Strain/stress hydrostatic pressure

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

For a fluid at rest (i.e., when all strain rates vanish), the stress tensor must reduce to that caused by hydrostatic pressure. The thermodynamic pressure, as defined... 

Incoherent Clusters. As described in Section B.l, for incoherent interfaces all of the lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface s core structure consists of all bad material. It is generally assumed that any shear stresses applied across such an interface can then be quickly relaxed by interface sliding (see Section 16.2) and that such an interface can therefore sustain only normal stresses. Material inside an enclosed, truly incoherent inclusion therefore behaves like a fluid under hydrostatic pressure. Nabarro used isotropic elasticity to find the elastic strain energy of an incoherent inclusion as a function of its shape [8]. The transformation strain was taken to be purely, dilational, the particle was assumed incompressible, and the shape was generalized to that of an... 

SMART MATERIALS. From a technical and simple point of view, a smart material is a material that responds to its environment in a timely manner. To expand on this definition, a smart material is one that receives, transmits, or processes a stimulus and responds by producing a useful effect, which may include a signal that the material is acting upon it. Stimuli may include strain, stress, temperature, chemicals, an electric field, a magnetic field, hydrostatic pressures, different types of radiation, and other forms of stimuli. Transmission or processing of the stimulus may be in the form of an absorption of a photon, of a chemical reaction, of an... 

A yielding criterion gives critical conditions (at a given temperature and strain rate) where yielding will occur whatever the stress state. Two main criteria, originally derived by Tresca and von Mises for metals, can be applied to polymers (with some modifications due to the influence of hydrostatic pressure) ... 

The influence of temperature and strain rate can be well represented by Eyring s law physical aging leads to an increase of the yield stress and a decrease of ductility the yield stress increases with hydrostatic pressure, and decreases with plasticization effect. Furthermore, it has been demonstrated that constant strain rate. Structure-property relationships display similar trends e.g., chain stiffness through a Tg increase and yielding is favored by the existence of mechanically active relaxations due to local molecular motions (fi relaxation). 

AGS may be expressed as proportional to compressive yield stress), yt (fracture strain), the plastic zone size, and the square of the concentration factor, K. The influence of hydrostatic pressure was taken into account with a modified von Mises criterion (Chapter 12). 

Crazes usually form under tensile stress when a critical strain is surpassed they do not occur under compressive stress applying hydrostatic pressure during tensile deformation can even inhibit their development. Crazes always nucleate preferentially at points of triaxial stress concentration. It is the dilatational strain which initiates crazes and cracks. 

Tphe literature is replete with examples showing that the application of hydrostatic pressure enhances the ductile behavior of strained amorphous polymers. In this paper we present a possible explanation of this effect and two experiments demonstrating the enhanced ductility of polymers under compressive shear stresses applied orthogonally to the plane of shear. 

Before concluding this discussion of cell walls, we note that the case of elasticity or reversible deformability is only one extreme of stress-strain behavior. At the opposite extreme is plastic (irreversible) extension. If the amount of strain is directly proportional to the time that a certain stress is applied, and if the strain persists when the stress is removed, we have viscous flow. The cell wall exhibits intermediate properties and is said to be viscoelastic. When a stress is applied to a viscoelastic material, the resulting strain is approximately proportional to the logarithm of time. Such extension is partly elastic (reversible) and partly plastic (irreversible). Underlying the viscoelastic behavior of the cell wall are the crosslinks between the various polymers. For example, if a bond from one cellulose microfibril to another is broken while the cell wall is under tension, a new bond may form in a less strained configuration, leading to an irreversible or plastic extension of the cell wall. The quantity responsible for the tension in the cell wall — which in turn leads to such viscoelastic extension — is the hydrostatic pressure within the cell. 

Eq. (5) shows that the strain rate increases rapidly near the die exit. There is therefore a very rapid rise in pressure near the die exit where the flow stress of the polymer is increasing with increasing plastic strain and with strain rate, and where the increasing plastic strain also gives rise to a larger effect of the hydrostatic pressure on the flow stress. 

If a solid is stressed beyond its elastic limit, it will acquire a permanent deformation. The deformation can be either brittle or ductile depending on (i) the material, (ii) the hydrostatic pressure, (iii) the temperature, and (iv) the strain rate. In general, a solid is more likely to deform in a brittle manner at low hydrostatic pressures, low temperatures, and at high strain-rates. Convesely, high hydrostatic pressures and temperatures and low strain-rates favor ductile deformation. 

When an elastic body is under the effect of a hydrostatic pressure, both the strain and stress deviatoric tensors are zero. Owing to the fact that in this case Yii = Y22 = Y33 < 11 = < 22 = < 33, Eq. (4.85) becomes... 

Figure 14.8 shows stress-strain curves for polycarbonate at 77 K obtained in tension and in uniaxial compression (12), where it can be seen that the yield stress differs in these two tests. In general, for polymers the compressive yield stress is higher than the tensile yield stress, as Figure 14.8 shows for polycarbonate. Also, yield stress increases significantly with hydrostatic pressure on polymers, though the Tresca and von Mises criteria predict that the yield stress measured in uniaxial tension is the same as that measured in compression. The differences observed between the behavior of polymers in uniaxial compression and in uniaxial tension are due to the fact that these materials are mostly van der Waals solids. Therefore it is not surprising that their mechanical properties are subject to hydrostatic pressure effects. It is possible to modify the yield criteria described in the previous section to take into account the pressure dependence. Thus, Xy in Eq. (14.10) can be expressed as a function of hydrostatic pressure P as... 

These authors envisaged the critical step in the yield process as being the nucleation under stress of small disc-sheared regions (analogous to dislocation loops) that form with the aid of thermal fluctuations. The model explains quantitatively the variation of the yield stress with temperature, strain rate and hydrostatic pressure, using only two parameters, the shear modulus of the material and the Burgers vector of the shared region which is a constant related to the molecular dimensions of the polymer. 

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

The structure of Equations (114) is specific for the strain/stress state in a sample under a hydrostatic pressure. 

For the case where cr, = cTj = cTj which is known as hydrostatic stress (the situation when pressure is applied on a glass embedded in a material of low elastic constants - glass piece in steatite or in AgCl in a high pressure cell or simply embedded in a liquid) - then there are no shear strains. The hydrostatic stress is simply the pressure, P and the volumetric strain is Cm so that bulk modulus is also defined as... 

Figure 2. The stress-strain dilatational behavior of a 63.5 ool % filled elastomer at a series of hydrostatic pressures...

The constant-temperature, constant-stress ensemble (NST) is an extension of the constant-pressure ensemble. In addition to the hydrostatic pressure that is applied isotropically, constant-stress ensemble allows you to control the xx, yy, zz, xy, yz, and zx components of the stress tensor (sometimes also known as the pressure tensor). This ensemble is particularly useful if one wants to study the stress-strain relationship in polymeric or metallic materials. 

In order to understand the behavior of composite propellants during motor ignition, we conducted a study of the mechanical and ultimate properties of a propellant filled with hydroxy-terminated polybutadiene under imposed hydrostatic pressure. The mechanical response of the propellant was examined by uniaxial tensile and simple shear tests at various temperatures, strain rates, and superimposed pressures from atmospheric pressure to 15 MPa. The experimentally observed ultimate properties were strongly pressure-sensitive. The data were formalized in a specific stress-failure criterion. 

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