Assessment of computational fracture models using Bayesian method
Introduction
In the past decades, numerous approaches to model material failure have been developed and successfully applied to study fracture in various materials for a variety of applications. They can be classified into discrete and continuous approaches to fracture [1]. The discrete approach to fracture requires two key ingredients, i.e. a computational method to capture the discontinuity in the displacement field and a fracture criterion which in turn is closely related to the underlying constitutive model. The most popular methods for discrete cracks are based on partition-of-unity (PU) enrichment. In PU enrichment, the original approximation is modified through enrichment functions. Furthermore, the jump in the displacement field is captured by introducing additional degrees of freedom into the variational formulation. Some of the popular methods to model cracks using the PU enrichment scheme include the extended finite element method (XFEM) [2], [3], [4], the generalised finite element method (GFEM) [5], [6], [7], meshfree/meshless methods (MM) [8], [9], [10], [11], [12], [13] and extended meshfree methods [14], [15], [16], the cracking particle method [17], the phantom node method (PNM) [18], [19], [20], and the extended isogeometric analysis (XIGA) [21], [22], [23], to name a few. Other popular computational methods for discrete fracture are cohesive elements [24], peridynamics [25], and dual-horizon peridynamics [26]. For quasi-brittle fracture, those approaches are commonly used in combination with so-called cohesive zone models, which regularize the underlying boundary value problem and ensure the correct energy dissipation at post-localization.
Classical representatives of continuous approaches to fracture are viscous, gradient based and nonlocal models, where the failure is described by a continuous damage variable in conjunction with a loading function and a damage law [27], [28], [29]. The phase field method [30], [31] has become another very popular continuous approach to model material failure and bears some similarities to gradient-based models, see for instance [32]. However, the thermodynamically consistent framework allows this approach to be extended to coupled problems with ease. In the phase field model, the evolution of the damage, i.e. the phase field, is obtained through the solution of an additional differential equation. The phase-field method has been implemented in the frameworks of finite element method (FEM) [33], [34], local maximum entropy (LME) [35] and Isogeometric analysis (IGA) [36]. Moreover, phase field models have been extended to study cohesive fracture [37], fracture in thin shells [35], ductile fracture [38], [39] and crack growth in multi-physics problems [40], [41], [42]. A similar approach to the phase-field model is the screened-poisson equation [43]. Further, continuum damage mechanics has been effectively applied to model fatigue damage nucleation under out of phase loading [44], [45], [46].
Computational modelling of fracture is a challenging problem due to the involved uncertainties in the model parameters. This type of uncertainty is an inherent feature associated with the model itself and/or its input parameters. The model uncertainty, also known as the epistemic uncertainty, is developed because of the associated simplifications of the model and the involved assumptions. On the other hand, the uncertainty in parameters, also known as aleatoric uncertainty, is related to the stochastic variance of the input parameters. In this context, introducing more factors to the model will increase the model complexity. This is expected to produce best fit of the model to the related physical behaviour and hence reduce the model uncertainty. However, in this case the uncertainties in model parameters will become more dominant. In light of this, the model with minimum total uncertainty (epistemic plus aleatoric uncertainties) is the most preferred. To the best knowledge of the authors, a general approach to evaluate the quality of different models for material failure accounting the model and parameter uncertainties is rare.
Therefore, in this study we propose a methodology to estimate the uncertainty in those models. The developed methodology has been tested on four different classes of computational approaches for fracture: (i) cohesive zone models in the context of the phantom node method, (ii) phase field method, (iii) non-local damage model and (iv) gradient-enhanced damage technique. These models have been selected due to their popularity and their easiness of applicability to different examples. Each approach are derived from different theory and assumptions as they well described and documented in the literature with different regularization type, different inputs, and consequently they have different uncertainties. The objective of the present work is to trace the current computational techniques in terms of uncertainties, which can serve as a platform for future investigations. Indeed, the evaluation and comparison of the performance of the selected numerical methods with respect to the experimental observations is definitely possible and the novelty of the present work is to confront two major challenges: (i) model uncertainties and (ii) model evaluation. To achieve the objective, sensitivity analysis is performed to quantify the influence of uncertainties in model parameters on the variation of the predictions. Then, the model uncertainty is assessed by comparing the prognoses of the addressed method classes to the measurements of a well established benchmark problem. The performance of the investigated models is compared to the results from the mode-I crack growth in concrete beams, using the well established three point bending experiment [47]. This is followed by the estimation of the model selection probability of each model class (computational method) using Bayesian method. To summarise, the present methodology is an efficient tool, considering the model uncertainties along with the uncertainties in the associated parameters.
The arrangement of the article is as follows. The motivation and objectives of the present study are summarised in the introduction Section 1 brief outline of the computational approaches for fracture to be evaluated, is provided in Section 2. The sensitivity analysis and the assessment methodology applied in the current study are discussed in Sections 3 Variance based sensitivity analysis, 4 Assessment method, respectively. The implementation details are addressed in Section 5. The key contributions along with the concluding remarks are presented in Section 6.
Section snippets
Selected computational methods for fracture
In this section, we briefly outline the models assessed with the proposed framework. The material is assumed to be isotropic. While the discrete approach to fracture assumes a linear elastic material model in the bulk, all continuous approaches for fracture are based on a damage model of the typewhere D is a scalar damage variable indicating the monotonic damage evolution; indicates the undamaged state whereas a completely damaged state, is the elasticity tensor of the
Variance based sensitivity analysis
Global sensitivity analysis (GSA) is a mathematical tool to quantify the uncertainty in the numerical models. The relative influence of each individual input parameter on the variance of the model output is expressed by a scaler value known as sensitivity index considering the variation of all the parameters at once [57]. In this context, the Sobol’s method [58] is one of the popular variance based GSA approach. The variance decomposition of the output is expressed the output as a sum of
Assessment method
A well developed model usually can be evaluated based on its theory. The actual state of the art of the mechanics of the physical phenomenon under study and the mathematical derivation should agree. On the other hand, the quality of the model prognosis is of high prominence, which should be taken into consideration while evaluating the model. The uncertainties are an inherent features of model’s prognoses. These uncertainties can be related to the model itself and/or to its input parameters.
Applications
The application of the developed sensitivity analysis and model quality evaluation of computational fracture models are discussed in this section. A set of four candidate damage numerical model classes, as discussed in Section 2, are considered for further investigation. The selected methods are: the cohesive zone model, phase field method, non-local damage model and gradient-enhanced damage model, hereafter, referred to as and , respectively. The bench mark problem to test the
Conclusion
Four different numerical model classes to model mechanics of material failure have been evaluated. The selected methods are: cohesive zone model in the context of phantom node method, phase field method, non-local damage model, and gradient-enhanced damage model. Uncertainties in modelling and the parameters were considered to assess the these numerical approaches, using the Bayesian method. The experimental results of mode-I crack growth in unreinforced concrete beam subjected to three point
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