General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects

The paper presents an approach for a general laminated shell geometry describable by orthogonal curvilinear coordinates. The theory includes a through-the-thickness parabolic distribu- tion of transverse shear stress.-Additionally, a simplified approach that allows large displacements and rotations is incorporated. The theory is cast into a displacement-based finite element formula- tion and then specialized to a cylindrical shell geometry. The theory is then applied to the problem of a transversely loaded isotropic deep arch, and results show a slightly more flexible response compared with published results that are based upon inextensible assumptions. This problem also indicates that the usual locking associated with shell elements is apparently eliminated.

Large displacement and rotational formulation for laminated shells including parabolic transverse shear

A two-dimensional finite element model of a mandibular quadrant was used to examine the stresses and displacements resulting from a 100 N load placed as follows: (i) distributed on the second molar, (ii) concentrated at 30 degrees to the vertical on the second molar, and (iii) distributed on the second premolar and second molar. Each tooth was supported by periodontal ligaments, cortical and cancellous bone. The modulus and Poisson's ratios for each material were selected from accepted values. The principal stresses were determined throughout the model, with special emphasis being placed for elements in the immediate vicinity of the teeth mentioned above. It was found that for the case where the load was uniformly distributed on the second molar the stresses sigma min were mostly compressive (-) in nature while sigma max were mostly tensile (+). In the case where the load was concentrated and at 30 degrees to the vertical, the magnitude of the stresses were three to five times greater and the pattern less uniform than those resulting from the distributed load. The concentrated load at 30 degrees to the vertical resulted in higher bending stresses than the distributed load and hence higher sigma max stresses.

Finite element analysis of a mandibular model

The main source of mechanical damping in laminated composite materials arises from the inelastic nature of the matrix and the relative slipping at the fiber/matrix interfaces. Damping in laminated composite materials is usually a function of many parameters including the volume fraction of the fibers, fiber diameter and fiber orientation relative to the axis of loading. Also the magnitude and frequency of the applied load and many environmental factors should be mentioned. Since the complex phenomenon of damping is difficult to incorporate into the structural dynamic analysis of laminated plates, a viscous damping approximation is employed here. Experimental data on specific damping capacity (SDC) of unidirectional composite beams is used to define an average modal loss factor associated with each mode of vibration of a laminated plate. These loss factors are then employed with a finite element analysis and a multidimensional definition of critical damping to form a Rayleigh damping matrix [ C ] as a linear combination of the stiffness and mass matrices. Realistic examples illustrate the importance of several parameters in the vibration of laminated composite plates with damping. Such an analysis can be useful in the development of new composite materials where high damping is one of the primary objectives.

Viscous damping approximation of laminated anisotropic composite plates using the finite element method

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a two-dimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2014.0494

On a consistent finite-strain plate theory based on three-dimensional energy principle

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/26435/0000523.pdf?sequence=1

On a formulation of a nonlinear theory of plates and shells with applications

http://deepblue.lib.umich.edu/bitstream/handle/2027.42/27484/0000527.pdf

On a nonlinear theory of plates and shells including consistent and inconsistent kinematics and the finite element method

http://deepblue.lib.umich.edu/bitstream/handle/2027.42/23401/0000346.pdf?sequence=1

Kamal A. Meroueh

Papers

On a formulation of a nonlinear theory of plates and shells with applications

On a nonlinear theory of plates and shells including consistent and inconsistent kinematics and the finite element method

The interface crack loaded in bending