Stress systems simple

In postulating a statistieal model for a statie stress variable, it is important to distinguish between brittle and duetile materials (Bury, 1975). For simple stress systems, i.e. uniaxial or pure torsion, where only one type of stress aets on the eomponent, the following equations determine the failure eriterion for duetile and brittle types to prediet the reliability (Haugen, 1980) ... 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

During service the impact behaviour of a plastic article will be influenced by the combined effects of the applied stress system and the geometry of the article. Although the applied stress system may appear simple (for example, uniaxial) it may become triaxial in local areas due to a geometrical discontinuity. Fig. 2.78... 

When a mbber block of rectangular cross-section, bonded between two rigid parallel plates, is deformed by a displacement of one of the bonded plates in the length direction, the rubber is placed in a state of simple shear (Figure 1.1). To maintain such a deformation throughout the block, compressive and shear stresses would be needed on the end surfaces, as well as on the bonded plates [1,2]. However, the end surfaces are generally stress-free, and therefore the stress system necessary... 

Maximum shear stress theory which postulates that failure will occur in a complex stress system when the maximum shear stress reaches the value of the shear stress at failure in simple tension. 

Show that regardless of the orientation of a straight dislocation line and its Burgers vector, there will exist a stress system that will convert the dislocation line into a helix whose axis is along the position of the original dislocation when the point-defect concentration is at the equilibrium value characteristic of the stress-free crystal. Use the simple line-tension approximation leading to Eq. 11.12. 

Although the presence of the reinforcing fibers enhances the strength and modulus properties of the base material, they also cause a complex distribution of stress in the materials. For example, even under simple tensile loading, a triaxial stress system is set up since the presence of the fiber restricts the lateral contraction of the matrix This system increases the possibility of brittle failure in the material. The type of fracture which occurs depends on the loading conditions and fiber matrix bonding. 

Figure 9.17 graphically shows a comparison of the four methods of analysis for a two-dimensional stress system (Xr = 0). The upper right quadrant represents tension for both fjc and fy. The upper left quadrant represents fy in tension and/r in compression, and the lower right quadrant represents in tension and fy in compression. The wlnl lines in the figure represent the locus of the conditions at which yield is assumed to begin according to the four theories. The square a-b-c-d represents the maximum stress theory. Point a represents equal tension in both the X and y perpendicular directions, both of which are considered to be equal to the yield-point stress obtained from a simple tensile lest. 

Using Eq. 2.60, the deviator matrix can be separated into five simple shear stress systems,... 

Thus it is possible, by using closed-form analyses of varying complexity, to predict the stresses in simple lap joints. (This approach is termed continuum mechanics.) In many instances, such solutions may be deemed acceptable. However, two problems still remain to be solved if it is required to predict the strength of real joints. These may be summarized as end effects and material non-linearity (adhesive and adherend plasticity). We will look first at end effects for linear elastic systems. 

Figure 3.5 shows this effect for an ordinary carbon steel and illustrates that the ductility of a material is affected by the type of stress system and the rate of application of this stress system. Between T2 and for example, the carbon steel displays ductile behavior in a simple uniaxial stress system (tensile test) or displays brittle characteristics at high rates of loading (impact test). Increasing either the strain rate or the complexity of the stress system moves the curve in Fig. 3.5 to the right. This amounts to an increase in the brittle transition temperature. Similar ductile or brittle behavior is observed above T4 and below Tj. 

In the tensile butt joint, shown in Fig. 3.3, the adhesive forms a thin disc between two (usually metallic) adherends. This appears to be a simple stress system, but it is far from this. Even if extensometry can be obtained to measure the true deflections across the glue-line, the radial... 

The situation is not so simple when these various parameters are time dependent. In the latter case, the moduli, designated by E(t)and G(t), are evaluated by examining the (time dependent) value of o needed to maintain a constant strain 7o- By constrast, the time-dependent compliances D(t) and J(t)are determined by measuring the time-dependent strain associated with a constant stress Oq. Thus whether the deformation mode is tension or shear, the modulus is a measure of the stress required to produce a unit strain. Likewise, the compliance is a measure of the strain associated with a unit stress. As required by these definitions, the units of compliance are the reciprocals of the units of the moduli m in the SI system. 

A piping system can be evaluated for its displacement and stress either by manual methods (charts, tables, hand calculation) or computerized solution. The latter has become the standard approach desktop and laptop computers can handle all but the most compHcated problems. Manual methods are used only for rough estimates on very simple systems. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

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