摘要
In this paper, we consider a linearly elastic shell, i.e. a three-dimensional linearly elastic body with a small thickness denoted by 2ε, which is clamped along its part of the lateral boundary and subjected to the regular loads. In the linear case, one can use the two-dimensional models of Ciarlet or Koiter to calculate the displacement for the shell. Some error estimates between the approximate solution of these models and the three-dimensional displacement vector field of a flexural or membrane shell have been obtained. Here we give a new model for a linear and nonlinear shell, prove that there exists a unique solution U of the two-dimensional variational problem and construct a three-dimensional approximate solutions UKT(x,ξ) in terms of U: We also provide the error estimates between our model and the three-dimensional displacement vector field :‖u-UKT‖1,Ω≤C∈r,r=3/2, an elliptic membrane, r = 1/2, a general membrane, where C is a constant dependent only upon the data‖u‖3,Ω,‖UKT‖3,Ω,θ.
In this paper, we consider a linearly elastic shell, I.e. A three-dimensional linearly elastic body with a small thickness denoted by 2ε, which is clamped along its part of the lateral boundary and subjected to the regular loads. In the linear case, one can use the two-dimensional models of Ciarlet or Koiter to calculate the displacement for the shell. Some error estimates between the approximate solution of these models and the three-dimensional displacement vector field of a flexural or membrane shell have been obtained. Here we give a new model for a linear and nonlinear shell, prove that there exists a unique solution U of the two-dimensional variational problem and construct a three-dimensional approximate solutions UKT(x,ξ) in terms of U:{ UKT(x,ξ):=U(x)+П1Uξ+П2Uξ2,П1U=-aαβ*▽βU3→eα-λ0γ0(U)→n,П2U=(1/λ+μ)*▽β(aαβλσ+γλσ(U))-bαβ*▽βU3)→eα+1/2λ0(ρKT0(U)-(1+λ0)Hγ0(U)-2β0(U))→n.We also provide the error estimates between our model and the three-dimensional displacement vector field:‖u-UKT‖1,(Ω)≤Cεr, r=3/2, an elliptic membrane, r=1/2, a general membrane,where C is a constant dependent only upon the data ‖u‖3,Ω, ‖UKT‖3,Ω, →θ.
基金
This work was supported by the National Natural Science Foundation of China(Grant Nos.10571142,10001028,50136030,50306019,40375010,10571148&10471110).
