Maximum normal stress

ABFC GDA. This last seems to well predict the behavior of cast iron that effectively has uneven properties in traction and compression. The equivalent stress amplitude derived from the maximum normal stress criterion knowing the three stress components Oa,y and is given by 

For a combination of cyclic traction and cyclic torsion Eq. (9.1) becomes 

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This optimum condition is designed to ensure that the botes of all components yield at the same time. If the cylinder is subjected to fatigue conditions, it has been suggested (39) that a better design criterion would arrange for the maximum normal stress, which controls fatigue crack propagation, to be the same in each component. 

By definition, a brittle material does not fail in shear failure oeeurs when the largest prineipal stress reaehes the ultimate tensile strength, Su. Where the ultimate eompressive strength, Su, and Su of brittle material are approximately the same, the Maximum Normal Stress Theory applies (Edwards and MeKee, 1991 Norton, 1996). The probabilistie failure eriterion is essentially the same as equation 4.55. 

The maximum normal stress at the point of contact, according to Hertz, is 1.5 times the average normal stress ... 

We can note that this curve is actually linear. Such linearity is also observed for other measurement points. A linear regression makes it possible to calculate Urn K and account for a maximum normal stress value at each measurement point. 

Fig. 5. Maximum normal stress (a) distributions for a 15.2 cm thick infinite slab and a 5.1 cm thick 0.3 m by 0.6 m sample of polymethacrylimide foam bonded to an aluminum plate (foam surface temp. = 317 K aluminum temp. = 20 K).

In a state of incipient failure, the yield locus is tangent to the Mohr circle. The Mohr circle graphically represents the equilibrium stress condition at a particular point at any orientation for a system in a condition of static equilibrium in a two-dimensional stress field. The equilibrium static conditions can also be applied to sufficiently slow steady flows. The maximum principal stress in Fig. 6.4(b) is called the unconfined yield strength. This is the maximum normal stress, under incipient failure conditions, at a point where the other principal stress becomes zero. Such a situation occurs on the exposed surface of an arch or dome in a feed hopper at the moment of failure see Fig. 7.5(b). In the analysis of bridging in feed hoppers, the unconfined yield strength becomes a very important parameter. The magnitude of the unconfined yield strength is determined by the YL and depends, therefore, on the consolidation pressure and time. 

As a result of the stresses acting on the screw flight, there will be a distribution of normal stresses in the flight that reaches a maximum value at the screw surface. The torque resulting from this normal stress distribution counteracts the torque resulting from t, and AP. If the maximum normal stress at the wall is Ty ,ax. then a torque balance per unit flight length yields ... 

When the material behavior is brittle rather than ductile, the mechanics of the failure process are much different. Instead of the slow coalescence of voids associated with ductile rupture, brittle fracture proceeds by the high-velocity propagation of a crack across the loaded member. If the material behavior is clearly brittle, fracture may be predicted with reasonable accuracy through use of the maximum normal stress theory of failure. Thus failure is predicted to occur in the multi-axial state of stress when the maximum principal normal stress becomes equal to or exceeds the maximum normal stress at the time of failure in a simple uniaxial stress test using a specimen of the sane material. 

Maximum normal stress theory (Lame-Navier) Failure occurs when the... 

Comparison Between Theory and Experiment Comparisons between theory and experiment have been made for many materials. Shown in Fig. 2.20 are the graphs in stress space for the equations for the three theories given above. Also shown is experimental data on five different metals as well as four different polymers. It will be noted that cast iron, a very brittle material agrees well with the maximum normal stress theory while the ductile materials of steel and aluminum tend to agree best with the von Mises criteria. Polymers tend to be better represented by von Mises than the other theories. 

These equations have been used to estimate the maximum shearing stresses in the attachment and the maximum normal stresses in the die when the silver-filled epoxy IP 670 and polyimide IP 680 adhesives are employed to bond silicon dice to ceramic substrates and to copper lead frames [4]. The temperature differential is assumed to be 150°C for the epoxy resin and 280°C for the polyimide (from - 50°C to their respective glass transition temperatures of 100 and 230°C). For the silicon-epoxy-ceramic assembly the maximum shearing stress in the attachment is 23.2 MPa while the maximum normal stress in the die o- iax reaches 73.2 MPa. On copper leadframe, whose CTE is three times larger, the values of o-max are 66.1 and 127.3 MPa, respectively. Although file elastic... 

The effect of the die size can be discussed with the assumption that small dice have a length culminating at 5-6 mm while large dice are 10 mm long, or larger. For small dice, the factors Xmax and Xmax which reflect the effect of die size on the maximum normal stresses and the maximum shear stresses, increase when the die size is increased. In contrast, the maximum stresses in the die are virtually independent from its size for large backside attached dice with stiff attachments (kl > 4). On the other hand, the maximum interfacial stresses become independent from the die size when kl > 2.5. This means that, with respect to these kl values, if... 

Figure 50 Stress distribution over the half-length of 1 cm dice bonded to ceramic substrate. Curves (a) and (a represent the maximum shearing stress in the attachment for 25 and 100 xm IP 670 adhesive layer, while curves (b) and (b show the maximum normal stress in the die for the same thicknesses.

The maximum normal stress yield strength (Ty to maintain the dimensionless feature of strain. Combining Eq. (9.58) with the Manson and Cofhn Eq. (6.10) it yields... 

It is usually desirable to run a simple bulk tensile test program and subsequently predict (calculate) shear properties from their tensile counterparts. This approach requires a clearly defined relationship between shear and tensile elastic limit and yield variables and material properties. The elastic limit and yield stress values can be related between tensile and shear conditions by using an appropriate failure criterion, such as maximum normal stress, maximum shear stress, and distortion energy criteria. A material parameter that needs to be converted in addition to the usual elastic properties is the viscosity coefficient. This can be done by using Tobolsky s (1960) assumption of equivalent relaxation times in shear and tension. Application of this assumption results in the relation ... 

In predicting limit (threshold) conditions, such as the elastic Kmit, yield, and failure conditions, classical failure criteria, such as the maximum normal stress criterion, maximum shear stress criterion and the distortion energy (von Mises) criterion can be employed. 

An estimate of the maximum normal stress can he obtained by modeling the desiccating graft in one dimension as a shrinking elastic beam bonded to a rigid surface, is shown below... 

Strain at fracture of 50% has been found with a permanent strain after elastic springback of only 10.7%. Although this amount of ultimate strain indicates ductile behavior, the nature of the fracture surface is typical of brittle fracture being oriented normal to the maximum normal stress as shown in Figure 2.2b. 

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