The Variational Approach
The basis of the variational approach to fracture mechanics relies on the association of a potential energy consisting of stored elastic energy, the work of external forces and the energy released through fracture to any crack and deformation configuration. A reference configuration \(\subset {\mathbb{R}}^{N}\), N = 2, of a homogeneous elastic body is considered which contains a generic crack as visualized in Fig. 1.
The total energy of the body is defined as
$${\mathcal{E}}\left( {f,K} \right) = \int\limits_{\varOmega \backslash K} {W\left( {F\left( x \right)} \right){\text{d}}x + \gamma {\mathcal{H}}^{N - 1} \left( K \right)}$$
(1)
where f is the body deformation, K \(\subset \overline{\varOmega }\) is the fractured zone, W: \({\mathbb{R}}^{NxN} \to {\mathbb{R}}\) is the stored energy function of a hyperelastic material, F is the deformation gradient, \(\gamma\) is the fracture energy, and \({\mathcal{H}}^{N - 1}\) is the Hausdorff measure of K which provides the measure of the length of the crack for sufficiently regular fractured zone. The first and the second terms on the right hand side of Eq. (1) represent bulk and surface energy of the body, respectively.
Equation (1) can be minimized, so crack growth is deduced by successive minimization of energy at fixed time steps. The minimization of Eq. (1) with respect to any kinematically admissible displacement and any set of crack curves introduces a high level of complexity for the variation calculus of free-discontinuity problems, particularly due to the presence of non-smooth values of the K parameter. In fact, unless topological constrains are added, it is not usually possible to deduce compactness properties from the only information that such kind of energies are bounded. Following the methodology proposed by De Giorgi and Ambrosio (1988), we introduce K as a set of discontinuity points S
f
of the function f and set the problems in a space of discontinuous functions. The weak formulation of the energy, by replacing the term K with a set of discontinuity points S
f
of deformation in a Sobolev space SBV (\(\varOmega ;{\mathbb{R}}^{N}\)), is therefore given by
$${\mathcal{E}}\left( f \right) = \int\limits_{\varOmega \backslash K} {W\left( {\nabla f} \right){\text{d}}x + \gamma {\mathcal{H}}^{N - 1} \left( {S_{f} } \right)} .$$
(2)
The presence of the term \({\mathcal{H}}^{N - 1} \left( {S_{f} } \right)\) creates challenges in terms of finite element discretization of the functional. To overcome such challenges, Eq. (2) has been approximated, in the sense of \(\varGamma\)-convergence (Bourdin et al. 2000), by a family of numerically more tractable functionals defined over a Generalized Sobolev space GSBV. GSBV consists of all functions whose truncations are in Sobolev space \(\left( {\varOmega ;{\mathbb{R}}^{N} } \right)\) and allows extending the definition of Eq. (2) to L1 functions, which are not of bounded variation. \(\varGamma\)-convergence makes it possible to achieve a variational convergence. This means if a minimizer \(\vartheta_{\varepsilon }\) for a function \({\mathcal{F}}_{\varepsilon }\) exists for every \(\varepsilon >\) 0 and if there is a sequence \(h \mapsto \varepsilon_{h}\) such that \(\varepsilon_{h} \to\) 0 and the corresponding \(\vartheta_{{\varepsilon_{h} }}\) converges to \(\vartheta\), then \(\vartheta\) is a minimizer for \({\mathcal{F}}_{\varepsilon }\). Based on the regularized formulation of the energy function for brittle fracture problems presented by Bourdin et al. (2000), an auxiliary variable s, which is called the damage parameter, is introduced. s is a regularized representation of the fractured zone defining the discontinuity set in Eq. (2). Therefore, a functional space X can be considered which its elements are pairs (f, s). Then, we take the functional X \(\to\) [0, + ∞] defined by
$${\mathcal{F}}\left( {\varvec{f},\varvec{s}} \right) = \left\{ {\begin{array}{*{20}l} {{\mathcal{E}}\left( \varvec{f} \right)} \hfill &\quad {{\text{if}}\,f \in {\mathcal{D}}, s \equiv 1} \hfill \\ { + \infty } \hfill & \quad {\text{otherwise}} \hfill \\ \end{array} } \right.$$
(3)
with \({\mathcal{E}}\left( \varvec{f} \right)\) as in the problem, \({\mathcal{D}}\) is the domain of the functions ∈ GSBV, the problem (\({\mathcal{P}}_{0}\)) defined as min{\({\mathbf{\mathcal{F}}}\left( {\varvec{f},\varvec{s}} \right):\left( {\varvec{f},\varvec{s}} \right) \in \varvec{X}\)} is considered. Damage parameter provides a picture of the damage state of the body; for an undamaged and intact body, s is equal to 1 everywhere, while it goes to zero in proximity of the discontinuity set S
f
.
The functional formulation for a generic p > 1 is expressed in the form provided by Ambrosio and Tortorelli (1992)
$${\mathcal{F}}_{\varepsilon } \left( {f,s} \right) = \int\limits_{\varOmega } {\left( {s^{2} \left( x \right) + \kappa_{\varepsilon } } \right)W\left( {F\left( x \right)} \right){\text{d}}x} + \gamma \int\limits_{{\varOmega^{\prime}}} {\left( {\frac{{\varepsilon^{p - 1} }}{p}\left| {\nabla s\left( x \right)} \right|^{p} + \frac{c}{{\varepsilon p^{\prime}}}\left( {1 - s\left( x \right)} \right)^{p} } \right){\text{d}}x}$$
(4)
where p is the pth power of the norm of the function defined in the Sobolev space, p′ = p/(p − 1), \(c = \left( {2\mathop \int \nolimits_{0}^{1} \left( {1 - t} \right)^{p/p'} {\text{d}}t} \right)^{{ - p^{'} }}\) is the normalization constant, \(\kappa_{\varepsilon }\) is a positive regularization parameter, and \(\varepsilon\) is related to the material length scale. Bulk and surface terms are two integrations over two different domains \(\varOmega\) and \(\varOmega^{\prime}\), physical domain and logical domain, respectively. \(\varOmega^{\prime}\) is defined as open set such that
$$\varOmega \subset \varOmega^{\prime},\quad \partial_{2} \varOmega \subset \partial \varOmega^{\prime},\quad \text{int} \,\,\partial_{1} \varOmega \cap \partial \varOmega^{\prime} = {\o}$$
where \(\partial_{1} \varOmega\) and \(\partial_{2} \varOmega\) are the two disjoint parts of the boundary of \(\varOmega\), and \(\text{int} \partial_{1} \varOmega\) is the interior of \(\partial_{1} \varOmega\) relative to \(\partial \varOmega\). The choice of the size of the logical domain is made on the consideration that it has to be big enough to avoid underestimation of the fracture energy when the crack reaches the boundary \(\partial_{1} \varOmega\).
For two-dimensional problem where p = 2, the total energy formulation for the body can be represented as
$${\mathcal{F}}_{\varepsilon } \left( {f,s} \right) = \mathop \int \limits_{{\varOmega_{0} }}^{{}} \left( {s^{2} \left( x \right) + \kappa_{\varepsilon } } \right)W\left( {\nabla_{0} f\left( {x_{0} } \right)} \right){\text{d}}x_{0} + \frac{\gamma }{2}\mathop \int \limits_{{\varOmega_{0}^{'} }}^{{}} \left( {\varepsilon \left| {\nabla_{0} s\left( {x_{0} } \right)} \right|^{2} + \frac{1}{\varepsilon }\left( {1 - s\left( {x_{0} } \right)} \right)^{2} } \right){\text{d}}x_{0}$$
(5)
where \(\varOmega_{0}\) and \(\varOmega_{0}^{'}\) represent initial unfractured and stress-free configuration of the body in the physical and logical domain, respectively.
Stored Energy Formulation
Following Del Piero et al. (2007), we used an isotropic, compressible neo-Hookean type stored energy model that is defined as
$$W\left( \varvec{F} \right) = \frac{\mu }{2}\left( {{\text{tr}}\varvec{C} - 2} \right) + \varPsi \left( J \right)$$
(6)
where \(\mu\) is the Lamé’s second parameter, and C is the right Cauchy–Green tensor. In the above equation, the first term represents the classical formulation of an incompressible neo-Hookean material and the second term is a convex function that is defined as (Del Piero et al. 2007)
$$\varPsi \left( J \right) = \left\{ {\begin{array}{*{20}l} {\frac{\lambda }{2}\left( {\ln J} \right)^{2} - \mu \ln J} \hfill & {0 \le J \le {\mathscr{j}}} \hfill \\ {\frac{\lambda }{2}\left( {\ln J} \right)^{2} - \mu \ln J + \left( {\lambda \ln J - \mu } \right)\left( {\frac{J - {\mathscr{j}}}{J}} \right)} \hfill & {J \ge {\mathscr{j}}} \hfill \\ \end{array} } \right.,$$
(7)
where \(\mu\) is the Lamé’s first parameter and \({\mathscr{j}} = {\text{e}}^{{{\raise0.7ex\hbox{${\left( {\lambda + \mu } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {\lambda + \mu } \right)} \lambda }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\lambda $}}}}\). Equation (7) is directly related to surface deformation, as it is function of the Jacobian of the deformation gradient (J). As J goes to zero the stored energy function goes to infinity, penalizing the extreme compression. To account for the tension–compression asymmetry of damage behavior of material, the methodology proposed in Li et al. (2016a, b) is followed, and the energy function is decomposed into two parts; a positive part which is considered to contribute to damage, and a negative part that resists to damage:
$$W\left( \varvec{F} \right) = W^{ + } + W^{ - }$$
(8)
where
$$W^{ + } = \left( {s^{2} + k_{\varepsilon } } \right) \left. W \right|_{J > 1}$$
(9)
$$W^{ - } = \left( {1 + k_{\varepsilon } } \right) \left. W \right|_{J < 1}$$
(10)
In the above equations, it can be noticed that the damage parameter appears only in the positive part of the energy function, the part associated with the elements that increase in surface (i.e., in the elements with J > 1), the value for damage in the elements is kept as calculated. The elements that decrease in surface (i.e., the elements with J < 1) do not contribute in damage. In this way, different behaviors for tension and compression are explicitly taken into account.
Numerical Solution Strategy
An approximation solution of the minimization of Eq. (5) is achieved using an iterative procedure, shown in algorithm 1, which consists of imposing a stationary condition to one of the deformation and damage variables, while keeping the other variable fixed. For all v ∈ W1,d (\(\varOmega ,{\mathbb{R}}^{n}\)), w \(\in\) W1,d (\(\varOmega_{0}\)), we look for a deformation that satisfies the stationary condition of
$$\delta {\mathcal{F}}_{\varepsilon } \left( {\varvec{f}_{n} ,s_{n - 1} } \right)\left[ {v,0} \right] = \mathop \int \limits_{{\varOmega_{0} }}^{{}} \left( {s_{n - 1}^{2} \left( {x_{0} } \right) + k_{{\mathcal{E}}} } \right)S\left( {\nabla_{0} \varvec{f}_{n} \left( {x_{0} } \right)} \right).\nabla_{0} v\left( {x_{0} } \right){\text{d}}x_{0} = 0,$$
(11)
and then for the scalar field s stationary condition of
$$\delta {\mathcal{F}}_{\varepsilon } \left( {\varvec{f}_{n} ,s_{n} } \right)\left[ {0,w} \right] = \mathop \int \limits_{{\varOmega_{0} }}^{{}} 2W\left( {\nabla_{0} \varvec{f}_{n} } \right)s_{n} w {\text{d}}x_{0} + G\mathop \int \limits_{{\varOmega_{0}^{'} }}^{{}} \varepsilon \nabla_{0} s_{n} \cdot \nabla_{0} w - \frac{{\left( {1 - s_{n} } \right)w}}{\varepsilon } {\text{d}}x_{0} .$$
(12)
where \(\varvec{S}\left( \varvec{F} \right) = \frac{\partial }{{\partial \varvec{F}}}W\left( \varvec{F} \right)\) is the first Piola–Kirchhoff stress tensor.
By taking the updated Lagrangian formulation of Eq. (11) and its linearization (Del Piero et al. 2007) the following is obtained
$$\mathop \int \limits_{{\varOmega_{n - 1} }}^{{}} \left( {s_{n - 1}^{2} + k_{\varepsilon } } \right)\left( {\left( {\det \varvec{F}} \right)^{ - 1} \left( {\varvec{I}{ \boxtimes }\varvec{F}} \right)\left[ {\varvec{S}\left( \varvec{F} \right)} \right] + \left( {\det \varvec{F}} \right)^{ - 1} \left( {\varvec{I}{ \boxtimes }\varvec{F}} \right)\frac{{\partial S\left( \varvec{F} \right)}}{\partial F}\left( {\varvec{I}{ \boxtimes }\varvec{F}} \right)^{T} \left[ {\nabla \overline{u}_{n} } \right]} \right) \cdot \nabla v\,{\text{d}}x = 0$$
(13)
where using Eq. (7)
$${\mathbf{S}}\left( \varvec{F} \right) = \left\{ {\begin{array}{*{20}l} {\mu \varvec{F} + \left( {\lambda \ln J - \mu } \right)\varvec{F}^{ - T} } \hfill & {0 \le J \le {\mathscr{j}}} \hfill \\ {\mu \varvec{F} + \lambda {\text{e}}^{{ - \left( {\lambda + \mu } \right)/\lambda }} J\varvec{F}^{ - T} } \hfill & {J \ge {\mathscr{j}}} \hfill \\ \end{array} } \right.$$
(14)
Applying the integration by parts, the final weak form of Eq. (12) is derived as
$$\mathop \int \limits_{{\varOmega_{0} }}^{{}} 2W\left( {\nabla_{0} \varvec{f}_{n} } \right)s_{n} w\,{\text{d}}x_{0} - G\mathop \int \limits_{{\varOmega_{0}^{'} }}^{{}} \left( {\varepsilon \Delta_{0} s_{n} - \frac{{1 - s_{n} }}{\varepsilon }} \right)w\,{\text{d}}x_{0} = 0$$
(15)
MATLAB Partial Differential Equation Toolbox together with the Newton–Raphson iteration scheme is used to solve the above equations. The iteration stops when two consecutive pairs of solution \(\left( {f_{n - 1} ,s_{n - 1} } \right)\) and \(\left( {f_{n} ,s_{n} } \right)\) are close enough according to an identified convergence criterion. In order to avoid the healing of the cracks, an approximation method was used to consider irreversibility condition for damage evolution. We followed the methodology proposed by Del Piero et al. (2007). Based on that, irreversibility condition of \(s_{n} \left( x \right) = s_{n - 1} \left( x \right)\) if \(s_{n} \left( x \right) > s_{n - 1} \left( x \right)\) is set, and the value of damage parameter associated with each point in the body cannot exceed the one calculated at the previous time step. We leave the development of more advanced and rigorous methods for incorporating the irreversibility condition to future works.
An adaptive h refinement strategy (Del Piero et al. 2007) is used to automatically refine the elements with values of \(s\) lower than the given thresholds. At new nodes generated through the remeshing strategy, the values of displacement and damage are calculated by linear interpolation from the existing nodes.
Stochastic Approach
In order to incorporate the effects of material heterogeneity in the computational model, a statistical technique is employed. The Weibull distribution function (Weibull 1939) is used to generate random distributions of the properties, as it has a simple structure and its applicability for modeling failure of brittle materials has been verified (Fang and Harrison 2002; Yang and Xu 2008; Gorjan and Ambrožič 2012). In this study, the fracture energy is considered as a random variable.
Cumulative density function (CDF) and probability density function (PDF) for Weibull distribution, plotted in Fig. 2, take the form of
$$P\left( \xi \right) = 1 - \exp \left( { - \left( {\frac{\xi }{{\xi_{0} }}} \right)^{m} } \right)$$
(16)
$$p\left( \xi \right) = \frac{{{\text{d}}P\left( \xi \right)}}{\xi } = \frac{m}{{\xi_{0} }}\left( {\frac{\xi }{{\xi_{0} }}} \right)^{m - 1} \exp \left( { - \left( {\frac{\xi }{{\xi_{0} }}} \right)^{m} } \right)$$
(17)
where \(\xi\) is the random parameter, \(m\) is the shape parameter, and \(\xi_{0}\) is the scale parameter.
In order to define both shape and scale parameters, maximum likelihood estimator (MLE) is used. Let \(\xi\)1, \(\xi\)2, … \(\xi\)
n
be a random sample of size \(n\) with a PDF \(f_{{\xi_{i} }} \left( {\xi_{i} ,m,\xi_{0} } \right)\), where \(\xi_{0}\) and \(m\) are unknown parameters. The method considers the joint density function for all observations. For an independent sample with known \(\xi_{0}\) and \(m\) likelihood, function \(L\left( {\xi_{0} , m , \xi_{1} , \ldots \xi_{n} } \right)\) is the joint density function of the \(n\) random variables defined as
$$L\left( {\xi_{0} , m , \xi_{1} , \ldots \xi_{n} } \right) = f\left( {\xi_{1} ,\xi_{2} , \ldots \xi_{n} |\xi_{0} ,m} \right) = \mathop \prod \limits_{i = 1}^{n} f_{{x_{i} }} \left( {\xi_{i} |\xi_{0} ,m} \right)$$
(18)
On the basis of the MLE, the maximum likelihood of \(\xi\) is achieved by maximizing \(L\) or, for convenience, its logarithm
$$\frac{{{\text{dLog}}L}}{{{\text{d}}m}} = 0,\quad \frac{{{\text{dLog}}L}}{{{\text{d}}\xi_{0} }} = 0.$$
(19)
In order to apply the MLE to estimate the Weibull parameters, we can substitute Eq. (17) into Eq. (18) and apply Eq. (19); after a straightforward calculation one gets the following pair of equations
$$\frac{{\mathop \sum \nolimits_{i = 1}^{N} \ln \left( {\xi_{i} } \right)\xi_{i}^{m} }}{{\mathop \sum \nolimits_{i = 1}^{N} \xi_{i}^{m} }} - \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \ln \left( {\xi_{i} } \right) - \frac{1}{m} = 0$$
(20a)
$$\xi_{0} = \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \xi_{i}^{m} }}{N}} \right)$$
(20b)
Equation (20a) is solved using the Newton–Raphson method to calculate the value of \(m\) as
$$m_{n + 1} = m_{n} - \frac{{f\left( {m_{n} } \right)}}{{f^{\prime}\left( {m_{n} } \right)}}$$
(21)
where \(n\) is the Newton–Raphson iteration number,
$$f\left( {m_{n} } \right) = {{\mathop \sum \limits_{i = 1}^{n} \xi_{i}^{m} \ln \left( {\xi_{i} } \right)} \mathord{\left/ {\vphantom {{\mathop \sum \limits_{i = 1}^{n} \xi_{i}^{m} \ln \left( {\xi_{i} } \right)} {\mathop \sum \limits_{i = 1}^{N} \xi_{i}^{m} - \frac{1}{m} - \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \ln \left( {\xi_{i} } \right)}}} \right. \kern-0pt} {\mathop \sum \limits_{i = 1}^{N} \xi_{i}^{m} - \frac{1}{m} - \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \ln \left( {\xi_{i} } \right)}}$$
(22)
and
$$f^{\prime}\left( {m_{n} } \right) = \mathop \sum \limits_{i = 1}^{n} \xi_{i}^{m} \left( {\ln \xi_{i} } \right)^{2} - \frac{1}{{m^{2} }}\mathop \sum \limits_{i = 1}^{n} \xi_{i}^{m} \left( {m \ln \left( {\xi_{i} } \right) - 1} \right) - \left( {\frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \ln \left( {\xi_{i} } \right)} \right)\left( {\frac{1}{N}\mathop \sum \limits_{i = 1}^{n} \xi_{i}^{m} \ln \left( {\xi_{i} } \right)} \right)$$
(23)
The calculated value for \(m\) is substituted into Eq. (20b) in order to obtain the scale parameter \(\xi_{0}\) and define the Weibull distribution. Then, the inverse cumulative distribution function is used to generate random realizations over the simulation domain that is derived as
$$P^{ - 1} \left( R \right) = \xi_{i} = - \xi_{0} \sqrt[m]{{\ln \left( {1 - \Re } \right)}}$$
(24)
where \(\Re\) is a random number between 0 and 1 representing the probability of an occurrence. The full stochastic procedure is summarized in Algorithm 2.
The scale parameter can be related to the size of the elements forming the material. It has been shown in Guy et al. (2012) that, for a specific range of material length scale and crack length over a domain, the material length scale itself can be used as scale parameter of the Weibull distribution. In our work, the material length scale is accounted in the generation of the hypothetical sample used successively to generate the shape and scale parameters by means of the MLE method. In this way, material length scale has been implicitly taken into account in the model.