Stress in-plane

According to Eqs. (13.145) and (13.148) the fracture stress in plane strain is a factor 1 /(1-v2) 1 /0.84 1.2 higher than in plane stress. Experimentally, however, the difference is much bigger. The reason for this discrepancy is that Griffith s equations were developed in linear fracture mechanics, which is based on the results of linear elasticity theory where the strains are supposed to be infinitesimal and proportional to the stress. 

Ox, Oy Geometrical normal stresses (in-plane stress condition)... 

For over three decades, there has been a continuous effort to develop a more universal failure criterion for unidirectional fiber composites and their laminates. A recent FAA publication lists 21 of these theories. The simplest choices for failure criteria are maximum stress or maximum strain. With the maximum stress theory, the ply stresses, in-plane tensile, out-of-plane tensile, and shear are calculated for each individual ply using lamination theory and compared with the allowables. When one of these stresses equals the allowable stress, the ply is considered to have failed. Other theories use more complicated (e.g., quadratic) parameters, which allow for interaction of these stresses in the failure process. 

I = stress index for various locations (see Table 11.5) cr = normal stress in plane being examined (psi) ov = tangential stress in plane being examined (psi)... 

It is useful to represent the yield criteria graphically. This is done for the case of plane stress deformation (0-3 = 0) in Fig. 5.26 where the stress axes are normalized with respect to the tensile yield stress. In plane stress the von Mises criterion (Equation 5.157) reduces to... 

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]. 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Dislocations are known to be responsible for die short-term plastic (nonelastic) properties of substances, which represents departure from die elastic behaviour described by Hooke s law. Their concentration determines, in part, not only dris immediate transport of planes of atoms drrough die solid at moderate temperatures, but also plays a decisive role in die behaviour of metals under long-term stress. In processes which occur slowly over a long period of time such as secondaiy creep, die dislocation distribution cannot be considered geometrically fixed widrin a solid because of die applied suess. 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

But we want the tensile yield strength, A tensile stress a creates a shear stress in the material that has a maximum value of t = a/2. (We show this in Chapter 11 where we resolve the tensile stress onto planes within the material.) To calculate cr from t,, we combine the Taylor factor with the resolution factor to give... 

Fig. 11.2. Shear stresses in a material have their maximum value on planes at 45 to the tensile axis.

Fig. 28.8. Exaggerated drawing of the deflections that occur in the loaded drum. The shaft deflects under four-point loading. This in turn causes the end plates to deflect out of plane, creating tensile (-r) and compressive (-) stresses in the weld.

Macrostrain is often observed in modified surfaces such as deposited thin films or corrosion layers. This results from compressive or tensile stress in the plane of the sample surface and causes shifts in diffraction peak positions. Such stresses can easily be analyzed by standard techniques if the surface layer is thick enough to detect a few diffraction peaks at high angles of incidence. If the film is too thin these techniques cannot be used and analysis can only be performed by assuming an un-... 

This is an alternative form of equation (2.91) and expresses the fundamental material parameter Gc in terms the applied stress and crack size. From a knowledge of Gc it is therefore possible to specify the maximum permissible applied stress for a given crack size, or vice versa. It should be noted that, strictly speaking, equation (2.96) only applies for the situation of plane stress. For plane strain it may be shown that material toughness is related to the stress system by the following equation. 

In order to describe completely the state of triaxial (as opposed to biaxial) stress in an anisotropic material, the compliance matrix will have 36 terms. The reader is referred to the more advanced composites texts listed in the Bibliography if these more complex states of stress are of interest. It is conventional to be consistent and use the terminology of the more general analysis even when one is considering the simpler plane stress situation. Hence, the compliance matrix [5] has the terms... 

The in-plane stiffness behaviour of symmetric laminates may be analysed as follows. The plies in a laminate are all securely bonded together so that when the laminate is subjected to a force in the plane of the laminate, all the plies deform by the same amount. Hence, the strain is the same in every ply but because the modulus of each ply is different, the stresses are not the same. This is illustrated in Fig. 3.19. 

Example 3.16 A unidirectional carbon hbre/PEEK laminate has the stacking sequence [O/SSa/—352]t- If it has an in-plane stress of = 100 MN/m applied, calculate the strains and curvatures in the global directions. The properties of the individual plies are... 

If the plies are each 0.1 mm thick, calculate the strains and curvatures if an in-plane stress of 1(X) MN/m is applied. 

A single ply glass/epoxy composite has the properties Usted below. If the fibres are aligned at 30° to the x-direction, determine the value of in-plane stresses, a, which would cause failure according to (a) the Maximum Stress criterion (b) the Maximum Strain criterion and (c) the Tsai-Hill criterion. 

A carbon/epoxy composite with the stacking arrangement [0/ - 30/30]j has the properties listed below. Determine the value of in-plane stress which would cause failure according to the (a) Maximum Strain (b) Maximum Stress and (c) Tsai-Hill criteria. 

This situation is sometimes referred to as plane stress because there are only stresses in one plane. It is important to note, however, that there are strains in all three co-ordinate directions. 

In the above situation, if the stresses are such that = 0 then this condition is referred to as plane strain. This is because strains are experienced in only one plane even though there are stresses in all three co-ordinate directions. 

It may be seen that when the moment is applied, the major Poisson s ratio v y corresponds as it should to the value when the in-plane stress, Ox, is applied. 

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