Gaussian curvature as an identifier of shell rigidity
In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn’s first (linear geometric rigidity estimate) and second inequalities on that kind of shells for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like and if the Gaussian curvature is negative, then the Korn constant scales like where is the thickness of the shell. These results have classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke, James and Müller for plates [References] (where they show that the Korn constant in the nonlinear Korn’s first inequality scales like ), extended to shells with nonzero curvature. We also recover the uniform Korn-Poincaré inequality proven for "boundary-less" shells by Lewicka and Müller in [References] in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents and in the present work appear for the first time in any sharp geometric rigidity estimate.
It is known, that the rigidity of a shell under compression is closely related to the optimal Korn’s constant in the nonlinear (in many cases linear) first Korn’s inequality-a geometric rigidity estimate for fields under the appropriate Robin type boundary conditions (or without them [References]) arising from the nature of the compression, e.g., [References,References]. In their celebrated work, Friesecke, James and Müller [References,References] derived a geometric rigidity estimate for plates, which gave rise to derivation of a hierarchy of plate theories for different scaling regimes of the elastic energy depending on the thickness of the plate [References]. This type of theories have been derived by Gamma-convergence and rely on -compactness arguments and of course the underlying nonlinear Korn’s inequality, which plays the main role. While the rigidity, and in particular buckling of plates has been understood almost completely [References,References,References], the rigidity and buckling of shells is less well understood. To be more precise, to our best knowledge only low energy scalings like where (which roughly speaking corresponds to bending, [References,References]), or very high energy profiles (which corresponds to stretching, where the limiting energy completely eliminates bending and captures only stretching of the shell) have been studied [References,References]. We also refer to the papers [References,References,References,References,References,References,References, References,References] for some results on shell deformation and theories. An interested reader can also check the work of Ciarlet for linearized rod, plate, shell and thin structures with junctions theories [References,References,References,References,References,References,References]. The gap in nonlinear shell theories is due to the lack of sharp111Although the same inequality (1.1) has been proven in [References], it is not sharp for shells with a uniformly nonzero principal curvature as the present work and [References] show. rigidity estimates for shells and -compactness. In particular, for cylindrical shells the sequence of buckling modes has the only weak limit zero as understood by Grabovsky and Harutyunyan in [References], thus one has to look for a Young measure as the limit to capture the oscillations developed by the deformation sequences. In a series of papers [References,References,References], Grabovsky and Harutyunyan studied the buckling of cylindrical shells under axial compression, where it became clear that even for simple geometries the problem may be unexpectedly rich and complex. It has been understood by Grabovsky and Truskinovsky in [References], see also [References], that in the case of shells, or simply any thin structures, when there is enough rigidity then actually the linear Korn’s first inequality can replace the nonlinear one in the buckling problem. Let us recall some of the known results to make things clear. The rigidity estimate of Friesecke, James and Müller mentioned above asserts the following: Assume is a bounded Lipschitz domain and let be a plate with base and thickness Then there exists a constant such that for every vector field there exists a constant rotation , such that
The linear Korn’s first inequality introduced by Korn [References,References] and proven by many different authors in different settings [References,References,References,References,References,References,References] reads a follows: Assume is a bounded Lipschitz domain and the closed subspace does not contain a rigid body motion with Then, there exists a constant such that for every vector field there holds:
where is the symmetrized gradient (the strain in linear elasticity). The subspace is identified by the boundary conditions satisfied by the displacement coming from the nature of the problem under consideration. It is known that then the dependence of the optimal constant called Korn’s constant, on the geometric parameters of the domain is crucial in the study of the buckling problem [References,References], namely the formula (1.4) below for the buckling load has been proven in [References]. Recall, that if the body is subject to dead load then the total energy of the deformation is given by
where is the elastic energy density of the body In their theory of buckling of thin structures, Grabovsky and Truskinovsky [References] assume that the thin domain is subject to the dead load where is the magnitude of the load. Then they introduce the Rayleigh quotient
which was studied further by Grabovsky and Harutyunyan in [References]. Here, is the linear elasticity matrix, is the Piola-Kirchhoff stress tensor and is the linear elastic strain. Define furthermore the constitutively linearized buckling load [References] as
where here is the closed subspace of identified by the boundary conditions fulfilled by the displacement Then the following theorem has been proven in [References]:
Theorem 1.1 (The critical load).
Assume that the quantity defined in (1.4) satisfies for all sufficiently small and
where is the Korn’s constant of the domain associated to the subspace Then is the critical buckling load and the variational problem (1.3) captures the buckling modes (deformations) too (see [References] for a precise definition of buckling modes).
As long as the task is to solve the variational problem (1.4), one clearly has to calculate the Korn’s constant as the numerator and the denominator in (1.3) are quadratic forms in the strain and the gradient respectively. All the Korn’s constants calculated prior to the work of Grabovsky and Harutyunyan [References], (where the authors calculate Korn’s constant for cylindrical shells), scale either like or The one for cylindrical shells calculated in [References] scales like which gave rise to the following very important question: What geometric quantities make shells more rigid than plates and how can one compare the rigidity of two different shells? It turns out, that the answer to the above question is encoded in the principal and Gaussian curvatures of the shell: the bigger the curvatures are the more rigid the shell is as will be clear from the analysis in this work. Mimicking cylindrical and conical shells, Grabovsky and Harutyunyan derived sharp Korn’s first inequalities for shells with one principal curvature vanishing and the other one having a constant sign in [References], where they prove that the Korn constant again scales like as for regular circular cylindrical shells. In this work we derive sharp Korn’s first inequalities for shells of positive or negative Gaussian curvature. It turns out, that for negative curvature the scaling of Korn’s constant is and for positive curvature it is , i.e., the zero and positive Gaussian curvature shells are the least and most rigid (under compression) ones respectively. All three exponents and are completely new in Korn’s inequalities222The exponent appeared first in the work of Grabovsky and Harutyunyan [References]. Let us mention, that a hint for the exponents and comes from the book of Tovstik and Smirnov [References], where the authors construct different Ansätze for shells, without proving any Ansatz-free lower bounds. Korn’s first inequalities mentioned above and proven in this work are the linear analogues of Friesecke, James and Müller rigidity estimate (1.1). This work opens up new ideas in the shell theory and also makes it clear that when dealing with a shell, it may be most convenient to work in the local coordinates where the principal curvatures of the shell pop up in the gradient structure of the vector field explicitly. Although it is hard to predict, we believe the analogous nonlinear estimates (with or without boundary conditions) will have the same scaling of the constant in terms of the shell thickness. That new nonlinear rigidity estimates will naturally give rise to shell theories being derived from 3-dimensional elasticity by -convergence like [References,References,References,References]. Let us now comment on the proof of our main results and also the relevant sharp rigidity estimates proven before. First of all it is worth mentioning that the only available constants (in terms of the thickness ) in the literature are and where is a constant and is the Korn constant of the domain (with or without boundary conditions), the constant in the rigidity estimate or in Korn’s first inequality. The lower bound proven for plates by Friesecke, James and Müller in [References] is universal333It is satisfied even for rods, which seem to be less rigid., in the sense that the same authors and Mora prove it for shells implicitly in [References], however, as mentioned before, it is not sharp for general shells. The main tool in their proof is the geometric rigidity estimate of Friesecke, James and Müller proven in [References], while the main technique is the localization technique, i.e., one divides the shell in many little cubes of size proves a local estimate on each of them, and sums them up, thus the curvature of the shell is not taken into account and does not play a role. This technique gives the universal lower bound and evidently can not give better such as where (also because of the simple reason of neglecting the curvature). Another interesting sharp Korn’s first inequality is proven by Lewicka and Müller [References], where the authors prove a uniform Korn-Poincaré estimate for shells that are made around a smooth boundary-less surface and for tangential vector fields with on An interesting aspect of the present work is that we also recover the Lewicka-Müller uniform Korn-Poincaré inequality, of course in the setting of uniformly positive or negative curvature444Of course there is no boundary-less surface in with uniformly negative Gaussian curvature. The statement for negative Gaussian curvature shells must be understood as for such shells with boundary (in the in-plane directions) and vector fields defined on them that satisfy zero or periodic boundary conditions in each of the principal directions (if the shell parametrization allows for talking about periodicity) on the thin faces of the shell. Roughly speaking, having no boundary is the same as having periodic boundary conditions. (see Theorem 3.2 and Remark 3.3), which seems to replace their "contraction type Schwartz inequality" hypothesis (see [References, (2.4) in Theorem 2.1]), that seems to be curvature and thickness-change related. Also, when proving a Korn inequality for a shell, then bounding the normal "out-of-plane" component is typically the most complex one, while in the presence of tangential boundary conditions it becomes trivial by the Poincaré inequality in the normal direction. Regarding our strategy of the proof, it turns out that for both positive and negative Gaussian curvatures, one can prove a Korn-Poincaré inequality that controls the norm of the displacement components in the principal "in plane" directions by the norm of the symmetrized gradient Moreover, it turns out that for positive curvatures, one can bound the normal component of the displacement by the norm of the symmetrized gradient as well. These are not possible for zero Gaussian curvature as the Ansätze in [References,References] show. Then utilizing these bounds we simplify the problem by "throwing away" the parts of the gradient that are of the order of the norm For the remaining part of the gradient we then prove a sharp first and a half Korn inequality (a Korn interpolation inequality introduced first in [References]) with a constant scaling like Finally, we combine the Korn-Poincaré inequality with the first-and-a-half Korn inequality to derive the sharp Korn’s first inequality. As already mentioned, the Ansatz for negative Gaussian curvature is due to Tovstik and Smirnov and can be found in the book [References], while the Ansatz for positive Gaussian curvature is Kirchhoff-like and is given by us. We mention in conclusion, that our analysis goes through for shells with nonconstant thickness as demonstrated in [References,References].
In this section we introduce notation and some definitions. In what follows we assume that the mid-surface of the shell under consideration is of class up to the boundary. Denote by and the coordinates on the mid-surface of the shell, such that constant and constant are the principal lines. Here, will denote the circumferential and –the longitudinal coordinates for cylindrically shaped shells and in particular, in the case of a straight circular cylinder, and are the standard cylindrical coordinates. Assume the mid-surface is given by the parametrization Then, introducing the normal coordinate , we obtain the set of orthogonal curvilinear coordinates on the entire shell given by
where is the outward unit normal. In this paper we will study shells of constant thickness around i.e., the domain
As mentioned in Remark 3.1, the present results hold for any surface with nonzero Gaussian curvature and for displacements satisfying some boundary conditions (depending on the geometry of the shell) on the thin faces of the shell However, in order to simplify the presentation of the analysis, we will assume in the sequel that the mid-surface is given by , and thus the shell is given by
i.e., we assume that the shell is cut along the principal directions of the mid-surface. Denote next
the two nonzero components of the metric tensor of the mid-surface and the two principal curvatures by and . The signs of and are choosen such that and are positive for a sphere. The four functions , , , and satisfy the Codazzi-Gauss relations (see e.g. [References,References]),
and define the Levi-Civita connection on the mid-surface of the shell via the following derivation formulas
Our convention is that inside the gradient matrix we will use for the partial derivative Given a weakly differentiable vector field , using the above formulas for the partial derivatives, we can compute the components of in the orthonormal basis , , to get
The gradient restricted to the mid-surface or the so called simplified gradient denoted by is obtained from (2.3) putting thus it has the form
The simplified gradient will be very useful in our analysis as it has the following features: it is simpler than the usual gradient and it is an approximation of to the order of due to the smallness of the variable In this paper all norms are norms and the Cartesian inner product of two functions will be given by
which gives rise to the norm . Denote by the mid-surface of the shell. The following conditions and follow from the fact that the surface is while conditions and will be assumed throughout this work.
The gradients and are bounded on i.e., for some
The gradients of the curvatures and are bounded on i.e.,
The functions and are uniformly positive and bounded on i.e., for some and This condition means that the mid-surface is non-degenerate.
The curvatures and are bounded on as follows:
To make it easier for reference, we combine all the inequalities in one equation:
3 Main results
Before formulating the results we have to set the boundary conditions the displacement is going to satisfy. We mimic the situation when the shell is a vase-like shell, i.e., it has no boundary in the principal direction and is cut at the principal lines in the principal direction. As mentioned in the introduction, when the load is applied to the top () of the shell and the bottom () is fixed, then zero boundary conditions must be imposed on some components of the displacement at and In the direction there are no boundary conditions imposed explicitly, but let us mention, that actually there are some and the condition is implicit, namely it is periodic Dirichlet boundary conditions on the faces and To keep things more general, we do not assume in this paper that the shell has no boundary in the direction, but we rather impose boundary conditions on the and thin faces of the shell too, namely we consider the following subspace of
It is also worth mentioning, that the analysis in this paper works for any shells (not necessarily cut in the principal directions) with a wide variety of boundary conditions such as Robin ones too, which can be checked easily. For instance instead of (3.1) one can assume that the conditions for all hold on the thin faces of the shell, i.e., Dirichlet boundary conditions on the in-plane components of
Next, let us mention that in what follows the constants will depend only on the quantities and i.e., the shell mid-surface parameters. As already mentioned in the abstract, we will work in the case when the Gaussian curvature has a constant sign. Our main results are sharp Korn’s first and second inequalities providing Ansatz free lower bounds for displacements satisfying the boundary conditions (3.1). We remark, that the Korn-Poincaré inequality proven in Section 4 as an auxiliary lemma can play a crucial role in the derivation of the analogous nonlinear geometric rigidity estimates in order to introduce "artificial" boundary conditions. This being said, Lemma 4.1 can be viewed as another central result of this work.
Theorem 3.2 (Korn’s second inequality).
Assume that the Gaussian curvature has a constant sign on the mid-surface , i.e., the principal curvatures and never vanish on Then there exists a constat such that Korn’s second inequality holds:
for all and
A useful remark concerning uniform Korn-Poincaré inequalities is as follows:
Remark 3.3 (Uniform Korn-Poincaré inequality).
Under the tangential boundary conditions on one of the faces the uniform Korn-Poincare inequality holds:
for all and
Theorem 3.4 (Korn’s first inequality).
Assume that the Gaussian curvature has a constant sign on the mid-surface , i.e., the principal curvatures and never vanish on Then the following are true:
If , then there exist constants such that if then
for all and
If then there exists a constant such that
for all and
Let us mention that the condition is purely of technical character and we believe that it can be removed, which will be task for future. Finally, we prove the sharpness of the estimates (3.2), (3.4) and (3.5).
4 The Korn-Poincaré inequality
One of the key estimates in the proof of Theorem 3.4 is the following Korn-Poincaré inequality for the simplified gradient.
There exists a constant such that the Korn-Poincaré inequality holds in the corresponding case:
for all and
If then there exists a constant such that if then
for all and
The Case The strategy here is to freeze the variable and to work solely with the lower right block of the symmetrized matrix We eliminate the component from the and entries of the symmetrized simplified gradient getting a suitable identity only in terms of and which will allow us to estimate the norms of and We fix any and work with the functions as functions of two variables and At the end we will integrate the obtained inequalities in to get estimates in terms of the full norm in Assume now is a smooth function depending only on yet to be chosen. We have that
thus we get subtracting
Next we have by integration by parts,
We have furthermore,
thus we get
Let us now manipulate the summand We utilize the equality,
to calculate as follows:
We have integrating by parts, that
thus we obtain from (4.6),
Finally, we get for from (4.4) the equality
where is a quadratic form in and Next, we aim to choose the function such that the form is either positive or negative definite. To that end we choose where is a big enough constant. With this choice of the coefficient of the function in the integrand in will become
hence due to the bounds (2.5), for bing enough it will have the sign of and bigger than in the absolute value uniformly on Similarly the coefficient of will have the opposite sign of and will be bigger than in the absolute value uniformly on Finally, as the function does not depend on then the coefficient of will be uniformly bounded by . We choose now the parameter such that
to ensure the estimate
which gives the desired estimate (4.1). The proof of the case is finished now.
The Case The proof of this case is more technical and consists of several steps. We again freeze the variable and work solely on the lower right block of the simplified gradient It will become clear from the proof, that basically the structure of the coefficients of the displacement components and their partial derivatives, that comes from the fact that one is dealing with a full gradient in the lower right block, does not play an important role, but we rather use only some partial information on them such as positivity and boundedness. The proof is divided into 3 steps that are done below.
Step 1: Bounding and in terms of each other. In the first step we prove that there exist two constants such that
For the proof we follow the calculation of the previous case, namely we recall the identity (4.9):
We are working in the case thus and have the same sign, which determines the sign of the coefficient of and in the integrand on the left hand side and on the right hand side of (4.13) respectively. We have by the bounds (2.5) and the Schwartz inequality, that there exists a constant such, that
We have like the previous case that,
thus in order to make the coefficients of and in (4.13) uniformly bounded from below by we have to choose where is a big enough constant. On the other hand the maximal value of must be controlled by its minimal value, so that can be eliminated from the inequality resulting from (4.13) and (4.14). We have, that
thus combining (4.13) and (4.14) we obtain the desired estimate (4.12) via the observation done in the case
Step 2: Bounding in terms of and In the second step we prove that there exists a constant such that
We aim to get an identity involving the component Namely, we have on one hand that
thus we get by the Schwartz inequality and the positivity of the Gaussian curvature that
for some constants We have on the other hand, that
Next we calculate by integration by parts,
We have again by integrating by parts,