Plane stress conditions

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... 

This plastic deformation is localised around the crack tip and is present in all stressed engineering materials at normal temperatures. The shape and size of this plastic zone can be calculated using Westergaards analysis. The plastic zone has a characteristic butterfly shape (Fig. 8.83). There are two sizes of plastic zone. One is associated with plane stress conditions, e.g. thin sections of materials, and the other with plane strain conditions in thick sections-this zone is smaller than found under plane stress. 

The elastic stress-strain relations for an orthotropic lamina under plane stress conditions are... 

K c Critical stress intensity factor in mode I and plane stress conditions... 

Figure 12.6 Unmodified Tresca and von Mises criteria in plane stress conditions (cr3 = 0).

Initiation is assumed to occur when a critical stress state is attained, according to one of the criteria reported in the literature as discussed in a preceding section. We choose to use the criterion formulated by Sternstein and Ongchin [24] to illustrate how it can be incorporated into the cohesive surface framework, which is flexible enough to account for another definition. The criterion is presented in Eq. 11 for a plane stress condition, with o the ma-... 

Kc and Gc are the parameters used in linear elastic fracture mechanics (LEFM). Both factors are implicitly defined to this point for plane stress conditions. To understand the term plane stress, imagine that the applied stress is resolved into three components along Cartesian coordinates plane stress occurs when one component is = 0. Such conditions are most likely to occur when the specimen is thin. 

Finally, stress whitening after neck formation in tensile bars of crystalline polymers imder plane stress conditions may be associated with some kind of craze-II-formation, in analogy to the corresponding observations in amorphous PC. 

Let us consider now another important problem consisting of an annular disc with internal and external radii represented, respectively, by i i and R2. The equation to be solved, according to the plane stress conditions, is... 

Aim of this paper is to investigate if the rate and temperature effects on the fracture parameters obtained by the EWF approach under plane stress condition are in some way related to the viscoelastic nature of the selected material (semicrystalline PET). 

Both criteria are exemplified in Table 2 and 3 for iPP/EPR-1 tested at room temperature. Table 2 shows (i) to be violated when the mode of failure is ductile (i.e at 0.001 m/s), whereas it remains valid, as expected, in case of brittle fracture (i.e 6 m/s). Table 3 highlights that plane stress conditions prevail roughly up to speeds higher than one decade of test speed tthan the ductile-brittle transition. 

Figure 8 shows a set of load-displacement curves for HM5411EA tested at 1 mm/s. Following the EWF procedure, the plot of the specific work of fracture, uy vs. ligament length, / is produced (Fig.9). It can be seen that linear approximation fits the data very well. From the Intercept between the fitted line and the y-axes, the value of 25.18 kJ/m is obtained for the essential work of fracture. This value represents fracture toughness under plane stress conditions. The slope of the linear fit represents the plastic work dissipation factor, Pwp, where is a shape factor associated with the shape and size of the plastic zone, and Wp is the plastic work dissipation per unit volume of material. The values of fiwp for all cases are given in Table 1.

This equation strictly applies to a surface crack of length c, or an interior crack of length 2r in a thin sheet. Since the surface of the material cannot support a stress normal to it, this condition corresponds to the plane stress condition (the stress is two-dimensional). In thick components, the situation is more complicated, but for brittle materials the two expressions vary slightly. 

The w can be obtained if 1/t ratio is large enough to ensure plane-stress condition in the ligament area and it is proved to be a material constant for a given sheet thickness [Mai and Cotterell, 1986 Mai et al., 1987 Mai and Powell, 1991]. With a reduction of 17t ratio, plastic constraint increases and the plane-stress/plane-strain fracture transition may occur at a certain 1/t ratio. Theoretical analysis shows that the specihc essential work of fracture method is equivalent to the J-integral method for all three fracture modes [Mai and Powell, 1991 Mai, 1993]. 

Equation 1 is for plane-stress condition which exists in thin plates and fracture results in shear lips or slanted fracture surfaces. For plane-strain condition. E is replaced by E/(i — y) in Equation 1. w herc u is Poisson s ratio. Plane strain describes the iriaxial state of stress that exists in thicker plates such that plastic deformation is constrained and fracture produces flat surfaces see Fig. 1. 

Figure 9.8 Dugdale s model for crack tip yielding under plane stress conditions. R, yielded zone length 5, crack tip opening. The shaded region is the yielded zone, while the arrows show the closure force on the hypothetical crack.

The fracture processes contain aspects of both plane stress and plane strain behaviour. The initial stages of crack growth will always be under plane strain conditions, because the yielded zone must grow to a certain size before plane stress conditions can develop. However, plane stress conditions always apply near the free surfaces. Consequently, plane stress shear lips are often observed at the edges of flat fracture surfaces. 

There is, however, no generally accepted theory for predicting the brittle-ductile transition or relating it to other properties of the polymer, although for some polymers it is closely related to the glass transition. The type of failure is also affected by geometrical factors and the precise nature of the stresses applied. Plane-strain conditions, under which one of the principal strains is zero, which are often found with thick samples, favour brittle fracture. Plane-stress conditions, xmder which one of the principal stresses is zero, which are often found with thin samples, favour ductile fracture. The type of starting crack or notch often deliberately introduced when fracture behaviour is examined can also have an important effect ... 

Assuming that plane-stress conditions apply, equation (8.19) shows that F = Aag = A 4yEf(nl), where Fb is the breaking force and / the length of the crack. Substitution leads to... 

The value of E depends upon the values of the elonents in the stress and strain tensors. Under plane stress conditions, one of die principal stresses is zero and E is equal to Young s modulus, E. However, under plane strain conditions, the strain in one of the principal axes is zero and E = E/(l — v ) where v is Poisson s ratio. For most polymers 0.3 < v < 0.5 and the values of both Gic and Kic invariably are much greater when measured in plane stress. For the purposes both of toughness comparisons and component design, die plane strain values of Gic and Kic are preferred because th are the minimum values fm any given material. In order to achieve plane strain conditions, the following criteria need to be satisfied ... 

Equations (6.9) or (6.4) and (6.10) provide the stiffnesses of a ply with hbers oriented at an angle 6 in the laminate coordinate system.The above discussion concentrated on plane stress conditions. In a three-dimensional situation, a 0° ply is dehned with nine stiffness parameters En, E22, E33, G12, G13, G23, J i2, 13, and i 23. Here, as before, 1 denotes the direction parallel to the hbers, 2 is perpendicular to the hbers, and 3 is the out-of-plane direction. Eor orthotropic plies, E22 = E33,... 

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

The stress field in the vicinity of a crack tip for an infinite plate containing a sharp, through-thickness, internal crack under plane stress conditions and uniaxially-applied tension (a) is expressed as... 

Barlat F, Lian K (1989) Plastic behavior and stretchability of sheet metals. Part I A yield function for orthotropic sheets under plane stress conditions. Int J Plasticity 5(l) 51-56... 

Tangential stress component at and shear stress neglected Plane stress condition (ae ... 

The fracture toughness, Kic, for the Mode I fracture test under the plane stress condition can be obtained from [23]... 

Again, once the critical energy release rate /nc is determined, the Mode 11 fracture toughness under the plane stress condition, Knc, can be found through the critical energy release rate as... 

---