Distribution of stresses

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

A FEM analysis was carried out and the predicted distribution of stresses on the pressure vessel compared with the stress distribution calibration using the SPATE technique. 

An experimental activity on the stress measurement of a pressure vessel using the SPATE technique was carried out. It was demontrated that this approach allows to define the distribution of stress level on the vessel surface with a quite good accuracy. The most significant advantage in using this technique rather than others is to provide a true fine map of stresses in a short time even if a preliminary meticolous calibration of the equipment has to be performed. 

For example, suppose we are interested in knowing the distribution of stress assoeiated with the tensile statie loading on a reetangular bar (see Figure 4.23). It is known from a statistieal analysis of the load data that the load, F, has a eoeffieient of variation Q = 0.1. The stress, L, in the bar is given by ... 

Suppose we are interested in knowing the distribution of stress assoeiated with the tensile statie loading on a reetangular bar (see Figure 1). The governing stress is given by ... 

The overall distribution of stresses and strains in the local and global directions is shown in Fig. 3.23. If both the normal stress and the bending are applied together then it is necessary to add the effects of each separate condition. That is, direct superposition can be used to determine the overall stresses. 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

This procedure demands a solution, based on numerical analysis and a rather long computer programme. Then, this method can be used only when detailed results are demanded, requiring the knowledge of the exact distribution of stresses and displacements between phases. 

The fiber fragment length can be measured using a conventional optical microscope for transparent matrix composites, notably those containing thermoset polymer matrices. The photoelastic technique along with polarized optical microscopy allows the spatial distribution of stresses to be evaluated in the matrix around the fiber and near its broken ends. 

Fig. 9.10. Deflection of a tube scanner. (A) Opposite and equal voltages are applied to the y electrodes of a tube scanner. The x, z electrodes are grounded. A positive stress (pressure) is generated in the upper quadrant, and a negative stress (tension) is generated in the lower quadrant. (B) At equilibrium, a distribution of stress and strain is established such that the total torque at each cross section is zero. This condition determines the deflection of the tube scanner in the y direction. (Reproduced from Chen, 1992, with permission.)...

By applying an ac voltage to only one of the quadrants, the distribution of stress on a cross section can be obtained using the conditions of zero force and zero torque, as described in Section 9.5. As shown in Fig. 9.14, if only the yi quadrant is activated, the stress distribution is... 

The explanation of the effect of secondary inclusions on the delocalization of shear banding is based on the concept of modification of the local stress fields and achieving favorable distribution of stress concentrations in the matrix due to presence of inclusions. This leads to a reduction in the external load needed to initiate plastic deformation over a large volume of the polymer. As a result, plastically deformed matter is formed at the crack tip effectively reducing the crack driving force. Above approximately 20 vol% of the elastomer inclusions. 

A conscious choice of such elements can be made but in general the equilibrium distribution of stress cannot be found except for particular geometries. The assumptions of uniform strain throughout the assembly or of uniform stress were respectively made by Voigt and by Reuss. Returning to the structures actually perceivable in polymers one may consider the spherulite in a semi crystalline polymer as being unsuitable as a RVE because the boundary is not included. However, an assembly of spherulites would be acceptable, since it would contain sufficient to make it entirely typical of the bulk and because such an assembly would have moduli independent of the surface tractions and displacements. The linear size of such a representative volume element of spherulites would be perhaps several hundred microns. 

An adhesive layer with appropriate thickness and elastic modulus is necessary to obtain reasonable distribution of stresses in the region of a weldbond joint. A thin adhesive layer of high elastic modulus improves the fatigue properties of weldbonded joints.43... 

The population of flaws can be characterized by a flaw spectrum which is defined by (a) the distribution of flaws in terms of their characteristic dimensions (for example, their lengths and radii of curvature) (b) the distribution in terms of mean spacing or distance between flaws and (c) the distribution in terms of anisotropy or directional preference with respect to some sample reference axis (e.g., an axis of orientation or draw direction). The interactive and collective probabilities which describe distributions (a), (b), and (c) comprise the flaw spectrum and through suitable analysis serve to determine, in principle, the distribution of stress concentration factors (SCF) in a given material. 

We now intend to examine the modifying effect of the elliptic hole on the distribution of stress in the plate. A straightforward but tedious solution of the Laplace equation for the displacement vector field leads (See section 1.2.2(a), for the equivalent solution of the scalar potential problem in a two-dimensional conductor with an elliptic dielectric hole inside), to the largest concentration of stress at the point C, where... 

Under conditions of high strain, deviations from the mechanical behavior predicted by the low strain analysis may occur, due to slip at the matrix-filler interface. Further, due to non-uniform distribution of stress and strain throughout the material may result in a more complex mechanical response to deformation at high strains. 

The brittle-ductile transition temperature depends on the characteristics of the sample such as thickness, surface defects, and the presence of flaws or notches. Increasing the thickness of the sample favors brittle fracture a typical example is polycarbonate at room temperature. The presence of surface defects (scratches) or the introduction of flaws and notches in the sample increases Tg. A polymer that displays ductile behavior at a particular temperature can break in the brittle mode if a notch is made in it examples are PVC and nylon. This type of behavior is explained by analyzing the distribution of stresses in the zone of the notch. When a sample is subjected to a uniaxial tension, a complex state of stresses is created at the tip of the notch and the yield stress brittle behavior known as notch brittleness. Brittle behavior is favored by sharp notches and thick samples where plane strain deformation prevails over plane stress deformation. 

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