobolev spaces

Hi'°(07) = {v G H 1(07) | v = 0 a.e. on r}, H(fi7) = Hi'°(07)5, H = || • U^)•

Due to presence of a rigid inclusion in the plate, restrictions of the functions describing displacements (W, w) and angles of rotation — to the domain wr satisfy a special kind of relations. We introduce the space which allows us to characterize the properties of volume rigid inclusion [27]

R(wr) = {Z|Z(x) = (bx2 + ci, —bxi + C2,a° + aixi + a2X2, —ai, —»2); x G }, (2)

where b, ci, c2, a°, ai, a2 G R. The condition of mutual nonpenetration of the opposite faces of the crack is given by [26]:

[W]v > |[-]v| on (3)

We formulate the contact problem of the plate with a rigid inclusion

in| n(x), (4)

where

Kr = {x = (W, w, -) G H(fi7) | [W]v > |[-]v| on Y, xU G RK)}

is the set of admissible functions. Note that the inclusion x G H(07) assumes that the homogeneous boundary-value conditions hold:

w = 0, - = W = (0, 0) on r. (5)

It can be shown that the set Kt is convex and closed in the Hilbert space H(07) [26]. Due to the estimate

B(07,x,x) < Ci||xllllxll,

where the constant ci > 0 is independent of x G H(07) and x G H(07), the symmetric bilinear form of B(07,x,x) is continuous with respect to H(07). The coercivity of the functional n(x) follows from the inequality

B(07,x,x) > clxl2, x G H(fi7), (6)

where the constants c > 0 independent of x (see [25]).

Remark 1. The inequality (6) yields the equivalence of the standard norm and the semi-norm determined by the bilinear form B(07, •, •) in the space H(07).

The above properties of energy functional n(x), bilinear form B(07, •, •), and set Kr allow one to establish the existence of a unique solution £r = (Ur, ur, 0r) G Kr for

the problem (4), see [22]. Symmetry and continuity of the bilinear form B(07, •, •) and the properties of the set Kr provide the equivalence of problem (4) to the variational inequality

Cr G Kr, B(ft7, Cr, x — Cr) > y F(x — Cr) dx Vx = (W, w, -) G Kr• (7)

3. Optimal control problem

We consider an arbitrary continuous functional G(x) : H(07) ^ R. It is possible to define the cost functional J : [r°, R] ^ R by the equality J(r) = G(Cr), where Cr is the solution of the problem (4).

The mentioned continuity property is fulfilled for many physically motivated functionals, for example, the following functional

Gi(x) = Hx — x^h^y )

characterizes the deviation of the generalized displacement vector from a given function x°.

Consider the optimal control problem:

Find r* G [r°,R] such that J(r*) = sup J(r). (8)

re[ro,R]

Theorem. There exists a solution to the optimal control problem (8).

Proof. Let {rn} be a maximizing sequence. By the boundedness of the interval [r°,R], we can extract a convergent subsequence {rnk} C {rn} such that

rnk ^ r* as k ^ to, r* G [r°,R].

Without loss of generality we assume that r„fc = r* for sufficiently large k. Otherwise there would exist a subsequence {rni} such that rni = r*, and therefore J(r*) is solution of (8).

Now we take into account Lemma 2, proved below: the solutions Ck of (4) corresponding to the parameters r„fc converge to the solution Cr* strongly in H(07) as k ^ to. This allows us to obtain convergence

J(r„fc) ^ J(r*),

indicating that

J (r*) = sup J (r). re[ro,R]

The theorem is proved.

4. Auxiliary lemmas

Now we have to justify auxiliary lemmas used within the proof of theorem. In establishing the proof, we needed Lemma 2; however before proceeding further we need first prove the following lemma.

Lemma 1. Let r* £ [ro, R) be a fixed real number and let {rn} С [r*, R] be a sequence of real numbers converging to r* as n ^ ж. Then for an arbitrary function n £ Kr* there exist a subsequence {rk} = {rnk} С {rn} and a sequence of functions {nk} such that щ £ Krk, к £ N, and щ ^ n weakly in H(fi7) as к ^ ж.

Proof. First of all we treat the simplest case of sequence {rnk} such that rnk = r*. It is obvious that the assertion of the lemma holds for nk = n, к £ N. Therefore, in what follows we assume that rn > r* for sufficiently large n. Denote by Z* the function describing the structure of П in ^r *, i.e.

C* /1 * I* 1 * i**i* I* * *\

= (6 x2 + c1; —6 xi + c2, a0 + axxi + a2x2, —a1; — a2)

in wr*. We extend the definition of Z* to the whole domain О by the equality:

Z*(x) = (6*x2 + c1, —6*x1 + c2, a0 + alxi + a*x2, —a*, —a*), x £

It is now necessary to fix an arbitrary value r £ (ro, R] and consider the following family of auxiliary problems:

Find nr £ Kr such that p(nr) = inf p(x), (9)

хек;

where p(x) = B(07, x — n, X — n),

Kr = {X = (W w) £ H(fi7) | x = n on y±, xL = Z*}.

It is easy to see that the functional p(x) is coercive and weakly lower semicontinuous on the space H). One can verify that the set Kr is convex and closed in H). These properties guarantee the existence of a solution nr of the problem (9). Besides, its solution is unique [20]. Since the functional p(x) is convex and differentiable on H), the problem (9) can be written in the equivalent form:

nr £ Kr, B(07,nr — n,x — nr) > 0 Vx £ K. (10)

Note that the solution nn of (10) for r = R belongs to all the sets Kr with r' £ (r0, R]. Substituting nn as the test functions into (10), we get

B(Q7, nr — n, nn) + B(Q7, n, nr) > B(Q7, nr, пг) Vr £ (ro, R].

Using the inequality (6) we obtain from this relation the following uniform upper bound:

llnrII < c Vr £ (ro,R]. This allows us to extract from the sequence {nr„} a subsequence {nk}, which is defined by equalities nk = nr„fc, k £ N (henceforth we define a sequence {rk} by the equality rk = rnfc) and {nk} weakly converges to some function j in H).

It is now necessary to show that rj = rj. Bearing in mind that the domain f2\cJr* is Lipschitzian, by construction we have inclusion (щ — i]) £ Дд(Г27\uJr*)5. Then, taking into account the properties of Jr* )5, we have the limit function {rj — rf)

belongs to the same functional space.

We consider now the functions of the form x± = nk ± a, where a is a function defined by zero extension of some arbitrary function a £ C0^(O7\ ^r * )5 into . It

is observed that for sufficiently large k we get the inclusion x± G Krk. Therefore we can substitute the elements of these sequences, {x+} and {x-}, as test functions into inequalities (10), revealing that

nk G K'k, ,nk — n, a) = 0. (11)

The function a is now fixed and by passing to the limit in (11) it is established that

-r,,a) = 0 VaeC0M(Si7\Ur*)5.

Hence, by the density of (il7\uJr*) in ii(j(il7\cjr*), we infer that ry — 17 = 0 in H^(il7\uJr*)5. Finally, by construction, the equality 77 = i] is satisfied in lor*. Consequently, j = n in H(07). Therefore, there is a sequence {nk} such that nk G Krk, k G N and nk — n weakly in H(07) as k — to. The Lemma is thus proved.

Now, we are in a position to prove an auxiliary statement which was used in the proof of the theorem.

Lemma 2. Let r* G [r°, R] be a fixed real number. Then Cr — Cr* strongly in H(07) as r — r*, where Cr = (Ur,ur,^r), Cr* = (Ur* , ur*, ^r*) are the solutions of (4), corresponding to parameters r G (r°, R], r* G [r°, R].

Proof. We proceed by contradiction. Assume that there exist a number e° > 0 and a sequence {rn} C (r°, R] such that rn — r*, llCn — Cr* H > e°, where C« = Crn, n G N are the solutions of (4), corresponding to rn.

Because of x° = 0 G Kr for all r G [r°, R], we can substitute x = x° in (7) for all r G [r°,R]. This provides

Cr G Kr, B(07, Cr, Cr) <y FCr dx Vr G [r°, R].

Oy

From this, using (6) we can deduce that for all r G [r°,R] the following estimate holds

llCr ll < c,

with some constant c > 0 independent of r. Consequently, replacing {C«} by its subsequence, if necessary, we can assume that {C«} converges to some C weakly in H (Q7).

Now we show that C G Kr*. Indeed, we have = Z« G R(^r„). In

accordance with the Sobolev embedding theorem [20], we obtain

— CUr* strongly in L2(^r*)5 as n — to, (12)

Cn|Y — CIY strongly in L2(7)5 as n — to. (13)

Choosing a subsequence, if necessary, we assume as n — to that C« — C

a.e. in ^r*.

This allows us to conclude that each of the numerical sequences {b„}, {cira}, {c2„}, {a°«}, {ai«}, {a2«} defining the structure of Z« in domains wrn is bounded. Thus, we can extract subsequences (retain notation) such that

b« —► b, a°„ —► a°, Ci„ —► Ci, ai„ —► ai, i = 1, 2, as n —► to.

We note that the sequence {rn} contains either a subsequence {rk} C {rn} converging to r* from the left or a subsequence {rk} C {rn} such that rk > r* for all k € N. Therefore, we can consider these cases separately. For the simple case, namely, if there exists a subsequence {rk} C {rn} such that rk > r* for all k € N, then the following strong convergence holds

£k Ur* ^ (bx2 + ci, -bxi + C2, ao + aixi + -ax, —a2) (14)

as k ^ to. Therefore, from (14) and (12) we obtain € R(wr*). Let us

now consider the more complicated another case. Suppose that the subsequence {rfe} C {rn} converges to r* from the left, i.e. rk < r* for all k € N and rk ^ r* as k ^ to. Under this assumption, for a fixed k' € N and the corresponding value r' = rk', we have

£k ^ (6x2 + ci, —bxi + C2, ao + aixi + a2X2, —ai, —a2) (15)

strongly in L2(wr')2 as k ^ to. It is possible to define a function l = ao + aixi + a2x2 in wr*. Taking into account the absolute continuity of the Lebesgue integral, we can choose a number k' € N large enough such that

\\l\\ls^Vv) < Ve, INIl^^Vv) < yfe. Further, using the triangle inequality, we get

Iluk — lIU2(^*\«r') < H^U2^*) + iKiil2^*\«r') < INIz^.W) + + ||Z||L2(c4VtVv) < 2Vi+ - w||L2(^);

therefore, we infer

I|uk — ) = |uk — l|L2(^r*\Wr') + I|uk — l|L,2(Wr' )

< (2Vi + IIuk - uWli^)? + I\uk - (16)

Thus, for sufficiently large k, the following estimates may be established:

IK - itII< Ve, IIuk - l\\L2M < A/6

allowing us to deduce that (16) is less than 10e and thus uk ^ l strongly in L2(wr*). Combining the last convergence and (12), we deduce the desired relation u|Wr* = l holds in ^r* .

It can be proved analogously that

U|^t* = 6(x2, —xi) + (ci,c2) a.e. in wr*, (|Wt* = (—ai, — a2) a.e. in wr*.

Thus, we conclude that € R(wr*); Therefore, in all possible cases we have

fUr* € R(wr*).

It remains to show that £ satisfies the inequality [Uv] > |[(v]| on 7. In view of (13), we can extract subsequences once again and obtain the following convergences £n|Y ^ , a.e. on both 7+ and 7-. Now we pass to the limit in the following inequalities as n ^ to

[U„v] > |[(nv]| on 7.

This leads to [Uv] > v]| on 7, that is £ G Kr*.

Our next goals are to prove that £ = £r* and establish the existence of a sequence £n = £rn, n = 1, 2,... of solutions strongly converging to £r* in H(07). Observe that, as rn — r*, there must exist either a subsequence {rni} such that rni < r* for all l G N or, if that is not the case, a subsequence {rnm}, rnm > r* for all m G N. For the first case we have the subsequence {rni} C (ro, R] with the property rni < r* for all l G N. For convenience, we denote this subsequence by {rn}. Since rn < r*, an arbitrary test function x G Kr* also belongs to the set Krn. Consequently, it is possible to pass to the limit as n — to in the following inequalities with the test function x G Kr*:

£n G Kr„, B(07,£„,x - £n) >y F(x - £n) dx Vx G Kr*.

Now, bearing in mind the weak convergence of {£n} to £ , the limiting inequality takes the form

B(07, £, x - £) > J F(x - £) dx Vx G Kr*.

Due to an arbitrariness of x G Kr* the last inequality is variational. A uniqueness of its solution yields the equality £ = £r*. To complete the proof for the first case, it suffices to show the strong convergence £n — £r* in H(07). By substituting x = 2£r and x = 0 into the variational inequalities (7) for r G [ro, R], we infer that

£r G Kr, B(07,£r>£r) = y F£rdx Vr G [ro,R]. (17)

Making use of (7), one can derive

£r G Kr, B(07,£r,x) >y Fx dx Vx G Kr, (18)

Oy

which hold for all r G [ro, R]. The equalities (17), together with the weak convergence £n — £r* in H(07) as n — to, imply that

lim B(07,£n,£n)= lim / F£n dx = F£r* dx = B(07, £r* ,£r*).

n—n—/ /

Since we have the equivalence of norms (see Remark 1 in Section 2), one can see that £n — £r* strongly in H(07) as n — to. Thus, in the first case we have obtained a contradiction to the assumption: ||£n - £r* || > e for all n G N.

We then proceed to the second case. For convenience we keep the same notation for the subsequence. In doing so, we have rn — r* and rn > r*. Taking into account the results at the beginning of the proof, we recall that £n — £ weakly in H(07) as

n — to. Therefore, making use of the first relation in (17), we get

lim ,fn,fn)= / Ff dx. (19)

Oy

Next, substituting x = fr' G Kr' C Kr, for arbitrary fixed numbers r, r' G (ro, R] such that r' > r, in (18) as a test function, we arrive at the inequality

B(Oy,fr,fr) > J Ffr dx.

Oy

We therefore conclude that for all rn and rm satisfying rn < rm the following inequality is fulfilled

B(07,fn ,fm) >J Ffm dx. (20)

Oy

Then, we fix an arbitrary value to in (20) and pass to the limit in the last relation as n — to, which provides

B(07 ,f,fm) ^ Ffm dx. (21)

Oy

Passing to the limit in (21) as m — to, yields

B(07> y Ff dx.

Oy

This inequality, the formula (19) and the weak lower semicontinuity of the bilinear form •) provide the following chain of relations

B(07,f,f) > f Ffdx = lim B(07,fn,fn) > B(07,f,f)

J n—w

Oy

indicating that

B(07,f,f)= lim B(07,£„,£„).

n—

Again, by the equivalence of norms (see Remark 1) that fn — f strongly in H(07) as n — to.

Now, let us prove that f = fr*. For this purpose we will analyze the variational inequality (7) and its limiting case. We can now apply the assertion of Lemma 1 to justify a passage to a limit in the variational inequalities. From Lemma 1, for any X G Kr* there exist a subsequence {r^} = {rnk} C {rn} and a sequence of functions {xk} such that xk G Krk and xk — X weakly in H(07) as k — to.

The properties established above for the convergent sequences {xk} and {fn} allow us to pass to the limit as k — to through following inequalities, derived from

(7) for rk and with the test functions

, £k,nk - £k) >/ F(nk - £k) dx.

As a result, we have

B(07,|,n - I) >y F(n - £) dx V n G Kr*.

The unique solvability of this variational inequality implies that £ = £r*.

Therefore, in either case, there exists a subsequence {rnk} C {rn} such that rk ^ r*, £k ^ £r* strongly in H(07), which is a contradiction. The Lemma is thus proved.

5. Conclusion

In this paper, we have analyzed a family of variational problems describing equilibrium of cracked plates with cylindrical inclusions of different radiuses r G [ro,R]. The existence of the solution to the optimal control problem (8) is proved. For that problem the cost functional J(r) is defined by an arbitrary continuous functional G : H(07 ) ^ R, while the radius r of the cylindrical rigid inclusion is chosen as the control parameter. Lemmas 1 and 2 establish a qualitative connection between the equilibrium problems for plates with rigid cylindrical inclusions of varying radiuses. These lemmas allow us to prove the strong convergence £r ^ £r* in the Sobolev space H(07), where {£r} are the solutions of (4) depending on the radius r. In the framework of developed methods the various cases of rigid inclusion shapes can be considered.

We have to mention that the problem of finding an explicit expression for the solution r* to (8) is quite difficult. Authors have no idea about reasonings that could help to solve this problem. The reasonable improvement of this result, in particular a determination of whether the solution r* is unique, poses an open question for future research.

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Submitted February 20, 2019 Revised February 28, 2019 Accepted March 1, 2019

Nyurgun P. Lazarev

M. K. Ammosov North-Eastern Federal University, 48 Kulakovsky Street, Yakutsk 677000, Russia nyurgunSngs.ru

Atusi Tani

Keio University, Department of Mathematics, 3-14-1 Hiyoshi, Yokohama, 223-8522, Japan [email protected] Petr V. Sivtsev

M. K. Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677000, Russia [email protected]