Accurate Closed-form Solution Research Articles

An approximate, analytical approach to the `HRR'-solution for sharp V-notches

The well-known so-called `HRR-solution' (Hutchinson, 1968 and Rice and Rosengren, 1968) considers the elasto-plastic stress field in a power-law strain hardening material near a sharp crack. It provides a closed form explicit expression for the stress singularity as a function of the power-law exponent `n' of the material, but the stress angular variation functions are not found in closed form. More recently, similar formulations have appeared in the literature for sharp V-notches under mode I and II loading conditions. In such cases not only is the angular variation of the stress fields obtained numerically, but so is the singularity exponent of the stress field. In the present paper, approximate but accurate closed form solutions are first reported for sharp V-notches with an included angle greater than π/6 radians. Such solutions, limited here to Mode I loading conditions, allow a very satisfactory estimate of the angular stress components in the neighbourhood of the notch tip, in the entire range of notch angles and for the most significant values of n (i.e. from 1 to 15). When the notch opening angle tends towards zero, and the notch approaches the crack case, the solution becomes much more complex and a precise evaluation of the parameters involved requires a best-fitting procedure which, however, can be carried out in an automatic way. This solution is also reported in the paper and its degree of accuracy is discussed in detail.

Mismatch effect on plastic yield loads in idealised weldments: I. Weld centre cracks

The accuracy of the defect assessment methods for strength mismatched structures is directly related to the accurate estimate of the (mismatch) yield load of the defective component of interest. This paper provides accurate closed form solutions of mismatch yield loads for two representative cracked plates, a middle crack tensile plate and a single edge cracked plate in pure bending. The weld configurations are idealised as a simple strip model, and the crack is assumed to be located in the centre of the weld metal. Solutions are given for two limiting out-of-plane constraint conditions, plane strain and plane stress. Comparisons with the results from detailed finite element limit analyses show excellent agreement, for almost all values of strength mismatch levels, ligament-to-weld width ratios, and crack lengths. Another important outcome is that, based on the proposed solutions, useful guidelines are given for assessing cracked structures with weldments, in the form of windows where the plastic yield load solutions of mismatched components can be obtained assuming the component can be wholly characterised by either base material or weld material properties. Such guidelines could be very useful when complex components with weldments are to be assessed, for which the plastic yield load solutions are not available or difficult to find.

Mismatch effect on plastic yield loads in idealised weldments: II. Heat affected zone cracks

In Part I, accurate closed form solutions of mismatch yield loads were given in terms of strength mismatch levels, ligament-to-weld width ratios, and crack lengths, for a middle crack tension (M(T)) plate and a single edge cracked plate in pure bending (SE(PB)). The weld configurations were idealised as a simple strip model, and the crack was assumed to be located in the centre of the weld metal. Solutions were given for two limiting out-of-plane constraint conditions, plane strain and plane stress. This paper is a continuation of Part I, providing similar solutions of mismatch yield loads when the crack is in the interface between the weld and base plate. As in Part I, useful guidelines are given for assessing welded structures with heat affected zone (HAZ) cracks, in the form of windows where the plastic yield load solutions of mismatched components can be obtained assuming the component is wholly made of either base material or weld material properties. Such guidelines could be very useful when complex components with weldments are to be assessed, for which the plastic yield load solutions are not available or difficult to find. Moreover, the effect of the crack location within the weld metal as well as of the HAZ properties on the mismatch yield load solutions is briefly discussed.

Analysis of a wedge impression test for measuring the interface toughness between films/coatings and ductile substrates

The adhesion between thin elastic films/coatings and ductile substrates can be measured using a wedge test, wherein a long sharp edge is impressed through the film into the substrate. The resulting plastic deformation causes delamination between the film and the substrate. The extent of the delamination can be correlated with the crack driving force to determine the toughness of the interface. Results are presented that relate the energy release rate and mode-mixity to the stresses: both residual and those induced by the impression. Numerical and asymptotic results are presented for the strain transferred to the coating. The numerical results verify that the asymptotics provide accurate closed-form solutions when the delaminations exceed about seven contact widths. An example is provided for a thermal barrier coating system.

Variational approach to the stress-transfer problem through partially debonded interfaces in a three-phase composite

In our previous study (Wu W, Verpoest I, Varna J. Compos Sci Technol 1998;58(12):1863–77) of the stress-transfer problem around a single fibre, we presented a variational approach based on the principle of minimum complementary energy, not only in the perfectly bonded zone but also in the zone with a discontinuous interface of a two-phase composite. This approach is adapted in this paper to derive an accurate axisymmetric analytical model for the description of the stress state around the fibre breaks and partially debonded interfaces of a three-phase composite with a fibre, coating and matrix. The debonded fibre/coating interface is treated as a special external boundary on which a presumed interfacial shear stress with some free parameters is specified. Once the parameters are given, minimisation of complementary energy can be applied for both debonded- and bonded-interface zones together, to extract the most accurate closed-form solution. The shear stress at the debonded interface (the right values of free parameters) is finally found by substituting the calculated radial stresses in the Coulomb friction law and minimising the discrepancy by a simple numerical iteration. As the minimisation procedure is applied for the both debonded and bonded zones simultaneously, the strong interaction of the two zones is correctly described. This model also includes the matrix axial stress non-uniformity in the radial direction. The stress profiles along both axial and radial directions are presented and closely compared with the results from a finite element model and both agree quite well. A number of applications of this model are also discussed.

A novel axisymmetric variational analysis of stress transfer into fibreS through a partially debonded interface

Most existing models for the problem of fibre/matrix stress-transfer through a partially debonded interface roughly solve the stress distribution in the debonded zone, neglecting the presence of the perfectly bonded zone. However the stress interactions between two zones is what makes the problem essentially different from the stress-transfer problem for a perfectly bonded interface. This paper suggests a variational approach based on the principle of minimum complementary energy not only in a perfectly bonded zone but also in a zone with a discontinuous interface. The debonded interface is treated as an external boundary on which a presumed interfacial shear stress is specified. A new analytical model, including stress non-uniformity in the radial direction and crack interaction, is derived to describe the stress state around fibre breaks and debonding tips in a single fibre embedded in an infinite matrix. For the presumed shear stress at the debonded interface the minimisation procedure renders the most accurate closed-form solution (under used assumptions) for both interactive zones. Finally, the ‘best’ shear stress distribution at the debonded interface is found by using Coulomb’s friction law and simple numerical iterations. The stress profiles along both axial and radial directions are presented and compared with results from a numerical model [1] available in the literature and also from finite-element analysis. Good agreements are achieved. Extensive applications of this approach and the derived model are also discussed.

ANALYTICAL MODELLING OF GROUND MOVEMENTS ASSOCIATED WITH TRENCHLESS PIPELAYING OPERATIONS.

The need for relatively simple, accurate closed form solutions to predict ground displacements around trenchless pipelaying operations is established. Three alternative analyses are described, all of which have been used, or suggested, for application to ground displacement around tunnels, The results of the theoretical analysis techniques are then compared with experimental data from a series of laboratory scale model tests of pipejacking and pipebursting operations. Finally, the applications and accuracies of the various methods are discussed in the conclusions. (A)

A close-form solution for the maximum-power operating point of a solar cell array

A solar cell array is inherently a nonlinear device consisting of several solar cell modules connected in series-parallel combinations to provide the desired DC voltage and current. At a fixed insolation level, the terminal voltage decreases nonlinearly as the load current increases. Due to this nonlinearity, it is difficult to determine analytically the operating point at which the output power is maximum; a condition required for maximum utilization efficiency of the array. Iterative techniques are normally used to determine this operating condition. These techniques are lengthy, time-consuming, and have to be repeated for any change in the array's parameters, temperature, and insolation level. In this paper, an accurate closed-form solution is derived in terms of the array's parameters and solar insolation level. This solution is useful to the system designer or researcher as a fast and accurate tool for system design and performance analysis.

Lateral-Distortional Buckling of Hollow Flange Beams

While the flanges of cold-formed hollow flange beams (HFBs) are very stiff torsionally, their webs are comparatively flexible, and may allow web distortion effects to reduce their resistances to lateral buckling. There is no simple formulation for predicting the effects of web distortion on the lateral buckling of HFBs. As a result, structural designers are unable either to check or to extend the available elastic buckling predictions; thus code writers are not able to provide explicit formulations for the effects of distortion. The conversion from elastic buckling to strength for HFBs is also questionable, since cold-formed beams have different stress-strain curves, residual stresses, and geometrical imperfections from those of hot-formed beams. This paper deals with these problems, first by finding a simple but sufficiently accurate closed-form solution for the effects of distortion on the elastic lateral buckling of simply supported HFBs in uniform bending, and then by developing an advanced theoretical method of predicting the effects of the stress-strain curve, residual stresses, and geometrical imperfections on the strengths of HFBs that fail by lateral-distortional buckling. Such a method allows the development of a conversion from elastic buckling to strength, which is appropriate for HFBs. The use of this conversion together with the simple approximation developed here for elastic lateral-distortional buckling provides a rational basis for the lateral buckling design of unbraced HFBs.

Probe spectroscopy in an inhomogeneously broadened three-level system saturated by an intense standing wave

We study inhomogeneously broadened three-level atoms ($\ensuremath{\Lambda}$ configuration). Two energy levels are strongly coupled by an intense standing wave. A weak laser field at an adjacent transition is used as a probe. The probe response is calculated by Green's functions involving continued fractions. The general solution is extendable to other three-level configurations as well to other types of multimode saturators. The connection of various approximate solutions to the exact one is discussed. We concentrate on the Doppler-free case where both transitions have nearly equal frequencies. For this we derive a reasonably accurate closed-form solution, the parameter dependencies of which are investigated in detail. In addition to its average changes the dynamics of the saturated transition plays an important role in the steady-state probe response. The spectra show sharp resonances which cannot be explained in terms of traveling wave superposition models. The width of the velocity distribution has a strong effect on the shape of the spectrum---some resonances are smeared away because of incomplete Doppler dephasing. The relaxation processes influence differently the resonances, which facilitates their interpretation.