Experimental determination of generalized stress intensity factors from full-field measurements
Introduction
The displacement and stress fields in the vicinity of a singular point can be described by a characteristic exponent and a coefficient representing the intensity of the field: the generalized stress intensity factor (GSIF). The GSIF of a singularity is a relevant parameter for failure prediction. In the well-known particular case of a crack in isotropic homogeneous media, and the GSIF corresponds to the usual stress intensity factor (SIF). The SIF can be computed using J-integral [28]. However, this method is specifically dedicated to cracks and cannot be extended to compute the GSIF of other singularities such as, for instance, a corner, a V-notch in a homogeneous material. Therefore, alternative approaches have to be employed to compute the GSIF. The numerical computation of 2D GSIF can be performed using a least-squares fitting procedure of Finite Element (FE) nodal displacements to the asymptotic fields in the vicinity of the singular point [2], [11], [16]. Lazzarin et al. [17] extracted the 2D opening and shear mode GSIFs of a V-notch based on the strain energy density computed considering the leading order terms of the William’s solution [40]. They highlighted the possibility of using relatively coarse meshes since this approach involves the mean value of the strain energy density that can be obtained from the nodal displacements. The most accurate method for GSIF computation is based on a path independent integral [14], [15], [18]. Although it can be employed to compute the SIF in the case of a crack (even in anisotropic media), this method differs significantly from J-integral which is strictly dedicated to SIF calculation for a crack. The GSIF use is particularly employed in Finite Fracture Mechanics (FFM) framework for crack initiation prediction in the vicinity of a singularity or a stress concentrator based on the matched asymptotic (MA) approach of the coupled criterion (CC) [8], [20], [21], [39]. This method has recently been extended to 3D [9], which was the missing tool for applying the MA approach of the CC in 3D [6], [23].
Although some methods exist to compute from experimental data the stress intensity factors at a crack tip in 2D or in 3D [24], [26], [30], [31], [33], [34], few authors estimated GSIFs from experimental data: Dunn et al. [11] carried out experiments on V-notch PMMA specimens with various opening angles and depths. They computed numerically 2D V-notch GSIFs for the critical loads measured experimentally and showed that the GSIF of the corner singularity was the relevant parameter to predict crack initiation. The GSIF was computed by least square fitting of the nodal displacements along the V-notch flanks from FE calculations. The critical GSIF was determined to be constant for a given V-notch angle whatever the notch depth, except may be for large V-notch angles. Labossiere and Dunn [16] performed four point bending tests on bimaterial specimens in order to study crack initiation occurring at the bimaterial interface corner under an opening mode. They obtained the 3D interface corner GSIF by means of a least-squares fitting procedure of the FE displacements fields computed on the whole structure in the vicinity of the singular point to the asymptotic displacement fields. Vicentini et al. [38] determined the opening and shear mode GSIFs at a bimaterial closed corner in Brazilian disk specimens by means of FE calculations based on the failure load measured experimentally. It can be noted that the previously cited work proposed indirect GSIF estimates based on experimental data, since these approaches rely on a FE model. Their main limitation is the representativeness of the FE model with respect to the experimental test in terms of specimen geometry or boundary conditions for instance. Moreover, these works only concerned pure opening mode configurations but such approaches would not allow identifying the GSIFs corresponding to different modes in a mixed mode loading configuration.
The objective of this work is the direct experimental determination of GSIFs from digital image correlation (DIC) displacement and strain fields. Tensile tests on PMMA square hole specimens are presented in Section 2. Section 3 describes the GSIF calculation method that is employed as a post-processing of either FE calculations or DIC fields. In Section 4, the coupled criterion for crack initiation prediction is recalled. Finally, GSIFs obtained from DIC or FE calculations are compared in Section 5.
Section snippets
Experiments
The material under investigation is a commercial extruded PMMA. Several rectangular specimens were manufactured using laser cutting to obtain square holes which sides are parallel to the specimen sides. The specimens were tested under uniaxial tension on a 20 kN Zwick machine. The choice of such a configuration is motivated by the fact that it leads to a mixed mode crack initiation, the mode mixity depending on the square hole size c. It is assumed here that c is small compared to the specimen
Singularity exponent, primal and dual fields
First, let us recall the generic form of Williams’ expansion in the vicinity of a V-notch [40]where is the solution to an elastic problem, holds for the rigid translation, r and are the polar coordinates emanating from the root of the V-notch. The exponent and its associated mode depend only on the local geometry and possibly local elastic properties while the GSIF k is a function of the global geometry of the specimen and the intensity of the applied load
The criterion
The coupled criterion (CC) aims at filling the gap of crack nucleation prediction in brittle material [20], which is not covered by classical Linear Elastic Fracture Mechanics [12], [13]. This approach, which allows the determination of the initiation loading level, crack length and crack angle, is briefly recalled here. A detailed description of this approach for mixed mode crack initiation can be found in [35], [10]. Basically, it lies on the simultaneous fulfilment of a stress and an energy
Results and discussion
The method presented in Section 3 is now applied to the experimental determination of GSIFs from displacement and strain fields measured by DIC.
Conclusion
Mixed mode failure is observed in PMMA specimens containing square holes under uniaxial tensile loading. The larger the hole size, the smaller the mode mixity defined as a normalized ratio between the GSIFs of the shear and opening modes. By means of DIC, we compute the displacement and strain fields around the square hole corners and determine the corresponding stress fields assuming a linear elastic isotropic material behavior which properties are also determined using DIC. By means of a path
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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