- Research article
- Open Access
Full thermo-mechanical coupling using eXtended finite element method in quasi-transient crack propagation
- Fakhreddine Habib^{1}Email authorView ORCID ID profile,
- Luca Sorelli^{2} and
- Mario Fafard^{1}
https://doi.org/10.1186/s40323-018-0112-9
© The Author(s) 2018
- Received: 9 February 2018
- Accepted: 20 June 2018
- Published: 3 July 2018
Abstract
This work aims to present a complete full coupling eXtended finite element formulation of the thermo-mechanical problem of cracked bodies. The basic concept of the extended finite element method is discussed in the context of mechanical and thermal discontinuities. Benchmarks are presented to validate at the same time the implementation of stress intensity factors and numerical mechanical and thermal responses. A quasi-transient crack propagation model, subjected to transient thermal load combined with a quasi-static crack growth was presented and implemented into a home-made object-oriented code. The developed eXtended finite element tool for modeling two-dimensional thermo-mechanical problem involving multiple cracks and defects are confirmed through selected examples by estimating the stress intensity factors with remarkable accuracy and robustness.
Keywords
- Thermo-mechanical
- Extended finite element method
- Full coupling
- Crack growth
- Stress intensity factors computation
- Quasi-transient
Background
The interest in fracture mechanics and its applications has gained considerable importance in recent years in various industries: aerospace engineering, automobile industry, civil engineering, etc. This attention is due to the high cost caused by the presence of cracks and defects, which require more energy, time, substantial efforts and dedicated strategies regarding intervention, maintenance, repair, etc. Practically, taking into account the real environmental conditions in service has become essential, when the material is subject to a significant gradient of temperature. For instance, temperature change in real structures, where the deformation are constrained, can engender a mechanical load and a high-stress concentration around crack tips. Subsequently, crack can propagate with a, a priori, not known orientation, direction, intensity etc. Since, cracks cannot be eliminated under any circumstances; this prompts engineers to guide our efforts towards winning strategies in prevention, design and especially analysis that can be provided by the tool of numerical modeling.
The numerical modeling of cracked domains using finite element method (FEM) has clearly stood aside for the eXtended finite element method (XFEM) in the last two decades. XFEM has been able to provide essential answers for several situations, where the FEM method becomes numerically very expensive to have an optimal convergence, such as singularities, strong discontinuities, high gradient, moving surfaces, etc. This technique allows, by prior knowledge of the physical behavior of the problem, to enrich the space of the solutions by non-polynomial asymptotic functions when it is a singularity and a jump-function when it comes to a discontinuity or a combination of both of them. The resulting approximation space has to reproduce the Partition of Unity (PU), Babuška and Melenk [1]. The first work that introduced enriched FEM was Belytchko and Black’s paper [2] which presented an implicit description of the crack with minimal remeshing. Moës et al. [3] improved this technique by incorporating a more suitable way to consider the discontinuities throughout the crack faces away from the crack tip by the generalized Heaviside function and branching functions for the near crack tip. Daux et al. [4] later extend the approach for multiple cracks and holes for the mechanical problem.
Sukumar et al. [5] used the XFEM to model fracture in three-dimensional by using the PU concept, where the two-dimensional asymptotic crack tip displacement fields were added to the FE approximation to account for the crack. The XFEM for non-planar cracks in three dimensions illustrating the crack geometry using two signed distance functions was presented by Moës et al. [6]. Sukumar and Prévost [7] extended XFEM for two-dimensional crack modeling in isotropic and bimaterial media and later to demonstrate the numerical modeling of stress intensity factors in crack growth problems in Sukumar, and Prévost [8]. Lee et al. [9] exposed a combination of the XFEM and the mesh superposition method for modeling of stationary and growing cracks, where a step function implicitly described the discontinuity on the PU, and the crack tip was modeled by superimposed quarter point elements on an overlaid mesh. Budyn et al. [10] displayed a model for multiple crack growth considering the junction of cracks in brittle materials using XFEM, which does not require remeshing as the cracks grow.
Other XFEM aspects have been addressed: In contact, Khoei et al. [11] used XFEM to model the frictional contact problem using the penalty method. Nistor et al. [12] developed an approach to couple the XFEM with the Lagrangian large sliding frictionless contact algorithm. An algorithm based on node-to-segment XFEM contact was presented by Khoei et al. [13] based on the XFEM to model the large deformation-large sliding contact problem using the penalty approach. In stabilization aspect, an XFEM pre-conditioner which stabilizes the enrichments by applying Cholesky decompositions to certain sub-matrices of the stiffness matrix was proposed by Béchet et al. [14]. Menk et al. [15] expose another pre-conditioning method suited for parallel computation. Also, another approach initially developed by Hansbo et al. [16] to simulate strong and weak discontinuities in solid mechanics. A similar method was used by Song et al. [17], named phantom nodes, for shear modeling dynamic crack and shear band propagation. Rabczuk et al. [18] developed a new crack tip element for the phantom node method suited for one-point quadrature scheme and can be used with other general quadrature schemes. XFEM numerical integration aspect is performed by Dolbow et al. [19] by using a sub-triangulation for computing the element area below and above the crack and to set criteria for node enrichment with discontinuity function. Laborde et al. [20] used a singular mapping for each sub-triangle and a bidirectional Gauss quadrature in each direction. In Ventura [21], the constructing sub-cells in the numerical integration of discontinuity functions is removed by defining an equivalent polynomial function. Schwarz–Christoffel conformal mapping was used to map an arbitrary polygon onto a unit disk by Natarajan et al. [22]. A fairly comprehensive review of the different aspects of XFEM was presented by Khoei [23]. All these advances in XFEM mentioned before are in the field of solid mechanics.
In this paper, the approach taken is based on a semi-implicit thermo-mechanical-crack-growth algorithm in which the combined full coupling thermal and mechanical responses have to be estimated beforehand. Then, the developed numerical fracture mechanics module takes those responses as inputs to evaluate the stress intensity factors, J-integral, the update of the crack in growth, etc. This actualization is done by an implicit description of the crack, using the Level-Set Method (LSM) presented firstly by Osher and Sethian [24]. The LSM provides a fundamental complementary to know when, where and how to enrich the crack by determining its relative position. Stolarska et al. [25] introduced an algorithm that combines the XFEM and LSM to model mechanical crack growth, where the LSM was used to model the crack surface and crack tip locations. Moreover, stress intensity factors (SIFs) computation, as the prime parameter of prediction, makes it possible to obtain an essential knowledge of the behavior of the crack. This evaluation enables to predict whether the structure becomes unsafe in service conditions, especially when it is in a thermo-mechanical context, where the spatial distribution of the mechanical stresses induced by the thermal field is unpredictable.
Interest in thermo-mechanical applications appeared later with Michlik and Berndt [26] presented an approach of thermo-mechanical XFEM analysis to account for the existence of cracks in thermal barrier coating for predicting an effective thermal conductivity and Young’s moduli of multi-layered. Duflot [27] used the XFEM for the analysis of steady-state thermally stressed, cracked solids in thermo-elastic problems, where he enriched both thermal and mechanical fields to represent the discontinuous temperature and displacement. Fagerstöm and Larsson [28] presented a thermo-mechanical fracture formulation based on discontinuous representation for temperature and displacements fields applicable to the fracture process zone into a cohesive zone. Zamani et al. [29] proposed a higher order XFEM to predict the SIFs for thermo-elastic with stationary cracks, The computation of SIF is extracted directly from the XFEM degrees of freedom. Zamani et al. [30], in a later work, implemented the XFEM to model the effect of the mechanical and thermal shocks on a body with a stationary crack. Lee et al. [31] presented an XFEM method for the analysis of heat conduction at submicron scales of geometrically complex nanostructured heterogeneous materials. Fan et al. [32] used XFEM to investigate the effect of thermally grown oxide on multiple surfaces cracking behavior in an air plasma sprayed thermal barrier coating system. Hosseini et al. [33] introduced a computational method based on the XFEM for fracture analysis of isotropic and orthotropic functionally graded materials (FGM) under mechanical and steady-state thermal loadings. Yu et al. [34] exploited XFEM for modeling the temperature field in heterogeneous materials, where the standard temperature field was enriched by using the level-set-based enrichment functions which model the interfaces. Macri et al. [35] presented a multiscale technique for modeling heterogeneous materials based on an enriched partition of unity that incorporates the thermal effects occurring on the microstructure into the global model for simulation. In Sapora et al. [36], an analogy between fracture and contact mechanics is proposed to investigate debonding phenomena at imperfect interfaces due to thermomechanical loading and thermal fields in bodies with cohesive cracks. From fracture mechanics point of view, Goli et al. [37] implemented the path-independent interaction integral in the context of the partition of unity for mixed mode adiabatic cracks under thermo-mechanical loadings particularly in orthotropic non-homogenous materials for a steady-state thermal problem. Bayesteh et al. [38] study a thermo-mechanical fracture of inhomogeneous cracked solids by the extended isogeometric analysis method, crack faces, and tip XFEM enrichment are incorporated into the non-uniform rational basis splines functions of isogeometric analysis (IGA) for static crack and steady-state thermal problem. Jia and Nie [39] adopted XFEM to analyze the interaction between a single or multiple macroscopic or microscopic inclusion and cracks for static crack and under the steady-state thermal problem. The work of Jaśkowiec [40] is concerned with modeling the heat flow through cracks in three-dimensional thermo-mechanical problems, the model for crack heat flow is combined with cohesive crack model. He et al. [41] established an XFEM thermo-elastic fracture problem for aluminium alloy metal inert gas welding, which includes a variable heat source with the initial and boundary conditions for a cracked plate structure. Li and Fish [42] developed a thermo-mechanical extended layerwise method for the laminated composite plates with delaminations and transverse cracks; transverse cracks are modeled using classical XFEM under pure mode-I. Recently, Zarmehri et al. [43] implemented XFEM to extract stress intensity factors for a stationary crack in an isotropic 2D finite domain under thermal shock, the coupled generalized thermo-elasticity theory employed is based on Green-Lindsay model.
Although the plethora of works has treated numerical thermo-mechanical analysis using classical XFEM recently, few works have employed the enhanced version of XFEM named XFEM-f.a. in order to ensure an optimal convergence through a geometrical enrichment regardless of the mesh size. This work aims at developing the complete full thermo-mechanical coupling using XFEM in adiabatic cracked media adopting a geometrical enrichment. The implementation was firstly validated for a single crack from the existing examples in the literature. Then, validation of the case of the combination between a hole and cracks, and the influence of crack size and a single hole size on the stress intensity factors, i.e., on the behavior of the rupture in a given structure, is performed. This case was investigated with the work of Prasad et al. based on the dual boundary element method for thermo-elastic crack problems [44]. Notably, the case of transient thermal loading and its impact on the SIFs profiles was treated, then a situation of the growth in mode-I was analyzed. Moreover, this work study the case of the thermo-mechanical propagation of multiple cracks in the presence of multiple holes in mixed mode.
This paper consists of six sections. The second one sketch the mathematical, physical and variational framework of the two-dimensional plane strain thermo-mechanical problem studied in a cracked medium. The third section intended for approximation spaces and the XFEM discretized forms of displacement and temperature fields as well as the full coupling XFEM matrices for each sub-problem part and the integration technique employed. Section four deals with the crack growth criterion assumed in this study, the form of the thermo-mechanical J-integral and the extraction of SIFs. Section five describes the specific numerical approach of a modified XFEM version involved. Several validation models from the literature are then considered for validation purpose; another benchmark with cracks and a manufacturing flaw idealized by a hole; an example of crack growth in mode-I, then in transient thermal loading; and lastly a mixed-mode crack growth model designed carefully for a specimen with multiple holes and cracks in the thermo-mechanical case. Finally, we conclude by a summary and some proposed extensions of this work.
Problem and variational formulations
Governing equations
Note that in case of several cracks, the mother crack \(\Gamma _{c}\) can be decomposed into many n adiabatic cracks, \(\Gamma _{c}=\bigcup ^{n}_{1}\Gamma _{c_{i}}\) such every \(i \mathrm{th}\mathrm{th}\) crack remains adiabatic \(q\cdot \;n=0\) and traction-free \(\sigma \cdot \;n=0\) on each \(\Gamma _{c_{i}}\), for any \(i\in \llbracket 1,n\rrbracket \). Henceforth, we will present the XFEM developments for the mother crack, which remains valid for all sub-cracks.
Variational form
XFEM approximation and numerical integration
Full coupled eXtended Finite Element form
\(\bullet \) Mechanical part:
Numerical integration
In split element each sub-element take 7 integration points per K, in total we obtain \(7*m_{e}\) points per split/vertex element. Also, we assume more integration points on the tip-elements to capture well numerically the singularity by 19 integration points per K; in total, we get \(18*m_{e}\) per tip element. To note that to refine the XFEM approximation on the elements of transition between fully enriched elements and standard ones, we keep the same treatment as tip element with a spider-web centered on the iso-barycenter of the element.
Enriched zone update in crack growth
In the initial state of propagation, the position of the crack is predefined \(\Gamma _{c}^{0}\); the mesh \(\mathcal {M}\) is properly generated, once and for all, without any change on it during the process of growth. Then, the relative position of the crack is implicitly identified by level-set functions, leaving aside its known Cartesian global position. Consequently, crack is recognized independently of the mesh definition, relatively, with respect to its nodal environment thanks to the signed-distance function. At this early stage, to write the discrete XFEM form of the displacement, Eq. (14), and the temperature, Eq. (16), fields, we ought to select the two kinds of enriched nodes families, Heaviside and tip enriched nodes. All the mesh nodes including those wholly enriched are roughly approximated by the standard shape functions. The nodes enriched by Heaviside function are described by the nodes forming the elements thoroughly crossed by the crack. The tip candidate nodes are such as those nodes composing the tip element for a topological enrichment; in geometrical enrichment on a given disk, \(\mathcal {D}\) of radius R and centered on the tip, the discrete set of tip nodes is formed by the intersection of the mesh \(\mathcal {M}\) and the disk \(\mathcal {D}\), \(\mathcal {M}\cap \mathcal {D}\). In geometrical enrichment instance, \(\mathcal {D}\) crosses undoubtedly, for a large radius, the already selected Heaviside nodes. As a result, these nodes should be enriched simultaneously by the combined form of branching function and Heaviside, \(F+H\). This selection configuration is performed for the initial increment and will identify the enriched-zone \(EZ^{0}\) related to the position of the crack \(\Gamma _{c}^{0}\) which is supposed to be unique. At the next increment, the same procedure is followed until a \((k\;-\;1)\mathrm{th}\) incremental crack is reached, \(\Gamma _{c}^{k-1}\). At the \(k\mathrm{th}\) increment, we suppose that the chosen crack growth criterion is satisfied, which allows extending the crack in the appropriate direction. This progress creates a new geometrical configuration of the crack and a new enriched-zone \(EZ^{k}\) to be identified. We proceed in the same way, by recurrence, as the \((k\;-\;1)\mathrm{th}\) increment. This time we find ourselves with two different configurations where some nodes in the previous increment that they were, for example, of Heaviside type become tip nodes, or standard nodes convert to Heaviside/tip nodes, or vice versa. The transport of nodal fields corresponding to temperature and displacements computed at the \((k\;-\;1)\mathrm{th}\) increment to the \(k\mathrm{th}\) one will be performed by an \(L^{2}\)-projection on the discrete space generated by the \(k\mathrm{th}\) recent configuration by means of least squares method. This strategy is possible since the different quantities are square-integrable, which allows ensuring the stability and efficiency of the used scheme. To note that, the advancement of the crack generates a new geometrical, topological and numerical reality of the thermo-mechanical problem resolution which requires a specific treatment at each increment. This results in an extensible and flexible set of degrees of freedom with the crack evolution; therefore, the linear system of discrete equations is also extensible and changes in dimension depending on the crack growth state, it can enlarge or diminish. On the other hand, the selection rule of the \(EZ^{k}\) nodes is independent of the previous configuration and related only to the relative position of the crack in its nodal environment at the actual increment. We are thus left with two different configurations of two successive increments. This process is repeated iteratively until the estimated or evaluated end of the propagation process.
Crack growth criterion and stress intensity factors evaluation
Propagation criterion and crack update
It has been shown that the use of level-set function plays an essential role in the implicit description of the crack and evaluation of enriched fields, mechanical Moës et al. [6] and thermal in this work. The crack is representing the zero-level set of a given function. The crack tip positions can be found by considering the intersection between zero-level contour and a second orthogonal level-set function Stolarska et al. [25] using the signed-distance function. The signed-distance in the level-set method is represented by a finite element approximation with the same mesh used for the mechanical and thermal problems. Adopting this representation makes the task easier when it is necessary to evaluate the level-set at element level by interpolation and when we need to compute its derivative which is well-defined by the derivative of shape functions.
On the other hand, different crack extension criteria exist in the literature and adequately ensures the crack progression, governed by fatigue law varieties. They are adapted to the crack progress when it is subjected to cyclic loading. The crack rate increment with respect to the loading cycle, i.e., speed growth, appeared in these laws and assumed to be, in general, a function which depends on the stress intensity factor range between two cycles and the stress ratio, Beden et al. [47]. The popular one is the classical law of Paris which is a version of the general law of fatigue, where the speed growth depends on the stress intensity factor range, and two constants, called constants of Paris law, that have to be identified for each specific material, Cherepanov et al. [48]. Its limitation lies in the fact that it requires a minimum stress intensity factor to ensure the propagation and does not take into account the stress ratio. Another version appeared later by Xiaoping et al. [49] that overcomes these limitations of the classical Paris law but requires three additional parameters more than the classical Paris law. All these models can ‘better’ capture the crack progress and monitor the history of the adapted crack increment for each promotion. They are more suited to fatigue propagation fashion and also require additional parameters related to the material that can be determined by fatigue tests. This last point may be a drawback for the attractiveness of these methods for the present work. However, the convergence of fixed crack increment method may be ‘lower’ in some cases, but with a suitable choice of the crack increment, which depends on the mesh and other parameters as cited previously, one can reach good results. Besides, fixed crack increment technique is more attractive; it needs less material parameters compared with the earlier mentioned laws. Its ability to obtain crack paths that coincide very well with reference solutions is investigated by Baydoun et al. [50].
Stress intensity factors evaluation
Numerical examples of thermo-mechanical analysis
A set of thermo-mechanical examples are herein discussed by considering a strong material discontinuity; for a static adiabatic crack and in propagation state of an isotropic material. Validation of the results is fulfilled by a comparison with the computation of the stress intensity factors which allows validating both mechanical and thermal responses as well as the quantification of the linear elastic fracture mechanics (LEFM) parameters. The computation domains chosen for the benchmarks are extracted from the literature and meshes are generated using Gmsh [52]. A hybrid object-oriented code has been developed in a monolithic multi-physical philosophy treating each step starting from the mesh generated from Gmsh, the definition of the enrichment-zone, the XFEM matrix computation blocks associated to each physical segment and to each coupled part, the computation of fracture mechanics quantities and post-processing context.
Material properties
Poisson ratio-\(\nu \) | \(0.3 \ [-]\) |
Young’s modulus-E | \(2.184*10^{5}\) [Pa] |
Thermal conductivity-k | 205 [\(\mathrm{W}\;\mathrm{m}^{-1}\;^\circ \)C\(^{-1}\)] |
Thermal expansion coefficient-\(\alpha \) | \(1.67*10^{-5}\) [\(^\circ \)C\(^{-1}\)] |
In this section, we present various examples to validate the thermo-mechanical model by XFEM-f.a. implementation, comparing with several benchmarks taken from the literature. The primary objective of all these cases is to investigate the accuracy and robustness of the numerical results. Then, we present an example of the cracked domain under transient-thermal load and in crack growth governed by mode-I. Finally, we design a model of crack propagation in mixed mode for a pure thermal loading, with round holes and multiple cracks.
Rectangular plate with a centered slope crack
A rectangular plate specimen with a centered inclined crack subjected to a pure thermal load is analyzed, with the dimensions 2L, 2W, the crack is defined with the half-length a and the slope is characterized by the \(\beta \) angle Fig. 4a. The displacements along the \(e_{y}\)-axis is fixed at the bottom extreme right corner, and the bottom left corner is clamped. Both right and left boards are completely insulated, an imposed temperatures of \(\pm \Theta _{0}\) are defined at the top and bottom sides, such \(\Theta _{0}=10\;^\circ \)C. We consider a uniform enrichment disks radius in the case where several crack-tips exist; hence, \(E_{R}\equiv E_{R}^{j}\). The radius of disks enrichment \(E_{R}\) is taken equal to \(0.15\;\mathrm{m}\) for both rectangular and square plates examples. We divide this example into two cases.
Normalized SIFs, various J-integral paths
Normalized SIFs for centred crack in a square plate, various a / W
Normalized SIFs for slope crack, \(\beta =30^\circ \) and various a / W
\(\varvec{\frac{a}{W}}\) | \(\varvec{K}_{\varvec{I}}^{\mathrm{norm}}\) | \(\varvec{K}_{\varvec{II}}^{\mathrm{norm}}\) | ||||||
---|---|---|---|---|---|---|---|---|
Present work | [54] | [44] | [27] | Present work | [54] | [44] | [27] | |
0.2 | 0.0021 | 0.002 | 0.002 | 0.0020 | 0.0301 | 0.030 | 0.030 | 0.0302 |
0.3 | 0.0069 | 0.008 | 0.006 | 0.0068 | 0.0484 | 0.048 | 0.048 | 0.0489 |
0.4 | 0.0152 | 0.015 | 0.014 | 0.0149 | 0.0640 | 0.064 | 0.064 | 0.0650 |
0.5 | 0.0269 | 0.027 | 0.026 | 0.0265 | 0.0773 | 0.076 | 0.076 | 0.0774 |
0.6 | 0.0408 | 0.041 | 0.040 | 0.0407 | 0.0872 | 0.086 | 0.087 | 0.0878 |
Normalized SIFs for slope crack at both tips, \(a/W=0.3\) and various \(\beta \)
\(\varvec{\frac{a}{W}}\) | \(\varvec{K}_{\varvec{I}}^{\mathrm{norm}}\) | \(\varvec{K}_{\varvec{II}}^{\mathrm{norm}}\) | ||||||
---|---|---|---|---|---|---|---|---|
Present work | [54] | [44] | [27] | Present work | [54] | [44] | [27] | |
\(0^\circ \) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0548 | 0.054 | 0.054 | 0.0546 |
\(15^\circ \) | 0.0036 | 0.0038 | 0.0036 | 0.0038 | 0.0533 | 0.054 | 0.054 | 0.0533 |
\(30^\circ \) | 0.0069 | 0.0071 | 0.0064 | 0.0068 | 0.0484 | 0.048 | 0.048 | 0.0489 |
\(45^\circ \) | 0.0075 | 0.0077 | 0.0071 | 0.0076 | 0.0413 | 0.042 | 0.041 | 0.0420 |
\(60^\circ \) | 0.0054 | 0.0053 | 0.0049 | 0.0054 | 0.0324 | 0.032 | 0.032 | 0.0322 |
\(75^\circ \) | 0.0012 | 0.0023 | 0.0010 | 0.0017 | 0.0181 | 0.018 | 0.018 | 0.0180 |
\(90^\circ \) | 0.0003 | 0.0000 | 0.0003 | 0.0000 | 0.0000 | 0.000 | 0.000 | 0.0000 |
Square plate with round hole and two cracks
In this example, we consider the case of a square plate with a hole and two cracks. The objective is: firstly, to show the influence of the presence of a hole, due to a manufacturing defect or willingly introduced into the material, on the stress intensity factors. Secondly, to examine the influence of radius of the fixed enriched zone on the SIFs. Thirdly, to show the influence of the characteristic length (h) on the convergence of SIFs when h goes to zero. The dimensions of the domain are chosen such \(L=0.5\;\mathrm{m}\), the hole is placed in the center of the plate defined by the radius R Fig. 11a. The two cracks are defined at the two ends, right and left, of the hole are centered (right and left cracks), with a length l. Half-length of the apparent crack is defined by \(a=l+R\). The bottom left corner is clamped and displacements along \(e_{y}\)-axis is fixed. The heat flux is zero at the right and left edges, an imposed temperature of \(\pm \Theta _{0}\) are defined at the top and bottom sides, such \(\Theta _{0}=10\;^\circ \mathrm{C}\).
Influence of enrichment radii on the computation of SIFs for square plate with round hole and two cracks, \(R/L=0.1\)
\(\frac{\varvec{l}}{\varvec{L}}\) | Normalized SIFs | Left crack | Right crack | ||||
---|---|---|---|---|---|---|---|
\(\varvec{E}_{\varvec{R}}\) | \(\varvec{E}_{\varvec{R}}\) | ||||||
0.2 | 0.15 | 0.1 | 0.2 | 0.15 | 0.1 | ||
0.5 | \(K_{II}^{\mathrm{norm}}\) | 0.219244 | 0.219256 | 0.219357 | \(-\) 0.219244 | \(-\) 0.219256 | \(-\) 0.219357 |
0.6 | \(K_{II}^{\mathrm{norm}}\) | 0.273149 | 0.273162 | 0.273056 | \(-\) 0.273149 | \(-\) 0.273162 | \(-\) 0.273056 |
Edge cracked strip under thermal loading
Thermal transient loading
Crack growth
Rectangular plate with two circular holes and multiple cracks
Conclusions
A new thermo-mechanical crack propagation model in a cracked body was presented which can be applied, for instance, to ensure the safety of structures subjected to thermal loading. The developed geometrical eXtended finite element method was successfully applied to model crack growth and achieving the expected optimal rate of convergence by confirming the benefit of the fixed enrichment area approach on the computation of stress intensity factor profile. Numerical development and various matrices in full coupling were presented for each sub-problems, mechanical and thermal, and for the full coupled XFEM part. The criteria for crack growth, as well as for the direction of the virtual crack extension are described, and their performance in the context of the XFEM is discussed. From three examples, various benchmarks result in a cracked domain are examined and validated from the existing results in the literature. The robustness and the accuracy of the model implementation to extract the thermo-mechanical responses and to compute the associated stress intensity factors for stationary crack, with and without holes, as well as the effect of crack length and hole position on the SIFs are proved. Furthermore, a quasi-transient load example governed by mode-I is presented and the contribution of this loading on the profile of the SIFs until reaching thermal equilibrium is analyzed. Finally, an example of multiple mixed-mode cracks growth and multiple holes that may be present as small flaws in the material manufacturing stage is examined; only the limiting cases of stable crack are discussed. When the heat flow is distributed by the presence of the cracks, we observe a high local intensification of thermal gradients followed by an intensification of thermo-mechanical stress around them, which may lead to the crack growth or inevitable collapse of the structure.
As outlook of future works, possible improvement of this study can be made by taking into consideration the mechanical contact aspect between the crack surfaces; this will be important to extend to study of the last example to simulate the complete process of growth. Another point can be viewed by holding the crack propagation in the overall dynamic of the whole problem and admitting a crack-pseudo-time-dependant; which make it possible to control the evolution of the crack with the transient loading. This case requires a sophisticated treatment of the stiffness matrix; K needs to be evaluated at two different times for two different configurations of the crack, \(K_{i}\) and \(K_{i+1}\).
Declarations
Author's contributions
All the authors have participated in the development of this work. All authors read and approved the final manuscript.
Acknowledgements
This work was carried out within the framework of a sub-project financed by: Aluminum Company of America (Alcoa), Natural Sciences and Engineering Research Council of Canada (NSERC), and the Fonds de recherche du Québec-Nature et technologies (FRQNT) through the Aluminium Research Centre-REGAL.
Competing interests
The authors declare that they have no competing interests. They are open to comments and constructive suggestions for possible future enhancements and extensions of this work.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
Aluminum Company of America (Alcoa), Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec-Nature et technologies (FRQNT).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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