# Micromechanics approach

Irrespective of the micromechanical stiffness approach used, the basic restrictions on the composite material that can be treated are [c.123]

Irrespective of the analysis approach, the representative volume element must be carefully defined and used. In fact, the representative volume element is crucial to the analysis and is the micromechanics analog of the free-body diagram in statics and dynamics. The representative volume element is of higher order than the free-body diagram because deformations and stresses are addressed in addition to forces. [c.125]

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., E2, arid

The division of micromechanics stiffness evaluation efforts into the mechanics of materials approach and the elasticity approach with its many subapproaches is rather arbitrary. Chamis and Sendeckyj [3-5] divide micro mechanics stiffness approaches into many more classes netting analyses, mechanics of materials approaches, self-consistent models, variational techniques using energy-bounding principles, exact solutions, statistical approaches, finite element methods, semiempirical approaches, and microstructure theories. All approaches have the common objective of the prediction of composite materials stiffnesses. All except the first two approaches use some or all of the principles of elasticity theory to varying degrees so are here classed as elasticity approaches. This simplifying and arbitrary division is useful in this book because the objective here is to merely become acquainted with advanced micromechanics theories after the basic concepts have been introduced by use of typical mechanics of materials reasoning. The reader who is interested in micromechanics should supplement this chapter with the excellent critique and extensive bibliography of Chamis and Sendeckyj [3-5]. [c.137]

There is much controversy associated with micromechanical analyses and predictions. Much of the controversy has to do with which approximations should be used. The Halpin-Tsai equations seem to be a commonly accepted approach. [c.157]

The micromechanical behavior of a lamina was treated in Chapter 3. Both a mechanics of materials and an elasticity approach were used to predict the fundamental lamina stiffnesses that were compared to measured stiffnesses. Mechanics of materials approaches were used to predict some of the fundamental strengths of a lamina. [c.332]

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. [c.362]

The whole gamut of topics from micromechanics and macromechanics through lamination theory and examples of plate bending, buckling, and vibration problems is treated so that the physical significance of the concepts is made clear. A comprehensive introduction to composite materials and motivation for their use in current structural applications is given in Chapter 1. Stress-strain relations for a lamina are displayed with engineering material constants in Chapter 2. Strength theories are also compared with experimental results. In Chapter 3, micromechanics is introduced by both the mechanics of materials approach and the elasticity approach. Predicted moduli are compared with measured values. Lamination theory is presented in Chapter 4 with the aid of a new laminate classification scheme. Laminate stiffness predictions are compared with experimental results. Laminate strength con- [c.539]

Irrespective of the micromechanical stiffness approach used, the basic restrictions on the composite material that can be treated are [c.123]

Irrespective of the analysis approach, the representative volume element must be carefully defined and used. In fact, the representative volume element is crucial to the analysis and is the micromechanics analog of the free-body diagram in statics and dynamics. The representative volume element is of higher order than the free-body diagram because deformations and stresses are addressed in addition to forces. [c.125]

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., Eg, v g. and G g for an orthotropic material are introduced. In Section 3.3, the elasticity approach to the micromechanics of material stiffnesses is addressed. Bounding techniques, exact solutions, the concept of contiguity, and the Halpin-Tsai approximate equations are all examined. Next, the various approaches to prediction of stiffness are compared in Section 3.4 with experimental data for both particulate composite materials and fiber-reinforced composite materials. Parallel to the study of the micromechanics of material stiffnesses is the micromechanics of material strengths which is introduced in Section 3.5. There, mechanics of materials predictions of tensile and compressive strengths are described. [c.126]

The division of micromechanics stiffness evaluation efforts into the mechanics of materials approach and the elasticity approach with its many subapproaches is rather arbitrary. Chamis and Sendeckyj [3-5] divide micro mechanics stiffness approaches into many more classes netting analyses, mechanics of materials approaches, self-consistent models, variational techniques using energy-bounding principles, exact solutions, statistical approaches, finite element methods, semiempirical approaches, and microstructure theories. All approaches have the common objective of the prediction of composite materials stiffnesses. All except the first two approaches use some or all of the principles of elasticity theory to varying degrees so are here classed as elasticity approaches. This simplifying and arbitrary division is useful in this book because the objective here is to merely become acquainted with advanced micromechanics theories after the basic concepts have been introduced by use of typical mechanics of materials reasoning. The reader who is interested in micromechanics should supplement this chapter with the excellent critique and extensive bibliography of Chamis and Sendeckyj [3-5]. [c.137]

There is much controversy associated with micromechanical analyses and predictions. Much of the controversy has to do with which approximations should be used. The Halpin-Tsai equations seem to be a commonly accepted approach. [c.157]

The micromechanical behavior of a lamina was treated in Chapter 3. Both a mechanics of materials and an elasticity approach were used to predict the fundamental lamina stiffnesses that were compared to measured stiffnesses. Mechanics of materials approaches were used to predict some of the fundamental strengths of a lamina. [c.332]

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. [c.362]

The whole gamut of topics from micromechanics and macromechanics through lamination theory and examples of plate bending, buckling, and vibration problems is treated so that the physical significance of the concepts is made clear. A comprehensive introduction to composite materials and motivation for their use in current structural applications is given in Chapter 1. Stress-strain relations for a lamina are displayed with engineering material constants in Chapter 2. Strength theories are also compared with experimental results. In Chapter 3, micromechanics is introduced by both the mechanics of materials approach and the elasticity approach. Predicted moduli are compared with measured values. Lamination theory is presented in Chapter 4 with the aid of a new laminate classification scheme. Laminate stiffness predictions are compared with experimental results. Laminate strength con- [c.537]

The identification of an apparent Bauschinger effect contribution to shock loading is not a new idea [44]-[46]. This concept has been utilized to explain the deviation of the shock unloading stress-strain path from that predicted by simple elastic-plastic theory [44]-[46]. In materials exhibiting ideal elastic-plastic behavior, the unloading path will consist of a purely elastic wave to the reverse yield surface and then an ideally plastic wave. In reality, experimentally measured unloading wave profiles show evidence of the onset of plastic flow occurring immediately upon release from the shock state. This results in a gradual transition to the fully plastic state without a clearly defined elastic component. This phenomena is termed quasi-elastic release. Utilization of the Bauschinger effect to explain this phenomena has been restricted to manipulation of the unloading stress-strain path until the wave profile was satisfactorialy reproduced. In the modeling work of Steinberg et al. [46] the transition from an ideal elastic-plastic release path to a curved quasi-elastic path is reproduced with the use of a variable effective shear modulus. While this modeling approach has duplicated the unloading profiles, it does not explain the micromechanisms of structure evolution, defect storage processes, and/or provide physical understanding of the shock-release process to attain predictive capability of material behavior. [c.206]

Halpin and Tsai [3-17] developed an interpolation procedure that is an approximate representation of more complicated micromechanics results. The beauty of the procedure is twofoldr-First. it is simple, so it can readily be used in the design process. Second, it enables the generalization of usually limited, although more exact, micromechanics results. Moreover, the procedure is apparently gnltp accurate if the fiber-volume fraction (Vf) does not approach one. [c.151]

The rule of mixtures is a very satisfactory approach to predicting the stiffness behavior of the composite material in the fiber direction. However, the analytical tools for prediction of the behavior transverse to the fiber direction simply do not work out well. The other analyses are not accurate enough to claim that micromechanics is a valid and effective design-analysis tool. Moreover, since the 1960s, we have changed from large-diameter, regular-array composite materials, such as boron-epoxy, when micromechanics was developed to small-diameter, irregular-array composite materials such as graphite-epoxy and Kevlar-epoxy. Thus, we simply cannot even begin to claim that the analyses that we formerly used for boron-epoxy, which were not very good then, are at all applicable to graphite-epoxy. [c.457]

Halpin and Tsai [3-17] developed an interpolation procedure that is an approximate representation of more complicated micromechanics results. The beauty of the procedure is twofolck-First. it is simple, so it can readily be used in the design process. Second, it enables the generalization of usually limited, although more exact, micromechanics results. Moreover, the procedure is apparently gnltp accurate if the fiber-volume fraction (Vf) does not approach one. [c.151]

The rule of mixtures is a very satisfactory approach to predicting the stiffness behavior of the composite material in the fiber direction. However, the analytical tools for prediction of the behavior transverse to the fiber direction simply do not work out well. The other analyses are not accurate enough to claim that micromechanics is a valid and effective design-analysis tool. Moreover, since the 1960s, we have changed from large-diameter, regular-array composite materials, such as boron-epoxy, when micromechanics was developed to small-diameter, irregular-array composite materials such as graphite-epoxy and Kevlar-epoxy. Thus, we simply cannot even begin to claim that the analyses that we formerly used for boron-epoxy, which were not very good then, are at all applicable to graphite-epoxy. [c.457]

Mechanics of composite materials (1999) -- [ c.122 , c.126 , c.127 , c.128 , c.129 , c.130 , c.131 , c.132 , c.133 , c.134 , c.135 ]

Machanics of composite materials (1998) -- [ c.122 , c.126 , c.127 , c.128 , c.129 , c.130 , c.131 , c.132 , c.133 , c.134 , c.135 ]