Failure criteria maximum strain criterion

As the second term is less than 1, a tensile failure in the 2-direction would be expected, (b) Maximum Strain Criterion From the data given... 

The denominators in the first two of Eqn (6.34) are the tensile or compressive ultimate strains in the corresponding direction depending on whethCT or ey are tensile or compressive, respectively. It should be noted that for a material that is linear to failure, the predictions from the maximum strain criterion would differ from those from the maximum stress criterion only by a factor depending on the Poisson s ratios and Vyx- As with the maximum stress criterion, the maximum strain criterion provides some feedback on the type of failure mode occurring. 

If we now allow for non-linear adhesive behaviour, the high adhesive stress concentrations predicted by the linear elastic analysis will be relieved to some extent. Figure 54 shows the predicted spread of the yield zone of adhesive at the tension end of a double-lap joint as the load is increased. As would be expected, plastic flow begins near the adherend corner and the load corresponds to a joint efficiency of 21%. Each subsequent load increment represents an increase in joint efficiency of 4 4%. When elastic perfectly-plastic behaviour is assumed for the adhesive, a maximum strain criterion for failure seems appropriate. In Fig. 55 the joint efficiency is plotted against the maximum principal strain in the adhesive at each end of a double-lap joint. Assuming a failure strain for the adhesive of 5%, the analysis predicts a joint efficiency of 31% for a double-lap joint compared with 16% predicted by the linear elastic analysis. Similarly, the non-linear analysis predicts an efficiency of 39% for the double-scarf joint compared with 20% predicted by the linear elastic analysis. Although the predicted efficiencies are almost doubled by allowing for non-linear behaviour in the adhesive, failure in the adhesive is still predicted to be more probable than failure in the adherends (Table 5). 

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 ... 

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. 

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. 

Figure 2-38 Maximum Strain Failure Criterion (After Tsai [2-21])...

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 ... 

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. 

A possible adjunct to the laminate design procedure is a specific laminate failure criterion that is based on the maximum strain criterion. In such a criterion, all lamina failure modes are ignored except for fiber failure. That is, matrix cracking is regarded as unimportant. The criterion is exercised by finding the strains in the fiber directions of each layer. When these strains exceed the fiber failure strain in a particular type of layer, then that layer is deemed to have failed. Obviously, more laminae of that fiber orientation are needed to successfully resist the applied load. That is, this criterion allows us to preserve the identity of the failing lamina or laminae so that more laminae of that type (fiber orientation) can be added to the laminate to achieve a positive margin of safety. 

The maximum principal strain criterion for failure simply states that failure (by yielding or by fracture) would occur when the maximum principal strain reaches a critical value (ie., the material s yield strain or fracture strain, e/). Again taking the maximum principal strain (corresponding to the maximum principal stress) to be 1, the failure criterion is then given by Eqn. (2.4). 

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. 

A macroscopic theory of strength is based on a phenomenological approach. No direct reference to the mode of deformation and fi acture is made. Essentially, this approach employs the mathematical theories of elasticity and tries to establish a yield or failure criterion. Among the most popular strength theories are those based on maximum stress, maximum strain, and maximum work. 

Kinloch(4) observed that the selection of appropriate failure criteria for the prediction of joint strength by conventional analysis is fraught with difficulty. The problem is in understanding the mechanisms of failure of bonded joints, and in assigning the relevant adhesive mechanical properties. Current practice is to use the maximum shear-strain or maximum shear-strain energy as the appropriate failure criterion. However, the failure of practical joints occurs by modes including, or other than, shear failure of the adhesive. This difficulty has led to the application of fracture mechanics to joint failure. 

P(l) When lap and strap joints are loaded simultaneously by more than one of the above listed loadings, the joint resistance shall be determined using the maximum strain failure criterion applied to the resultant shear strain vector in the adhesive. 

Generally, as the adhesive seems to be mainly a strain limited material the best failure criterion for a combined loading would be some strain based failure criterion such as the maximum strain criterion. 

The material will have a given strength expressed as stress or strain, beyond which it fails. In order to postulate the failure, it is necessary to have a failure criterion with an associate theory to be able to effect a satisfactory design. Such theories include maximum stress, maximum strain, Tsai-Hill (based on deviatoric strain energy theory) and Tsai-Wu (based on interactive polynomial theory). The Tsai-Wu theory is the most commonly used. 

Other advantages of the Sc test are (1) Sc values for different materials could include values on both sides of the acceptability criterion, thus providing a tool for research on improved steels since one would have a measure of the nearness to the desired goal (2) the test evaluates the effect of plastic strain, a critical factor in SSC failures and (3) the small area of maximum strain allows evaluating SSC susceptibility of localized areas and minimizes inclusion-related hydrogen effects. A major limitation of the Sc test is... 

The large strain response in the glassy or semicrystalline state is that of a nonlinear viscoelastic solid. However, both engineering and theoretical approaches to plasticity in polymers have largely developed as an independent discipline, in which (Ty plays a central role, in spite of its somewhat arbitrary definition (indeed it is not always possible to associate cty with a maximum in the force-deformation curve [5]). This is because in practice the yield point, rather than the ultimate strength, is usually considered to be the failure criterion for ductile materials. 

For over three decades, there has been a continuous effort to develop a more universal failure criterion for unidirectional fiber composites and their laminates. A recent FAA publication lists 21 of these theories. The simplest choices for failure criteria are maximum stress or maximum strain. With the maximum stress theory, the ply stresses, in-plane tensile, out-of-plane tensile, and shear are calculated for each individual ply using lamination theory and compared with the allowables. When one of these stresses equals the allowable stress, the ply is considered to have failed. Other theories use more complicated (e.g., quadratic) parameters, which allow for interaction of these stresses in the failure process. 

Of the many theories developed to predict elastic failure, the three most commonly used are the maximum principal stress theory, the maximum shear stress theory, and the distortion energy theory. The maximum (principal) stress theory considers failure to occur when any one of the three principal stresses has reached a stress equal to the elastic limit as determined from a uniaxial tension or compression test. The maximum shear stress theory (also called the Tresca criterion) considers failure to occur when the maximum shear stress equals the shear stress at the elastic limit as determined from a pure shear test. The maximum shear stress is defined as one-half the algebraic difference between the largest and smallest of the three principal stresses. The distortion energy theory (also called the maximum strain energy theory, the octahedral shear theory, and the von Mises criterion) considers failure to have occurred when the distortion energy accumulated in the part under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension or compression test. 

The stress distribution given by Eq. 15.1 is shown in Fig. 15.1 for a vessel with r /fj = 2.2, The maximum stress is in the hoop direction and is at the inner surface where r = r. As the pressure is increased, the stresses increase until they reach a maximum limiting stress where failure is assumed to occur. For thin vessels the ASME Code assumes that failure occurs when the yield point is reached. This failure criterion is convenient and is called the maximum principal stress theory. In thick vessels the criterion usually applied for ductile materials is the energy of distention theory. This theory states that the inelastic action at any point in a body under any combination of stresses begins only when the strain energy of distortion per unit volume absorbed at the point is equal to die strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under a state of uniaxial stress as occurs in a simple tension test. The equation that expresses this theory is given by... 

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