Ronald C. Averill

Static and dynamic response of moderately thick laminated beams with damage

Abstract The bending and vibration response of thin and moderately thick laminated beams with ply damage or delaminations is studied using a new discrete-layer laminated beam finite element. The two-noded C° element is based on a new generalized form of the first-order zig-zag laminate theory, and has only four degrees of freedom per node, regardless of the number of layers in the laminate. The element formulation employs the penalty-function concept in conjunction with an interdependent element interpolation scheme, making the element very accurate and robust for application to thick and thin laminated beams. The element stiffness coefficients are exactly integrated without giving rise to shear locking, and a consistent force and mass matrix is derived. The compliant layer concept is employed to simulate delaminations. Comparison of numerical results using the current model with predictions of exact elasticity solutions demonstrates that the current model is capable of accurately simulating the response of thin and moderately thick laminates that contain damage. Additional comparisons with equivalent single-layer theories indicate that the layerwise construction and kinematics must be explicitly taken into account when modelling laminates whose layers have drastically different stiffness properties, as is the case when damage is present.

First-order zig-zag sublaminate plate theory and finite element model for laminated composite and sandwich panels

A refined laminated plate theory and three-dimensional finite element based on first-order zig-zag sublaminate approximations has been developed. The in-plane displacement fields in each sublaminate are assumed to be piecewise linear functions and vary in a zig-zag fashion through-the-thickness of the sublaminate. The zig-zag functions are evaluated by enforcing the continuity of transverse shear stresses at layer interfaces. This in-plane displacement field assumption accounts for discrete layer effects without increasing the number of degrees of freedom as the number of layers is increased. The transverse displacement field is assumed to vary linearly through-the-thickness. The transverse normal strain predictions are improved by assuming a constant variation of transverse normal stress in each sublaminate. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of an eight-noded brick, allowing the thickness of the plate to be discretized into several elements, or sublaminates, where each sublaminate can contain more than one physical layer. Each node has five engineering degrees of freedom, three translations and two rotations. Thus, this element can be conveniently implemented into general purpose finite element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of an interdependent interpolation scheme and consistent shear strain fields. Numerical performance of the current element is investigated for a composite armored vehicle panel and a sandwich panel. These tests demonstrate that the element is very accurate and robust.

Development of simple, robust finite elements based on refined theories for thick laminated beams

An approach for developing accurate, simple and robust two-noded C0 finite elements based on shear deformable and layerwise (zig-zag) laminated beam theories is presented. First, a new generalized form of high-order shear deformation and zig-zag laminate theories is described that allows the continuity requirement on the transverse deflection degree freedom to be reduced. An associated finite element model is then developed that employs the penalty function concept in conjunction with an interdependent element interpolation scheme, making the element very accurate and robust for application to thick and thin laminated beams. The element stiffness coefficients are exactly integrated without giving rise to shear locking, and a consistent force vector is derived. Comparison of numerical results using the refined laminate models with exact elasticity solutions demonstrates that the current model is capable of accurately simulating the response of thin and moderately thick laminates, with or without damage.

An improved theory and finite-element model for laminated composite and sandwich beams using first-order zig-zag sublaminate approximations

Abstract A new beam finite element based on a new discrete-layer laminated beam theory with sublaminate first-order zig-zag kinematic assumptions is presented and assessed for thick and thin laminated beams. The model allows a laminate to be represented as an assemblage of sublaminates in order to increase the model refinement through the thickness, when needed. Within each sublaminate, discrete-layer effects are accounted for via a modified form of DiSciuvas linear zig-zag laminate kinematics, in which continuity of interfacial transverse shear stresses is satisfied identically. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of a fournoded rectangle, allowing the thickness of the beam to be discretized into several elements, or sublaminates, if necessary, to improve accuracy. Each node has three engineering degrees of freedom, two translations and one rotation. Thus, this element can be conveniently implemented into general purpose finite-element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of a consistent interdependent interpolation scheme. Numerical performance of the current element is investigated for an arbitrarily layered beam, a symmetrically layered beam and a sandwich beam with low and high aspect ratios. The comparisons of numerical results with elasticity solutions show that the element is very accurate and robust.

A penalty-based finite element interface technology

Abstract An effective and robust interface element technology able to connect independently modeled finite element subdomains is presented. This method has been developed using the penalty constraints and allows coupling of finite element models whose nodes do not coincide along their common interface. Additionally, the present formulation leads to a computational approach that is very efficient and completely compatible with existing commercial software. A significant effort has been directed toward identifying those model characteristics (element geometric properties, material properties and loads) that most strongly affect the required penalty parameter, and subsequently to developing simple “formulae” for automatically calculating the proper penalty parameter for each interface constraint. This task is especially critical in composite materials and structures, where adjacent subregions may be composed of significantly different materials or laminates. This approach has been validated by investigating a variety of two-dimensional problems, including composite laminates.

Finite element analysis of textile composite preform stamping

The forming or draping of a textile composite preform may result in large changes in the fibrous microstructure of the preform. This change in the local fiber orientation leads to significant changes in the fabric permeability as well as the mechanical properties of the ensuing composite structure. Therefore, this change in orientation of the tows of the preform needs to be known accurately to calculate the various effective properties of the composite. A new finite element approach for stamping analysis of a plain-weave textile composite preform has been developed. This model is simple, efficient and can be used in the existing finite element codes. The model represents the preform as a mesh of 3-D truss elements and 3-D shell elements. The truss elements model the tows, which are allowed to both scissor and slide relative to one another. The shell elements represent a fictitious material that accounts for inter-tow friction and fiber angle jamming. The model takes into account large strains and large deformations. In-plane uniaxial tension tests have been performed on plain-weave specimens for determining the constitutive law of the transforming medium and to show the inter-tow sliding. Application of the model is demonstrated by simulating the stamping of a preform by a spherical punch. The results from the simulation show good correlation with results from the experiments.

Optimal design of flywheels using an injection island genetic algorithm

This paper presents an approach to optimal design of elastic flywheels using an Injection Island Genetic Algorithm (iiGA), summarizing a sequence of results reported in earlier publications. An iiGA in combination with a structural finite element code is used to search for shape variations and material placement to optimize the Specific Energy Density (SED, rotational energy per unit weight) of elastic flywheels while controlling the failure angular velocity. iiGAs seek solutions simultaneously at different levels of refinement of the problem representation (and correspondingly different definitions of the fitness function) in separate subpopulations (islands). Solutions are sought first at low levels of refinement with an axi-symmetric plane stress finite element code for high-speed exploration of the coarse design space. Next, individuals are injected into populations with a higher level of resolution that use an axi-symmetric three-dimensional finite element code to “fine-tune” the structures. A greatly simplified design space (containing two million possible solutions) was enumerated for comparison with various approaches that include: simple GAs, threshold accepting (TA), iiGAs and hybrid iiGAs. For all approaches compared for this simplified problem, all variations of the iiGA were found to be the most efficient. This paper will summarize results obtained studying a constrained optimization problem with a huge design space approached with parallel GAs that had various topological structures and several different types of iiGA, to compare efficiency. For this problem, all variations of the iiGA were found to be extremely efficient in terms of computational time required to final solution of similar fitness when compared to the parallel GAs.

Optimal design of laminated composite structures using coarse-grain parallel genetic algorithms

Abstract A new coarse-grain parallel architecture for genetic algorithms, called island injection genetic algorithms, is implemented for the optimal design of laminated composite structures. This approach represents the design at various levels of refinement in subpopulations on separate computational nodes, then seeks good designs at each level of resolution and injects these good solutions into a population (node) with higher resolution to “fine-tune” the design. Numerical results are presented for the optimal design of a laminated composite beam to maximize its capacity to absorb mechanical energy without fracture. It is shown that super linear speedup can be achieved by using the injection algorithm.

Using genetic algorithms to design laminated composite structures

By narrowing the search space, genetic algorithms facilitate the design of 2D and 3D laminated composite beams. Adapting these algorithms for parallel distributed processing can improve their efficiency and accuracy. >

Parallel Genetic Algorithms in the Optimization of Composite Structures

Genetic Algorithms (GAs) are a powerful technique for search and optimization problems, and are particularly useful in the optimization of composite structures. The search space for an optimal composite structure is generally discontinuous and strongly multimodal, with the possibility for many local sub-optimal solutions or even singular extrema. These facts severely limit gradient-type approaches to optimization, bringing this broad class of problems under scrutiny for application of GAs. Examples described here of the successful use of parallel GAs to design composite structures by the authors include energy-absorbing laminated beams [1], airfoils with tailored bending-twisting coupling [2], and flywheel structures [3]. Optimal design of laminated composite beams was performed using a GA with a specialized finite element model to design material stacking sequences to maximize the mechanical energy absorbed before fracture. An initial GA approach to the optimal design of a specialized, idealized composite airfoil is now being refined for a practical application. The optimum stacking sequence to produce a desired twisting response while minimizing weight, maximizing in-plane stiffness and maintaining acceptable stress levels is determined. The GA has also been used to maximize the Specific Energy Density (SED) of composite flywheels. SED is defined as the amount of rotational energy stored per unit mass. Optimization of SED was achieved by allowing the GA to search for various flywheel shapes and allowing the GA to pick material sequences along the radius of the flywheel.