Stress distribution: Poisson’s ratio

Other relevant articles include Stress distribution bond thickness. Stress distribution Poisson s ratio. Stress distribution stress singularities. Stress distribution shear lag solution and Stress distribution mode of failure. 

At first sight, it might seem a simple test to analyse with uniform tensile stress throughout the adhesive layer. In practice, the stress distribution is not uniform the disparity of modulus and Poisson s ratio between the cylinders and the adhesive means that shear stresses are introduced on loading (see Stress distribution Poisson s ratio). Thus, the failure stress is not independent of the dimensions of the joint. 

Stress distribution Poisson s ratio D A DILLARD Poisson contraction and stress distribution... 

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

For the calculation of the thermal shock-induced stresses, we consider the plate shown in Fig. 15.1 with Young s modulus E, Poisson s ratio v, and coefficient of thermal expansion (CTE) a, initially held at temperature /j. If the top and bottom surfaces of the plate come into sudden contact with a medium of lower temperature T they will cool and try to contract. However, the inner part of the plate initially remains at a higher temperature, which hinders the contraction of the outer surfaces, giving rise to tensile surface stresses balanced by a distribution of compressive stresses at the interior. By contrast, if the surfaces come into contact with a medium of higher temperature Tm, they will try to expand. As the interior will be at a lower temperature, it will constrain the expansion of the surfaces, thus giving rise to compressive surface stresses balanced by a distribution of tensile stresses at the interior. 

A recent analysis [5] has developed that used previously [4] and has included the lateral stress distribution in the beam beyond the crack tip. For anisotropic composites with a transverse modulus E2, a shear modulus /land Poisson s ratio v we have,... 

As in true strain, the expression above takes into account cross-sectional area changes in a certain cohesive structure (or cake) of powdered material. If the material is isotropic, another possible expression that includes the Poisson s ratio p (the ratio of transverse strain and axial strain resulting from uniformly distributed axial radial stress during static compression of the material in absolute value). The Poisson s ratio, or bulk modulus, permits prediction of the transverse contraction or expansion that occurs when a stress is applied longitudinally. 

Fig. 2a Distribution of the first principal stress at the surfece of a typical notched ball specimen at a load of 1 N (ball diameter 31.75 mm notch length 80 % of the diameter, notch width 8.5 % of the diameter, fillet radius 0.675 mm Poisson s ratio 0.27). a) Overview on the specimen s surface, and b) path along the equatorial plane (path 1-3) and the plane perpendicular to the equatorial plane (path 1-2).

As Poisson s ratio v for a rubber = 0.499, this simplifies to E = 3G. Figure 4.2 shows a finite element simulation of the compression of a rubber cube by 25%. There is a relatively low compressive stress in the region that has bulged out at the sides of the block. The compressive stress distribution has a peak near the edges of the surface bonded to the metal plate, especially at the corner. There is a subsidiary stress maximum at the centre of the cube. 

Another factor of anisotropic design analysis is greater dependence of stress distributions on materials properties. For isotropic materials, whether elastic, viscoelastic, etc., static values often result in stress fields which are independent of material stiffness properties. In part, this is due to the fact that Poisson s ratio is the only material parameter appearing in the compatibility equations for stress. This parameter does not vary widely between materials. However, the compatibility equations in stress for anisotropic materials depend on ratios of Young s moduli for different material axes, and this can introduce a strong dependence of stress on material stiffness. This approach can be used in component design, but the product and material design analysis become more closely related. 

During the finite element analysis, the parameters of slopes be taken as h (height) = 10 m, E (elastic modulus of the soil) = 20 MPa, v(Poisson s ratio) = 0.32, (unit weight) = 18 kN/m or (dip angle) = 30°-80°. In order to obtain the shear stress distributions under the slope s top face and under the slope s incline face respectively, seven stress paths have been defined in slope, their position shown in Figure 2. 

In this approach, no assumptions are made about the stress and strain distributions per unit volume. The specific fiber-packing geometry is taken into account, as is the difference in Poisson s ratio between the fiber and matrix phases. The equations of elasticity are to be satisfied at every point in the composite, and numerical solutions generally are required for the complex geometries of the representative volume elements. Such a treatment provides for tighter upper and lower bounds on the elastic properties than estimated by the rule of mixtures, as is describe in the references used in this section. 

In the literature [1,2,3], some analytical models have been developed to describe the compression behavior of closed-cell foams. Christensen [2 introduced a model to derive the Youngs s modulus E, the Poisson s ratio V and the compressive strength cr, which denotes the beginning of the collapse plateau, of a foam from the corresponding material properties Eg, Lij of the solid material. In this model, it is assumed that the material of the foam is distributed randomly in a three-dimensional network as thin membranes under plane stress conditions. By the use of... 

At the substrate-adhesive interface both the shear and tensile stresses reach a maximum at the free edge of the adhesive bond. Harrison and Harrison used a finite element analysis to determine the effect of varying Poisson s ratios on interfacial shear strengths.Rubbery materials can distribute the stress over larger areas while materials with lower Poisson s ratios produce greater interfacial shear stresses. 

The calculated temperature distribution is subsequently used to simulate the thermal deformation and, consequently, the resulting stresses and strains within the welded material. To this end the (thermo-) mechanical materials data, viz. a (cf. Figure 12, lower right), E (cf. Figure 15, right), the stress-strain curves (cf. Figure 13), and Poisson s ratio v = 0.3 are applied. Furthermore (linear) 8-node hexahedral elements are chosen. Note, that during... 

For plane strain and a small value of (d/a) the normal stress adopts a parabolic distribution (see Johnson, 10), providing that the Poisson s ratio is not close to 0.5. For lubricated conjunctions formed between a thin, soft layer of elastic material and a rigid plane, the elastic deformation is very similar to the parabolic profile for dry contact, with the typical build-up of pressure on the inlet side associated with lubricant entrainment superimposed. It is therefore convenient to consider plane strain and to determine the local deformation associated with the local pressure on the basis of a constrained column model (Medley et al 6) in lubrication analysis. For a local pressure (p), a soft layer of thickness (d) having a modulus of elasticity (E) and a Poisson s ratio (v), the normal deflection is thus given by. 

In the stress analysis of composite materials, the parameters are as follows. The elastic modulus of resin matrix = 1.67 GPa, Poisson s ratio = 0.2, yield strength = 3.5 MPa the elastic modulus of the SiC whisker Ef = 410 GPa, Poisson s ratio Vf = 0.17 the exterior stress o is set to 0.8 Mpa and the volume fraction of the whisker in the composite material is 12.5%. The distribution curve of the axial stress of composite material reinforced with whiskers with different L/D ratios is shown in Figure 4.15. In the figure, Zj(//) stands for the axial... 

We will first consider the simple case of a rubber disk confined between two rigid plates and subjected to tension or compression as seen in Fig. 13. There are several solutions based on different boundary conditions and simplifying assumptions. We will see that the extent of constrainment of the elastomer is a key property governing the stress distribution and apparent stiffness of the joint. The amount of constrainment is given by the height to diameter ratio, the Poisson s ratio of the elastomer, and the friction at the substrate-elastomer surfaces as seen in Fig. 14. 

Closed form solutions have been derived which neglect outer edge effects, but can account for full adhesion or high friction, zero adhesion and intermediate levels of adhesion at the substrate. The aforementioned derivations are made possible by the simplifying assumption that Poisson s ratio for the elastomer is 0.5. Such solutions are useful for determining apparent stiffness and stress distributions in the interior of the joint, as we shall see in the following paragraphs. 

In Fig. 14, center, we see that the shear stresses are equivalent to the solution by Gent (Fig. 14, left), up to the point that slippage occurs, known as the critical radius. In addition, the normal stresses are lower and vanish beyond the critical radius, as compared to the perfectly adhering solution. The above derivation assumed that Poisson s ratio equals 0.5. Of course, real elastomers are not incompressible. Lai et al. have studied the effect of compressibility on the stress distributions in thin elastomeric blocks and annular bushings [9]. For the special cases of an infinite strip of finite width the following closed form solution for the effective modulus was obtained ... 

All the above analyses, with the exception of those by Kuenzi and Stevens (1963) and by Alwar and Nagaraja (1976a) make the assumption that the adherends are infinitely stiff compared with the adhesive. Although the adherends will normally have elastic moduli at least an order of magnitude greater than those of the adhesive, there will always be some Poisson s ratio strain in the adherends when the joint is loaded. Also, the interface does not remain plane under load since the stresses in the adhesive are not uniform. None of the above analyses makes any reference to the radial and circumferential stresses in the adhesive, except for that by Kuenzi and Stevens (1963), which makes the unrealistic assumption that the axial stress distribution in the adhesive is uniform. As in the case of torsion specimens, the presence of a spew fillet may affect the stress distribution, and none of the above analyses takes this into account. 

When a complex joint is to be introduced in a structure, the ideal situation is to test that specific joint. However, this approach is very expensive. Before real joints or prototypes are built, the designer should first come up with a good prediction of the failure load based, among other things, on the basic mechanical properties of the adhesive. The basic properties can mean the elastic properties, such as the Young s modulus and the Poisson s ratio in case the analysis is linear elastic. However, for the more realistic theoretical methods that take into account the nonlinear behavior of the adhesive, the yield stress, the ultimate stress, and the failure strain are necessary. The stress-strain curve of adhesives is necessary for designing adhesive joints in order to compute the stress distribution and apply a suitable failure criterion based on continuum mechanics principles. 

O Figure 22.6a shows the geometry for the tensile test. In the tensile test, two rigid cylinders are bonded end to end with adhesives, and the joint is tested by applying tensile forces on both cylinders and measuring the resulting stress-strain curves. Though the test method looks quite simple, special care has to be paid for the data analysis. In practice, the stress distribution at the adhesive-substrate interface is not uniform (see Fig. 22.6b), but depends on the elasticity (modulus and Poisson s ratio) of the cylinder and the adhesive, and also on the aspect ratio (thickness/diameter) of the adhesive (Adams et al. 1978 Anderson et al. 1977). Due to this reason, the tensile test has to be performed under (at least) controlled thickness conditions. 

Even though the Biot number takes into account a characteristic dimension of the object, the behavior towards thermal shock is also influenced by its geometry. The latter parameter has an influence over the distribution of stresses in the voliune, which can be deciphered indirectly with the help of a function of Poisson s ratio, f(v) ... 

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