Plane stress, defined

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... 

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

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... 

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... 

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... 

Eq. (13.145) is called the Griffith equation for thin sheets, i.e. in plane stress, where Gic, known as the fracture energy, has replaced 2y. A frequently used parameter in plane stress is the critical stress energy factor Kq, which in the case of a wide sheet (plane stress) is defined as... 

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. 

If we eliminate the unknown ns via the constraint n n = 1, then minimizing of eqn (2.58) with respect to i and 2 results in two algebraic equations in these unknowns. What emerges upon solving these equations is the insight that the planes of maximum shear stress correspond to the set of 110 -type planes as defined with respect to the principal axes, and that the associated shear stresses on these planes take the values r = (ai — o ), which are the differences between the and principal stresses. 

For two-dimensional problems, two special cases are considered namely, p/anc stress and plane strain. For the case of plane stress, only the in-plane (e.g., the xy-plane) components of the stresses are nonzero and for plane strain, only the inplane components of strains are nonzero. In reality, however, only the average values of the z-component stresses are zero in the plane stress cases. As such, this class of problems is designated by the term generalized plane stress. The conditions for each case will be discussed later. It is to be recognized that, in actual crack problems, these limiting conditions are never achieved. References to plane stress and plane strain, therefore, always connote approximations to these well-defined conditions. 

In plane strain problems, the displacements that exist in a particular direction are assumed to be zero. If this direction is Xy it follows from the definition of strain (Eq. (2.14)) that e,3=e23=e33=0, i.e., the strains are two-dimensional. As an example, consider the problem shown in Fig. 4.13 a knife edge indenting a thick block of material. Most of the displacements are occurring in the x and directions, i.e., the material is being pushed downwards or sideways. The only exceptions are in the vicinity of the front and back surfaces, where displacements in the Xj direction are possible. Overall, the components of the displacement vector at any point can be assumed to be independent of Xj. From Hooke s Law, the assumption that 3=e23 33 plies that <7 3=cr23=0. As with plane stress, only the stress components and are needed to define the... 

In Chapter 2, stress and strain were defined, the compatibility and equilibrium equations were introduced and the relationship between stress and strain was defined. Thus, any solution that satisfies all these equations and the appropriate boundary conditions will be the solution that gives the stress and strain distribution for a particular loading geometry. For the most general problems, the scientific process can be difficult but for plane stress and plane strain problems in elastically isotropic bodies the solution involves a single differential equation. 

Various theories have been put forward to define the failure condition and the direction of crack propagation. For example, if failure occurs at a critical value of G, one can use Eq. (8.20) (plane stress) to write... 

Banabic et al. (2005) proposed a new expression of the plane stress potential. An improvement of this criterion has been implemented in the finite element commercial code AUTOFORM version 4.1 (Banabic and Sester 2012). The equivalent stress is defined by the following formula ... 

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

Equation 5.21 holds for plane-stress conditions at the crack tip, which are most likely to occur if the sheet is thin (we discuss later what thin implies in this context). For plane-stress, then, the operative fracture parameter is G which is known as the firacture energy. A related (and more useful) parameter is the plane-stress critical stress intensity factor K, (5.N.6), which is defined, in the case of a wide sheet, by... 

There is a critical specimen width B above which this approximation can be used. This critical width is reached when the whole specimen cross-section is under the conditions of plane stress and thus the measured fracture toughness reaches its upper plane stress value G. Mathematically, critical width is defined using critical stress intensity factor as ... 

In Eq. (1.17), the indicated stress is acting at a point on an inclined plane, reproduced here as Eq. (1.22). An inclined plane is defined by its directional cosines in the absence of shear stress and, if the stress is normal (as shown in Fig. 1.15), it is a principal stress . It follows that... 

By equating the partial differentials of Eq. (1.26) to zero, the positions of the planes are defined by where the shear stress reaches its extreme values, namely its... 

Plane strain defines a triaxial stress state at the crack tip, which is, for example, the case for a thick block of material. Plane stress is typical for thin samples where there is no stress acting normal to the plane of the sample. 

We now want to estimate the change dU ) in the stored elastic strain energy during crack propagation. To do this, we again consider the case of vanishing external work and assume a constant stress cr and a state of plane stress. Furthermore, we define a state 1 in which a stress (Tr is applied to the crack surfaces (see figure 5.7). At ctr = a, the crack is completely closed... 

Because a principal plane is defined as one on which there is only a normal stress, the traction vector and the unit normal to that plane must be in the same direction ... 

Fig. 5.32 Envelopes defining crazing and yield for an amorphous polymer undergoing plane stress deformation. (After Sternstein and Ongchin (1969), ACS Polymer Reprints, 19, 1117.)...

Fig. 5.32 Envelopes defining crazing and yield for an amorphous polymer undergoing plane stress deformation.

The state of stress or strain within a test piece may be defined by considering an infinitesimally small cubic volume element with edges parallel to orthogonal axes x, y and z. The components of stress (defined as force per unit area) acting normally on planes perpendicular to the directions x, y and z are customarily represented by yy and respectively. The stresses acting on planes perpendicular to x and in the direction of z are the shear stress components = a y and The stress-induced displacement of a point (x,y,z) within a material can be resolved into components M, V and w parallel to the axes x, y and z. The normal strains are defined by... 

When elastic terms are neglected the shear stress relevant to Fig. 4.1 is the viscous force along the a -direction per unit area in a plane normal to the 2 -axis this is simply i3. The apparent viscosity t7(, 0) is defined in the usual way for such a geometry by (cf. de Gennes and Frost [110, p.211], who use the transpose of the viscous stress defined in this text)... 

The material parameter E is given by E = E for plane stress conditions and E = E/(l -v ) for plane strain conditions. These conditions are defined as follows ... 

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