Formulation of Shell Elements Based on the Motion Formalism

Round 1

Reviewer 1 Report

This is a very well written paper which presents an interesting formulation for shell models. The developments follow a rigorous approach with strong argumentations justifying the approach. The results confirm the consistency of the method.

Nevertheless, I have two important remarks.

1. Obviously, the paper is strongly inspired from the PhD thesis of one of the authors (Mr. Sonneville) which was published several years ago. The position of the paper with respect to this previous work should definitely be clarified in the paper. Do the authors claim any originality with respect to this previous work? Or do they only want to give a fresh look on this previous work, even though there is no methodological innovation?

2. The problem of locking in finite element discretization of nonlinear structures is briefly mentioned in the introduction but is not addressed later in the paper. Even though the results tend to show that there is no locking issue, in the end, the paper does not really underline the advantages of the proposed method on this problem. Considering the importance of this issue in shell discretization, this point definitely needs to be addressed.

 

Author Response

The authors thank the reviewer for their insightful comments. 

Comments 1. Obviously, the paper is strongly inspired from the PhD thesis of one of the authors (Mr. Sonneville) which was published several years ago. The position of the paper with respect to this previous work should definitely be clarified in the paper. Do the authors claim any originality with respect to this previous work? Or do they only want to give a fresh look on this previous work, even though there is no methodological innovation?

Reply: This is clearly an oversight from our part. The following paragraph was added to the paper's introduction section to contrast the present work to that presented in Dr. Sonneville's doctoral thesis.

In his doctoral thesis, Sonneville~[32] presented a shell element based on the concept of local frames but key elements of his development differ from those presented here. First, this paper treats kinematics via dual orthogonal rather than homogeneous transformation matrices. Second, the present paper derives a closed-form solution of the implicit interpolation scheme for motion, leading to a closed-form expression for the interpolated strain field that simplifies the formulation considerably. In contrast, the thesis relies on an implicit interpolation scheme that has to be solved numerically at each Gauss point. Third, the thesis developed a consistent interpolation formula for the velocity field that satisfies the space-time Lie bracket identically; the simpler interpolation scheme developed in this paper proves to be effective. Fourth, the present paper treats body attached frames and change of frame operations in a consistent, streamlined manner; in contrast, the thesis uses different approaches for the representation of the two kinematic entities. Finally, the present paper presents realistic multibody applications whereas the doctoral thesis tackled benchmark examples only.

Comment 2. The problem of locking in finite element discretization of nonlinear structures is briefly mentioned in the introduction but is not addressed later in the paper. Even though the results tend to show that there is no locking issue, in the end, the paper does not really underline the advantages of the proposed method on this problem. Considering the importance of this issue in shell discretization, this point definitely needs to be addressed.

Reply. This is another oversight from out part. The following paragraph was added to the paper's at the end of the first example.

Bucalem and Bathe~[54] present an extensive review of the literature concerning the shear and membrane locking phenomena that plague shell elements based on classical kinematic descriptions, together with the numerous numerical techniques that have been proposed to remedy the problem. This first example reveals an important property of the proposed shell element: although no particular numerical technique was used to alleviate locking, the element appears to be locking free. This important property stem from the manner in which strain components are evaluated. For classical kinematics formulations, transverse shear strain components are the difference between the slope of the plate and the rotation of the normal material line. Because these two quantities are interpolated to different orders, the shear strain component cannot vanish over the element, as expected for thin plates, leading to locking. In the present formulation, all strain components are interpolated simultaneously using eq. (43), which does not involve differences between quantities interpolated to different orders, thereby relieving the locking phenomenon. While the proposed element does not lock, its rate of convergence decreases when dealing with very thin plates.

The following item was added to the list of advantageous features of the proposed element presented in the conclusion section.

(8) The element appear to be locking free.

Reviewer 2 Report

The paper deals with an analysis of plate and shell structures, by means of finite elements, based on a motion formalism that takes into account the kinematics of flexible multibody systems, by coupling their displacement and rotation components, and by solving the tensors components in local frames.After a deep introduction, where it is defined the context in which the analysis is positioned, the objectives of the study are very clearly defined. The approach employed is already presented in literature, but just for rigid multibody systems, or for beams, and it still is not commonly utilized in the context of mechanics and structural mechanics.  The main focus of the paper is to overcome this gap. The analysis is rigorous, well exposed, the text is well written. The figures, as well as the captions, are incisive, and help the reader in understanding the full text. Bibliography is adequate, and opportunely cited in the paper. In the sections, the main results are well explained, and in the conclusions they are summarized in a very precise way. For all these reasons, it is believed that the manuscript can be accepted for publication in the present form.

Author Response

The authors thank the reviewer for their insightful comments.

The reviewers did no request changes to the papers.

Reviewer 3 Report

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. Mathematical formulations are well and clear described, with interesting implementation in four examples with gradual complexity. The conclusions is supported by the results. It is a interesting and very well written paper.   

Author Response

The authors thank the reviewer for their insightful comments. 

The reviewers did not request changes to the paper.

 

Round 2

Reviewer 1 Report

the authors have properly answered my remarks. I am pleased to recommend the publication of the paper in its present form.