Residual stress as a fracture toughening mechanism: A Phase-Field study on a brittle material

Highlights

• A Phase-Field model simulated crack propagation in a residually stressed material.

• Residual stress can be thought of as a toughening mechanism.

• Cracks can be arrested thanks to residual stress, even under force control loading.

• Some light on the residual stress role on crack propagation path was shed.

Abstract

Recent engineering design practice for materials and structures relies more and more on damage-tolerant criteria. Such a design approach is attained mainly by employing materials showing a certain level of fracture toughness.

This work aims to explore a way to generate fracture toughness in materials that intrinsically shows no toughness at all, i.e. brittle materials. The key idea lies in the introduction of inelastically deformed sub-regions (e.g. circular inclusions) in the base material, which inevitably generate a residual stress field.

To accomplish this purpose, the advanced Phase-Field method coupled with the eigenstrain theory is employed, respectively to simulate the crack propagation behavior and to introduce a residual stress field in a pre-notched sample. Information about crack propagation and displacement externally imposed is used to obtain the resistance curve (R-curve) for several configurations.

One of the main findings of this research regards the possibility of originating fracture toughness in intrinsically brittle materials upon appropriate positioning of one inclusion - containing a certain amount of inelastic deformation – with respect to a notch tip. This result demonstrates that accurate design of residual stress is crucial to attaining unprecedented material or structure performance, and the method shown here represents a valid tool to exploit this advanced design capability.

Introduction

Failure of engineering materials is frequently caused by nucleation and subsequent propagation of one or more cracks. In most cases, failures of materials are triggered by the presence of stresses exceeding a certain threshold evaluated through an appropriate failure criterion, for the specific material. The recent practice in engineering design is diverting more and more towards the use of damage-tolerant materials (or components), which facilitates structural integrity inspection processes and therefore it prevents catastrophic failures. In other words, materials showing a high level of fracture toughness are sought, although searching for a good compromise with the strength of a material is not a trivial task [1]. Ideally, under quasi-static or fatigue loading modes, cracks should propagate stably so that their detection can occur before complete failure is reached. After detection, there exist a number of ways to take actions and mitigate the issue, for instance by replacing the part or by repairing [2], [3] using the stop-hole technique [4], employing interference fit [5], introducing patches [6], via Laser Metal Deposition [7], cold expansion [8], infiltration plating [9] and other methods. Some of these techniques - particularly the latter two outlined methods - rely on the modification of the mean stress during cyclic loading due to the combined effect of crack closure and residual stress (RS), similarly with what has been extensively seen in ductile materials when an overload is applied during cyclic loading [10], [11], [12].

Unfortunately, not all the engineering materials are intrinsically tough or show a certain degree of ductility, hence, ensuring continuous monitoring of material damage becomes a rather challenging task. For example, brittle or semi-brittle ceramics are often used in high-temperature applications due to the limited number of alternative candidate materials which can perform well under this loading condition. More examples can be found in some scenarios where material embrittlement is present, e.g. low-temperature conditions, hydrogen diffusion, etc.

An accepted theory that establishes the conditions to meet in order to achieve crack stop in brittle propagation is called crack arrest [13], which is valid for a monotonic quasi-static loading test but it is found very useful also for more complex loading manners. This theory simply relies on the assumptions that a cleaving crack arrests when the crack driving force of the growing crack front falls below the crack arrest toughness value of the material. The great advantage of such a theory is that, although a crack propagates in an unstable manner, a dynamic analysis of the propagating crack can be replaced by a static simulation without the introduction of relevant errors [14]. Therefore, in the context of brittle materials, a propagating crack can be arrested provided that either the fracture toughness of the material increases locally or the crack driving force is reduced, for instance, by lowering the stress baseline through the presence of RS. The same concept, but using a probabilistic approach, has also been proposed [15], [16].

At the microscopic length scale it is well known how the presence of inclusions (or sub-domains) - presenting different properties with respect to the matrix - within the microstructure can modify materials strength and ductility, and in turns toughness [17], [18], [19], [20]. On the other hand, macroscale inclusions can be thought as of a well-defined sub-domains within a host matrix which have undergone initial inelastic deformation or, more in general, even having different material properties. This inelastic deformation may correspond to plastic deformations, microstructure phase-change, etc.

First studies on the effect of the presence of inclusions within a homogeneous material can be undoubtedly attributed to Eshelby [21]. In his works, Eshelby considered a sub-volume undergoing a uniform permanent deformation also known as eigenstrain [22], [23], [24], [25], [26], [27], within a homogeneous linear elastic solid. As a consequence of this uniform expansion or shrinkage, the material surrounding the inclusion experiences deformation of elastic nature and therefore origination of stress. The problem solved by Eshelby was actually the solution for the stress, strain and displacement field in both the inclusion and matrix. When dealing with failure problems, the presence of inclusions and thus RS plays an important role in affecting the crack propagation rate and stability. Therefore, the mutual interplay between inclusion and propagating cracks is extremely important [28], particularly when damage-tolerant materials or structures are sought. More recent studies have focussed their attention on the capability of RS in hampering crack nucleation and propagation in brittle materials, for instance by designing RS profiles to arrest crack propagation in glass [29] and a combination of RS and interfaces [30].

In the last decades, a lot of effort has been put by the fracture mechanics scientific community on the development of methods able to model and predict crack propagation through materials. The main challenge lays in the analysis of materials presenting heterogeneities, for instance: interfaces, inclusions, residual stress, graded materials. Relying on analytical solutions for this class of problems is very infrequent [31], [32], while generic numerical approaches seem to be more versatile in some cases. Indeed, numerical algorithms have been developed for the Stress Intensity Factor (SIF) of a crack in proximity or in front of an inclusion [33]. Nevertheless, only the use of Finite Element Methods (FEM)[34], which is itself a numerical approach, can cope with the myriad of complex problems seen in practical engineering and scientific applications, particularly when coupled with experimental observations [35]. In particular, to deal with the problem of propagating cracks, several methods have been developed, for example, node release techniques [36], domain remeshing [37], discrete eXtended FEM (XFEM) [38], [39], [40] & regularised XFEM [41], Cohesive Zone Model (CZM) [42], [43], [44], Peridynamics [45] and Phase-Field [46], [47]. Some of these techniques present some difficulties associated with the discrete modelling nature of the crack and the problems related to the intersection of cracks or crack branching [48], and for these reasons, distinctive attention is lately being put on the development of a variational approach based on the energetic Griffith theory [49], the Phase-Field Model (PFM). PFM is characterised by its diffusive description of the crack by a scalar phase-field that discriminates the damaged or broken material from the undamaged material. Such a diffusive description of the crack is essentially defined by a length scale $l_0$. In this way, there are no particular computational difficulties in dealing with spatial derivatives. Over the last 20 years, the PFM has been developed for several material behaviours and is often coupled even with different physics. Starting from the brittle formulation [50], [51], [52], PFM has been implemented for ductile materials [53], [54], functionally graded materials [55], cohesive fracture [56] dynamics brittle fracture [57], composites [58], fatigue crack growth [59], hydrogen assisted cracking [60], heterogeneous materials [61], hydraulic fracture [62], phase transformation and crack interaction [63], three-dimensional problems [64] and others.

In the present paper, the PFM is employed for the first time to account for the presence of RS in a brittle material. This analysis was made possible thanks to the incorporation of inelastic deformation (eigenstrain) within a well-defined material sub-domain (inclusion). By using this novel approach, a calculation framework of the crack growth resistance curve (R-curve) was developed and the influence of RS on the overall fracture toughness was assessed. Several eigenstrain magnitudes were prescribed within inclusions placed ahead of a pre-notched sample at three relevant different positions. The simulation of crack propagation under quasi-static displacement-controlled loading was performed by assuming identical material properties for both the matrix and the inclusion regions. The R-curves were obtained through the analysis of the SIF for a specified 2D plane-stress sample provided by the literature, from the displacement vs. crack length information provided by the PFM simulation results. The outcome of the numerical investigation is thoroughly discussed, and conclusions are drawn with special attention to those scenarios that revealed beneficial effects on the fracture toughness of the sample.