Determining Shear Stress Distribution in a Laminate

A "simplified shear solution" method approximates the through-thickness shear stress distribution within a composite laminate based on an extension of laminated beam theory. The method does not consider the solution of a particular boundary value problem; rather, it requires only knowledge of the global shear loading, geometry, and material properties of the laminate or panel. It is thus analogous to lamination theory in that ply-level stresses can be efficiently determined from global load resultants at a given location in a structure and used to evaluate the margin of safety on a ply-by-ply basis. The simplified shear solution stress distribution is zero at free surfaces, continuous at ply boundaries, and integrates to the applied shear load. The method has been incorporated within the HyperSizer commercial structural sizing software to improve its predictive capability for designing composite structures. The HyperSizer structural sizing software is used extensively by NASA to design composite structures. In the case of through-thickness shear loading on panels, HyperSizer previously included a basic, industry-standard, method for approximating the resulting shear stress distribution in sandwich panels. However, no such method was employed for solid laminate panels. The purpose of the innovation is to provide an approximation of the through-thickness shear stresses in a solid laminate given the through-thickness shear loads (Qx and Qy) on the panel. The method was needed for implementation within the HyperSizer structural sizing software so that the approximated ply-level shear stresses could be utilized in a failure theory to assess the adequacy of a panel design. The simplified shear solution method was developed based on extending and generalizing bi-material beam theory to plate-like structures. It is assumed that the through-thickness shear stresses arise due to local bending of the laminate induced by the through-thickness shear load, and by imposing equilibrium both vertically and horizontally, the through-thickness shear stress distribution can be calculated. The resulting shear stresses integrate to the applied shear load, are continuous at the ply interfaces, and are zero at the laminate-free surfaces. If both Qx and Qy shear loads are present, it is assumed that they act independently and that their effects can be superposed. The calculated shear stresses can be rotated within each ply to the principal material coordinates for use in a ply-level failure criterion. The novelty of the simplified shear solution method is its simplicity and the fact that it does not require solution of a particular boundary value problem. The advantages of the innovation are that an approximation of the though-thickness shear stress distribution can be quickly determined for any solid laminate or solid laminate region within a stiffened panel

Analysis of Fiber Clustering in Composite Materials Using High-Fidelity Multiscale Micromechanics

A new multiscale micromechanical approach is developed for the prediction of the behavior of fiber reinforced composites in presence of fiber clustering. The developed method is based on a coupled two-scale implementation of the High-Fidelity Generalized Method of Cells theory, wherein both the local and global scales are represented using this micromechanical method. Concentration tensors and effective constitutive equations are established on both scales and linked to establish the required coupling, thus providing the local fields throughout the composite as well as the global properties and effective nonlinear response. Two nondimensional parameters, in conjunction with actual composite micrographs, are used to characterize the clustering of fibers in the composite. Based on the predicted local fields, initial yield and damage envelopes are generated for various clustering parameters for a polymer matrix composite with both carbon and glass fibers. Nonlinear epoxy matrix behavior is also considered, with results in the form of effective nonlinear response curves, with varying fiber clustering and for two sets of nonlinear matrix parameters

The Equivalence of the Radial Return and Mendelson Methods for Integrating the Classical Plasticity Equations

The radial return and Mendelson methods for integrating the equations of classical plasticity, which appear independently in the literature, are shown to be identical. Both methods are presented in detail as are the specifics of their algorithmic implementation. Results illustrate the methods' equivalence across a range of conditions and address the question of when the methods require iteration in order for the plastic state to remain on the yield surface. FORTRAN code implementations of the radial return and Mendelson methods are provided in the appendix

FORTRAN Versions of Reformulated HFGMC Codes

Several FORTRAN codes have been written to implement the reformulated version of the high-fidelity generalized method of cells (HFGMC). Various aspects of the HFGMC and its predecessors were described in several prior NASA Tech Briefs articles, the most recent being HFGMC Enhancement of MAC/GMC (LEW-17818-1), NASA Tech Briefs, Vol. 30, No. 3 (March 2006), page 34. The HFGMC is a mathematical model of micromechanics for simulating stress and strain responses of fiber/matrix and other composite materials. The HFGMC overcomes a major limitation of a prior version of the GMC by accounting for coupling of shear and normal stresses and thereby affords greater accuracy, albeit at a large computational cost. In the reformulation of the HFGMC, the issue of computational efficiency was addressed: as a result, codes that implement the reformulated HFGMC complete their calculations about 10 times as fast as do those that implement the HFGMC. The present FORTRAN implementations of the reformulated HFGMC were written to satisfy a need for compatibility with other FORTRAN programs used to analyze structures and composite materials. The FORTRAN implementations also afford capabilities, beyond those of the basic HFGMC, for modeling inelasticity, fiber/matrix debonding, and coupled thermal, mechanical, piezo, and electromagnetic effects

Determination of the Shear Stress Distribution in a Laminate from the Applied Shear Resultant--A Simplified Shear Solution

The simplified shear solution method is presented for approximating the through-thickness shear stress distribution within a composite laminate based on laminated beam theory. The method does not consider the solution of a particular boundary value problem, rather it requires only knowledge of the global shear loading, geometry, and material properties of the laminate or panel. It is thus analogous to lamination theory in that ply level stresses can be efficiently determined from global load resultants (as determined, for instance, by finite element analysis) at a given location in a structure and used to evaluate the margin of safety on a ply by ply basis. The simplified shear solution stress distribution is zero at free surfaces, continuous at ply boundaries, and integrates to the applied shear load. Comparisons to existing theories are made for a variety of laminates, and design examples are provided illustrating the use of the method for determining through-thickness shear stress margins in several types of composite panels and in the context of a finite element structural analysis

Micromechanics Modeling of Composites Subjected to Multiaxial Progressive Damage in the Constituents

The high-fidelity generalized method of cells composite micromechanics model is extended to include constituent-scale progressive damage via a proposed damage model. The damage model assumes that all material nonlinearity is due to damage in the form of reduced stiffness, and it uses six scalar damage variables (three for tension and three for compression) to track the damage. Damage strains are introduced that account for interaction among the strain components and that also allow the development of the damage evolution equations based on the constituent material uniaxial stress strain response. Local final-failure criteria are also proposed based on mode-specific strain energy release rates and total dissipated strain energy. The coupled micromechanics-damage model described herein is applied to a unidirectional E-glass/epoxy composite and a proprietary polymer matrix composite. Results illustrate the capability of the coupled model to capture the vastly different character of the monolithic (neat) resin matrix and the composite in response to far-field tension, compression, and shear loading

MAC/GMC Code Enhanced for Coupled Electromagnetothermoelastic Analysis of Smart Composites

Intelligent materials are those that exhibit coupling between their electromagnetic response and their thermomechanical response. This coupling allows smart materials to react mechanically (e.g., an induced displacement) to applied electrical or magnetic fields (for instance). These materials find many important applications in sensors, actuators, and transducers. Recently interest has arisen in the development of smart composites that are formed via the combination of two or more phases, one or more of which is a smart material. To design with and utilize smart composites, designers need theories that predict the coupled smart behavior of these materials from the electromagnetothermoelastic properties of the individual phases. The micromechanics model known as the generalized method of cells (GMC) has recently been extended to provide this important capability. This coupled electromagnetothermoelastic theory has recently been incorporated within NASA Glenn Research Center's Micromechanics Analysis Code with Generalized Method of Cells (MAC/GMC). This software package is user friendly and has many additional features that render it useful as a design and analysis tool for composite materials in general, and with its new capabilities, for smart composites as well

HFGMC Enhancement of MAC/GMC

Additional information about a mathematical model denoted the high-fidelity generalized method of cells (HFGMC) and implementation of the HFGMC within version 4.0 of the MAC/GMC software has become available. MAC/GMC (Micromechanics Analysis Code With Generalized Method of Cells) was a topic of several prior NASA Tech Briefs articles, version 4.0 having been described in "Comprehensive Micromechanics-Analysis Code - Version 4.0" (LEW-17495-1), NASA Tech Briefs, Vol. 29, No. 9 (September 2005), page 54. MAC/GMC predicts elastic and inelastic thermomechanical responses of composite materials. MAC/GMC utilizes the generalized method of cells (GMC) - a model of micromechanics that predicts macroscopic responses of a composite material as functions of the properties, sizes, shapes, and responses of its constituents (e.g., matrix and fibers). The accuracy of the GMC is limited by neglect of coupling between normal and shear stresses. The HFGMC was developed by combining elements of the GMC and a related model, denoted the higher-order theory for functionally graded materials (HOTFGM), that can account for this coupling. Hence, the HFGMC enables simulation of stress and strain with greater accuracy. Some alterations of the MAC/GMC data structure were necessitated by the greater computational complexity of the HFGMC

Investigating the Impact of Target Firm Members’ Trust Antecedents on the Perofrmance of Cross-border Acquisition (CBA)

Cross-border acquisitions (CBAs), as organisational entities which are formed and controlled by foreign firms, are one of the most effective methods of expansion available to international firms. Cross-border acquisition is a highly complex and multi-faceted business model which presents several challenges for management. Over the past several years, CBA has become a more frequent target for research in business, and has received significant attention from academia and practitioners alike. The aim of this research is to investigate the factors that impact interorganizational trust and as an outcome, the performance of CBAs in Iraq and the United Arab Emirates (UAE). According Fadol and Sandhu, (2013), Abosag and Lee, (2012), there are few studies that focused on alliances in Middle Eastern countries and more studies on these areas are required. The empirical study is preceded by a review of the existing literature that led to the development of a conceptual framework for this research. The target population samples for this research were collected from four target firm’s employees in Iraq and the UAE, where those firms were identified as having acquired by firms from Western Europe and Asia. The research tool used to collect the necessary data was a survey questionnaire, and the data analysis was undertaken using SPSS and Structural Equation Modelling (SEM). The findings of this study identify that relationship history, inter-firm distance, and integration approach factors have a positive impact on the level of target firm members’ trust; while factors such as collaboration history, complementarity, and autonomy preservation were not found to have a positive and significantly related to the trust. Moreover, Country risk was found to have a negative and significant relationship with the trust. Religion was not found to be positive and significantly related to the trust. Furthermore, positive relationship between trust and performance and between commitment and performance were found. The results of this study offer a framework that identifies the key factors in the development of trust within CBAs and demonstrates that this trust and it’s antecedents have a significant impact on firm performance in Middle Eastern countries such as Iraq and UAE. The findings of this study offer us implications for meaningful managerial practice in selecting cross-border competitors as target firms

Micromechanics of Spray-On Foam Insulation

Understanding the thermo-mechanical response of the Space Shuttle External Tank spray-on foam insulation (SOFI) material is critical, to NASA's Return to Flight effort. This closed-cell rigid polymeric foam is used to insulate the metallic Space Shuttle External Tank, which is at cryogenic temperatures immediately prior to and during lift off. The shedding of the SOFI during ascent led to the loss of the Columbia, and eliminating/minimizing foam lass from the tank has become a priority for NASA as it seeks to resume scheduled space shuttle missions. Determining the nature of the SOFI material behavior in response to both thermal and mechanical loading plays an important role as any structural modeling of the shedding phenomenon k predicated on knowledge of the constitutive behavior of the foam. In this paper, the SOFI material has been analyzed using the High-Fidelity Generalized Method of Cells (HFGMC) micromechanics model, which has recently been extended to admit a triply-periodic 3-D repeating unit cell (RUC). Additional theoretical extensions that mere made in order to enable modeling of the closed-cell-foam material include the ability to represent internal boundaries within the RUC (to simulated internal pores) and the ability to impose an internal pressure within the simulated pores. This latter extension is crucial as two sources contribute to significant internal pressure changes within the SOFI pores. First, gas trapped in the pores during the spray process will expand or contract due to temperature changes. Second, the pore pressure will increase due to outgassing of water and other species present in the foam skeleton polymer material. With HFGMC's new pore pressure modeling capabilities, a nonlinear pressure change within the simulated pore can be imposed that accounts for both of these sources, in addition to stmdar&-thermal and mechanical loading; The triply-periodic HFGMC micromechanics model described above was implemented within NASA GRC's MAC/GMC software package, giving the model access to a range of nonlinear constitutive models for the polymeric foam skeleton material. A repeating unit cell architecture was constructed that, while relatively simple, still accounts for the geometric anisotropy of the porous foam microstructure and its thin walls and thicker edges. With the lack of reliable polymeric foam skeleton materia1 properties, many simulations were executed aimed at backing out these material properties. Then, using these properties, predictions of the thermo-mechanical behavior of the foam, including calculated internal applied pressure profiles, were performed and compared with appropriate experimental data