A Justification of Two-Dimensional Nonlinear Viscoelastic Shells Model
By applying formal asymptotic analysis and Laplace transformation, we obtain two-dimensional nonlinear viscoelastic shells model satisfied by the leading term of asymptotic expansion of the solution to the three-dimensional equations.
In the case of pure nonlinear elasticity, Ciarlet and his collaborators have studied membrane shells, flexural shell and Koiter shell (see  and the references therein). The linear viscoelasticity was studied in [2–5], and Li [6–8] studied the global existence and uniqueness of weak solution, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Kármán shallow shells. Xiao studied the time-dependent nonlinear elastic shells by the method of asymptotic analysis (see ).
Motivated by the above work, we deal with nonlinear viscoelastic shells and give the identification of two-dimensional variation problem satisfied by the leading term of of asymptotic expansion of the solution to the three-dimensional equations. The main contributions of this paper are the following: (a) the problem considered in this paper is nonlinear viscoelastic shells, to our knowledge this model has not been considered; (b) applying Laplace transformation, we overcome the difficulties caused by the integral term in the model; (c) the calculation and derivation are precise.
2. Preliminaries and Main Results
We use the following conventions and notations throughout this work: Greek indices and exponents (except ) belong to the set , Latin indices and exponents (except when otherwise indicated, as, e.g., when they are used to index sequences) belong to the set , and the summation convention with respect to the repeated indices and exponents is systematically used. The sign indicates that the right-hand side defines the left-hand side.
Let be a bounded connected open set with a Lipschitz boundary , let denote a generic point in the set , and let . Let be an injective mapping of such that the two vectors are linear independent at all points . They form the covariant basis of the tangent plane to the surface at the point ; the two vectors of the same tangent plane defined by the relations constitute its contravariant basis. We also define the unit vector which is normal to the at the point .
One then defines the first fundamental form, also known as metric tensor or ), the second fundamental form, also known as the curvature tensor or , and the Christoffel symbols of the surface by setting (whenever no confusion should arise, we henceforth drop the explicit dependence on the variable ) Note the symmetries , and . The area element along is , where . All the functions defined above are at least continuous over the set . In particular, there exists a constant such that .
In addition, let the covariant derivatives and the covariant components of the third form of the surface be defined by
For each , we consider a shell with thickness and middle surface , whose lamé relaxation modules and are independent of . We define the sets where and . Note that constitutes a partition of the boundary of the set . Let denote a generic point in the set , and let ; hence and .
We then define a mapping by then there exists such that for all the mapping is an injective mapping and the three vectors are linear independent for each . The injectivity of the mapping ensures that the physical problem described below is meaningful.
The three vectors form the covariant basis at the point , and the three vectors defined by form the contravariant. We define the metric tensor or and the Christoffel symbols of the manifold by setting (we omit the explicit dependence on ) Note the symmetries The volume element in the set is , where .
For each , the set is the reference configuration of a viscoelastic shell with middle surface and thickness . We assume that the material constituting the shell is homogeneous isotropic and is of a nature state, so that the material is characterized by its two lamé relaxation modules and . Under the action of forces, the shell undergoes a displacement field.
Let in terms of curvilinear coordinates of the reference configuration . Then, the covariant displacement field satisfies the following three-dimensional equations (c.f. [1, 10]): where the symbol denotes the subspace of such that there exists a constant such that the functions vanish as . And, designate the contravariant components of the three-dimensional elasticity tensor, designate the strains in the curvilinear coordinates associated with an arbitrary displacement field of the manifold , and, finally, and denote the contravariant components of the applied body and surface force densities, respectively, applied to the interior of the shell and to its “uper” and “lower” faces and , and designate the area element along . We thus assume that there are no surface forces applied to the portion of the lateral face of the shell.
We record in passing the symmetries and the relation
Our final objective consists in showing, by means of the method of formal asymptotic expansions that, if the data are of an appropriate order with respect to as , the above three-dimensional problems are “asymptotically equivalent” to a “two-dimensional problem posed over the middle surface of the shell.” This means that the new unknown should be , where are the covariant components of the displacement of the middle surface . In other words, is the displacement of the point .
“Asymptotic analysis” means that our objective is to study the behavior of the displacement field as , an endeavour that will be a behavior as of the covariant components of the displacement field, that is, the behavior of the unknown of the three-dimensional shell problem.
Since these fields are defined on sets that themselves vary with , our first task naturally consists in transforming the three-dimensional problems into problems posed over a set that does not depend on .
Furthermore, we transform problem (2.7) into an equivalent problem independent of , posed over the domain.
Let , and , and let denote a generic point in . With each point , we associate the point through the bijection ; we thus have and . Let and the vector fields appearing in the three-dimensional problem (2.7) be associated with the functions and the scaled vector fields defined by Functions and are defined by setting Then the scaled unknown defined above satisfies (c.f. ) where
The functions are called the contravariant components of the scaled three-dimensional elasticity tensor of the shell. The functions are called the scaled strains in the curvilinear coordinates because they satisfy Note that the above definitions likewise imply that
For notational brevity, the point of some functions is suppressed where no confusion can arise.
The following two requirements constantly guide the procedures of the formal asymptotic analysis. The first requirement asserts that no restriction should be imposed on the applied forces entering the right-hand side of the equations used for determining the leading term. The second requirement asserts that, by retaining only the linear terms in any relation satisfied by terms of arbitrary order in the formal asymptotic expansion of the scaled unknown , a relation of linear theory should be recovered. For brevity, we will call it “linearization trick” (see ).
Theorem 2.1. Assume that the scaled unknown satisfying problem (2.15) admits a formal asymptotic expansion of the form
with and . Then in order that no restriction be put on the applied forces and that the linearization be satisfied, the components of the applied forces must be of the form
where the functions and are independent of .
This being the case, the leading term is independent of the transverse variable and satisfies the following two-dimensional variation problem: where (recall that : denote Laplace transformation of , respectively, and denotes the inverse Laplace transformation.
Lemma 2.2. For small it is not difficult to verify the following relations: where where where
Lemma 2.3 (see ). Let be a domain in , and let be an injective mapping such that the two vectors are linear independent at all points of . The derivatives of the vectors of the covariant and contravariant basis are given by the formulas of Gauss and Weingarten
Lemma 2.4. Let be a function such that for all satisfing on . Then, .
Proof. Thanks to Theorem 3.4-1 in .
Lemma 2.5. Assume that the scaled unknown satisfying (2.15) admits for each a formal asymptotic expansion of the form with for some integer . Then, .
Proof. The proof is broken into seven parts. Before beginning the proper induction in (iv), we record several useful preliminaries.
(i) Let the functions be defined as in Lemma 2.2. Then, for any symmetric matrices and ,
This formula, which immediately follows from the definitions, will be constantly put to use in the ensuing arguments.
(ii) Let . Then, for any and any matrix , Given any and any matrix , let and let denote the th Cartesian component of the vector . We thus have Hence, and for the three vectors are linear independent.
(iii) Assume that the formal asymptotic expansion of the scaled unknown is of the form with and .
Together with the asymptotic behavior of the functions and as , such an expansion induces specific formal asymptotic expansions of the various functions appearing in the formulation of problem (2.15) where, by definition, , and designate for each the coefficient of in the induced expansions of , and .
Note in passing that, while the functions factorizing the powers of are by definition independent of , they are dependent on one or several terms . In this respect, particular caution should be exercised as regards this dependence. For instance, that is, the factor of in depends on but the one in depends also on .
Likewise, it should be remembered that the expression of some factor may differ according to which value of is considered, for instance, where
We are now in a position to start the cancellation of the factors of the successive powers of found in the variational equations of problem (2.15) when is replaced by its formal expansion. In what follows, designates for any integer the linear form defined by
(iv) Assume that . Since the lowest power of in the left-hand side is , we are naturally led to first try
Comparing the coefficients of in (2.15) and using Lemma 2.2 and (2.36), we get the equations for all . Since we must have for all that are independent of . Consequently, the first requirement (that there be no restriction on the applied forces) implies that we must let
By recalling (2.42)–(2.45), we have that is, Therefore, which implies
Letting in (2.49) shows that
Since the symmetric is positive definite, we conclude that that is, is independent of . Inserting (2.51) into (2.43) yields A usual, any function defined on that is independent of is identified with a function defined on , and (2.36) and (2.52) imply
Noting (2.36) and (2.51), we also have Since (the leading term in the formal expansion of is order of ) and (since , each factor of in the expansion of vanishes because it contains some derivative and the leading term in the expansion of is of order strictly higher than , our next try is thus Comparing the coefficient of in (2.15) then yields equations (the functions are defined in Lemma 2.2) for all . But since (2.55), we must let and (first requirement) and accordingly try In which case the cancellation of the coefficient of in (2.15) yields the equations for all . But since (2.54), we must let and (first requirement).
(v) Assume that . Our next try being thus the cancellation of the coefficient of in the variational equations of problem (2.15) then yields the equations for all , where the functions being those defined in (2.36).
Letting be independent of then shows that we must let and ; hence, for all . Let the field be defined for all by Then, because both and are assumed to be in the space .
Furthermore, , so that
Using Lemma 2.2 and (2.65), we get
Since (by (ii)) (the matrix is positive definite), in a similar way as in (iv), we can obtain from (2.63) that for all .
Letting in (2.69) and noting (2.66)–(2.68), we conclude that hence (the matrix is positive definite)
In particular then, (by (2.62)) and thus (the matrix is positive definite)
(vi) Assume that (the case is considered separately, c.f. (viii)). Our next try being thus the cancellation of the coefficient of in the variational equations of problem (2.15) then yields the equations (note that two terms are needed here from the expansions of the functions , c.f. Lemma 2.2)) for all , where (by (2.62) and (2.72)) the last expression of being valid only if (the expressions of are not needed since by Lemma 2.2 and by (2.71)).
Noting that if , we thus conclude that the variational equations (2.74) reduce to for all that are independent of .
Since each term in the sum is cubic with respect to the functions , the linearization trick (second requirement) implies that for all that are independent of . Hence, we must let and . Hence
In a similar way as in (iv), we can obtain from (2.77) that