Modulus of elasticity in shear

Metal-vacuum-metal tunneling 49—50 Method of Harris and Liebsch 110, 123 form of corrugation function 111 leading-Bloch-waves approximation 123 Microphone effect 256 Modified Bardeen approach 65—72 derivation 65 error estimation 69 modified Helmholtz equation 348 Modulus of elasticity in shear 367 deflection 367 Mo(lOO) 101, 118 Na-atom-tip model 157—159 and STM experiments 157 NaCl 322 NbSej 332 NionAu(lll) 331 Nucleation 331... 

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

When torque is applied, the modulus of elasticity in shear (shear modulus) can be estimated by using Poisson s ratio ... 

The ratio of stress to strain in a given material. The strain may be a change in length, a twist or shear, or a change in volume. Modulus of elasticity in this scientific sense must not be confused with the term modulus which has a particular significance in rubber technology. 

Therefore, the main flow properties of plastics in the widest sense are influenced by the mean molar mass. These properties include melt viscosity, modulus of elasticity and shear modulus above the glass transition range, creep behavior, stress cracking behavior, strain at break, mechanical strength, solubility and swelling behavior, etc. 

The calculated results can only be accurate when substance laws applied describe the relevant material behavior with sufficient accuracy. Also, the required material parameters, for example modulus of elasticity E, shear modulus G, transversal contraction number must be available plotted against elongation, time, rate, temperature, as well as the effects of mediums or radiation as applicable. Most commercially available databases are still deficient in this respect. 

The four independent constants in the stress/strain equations are Ei, the modulus of elasticity in the fibre direction, E2, the modulus in the transverse direction, v, Poisson s ratio, and Gu, the in-plane shear modulus. The unidirectional lamina plays an essential role in structural engineering, for... 

Because of the complicated phase morphology and the large difference in the properties of the hard and soft phases, it has not yet been possible to exactly calculate the modulus of elasticity, the shear modulus, or the melt... 

Difference in modulus of elasticity and shear strengths of adhesive and adherends... 

For isotropic materials, such as mat-reinforced construction, if E is the modulus of elasticity in any reference direction, the modulus Ei at any angle to this direction is the same, and the ratio ExIE is therefore unity. Poisson s ratio v is similarly a constant in all directions, and the shearing modulus is G = EI2 + v). If V/ for example, is 0.3, then GIE = 0.385 at all angles. These relationships are shown in Figure 6-43. 

Stiffness is defined in terms of modulus of elasticity or shear modulus. As defined in elasticity theory, the modulus of elasticity links normal stress to strain (t= ex E) and the shear modulus links shear stress to shear strain (t= /xG). 

An element will have hoth geometric and material properties. Spatially, an element is defined by its nodes however, additional geometric input is usually required for line and surface elements. For structural analysis the minimum material property is the modulus of elasticity. In most cases, Poisson s ratio or shear modulus must also be specified. If an orthotropic material is used then the orientation of the material must be specified as well as the elastic constants relative to each principal axis. If post-yield behavior is to be modelled then an elasto-plastic material model must be applied and the yield and hardening behavior defined. Constitutive adhesive and sealant models are discussed in more detail in O Chap. 23. Additional material properties will also be required for dynamic or thermal analysis. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

The ratio (p/G) has the units of time and is known as the elastic time constant, te, of the material. Little information exists in the published literature on the rheomechanical parameters, p, and G for biomaterials. An exception is red blood cells for which the shear modulus of elasticity and viscosity have been measured by using micro-pipette techniques 166,68,70,72]. The shear modulus of elasticity data is usually given in units of N m and is sometimes compared with the interfacial tension of liquids. However, these properties are not the same. Interfacial tension originates from an imbalance of surface forces whereas the shear modulus of elasticity is an interaction force closely related to the slope of the force-distance plot (Fig. 3). Typical reported values of the shear modulus of elasticity and viscosity of red blood cells are 6 x 10 N m and 10 Pa s respectively 1701. Red blood cells typically have a mean length scale of the order of 7 pm, thus G is of the order of 10 N m and the elastic time constant (p/G) is of the order of 10 s. 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

For a Hookian material, the concept of minimum strain energy states that a material fails, for example cell wall disruption occurs, when the total strain energy per unit volume attains a critical value. Such an approach has been used in the past to describe a number of experimental observations on the breakage of filamentous micro-organisms [78,79]. Unfortunately, little direct experimental data are available on the Young s modulus of elasticity, E, or shear modulus of elasticity G representing the wall properties of biomaterial. Few (natural) materials behave in an ideal Hookian manner and in the absence of any other information, it is not unreasonable to assume that the mechanical properties of the external walls of biomaterials will be anisotropic and anelastic. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

The modulus of elasticity of a material it is the ratio of the stress to the strain produced by the stress in the material. Hooke s law is obeyed by metals but mbber obeys Hooke s law only at small strains in shear. At low strains up to about 15% the stress-strain curve is almost linear, but above 15% the stress and strain are no longer proportional. See Modulus. 

Therefore, the unidirectional translaminar (i.e. through-thickness) shear strength can be obtained for the maximum load and the in-plane shear modulus of elasticity, Gu, taken from the initial linear portion of the unidirectional shear stress-shear strain (ti2 - y 2) curve ... 

When there is no volume change, as when an elastomer is stretched, Poisson s ratio is 0.5. This value decreases as the Tg of the polymer increases and approaches 0.3 for rigid solids such as PVC and ebonite. For simplicity, the polymers dealt with here will be considered to be isotropic viscoelastic solids with a Poisson s ratio of 0.5, and only deformations in tension and shear will be considered. Thus, a shear modulus (G) will usually be used in place of Young s modulus of elasticity E Equation 14.2) where E is about 2.6G at temperatures below Tg. 

Silvery-white lustrous metal face-centered cubic crystal structure ductile ferromagnetic density 8.908 g/cm at 20°C hardness 3.8 Mohs melts at 1,455°C vaporizes at 2,730°C electrical resistivity 6.97 microhm-cm at 20°C total emissivity 0.045, 0.060 and 0.190 erg/s.cm2 at 25, 100 and 1,000°C, respectively modulus of elasticity (tension) 206.0x10 MPa, modulus of elasticity (shear) 73.6x10 MPa Poisson s ratio 0.30 thermal neutron cross section (for neutron velocity of 2,200 m/s) absorption 4.5 barns, reaction cross section 17.5 barns insoluble in water dissolves in dilute nitric acid shghtly soluble in dilute HCl and H2SO4 insoluble in ammonia solution. Thermochemical Properties... 

A bright white metal soft and ductile body-centered cubic structure index of refraction 3.03 density 5.96 g/cm melts at 1,910°C vaporizes at 3,407°C electrical resistivity, 18.1 microhm-cm at 0°C and 20.1 microhm-cm at 25°C magnetic susceptibility 1.4x10 cgs units modulus of elasticity 18-19x10 psi shear modulus 6.73xl0 psi Poisson s ratio 0.36 thermal neutron absorption cross section 5 barns/atom insoluble in water, dilute sulfuric acid, and hydrochloric acid at all concentrations soluble in nitric acid, aqua regia, and concentrated sulfuric acid insoluble in alkalies. 

Figure 7.6 Models fitted for the effect of the number of freeze/thaw cycles on the apparent moduli of elasticity in the compression (Ec) a-tid tensile tests (Et), and on the modulus of rigidity (Cs) in the shear test.

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