Boundary element method: cells with embedded discontinuity modeling the fracture process zone in 3D failure analysis

Abstract

Damage and failure in quasi-brittle materials such as rocks, concrete, and ceramics, have a complex non-linear behavior due to their heterogeneous character and the development of a fracture process zone (FPZ), formed by micro-cracking around the tip of an induced or pre-existing flaw. A softening behavior is observed in the FPZ, and the linear elastic fracture mechanic (LEFM) cannot correctly reproduce the stress field ahead of the crack tip. The existence of the FPZ may be the intrinsic cause of the size effect. An appropriate modeling of this process zone is mandatory to reproduce accurately the failure propagation and consequently, the structural behavior. Different from most of the domain numerical techniques, the boundary element method (BEM) requires (besides the boundary division into elements) only the discretization of a small region where dissipative effects occur. Cells with embedded continuum strong discontinuity approach (CSDA), placed in the region where the crack is supposed to occur, are capable of capturing the transition of regimes in the failure zone. Numerical bifurcation analysis, based on the singularity of the localization tensor, is used to determine the end of the continuum regime. Weak and strong discontinuity regimes are associated with diffuse micro-cracks (strain discontinuity) and macro-crack (displacement discontinuity). A variable bandwidth model is used during the weak discontinuity regime to represent the advance of micro-cracks density and their coalescence. Continuum and discrete cohesive isotropic damage models are used to describe the softening behavior. Analysis of three-dimensional problems with single crack in standard and mixed fracture modes, using this transitional approach and the BEM cells is firstly presented in this work. Experimental reference results are used to attest the capability of the approach in reproducing the structural behavior during crack propagation. Some necessary advances required for its applications for general complex structural problems are pointed out.

Appendix A: Interface constitutive models

1.1 Isotropic damage constitutive model

An isotropic damage constitutive model, in which damage evolution occurs preferentially under traction states, is used in this work. This is used to represent damage dissipation in finite regions of a solid domain, over the discontinuity surface. This model, used in Oliver et al. (2006) and Peixoto et al. (2018), is briefly stated by the following expressions:

$$\begin{aligned}&\text {Free energy:} \psi (\epsilon _{ij},r) = \big [1-D(r)\big ]\psi _o(\epsilon _{ij}) ,{\hspace{7pt}}\nonumber \\&\quad \psi _o(\epsilon _{ij}) = \dfrac{1}{2} \epsilon _{ij} E^{o}_{ijkl} \epsilon _{kl} \end{aligned}$$

(A1a)

$$\begin{aligned}&\text {Constitutive equation:} \hspace{10pt} \sigma ^{{\mathcal {S}}}_{ij} = \dfrac{\partial \psi (\epsilon _{ij},r)}{\partial \epsilon _{ij}} \nonumber \\&\quad = (1-D)E^{o}_{ijkl}\epsilon _{kl} = E_{ijkl}\epsilon _{kl} \end{aligned}$$

(A1b)

$$\begin{aligned}&\text {Damage variable:} \hspace{10pt} D \equiv D(r) = 1 - \dfrac{q(r)}{r} \text {,} D \in [0,1] \end{aligned}$$

(A1c)

$$\begin{aligned}&\text {Internal variable evolution law:} \hspace{10pt} \nonumber \\&\quad {\dot{r}} \text {,} \hspace{10pt} {\left\{ \begin{array}{ll} r \in [r_{o}, \infty ) \text {,} \\ r_{o} = r|_{t=0} = \frac{f_{t}}{\sqrt{E}} \end{array}\right. } \end{aligned}$$

(A1d)

$$\begin{aligned}&\text {Damage criterion in strain space } {\mathbb {E}}_\epsilon \text {:} \nonumber \\&\quad \hspace{10pt} {\bar{F}}(\epsilon _{ij}, r) \equiv \tau _{\epsilon } - r \end{aligned}$$

(A1e)

$$\begin{aligned}&\text {Loading-unloading conditions:} \hspace{10pt}\nonumber \\&\quad {\bar{F}} \leqslant 0 \text {,} \hspace{7pt} {\dot{r}} \geqslant 0 \text {,} \hspace{7pt} {\dot{r}}{\bar{F}} = 0 \text {,} \hspace{7pt} {\dot{r}}\dot{{\bar{F}}} = 0 \end{aligned}$$

(A1f)

$$\begin{aligned}&\text {Softening law :} \hspace{10pt} {\dot{q}} = H(r){\dot{r}} \text {,} \hspace{10pt} (H=q'(r) \leqslant 0) \text {,} \hspace{10pt}\nonumber \\&\quad {\left\{ \begin{array}{ll} q \in [0,r_{o}] \text {,} \\ q|_{t=0} = r_{o} \end{array}\right. } \end{aligned}$$

(A1g)

where r is the strain-like scalar internal variable, q is the stress-like internal variable, H is the hardening/softening modulus, $E_{ijkl}$ is the secant constitutive tensor. The value $r_{o}$ is the threshold of the initial elastic domain, characterized in terms of the uniaxial elastic strength $f_t$ and the elasticity modulus E.

For the damage criterion (Eq. A1e), this work uses the equivalent strain expression presented in Oliver et al. (2006). This model seems to be suitable for the representation of brittle or quasi-brittle materials, since it allows the degradation to occur only in tensile states. The expression is given as:

$$\begin{aligned} \tau _{\epsilon } = \sqrt{\epsilon ^{+}_{ij}E^{o}_{ijkl}\epsilon _{kl}} \end{aligned}$$

(A2)

in which, the tensor $\epsilon ^{+}_{ij}$, referring to a coordinate system aligned with the principal strain directions, is given by

$$\begin{aligned} \varvec{\epsilon ^+} = \sum _{k=1}^{3} \langle \epsilon _{k} \rangle {\hat{\textbf{e}}}_{k} \otimes {\hat{\textbf{e}}}_{k} \end{aligned}$$

(A3)

where $\epsilon _{k}$ represents the k-th principal strain, and ${\hat{\textbf{e}}}_k$ is the unit vector in the corresponding principal direction, and $\langle \cdot \rangle$ refers to the Macaulay brackets.

Regarding the relationship between stress and strain rates, it is given through a constitutive tangent tensor $E^t_{ijkl}$ as follows:

$$\begin{aligned} {\dot{\sigma }}^{{\mathcal {S}}}_{ij} = E_{ijkl}^{t} {\dot{\epsilon }}_{kl} = E_{ijkl} {{\dot{\epsilon }}}_{kl} + {\dot{E}}_{ijkl} \epsilon _{kl} \end{aligned}$$

(A4)

For unloading (or neutral load) ${\dot{D}}={\dot{r}}=0$, then $E_{ijkl}^{t} = E_{ijkl}$. For inelastic loading ${\dot{r}}=\dot{\tau _{\epsilon }}$, the expression for the constitutive tangent tensor is given by:

$$\begin{aligned} {E}_{ijkl}^t = E_{ijkl} - \left( \dfrac{\partial D}{\partial r} \right) \left( \dfrac{\partial \tau _\epsilon }{\partial \epsilon _{kl}} \right) {E}_{ijrs}^o \epsilon _{rs} \end{aligned}$$

(A5)

1.2 Cohesive constitutive model associated to the continuum isotropic damage

The strong discontinuity kinematics, when applied to a continuum constitutive model ($\sigma _{ij} - \epsilon _{ij}$) equipped with a softening rule, induces a corresponding cohesive (or discrete) model. Therefore, for an instant after the strong discontinuity onset, $t = t_{SD}$, a relationship between stresses ($\sigma _{ij}$) with discontinuity traction components ($t_i$), and strains ($\epsilon _{ij}$) with displacement jumps in ${\mathcal {S}}$ ($\Delta [\![u_i]\!]= [\![u_i]\!]({\textbf{x}},t) - [\![u_i]\!]({\textbf{x}},t_{SD})$). A complete analysis that leads to the following expressions can be seen in Oliver (2000). The expressions are given in the sequence:

$$\begin{aligned}&\text {Free energy:} \hspace{10pt} {\left\{ \begin{array}{ll} {\hat{\psi }} (\Delta [\![u_i]\!],\omega ) = [1 - \omega (\Delta \alpha )]{\hat{\psi }}_o(\Delta [\![u_i]\!]) \text {;} \\ {\hat{\psi }}_o (\Delta [\![u_i]\!]) = \dfrac{1}{2} \Delta [\![u_i]\!] Q^{e}_{ij} \Delta [\![u_j]\!] \end{array}\right. } \end{aligned}$$

(A6a)

$$\begin{aligned}&\text {Constitutive equation:} \hspace{10pt} t_i = \dfrac{\partial {\hat{\psi }} (\Delta [\![u_i]\!],\omega )}{\partial (\Delta [\![u_i]\!])}\nonumber \\&\quad = (1 - \omega ) Q^{e}_{ij} \Delta [\![u_j]\!] \end{aligned}$$

(A6b)

$$\begin{aligned}&\text {Damage variable:} \hspace{10pt} \omega \equiv \omega (\Delta \alpha ) \nonumber \\&\quad = 1 - \dfrac{q^*(\Delta \alpha )}{\Delta \alpha } \text {,} \hspace{10pt} \omega \in (-\infty ,1] \end{aligned}$$

(A6c)

$$\begin{aligned}&\text {Internal variable evolution law:} \hspace{10pt} \dfrac{\partial (\Delta \alpha )}{\partial t}\nonumber \\&\quad = {\dot{\alpha }} \text {,} \hspace{10pt} \Delta \alpha \in [0, \infty ) \end{aligned}$$

(A6d)

$$\begin{aligned}&\text {Damage criterion:} \hspace{7pt} {\bar{G}}(\Delta [\![u_i]\!],\Delta \alpha ) \equiv \tau _{\Delta [\![u]\!]} - \Delta \alpha \nonumber \\&\quad = \sqrt{\Delta [\![u_i]\!] Q^{e}_{ij} \Delta [\![u_j]\!]} - \Delta \alpha \end{aligned}$$

(A6e)

$$\begin{aligned}&\text {Loading-unloading conditions:} \hspace{10pt} {\bar{G}} \leqslant 0 \text {,} {\dot{\alpha }} \geqslant 0 \text {,} {\dot{\alpha }}{\bar{G}}\nonumber \\&\quad = 0 \text {,} {\dot{\alpha }}\dot{{\bar{G}}} = 0 \end{aligned}$$

(A6f)

$$\begin{aligned}&\text {Softening law:} \hspace{8pt} {\dot{q}}^* = H^*{\dot{\alpha }} \text {,} \nonumber \\&\quad \hspace{8pt} (H^* = \frac{1}{h}H < 0) \text {,} \hspace{6pt} {\left\{ \begin{array}{ll} q^* \in [0,q_{SD}] \text {,} \\ q^*|_{t=t_{SD}} = q_{SD} \end{array}\right. } \end{aligned}$$

(A6g)

The correspondence between parameters, from the continuum model and from the cohesive model, is shown in Table 2 where $Q^{e}_{ij}$ is the elastic localization tensor ($Q^{e}_{jk} = n_i E^{o}_{ijkl} n_l$); $\Delta \alpha$ is the internal cohesive variable, from the onset of the strong discontinuity ($\alpha |_t - \alpha |_{t_{SD}}$); $\omega (\Delta \alpha )$ is the cohesive damage variable; $q^*(\Delta \alpha )$ is the softening law; $H^*$ is the cohesive softening modulus; and ${\bar{G}}$ represents the damage criterion.