Plane-stress

The distance from the crack tip, along the X-axis, at which the von Mises equivalent stress falls below the yield stress, defines the size of the plastic zone, r. For the plane stress case of unconstrained yielding, which corresponds to the free surface of the specimen in Figure 4, this gives... 

One aspect of pressure vessel design which has received considerable attention in recent years is the design of threaded closures where, due to the high stress concentration at the root of the first active thread, a fatigue crack may quickly initiate and propagate in the radial—circumferential plane. Stress intensity factors for this type of crack are difficult to compute (112,113), and more geometries need to be examined before the factors can be used with confidence. 

The four quantities E, 12 sufficient to define the stress—strain law for a unidirectional or woven fiber ply under plane stress, loaded... 

In the case of metals, R is mainly plastic energy associated with the formation of a crack tip plastic zone. It is obvious from Eq. 9 that, for plane stress,... 

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. 

This is for plane stress and so for the plane strain situation... 

Consider the situation of a thin unidirectional lamina under a state of plane stress as shown in Fig. 3.9. The properties of the lamina are anisotropic so it will have modulus values of E and Ei in the fibre and transverse directions, respectively. The values of these parameters may be determined as illustrated above. 

Fig. 3.9 Single thin lamina under plane stress...

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... 

Note that if both plane stresses and moments are applied then the total stresses will be the algebraic sum of the individual stresses. 

This equation may be utilised to give elastic properties, strains, curvatures, etc. It is much more general than the approach in the previous section and can accommodate bending as well as plane stresses. Its use is illustrated in the following Examples. 

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. 

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. 

STRESS-STRAIN RELATIONS FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL... 

For a unidirectionally reinforced lamina in the 1-2 plane as shown in Figure 2-7 or a woven lamina as in Figure 2-1, a plane stress state is defined by setting... 

For orthotropic materials, imposing a state of plane stress results in implied out-of-plane strains of... 

For plane stress on isotropic materials, the strain-stress relations are... 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

For plane stress in the 1-2 plane of a unidirectional lamina with fibers in the 1-direction, < = T. 3 = r23 = 0- However, from the cross section of such a lamina in Figure 2-39, Y = Z from the obvious geometrical symmetry of the material construction. Thus, Equation (2.126) leads to... 

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... 

The stress-strain reiations in principal material coordinates for a lamina of an orthotropic material under plane stress are... 

For plane stress on an orthotropic lamina in principal material coordinates. 

Derive the thermoelastic stress-strain relations for an orthotropic lamina under plane stress, Equation (4.102), from the anisotropic thermoelastic stress-strain relations in three dimensions. Equation (4.101) [or from Equation (4.100)]. 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are... 

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