# Elasticity approach

It should be pointed out that the view of the glass transition temperature described above is not universally accepted. In essence the concept that at the glass transition temperature the polymers have a certain molecular orientation time is an iso-elastic approach while other theories are based on iso-viscous. [c.46]

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 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

ELASTICITY APPROACH TO STIFFNESS [c.137]

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]

In this section, first, the interlaminar stress in the simple case of the free edges of an angle-ply laminate will be identified. Then, the concept of interlaminar stresses will be described with an elasticity approach. Next, experimental verification of the theory is offered. Then, a cross-ply laminate will be analyzed, followed by a mixed-angle laminate. Finally, the interaction of interlaminar stresses and stacking sequence and their influence on laminate strength will be examined along with some suggestions for how to suppress free-edge delamination. [c.262]

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are [c.264]

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 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 (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 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]

ELASTICITY APPROACH TO STIFFNESS [c.137]

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]

In this section, first, the interlaminar stress in the simple case of the free edges of an angle-ply laminate will be identified. Then, the concept of interlaminar stresses will be described with an elasticity approach. Next, experimental verification of the theory is offered. Then, a cross-ply laminate will be analyzed, followed by a mixed-angle laminate. Finally, the interaction of interlaminar stresses and stacking sequence and their influence on laminate strength will be examined along with some suggestions for how to suppress free-edge delamination. [c.262]

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are [c.264]

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 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]

SMD is a novel approach to the study of the dynamics of binding/unbinding events in biomolecular systems and of their elastic properties. The simulations reveal the details of molecular interactions in the course of unbinding, thereby providing important information about binding mechanisms. The advantage of SMD over conventional molecular dynamics is the possibility of inducing relatively large conformational changes in molecules on nanosecond time scales. Other methods, such as umbrella sampling, free energy perturbation (McCammon and Harvey, 1987), and weighted histogram analysis (Kumar ct ah, 1992), aiming at the determination of the energy landscapes, typically involve small conformational changes and require extensive computations to achieve accuracy. [c.59]

In a significant number of polymer processes the influence of fluid elasticity on the flow behaviour is small and hence it is reasonable to use the generalized Newtonian approach to analyse the flow regime. In generalized Newtonian fluids tire extra stress is explicitly expressed in terms of velocity gradients and viscosity and can be eliminated from the equation of motion. This results in the derivation of Navier-Stokes equations with velocity and pressure as tire only prime field unknowns. Solution of Navier-Stokes equations (or Stokes equation for creeping flow) by the finite element schemes is the basis of computer modelling of non-elastic polymer flow regimes. In contrast, in viscoelastic flow models the extra stress can only be given through implicit relationships with the rate of strain, and hence remains as a prime field unknown in the governing equations. In this case therefore, in conjunction with the governing equations of continuity and momentum (generally given as Cauchy s equation of motion) an appropriate constitutive equation must be solved. Numerical solution of viscoelastic constitutive equations has been the subject of a considerable amount of research in the last two decades. This has given rise to a plethora of methods [c.79]

The plane of the rotor blade cross-section representing the flow field configuration at the start of mixing in a partially filled single-blade mixer is shown in Figure 5.1. Initial distribution of the compound inside the mixer chamber corresponds to a fill factor of 71 per cent and is chosen arbitrarily. It is evident that the flow field within this domain should be modelled as a free surface regime with random moving boundaries. Available options for the modelling of such a flow regime are explained in Chapter 3, Section 5. In this example, utilization of the volume of fluid (VOF) approach based on an Eulerian framework is described. To maintain simplicity we neglect elastic effects and the variations of compound viscosity with mixing, and focus on the simulation of the flow corresponding to a generalized Newtonian fluid. In the VOF approach [c.142]

As discussed in the previous chapters, utilization of viscoelastic constitutive equations in the finite element schemes requires a significantly higher computational effort than the generalized Newtonian approach. Therefore an important simplification in the model development is achieved if the elastic effects in a flow system can be ignored. However, almost all types of polymeric fluids exhibit some degree of viscoelastic behaviour during their flow and deformation. Hence the neglect of these effects, without a sound evaluation of the flow regime characteristics, which may not allow such a simplification, can yield inaccurate results. [c.150]

Tlie focus of discussions presented so far in this publication has been on the finite element modelling of polymers as liquids. This approach is justified considering that the majority of polymer-fonning operations are associated with temperatures that are above the melting points of these materials. However, solid state processing of polymers is not uncommon, furthermore, after the processing stage most polymeric materials are used as solid products. In particular, fibre- or particulate-reinforced polymers are major new material resources increasingly used by modern industry. Therefore analysis of the mechanical behaviour of solid polymers, which provides quantitative data required for their design and manufacture, is a significant aspect of the modelling of these materials. In this section, a Galerkin finite element scheme based on the continuous penalty method for elasticity analyses of different types of polymer composites is described. To develop this scheme the mathematical similarity between the Stokes flow equations for incompressible fluids and the equations of linear elasticity is utilized. [c.183]

In (a), an ion and a gas atom approach each other with a total kinetic energy of KE, + KEj. After collision (b), the atom and ion follow new trajectories. If the sum of KE, + KEj is equal to KE3 + KE4, the collision is elastic. In an inelastic collision (b), the sums of kinetic energies are not equal, and the difference appears as an excess of internal energy in the ion and gas molecule. If the collision gas is atomic, there can be no rotational and no vibrational energy in the atom, but there is a possibility of electronic excitation. Since most collision gases are helium or argon, almost all of the excess of internal energy appears in the ion. [c.374]

An important fact to remember about the field of thermodynamics is that it is blind to details concerning the structure of matter. Thermodynamics are concerned with observable, measurable quantities and the relationships between them, although there is a danger of losing sight of this fact in the somewhat abstract mathematical formalism of the subject. In discussing elasticity in Chap. 3, we took the position that entropy is often more intelligible from a statistical, atomistic point of view than from a purely phenomenological perspective. It is the latter that is pure thermodynamics the former is the approach of statistical thermodynamics. In this chapter, too, we shall make extensive use of the statistical point of view to understand the molecular origin of certain phenomena. [c.506]

Hydrogels. A new approach has been taken to produce organic—inorganic hybrid interpenetrating networks using the sol—gel process, as described previously for PTMO, and y-radiation (69—70). By swelling existing organic—inorganic networks with either methacrylic acid or A/-vinylpyrrohdinone and subsequent y-polymerization of the monomers in situ IPNs which exhibit some degree of hydrogel behavior, ie, the abiUty to absorb significant amounts of moisture or preferential swelling dependent upon pH, are produced, and they possess the optical transparency of the original gels. Additionally, very significant increases in the strength (up to a factor of 50) and elongation at break, as well as the elastic and dynamic storage modulus were induced. [c.331]

The Monte Cado methods, appUed to ion—soUd interactions, have a number of distinct advantages over analytical calculations based on transport theory. The MC approach aUows for a more rigorous treatment of elastic scattering and of the determination of angular and energy distributions. A number of MC codes have been developed over the years (43). The various MC programs differ primarily in their basic treatment of elastic scattering. The program TRIM (transport of ions in matter) (9,10) is the most commonly cited for range and damage distributions in amorphous materials. TRIM can also simulate sputtering processes. The program provides high computer efficiency, and the agreement between TRIM and experimental data is exceUent. The induence of the crystal lattice on the range and damage distributions has been accounted for in the MC program caUed Madowe (44—46). [c.397]

Other products, such as windows, may or may not use the powder compound, twin-screw extmder, and vacuum-sizing approach. For windows, this is a much more complex operation than for pipe, requiring large investments to develop and operate the process. Cubed compound, where the PVC grains are already broken down, can be mn faster on simpler single-screw extmders because lower melt temperatures are typical. Because elastic swell is reduced at lower melt temperatures, die design is simpler. Either vacuum sizing or air sizing/cooling is possible. The products are designed to meet appropriate ASTM standards. Products include siding, soffits, gutters and down spouts, windows (including aH-vinyl windows and vinyl-protected wood windows), and door-glazing apphcations and garage doors. [c.507]

A more practical approach for quantifyiag the conditions required for fracture uses a stress intensity criterion instead of an energy criterion. Using linear elastic theory, it has been shown that under an appHed stress, when the stress intensity K, [c.318]

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]

In AFM, the relative approach of sample and tip is nonnally stopped after contact is reached. Flowever, the instrument may also be used as a nanoindenter, measuring the penetration deptli of the tip as it is pressed into the surface of the material under test. Infomiation such as the elastic modulus at a given point on the surface may be obtained in tliis way [114], altliough producing enough points to synthesize an elastic modulus image is very time consuming. [c.1700]

A striking example of the nse of the AFM as an indenter is provided by Bumliam and Colton [184] ( figure BL19.35L where the differences (at <100 mn indentation) between the plastic behaviour of gold at 20 pN load and the elastic behaviour of graphite and elastomer at lower loads are clearly observable. The importance of surface forces in measuring mechanical properties at low load has been demonstrated by Salmeron et al [184], who showed that tip-sample adhesion can seriously perturb hardness measurements, not only for clean surfaces, but also for those covered by a layer of contamination. These authors also suggested that initial passivation of the surface (e.g., by snlfidation) might be an effective approach to overcoming these artefacts. [c.1712]

Of tire several trapping possibilities described in tire last section, by far tire most popular choice for collision studies has been tire magneto-optical trap (MOT). An MOT uses spatially dependent resonant scattering to cool and confine atoms. If tliese atoms also absorb tire trapping light at tire initial stage of a binary collision and approach each otlier on an excited molecular potential, tlien during tire time of approach tire colliding partners can undergo a fine-stmcture-changing collision (FCC) or relax to tire ground state by spontaneously emitting a photon. In eitlier case, electronic energy of tire quasimolecule converts to nuclear kinetic energy. If botli atoms are in tlieir electronic ground states from tire beginning to tire end of tire collision, only elastic and hyperfine changing (HCC) collisions [c.2472]

Here /, r and, v are unequal integers in the set 1, 2, 3. As already mentioned, in the thin-layer approach the fluid is assumed to be non-elastic and hence the stress tensor here is given in ternis of the rate of deforaiation tensor as r(p) = riD(ij), where, in the present analysis, viscosity p is defined using the power law equation. The model equations are non-dimensionalized using [c.177]

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. [c.69]

Khludnev A.M. (1983) A contact problem of a linear elastic body and a rigid punch (variational approach). Appls. Maths. Mechs. 47 (6), 999-1005 (in Russian). [c.378]

The new approach to crack theory used in the book is intriguing in that it fails to lead to physical contradictions. Given a classical approach to the description of cracks in elastic bodies, the boundary conditions on crack faces are known to be considered as equations. In a number of specific cases there is no difflculty in finding solutions of such problems leading to physical contradictions. It is precisely these crack faces for such solutions that penetrate each other. Boundary conditions analysed in the book are given in the form of inequalities, and they are properly nonpenetration conditions of crack faces. The above implies that similar problems may be considered from the contact mechanics standpoint. [c.394]

A combination of siUca derived from siUcon alkoxide and different types of polymers leads to transparent composites having microphase (in the range of 10—100 nm) separation, a higher elastic modulus, and a greater strength than the polymer. Sol—gel siUca-PDMS (polydimethyl sHoxane) hybrids are prepared by in situ precipitation of sol—gel-derived siUca in a swollen PDMS network. The process includes the preparation of a cross-linked PDMS, swelling of the network in tetraethyl orthosiUcate, and precipitation of the siUca by introducing the PDMS filled with TEOS in an aqueous solution containing ethylamine. Mechanical performance of these composites is better than PDMS filled with fumed siUca as in conventional PDMSs. Another approach to the preparation of sol—gel siUca-PDMS hybrids uses the interaction between silanol-terrninated PDMS and silanol groups formed during hydrolysis of TEOS (54). This gives a more homogeneous stmcture, generated by covalent bonds between the siUca network and PDMS chains. The material is optically transparent, showing the absence of macrophase separation, and exhibits a high degree of flexibiUty. [c.260]

In the early stages of growth for high-shear mixers, the Stokes analysis in its present form is inappheame. Freshly formed, uncom-pac ted granules are easily deformed, and as growth proceeds and consolidation of granules occur, they will surface harden and become more resistant to deformation. This increases the importance of the elasticity of the granule assembly. Therefore, in later stages of growth, older granules approach the ideal Stokes model of rigid, elastic colh-sions. For these reasons, the Stokes approach has had reasonable success in providing an overall framework with which to compare a wide variety of granulating materials [Ennis, Powder Tech., 88, 203 (1996)]. In addition, the Stokes number controls in part the degree of deformation occurring during a collision since it represents the importance of collision kinetic energy in relation to viscous dissipation, although the exact dependence of deformation on St is presently unknown. [c.1884]

See pages that mention the term

**Elasticity approach**:

**[c.148] [c.148] [c.2205] [c.444] [c.184] [c.408]**

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

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