Neutral plane

El theory In each case displacing material from the neutral plane makes the improvement in flexural stiffness. This increases the El product that is the geometry material index that determines resistance to flexure. The El theory applies to all materials (plastics, metals, wood, etc.). It is the elementary mechanical engineering theory that demonstrates some shapes resist deformation from external loads. 

From Figure 11.3, under free convection, there will be a height in the vent at which the flow is zero, N this is called the neutral plane. The pressure difference across the vent from inside (i) to outside (o) can be expressed above the neutral plane as... 

The two-zone model gives the result in terms of the neutral plane height (//rl) and the layer height (Hs) for a doorway vent of area A0 and height //,. The ambient to room temperature ratio is designated as 0 = T0/T ... 

A 500 kW fire in a compartment causes the smoke layer to reach an average temperature of 400 °C. The ambient temperature is 20 °C. Air and hot gas flow through a doorway at which the neutral plane is observed to be 1 m above the floor. A narrow slit 1 cm high and 3 m wide is nominally 2 m above the floor. 

Fig. 9.8. Deflection of a bimorph. Two long, thin plates of piezoelectric material are glued together, with a metal film sandwiched in between. Two more metal films cover the outer surfaces. Both piezoelectric plates are poled along the same direction, perpendicular to the large surface, labeled z. (A) By applying a voltage, stress of opposite sign is developed in both plates, which generates a torque. (B) The bimorph flexes to generate a stress to compensate the torque. The neutral plane, where the stress is zero, lies at hi i from the central plane.

The neutral plane, where the stress is zero, lies beyond the periphery of the tube ... 

The neutral plane (point N in Figure 7.10) is the point at which the subtended angle is 9 = Off, and can be found in several ways. A precise method is to balance the entry and exit forces pressures i.e., Pe = Px and solve for H at the neutral plane, Hn, which can then be substituted into Eq. (7.7) to solve for Off. Alternatively, there are empirical relations for Off. Consider the cold-rolling of an aluminum strip that is rolled from 4 mm to 3.3 mm in thickness with a roller 500 mm in diameter. The coefficient of friction is 0.06. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

Fig. 6.29 A bimorph under an applied field (a) the two free halves of the bimorph (b) the free halves in a field (c) behaviour when the two halves shown in (a) are joined to form a bimorph and the neutral planes have lengths equivalent to those in (b).

Figure 3.39 (page 114) shows the phase velocities of the waves as a function of the product k4, where k, = 27t/A, A, is the wavelength of the bulk transverse (shear) wave in the medium of which the plate is made, and d is the plate thickness. The waves divide naturally into two sets symmetric waves (denoted by So, S],. ..) whose particle displacements are symmetric about the neutral plane of the plate, and antisymmetric waves (Aq, A, . ..), whose displacements have odd symmetry about the neutral plane. Figure 3.38 shows that for sufficiently thin plates (M < 1-6), only two waves exist — the lowest-order symmetric mode (Sq) and the lowest-order antisymmetric mode (Aq). These are the modes shown earlier in Figure 2.0d. The plate mode that we will emphasize here is the Ao mode, in which the elements of the plate undergo flexure as the wave propagates. The shape of a plate during propagation of this flexural mode has been likened to that of a flag waving in the wind.

The properties desired of an ideal spacer layer are that it be stiff in shear, but that the spacer itself contribute minimally to the bending stiffness of the base structure, shifting the neutral plane as little as possible. We note that for the spaced constrained layer, the combined function of the viscoelastic layer and spacer is to provide a thick, dissipative and appropriately stiff (in shear) layer between the constraining and base layers. Therefore the order of the viscoelastic and spacer elements is arbitrary and they may be subdivided as long as the desired properties are preserved. These possibilities give additional freedom in adapting viscoelastic materials for effective damping. 

The horizontal shear stress is a maximum at the mid-depth of the beam (known as the neutral plane beeause here the bending stress is zero) and the shear stress falls to zero at the upper and lower surfaces. The shear stress in the neutral plane is ... 

In a solid beam, the compressive and tensile stresses are not confined to the surfaces. The compressive stress in a section is highest at the upper surface and gradually diminishes to zero at the neutral plane. Similarly, the tensile stress is highest on the lower surface and diminishes to zero at the neutral plane (Figure 10.6a). While the beam deforms elastically, the compressive and tensile stresses increase proportionately with distance from the neutral plane. The compressive stress at a distance, d, above the neutral plane will be the same as the tensile stress at a distance, d, below the neutral plane. Further, as the modulus of elasticity is the same in compression and tension, the strain at both positions will be similar. Simple beam theory assumes that the beam behaves elastically until failure. However, the limit of proportionality in compression is quite low and once exceeded the fibres near the upper surface will start to buckle, crash, and strain at a greater rate while... 

The deterioration of the electrical performances in the SiNx-passivated TFT under mechanical stress was more severe than that in the acryl-passivated TFT. Under an outward bending, the bending momentum elongates the TFT-films in the upper part of TFT on a metal substrate and compresses the flexible metal substrate in the lower part. Neutral plane is free from any stress between the elongated and the compressed part. The Young s modulus of acrylic polymer (3.2 GPa) is lower than that of SiNx film (183 GPa). As the 3 jmq-thick acrylic polymer was employed as the passivation layer in place of 0.3 jMn-thick SiNx film, a total thickness of TFT-films was increased. Therefore, the position of the neutral plane may shift from mid-surface toward the TFT-films. It accompanies with decrease of stress applied on the TFT-films. Hence, the acryl-passivated TFT could endure mechanical stress better than the SiNx-passivated TFT. 

TFTs in the bended condition (R = 5 mm) were measured at arbitrary intervals as shown in Fig. 11 (left) and 12. A zero hour duration time meant that TFTs were measured as flattened before bending them. The mobility change ( ife/nfeo) of 0.92, the subthreshold slope change (SS/SSo) of 1.04, and the threshold voltage shift (AVth) of 1.03 are almost same as those of TFT employing the single acryl passivation. As the 50 nm-thick SiNx and 3 )can-thick acrylic polymer were employed as the passivation, the position of the neutral plane may shift from mid-surface toward the TFT-films. These results are similar to that of a single acrylic polymer passivation. 

The structure and stability of free Ce02 surfaces has been considered theoretically using simulation techniques based upon interatomic potentials and molecular dynamics. For surfaces examined, it is found that the stability increases as (310) < (110) < (111). The (111) surface is predicted to terminate in an anionic layer of a neutral three plane repeat unit while (110) terminates in a neutral plane of mixed cations and anions. Minor inward relaxation of the outermost anionic layer (0.005... 

Such two-dimensional pair correlation functions for mobile ions adsorbed on charged planes have recently been investigated theoretically in Ref. 42. The mean separation d, = ( < 0/s )l/2 of w-valent ions on the completely neutralized plane of surface charge density s is identified as the important scaling length. The main predictions for the strongly coupled regime =... 

As discussed in Section 8.2 (Fig. 8.3), interparticle and wall friction, force dissipation from particle to particle, and sliding under shear cause differences in density distribution in a compact that is produced in a die with one-sided compression by one of the punches. To obtain a somewhat more uniform structure, compaction can be carried out by both punches. If both move at the same rate and for the same stroke length, thus exerting identical forces, and assuming uniform filling of the die, a mirror image of the density distributions that were shown in Fig. 8.3 (Section 8.2) develops along a neutral plane which, under those conditions, is located in the middle of the compact. Machines that operate in this manner may be identified as presses with double pressure (see also below). 

Fig. 8.99 Sketches describing possibilities to influence the location of the "neutral plane in upper punch pressing and controlled withdrawal die [B.25, B.42],...

Particularly in complex parts, it is also necessary to control the position of the neutral plane, the low density zone that is approximately perpendicular to the direction of pressing. This is achieved by the relative motions of the tooling members (Fig. 8.99). It is also important to understand that, particularly under pressure, particles will not move from one level or position in the developing structure of a part to another one. As a consequence, if parts are pressed that feature more than one level, separate pressing forces must be applied simultaneously for each level. As a result, neutral planes will exist for each part level (Fig. 8.100). 

Double pressing , that is, the densification of particulate solids by the movement of both the top and bottom punches, is used to overcome at least part of the uneven density distribution caused by unidirectional pressing. If both punches move towards each other with the same speed, exerting the same force, a mirror image of the density distributions across a neutral axis , which, in this case, is in the middle of the body, will be created (Fig. 6.7-23). If in addition, the die walls move with or to some extent in regard to the travel of the punches (withdrawal die), the location of the neutral plane... 

Fig. 6.7-24 Drawings describing ways of influencingthe location ofthe neutral plane in unidirectional (upper punch) pressing by controlled die withdrawal [B.28, B.48, B.97]...

Fig. 6.7-24 Drawings describing ways of influencing the location of the neutral plane" in unidirectional (upper prmch) pressing by controlled die withdrawal [B.28, B.48, B.97] Fig. 6.7-25 Different neutral planes" in single- and multi-level parts [B.28, B.48, B.97] Fig. 6.7-26 Diagram of the differences between dry- and wet-bag pressing [B.13a, B.48, B.97]...

Figure 6.9 Horizontal convective flow in an electrophoresis cell observations must be made at the level of one of the neutral planes.

When a hber is bent (see Figure 8-21), the outer layers of the arc that is formed (A) are stretched, and the inner layers (C) are compressed. A region in the center, the neutral plane (B), is unchanged in length. Stiffness is simply the resistance to bending and is a fundamental hber property [64] that requires more attention than it has received. 

ZnO surfaces are more complex than those of the rock-salt type oxides Uke MgO and NiO. ZnO crystalhzes in the wurtzite structure in which each Zn cation is tetrahedrally coordinated to four O anions and vice versa [105]. This crystal structure has no inversion center. The most important low-index surface planes are two polar planes, the Zn-terminated ZnO(OOOl) and 0-termi-nated ZnO(OOO-l) plane, and two neutral planes, ZnO(lO-lO) and ZnO(l 1-20). According to Nosker et al. [106] and Tasker [107], the two polar surfaces are thermodynamically unstable, however, they can be easily prepared and characterized experimentally, and do even show rather regular (1x1) LEED patterns [108]. This indicates that they are not stabilized by major reconstructions or other modifications. Therefore, it was believed for a long time that both polar surfaces exist in an unreconstructed bulk-Hke trimcation. Several contradicting proposals have been made to explain how the stability of the polar un-... 

Stable ZnO surfaces could be achieved Metallization at the two surfaces [109, 110], large reconstructions which might be even different on the Zn- and the 0-terminatedsurfaces (triangular islands on the (0001) plane [111, 112],(1x3) reconstructions on the (000-1) plane [113-115]), adsorption of charged species such as on the O-ZnO(OOO-l) plane or OH" ions on the Zn-ZnO(OOOl) plane [115]. The two neutral planes, ZnO(lO-lO) and ZnO(ll-20), on the other hand, are thermodynamically stable and need not be reconstructed. We will not present all the details here this discussion is by no means finished at present. 

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