Smooth Generalized/eXtended FEM approximations in the computation of configurational forces in linear elastic fracture mechanics

Abstract

The computation of crack severity parameters in the linear elastic fracture mechanics (LEFM) modeling is strongly dependent on the local quality of the approximated stress fields right at the crack tip vicinity. This work investigates the behavior of extrinsically enriched smooth mesh-based approximations, obtained via $C^{k}$-GFEM framework (Duarte et al. in Comput Methods Appl Mech Eng 196:33–56, 2006), in the computation of $\mathcal {J}$-integral in both pure mode I and mixed-mode loadings for two-dimensional problems of the LEFM. The method of configurational forces is used for this purpose as shown in Steinmann et al. (Int J Solids Struct 38:5509–5526, 2001), for instance, by performing some adaptations according to Häusler et al. (Int J Numer Methods Eng 85:1522–1542, 2011). As such method provides vector quantities, it is also possible to compute the angle $\theta _{{\mathrm{ADV}}}$ of probable crack advance. The $C^{k}$-GFEM is quite versatile and shares similar features with the standard FEM regarding the domain partition and numerical integration (Mendonça et al. in Finite Elem Anal Des 47:698–717, 2011). The tests were conducted using three-noded triangular element meshes and numerical integrations were performed using only global coordinates. The evaluations combined different schemes of polynomial and discontinuous/singular (Moës et al. in Int J Numer Methods Eng 46:131–150, 1999) enrichments. The use of a smooth partition of unity (PoU) can influence the accuracy of computed crack severity parameters. The configurational forces computation is favored by the smoothness, reducing the dependence on the way the crack severity parameters are evaluated.

Similarities between the material force method and conventional ways to compute crack severity parameters

In LEFM, the stress field around a crack tip can be mainly described in terms of a single parameter, the stress intensity factor, which depends on the opening modes (Anderson 2005; Kuna 2013). The direct methods, also called field variable methods, compute the stress intensity factors from displacement or stress extrapolations near the crack tip (see Qian et al. (2016) for a list of references about such methods). There are other methods which consider the stress solution globally, i.e., by integral quantities related to the energy of the solution and, hence, are classified as energetic or indirect methods. These last methods are designed to compute the energy release rate $\mathcal {G}$, and then the stress intensity factors are obtained from it.

One of the widely used energetic method is the J-integral. The classical $\mathcal {J}$-integral, in its contour version (Cherepanov 1967; Rice 1968), provides the energy release rate by a path-independent integral, applicable to both linear and non-linear problems. For 3D problems, the contour integral becomes a surface integral and its computational implementation may involve some difficulties.

A well-known modification to simplify the computation of the $\mathcal {J}$-integral is its domain version (Shih et al. 1986; Moran and Shih 1987), called Equivalent Domain Integral (EDI) method, which is applicable to time dependent or independent problems with linear or non-linear stress-strain relationships. The EDI is expressed as

$$\begin{aligned} \mathcal {J} = \int _{\varOmega ^{*}} \left( \sigma _{ij} \dfrac{ \partial u_{j} }{ \partial x_{1} } - \mathfrak {W} \ \delta _{1i} \right) \dfrac{ \partial q }{ \partial x_{1} } d \varOmega \end{aligned}$$

(37)

using indicial notation, with $\sigma _{ij}$ being cartesian components of stress, $u_{j}$ being the displacement components and $\mathfrak {W}$ the strain energy density per unit of volume. In (37), the 1-direction is tangent to the crack faces, and the function q is an arbitrary, but smooth, scalar function that should vary from 0 to 1, going from the outer boundary to the inner one of $\varOmega ^{*}$. Notably, its implementation is simpler than the original version of Rice (1968), even in 3D problems, it is theoretically insensitive to the actual selection of the prescribed virtual material displacement and to the integration domain (Steinmann et al. 2001). In contrast, in GFEM / XFEM implementations, the results are generally sensitive to the enrichment strategy and to the selection of extraction domains (Qian et al. 2016). It is interesting to note the similarity between the terms in parentheses in (37) and (18), even though different notations have been used in these equations.

The EDI is strongly related to the prior Virtual Crack Extension (VCE) method, in a continuum form (deLorenzi 1982, 1985; Lin and Abel 1988), in which the energy release rate $\mathcal {G}$ is obtained by

$$\begin{aligned} \mathcal {G} = \dfrac{ 1 }{ \varDelta a } \int _{\varOmega ^{*}} \left( \sigma _{ij} \dfrac{ \partial u_{j} }{ \partial x_{1} } - \mathfrak {W} \ \delta _{1i} \right) \dfrac{ \partial \varDelta x_{1} }{ \partial x_{i} } d \varOmega \end{aligned}$$

(38)

where $\varDelta a$ is the virtual crack extension and $\varDelta x_{1}$ is the material point translation, that vary from 0 to $\varDelta a$ between the outer and inner boundary of $\varOmega ^{*}$, the integration domain. Additionally, $\partial \varDelta x_{1} / \partial x_{i} = 0$ outside this domain. The VCE (Hwang et al. 1998) is an attempt to generalize the Stiffness Derivative (SD) method (Parks 1974; Hellen 1975).

In the SD method, in turn, the energy release rate is obtained by the change in the global potential energy due to an increment in the crack extension, explicitly in terms of a discretized formulation. The total potential energy of a discretized system $\widetilde{\varPi } = \varPi ( \widetilde{\varvec{u}} )$, given by

$$\begin{aligned} \widetilde{\varPi } = \dfrac{1}{2} \varvec{U}^{T} \varvec{K} \varvec{U} - \varvec{U}^{T} \varvec{F} \end{aligned}$$

(39)

is differentiated with respect to the crack extension a, considering fixed load (Irwin 1956)

$$\begin{aligned} \mathcal {G}= & {} - \left( \dfrac{ \partial \widetilde{\varPi } }{ \partial a } \right) _{load} = - \dfrac{ \partial \varvec{U}^{T} }{ \partial a } \left\{ \varvec{K} \varvec{U} - \varvec{F} \right\} \nonumber \\&- \dfrac{1}{2} \varvec{U}^{T} \dfrac{ \partial \varvec{K} }{ \partial a } \varvec{U} + \varvec{U}^{T} \dfrac{ \partial \varvec{F} }{ \partial a } \end{aligned}$$

(40)

where the first term on the right hand side is zero due to the equilibrium condition, and considering absence of tractions on the crack faces the third term also vanishes. The vector $\varvec{U}$ contains the nodal parameters related with the displacement field [see (10)], and $\varvec{K}$ and $\varvec{F}$ are the stiffness matrix and the load vector, respectively, resulting from the computation of (7) and (8) in the discretized version.

Hence, the energy release rate $\mathcal {G}$ is given by

$$\begin{aligned} \mathcal {G} = - \dfrac{1}{2} \varvec{U}^{T} \dfrac{ \partial \varvec{K} }{ \partial a } \varvec{U} \end{aligned}$$

(41)

noting that the minus signal is in accordance with the Griffith’s physical reasoning (Rice 1968) since such portion of potential energy goes into dissipation, and, therefore, the crack cannot self-repair (Maugin 1993, 1995).

Banks-Sills and Sherman (1992) and Giner et al. (2002) showed the equivalence between the Stiffness Derivative Method and the Equivalent Domain Integral method (EDI) implemented with isoparametric finite elements.

Although the energetic methods yield more accurate estimates, one major drawback related to them is the need of uncoupling the strain energy release rate contributions from each crack opening mode in mixed-mode problems. In this regard, field decomposition technique (Ishikawa et al. 1979) consists in separating the displacement and stress fields into their symmetric and antisymmetric parts, in order to uncouple the modes I and II of a mixed-mode problem in LEFM.

The EDI method yields good results, but Giner et al. (2002) showed that the SD method gives more accurate estimates. The reason may be due to the fact that SD method only explicitly takes into account the discretized displacement field. The EDI method needs both the decomposition of displacements and stress, whereas the discrete analytical approach of the SD method of Giner et al. (2002) needs only the decomposition of the displacement field and performs better in computations of stress intensity parameters when using arbitrary meshes.

As the VCE concept, the SD can be interpreted as a Shape Design Sensitivity Analysis. A crack extension increment $\partial a$ in (41) and (40) may be interpreted as a change in a design variable, the crack length a, which causes a shape change and a corresponding variation of the structural response.

For the discrete semi-analytical method of Giner et al. (2002) for shape design sensitivity analysis, the sensitivity with respect to the spatial coordinates is treated as a velocity field in analogy with the continuum mechanics. Such velocity field must be as regular as the displacement field of the discretized equilibrium problem, and the velocity field is required to depend linearly on the shape design variable, the crack advance. Therefore, only the velocity component in the tangent direction of the crack faces are non zero.

In the work of Waismann (2010), the crack is embedded in the mathematical formulation of the stiffness matrix, and thus, the derivatives may be computed directly. In order to extract the mixed-mode strain energy release rate, a mutual potential representation based on Betti’s reciprocal theorem is employed.

From this observation, the node positions $\varvec{x}_{\alpha }$ in a finite element mesh can be viewed as design variables and, therefore, it is possible to apply techniques from variational design sensitivity analysis to mesh optimization (Materna and Barthold 2008). Thus, the relation between the design sensitivity analysis and the configurational mechanics may be noted (Mueller and Maugin 2002; Gurtin 2000; Braun 2005, 2007; Miehe et al. 2007; Miehe and Gürses 2007; Steinmann et al. 2009). In other words, the first sensitivity of the energy functional $\widetilde{\varPi }$ with respect to changes in the design leads to the well-known weak form of the material or configurational forces equilibrium, whereas the second sensitivity of $\widetilde{\varPi }$ provides information about the sensitivity of the energy release rate $\mathcal {G}$.

Consequently, as configurational forces $\varvec{G}_{\alpha }$ (26) are related to the energy release rate resulting from a configurational change (Mueller and Maugin 2002; Gross et al. 2002; Kienzler and Herrmann 2000; Gurtin 2000; Steinmann et al. 2009), since it can be shown that its physical meaning is

$$\begin{aligned} \varvec{G}_{\alpha } = - \dfrac{ \partial \widetilde{\varPi } }{\partial \varvec{x}_{\alpha } } \end{aligned}$$

(42)

with $\varvec{x}_{\alpha }$ being the position of the node $\alpha$. If the node position moves in the opposite direction of the configurational force the total potential energy will be smaller, then the nodal configurational force $\varvec{G}_{\alpha }$ can be understood as

$$\begin{aligned} \varvec{G}_{\alpha } = - \dfrac{ \partial }{ \partial \varvec{x}_{\alpha } } \left[ \dfrac{1}{2} \varvec{U}^{T} \varvec{K} \varvec{U} - \varvec{U}^{T} \varvec{F} \right] = - \dfrac{1}{2} \varvec{U}^{T} \dfrac{ \partial \varvec{K} }{ \partial \varvec{x}_{\alpha } } \varvec{U}.\nonumber \\ \end{aligned}$$

(43)