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PDF | The construction of solutions of singularly perturbed systems of equations and boundary value problems that are characteristic for the mechanics of.
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Asymptotic methods in mechanics of solids

Problem URL. Describe the connection issue. SearchWorks Catalog Stanford Libraries. Asymptotic methods in mechanics of solids. Responsibility Svetlana M. Bauer, Sergei B. Filippov, Andrei L. Smirnov, Petr E. Tovstik, Remi Vaillancourt. Series International series of numerical mathematics ; v. Online Available online. SpringerLink Full view. Science Library Li and Ma. More options. Find it at other libraries via WorldCat Limited preview. Contributor Bauer, Svetlana M. Filippov, Sergei B. Smirnov, Andrei L.

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Cheng et al. The present study uses the procedure of asymptotic splitting Eliseev, , , which has already been systematically applied to the theories of thin rods with a non-homogeneous cross-section Yeliseyev and Orlov, , of thin-walled rods of open profile Eliseev, ; Vetyukov, , and of homogeneous thin plates Eliseev, In comparison to the above mentioned works featuring formal asymptotic expansions, the advantages are the following:.

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An especially strong point of the present study is the analysis of the edge layer. Matching two asymptotic expansions, derived for the inner part of the plate and near its contour, we arrive at four classical boundary conditions. The theoretical basis of the paper is particularly important for modeling and development of smart structures, which feature piezoelectric sensors and actuators, see e.

Reddy , Nader To this end the present analysis is accomplished by accounting for the piezoelectric effect in the material of the structure, which results in a plate theory with additional field variables, namely voltage and free charge on the surface. The closed system of equations includes additional relations for those variables, i.

Application of the results of the analysis in comparison with a numerical three-dimensional solution is demonstrated on a sample problem, which experimentally proves the asymptotic nature of the presented theory. A plate is a three-dimensional body with the position vector of a point. Due to the structure of our problem, it appears to be convenient to separate the in-plane and out-of-plane parts of vectors and tensors. Now the magnitudes of z and x in 9 have formally the same order. Another possible argumentation to 10 is that we seek for solutions, which vary in the plane much slower than over the thickness i.

We seek for the unknown field of stresses in the form of a power series in the small parameter:. It means that we are interested rather in the convergence of the solution to the principal terms than in the convergence of the series itself. The simple advantage of the dimensionless formal small parameter in comparison to e. The formal mathematical proof of the approach is yet incomplete in the sense that we do not show the existence of the solution in the form However, already published results indicate its validity; for an additional discussion see Yeliseyev and Orlov , Vetyukov Now we can write the balance equations with the small parameter.

Using 10 in 2 , we obtain. Substituting the series expansion 11 in 12 , we begin the asymptotic procedure. The results are:. Now we proceed to the next term, taking 13 into account. The standard notation for the mean value over the thickness is introduced in 15 and used in the following.

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In order to obtain the second classical equation of balance of moments, the third step of the asymptotic procedure is needed. The form of the vector of transversal forces Q and the corresponding balance equation are in correspondence with the classical plate theory.

We proceed by using 14 :. Classically e. We use the condition 6 and the differential operator in the form Balancing the in-plane and out-of-plane components of the resulting equation, we arrive at. Due to the linearity of the relation between strains and stresses, the field of strains must have the same asymptotic behavior with respect to the formal small parameter:. From the first relation in 21 we immediately conclude, that the plane part of the principal term of the strain tensor is linearly distributed over the thickness:. In the following they will be related to the corresponding classical plate strain measures, namely to the curvature and to the reference surface strain.

The second and the third equation in 21 lead to additional conditions for the newly introduced plate strain measures:. The first relation here is the condition of compatibility for the bending strain measure, which is in correspondence with the more general compatibility condition for curved shells Eliseev, ; Vetyukov, The second equality is a well-known condition of compatibility for the plane problem of elasticity. The out-of-plane components of the principal term of the strain tensor, which are coupled with the minor terms in the series expansion 22 , will be determined in the following section from the results of the first step of the procedure for stresses.

Deriving elastic relations in terms of the theory of plates can be done in a straightforward manner, which is very much similar to the existing approximation techniques. To this end we consider the elastic relation 4 , written for the principal terms, together with the conditions Solving this linear system we obtain the out-of-plane components of the strain tensor and the in-plane stresses:.

Integrating over the thickness according to 15 and 19 , we arrive at the linear relations. Krommer The constitutive relation has the following structure:. From 25 it then follows. The elastic relations 26 complete the system of equations of statics for a plate, the other two groups of relations being the balance Eqs.

The picture is however yet incomplete as long as the field of displacements and the boundary conditions in terms of the plate theory are not analyzed. We study the asymptotic behavior of the displacement vector u 3 in the three-dimensional problem. The consequences of the kinematic relation 3 for the given asymptotic expansion of the strain tensor 22 , 23 will be analyzed.

Asymptotic methods in mechanics of solids

We rewrite 3 in components with the small parameter:. Substituting 22 , 23 and 30 in 29 , we begin the asymptotic procedure. The leading term in the series expansion 30 is a function of the in-plane coordinates. Together with the third equality this results in.

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The in-plane displacement of the plate model u x is asymptotically smaller than the lateral displacement w. The last step is the most important one as it produces actual kinematic relations for the plate model. Only the first of the kinematic relations 29 was applied at the third step. The other two also have a non-zero right-hand side, and it is the role of the lower order terms starting with u 2 to fulfill these conditions. For the consistency of the theory it is important that all three-dimensional equations will be satisfied, but the exact look of the lower order correction terms is not of interest from the asymptotic point of view.

It is therefore necessary to consider another expansion of the solution of the three-dimensional problem, which is valid in a thin domain close to the contour of the plate and which allows satisfying the boundary conditions. In the literature on solid mechanics this small domain is traditionally called edge layer Nayfeh, ; Dauge and Gruais, , and the expansion inside this layer is called inner expansion.

Both expansions represent the same solution of the three-dimensional problem, but in different regions of the plate. Therefore the inner expansion must turn into the outer one as we move away from the contour. A special procedure, which is called matching, is intended to provide such equivalence of expansions Nayfeh, Matching allows to completely determine the inner expansion and, what is more important, to provide boundary conditions for the outer one.

In the following we demonstrate the method of matched expansions for the case of a straight edge of the plate. Static boundary conditions for the outer expansion are obtained, which can directly be interpreted in terms of the plate theory. The three-dimensional solution is assumed to be changing rapidly over the thickness of the plate and in the direction of the normal n inside the domain.

Strain energy and strain energy density introduced

Relations 2 and 6 remain valid and we again use expansions 11 and The static boundary conditions on the edge of the plate are the following:. It allows to write 2 and 6 in the following way:. For the sake of clarity let us now restrict the consideration to a plate with isotropic homogenous material properties.

Asymptotic Methods in Mechanics

We write the material constitutive law in the following form:. For the components 39 this material law can be written in the following form:. The complete system of equations consists of 40 , 41 , 43 together with the boundary conditions on the upper and lower surfaces 2 and on the edge Using the asymptotic expansions 11 and 22 , we obtain a set of problems for each power of the small parameter. Goldenveizer, The first one is the antiplane strain problem:. The problems are nevertheless uncoupled and can be solved independently in the considered case of the material isotropy.

The general solution of 44 is.