1 Introduction
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First, the derivation of well-posed theoretical formulations for describing the forward model. Hereby, variational phase-field modeling is considered, which is a regularized approach to fracture with a strong capability to simulate complex failure processes. This includes crack initiation (also in the absence of a crack tip singularity) [16‐18], propagation, coalescence, and branching, without additional ad-hoc criteria [8, 19, 20]. Moreover, the boundary value problem can be solved using standard finite element approximation spaces. A summary of multiphysics phase-field fracture models including algorithmic treatments is outlined in [21]. We further note that, in addition to the finite element method, meshless methods in [22, 23], as well as isogeometric analysis (IGA) in [24] and virtual element method (VEM) in [25] can be used in phase-field models.
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The second challenge is to elucidate the backward model in order to estimate the model parameters and other univariate quantities of interest. A Bayesian estimation model (as an inverse model) is here used for the ductile fracture problem to provide accurate knowledge regarding the effective mechanical parameters.
1.1 Ductile phase-field fracture as a forward model
1.2 Bayesian inversion as a backward model
1.3 Physical interpretation of the ductile parameters
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The shear modulus \(\mu \) is a positive constant, smaller than K, which indicates the response of the solid to shear stress (the ratio of shear stress to shear strain). Large shearing stresses give rise to flow and permanent deformation or fracture. See, e.g., [66, 68, 69].The elastic properties of the solid can be alternatively described in terms of the Young’s modulus and the Poisson’s ratio, or any other pair of Lamé’s parameters. In our previous work [56], it was reported that due to the boundness of the Poisson’s ratio (\(-1<\nu <\frac{1}{2}\)) and Lamé’s first parameter (\(\lambda >\frac{2\mu }{3}\)), the Poisson’s ratio and Lamé’s first parameter are not appropriate for Bayesian inference. Hence, for the elasticity identification, K and \(\mu \) are selected, where \(K>0\) and \(\mu >0\).
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The Griffith’s energy release rate \(G_c\) indicates the necessary energy (absolutely positive) to drive crack growth in elastic media. It measures the amount of energy dissipated in a localized fractured state, and therefore has units of energy per unit area. The energy rate is directly related to the toughness, indicating that in tougher materials, more energy is required to initiate fracture [70‐72].
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The yield stress \(\sigma _Y\) denotes the stress related to the yield point (the starting point of plasticity) where the material starts to deform in the plastic regime [69].
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The critical value \(\alpha _{\mathrm{crit}}\) stems from a physical assumption that fracture evolution is promoted once a threshold value for the accumulated plastic strain has been reached [72].
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The specific fracture energy \(\psi _c\) characterizes the dissipated energy during a complete damage process in a homogeneous volume element [76]. This property is related to Griffith’s energy release rate and can be interpreted as the amount of strain energy density (strain on a unit volume of material) that a given material can absorb before it fractures [77].
2 Phase-field modeling of ductile fracture in anisotropic elastic-plastic materials
2.1 Basic continuum mechanics
2.2 Energy quantities and variational principles
2.2.1 Elastic contribution
2.2.2 Fracture contribution
2.2.3 Plastic contribution
2.3 Stationarity conditions and governing equations
2.3.1 Elasticity
2.3.2 Fracture
2.3.3 Plasticity
2.4 Specific models revisited
2.4.1 Local plasticity with \(G_c\) based fracture criterion: Model 1 (\({\mathcal {M}}_1\))
Model property | \({\mathcal {M}}_1\) | \({\mathcal {M}}_2\) | \({\mathcal {M}}_3\) |
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Elastic degradation \(g_e\) | \((1-d)^{2{\alpha }/{\alpha _{\mathrm{crit}}}}\) | \((1-d)^2\) | \((1-d)^2\) |
Fracture constant \(g_f\) | \(G_c\) | \(2\,l_f c_f{\psi _c}\) | \(l_f c_f{w_0}\) |
Plastic degradation \(g_p\) | 1 | \((1-d)^2\) | \((1-d)^2\) |
Local fracture energy \(\omega \) | \(d^2\) | d | d |
Crack viscosity \(\eta _f\) | 0 | \(\ge 0\) | 0 |
Plastic length-scale \(l_p\) | 0 | 0 | \(\ge 0\) |
Driving scaling factor \(\zeta \) | 1 | \(\ge 0\) | \(\ge 0\) |
2.4.2 Local plasticity with \(\psi _c\) based fracture criteria: Model 2 \(({\mathcal {M}}_2)\)
2.4.3 Non-local plasticity with \(w_0\) based fracture criteria: Model 3 (\({\mathcal {M}}_3\))
3 Parameter estimation based on Bayesian inference
3.1 Metropolis and Metropolis–Hasting algorithms
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A proposal \(\chi ^{\star }\) that results in \(\pi (m|\chi ^{\star })>\pi (m|\chi ^{j-1})\) entails a small sum of squared error and thus leads to candidate acceptance.
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A proposal \(\chi ^{\star }\) that leads to \(\pi (m|\chi ^{\star })<\pi (m|\chi ^{j-1})\) entails a higher sum of squared error and the proposal may be rejected.
3.2 Delayed rejection adaptive metropolis (DRAM)
3.3 MCMC with ensemble-Kalman filter
3.4 Bayesian inversion for ductile phase-field fracture
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Based on the load-displacement curve: this approach allows us to observe the crack behavior in all time steps up to complete failure. At time-step n, the load-displacement curve can be computed aswhere \({\varvec{n}}\) is the outward unit normal on the surface, defined in (11). The main advantage of working with this curve is its easiness, since it involves a one-dimensional parameter. However, it is sensitive to the mesh size and the length scale; therefore, a sufficiently small (and thus more computationally expensive) mesh size is needed.$$\begin{aligned} F_n=\int _{\partial _D{\mathcal {B}}} \varvec{n}\cdot \varvec{\sigma } \cdot \varvec{n}\,da, \end{aligned}$$(103)
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Based on the point-wise primary fields: this approach monitors the crack behavior, the displacement, and the equivalent plastic strain in the entire geometry. Here, the Bayesian setting strives to find the inferred parameters \(\chi ^\star \) which minimizewhere \(L^2\)-norm can be used, and \(\bar{\varvec{u}}\), \(\bar{d}\), and \(\bar{\alpha }\) are the experimental data throughout geometry with respective measurement errors \(\varepsilon _1\), \(\varepsilon _2\), and \(\varepsilon _3\). This method is informative and provides precise information since the displacement and phase-filed in the entire geometry are considered. However, it is difficult and perhaps even impossible to obtain the measured data from actual experiments in a point-wise manner. Furthermore, a small mesh size must be chosen in numerical simulations to guarantee accurate estimations in the whole geometry, further rendering the method computationally prohibitive.$$\begin{aligned}&\Vert \bar{\varvec{u}}(\varvec{x})-\varvec{u}(\varvec{x},\chi ^\star ) -\varepsilon _1{\mathcal {I}}\Vert ^2+\Vert \bar{d}(\varvec{x})-d(\varvec{x},\chi ^\star ) -\varepsilon _2{\mathcal {I}}\Vert ^2\\&\quad +\Vert \bar{\alpha }(\varvec{x})-\alpha (\varvec{x}, \chi ^\star )-\varepsilon _3{\mathcal {I}}\Vert ^2, \end{aligned}$$××
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Based on the point-wise phase-field propagation: this approach is less complex than the previous method. Reliable experimental values of the crack path can be obtained using X-ray or \(\mu \)-CT scan in two- or three-dimensional problems; see, e.g., [108]. However, the computational issue regarding mesh sensitivity persists.
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Based on snapshots of proper orthogonal decomposition (POD): this approach employs a reduced order method (ROM) to reduce the computational complexity. Here, the snapshots of the solution (using measurements of the crack phase-field) are used to construct the POD basis [109]. If an efficient ROM is used, the computational complexity can be reduced significantly. Similarly, a Global-Local approach [110] can be employed to reduce the computational complexity of the forward model.
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Based on the effective stress-strain response: this approach explains the relation between \(\varvec{\varepsilon }\) and \(\varvec{\sigma }\). It provides useful information regarding different material properties such as bulk modulus, hardening, and yield strength. Therefore, considering the availability of measurements, it entails an instructive procedure. Nevertheless, the computational costs, i.e., the effect of the mesh size and length scale, must be taken into account.
Parameter | Name | Unit | Value |
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\(\mu \) | shear modulus | \(\mathrm{MPa}\) | \(\mathrm{BI}\) |
K | bulk modulus | \(\mathrm{MPa}\) | \(\mathrm{BI}\) |
H | hardening modulus | \(\mathrm{MPa}\) | \(\mathrm{BI}\) |
\(\sigma _Y\) | yield stress | \(\mathrm{MPa}\) | \(\mathrm{BI}\) |
\(\alpha _{\mathrm{crit}}\) | hardening critical value | – | \(\mathrm{BI}\) |
\(\psi _c\) | specific fracture energy | \(\mathrm{MPa}\) | \(\mathrm{BI}\) |
\(G_c\) | Griffith’s energy release rate | \(\mathrm{MPa \,\, mm}\) | \(\mathrm{BI}\) |
\(w_0\) | specific fracture toughness | \(\mathrm{MPa}\) | \(\mathrm{BI}\) |
\(\zeta \) | driving scaling factor | – | \(\mathrm{BI}\) |
\(\chi _a\) | stiffness parameter | – | \(\mathrm{BI}\) |
\(\eta _f\) | crack viscosity | \(\hbox {N}/\hbox {m}^{2}\hbox {s}\) | \(10^{-9}\) |
\(\eta _p\) | plasticity viscosity | \(\hbox {N}/\hbox {m}^{2}\hbox {s}\) | \(10^{-9}\) |
\(\kappa \) | stabilization parameter | – | \(10^{-8}\) |
\(l_d\) | fracture length-scale | \(\hbox {mm}\) | \(10^{-8}\) |
\(l_p\) | plastic length-scale | \(\hbox {mm}\) | \(10^{-8}\) |
\(\phi \) | fiber rotation | degree | \([30^{\circ }, 45^{\circ }, 60^{\circ }]\) |
4 Numerical examples
Parameter | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) |
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Min | 150 | 20,000 | 40,000 | 275 | 5 | 0.05 | 20 | 20 | 0.5 | 0.25 |
Max | 375 | 40,000 | 100,000 | 400 | 15 | 0.2 | 60 | 60 | 2.5 | 10 |
Initial | 220 | 25,000 | 80,000 | 350 | 12 | 0.12 | 30 | 25 | 1.2 | 2 |
Model | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) |
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\({\mathcal {M}}_1\) | 240 | 27 212 | 71 527 | 335 | 10.5 | 0.11 | – | – | – | – |
\({\mathcal {M}}_2\) | 241 | 27 120 | 71 245 | 330 | – | – | 41 | – | 0.98 | |
\({\mathcal {M}}_3\) | 248 | 26 699 | 74 500 | 328 | – | – | – | 38 | 1.25 | 1.02 |
4.1 Example 1: Asymmetrically I-shaped specimen under tensile loading
4.2 Example 2: I-shaped tensile specimen for anisotropic ductile fracture
\(\psi _c\) | 25 | 35 | 45 | 55 | 65 |
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\(\alpha _{\mathrm{crit}}\) | 0.065 | 0.092 | 0.12 | 0.136 | 0.15 |
\(G_c\) | 8.25 | 10.1 | 12 | 15 | 18 |
\(w_0\) | 20.8 | 29.1 | 38 | 48 | 55 |
\(l_p\) | 1.05 | 1.11 | 1.25 | 1.4 | 1.6 |
Parameter | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) | \(\chi _a\) |
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min | 150 | 20 000 | 40 000 | 275 | 5 | 0.01 | 10 | 10 | 0.5 | 0.25 | 10 |
max | 375 | 40 000 | 100 000 | 400 | 15 | 0.2 | 60 | 60 | 10 | 10 | 100 |
Model | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) | \(\chi _a\) |
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\({\mathcal {M}}_1\) | 265 | 26 050 | 94 010 | 355 | 11.6 | 0.038 | – | – | – | – | 50 |
\({\mathcal {M}}_2\) | 245 | 26 100 | 92 100 | 354 | – | – | 25.25 | – | 1.01 | 52 | |
\({\mathcal {M}}_3\) | 220 | 26 300 | 88 950 | 340 | – | – | – | 22 | 8.29 | 1.6 | 55 |
4.3 Example 3: Flat I-shaped Al-5005 test under tensile loading
Parameter | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) |
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Min | 10 | 20 000 | 40 000 | 50 | 100 | 0.001 | 5 | 5 | 0.001 | 1 |
Max | 50 | 40 000 | 100 000 | 200 | 300 | 0.1 | 25 | 25 | 10 | 20 |
Model | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) |
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\({\mathcal {M}}_1\) | 30 | 26 500 | 73 500 | 115 | 248 | 0.0142 | – | – | – | – |
\({\mathcal {M}}_2\) | 15 | 26 200 | 73 700 | 116 | – | – | 13.4 | – | 2 | |
\({\mathcal {M}}_3\) | 15 | 30 100 | 75 050 | 112 | – | – | – | 10.3 | 0.0018 | 15 |
4.3.1 Convergence performance of the MCMC methods
4.4 Example 4: Sandia fracture challenge
Parameter | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) |
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Min | 5 | 20,000 | 40,000 | 150 | 100 | 0.001 | 10 | 10 | 5 | 1 |
Max | 20 | 40,000 | 100,000 | 300 | 400 | 0.3 | 50 | 50 | 30 | 20 |
Model | H | \(\mu \) | K | \(\sigma _Y\) | \(G_c\) | \(\alpha _{\mathrm{crit}}\) | \(\psi _c\) | \(w_0\) | \(l_p\) | \(\zeta \) |
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\({\mathcal {M}}_1\) | 9.8 | 25,500 | 67,500 | 195 | 360 | 0.2 | – | – | – | – |
\({\mathcal {M}}_2\) | 10 | 26,200 | 67,200 | 194 | – | – | 35 | – | 8 | |
\({\mathcal {M}}_3\) | 10.1 | 25,900 | 67,800 | 195 | – | – | – | 28 | 9.64 | 15 |
5 Conclusion
Acknowledgements
DFG-SPP 2020
within its second funding phase. T. Wick and P. Wriggers were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD, EXC 2122
(project number: 390833453).