Korn inequalities for shells with zero Gaussian curvature
We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.
Classical first and second Korn inequalities have been known to play a central role in the theory of linear elasticity and recently they have found very important applications in the problems of buckling of slender structures [6, 4, 5]. Let us recall the classical first and second Kohn inequalities, that actually date back to 1908, [8, 9]. To that end we denote
be the set of all infinitesimal motions, i.e., a Lie algebra of the group of all Euclidean transformations (rigid body motions). Let be an open connected subset of and . We denote333We reserve more streamlined notations and for “simplified” gradient and symmetrized gradient, respectively, that will be our main characters in the technical part of the paper. by and the gradient and the symmetric part of the gradient, respectively, of a vector field . It is well-known that in (in the sense of distributions) if and only if . This is an immediate consequence of a simple observation (also very well-known) that all partial derivatives of the gradient can be expressed as a linear combination of partial derivatives of the symmetric part of the gradient :
The classical first Korn inequality (e.g., as stated in ) quantifies this result by describing how large must be if lies in a closed subspace that has trivial intersection with . If is a Lipschitz domain, then there exists a constant , such that for every
where is the -norm. The Korn constant measures the distance between the subspace and . The classical second Korn inequality asserts that the standard norm topology can be equivalently defined by replacing with :
Originally, Korn inequalities were used to prove existence, uniqueness and well-posedness of boundary value problems of linear elasticity (see e.g., [11, 1]). Nowadays, often, as in our particular case, it is the best Korn constant in the first Korn inequality that is of central importance (e.g., [2, 12, 13, 15, 16, 10]). Specifically, we are interested in the asymptotic behavior of the Korn constant for shells with zero Gaussian curvature as a function of their thickness for subspaces of functions satisfying various boundary conditions at the thin edges of the shell. In [6, 5] we have shown that represents an absolute lower bound on safe loads for any slender structure. For a classical circular cylindrical shell we have proved in  that for a broad class of boundary conditions at the thin edges of the shell.
The motivation for this work comes from the fact that the experimentally measured buckling loads of axially compressed cylindrical shells behave in a paradoxical way, dramatically disagreeing with predictions of classical shell theory. The universal consensus is that such behavior is due to the extreme sensitivity of shells to imperfections of shape and load. This study is a part of rigorous analytical investigation of the influence of small changes in shape on the structural behavior of cylindrically-shaped shells. It looks like (and this will be addressed in future work) the determining factor of the effect of shape imperfections is the Gaussian curvature of the actual imperfect shell as the Ansatzen suggest in . In this paper we show that if the shell has a vanishing principal curvature (yielding zero Gaussian curvature), like circular cylindrical shells, then the scaling of the Korn constant will remain unaffected, provided the non-zero principal curvature has a constant sign. Our analysis also shows that if both principal curvatures are zero on any open subset of the shell’s mid-surface, then . We conjecture that for shells of uniformly positive Gaussian curvature, while if the Gaussian curvature is negative on any open subset of the shell’s middle surface, as suggested by test functions constructed in . These conjectures will be addressed elsewhere.
The goal of this paper is to show that the tools developed in  for circular cylindrical shells, and extended and developed further in , possess enough flexibility to be applicable to a wide family of shells and in particular cylindrically-shaped shells (the ones, that have no boundary in one of the principal directions). The main idea is to first prove an inequality that is a hybrid between the first and second Korn inequalities (we call it “first-and-a-half Korn inequality” for this reason) by “assembling” it from its two-dimensional versions corresponding to cross-sections of the shell by curvilinear coordinate surfaces. The first Korn inequality is then a consequence of the first-and-a-half Korn inequality and an estimate on the normal component of . We believe that this general methodology will work for broad classes of shells, even though the Gaussian curvature does affect the validity of some of the technical steps in the proof, which must be adjusted to the particular case. In particular, the assumption of zero Gaussian curvature is essential for all main results in Section 3.
Consider a shell whose mid-surface is of class Suppose and are coordinates on the mid-surface of the shell, such that constant and constant are the lines of principal curvatures. Here will denote the circumferential coordinate and –the longitudinal for cylindrically shaped shells. In the case of a straight circular cylinder and are the standard cylindrical coordinates. Let be the position vector of the shell’s mid-surface. Introducing the normal coordinate , we obtain the set of orthogonal curvilinear coordinates , related to Cartesian coordinates via
where is the outward unit normal, and is the position vector of a point in space with coordinates . In this paper we will study shells of uniform thickness , given in coordinates by
the two nonzero components444The principal directions are mutually orthogonal. of the metric tensor of the middle surface. The two principal curvatures will be denoted by and . Their signs are chosen in such a way that and are positive for a barrel-shaped shells, like a sphere. The four functions , , , and satisfy the Codazzi-Gauss relations (see e.g. [17, Section 1.1])
and define the Levi-Civita connection on the middle surface of the shell via the following derivation formulas
Using these formulas we can compute the components of in the orthonormal basis , , :
We require that , and be strictly positive functions of their variables on the mid-surface of the shell. Hence, for shells of zero Gaussian curvature the formula for simplifies:
In the case of shells the thickness variable is uniformly small. We therefore introduce the simplified gradient
We note in (2.6) the components , and are still functions of .
To be more specific we give two examples of zero Gaussian curvature shells: cylinders and cones. A cylinder is described by a simple, smooth closed curve of length in the -plane. Let , be the position vector of this curve, parametrized by its arc-length. The position vector of the middle surface of the shell is then given by , where is the unit vector perpendicular to the =plane, i.e. the unit vector in the -direction. It is easy to verify that constant and constant are lines of curvature and is the curvature of curve in the plane, whose sign is chosen to be positive for a circle.
A second example is a cone with vertex at the origin. A cone is described by a a simple, smooth closed curve of length lying in the northern hemisphere of a unit sphere centered at the origin. Let , be the arc-length parametrization of this curve. In this case the middle surface of the shell is given by . Once again, it is easy to verify that constant and constant are lines of curvature and
where is the triple-product of 3 vectors in space. We summarize the data for cylinders and cones in Table 1.
In this paper all norms are norms. However because of the curvilinear coordinates we will use several different flavors of the inner product and the corresponding norm. For we define the inner product
which gives rise to the norm .
In cross-sections =constant we use
We will also use the Euclidean version of the norm on cross-sections
where and , , are the corresponding limits of integration. In each case it will be clear which variable , or plays the role of the fixed variable . Of course, due to uniform positivity of and the norms and are obviously equivalent. In particular, all inequalities involving one type of norm will also be valid for another. Finally, all constants that are independent of and will be denoted by . Once this is understood, such abuse of notation does not lead to any ambiguity.
3 Main results
We formulate our Korn inequalities for vector fields satisfying specific boundary conditions at the two edges of the shell. We define
We state our main results as a sequence of related theorems.
Suppose on . Then there exist a constant , independent of , such that for every
for all .
If additionally, the curvature vanishes on an open subset of the middle surface of the shell then, according to Theorem 3.3 the bound in (3.4) is asymptotically sharp as . If the curvature does not change sign (i.e. stays uniformly positive for cylindrically-shaped shells), then the first Korn inequality (3.4) can be improved.
Suppose that and on . Then there exist a constant , independent of , such that for every inequalities (3.3) and
hold for all .
Theorem 3.3 (Existence of optimal ansatz).
Suppose that assumptions of Theorem 3.1 are satisfied. Suppose, additionally that the curvature vanishes on an open subset of the middle surface of the shell. Then there exist functions and a constant , independent of , for which
Suppose that assumptions of Theorem 3.2 are satisfied. Then there exist functions and a constant , independent of , for which
Our results are formulated for shells cut along the coordinate surfaces. However, they are also valid for any shell , bounded by the surfaces , where the spaces , are defined by (3.1), (3.2), respectively, except the indicated components of vanish on the surfaces , instead of . This is because there exists shells , such that the shells are bounded by surfaces constant. But then the ansatz from Theorem 3.3 supported in gives an upper bound on the Korn constant of that scales as (or as ). At the same time every function in or of can be extended (by extending the relevant components of by zero) to a function in or of , giving the lower bound on the Korn constant that scales as (or as ).
Note, that the periodicity condition in the direction in (3.1)-(3.2) will be automatically satisfied for cylindrically-shaped shells. However, it is easy to check, that our main results hold (the proofs below go through) for different class or displacements satisfying Robin boundary conditions on the thin faces of the shell. For instance, one can assume the condition
on the thin edges of the shell in the direction instead of the perioditicy of the and the components in (3.1) and (3.2). Note, also, that the boundary condition (3.8) is of Dirichlet type, but it can be rewritten as a Neumann condition on the thin edges of the shell in the direction.
Our the strategy is to prove first-and-a-half Korn inequality for the simplified version of , given by (2.6)
We then show that (4.1) implies (3.3). In order to prove (4.1) we apply the method, introduced in , of assembling (4.1) from the analogous two-dimensional inequalities corresponding to the three coordinate surface cross-sections of the shell. Most of the proof is done under the common assumptions of Theorems 3.1 and 3.2. In other words, we assume that and we do not make any assumptions on the sign of , until we say otherwise.
4.1 The cross-section
The Korn-type inequality corresponding to cross-section involves , , and components of the gradient. The first-and-a-half Korn inequality in this case is stated in the following lemma.
we conclude that it is sufficient to prove
We have, that
since at and in both and .
Let us estimate . The idea is to observe that in both and
and then express and in terms of and , respectively. Thus,
Applying the Schwartz inequality we obtain
By the Poincaré inequality
4.2 The cross-section
Let us show that Lemma 4.2 is an immediate consequence of the same two-dimensional inequality in Cartesian coordinates, proved in [3, Theorem 3.1]. It states that if , satisfies , in the sense of traces, then
Proof of Lemma 4.2.
We first prove inequality (4.7) for each fixed :
which becomes equality if . Let
Then, by (4.8), applied to , we obtain
By uniform positivity and boundedness of , norms and are equivalent. Hence,
Applying (4.10), we prove the lemma, if we show that
By the Poincaré inequality , so that
It remains to observe that
4.3 The cross-section
As before, we will show that that (4.12) is a consequence of a two-dimensional Korn-type inequality. However, before we can proceed with this strategy, we observe that the term with in the -component of can be easily discarded due to the Poincaré inequality (4.6). Indeed, suppose we have proved (4.12), where is replaced with
Next we prove the two-dimensional Korn-type inequality
This lemma is proved in Section 5. If we apply this lemma to (for each fixed value of ) and then integrate over , we obtain the inequality (taking into account the equivalence between the curvilinear norm (2.7) and the Euclidean norm)
By the Poincaré inequality and (4.2) we obtain
If then we can just use the Poincaré inequality and (4.7)
Under the assumptions of Theorem 3.2 the Poincaré inequality cannot be used. Instead we estimate expressing in terms of :
Multiplying both sides of (4.17) by and integrating by parts (without writing integrals), we obtain
Finally, replacing in the formula above by
and using periodicity in we obtain the estimate, taking into account the equivalence of and norms
Hence, using the inequality (several times) and replacing by its estimate from (4.15), we obtain the bound.
4.4 Conclusion of the proof
Combining the estimates (4.2), (4.7) and (4.12) we arrive at (4.1). However, and are the simplified versions of and . Thus, we need to show that (3.3) follows from (4.1). Under the assumptions of Theorems 3.1 it is a consequence of (4.16), while under the assumptions of Theorems 3.2 it is a consequence of (4.18).
The main observation in either case is that components of and are multiples of one another with coefficients that are independent of . Thus, by direct calculation, we estimate
from which we get additionally,