# Mechanics of materials approach

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

MECHANICS OF MATERIALS APPROACH TO STIFFNESS [c.126]

Use a mechanics of materials approach to determine the apparent Young s modulus for a composite material with an inclusion of arbitrary shape in a cubic element of equal unit-length sides as In the representative volume element (RVE) of Figure 3-17. Fill in the details to show that the modulus is [c.135]

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 constituent material properties are substituted in Equations (3.61) and (3.57) to obtain the upper bound on E of the composite material and in Equation (3.47) to obtain the lower bound on E. In addition, the mechanics of materials approach studied in Problems 3.2.1 through [c.158]

MECHANICS OF MATERIALS APPROACH TO STRENGTH 3.5.1 Introduction [c.163]

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]

MECHANICS OF MATERIALS APPROACH TO STIFFNESS [c.126]

Use a mechanics of materials approach to determine the apparent Young s modulus for a composite material with an inclusion of arbitrary shape in a cubic element of equal unit-length sides as In the representative volume element (RVE) of Figure 3-17. Fill in the details to show that the modulus is g F/A F/(1[L]x1[L]) e /L [c.135]

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 constituent material properties are substituted in Equations (3.61) and (3.57) to obtain the upper bound on E of the composite material and in Equation (3.47) to obtain the lower bound on E. In addition, the mechanics of materials approach studied in Problems 3.2.1 through [c.158]

MECHANICS OF MATERIALS APPROACH TO STRENGTH 3.5.1 Introduction [c.163]

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 mechanics of materials (or strength of materials or resistance of materials) approach embodies the usual concept of vastly simplifying assumptions regarding the hypothesized behavior of the mechanical system. The elasticity approach actually is at least three approaches (1) bounding principles, (2) exact solutions, and (3) approximate solutions. [c.122]

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 mechanics of materials (or strength of materials or resistance of materials) approach embodies the usual concept of vastly simplifying assumptions regarding the hypothesized behavior of the mechanical system. The elasticity approach actually is at least three approaches (1) bounding principles, (2) exact solutions, and (3) approximate solutions. [c.122]

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]

Of the various physical properties, it is the mechanical properties that make metallic glasses so unique when compared to their crystalline counterparts. A metallic glass obtains its mechanical strength ia quite a different way from crystalline alloys. The disordered atomic stmcture iacreases the resistance to flow ia metallic glasses so that these materials approach their theoretical strength. Strengths of E/50 where E is Young s modulus, are common (Table 2). An attractive feature is that metallic glasses ate equally strong ia all directions because of the random order of their atomic stmcture. [c.340]

More than 95% of the base material used in paper and board manufacture (4) is fibrous. Whereas a large (90%) percentage originates from wood (qv), the filler contents of some grades of paper approach 30%. Many tree species encompassing both hardwood and softwood are used to produce pulp. In addition to the large number of wood types, there are many different manufacturing processes involved in the conversion of wood to pulp. These range from mechanical processes, by which only mechanical energy is used to separate the fiber from the wood matrix, to chemical processes, by which the bonding material, ie, lignin (qv), is removed chemically. Many combinations of mechanical and chemical methods also are employed (see PuLP). Pulp properties are deterrnined by the raw material and manufacturing process, and must be matched to the needs of the final paper product. [c.1]

A positive result in a mutagenicity test system is not necessarily a directly usable end point in toxicity evaluation. There is general agreement that materials exhibiting a mutagenic potential, particularly by an in vivo approach, need to be reviewed in particular with respect to their possible genetic, teratogenic, and direct carcinogenic activity. The relationship between mutagenicity tests, both in vivo and in vitro and the abiHty of a chemical to produce genetically transmitted adverse effects in the progeny of exposed individuals is unclear and the subject of much debate and research. A variety of mechanisms may be concerned in the induction of teratogenic effects by differing materials, of which one is mutagenicity. Thus, a material that exhibits a mutagenic potential in appropriate tests should be suspected of being teratogenic, and appropriate laboratory studies may be required. However, it is of the utmost importance to be aware that many other mechanisms for teratogenesis exist, and a material devoid of mutagenic potential may not necessarily be devoid of teratogenic potential. Perhaps the most common appHcation of mutagenicity studies is to assess the carcinogenic potential of materials. There are correlative relations between mammalian carcinogens and mutagens (125). The latest thought is that cancer may result from the sequential or simultaneous genetic damage to several genes which govern ceU growth, death, and maturation (126). Also, there is a considerable body of evidence to indicate that an early critical and irreversible stage in carcinogenesis is the induction of a mutagenic event in the affected ceU line. For these reasons, materials which have been shown to be mutagens are suspect of being chemical carcinogens and this may necessitate, and assign a priority for, chronic exposure studies. However, this facet of mutagenicity testing requires considerable caution in its appHcation. There is a need to look at the effect of metaboHc activation, both in vivo and in vitro, the nature of the end point indicating a mutagenic potential the potency of a material (or metaboHte) in inducing events characteristic of chemical carcinogenesis the possible influence of route on in vivo tests and to determine that any response was not due to the presence of contaminants. [c.237]

Low temperature sol-gel technology is promising approach for preparation of modified with organic molecules silica (SG) thin films. Such films are perspective as sensitive elements of optical sensors. Incorporation of polyelectrolytes into SG sol gives the possibility to obtain composite films with ion-exchange properties. The addition of non-ionic surfactants as template agents into SG sol results formation of ordered mechanically stable materials with tunable pore size. [c.317]

Non-complex and/or non-critical applications in mechanical design can also make use of probabilistic design techniques and justify a more in-depth approach if the benefits are related to practitioners and customers alike. Surveys have indicated that many products in the industrial sector have in the past been overdesigned (Kalpakjian, 1995). That is, they were either too bulky, were made of materials too high in quality, or were made with unwarranted precision for the intended use. Overdesign may result from uncertainties in design calculations or the concern of the designer and manufacturer over product safety in order to avoid user injury or [c.134]

The key feature of the mechanics of materials approach is that certain simplifying assumptions must be made regarding the mechanical behavior of a composite material in order to get an effective solution. Each assumption must be plausible, i.e., there must be a reason why the assumption might be true (in mechanics, assumptions cannot be arbitrary ). The most prominent assumption is that the strains in the fiber direction of a unidirectional fiber-reinforced composite material are the same in the fibers as in the matrix as shown in Figure 3-5. If the strains were not the same, then a fracture between the fibers and the matrix is implied. Thus, the assumption has a plausible reason. Because the strains in both the matrix and fiber are the same, then it is obvious that sections normal to the 1-axis, which were plane before being stressed, remain plane after stressing. The foregoing is a prominent assumption in the usual mechanics of materials approaches such as in beam, plate, and shell theories. We will derive, on that basis, the mechanics of materials predictions for the apparent orthotropic moduli of a unidirectionally fiber-reinforced composite material, namely, E., Eg, v.,2, and G.,2. Note that the basis for the simplifying assumptions for each prediction is firm, and not wishful thinking. [c.126]

The apparent Young s modulus, E2, of the composite material in the direction transverse to the fibers is considered next. In the mechanics of materials approach, the same transverse stress, 02, is assumed to be applied to both the fiber and the matrix as in Figure 3-9. That is, equilibrium of adjacent elements in the composite material (fibers and matrix) must occur (certainly plausible). However, we cannot make any plausible approximation or assumption about the strains in the fiber and in the matrix in the 2-direction. [c.129]

The in-plane shear modulus of a lamina, G12. is determined in the mechanics of materials approach by presuming that the shearing stresses on the fiber and on the matrix are the same (clearly, the shear deformations cannot be the samel). The loading Is shown in the representative volume element of Figure 3-15. By virtue of the basic presumption, [c.133]

The key feature of the mechanics of materials approach is that certain simplifying assumptions must be made regarding the mechanical behavior of a composite material in order to get an effective solution. Each assumption must be plausible, i.e., there must be a reason why the assumption might be true (in mechanics, assumptions cannot be arbitrary ). The most prominent assumption is that the strains in the fiber direction of a unidirectional fiber-reinforced composite material are the same in the fibers as in the matrix as shown in Figure 3-5. If the strains were not the same, then a fracture between the fibers and the matrix is implied. Thus, the assumption has a plausible reason. Because the strains in both the matrix and fiber are the same, then it is obvious that sections normal to the 1-axis, which were plane before being stressed, remain plane after stressing. The foregoing is a prominent assumption in the usual mechanics of materials approaches such as in beam, plate, and shell theories. We will derive, on that basis, the mechanics of materials predictions for the apparent orthotropic moduli of a unidirectionally fiber-reinforced composite material, namely, E., Eg, v.,g, and G g. Note that the basis for the simplifying assumptions for each prediction is firm, and not wishful thinking. [c.126]

The apparent Young s modulus, E2, of the composite material in the direction transverse to the fibers is considered next. In the mechanics of materials approach, the same transverse stress, 02, is assumed to be applied to both the fiber and the matrix as in Figure 3-9. That is, equilibrium of adjacent elements in the composite material (fibers and matrix) must occur (certainly plausible). However, we cannot make any plausible approximation or assumption about the strains in the fiber and in the matrix in the 2-direction. [c.129]

The in-plane shear modulus of a lamina, G 2i is determined in the mechanics of materials approach by presuming that the shearing stresses on the fiber and on the matrix are the same (clearly, the shear deformations cannot be the samel). The loading Is shown in the representative volume element of Figure 3-15. By virtue of the basic presumption, [c.133]

The problem of determining exact solutions to various cases of elastic inclusions in an elastic matrix is very difficult and well beyond the scope of this book. However, it is appropriate to indicate the types of solutions that are available and to compare them with the mechanics of materials results (in a later section). As in many other elasticity problems, the Saint-Venant semi-inverse method is prominent among the available techniques. In brief, the semi-inverse method consists of dreaming up or assuming a part of the solution, i.e., some of the components of stress, strain, or displacement, and then seeing if the assumed solution satisfies the governing differential equations of equilibrium and the boundary conditions. The assumed solution must not be so rigorously specified that the equilibrium and compatibility equations cannot be satisfied. As an example, the assumption that plane sections remain plane Is a semi-inverse method approach. In combination with the bounding theorem of elasticity, the semi-inverse method is quite effective. [c.145]

The problem of determining exact solutions to various cases of elastic inclusions in an elastic matrix is very difficult and well beyond the scope of this book. However, it is appropriate to indicate the types of solutions that are available and to compare them with the mechanics of materials results (in a later section). As in many other elasticity problems, the Saint-Venant semi-inverse method is prominent among the available techniques. In brief, the semi-inverse method consists of dreaming up or assuming a part of the solution, i.e., some of the components of stress, strain, or displacement, and then seeing if the assumed solution satisfies the governing differential equations of equilibrium and the boundary conditions. The assumed solution must not be so rigorously specified that the equilibrium and compatibility equations cannot be satisfied. As an example, the assumption that plane sections remain plane Is a semi-inverse method approach. In combination with the bounding theorem of elasticity, the semi-inverse method is quite effective. [c.145]

In pattern recognition analysis, each AE hit is represented by a pattern vector, composing of representative AE features. The technique quantify the similarity or dissimilarity of pattern vectors/AE hits and assists the AE analyst to find statistical correlation between signal groups in a multi-dimensional feature space and establish the signature of AE, i.e. to discover a characteristic set of reproducible attributes of AE signals capable to describe the evolution of damage stages and/or to correlate with the various failure mechanisms. Such an approach, enables the AE analyst to overcome difficulties encountered due to human incapability of representing and visualising data in more than three dimensions. Furthermore, it does not require previously known signals from each failure mechanism, which for composite materials it is doubtful if they can be simulated in a reliable way by model specimens. [c.39]

The problem of fatigue and fracture prediction for construction materials practically always run into difFiculties connected with low cost effectiveness of experiments, which cannot create the full spectrum of loads and actions existing in reality during service time of the material or structure. The main experiments of failure and fraeture testing use short-time loads and actions. The results of these tests serve for long-time forecast of the material or structure behavior. Such a methodic of using data of short time testing for long-time prediction usually contain a considerable error of fatigue and fracture forecast. It can be explained by the fact that a lot of basic characteristics of materials such as strength, elasticity, plastic flow, micro relaxation, and crack growth process are not able to manifest themselves fully during short-time testing. Fatigue and fracture are non-stationary stochastic processes as during the material or structure service time there are a lot of various unidentified actions bringing about another source of forecast errors. It is especially significant for brittle and elasto-plastic materials, where thermodynamic balance between elastic and plastic properties in time has considerable differences for short and long-time aetions. The majorities of known approaehes, which are used for fatigue physical tests, have deterministic character or use linear stochastic models for materials characterization. This problem is described by differences in building up models for various materials, which have brittle or elasto-plastic properties. It is necessary to note that sometimes the same materials may have different mechanical properties due to impact of different physical actions, such as high temperature, radiation, cryogenic process, etc. That is why it seems to be attractive to find a way for possibility of control and monitoring of materials and structures during their service time. For these purposes it is proposed to use trying loads of materials or structural elements with registration of processes of the material reaction as the AE process. It gives an opportunity to use a system approach for building up models for analysis of the material properties using a short-time behavior. [c.187]

The refractory hard materials used for coatings on cutting tools are generally britde and hence not tough. The fracture mechanism consists of crack initiation at stress concentrations and its rapid propagation to failure (see Fracture mechanics). By arresting the propagation of the cracks, it is possible to increase the toughness of these hard coatings significantly without compromising on hardness. This is accompHshed by applying multiple nanolayer coatings of alternating hard and tough materials (see Nanotechnology (see Supplement)). Investigations have been directed at improving the properties of materials significantly by reducing the microstmctural or spatial scale of a material system to nanometer dimensions (100,101). In this approach, a crack initiated in any hard layer is stopped when it reaches the tough layer, faciHtating higher toughness (100—103). The number of nanolayers can be several hundred in contrast to the few layers used on cutting tools prepared by CVD techniques. [c.211]

Both the metaboHsm of a material and its potential to cause toxic injury may vary with the route of exposure, although the magnitude of the dose and duration of dosing may influence this relationship. For example, materials that are metaboHcally activated by the Hver are likely to exhibit a comparatively greater degree of toxicity when given peroraHy than when absorbed in the lung or across the skin. This is largely related to the anatomical routes of transport. Thus, the greatest proportion of material absorbed from the gastrointestinal tract passes via the portal vein direcdy to the Hver. In contrast, materials absorbed as a result of respiratory exposure or skin contact initially pass to the lung and then into the systemic circulation, with only a small fraction of the cardiac output being deHvered to the Hver through the hepatic artery (Fig. 3). By similar reasoning, materials that are detoxified by the Hver may be significantly less toxic by swallowing than by either inhalation or penetration across the skin. An example of the influence of route on toxicity is presented in Table 3. When assessing the relevance of metaboHsm in acute toxicity testing, and particularly when comparing toxicity by different routes of exposure, both the magnitude of the dose and the time period over which it is given must be considered. For example, when a single large dose (a bolus) of a metaboHcally activated material is given by gavage, it may be almost completely metabolized, resulting in the rapid development of acute toxic injury. When the same material is given orally at a slower rate, eg, by continuous inclusion in the diet, then there is a slow and continual absorption and metaboHsm of the material, and in these circumstances the rate of generation of the toxic species may approach that which occurs from the continuous absorption resulting from persistent exposure to an atmosphere of the material. The influence of dose magnitude—time relationships also apply to the interpretation of results with materials detoxified by the Hver. With such materials, slow continuous peroral adininistration of the material results in slow titration to the Hver and a high proportion of the material being detoxified. In this instance, the anticipated differential toxic effect between the oral and inhalation routes of exposure occurs. However, if a bolus of the material is introduced into the stomach, then the endogenous hepatic detoxification mechanisms may be exceeded, and unmetaboHzed material may enter the systemic circulation and initiate toxic injury. [c.231]

See pages that mention the term

**Mechanics of materials approach**:

**[c.143] [c.143] [c.441] [c.2425] [c.162] [c.10]**

Mechanics of composite materials (1999) -- [ c.0 ]

Machanics of composite materials (1998) -- [ c.0 ]