# Mechanics of materials approach of stiffness

MECHANICS OF MATERIALS APPROACH TO STIFFNESS [c.126]

MECHANICS OF MATERIALS APPROACH TO STIFFNESS [c.126]

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]

The mechanics of materials approach to the estimation of stiffness of a composite material has been shown to be an upper bound on the actual stiffness. Paul [3-4] compared the upper and lower bound stiffness predictions with experimental data [3-24 and 3-25] for an alloy of tungsten carbide in cobalt. Tungsten carbide (WC) has a Young s modulus of 102 X 10 psi (703 GPa) and a Poisson s ratio of. 22. Cobalt (Co) has a Young s modulus of 30x 10 psi (207 GPa) and a Poisson s ratio of. 3. [c.158]

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]

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]

The mechanics of materials approach to the estimation of stiffness of a composite material has been shown to be an upper bound on the actual stiffness. Paul [3-4] compared the upper and lower bound stiffness predictions with experimental data [3-24 and 3-25] for an alloy of tungsten carbide in cobalt. Tungsten carbide (WC) has a Young s modulus of 102 X 10 psi (703 GPa) and a Poisson s ratio of. 22. Cobalt (Co) has a Young s modulus of 30x 10 psi (207 GPa) and a Poisson s ratio of. 3. [c.158]

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]

In Chapter 1 the general mechanical properties of plastics were introduced. In order to facilitate comparisons with the behaviour of other classes of materials the approach taken was to refer to standard methods of data presentation, such as stress-strain graphs, etc. However, it is important to note that when one becomes involved in engineering design with plastics, such graphs are of limited value. The reason is that they are the results of relatively shortterm tests and so their use is restricted to quality control and, perhaps, the initial sorting of materials in terms of stiffness, strength etc. Designs based on, say, the modulus obtained from a short-term test would not predict accurately the long-term behaviour of plastics because they are viscoelastic materials. This viscoelasticity means that quantities such as modulus, strength, ductility and coefficient of friction are sensitive to straining rate, elapsed time, loading history, temperature, etc. It will also be shown later that the manufacturing method used for the plastic product can create changes in the structure of the material which have a pronounced effect on properties. The behaviour of the moulded product may therefore be different from the behaviour of a moulded test-piece of the same material. [c.41]

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]

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]

Most measurements of the strength of a material are based on uniaxial stress states. However, the general practical design problem involves at least a biaxial if not a triaxial state of stress. Thus, a logical method of using uniaxial strength information obtained in principal material coordinates is required for analysis of multiaxial loading problems. Obtaining the strength characteristics of a lamina at all possible orientations is physically impossible, so a method must be determined for obtaining the characteristics at any orientation in terms of characteristics in the principal material coordinates. In such an extension of the information obtained in principal material coordinates, the well-known concepts of principal stresses and principal strains are of no value. A multitude of possible microscopic failure mechanisms exists, so a tensor transformation of strengths is very difficult. Moreover, tensor transformations of strength properties are much more complicated than the tensor transformation of stiffness properties. (The strength tensor, if one even exists, must be of higher order than the stiffness tensor.) Nevertheless, tensor transformations of strength are performed and used as a phenomenological failure criterion (phenomenological because only the occurrence of failure is predicted, not the actual mode of failure). A somewhat empirical approach will be adopted the actual failure envelopes in stress space will be compared with simplified failure envelopes. [c.102]

Most measurements of the strength of a material are based on uniaxial stress states. However, the general practical design problem involves at least a biaxial if not a triaxial state of stress. Thus, a logical method of using uniaxial strength information obtained in principal material coordinates is required for analysis of multiaxial loading problems. Obtaining the strength characteristics of a lamina at all possible orientations is physically impossible, so a method must be determined for obtaining the characteristics at any orientation in terms of characteristics in the principal material coordinates. In such an extension of the information obtained in principal material coordinates, the well-known concepts of principal stresses and principal strains are of no value. A multitude of possible microscopic failure mechanisms exists, so a tensor transformation of strengths is very difficult. Moreover, tensor transformations of strength properties are much more complicated than the tensor transformation of stiffness properties. (The strength tensor, if one even exists, must be of higher order than the stiffness tensor.) Nevertheless, tensor transformations of strength are performed and used as a phenomenological failure criterion (phenomenological because only the occurrence of failure is predicted, not the actual mode of failure). A somewhat empirical approach will be adopted the actual failure envelopes in stress space will be compared with simplified failure envelopes. [c.102]

Mechanical Properties. Along with thermal properties, mechanical properties are key to the utihty of engineering plastics. In the majority of apphcations the materials are subject to mechanical stress, often in conjunction with thermal stress. Stiffness and impact strength are among the most desirable properties of a plastic. The first allows for easy and economical design, coming closest to the classical design tradition based on metal properties the second guarantees that the inevitable sudden stresses which everyday articles experience in everyday use will not destroy a given article or prematurely shorten its usefijl lifespan. Frequently, fillers such as glass and other fibers are compounded into plastics to increase their stiffness. Especially in the case of brittle resins this also results in an increase in the tensile and impact strength. To improve impact, special mbber additives of small particle si2e are employed. Typically these impact modifiers are produced by emulsion polymeri2ation and they are almost always copolymers whose outermost layer is composed of a polymer designed to be compatible with the future host resin. Another approach frequently used in impact modification is the chemical reaction of an elastomer, eg, ethylene—propylene—diene, suitably modified to react with the amine functional groups of the polyamide molecule. Impact modifiers are beheved to act by reducing formation of large cracks through cra2e formation, thereby absorbing energy. [c.265]

Mechanics of composite materials (1999) -- [ c.123 , c.126 , c.127 , c.128 , c.129 , c.130 , c.131 , c.132 , c.133 , c.134 , c.135 , c.136 , c.158 , c.159 , c.160 , c.161 , c.162 , c.163 ]

Machanics of composite materials (1998) -- [ c.123 , c.126 , c.127 , c.128 , c.129 , c.130 , c.131 , c.132 , c.133 , c.134 , c.135 , c.136 , c.158 , c.159 , c.160 , c.161 , c.162 , c.163 ]