Plane stress states

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... 

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... 

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

In this section, an analytical solution to calculate residual stresses in an FGM disk is discussed, based on simple linear elastic plate theories of classical mechanics, and used for the calculation of residual stresses in a plane stress state. An equi-biaxial stress analysis differs from a plane stress state by simply replacing the Young s modulus A by the corresponding biaxial modulus E = E/( 1 - v). In this way, the residual thermal stress can be calculated in the centre of the FGM disk, far enough away from the free edges where a complex stress state is present. 

In a simple model for this case, which, as in the 3-D case, ignores fiber straightening and anisotropy of the fibrous network, a plane stress version of Eqn. (35) can be developed. As such, it can only be used for plane stress states. Consider the x-y plane to be that in which the fibers are woven or the whiskers are lying. The strain rates in this plane are taken to be homogeneous throughout the composite material and crzz, axz and ayz are taken to be zero. The resulting law is... 

M. Arcan, Z. Hashin, and A. Voloshin, A Method to Produce Uniform Plane-Stress States with Applications to Fiber-Reinforced Materials, Exp. Mech., 18[4], 141-146 (1978). 

We shall see that for a prismatic or cylindrical body with the same symmetry as in the case of plane strain and loaded normal to the z axis but now with its ends load-free, a plane stress problem is obtained in which the nonzero stresses vary with z. Strictly speaking, a true plane stress state is present only in thin plates with the main surfaces load-free and with external forces z-independent but symmetrically distributed through its thickness. 

In what follows, we first describe an experiment done by "disk-bend" testing. This method of testing is advantageous to the matter concerned because it allows us to make an equibiaxial plane-stress state of loading to be met in individual "layers" constituting an FGM. However, its application to materials exhibiting inelastic deformation has not been established yet. Under the circumstances, we devise a new method for the analysis of test data. 

In the plane stress state of bioceramic coatings is general equation to determine the stress state is obtained as... 

The development of stresses in the scale is caused by various mechanisms which are briefly considered in the following. The relation between the stress, elastic strain, 8el, within the alumina scale is given by the Hooke s law. The elastic properties of the polycrystalline scale are assumed to be isotropic with E0 as Young s modulus and i as Poisson s ratio. Because of the free surface of the scale, a plane stress state in the scale is supposed with = 0. z is the direction perpendicular to the film plane, and x,v are the in-plane coordinates. The x-component of the stress tensor is then given by... 

Zinoviev PA, Grigoriev SV, Lebedeva OV, Tairova LP. The strength of multilayered composites under a plane-stress state. Compos Sd Technol 1998 58 1209-33. 

Anderson YaA, Limonov VA, Tamuzh VP. Effect of phase asynchronism on the fatigue resistance of laminated fiber composites in a plane stress state. Mech Compos Mater 1992 27(5) 521-9. 

A similar expression with double prime defines C". A plane stress state can be described... 

GR X E, for plane stress states (where E is Young s modulus)... 

For polymers that fail as a result of formation of so-called silver cracks the coefficient a is customarily considered as equal to 1. In this case, calculation using Equation (3.6) and using the characteristics Gic and 6ic for composition 5 (see Table 3.1) gives n po.2 = 1-6 X 10 MPa. The adhesive interlayer yield limit determined experimentally at deformation rates e = 1 x 10 —1 x 10 s changes within the range (2.2-2.6) x 10 MPa. If the relationship a = 7t/4 is assumed for the plane stressed state, the estimated value of o-po.2 = 2.0 x 10 MPa will come close to that found experimentally. 

The results presented in Fig. 11.21 are very meaningful as they clarify some of the key issues concerning the contribution of void formation to toughness in polymer blends. It appears that the cavitation is extremely important in notched specimens because it allows the blend to yield under plane-strain conditions at stiU moderate stresses due to increased sensitivity to the mean stress. It implies that this modification of yielding does not result from eliminating geometrical constraints and converting a state of plane-strain to plane-stress state, as it has been frequently postulated in the past (Bucknall and Paul 2009). 

Assumptions of the model (a) plane stress state, (b) The surrounding rocks are uniform and elastic isotropic materials, (c) the fault is assumed as contact element with no thickness., and (d) Mohr-Coulomb failure criterion is chosen for rocks. 

On the other hand, PP shows shear yielding without craze formation and yielding behavior (i.e. necking) with a plateau in a plane stress state. The stress for yielding is also affected by the density of tie molecules because the tie molecules are pulled out from the lamellar fragment during the yielding process [4]. 

Choice of the appropriate size and thickness of the sample depends on several factors, inclnding the morphology of the pol5uneric material, sample shape or geometry, and stress state (eg, plane strain or plane stress state). Therefore, the various techniques mentioned above allow the stndy of very different polymers. 

For two dimensional problems we can consider two idealized states the plane strain state where s = Sxz = yi = and the plane stress state in which = Oxz = [Pg.52]

If we reduce the number of variable principal stresses to two, for example, by considering a plane stress state, the yield surface becomes a line. As in the general case, yielding occurs if the stress state reaches this line as sketched in figure 3.17. Figure 3.18 illustrates how the yield surface can be constructed from /([Pg.85]

In figure 3.25, the different yield criteria are compared for a plane stress state. The modified yield criteria lead to a compressive yield strength that surpasses the tensile yield strength. 

A component made of a polycrystalline aluminium alloy with yield strength RpO.2 = 200 MPa is loaded in a plane-stress state. The stress components are an = 155 MPa, a22 = 155 MPa, and T12 = 55 MPa. 

Matrix the ligament between the particles (including interphase) should stay locally under plane stress state in order to favor shear yielding, plastic deformation. 

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