Maximum Stress Failure Criterion

In the maximum stress failure criterion, each and every one of the stresses in principal material coordinates must be less than the respective strengths otherwise, fracture is said to have occurred. That is, for tensile stresses, 

Note that the shear strength is independent of the sign of dis- 

Then by inversion of Equation (2.117) and substitution of Equations (2.114)-(2.116), the maximum uniaxiai stress, a, is the smailest of 

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Figure 2-37 Maximum Stress Failure Criterion (After Tsai [2-21])...

The maximum strain failure criterion is quite similar to the maximum stress failure criterion. However, here strains are limited rather than stresses. Specifically, the material is said to have failed if one or more of the following inequalities is not satisfied ... 

The only difference between the maximum strain failure criterion. Equation (2.125), and the maximum stress failure criterion, Equation (2.118), is the inclusion of Poisson s ratio terms in the maximum strain failure criterion. 

As with the maximum stress failure criterion, the maximum strain failure criterion can be plotted against available experimental results for uniaxial loading of an off-axis composite material. The discrepancies between experimental results and the prediction in Figure 2-38 are similar to, but even more pronounced than, those for the maximum stress failure criterion in Figure 2-37. Thus, the appropriate failure criterion for this E-glass-epoxy composite material still has not been found. 

One advantage of the maximum stress failure criterion is that it gives an indication of the type of failure mode. By checking which of Eqn (6.33) is satisfied, one can determine whether the fibers or the matrix failed and whether it is a tension, compression, or shear failure. Of course, this is a macroscopic assessment as the failure criterion in this form does not allow a more detailed determination of failure initiation. For example, a shear failure determined by the last of Eqn (6.33) cannot be traced down to a matrix failure due to local tension or compression nor can it be determined whether the matrix crack was parallel or perpendicular to the fibers. 

Another three-dimensional axisymmetric stress distribution for the stress around fiber breaks was obtained by Naim [93] using variational mechanics. In this study, breaks interaction was also included and it was assumed that both fiber and matrix were linearly elastic and a perfect adhesion at the fiber-matrix interface. To account for the stress singularity at the matrix crack tip of the fiber break, the matrix plastic-model was also included. Imperfect adhesion to mimic a failed fiber-matrix interface was added to this model to study the mechanism of interfacial failure, that is, the stress conditions that cause the extent of interfacial failure or its increase. It was suggested that due to the complexity of the multi-axial stress state, a simple maximum stress failure criterion was unrealistic and an energy release rate analysis was necessary to calculate the total energy release rate associated with the growth of interfacial damage. 

The following sections will describe methods used to calculate the stresses from each loading source. Using the maximum stress failure criterion, stress states of each ply of a laminate will be evaluated to determine whether or not cracking will occur. 

The maximum stress failure criterion is applied to investigate ply level damage related to transverse matrix cracking and longitudinal tensile and compressive failure. 

The Tsai-Hill failure criterion appears to be much more applicable to failure prediction for this E-glass-epoxy composite material than either the maximum stress criterion or the maximum strain failure criterion. Other less obvious advantages of the Tsai-Hill failure criterion are ... 

Tsai—Hill The maximum stress and maximum strain failure criteria consider each stress component individually. This is a simplification. Test results show that if more than one stress is present in a ply, they can combine to give failure earlier (or later) than the maximum stress or maximum strain failure criterion would predict. One example that shows this effect is the case of a unidirectional ply under shear on which a tensile or a compressive stress is applied parallel to the fibers. The situation is shown in Eigure 6.7. 

During the process of stress wave propagation, tensile stresses or shear stresses do occur and cause rock material to fail in tension or in shear. Therefore, a modified principal stress failure criterion is applied to determining material status, which is suitable for describing material tensile failure or shear failure. The modified principal stress failure criterion dictates that when the major principal stress or the maximum shear stress in an element exceeds material tensile or shear strength, the element fails. After an element has failed, it will not be able to sustain any tensile and shear loadings, but it is still able to sustain compressive loading. The normal compressive stresses, and cr, of a failed ele-... 

Accordingly, and based on a maximum shear stress failure criterion (for justification see below), the following equation was appHed [17] ... 

Fig. 3.5. Failure in multiaxial stress, o PMMA tubes (Broutman et al., 1231), a 6 PA tubes, A buckling (Ely, 1241), x PUR tubes (Lim, 1221), SBR membranes (Dickie et al., (251) ------maximum strain failure criterion, - - - octahedral shear stress failure criterion.

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

Subsection A This subsection contains the general requirements applicable to all materials and methods of construction. Design temperature and pressure are defined here, and the loadings to be considered in design are specified. For stress failure and yielding, this section of the code uses the maximum-stress theory of failure as its criterion. 

The decrease in with crack depth for fracture of IG-11 graphite presents an interesting dilemma. The utihty of fracture mechanics is that equivalent values of K should represent an equivalent crack tip mechanical state and a singular critical value of K should define the failure criterion. Recall Eq. 2 where K is defined as the first term of the series solution for the crack tip stress field, Oy, normal to the crack plane. It was noted that this solution must be modified at the crack tip and at the far field. The maximum value of a. should be limited to and that the far... 

If the fibres are aligned at 15° to the jc-direction, calculate what tensile value of Ox will cause failure according to (i) the Maximum Stress Criterion (ii) the Maximum Strain Criterion and (iii) the Tsai-Hill Criterion. The thickness of the composite is 1 mm. 

A single ply glass/epoxy composite has the properties Usted below. If the fibres are aligned at 30° to the x-direction, determine the value of in-plane stresses, a, which would cause failure according to (a) the Maximum Stress criterion (b) the Maximum Strain criterion and (c) the Tsai-Hill criterion. 

For E-glass-epoxy, the Tsai-Hill failure criterion seems the most accurate of the criteria discussed. However, the applicability of a particular failure criterion depends on whether the material being studied is ductile or brittle. Other composite materials might be better treated with the maximum stress or the maximum strain criteria or even some other criterion. 

The results of the experiments were analysed according to continuum mechanics as v/ell as fracture mechanics principles. The evaluation of the stress at failure as well as the energy released are used to evaluate the validity of, respectively, a maximum stress criterion energy approach as a failure criterion. 

The total stress at the occurrence of the first transverse crack is the principal stress, and it is therefore possible to use a Maximum Stress Criterion as a failure criterion. It is simply performed by comparing the stress at the occurrence of the first transverse crack to the strength of the layer. In other words, failure occurs when ... 

The maximum shearing stress criterion for failure simply states that failure (by yielding) would occur when the maximum shearing stress reaches a critical value (i.e., the material s yield strength in shear). Taking the maximum and minimum principal stresses to be and 03, respectively, then the failure criterion is given by Eqn. (2.3), where the yield strength in shear is taken to be one-half that for uniaxial tension. 

This failure criterion is given in terms of the octahedral shearing stress, ft is identical to the maximum distortion energy criterion, except that it is expressed in stress versus energy units. The criterion, expressed in terms of the principal stresses, is given in Eqn. (2.9). 

Several things immediately become obvious from Eqns. (7.10) and (7.11) (i) A specific, independent failure criterion is used, and the crack size for failure, Uf, is a function of the fracture toughness and the maximum applied stress,... 

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