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More accurate determination of residual strength for components with surface cracks

In 1976, based on the so-called strip yield model, which is developed starting from the endless plate with the through-crack, the R6 method has been introduced (also known as FAD = Failure Assessment Diagram). It was assigned by SINTAP (Structural Integrity Assessment Procedures for European industry) also to the case of structures with the surface cracks without relevant proof in this regard. Unfortunately, there is no theoretical or empirical evidence for the validity of this solution for surface cracks. Numerous experimental results show the opposite which is why even a separate standard was developed in the USA for the determination of K_Ie (ASTM-E-740).
Also our experimental results within of the from ESA (European Space Agency) financed project have shown a decisive difference in the behaviour of specimens with through and surface cracks. Therefore, for the treatment of surface cracks in elastic-plastic region new methodology was developed and successfully used for the first time for the Ariane 5 structure design.
The stress intensity along the contour of the surface cracks is uneven, where the unevenness even further increases under elasto-plastic loading conditions. Since the crack opening is handicapped locally, the failure occurs only after the stress intensity redistribution. In fact, the failure is dependent on the total energy, i.e. the area of the crack and not on local values. By contrast, in the case of a through crack, two crack tips exist (even with the same values for K, or J), which in case of failure behave more or less in the same way.
Because the deformation and failure are controlled by the energy in cross section, for a suitable component evaluation, the consideration of two parameters that are under general conditions representative for the stored energy - stress and strain in the critical net section with the crack - is necessary.
The adequate FAD should fulfil all these requirements. FAD is a plot of the failure envelope of the cracked structure, defined in terms of two parameters:

K_r.....The ratio of the applied linear elastic stress intensity factor, K_I, to the materials fracture toughness,, K_mat

L_r.....The ratio of the applied stress to the stress at which the plastic yielding of the cracked structure begins.

More general, the failure envelope for the typical procedure is dependent only on the material's tensile properties and based on J-integral solution is defined by

For the evaluation we rearrange the well-known LEFM solution for J-Integral

as the product between two K-solutions: one for stress and another for strain:

By the extension the same solution can be applied in all area between "elastic" and "plastic".

Constitutive laws based on classical plasticity generally define total strain as the sum of its elastic and inelastic components, with independent constitutive relationships describing each.

Therefore

Here the elastic part of J-integral does not consider the Irwin's plastic correction. In fact this correction adulterate the real relationship and cannot be recommended. If we consider that the PZ represents, under the given conditions, the ability of the material to reduce the stress peaks and thus to weaken the crack effect this correction is not adequate. As well known, the ductile materials are able (due to plasticity) to reduce the crack (or notch) effect (stress redistribution, blunting), where the Irwin's plasticity correction goes exactly in another direction (increase of crack size and more severe stress intensity values). On top of it, under the plane strain conditions, the correction is significantly smaller than under plane stress conditions, even though the corresponding situation becomes more severe. Clear, approaching the yield strength the inelastic part of strain becomes significant compared to the elastic one, and this is considered by the second term in the above formula.

Typical solution for surface crack has the form

or for strain

After substitution in (2)

Here is (7)

It follows based on (6) and (9):

As usual, the actual strain is evaluated according to the stress-strain curve of material. In our case, based on engineering curve. For this purpose use of well known Ramberg-Osgood approximation is typical

Introduction of

leads to the solution for FAD based on Stress-Strain curve:

For the comparison the R6 solution:

can also be written as

and wail (4)

Compared to the solution (10), considering (12), additional reduction factor in denominator appears

According to the (1), this factor means an additional increase of plastic portion of the J-Integral. It is based on relationship

Where the value of total J-integral includes Plastic Zone Correction (see above discussion).

A comparison with the R6 method is shown in the following illustration. By our formel

-Method no continual residual strength reduction (based on K_IC, J_IC, etc.) originates in the linear-elastic range (up to 60 -70 % of yield strength). The reduction in the plastic region below and above yield strength is determined by the intensity of the strains in the cross section weakened by crack. This corresponds to the observed behaviour. Based on the R6 curve, the measures of the material resistance K_Ic, J_Ic, etc. are only unchanged for = O in the cross-section which is hardly the case. Furthermore, the additional margin of safety is only effective below yield strength, where above yiled strength the R6 curve produce even the non-conservative results.

In addition to the stress-strain approximation other important conditions of the new method are:

The surface crack is represented by its area.

All relevant variables are calculated for the cross section weakened by the crack.

Stress-strain behaviour of the material is represented by the measured engineering curve.

Proof of the new Method

The proposed stress-strain method is proved for the application at the most critical thin-walled components with surface defects through numerous tests (carried out also on component-like specimens).

The following diagram shows the same trend as on the FAD above. Obviously, this indicates a systematic deviation from the experimental results, which lies in the structure of R-6/Option 2 formula of FADs and this make the need for the above described addition in (16) questionable.

Increased accuracy of the method both below and above the yield strength in comparison to existing methods (R-6, FAD, etc.) avoid the unnecessary conservatism by the design in this most important area for the applications.

In general, the failure of components in engineering structures and the structures themselves can usually be traced to surface cracks. Therefore, surface cracks dominate the practical application. Especially in thin-walled structures is the failure of ligament (residual cross-section at the place of surface crack) usually the main cause for the failure of the structure. This means that despite the small dimensions and the approaching to collapse conditions, the critical cross-sectional is effectively weakened by the presence of the crack and therefore decisive for the failure. The investigation of the relationships for the evaluation of the limit load of the ligament of surface crack is therefore an important project for the conventional failure prediction, especially for the design of pressurized components.

Finally, one more note !

Unfortunately, in SINTAP also other weaknesses of the existing standards were taken carelessly, as, for example, the treatment of crack interaction. In none of the cases the crack closure effects of the material between the two basic cracks has been considered. The result is, as selected example shows, whenever s is smaller (less closure effects as well) the resulting crack will be evaluated as less dangerous. On the other hand, for the case s = a₁ + a₂ the resulting crack reaches its maximum size even though right after that (s > a₁ + a₂) the interaction of the two cracks will be ignored.