On the influence of T-Stress on photoelastic analysis under pure mode II loading

. According to the classical definition for in-plane modes of crack deformation, the constant stress term T exists only in the presence of mode I. However, recent studies show that this term can exist in mode II conditions as well, and significantly affect the elastic stress field around the crack tip. These effects can be visualized using the experimental method of photoelasticity. Based on the analytical studies, presence of the T-stress in mode II cracks transforms the isochromatic fringe patterns from symmetric closed loops to asymmetric and discontinuous shapes. In this paper, presence of the T-stress in mode II cracks and its effects on the fringe patterns is experimentally investigated. The test specimens are Brazilian disks containing very sharp central cracks: experimental results indicate that these specimens contain negative values of T-stress. Experimental values are then compared to numerical results. To better understand the differences between experimental and numerical values, a thee dimensional analysis is performed with the finite element method: results show the influence of the real geometry of the crack front on the stress intensity factors. ABSTRACT . A recent criterion based on the local strain energy density (SED) averaged over a given con-trol volume is applied to well-documented experimental data taken from the literature, all related to steel welded joints of complex geometry. This small size volume embraces the weld root or the weld toe, both re-gions being modelled as sharp (zero notch radius) V-notches with different opening angles. The SED is evaluated from three-dimensional finite element models ABSTRACT . Synchrotron light microtomography has proved to be particularly efficient in order to analyze the microstructural characteristics in terms of reinforce fibre distribution and orientation in glass fibre reinforced composites. The spatial distribution of fibre within the polymeric matrix could be detected even in case of fibre characterized by a


INTRODUCTION
Many structural materials are subjected to crack forming and propagation during their service life.These cracks influence the stress distribution in the component and can result in significant decrease of its strength.Because of the importance of safety and reliability, the crack problem has been of interest to a large number of researchers.Elastic stress field around a crack tip is usually written as a set of infinite series expansions as [1]: ( ) RIASSUNTO: Dalla definizione classica dello stato di sollecitazione elastica in prossimità dell'apice di una cricca, il termine T costante nello sviluppo in serie del fattore di intensificazione degli sforzi esiste solo in presenza del modo I di carico.Tuttavia, recenti studi mostrano che il T-stress può esistere anche in condizione di modo II, e modificare significativamente il campo di sforzi elastici presenti nell'intorno dell'apice della cricca.Questi effetti possono essere visualizzati e testati sperimentalmente col metodo della fotoelasticità.In questo lavoro è proposto uno studio sull'influenza del T-stress in cricche sollecitate secondo il modo II e i suoi effetti sul campo di frange visibili sperimentalmente.I provini utilizzati sono dischi, chiamati Brazilian disks, al cui interno sono contenute cricche centrali da analizzare: i risultati sperimentali indicano che questi tipi di provini contengono valori negativi di T-stress.I valori ottenuti sperimentalmente sono poi confrontati con i risultati di simulazioni numeriche.Per meglio interpretare le differenze tra valori sperimentali e numerici, sono inoltre state eseguite analisi FEM 3D: i risultati mostrano l'influenza della reale geometria del fronte sui valori dei fattori di intensificazione degli sforzi.
ABSTRACT.According to the classical definition for in-plane modes of crack deformation, the constant stress term T exists only in the presence of mode I.However, recent studies show that this term can exist in mode II conditions as well, and significantly affect the elastic stress field around the crack tip.These effects can be visualized using the experimental method of photoelasticity.Based on the analytical studies, presence of the T-stress in mode II cracks transforms the isochromatic fringe patterns from symmetric closed loops to asymmetric and discontinuous shapes.In this paper, presence of the T-stress in mode II cracks and its effects on the fringe patterns is experimentally investigated.fter thermal er 50mm from ong with the d for a calibr according to uding loading , the fringe .Fig. 5     Crack curvature radius through the thickness is 10 mm, and the maximum extension of the crack is 2a=96 mm, indicated with a black thicker line in Fig. 9.The angle α between the direction of application of the compressive force (F=375N) and the crack line is 25.4°.This angle is chosen according to [4] in order to obtain pure mode II on the crack, considering the problem in 2D plane stress state.Displacements of the nodes in which the force is applied, are forced to be in line with the loading direction.
Since the results in terms of K II are equivalent considering both the crack tips, only for one of them the mesh has been refined in the circumferential direction.In this way, it is possible to reduce the analysis run time, without loosing accuracy in the final result.
The material of the disk in numerical model is the polycarbonate, with elastic modulus of E=2480MPa and Poisson's ratio ν=0.38, according to [11].Solid elements used for the modeling have a shape function of the second order, with a midside node in each edge.This choice allows having more nodes despite a not excessively refined mesh.Moreover, the use of quadratic element is necessary to use the quarter point technique [17,18], that is to move the midside nodes next to the tip to ¼ of the edge length, which results in a better stress gradient in this area with singularity in the crack tip.Since good results are achievable with these elements even if the singularity is not well modeled on lines other than elements edges [19,20], no collapsed element is used.
It should be mentioned that to get better results in Jintegral evaluation and consequently on stress intensity factors assessment, mesh directions should always be perpendicular to the crack front [21], avoiding distorted elements.However, the circular shape of the crack front causes a particular pattern for the mesh through the specimen thickness.As shown in Fig. 8, in the upper part form point A to B, the mesh is more regular and the elements of this region describe the radial directions per-pendicular to the crack front.In the lower part, the arc geometry makes it impossible to draw a regular mesh, and the normal to the crack front is not coincident with the mesh direction.Numerical results are obtained starting from node 1 corresponding to point A to node 33 that is point C in Fig. 8. Convergence of J-integral and stress intensity results is obtained at the third contour.The trend of stress intensity factors can be graphically observed in Fig. 10 in function of the node distance from the surface.However, the stress intensity factor values obtained near to point C should not be taken into consideration, since elements present a high level of distortion producing low accuracy in the results.Values of the first three nodes are moreover invalid in the discussion, since the third contour integral cannot be calculated and results are infected by the presence of the surface border.

DISCUSSION
The semi-natural cracks created with a mechanical shock after making brittle the polycarbonate in the liquid nitrogen, have a nonlinear curved tip through the thickness.
When the specimen containing such a crack is subjected to mode II loading condition, the global deformation of the crack front is in-plane sliding in X-direction.However, considering local coordinate systems n-t moving along the crack tip curve (see Fig. 11), the global displacement of the crack tip points will have two components.The normal component in n-direction leads to mode II; and the tangential component in t-direction implies that there is also mode III deformations in local view.In order to find the effect of specimen thickness on the numerical results, they can be compared with the previous results [4] obtained from 2D finite element modeling.For this aim, a new parameter K IIeq is defined as: Figure 10.Stress intensity factors along the crack front in function of the depth.(-•-K I , -♦-K II , -▲-K III , -x-K IIeq , __ K II-2DFEM [10]).which presents the equivalent mode II stress intensity factor in X-direction of the global coordinate system.It can be noticed from Fig. 10 that K I is negligible with respect to K II and K III for all the considered nodes.Also, K III is initially less than K II .Increasing the curvature that is going toward points B and C, K III values are increasing and finally becoming more than K II values.However, values of K IIeq remain about constant, except from surface nodes which are not valid as described before.The observed difference between K IIeq and the result of 2D model [4] shows that the thickness of specimen affects the ideal plane stress conditions and leads to some errors in the photoelastic experiment results.

CONCLUSION
In this research, presence of the T-stress and its effects on the elastic stress field around a mode II crack tip were experimentally studied.Very sharp cracks were created in polycarbonate sheets by using a new method with different steps.The cracks obtained in this way are completely sharp, but the crack tip has a curved shape through the thickness of the specimen.Specimens were cut in the form of centrally cracked Brazilian disk specimens.Photoelastic experiments were conducted on these specimens subjected to mode II loading conditions, to determine from the isochromatic fringe patterns the crack parameters K I , K II , and T by using computer codes developed with the MATLAB software.Experimental results revealed that the specimens had negative T-stresses in mode II condition.The experimental results were consistent very well with numerical bidimensional predictions in that the T-stress significantly affects the symmetric shape of the fringe loops, and causes the loops to become asymmetric and discontinuous along the crack edges.However, there were some minor errors which could be related to the curved shape of the crack front through the specimen thickness.The effect of crack tip curvature on the crack parameters was also investigated by developing a 3D finite element model.The crack front was assumed to be in a circular arc form and, even if it is not com-pletely correspondent to the real situations, aim of this model is to get an indication about the stress intensity factors trend along a non-straight crack front.
The numerical results show that though the global deformation of the crack is in-plane sliding (mode II), in local coordinates there are two shear components which are parallel and perpendicular to the crack front.That is, the crack tip points are subjected to a combination of mode II and mode III deformations.This local mixed mode condition can lead to some errors in the experimental results, which can be a source of difference of experimental results compared to the values of the finite element model.

Figure 8
Figure 8. M hicker front and enable to get an indication about the stress intensity factors trend along a non-straight crack front.