Anthony N. Palazotto

Nonlinear analysis of shell structures

The increasing use of composite materials requires a better understanding of the behavior of laminated plates and shells. Large displacements and rotations, as well as shear deformations, must be included in the analysis. Since linear theories of shells and plates are no longer adequate for the analysis and design of composite structures, more refined theories are now used for such structures. This text develops, in a systematic manner, the overall concepts of the nonlinear analysis of shell structures. The authors start with a survey of theories for the analysis of plates and shells with small deflections and then lead to the theory of shells undergoing large deflections and rotations applicable to elastic laminated anisotropic materials. Subsequent chapters are devoted to the finite element solutions and include test case comparisons. The book is intended for graduate engineering students and stress analysts in aerospace, civil, or mechanical engineering.

Low-velocity impact damage initiation in graphite/epoxy/Nomex honeycomb-sandwich plates

Abstract Low-velocity impact and static indentation tests on sandwich plates composed of 4- to 48-ply graphite/epoxy cross-ply laminate facesheets and Nomex honeycomb cores have been performed to characterize damage initiation as a function of facesheet thickness and loading rate. Force histories during low-velocity impact are measured by using an instrumented impactor and integrated to produce energy histories. Energy histories are shown to reveal damage initiation. Static indentation tests show damage that is similar to that produced by low-velocity impact. The force at which damage initiates is shown to be lower for static tests than for low-velocity impact tests, and differences between equilibrium curves for the two types of loading are discussed. The difference between static and low-velocity impact tests is greater for plates with thicker facesheets. This may indicate a limitation of the applicability of the common assumption that low-velocity impact is a quasi-steady process.

FINITE ELEMENT ANALYSIS OF LOW VELOCITY IMPACT ON COMPOSITE SANDWICH PLATES

Abstract The response of composite sandwich plates to low-velocity impact is predicted by a displacement-based, plate bending, finite element algorithm. Fifth order Hermitian interpolation allows three-dimensional equilibrium integration for transverse stress calculations to be carried out symbolically on the interpolation functions so that transverse stresses within the elements are expressed directly in terms of nodal quantities. Nomex honeycomb sandwich core is modeled using an elastic–plastic foundation and contact loading is simulated by Hertzian pressure distribution for which the contact radius is determined iteratively. Damage prediction by failure criteria and damage progression via stiffness reduction are employed. Comparison to experimental low-velocity impact and static indentation data shows the ability to model some of the important features of static indentation of composite sandwich structures. In particular, the slope of the load displacement curve (stiffness), including contact, before damage is well represented. Core failure load is predicted by the analysis within 10% of the experimental value.

Vibration and buckling characteristics of composite cylindrical panels incorporating the effects of a higher order shear theory

Abstract An analytical study is conducted to determine the fundamental frequencies and critical buckling loads for laminated anisotropic circular cylindrical shell panels, including the effects of transverse shear deformation and rotary inertia, by using the Galerkin technique. A linearized form of Sanders shell strain-displacement relations are derived, which include a parabolic distribution of transverse shear strains. The theory is valid for laminate thickness to radius ratios h / R of up to 1/5. Higher order constitutive relations are derived for the laminate. A set of five coupled partial differential equations of motion and boundary conditions are derived and then solved using the modified Galerkin technique. Simply supported and clamped boundary conditions are investigated. Comparisons are made with the Donnell shell solutions. The effects of transverse shear deformation and rotary inertia arc examined by comparing the results with classical solutions, where applicable. The radius of curvature is varied to determine the effects of membrane and bending coupling. The theory compares exactly with the Donnell solutions, which are valid up to 1/50. As expected, as length to thickness ratios are reduced, shear deformation effects significantly lower the natural frequencies and buckling loads. Analysis also shows that rotary inertia effects are very small. Finally, as h / R is varied from 0 (flat plate) to 1/5. (maximum limit), the frequencies and buckling loads increase due to membrane and bending coupling.

Large-deformation analysis of flexible beams

Presented here are numerical verifications of a geometrically-exact curved beam model which fully accounts for large rotations, large displacements, initial curvatures and extensionality. A multiple shooting method is used to solve the two-point boundary-value problem of flexible beams undergoing large elastic rotations and displacements in three-dimensional space. Numerically exact large static deformations of eight beams subjected to different loading and/or boundary conditions are obtained. These solutions can be used to verify the performance of general finite-element codes in analyzing large structural deformations.

Low velocity impact damage characteristics of Z-fiber reinforced sandwich panels — an experimental study

Experimental results related to the initiation of damage in Z-fiber truss-reinforced sandwich panels are presented. Two different orientations of pin angles are considered. A pendulum-type impactor is used to impart very low levels of impact energy. Threshold energy levels at which the observable damage initiation occurs are determined. Nondestructive evaluation studies conducted prior to the impact are compared with the post-impact status of the panels for assessing the type and magnitude of the damage. These studies include ultrasonic inspection, microscopic inspection and acoustic emission tests. Damage is also evaluated by carrying out vibration tests before and after impact. Static indentation tests are carried out to compare with low velocity impact tests. Effects specific to the impact phenomena are studied by comparing the load displacement characteristics. Compression after impact characteristics are also evaluated.

A higher-order sandwich plate theory accounting for 3-D stresses

In this paper we extend a layerwise higher-order shear-deformation theory to model a sandwich plate impacting with an elastic foundation at a low velocity. A new concept of sublaminates is introduced, and the new sandwich plate theory satisfies the continuity conditions of interlaminar shear and normal stresses, accommodates the normal and shear stresses on the bonding surfaces, and accounts for non-uniform distributions of transverse shear stresses in each layer. Moreover, the use of sublaminates enables the modeling of shear warpings that change with the spatial location, vibration frequency, and loading and boundary conditions. A finite-element model based on this sandwich plate theory is derived for performing direct transient analyses to predict the initiation and location of critical matrix crack and the threshold of impact damage. Moreover, analytical shear warping functions, shear coupling functions, and normal strain functions due to in-plane stretching, bending, transverse shearing, and surface loading are presented.

Nonlinear finite element analysis of thick composite plates using cubic spline functions

A nonlinear, thick, composite plate element is developed in which the usual Kirchhoff hypothesis of plane sections remaining plane and undeformed after loading is abandoned. The displacement field is characterized by the sum of displacements with respect to a reference surface and displacements through the thickness. The through-the-thickness deformations are modeled by imposing a cubic spline function and allowing the rotations at interlaminar boundaries to be degrees of freedom in the element. The theory is developed by considering the Lagrangian strains in conjunction with the second Piola-Kirchhoff stress. This formulation leads to a quasi-threedimensional element that encompasses large displacements with moderately large rotations but is restricted to small strains. Comparisons of linear and nonlinear thick orthotropic plate solutions with those of previously published analytical and numerical results show the validity of the method.

Kelvin-Voigt vs fractional derivative model as constitutive relations for viscoelastic materials

Three simple constitutive relationships are studied for application to viscoelastic materials. Experimental results for both a rubbery and a glassy viscoelastic material are fit by the three schemes. The Kelvin-Voigt scheme is shown to be adequate in only limited frequency ranges. A three-parameter fractional order constitutive relationship provides a substantially better model over a much larger bandwidth. A four-parameter fractional model improves on the accuracy in materials with significant glassy regions

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

Abstract 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.