Kaitai Li and Xiaoqin Shen
1 School of Mathematical Sciences and Statistics,Xi’an Jiaotong University,Xi’an,710049,P.R.China.
2 School of Sciences,Xi’an University of Technology,Xi’an,710048,P.R.China.
Abstract.In this paper,a mixed tensor analysis for a two-dimensional(2D)manifold embedded into a three-dimensional(3D)Riemannian space is conducted and its applications to construct a dimensional splitting method for linear and nonlinear 3D elastic shells are provided. We establish a semi-geodesic coordinate system based on this 2D manifold,providing the relations between metrics tensors,Christoffel symbols,covariant derivatives and differential operators on the 2D manifold and 3D space,and establish the Gateaux derivatives of metric tensor,curvature tensor and normal vector and so on,with respect to the surface →Θ along any direction→η when the deformation of the surface occurs.Under the assumption that the solution of 3D elastic equations can be expressed in a Taylor expansion with respect to transverse variable,the boundary value problems satisfied by the coefficients of the Taylor expansion are given.
Key words:Dimensional splitting method,linear elastic shell,mixed tensor analysis,nonlinear elastic shell.
A shell is a three-dimensional(3D)elastic body that is geometrically characterized by its middle surface and its small thickness.The middle surface ℑ is a compact surface in ℜ3that is not a plane(otherwise the shell is a plate),and it may or may not have a boundary.For instance,the middle surface of a sail has a boundary,whereas that of a basketball has no boundary.
At each point s ∈ℑ,let n(s)denote a unit vector normal to ℑ. Then the reference configuration of the shell,(that is,the subset of ℜ3that occupies before forces are applied to it),is a set of the formwhere the function e:ℑ→ℜ is sufficiently smooth and satisfiesfor all s∈ℑ.Additionally,ε>0 is thought of as being‘small’compared with the‘characteristic’length of ℑ(its diameter,for instance).If e(s)=ε,∀s∈ℑ,the shell is said to have a constant thickness of 2ε.If e is not a constant function,the shell is said to have a variable thickness.
The theory of elastic plates and shells is one of the most important theories of elasticity.Thin shells and plates are widely used in civil engineering projects as well as engineering projects.Examples include aircraft,cars,missiles,orbital launch systems,rockets,and trains.
Considerable work on the subject was conducted by the Russian scholars A.I.Lurje(1937),V.Z.Vlasov(1944)and V.V.Novozhilov(1951)after the pioneer idea of Love.However,now it appears necessary to improve the mathematical understanding of the classical plate and shell models pioneered by these scholars.The reason for this is that the precision required in aircraft and spacecraft projects has intensified with the advent of powerful electronic computers.The goal therein is,on the one hand,to develop better finite approximations on elements and,on the other hand,to refine theoretical models when necessary.
A.L.Gol’denveizer(1953)first put forward the ideas of conducting an asymptotic analysis based on the thickness of a shell or plate.New formulations for shells and plates were obtained by relaxing the constitutive relations and reinforcing the equilibrium that must be satisfied.Even in presenting a much more detailed analysis than that offered by his predecessors,no mathematical justification was given by Gol’denveizer.As such,a large number of difficulties were still to be overcome.At the same time,the weak formulations of J.L.Lions and mixed formulations of Hellinger-Reissner and Hu-Washizu(1968)appeared to provide alleviation.Of particular significance was the work of J.L.Lions(1973)on singular perturbation,which provided the tools and explanation for what happens when the asymptotic method is applied to plates,shells,and beams.The mathematical foundation of elasticity can be found in[4,6].The asymptotic method was revisited in a functional framework proposed by Li,Zhang and Huang[1],P.G.Ciarlet[3,4]and M.Bernadou[5].Their work made convergence and error analysis possible therein for the first time.
The 3D models are derived directly from the principles of equilibrium in classical mechanics,and are viewed as singular perturbation problems dependent upon the small parameter ε(the half-thickness of the shell).2D models are obtained by making some additional hypotheses that are not justified by physical law.Our aim here is to justify mathematically the assumptions that formulate the basis of a 2D model of elasticity,and whose solution approaches 3D displacement better than the solution of classical models.
This paper is organized as follows:in Section 2,we present a mixed tensor analysis on 2D manifolds embedded into 3D Euclidian space;in Section 3,we provide the exchange of tensors and curvature after the deformation of curvatures;in Section 4,we give differential operators in 3D Riemannian space under semi geodesic coordinate system;in Section 5,asymptotic forms of 3D linear and nonlinear elastic operators with respect to transverse variable ξ are derived;in Section 6,a specific shell as an example,is provided.
Tensor analysis in Riemannian space can be found in[2]. Here it presents for mixed tensor analysis on a 2D manifold embedded into a 3D Riemannian space,in which we provide basic theorem and formula and gives proof in details.
Let us consider elastic shells,which is assumed to be a St Venant-Kirchhoff material and homogeneous and isotropic.Hence this material is characterized by its two Lam´e constants λ>0 andµ>0 which are thus independent of its thickness.
An elastic shell whose reference configuration(3D-Euclidean space)consists of all points within a distance ≤ε from a given surface ℑ ⊂E3and ε >0 which is thought of being small.The 2D manifold ℑ⊂E3is called the middle surface of the shell,and the parameter ε is called the semi-thickness of the shell.The surface ℑ can be defined as the image →θ of the closure of a domain ω ⊂R2,where →θ:¯ω →E3is a smooth injective mapping.Let→n denote an unit normal vector along ℑ and let
The pair(x1,x2)is usually called Gaussian coordinate on ℑ,and(x1,x2,ξ)is called semigeodesic coordinate system(S-Coordinate System)(if E3is a Riemmannian space andℑ is a 2D manifold). The boundary of shellconsists as follows: ·top surface Γt=ℑ×{+ε},
·bottom surface Γb=ℑ×{-ε},
·lateral surface Γl=Γ0∩Γ1:Γ0=γ0×{-ε,+ε},Γ1=γ1×{-ε,+ε},where γ=γ0∪γ1is the boundary of ω:γ=∂ω.
In what follows,Latin indices and exponent:(i,j,k,···)take their values in the set{1,2,3}whereas Greek indices and exponents(α,β,γ,···,)take their values in the set{1,2}. In addition,Einstein’s summation convention with respect to repeated indices and exponent is used.
It is well known that the covariant and contravariant component of the metric tensors on the surface ℑ are given by
Furthermore,it is well known that the contravariant components of bαβ,cαβare given by
which will play important role in the followings.In the same season we have to introduce permutation tensors in ℜ3and on ℑ which are given by
where g=det(gij)and gijis metric tensor of ℜ3.
Similarly
Let H and K denote mean curvature and Gaussian curvature respectively:
Then the determinant of third fundamental tensor cαβis
Using permutation tensors in ℜ3and on ℑ the following relationships are held
and
The following lemma present fundamental formula concerning many basic tensors of order two on the surface.
Lemma 2.1.The third fundamental tensor is not independent of the first and second fundamental tensors aαβ,bαβ.There are following relationships
Besides,there are relationships between matrices bαβ,cαβand its inverse matricesas follows
Furthermore,
Proof.First,we prove(2.11).For the simplicity,let r=→θ and n denote the unite normal vector to surface ℑ.(2.3)shows bαβ=-nαrβ,therefore
In terms of formula in vector analysis
it yields that
On the other hand,applying Weingarten formula
and permutation tensor
we can obtain
This is the first part of(2.11).
Next we prove last two formula in(2.11).Remember that
Therefore,
This is the forth part of(2.11).Using εαλaανaλµ=ενµwe derive
This is the fifth part of(2.11).
Applying εβλto contraction of indices of tensor for both sides of above equality
This is the third part of(2.11).
Next,we prove second of(2.11).To do that,combing the first and third part of(2.11),we have
From this it yields the second part of(2.11).
Next we prove(2.9).To do that by contraction of tensor indices for the second part of(2.11)with aλσwe have
Because of
we can obtain the first part of(2.9).
Using the trick of tensor indices left leads to the second part of(2.9).
In order to prove the third part of(2.9),multiplying both sides of tensor index of the first part of(2.9)by εαλεβσ,then using(2.7),we derive
This is the third part of(2.9).
Applying trick of tensor index left,the second part of(2.11)can be rewritten as
multiplying both sides of above equality,by bλσand usingit is easy to yield
On the other hand,the first part of(2.9)can be rewritten in mixed tensor formulae
Consequently,we have
Combing(2.14)and(2.15),we prove the fourth part of(2.9).
Because of the third and fourth part of(2.9)
it is easy to derive the fifth part of(2.9).Now we prove(2.10).With
it yields the first part of(2.10):
Similarly,
This is the second part of(2.10).
On the other hand,with(2.10)we derive
This is the third part of(2.10).
Next we prove(2.12).Repeatedly using(2.9)
Then,we have
Those are(2.12),thus,we complete our proof.
In the following sections,we consider the metric tensor of 3D Euclidean space under semi-geodesic coordinate(x1,x2,ξ).
Lemma 2.2.Under semi-geodesic coordinate system,the covariant componentsof metric tensor of 3D Euclidian space E3are the polynomials of two degree with respect transversal variable ξ and can be expressed by means of the first,second and third fundamental form of surfaceℑ:
where
κλ,λ=1,2 are the principle curvatures of ℑ.
Its contravariant componentsare the rational functions of transverse variable ξ and can be expressed by inverse matrices
where x=(x1,x2)and
In particular,gαβadmits a Taylor expansion
Proof.Let ∀(x,ξ)∈E3
and its derivatives
It is obvious that they are independent on Ω.Hence
Combing the first part of(2.9),we derive
Since|n|2=1,nα(x,ξ)·n(x,ξ)=0,we claim
Consider determinant
With(2.8)
it infers
From(2.8)and(2.9)
we obtain
We complete the proof of(2.16).
In order to prove(2.18),we observe
and
Then
Similarly
With(2.21),we have
Using(2.12),we obtain
Making Taylor expansion
gives(2.20).The proof is completed.
Since ℑ is as a 2D manifold embedded in 3D Euclidian space E3,we need consider the mixed tensor,in particular,mixed covariant derivative for the mixed tensor.Lemmas 2.2 and 2.3 provide the relations between the tensors in E3and those on ℑ.What follows,we consider others relations,for example,let,anddenote Christoffel symbols and covariant derivative in E3and on ℑ respectively,
Then we have
Lemma 2.3.Under S-coordinate system,Christoffel symbolsin E3can be expressed by means of Christoffel symbolsof ℑ
where
Proof.With Weingarten formula and Gaussian formulae
we have
Therefore,
Here we used Gadazzi formula and the covariant derivative of bαβ
where
We complete the proof of first part of(2.26).
Similarly,
Next we prove(2.27).In deed,from the first part of(2.26),we derive
As the covariant derivative of metric tensor is vanished,from Godazzi formula,we have
It can also be expressed by
Taking(2.19)into account,simply computation shows
Therefore,
This is the first part of(2.27).In addition,
Other conclusions can be proved easily.
As we all know,the covariant derivatives of tensor in E3and ℑ are defined by
Since ℑ is embedded into E3,there are relations between covariant derivatives of tensor at different level.
Lemma 2.4.Under S-coordinate system covariant derivative of a vectorin E3can be expressed by derivatives of its components on the tangent space at ℑ.Furthermore it is a rational function of transversal variable ξ
which admit to make Taylor expansion with respect to transversal variable ξ
where
Proof.Note that(2.34)can be easly derived from(2.33)and(2.27).In order to consider Taylor expansion,it is value to mention that when ξ is small enough,function θ-1can be made Taylor expansion
Since
it immediately yields(2.35)and(2.36).Remind that we have to compute divergence.In fact,
Applying Godazzi formula and Lemma 2.1,
we have
Combining above results,it is easy to obtain(2.35)-(2.37).The proof is completed.
Following lemma is very useful throughout this paper which indicates the relations betweenand so on.
Lemma 2.5.The following formulae are valid
Proof.Repeatedly using Lemma 2.1,we have
Since
we have,
Next,we compute
Since
this follows from the first part of(2.12)that
we obtain
Combining above results we get the forth part of(2.41).
Taking the first and second part of(2.41)into account,
It yields seventh of(2.41)
Next we prove the fifth part of(2.41).Note that
By(2.12)
and θ=1-2Hξ+Kξ2and(2.9)
simple calculation shows that
This is the fifth part of(2.41).
In order to prove the eighth part of(2.41),bywe obtain
Next,we prove sixth of(2.41).Indeed,from fourth of(2.41)it gives that
This is sixth of(2.41).
Finally,we have to prove the night part of(2.41). From(2.18),(2.28)and Godazzi formulait leads to
Therefore,
Since covariant derivative of metric tensor is vanished,from Lemma 2.1,with bβσbβσ=it is not difficult to obtain
Indeed,
but
hence
Finally we obtain
The proof is completed.
In what follows,we consider strain tensor eij(u)in E3associated with displacement vector u
We use derivative of metric tensor being vanished
which will be frequently used throughout this paper.
The contravariant component of vector u:uk=gkjujinstead of covariant component of vector.The strain tensor on ℑ associated with displacement vector u:
The contravariant components of strain tensor are defined by
We also consider Green St.Vennant strain tensor Eij(u)of the displacement u for nonlinear elastic case
Lemma 2.6.Under S-coordinate system the strain tensor and Green St.Vennant strain tensor are the polynomials of degree two with respect to transverse variable ξ
where
where the strain tensors on the two-dimensional manifold S are given as:
where
Remark 2.1.Since displacement vector u3=0 andat tangent space of 2D manifoldℑ,the strain tensor is given bywhile displacement vector are in 3D space,the strain tensor of displacement vector restrict on the ℑ will be given by γαβ(u).
Proof.
Since of Godazzi formula and vanishing of covariant derivatives of metric tensor,we get
Next,we prove the second part of(2.46).With(2.41)and Godazzi formula,we have
It infers that
In what follows,we prove
Indeed,by(2.28)we derive
Taking into account of
and(2.54),we immediately obtain our conclusion.By above discussion,it infers that
where
with
Applying(2.36)immediately derive formula(2.50).
By similar manner,the Calculations show that
where
Furthermore,from(2.4)and(2.34)we have
where
Our proof is completed.
We assume that the elastic material constituting the shell are isotropic and homogenous.The contravariant components of elasticity tensor are given by
where(λ≥0,µ>0)are the elastic coefficient constants.Let
Lemma 2.7.The elasticity tensor are a rational polynomials with respect to ξ in the S-coordinate system:
where
Proof.From(2.55)and(2.18),simple calculation show our results.The proof is complete.
In what follows,we introduce following invariant scale function
In this section,we have to study the exchange of geometry of the surface in ℜ3when the surface occurs deformation.We will give the formula for the exchange of metric tensor,curvatures tensor and normal vector to the surface.
Let ω ⊂ℜ2be a compact domain and a immersionis smooth enough,the middle surface ℑ of shell defined as the imageThe deformation of a surface means that at each point on the surface bears a small displacement →η and new surface after deformation denote ℑ(→η)as the image
For simplicity,later on,we denote the Gateaux derivative of a geometric tensor(for example,metric tensor)with respect to surface →Θ along director →η
Theorem 3.1.Assume that surface ℑ is burned a deformationThen following formulae hold
The Gateaux derivative of Riemann curvature with respect to surface ℑ along direction →η is given by
where
Proof.(i)Preliminary
Assume that the displacement vector and base vectors of S-coordinate system at ℑ
are given.Then
Proposition 3.1.The followings on the ℑ are valid
Proof.Indeed,using Gaussian and Weingarten’s formula,reads
From above formula(3.7)is obtained.
What follows that we prove(3.8),In fact,by virtue of Weingarten’s and Gaussian formula
and(see[2])
Therefore,
Observe that
Taking into account of
from(3.11)it infers(3.8)immediately.
(ii)Metric tensor and its determinanta(η)=det(aαβ(η)).
Proposition 3.2.The Gateaux derivatives for metric tensors and its determinant with respect to ℑ along director →η are given by the followings
Furthermore,
Proof.The deformed surface ℑ(η)define as the imageAssume vectors
are linearly independent at all points ofIt is obvious that if the vectoris small enough,e(η)can be as base vectors of two dimensional manifold ℑ(η).So that aαβ(η)=eα(η)eβ(η)are covariant components of metric tensor of ℑ(η)which is nonsingular matrix.Indeed
By(3.7),
According to the calculation’s principle of the determinant for a matrix
and the formula
we assert
Therefor there exist contravariant components of metric tensor
where the permutation tensor is defined by
Let the normal unite vector be
and the contravariant base vector
From this,of course,it infers
The second and third terms of the above equality are two degree of η,so that with(3.13)
On the other hand
The proof of Proposition 3.2 is complete.
(iii)Second fundamental form and unit normal vector n(η)to ℑ(η)
Proposition 3.3.The Gateaux derivatives for unit normal vector with respect to ℑ along director →η are given by the followings
From(3.7),(3.10)and the formula
it infers
Here we used formula
In a similar manner,using(3.8)gives
Substituting above equalities into(3.15)leads to sixth formula of(3.14).
Using sixth formula of(3.14),
Hence we assert
Proposition 3.4.The Gateaux derivatives for curvature tensors and its determinant with respect to ℑ along director →η are given by the followings
where the tensors of order two are defined by
Proof.According to the Gaussian formula
we have
On the other hand,the following formula is held
Indeed,by the Weingarten’s and Gaussian formula(3.9)
Hence,we conclude(3.19)by virtue of(3.17).Using the Weingarten’s formula(3.9)with(3.19),we assert
Coming back(3.18)with(3.7)and(3.9)shows
It can be rewritten in
These are the first second of(3.16).Next we consider the contravariant component
Using above,(3.12)and(3.16),and link chain of derivative
Note that
If b=det(bαβ)/0 then
Hence
To sum up,it completes our proof.
Proposition 3.5.The Gateaux derivatives for(H,K,cαβ)with respect to ℑ along director→η are given by the followings
Proposition 3.6.The Gateaux derivative of Riemannian curvature tensor with respect to surface ℑ(η)is give by
Indeed,the Riemannian curvature tensor of surface ℑ(η)is given by
Applying Proposition 3.4 immediately yields to(3.21).
(iv)Symmetry of indices forραβ(η)and(η)
Let us define the tensor ραβ(η)of order two and(η)of order three generated by the displacement vector
Proposition 3.7.The tensors ραβ(η)and(η)are symmetric tensors with respect to index(α,β):
and have equivalent form
Proof.First,we prove(3.22)and(3.23).Indeed,by virtue of(2.36)and(3.17),
since η3is looked as a scale function define on ℑ and Goddazi formula we claim
From this and(2.47),it implies(3.22).Next we prove(3.23).In fact,by(2.29),we claim
Since the Godazzi formula and the covariant derivative of metric tensor being vanishing,
In addition,by virtue of the Ricci formula
Hence
Finally,we end our proof for Theorem 3.1.
Hoge-Laplave operator under S-coordinate system
It is well known that for the Navier-Stokes equations in fluid mechanics or the Lamee-Navier equations in elastic mechanics,their principle part contain the divergence of the strain tensor for velocity vector or displacement vector.In the Riemannian space,they do not have interchangeability with Leray projector on the divergence free subspace Kerdiv,but it is possible to make interchange with the Hoge-Laplave operator.In order to make mix with either self,denote ΔHby
where d and δ are the exterior differential operator and the supper differential operator,respectively.According to the Weitzenbock formula,it is equal to Bochner-Laplace(traci-Laplace)plus Ricci operator when it acts on vector field,i.e
where Bochner-Laplace operator is defined by
It is well known that the conservation of the energy-momentum in physics concern divergence of enrgy-momentum tensor,the constitutive equation in continuum also contain the relationship between strain tensor and stress tensor.It is natural that the divergence of strain tensor play important role. The relationship of strain tensor,Bochner-Lpalace operator and Ricci operator in Riemannian space are given by
Combining above notations,the divergence of the strain tensor in higher space and two dimensional surface are given by
respectively. It is obvious that it is enough to compute the Bochner-Laplace operator when we have to compute the divergence of strain tensor.
By the way in 3D-Euclidean space E3,following is given in terms of the operator rotrot to compute divergence of strain tensor
The following theorem gives the expansion with respect to the transverse variable ξ for the Riemmannian curvature and the Ricci curvature tensors.
Theorem 4.1.Under S-coordinate in the 3D-Riemmannian space,the Riemannian curvature tensor is a polynomial of degree two with respect to the transverse variable ξ
where
Proof.At the first,we give the expression for the Riemann curvature tensor of four order covariant components under S-coordinate.To do that,according to
and(2.16)-(2.17),we have
By applying
we have
According to the formula for the Riemann curvature in 2D Surface([1])
Therefore
By appling(2.27)and(2.28)
it yields
Using
and(4.14),we have
where we have used the following two equalities
Finally,
it still possess anti-symmetric of indices.Combing(4.14)and(4.15)with(4.12)yields
Substituting(4.11),(4.12)and(4.16)into(4.9)leads to
Owing to the anti-symmetric of index of Riemann curvature tensor
the sum of first three terms in(4.17)equal to zero
Then(4.17)becomes
It can be also expressed by
Let us define a tensor of four order covariant components
Then(4.17)becomes
The remaining is to prove formula(4.8)
Using
and the Godazzi formula
we obtain
On the other hand,by(2.41)
Substituting into(4.24)leads to
Combining(4.22),(4.24)and(4.25)yields
By symmetry and anti-symmetry of indices for Rimemann curture tensor,we obtain
In what follows,
Theorem 4.2.Under the S-coordinate in the 3D-Riemmannian space,the Ricci curvature tensor is a rational polynomial of degree two with respect to the transverse variable ξ whose Taylor expansion is given by
where Hα=∂αH,
Proof.Applying Lemmas 2.2,2.3 and 2.5,and g=θ2a,we have
On the other hand,
Similarly,
Note that
we have
Furthermore,
Thanks to
Finally
In addition
It follows from
that
Since
we obtain
The proof is complete.
Next we consider the relationships of covariant derivatives of order two in 3D-space and on the two dimensional manifolds which are necessary for studying differential operators on the manifolds.
Lemma 4.1.There are relationships between the covariant derivatives of two order of the vectors in 3D space and on two dimensional manifold
where
and
Proof.In order to prove Lemma 4.1,we repeatedly and alternately to apply Lemmas 2.1-2.7.Indeed,for example,according to the definition of covariant derivative of second order tensor
Making rearrangement to obtain
Next we consider
Below we prove
In fact,in view of(2.41)
In addition,we have
Using the Godazzi formula
we assert
Substituting above formula into(4.35)leads to
From this it yields(4.34).Finally we obtain
Combing(4.33)and(4.36)we claim
Next we consider
Owing to(2.27):
it infers
The following equality is very useful later on
To obtain that,we first show that
On the other hand,
so that
This infers(4.41).Coming back to(4.40)
In addition,in view of(4.39)we have
Combining(4.42)and(4.43)gives
Next we compute
Finally we find
Straightforward calculasions show
Howerver
Therefore
On the other hand
Combining(4.46)and(4.47)leads to
Owing to
and applying(4.46),(4.45)becomes
By a similar manner,we find
Note that
we have
With(4.45)we assert
where
To sum up we verify(4.30).
Theorem 4.3.Under the S-coordinate system in the 3D Riemannian space,the Bochner-Lplace operator acting on a vector field can be expressed in a rational polynomial with respect to the transverse variable(the length of geodesic curve)ξ.
where
Remark 4.1.The Taylor expansions in(4.53)are given by
Proof.Applying Lemma 2.2 and Lemma 4.1
By virtue(4.31)and Lemma 2.5
Applying(4.46)and Lemma 2.1it yield
Hence,we obtain the 2nd and 3rd parts of(4.53).
Theorem 4.4.Under the S-cooedinate system in the 3D Riemann space,the Betrimi-Laplace operator Δ=gij∇i∇jis a polynomial with respect to transfers variable ξ,which can be made Taylor expansion with respect to ξ,i.e.for a two times differential function φ,
Proof.Indeed,by(2.34),
Since
it is not difficult to prove that
Therefore
The proof is completed.
As well know that the initial and boundary value problem of linearly elastic mechanics are give by
Since Aijklis defined by(2.55)and gklekl(u)=divu,we claim
Therefore
Applying the Ricci formula
if E3is Euclidian space,Furthermore
where Rmkare Ricci curvature tensor.
In this case,the linear elasticity operator is given by
Finally elasticity equations in Euclidian space are given by
where the lateral surface Γ0=Γ01∪Γ02,σ is a stress tensor.
Theorem 5.1.Under the S-coordinate system in E3,Eq.(5.1)can be expressed as
in details,
where
Matrices dijare given by
Remark 5.1.Taylor expansions of(5.3)are given by
Proof.At first we prove(5.2).To do that,since(4.30)and(2.35),rewritten(5.1)into components form
In addition,
Therefore
Substituting(5.6)leads to(5.2),we end our proof.
Theorem 5.2.Under the S-coordinate system in E3,if the solution of(5.1)in neighborhood of surface ℑ and right hand f can be made Taylor expansions with respect to transverse variable ξ
then the linear elasticity operators cam be made Taylor expansions as
where u0(x),u1(x),u2(x)satisfy following boundary value problems
with boundary conditions in(5.1)on the boundary γ1=Γ02∩{ξ=0}of middle surface
where
Proof.Consider(5.2)
Using
gives
Denote
Taking(5.18)-(5.21)into account,(5.17)can be made expansion
So that we obtain following equations
By similar manner,we assert
Substituting(5.10)into(5.23)leads to
Hence it can be expressed as
Consequently,we obtain following equations
where
We then complete our proof.
Theorem 5.3.Under S-coordinate system in E3,if(5.5)is satisfied then linearly stress tensor σij(u)=Aijklekl(u)can be made Taylor expansion with respect to transverse variable ξ
where
where
The boundary conditions on top and bottom surface of shell are given by
Proof.As well known that linearly stress tensor σij(u)of isotropic linearly elastic materials corresponding to displacement vector u is given by
Elastic coefficient tensor of isotropic linearly elastic materials is given by(2.55)and can be made Taylor expansion with respect to transverse variable ξ by(2.57). In addition,owing to Lemma 2.6 andare linear form for u,therefore
Note non vanishing Aijkmby Lemma 2.7
where ck(ξ)defined by(2.34).Thanks to(2.20)
We assert that
From this it yields(5.29).Moreover
where ck(ξ)are defined by(2.34)
Hence
Finally,boundary conditions on the top and bottom of shell are nature boundary conditions
Note,in Theorem 3.1,displacement →η=ε→n=(0,0,1)at top surface of shell,it yields from(3.2)
Therefore,by Theorem 5.2,we have
Similarly,since n(-εn)=-n,hi=gjmσij(u)nm(-εn)on the bottom surface of shell,so that
This completes our proof.
In this section we study nonlinearly elastic equations for isomeric and isomorphic St.Venant-Kirchhoff materials.Nonlinearly elastic equations for 3D elastic shell are given by:findsuch that:
where stress tensor σij,second Piola-Kirchhoff stress tensor Σijand first Piola-Kirchhoff stress tensorare given respectively by:
where eij(u),Eij(u)are strain tensor(2.42)and Green-St Venant strain tensor(2.45).
In the followings,we have to consider first Piola-Kirchhoff stress tensor.The covariant derivatives of first Piola-Kirchhoff stress tensor are given by
where we consider the materials are isomeric and isomorphic,and covariant derivatives of metric tensor in Euclisean space are vanising(i.e.∇jAkjlm=0).As well known that linearly elastic operator
Therefore nonlinearly elastic operator is given by
As well known that isotropic and homogenous elstic coefficient tensor of four order are given by(2.55)and satisfiy(2.57),in particular all components Aijlmare vanissh except
In addition
So that it assert that
It can be reads as
According Ricci Theorem and Riemannian Curvature tensor are vanish in Euclidian space,therefore
and symmetry of indices of strain tensor,previous equality becomes
Substitute it into(6.5)leads to
Since
Hence
Reads otherwise
Furthermore,by similar manner,we assert
Substitutin it into(6.7)leads to
Next we need to consider relationship of covariant derivatives of nonlinear stain tensor tensor.
Lemma 6.1.The covariant derivatives of symmetric tensor of stain tensor Dijin S-coordinate system are given by
where
Proof.According to definition of covariant derivative,
By virtue of Lemma 2.3
Therefore,since symmetry with respect to subscript of Dij=Dji,
In similar manner,by Lemma 2.3
Other equalities of(6.9)can be obtain in same way.End our proof.
Lemma 6.2.Assume that the elastic materials is isomeric and isomorphic St.Venaut-Kirchhoff materials. Then under S-coordinate system nonlinear elastic operator defined by(6.1)can be expressed as
where Eij(u)is Green-St-Vennant strain tensor defined by(2.45)and
Proof.Taking(6.9)into account,we claim
Similarly,applying Lemma 4.1 and Theorem 4.3 and Theorem 5.1,we assert
where
Owing to(5.2)
with(6.8),(6.14)and(6.15),we obtain
which can be rewritten as
where
Third component is given by
which also can be expressed as
where
To sum up,nonlinearly elasticity operators can be written as
where
Hence we complete our proof.
Theorem 6.1.Assume that solution u of nonlinearly elastic operators defined by(6.11)and right hand term f can be made Taylor expansion with respect to ξ:
Then nonlinearly elastic operators defined by(6.11)under the S-coordinates can be made Taylor expansion
and(u0,u1,u2)satisfy approximately following boundary value problems
with boundary conditions
where
Proof.As well known that the nonlinearly elastic operators are given by(6.11),
As well known all coefficients can be made Taylor expansions with respect to transverse variable ξ if taking(6.19)into account
Therefore
Next we consider Didefined by(6.18).To do that let remember bilinear and symmetric form D(u)defined by(2.48)
Denot e
The n
In addition,let
Then
Let us come back to(6.18).Note that
where
Futhermore,in view of(2.34)and(6.19)we claim
where
Owing to(6.36)and(6.32)we assert
Substituting(6.38)and(6.26)into(6.25)leads to
This is(6.20).
Next we consider expansion for bilinearly symmetric form. Note Green St-vennen strain tensor Eij(u)of vector u is defined by(2.45)
and satisfies following formula(see Lemma 2.6)
which are polynomials of two degree with respect to transverse varialble ξ,and its coefficients do not contain three dimensional cavariant derivativesof displacement vector u but contain two dimensional derivativeson the surface.Therefore if displacement vector satisfies Taylor expansion(6.19),then we claim
Let us denote
where
Finally,the coefficients in(6.26)are given by using(6.12),(6.41),(6.42)and(6.44).
Next we have to give boundary conditions
where σ are the first Piola-Kirchhoff stress defined by(6.2)
The normal vector n on γ1is n-nαeα,
Hence boundary conditions are given by
The proof is completed.
Theorem 6.2.Assume that solution u of nonlinearly elastic operators defined by(6.11)and right hand term f can be made Taylor expansion with respect to ξ(6.19).Then the first Piola-Kirchhoff stress
defined by(6.2)on ℑ under the S-coordinates can be made Taylor expansion
where
Proof.As well known that according to(6.2)the first Piola Kirchhoff stress tensor are given by
Then using(2.57),(6.51)and(6.52)we can obtain Taylor expansion for σij. By similar manner we also can obtain for other expansions.The proof is completed.
Let us consider the hemi ellipsoid shell.As well known that parametric equation of the ellipsoid be given by
where(i,j,k)are Cartesian basis,(φ,θ)are the parameters and(x1=φ,x2=θ)are called Guassian coordinates of ellipsoid.The base vectors on the ellipsoid
The metric tensor of the ellipsoid is given by
Owing to
the coefficients of second fundamental form,i.e.curvature tensor of ellipsoid
Consequently,curvature tensor,mean curvature and Gaussian curvature are given by
Semi-Geodesic coordinate system based on ellipsoid ℑ
Note that
The radial vector at any point in ℜ3
We remainder to give the covariant derivatives of the velocity field,Laplace-Betrami operator and trac-Laplace operator.To do this we have to give the first and second kind of Christoffel symbols on the ellipsoid ℑ as a two dimensional manifolds
Then covariant derivatives of vector u=uαeα+u3n on the two dimensional manifold ℑ
As well known that displacement vector u=(u1,u2,u3)on middle surface of shell has three components,third component u3looked as scale function,is a Laplace-Betrami operator on ℑ which is given by
The trace-Laplace operatoe on ℑ is given by
Next let return to stationary equations(5.6)with boundary value
and consider associate variational formulations for(u,u1),find(,i=1,2,3)∈(ω)3×(ω)3,such that
Note that(5.13)shows that
Note that integrals by applying the Gaussian theorem become
Let us denote
By similar manner
In addition,
Therefore,we assert
The variational problem(7.13)can be rewritten as,findsuch that
Taking(7.5),(7.15)and Remark 5.1 into account,simple calculations show that
Substituting(7.22)into(7.18)and(7.20)leads to
The bilinear form of the variational problem(7.21)is given.
Acknowledgements
This research was supported by the National Natural Science Foundation of China(NSFC)(NO.11571275,11572244)and by the Natural Science Foundation of Shaanxi Province(NO.2018JM1014).
Journal of Mathematical Study2018年4期