Stress concentration factors

The stress distribution in a component can be visualised using so-caUed stress trajectories. These trajectories always run in the direction of the maximum principal stress. Their distance is inversely proportional to the stress so that the stress trajectory density is a measure of the locally acting stress. Each abrupt change in cross section deflects the stress trajectories which then move closer together. Thus, a local stress concentration arises. 

The term stress trajectory is due to the fact that the stress distribution in a component is analogous to the velocity distribution of a laminar, frictionless fluid. The stress concentration at changes in the geometry corresponds to the disturbed flow of the fluid at similar geometries. Different from fluid flows, stress trajectories cannot become turbulent. 

Fig- 4.1. Stress trajectories in notched components. The stress trajectories are aligned with the maximum principal stress, their density is a measure of the stress level. At the notch root, there is a stress concentration in both geometries 

In the centre of the specimen at the position of the notch, the stress is smaller than the net-section stress (Tnss- The reason for this is that the total force transferred through the cross section does not change and that the stress at the notch root is larger than (Tnss- 

In the context of notches, the stress concentration is always related to the net-section stress Tnss- If the stress concentration at the notch root has to be compared to the nominal stress far away from the notch or, more precisely, (Tnooi infinitely far away from the notch), the increase in stress due to the reduced cross section and the stress concentration at the notch root have to be multiplied. 

---

Division 2. With the advent of higher design pressures the ASME recognized the need for alternative rules permitting thinner walls with adequate safety factors. Division 2 provides for these alternative rules it is more restrictive in both materials and methods of analysis, but it makes use of higher allowable stresses than does Division 1. The maximum allowable stresses were increased from one-fourth to one-third of the ultimate tensile stress or two-thkds of the yield stress, whichever is least for materials at any temperature. Division 2 requkes an analysis of combined stress, stress concentration factors, fatigue stresses, and thermal stress. The same type of materials are covered as in Division 1. 

Figure 4.20 Values of stress concentration factor, Kt, as a function of radius, r, with 3a limits for a circumferentially notched round bar in tension [d A/(0.5, 0.00266) inches, = 0.00333 inches] (adapted from Haugen, 1980)...

K = actual stress concentration factor for static loading... 

Peterson, R.E., Stress Concentration Factors, John Wiley Son, 1953. 

The parameter (1 -f l ajr ) is commonly termed the stress concentration factor K,) and for a hole where a = r then K, = 3, i.e. the stresses around the periphery of the hole are three times as great as the nominal stress in the material. 

The stress intensity factor is a means of characterising the elastic stress distribution near the crack tip but in itself has no physical reality. It has units of MN and should not be confused with the elastic stress concentration factor (K,) referred to earlier. 

It has been shown that for acrylic, glass-filled nylon and methyl pentene there is reasonable correlation between the reciprocal of the stress concentration factor, K, and impact strength. However, for PVC good correlation could only be achieved if the finite dimensions of the sample were taken into account in the calculation of stress concentration factor. 

For the purposes of performing an impact test on a material it is proposed to use an elastic stress concentration factor of 3.5. If the notch tip radius is to be 0.25 mm estimate a suitable notch depth. 

The second special case is an orthotropic lamina loaded at angle a to the fiber direction. Such a situation is effectively an anisotropic lamina under load. Stress concentration factors for boron-epoxy were obtained by Greszczuk [6-11] in Figure 6-7. There, the circumferential stress around the edge of the circular hole is plotted versus angular position around the hole. The circumferential stress is normalized by a , the applied stress. The results for a = 0° are, of course, identical to those in Figure 6-6. As a approaches 90°, the peak stress concentration factor decreases and shifts location around the hole. However, as shown, the combined stress state at failure, upon application of a failure criterion, always occurs near 0 = 90°. Thus, the analysis of failure due to stress concentrations around holes in a lamina is quite involved. 

Fracture is caused by higher stresses around flaws or cracks than in the surrounding material. However, fracture mechanics is much more than the study of stress concentration factors. Such factors are useful in determining the influence of relatively large holes in bodies (see Section 6.3, Holes in Laminates), but are not particularly helpful when the body has sharp notches or crack-like flaws. For composite materials, fracture has a new dimension as opposed to homogeneous isotropic materials because of the presence of two or more constituents. Fracture can be a fracture of the individual constituents or a separation of the interface between the constituents. 

The stress-intensity factors are quite different from stress concentration factors. For the same circular hole, the stress concentration factor is 3 under uniaxial tension, 2 under biaxiai tension, and 4 under pure shear. Thus, the stress concentration factor, which is a single scalar parameter, cannot characterize the stress state, a second-order tensor. However, the stress-intensity factor exists in all stress components, so is a useful concept in stress-type fracture processes. For example. 

ERENCES section at the end of this book provide detailed analysis of these stress-concentration factors and other load-bearing parameters. 

Sharp corners become stress concentrators. The stress-concentration factor... 

This result indicates that the stress necessary to cause brittle fracture is lower, the longer the existing crack and the smaller the energy, P, expended in plastic deformation. The quantity Of represents the smallest tensile stress that would be able to propagate the crack of length 2 L. The term Of (tt L)°5 is generally denoted by the symbol K and is known as the stress-intensity factor (for a sharp elastic crack in an infinitely wide plate). Fracture occurs when the product of the nominal applied stress and the stress concentration factor of a flaw attains a value equal to that of the cohesive stress. 

There are two junctions in a torispherical end closure that between the cylindrical section and the head, and that at the junction of the crown and the knuckle radii. The bending and shear stresses caused by the differential dilation that will occur at these points must be taken into account in the design of the heads. One approach taken is to use the basic equation for a hemisphere and to introduce a stress concentration, or shape, factor to allow for the increased stress due to the discontinuity. The stress concentration factor is a function of the knuckle and crown radii. 

This equation will only apply at points away from the cone to cylinder junction. Bending and shear stresses will be caused by the different dilation of the conical and cylindrical sections. This can be allowed for by introducing a stress concentration factor, in a similar manner to the method used for torispherical heads,... 

Stresses can can be concentrated by various mechanisms. Perhaps the most simple of these is the one used by Zener (1946) to explain the grain size dependence of the yield stresses of polycrystals. This is the case of the shear crack which was studied by Inglis (1913). Consider a penny-shaped plane region in an elastic material of diameter, D, on which slip occurs freely and which has a radius of curvature, p at its edge. Then the shear stress concentration factor at its edge will be = (D/p)1/2.The shear stress needed to cause plastic shear is given by a proportionality constant, a times the elastic shear modulus,... 

The stress needed to move a dislocation line in a glassy medium is expected to be the amount needed to overcome the maximum barrier to the motion less a stress concentration factor that depends on the shape of the line. The macro-scopic behavior suggests that this factor is not large, so it will be assumed to be unity. The barrier is quasi-periodic where the quasi-period is the average mesh size, A of the glassy structure. The resistive stress, initially zero, rises with displacement to a maximum and then declines to zero. Since this happens at a dislocation line, the maximum lies at about A/4. The initial rise can be described by means of a shear modulus, G, which starts at its maximum value, G0, and then declines to zero at A/4. A simple function that describes this is, G = G0 cos (4jix/A) where x is the displacement of the dislocation line. The resistive force is then approximately G(x) A2, and the resistive energy, U, is ... 

In contrast to the impact tests, these can be analysed toughness is reported as the critical energy release rate (7, or the stress concentration factor K Values may tange from 5000 J. nr for a tough nylon or polycarbonate down to 350. J/m lor buttle unmodified polystyrene. The values can be sensitive to rale and temprature... 

Based on transverse stress concentration factor of 2.40. HBE high ion beam energy LEB low ion beam energy. 

Layup Young s modulus (GPa) Measured stress concentration factor Predicted stress concentration factor Notched strength, [Pg.344]

Consider the same unidirectional lamina with the stresses now applied perpendicular to the fiber axis as shown in Fig. 12. The local stress at the fiber matrix interface can be calculated and compared to the nominally applied stress on the whole lamina to give K, the stress concentration factor. The plot of the results of this analysis shows that the interfacial stresses at the point of maximum principal stress can range up to 2.6 times the applied stress depending on the moduli of the constituents and the volume fraction of the reinforcement. For a typical graphite-epoxy composite, with a modulus ratio of 70 and a volume fraction of 70 % the stress concentration factor at the interface is about 2.4. That is, the local stresses at the interface are a factor of 2.4 times greater than the applied stress. 

The assumptions used in the model to predict the stress concentration factor assumed homogeneity of the fiber and the matrix up to the fiber-matrix interface. This assumption is not correct in all cases. While glass may be homogeneous in all directions, graphite and aramid are not. The tranverse modulus of graphite varies with the tensile... 

Fig. 12. A unidirectional lamina under transverse tension. The points of stress concentration are at the dots. The micromechanical analysis shows that the stress concentration factor increases with volume fraction of fiber and fiber to matrix modulus ratio. From Adams et al.70)...

Imposing a shear stress parallel to the fiber axis of a unidirectional composite creates an interfacial shear stress. Because of the disparity in material properties between fiber and matrix, a stress concentration factor can develop at the fiber-matrix interface. Linder longitudinal shear stress as shown by the diagram in Fig. 13, the stress concentration factor is interfacial. The analysis shows that the stress concentration factor can be increased with the constituent shear modulus ratio and volume fraction of fibers in the composite. Under shear loading conditions at the interface, the stress concentration factor can range up to 11. This is a value that is much greater than any of the other loadings have produced at the fiber-matrix interface. 

Klc = critical stress intensity factor in mode I, MPam1/2 Km = stress concentration factor Me = average molar mass between crosslinks, kg mol 1 Mn = number-average molar mass, kg mol 1 p = hydrostatic pressure, Pa Tg = glass transition temperature, K... 

---