We consider the dynamical evolution of a thin rod described by an appropriately scaled wave equation of nonlinear elasticity. Under the assumption of well-prepared initial data and external forces, we prove that a solution exists for arbitrarily large times, if the diameter of the cross section is chosen sufficiently small. The scaling regime is such that the limiting equations are linear.
In this contribution we study the existence of strong solutions to the following wave equation of nonlinear elasticity
together with suitable initial and boundary conditions, which will be specified later. Here is the reference domain of the rod with thickness proportional to h, is the deformation of the rod and describes a given external volume load with a certain dependence on . Moreover, is the cross section of the rod, which is assumed to be a smooth and bounded domain, , and is an elastic energy density, which satisfies standard assumptions on smoothness, coercivity and frame invariance, cf. Section 2.2 below for the details.
The goal of the contribution is to show existence of unique strong solutions of (1.1) for times for any and sufficiently small together with periodic boundary conditions with respect to and natural boundary conditions on and suitable (well-prepared) initial data and forces of order with . To this end a suitable scaling of time and in is needed.
In order to motivate the scaling used in the following we briefly summarize known results in the time independent situation. The starting point of the analysis is the nonlinear elastic energy
where is the deformation of the rod. In order to discuss the asymptotics for it is convenient to rescale the energy to an h-independent reference domain . Then we obtain
with , and . The limit of this energy depends on the scaling property of and thus on the scaling of the elastic energy with respect to . An in depth analysis of the convergence properties in the sense of Γ-convergence can be found in [11,12] and in case of a curved reference configuration see for instance [14,15]. In case of periodic boundary conditions we refer to [3].
In detail energies of order for correspond to being of order . The choice of and leads to a von Kármán limiting energy and a linearised theory, respectively. Deformations of this scaling behaviour are close to a rigid motion. For completeness we note that the limit energies for are derived as
where
Here , are the limits of appropriately scaled means of and is the -lower left submatrix of the limit of an approximating rotation. The matrix is given by
Moreover, is defined by
with , the quadratic form of linearised elasticity. But the precise form of the limit energy will not be needed in the following.
In our work we study the case in the time dependent situation. We note that most of the results in the following hold also true in the case . But for technical reason we can prove the main result only in the case . But we expect that a similar result holds true in the case . The basic equations from continuum mechanics emerge from the balance of linear and angular momentum and the balance of energy. Formally the evolution preserves the total energy
if is independent of t and appropriate boundary conditions are satisfied.
According to the scaling behaviour of the S-means of we expect
Moreover, the Γ-convergence results suggest . For simplicity we choose . For some results with normal and tangential forces we refer to [8]. In order to balance the kinetic and elastic part of the total energy we rescale the time via . Hence, we obtain
for the total energy. This leads to the scaled evolution equation
with , where for , and which can be obtained from (1.1) by the previous scaling in space and time. Furthermore, we assume homogeneous Neumann boundary conditions on the outer surface, periodicity on the end faces of Ω and suitable initial conditions. The main result of the contribution is that we are able to show that for well-prepared initial data and any there exists an such that strong solutions exist on for all .
For a general introduction to elasticity theory we refer to [4] and for a different approach on the behaviour of thin rods using unfolding methods see [6]. In the case of thin plates and a similar long time existence results was shown in [1]. Furthermore we want to mention the results on convergence of weak solutions in the dynamical setting for plates [2] and shells [13], where in the latter the large times existence is still an open problem to the best of the authors’ knowledge.
The strategy to prove the long time existence result is as follows: In a first step we use the methods of [7] in order to establish short time existence for , where the time of existence depends on h. As the results are proved for a sufficiently smooth bounded domain, we cannot directly apply the results in [7]. The periodic boundary conditions on the end faces of the rod and homogeneous Neumann boundary conditions on the outer surface however admit applying many arguments of the proofs. More details are given in Section 3.1 and the Appendix.
The main difficulty is therefore to prove the existence of a uniform lower bound for T with respect to . As the global strategy is quite similar to the one used in [1], we want to give a short overview on the main new difficulties of this work. Generally we use an energy method in order to obtain enough regularity bounds for the solution. With this we can then conclude for well-prepared initial data that the solution has to exist for some short time interval. Using a proof towards contradiction we obtain that for an arbitrary there exists some such that for all the solution exists on .
In order to do so we use the classical energy ansatz and differentiate (1.2) with respect to and test with to obtain
where is some remainder term. We use the properties of to apply a Gronwall and absorption argument for the remainder. More precisely the following bound is crucial
with . Unfortunately, the boundary conditions do not prevent large rotations around the -axis. Therefore the symmetric gradient can not bound the full gradient, which is reflected in an adapted Korn inequality for thin rods
where and independent of . We overcome this problem by using the balance of angular momentum for the solution to the non-linear equation. For these arguments we have to restrict to the case (instead of ).
On the technical level the most difficulties arise from the fact that the cross section of the rod is two dimensional. As a result we need higher regularity bounds in order to obtain . Due to the boundary conditions on the end faces of the rod, we can easily derive the equation in direction and in time. To obtain bounds on the second scaled gradient in direction we derive a lower dimensional system of the from
as we can not algebraically solve for the higher order terms. Here is a suitable restriction for a two dimensional system and arbitrary.
The structure of this contribution is as follows: In Section 2 we introduce the notation and some needed auxiliary results. Most important some specific bounds for the strain energy density W and Korn’s inequality in thin rods. Section 3 is devoted to the main result, which is stated in Section 3.1 and proven in Section 3.3. The analysis of the linearised system is done in Section 3.2. In the appendix we discuss the existence of classical solutions for fixed .
Preliminaries and auxiliary results
Notation
The natural numbers without zero are denoted by and . For any we denote the norm on , and the absolute value by . With , and , for , we denote the classical Lebesgue and Sobolev spaces for some bounded, open set . Throughout the paper , , denotes the space of all n-linear mappings for a vector space V. As common we will use the classical identification of with , i.e., is identified with such that
where is the standard inner product on . Similarly, is identified with defined by
In anticipation of the scaled Korn inequality stated in Section 2.3 we introduce a scaled inner product on
for all A, and . The corresponding norm is denoted by . For we define the induced scaled norm by
As for all it follows that for all and .
The scaled -spaces are defined as follows
if , where is measurable. Thus for all . The scaled norm for is defined in the same way
and the inequality holds the other way round
Throughout this work we denote by a smooth domain and for and some length in . As an abbreviation we will write . We assume that S satisfies ,
and
where . This can always be achieved via a scaling, translation and rotation. Furthermore, we denote by the scaled gradient defined as
The respective scaled and unscaled gradients in only direction are denoted as follows
The standard notation and is used for -Sobolev spaces of order with values in and some space X, respectively. Moreover, we denote for
A subscript on a function space will always indicate that elements have zero mean value, e.g., for we have
where is open and bounded. In various estimates we will use an anisotropic variant of , as we will have more regularity in lateral direction. Therefore we define
where , the inner product is given by
Furthermore we will use the scaled norms
for and and . As an abbreviation we denote for the symmetric scaled gradient by and .
The space of periodic functions can be defined in an equivalent way, which is in some situations more convenient
equipped with the standard -norm. As the maps and are isomorphisms, we identify with . With this definition we obtain immediately that is dense in , because, as S is smooth, there exists an appropriate extension operator and thus we can use a convolution argument. The following lemma provides the possibility to take traces for , more precisely
The operator,is well defined and bounded.
This is an immediate consequence of the embedding
where is the space of all uniformly continuous functions for some Banach space X. □
The strain energy density W
We investigate the mathematical assumptions and resulting properties of the strain-energy density W in three dimensions. Assume satisfies the following conditions:
for some ;
W is frame-invariant, i.e., for all and ;
there exists such that for all and for every .
First of all we note that W has a minimum at the identity, as and for all . Hence, we have for all . With the frame invariance symmetry of the Piola–Kirchhoff stress follows, i.e., for all : . Moreover, using the frame invariance we can deduce
for some and all .
Using the identification (2.1), we can find , for such that
for all . Therefore
for and . Hence, we obtain with (2.6) that . In order to see this we choose , and , , respectively. Thus either or and with the symmetry property of Remark 2.2 it follows
For later use we introduce
Let be defined by . The results of Remark 2.2 therefore hold for as well, i.e.,
and
The following lemma provides an essential decomposition of .
There is some constant,andsuch that for allwithwe havewhere
There exist C,such thatfor all,,,andandfor all,,,andandfor all,,,and.
The inequalities follow directly from Lemma 2.4 and Hölder’s inequality. □
Korn’s inequality in thin rods
In order to derive sharp estimates based on the linearised system, we need a good understanding on how the scaled gradient of a function can be bounded by the scaled symmetric gradient . As rigid motions for arbitrary are admissible functions in we can not expect that the full scaled gradient is bounded by . Moreover, a quantitative, sharp understanding of the dependency of a possible prefactor from the small parameter h is essential.
There exists a constantsuch that for allandwherewith.
The proof is similar to [1, Lemma 2.1] and is done in [3, Lemma 2.4.4] □
(Korn inequality in integral form).
For alland, there exists a constant, such thatwhere.
First we note that we can reduce to the case of mean value free functions. If (2.15) holds for mean value free functions and , we can consider . Then it follows
and thus (2.15) holds for u, as .
In the following we will argue by contradiction and therefore assume that (2.15) does not hold. Thus we can find a monotone sequence for and such that
For sake of readability, we write h instead of in the following calculations. From (2.16) it follows
which implies, using Lemma 2.6
Thus is bounded in and therefore bounded in . Using a subsequence, also denoted by , it follows for . As a consequence of (2.14) the structure of is given by
where as, in and
Define now
with and . Then
in and . Thus using the Poincaré inequality it follows
Thus, there exists a subsequence in and in . Choose and such that . Then
From the above we know
in and thus a subsequence converges pointwise almost everywhere. For the right hand side we have that for in because of
where we used Hölder’s inequality and the dominated convergence theorem for
Thus the mean value satisfies in . Hence, as S is a domain, is independent of . Similarly one can show that is independent of . Furthermore
where the right-hand side converges to the constant in . Since was chosen arbitrarily it follows
where is a constant. Applying the same argument to it follows that
Hence
as . But this contradicts (2.16). □
Later we will need the Korn inequality in two dimensions without scaling while analysing a stationary problem associated with the linearised equation.
(Korn inequality in two dimensions).
There exists a constantsuch that for allwhere in this situation.
We can deduce (2.17) from Lemma 2.7. Let and define by
Then it follows from (2.15) for applied to
□
Large time existence for the non-linear system
Main result
We consider the following system:
where for convenience . As an abbreviation we denote in the following . Note that satisfies the Legendre–Hadamard condition and Lemma 2.4 holds. The main result is:
Let,,,,and,such thatwhereMoreover we assume for the initial dataand for the right hand sideuniformly in. Then there existsanddepending only on M and T such that for everythere is a unique solutionof (
3.1
)–(
3.4
) satisfyinguniformly in.
For fixed short time existence is already known via the methods of [7] if the fixed time of existence is replaced by some h dependent maximal time . Hence only the uniform estimates for and that T does not depend on h has to be shown. In detail, one obtains from [7]:
Let the assumption of Theorem
3.1
hold true. Then for anythere exists a neighbourhoodof 0 and somesuch that (
3.1
)–(
3.4
) has a unique solution. If, then eitheris not precompact inor
We will give a more precise explanation on how the results of [7] are applied to our situation in the appendix. At this point however we want to mention that the neighbourhood can be chosen as
where is sufficiently small. With this it follows that as long as is satisfied the necessary weak coercivity holds, cf. Section 3.2.
The strategy for proving the main result is as follows. In a first step we will derive precise estimates for solutions of the linearisation of (3.1)–(3.4) under the assumption that is small in appropriate norms. To this end we use the natural boundary conditions, differentiate tangentially and utilize the central estimate
proven below. By differentiating (3.1) in time and we obtain that the respective derivative of solves now the linearised system. Applying then the results of the first step we can deduce with the balance of angular momentum that the solutions are uniformly bounded in h if the initial values and external force are sufficiently small.
We want to show h-independent estimates for solutions of the linearised system. For this we assume that satisfies for
where , with chosen later appropriately small. With this it follows that satisfies
and
Here is independent of h, R and . In the following we assume that is chosen so small such that and Lemma 2.4 is applicable.
Using Corollary 2.5, we obtain
uniformly in , and . Thus it follows
The structure of this subsection is that we will start with a general lemma providing a bound for derivatives in z of in the case that satisfies (3.16). To obtain bounds on higher derivatives we investigate the static problem and apply these subsequently to the evolution equation. This approach leads to uniform estimates for the solution of the linearised system.
Let (
3.16
) hold true and letand. Then there is someindependent of t, R, h such thatforand,.
If , we obtain by (3.16) and (2.10)
If , it follows for j, chosen correctly
For the first term we use (2.11)
and as
Finally for and j, k and such that we have
The fifth order term can be estimated in the same way as in the case of . As and , it follows with the Hölder inequality that
For the last term we use (2.10)
□
The first step to obtain higher regularity is done in the following theorem.
Assumesatisfies (
3.16
) and. Then there existandsuch that, ifsolves for someandthen
We start proving the result in the case . Using the fundamental theorem of calculus it follows
and
where denotes the kth column of . Moreover
where we have used that and is the outer unit normal on . Hence, we know that
solves for almost all the system
with ,
and
satisfying
for almost all . Then due to the inequality of Lemma 3.6 below it follows
for a.e. . Using the generalised Poincaré’s inequality
we obtain with the boundedness of
for such that in . Such an exists using a classical extension operator , which is right inverse to . Applying the preceding inequality on we deduce
Using Gauss’ theorem and (3.26) leads to
Integration with respect to yields
We can now estimate each term separately, beginning with . As is a linear combination of terms involving only it follows
where we used Korn’s inequality and the fact that φ is L-periodic in direction. Next we have
where for some suitable . Thus it follows with the identification of with via the standard scalar product
As and it follows
where we used Hölder inequality, the embedding and due to (3.16). Analogously using and it follows
Finally as is bounded, we obtain
Altogether we can conclude
From the definition of it follows
The first term on the right hand side is a linear combination of . Hence
The second term can be bounded analogously to (3.30)
Hence as we have
Due to the structure
for , it follows
Hence, by plugging all inequalities into (3.31) and applying Korn’s inequality
Using an absorption argument for sufficiently small and the structure in (3.32) it follows
as
and holds. Finally, (3.23) for follows from
In the case of we prove (3.23) in two steps. The first step consists in differentiating (3.22) in direction of and with (3.23) for we obtain
In the second step we apply classical higher regularity theory similarly to Lemma 3.6 below, see for instance [10, Theorem 4.18], and analogous inequalities as in the first case. □
Let the assumptions of Theorem
3.5
be satisfied andbe as in (
3.25
) a solution of (
3.26
). Then for almost allit holdsfor someindependent of φ,,,and.
As we have , one can test the equation (3.26) with to obtain
for almost all . Using now the Legendre–Hadamard condition, Korn’s inequality in two dimensions and Poincaré’s inequality it follows, due to the fact that is mean value free
Here we used that . Thus applying Young’s inequality and an absorption argument we are led to
For higher regularity we apply standard results, found in [5] or [10]. When considering the data one notices that and for almost every holds. This is a consequence on one hand from the assumptions on and . On the other hand we know that due to (3.28). Moreover, because it follows . The Legendre–Hadamard condition of is inherited from . Finally due to Korn’s inequality in two dimensions and Poincaré’s inequality with mean value we can conclude that , defined as
is weakly coercive on . Hence we obtain
Putting the above inequalities together leads to the desired result. □
In case we have homogeneous Neumann boundary conditions, we can refine Theorem 3.5 in the following way.
Assumefulfills (
3.16
) and. Ifsatisfiesfor some, then it holdsfor someindependent w, f, h, t, and.
Applying Theorem 3.5 with , and we are lead to
Consequently we want to eliminate the second term on the right hand side. For it follows from (3.34) that
for . Now we want to choose where and . First we start with . Periodicity of , w and , a Taylor expansion and creation of a part lead to
Here we used that
due to (2.6). Moreover, because of the specific construction of we can deduce
Hence,
Now, by the Hölder, Young and Poincaré inequality
for any . Secondly with Corollary 2.5, (3.17) and , we obtain
and
For sufficiently small and it follows
In order to use , we exploit the density of in . For we can use as a test function and obtain
Integration by parts, the periodicity and the homogeneous Neumann boundary conditions lead to
The density of implies now that the latter equation holds for as well. Thus
as . Using , we can conclude
In the preceding calculation Hölder’s and Young’s inequality are used as well as Lemma 3.4. Hence, by Korn’s inequality and (3.38)
Choosing sufficiently small leads to
Combining (3.38) and (3.39) with (3.23) the case follows.
For it remains to estimate . The bound can be seen as follows: Analogously as above we can choose first for . Thus, twice integration by parts leads to
Then the density of implies that the latter equality holds for . Using
By virtue of Lemma 3.4, we obtain
Finally, choosing ϵ and small, using an absorption argument and applying (3.38) and (3.39), leads to the desired result. □
Before we use the preceding Corollary 3.7 to obtain higher regularity bounds for (3.12)–(3.15), we state standard first order energy estimate for the solution of the dynamic system.
Let,,be given, whereis chosen small enough, but independent of h. Furthermore, assume thatsatisfies (
3.16
). For every,andthere exists a unique solutionof the system (
3.12
)–(
3.15
) satisfyingwhereis independent of h, w,,, f,and
The proof uses classical energy estimates and can be found in [3], Lemma 5.2.3. □
The second order regularity bounds are given in the following theorem.
Let,,be given, whereis chosen small enough, but independent of h. Furthermore, letsatisfy (
3.16
). Assumeis the unique solution of the system (
3.12
)–(
3.15
) for some,and, then there exist constants,(independent of h, T, R, w, f,) such thatholds, whereand
Differentiating (3.12) with respect to t and testing the result with yields
As
and
it follows
Due to (3.20) and Korn’s inequality
Using the definition of and Korn inequality, it follows
Moreover,
Putting everything together, using the coercivity (3.19) of and Young’s inequality we obtain
uniformly in . We use an absorption argument for
and the fact that with . Now, due to (3.35)
Applying
for f, we arrive at
where we used again. Hence, due (3.40) and choosing sufficiently small
where . As , there exists , such that
because of with . The Lemma of Gronwall yields then
□
The existence of a unique solution for (3.12)–(3.15) under the conditions of Theorem 3.9 follows from classical PDE theory. For higher regularity one would then apply hyperbolic regularity theory, cf. [16] and [9, Chapter 5]. Necessarily we need at this point that suitable compatibility conditions hold. As in the proof of the main result solutions of the linearised system are obtained via differentiation, we will not show the details here.
In order to prove the main theorem via contradiction, we need that is regular enough, to avoid blow ups. In our situation we need the following regularity theorem.
Let,,be given, whereis chosen small enough, but independent of h and letsatisfy (
3.16
). Assumeto be the unique solution of the linearised system (
3.12
)–(
3.15
) for some,and. Then there exist constants,depending on T such that
Due to Lemma 3.8 and Theorem 3.9 it remains to bound only the third order terms.
Differentiation with respect to of the system (3.12)–(3.13) leads to
where . First we want to apply Theorem 3.5 with and
where we used the convention . Then we obtain
Moreover, as is uniformly bounded and satisfies (3.16)
and
Hence
for almost every . With this we can follow a similar argument as in the proof of Theorem 3.9. For this we differentiate the equation for in time and test with . Then it follows
because all boundary integrals disappear due to periodicity of and , respectively, and the Neumann boundary conditions. With this we can follow a similar strategy as in Theorem 3.9 to obtain an analogous result. Finally using
for we obtain the claimed inequality. □
Before we start the proof of the main theorem, we will prepare some bounds on the rotation of the solution around the -axis. More precisely we need to bound the following quantity
We can transform the system (3.1)–(3.4) via , . Hence solve the equation
with . Due to the frame invariance the Piola–Kirchhoff stress fulfills the appropriate symmetry condition and one can apply the balance law of angular momentum
With the transformation formula applied for we conclude with
We can restrict to just the first component, as only rotations around -axis have to be controlled for the use of Korn’s inequality. For the first component we have
because on and as does not depend on it follows that is L-periodic in the -direction. Using this and for all we deduce
Thus we have
for almost all . With this we can later bound for uniformly in . We note that with (3.43) it follows
Without loss of generality we will assume that . This is possible as we can perform a rescaling in h and t by , changing only M by a T-depending factor. Furthermore we assume that is sufficiently small, such that all results of Section 3.2 are applicable.
The assumptions (3.5)–(3.7) and (3.8) imply
for with some universal constant . We choose small enough such that holds, where is chosen as in Corollary 3.11. Let be the solution of (3.1)–(3.4) from Theorem 3.2. Then there exists some maximal such that
This maximum exists since as, if (3.48) holds, the set is precompact in , cf. the Appendix. Moreover it holds
This can be seen by using (3.48), as it follows
and
for all . Moreover as long as (3.48) is valid, satisfies (3.16) and all the results of Section 3.2, especially Corollary 3.11, are applicable. We want to reduce to the case that is mean value free. Hence we assume in a first step that
for all . Using (3.1)–(3.4), we obtain that , , solves
with and . Hence with Theorem 3.11 and (3.47) it follows
where we used and note that can be estimated by
due to Korn’s inequality. Now we want to apply (3.44)–(3.46) in order to bound the rotational part of . It follows for
Due to the assumptions on initial values and the external force , we obtain that
and
Moreover for we deduce with the Cauchy–Schwarz and Poincaré inequality
as for . Similarly we obtain
and
Altogether this leads to
Thus we have shown
Moreover, exploiting
we obtain with (3.6)
and
due to (3.5), (3.7) and (3.50). Hence we deduce
As we can find such that
uniformly in .
Now we have to consider the case that the force or the initial data is not mean value free. In this case we define
Then solves
where we subtracted from their mean values to obtain .
Then it holds for
as, integration of the nonlinear equation (3.1) implies with the boundary and periodicity condition, (3.2) and (3.3), respectively
Moreover (3.7) and (3.9) is fulfilled for , and , because . Deploying the fact that the initial data is only changed by a constant, (3.6) holds for the new initial values. With and triangle inequality it follows (3.8) with instead of M, for some independent of , h and M. In the same way one can deal with in (3.16). Hence, as for (3.6), we obtain that (3.5) holds with M replaced by . Thus we can apply (3.51) for and get
From the definition of it follows
Because of , we have
Lastly for we deduce from and (3.9) that
Thus it follows
for small. □
Footnotes
Acknowledgements
Tobias Ameismeier was supported by the RTG 2339 “Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). The support is gratefully acknowledged.
Existence of classical solutions for fixed h > 0
In this appendix we want to give more details on how the existence result of [7] is applied in the regarded situation. First we shortly summaries the assumptions and equation considered in [7] and the main result [7, Theorem 1.1], which we want to apply. Second we give some remarks on how our system is obtained and why the assumptions assumed in Theorem 3.1 are sufficient.
In [7] a quasi-linear hyperbolic equation of the form
is considered, where , , is a bounded domain with boundary of class , and ν is the outer normal. Moreover and , with . For convenience we state the assumptions made in [7] in a slightly simplified version.
We assume that , and let U be an open neighbourhood of in . Moreover assume F, , and define
in U for all , .
For any , and with there exists and such that for all the inequality
For any there exists a such that for all the inequality
holds.
We suppose that the compatibility condition holds up to order s.
Under these assumptions the following theorem holds:
In the situation this paper the considered domain Ω is not sufficiently smooth, but due to the periodic boundary condition on the end faces of Ω the equations (3.1)–(3.4) are equivalent to the equations on the manifold . This is a bounded manifold with smooth boundary, as S is a domain. The ideas of [7] are similar as in Section 3.2, using differentiation in time and applying results from the elliptic theory.
Choosing , , and
for . Then we obtain the symmetry condition A2. As the assumption A4 is fulfilled, with . Moreover the compatibility assumptions of Theorem 3.1 imply A5. For the first assumption we choose . Then the initial data is sufficiently regular and as does not depend on the prescribed regularity is sufficient. Lastly we can choose U as
for some sufficiently small ε. This is indeed an applicable neighbourhood as for small , it holds for all , as . Finally due to Lemma 2.4, (3.11) and Korn’s inequality we obtain that the coerciveness assumption A3 is satisfied.
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