Challenges, Tools and Applications of Tracking Algorithms in the Numerical Modelling of Cracks in Concrete and Masonry Structures

Abstract

The importance of crack propagation in the structural behaviour of concrete and masonry structures has led to the development of a wide range of finite element methods for crack simulation. A common standpoint in many of them is the use of tracking algorithms, which identify and designate the location of cracks within the analysed structure. In this way, the crack modelling techniques, smeared or discrete, are applied only to a restricted part of the discretized domain. This paper presents a review of finite element approaches to cracking focusing on the development and use of tracking algorithms. These are presented in four categories according to the information necessary for the definition and storage of the crack-path. In addition to that, the most utilised criteria for the selection of the crack propagation direction are summarized. The various algorithmic issues involved in the development of a tracking algorithm are discussed through the presentation of a local tracking algorithm based on the smeared crack approach. Challenges such as the modelling of arbitrary and multiple cracks propagating towards more than one direction, as well as multi-directional and intersecting cracking, are detailed. The presented numerical model is applied to the analysis of small- and large-scale masonry and concrete structures under monotonic and cyclic loading.

Appendix: Continuum Damage Mechanics Model

1.1 Constitutive Equation

The constitutive model used in this work is a continuum damage model that distinguishes between tensile ($d^+$) and compressive damage ($d^-$). The constitutive equation is

$$\begin{aligned} {\varvec{\sigma }}= (1-d^+) \, {\bar{\varvec{\sigma }}}^{+}+ (1-d^-) \, {\bar{\varvec{\sigma}}}^{-}. \end{aligned}$$

(25)

The effective stresses ($\bar{\varvec{\sigma }}$) are computed adopting a strain equivalent hypothesis [284, 285] as

$$\begin{aligned} \bar{\varvec{\sigma }}&= \varvec{C_0}:\varvec{\varepsilon }^e \end{aligned}$$

(26)

$$\begin{aligned} \varvec{\varepsilon }^e&= \varvec{\varepsilon }- \varvec{\varepsilon }^i \end{aligned}$$

(27)

where $\varvec{C_0}$ is the 4th order isotropic elastic constitutive tensor, while $\varvec{\varepsilon }$, $\varvec{\varepsilon }^e$ and $\varvec{\varepsilon }^i$ are second order tensors representing the total, the elastic and the irreversible strains, respectively. The split of the effective stress tensor into a positive ($\varvec{\bar{\sigma }}^{+}$) and a negative part ($\varvec{\bar{\sigma }}^{-}$) is performed according to Faria et al. [13, 286] as

$$\begin{aligned} \bar{\varvec{\sigma }}^{+}&= \sum _{j=1}^{3} \langle \bar{\sigma }_j \rangle \, {\mathbf {p_j}} \otimes {\mathbf {p_j}} \end{aligned}$$

(28)

$$\begin{aligned} \varvec{\bar{\sigma }}^{-}&= \bar{\varvec{\sigma }}- \varvec{\bar{\sigma }}^{+}. \end{aligned}$$

(29)

In the above equations, $\bar{\sigma }_j$ is the principal effective stress corresponding to the eigenvector ${\mathbf {p_j}}$ of the effective stress tensor and the symbols $\langle \cdot \rangle$ are the Macaulay brackets ($\langle x \rangle =x$, if $x \ge 0$ ,$\langle x \rangle =0$, if $x < 0$).

1.2 Damage Criteria

Loading, unloading and reloading conditions are distinguished with the use of two scalar positive quantities, one for tension $\tau ^+$ and a second for compression $\tau ^-$, termed as equivalent stresses. Their values are defined according to the following functions proposed by Petracca et al. [287, 288]

$$\begin{aligned} \tau ^+&= H_0\left[ \bar{\sigma }_{max} \right] \left[ \frac{1}{1-a} \left( a \bar{I_1} + \sqrt{3\bar{J_2}} + b \langle \bar{\sigma }_{max} \rangle \right) \frac{f^+}{f^-}\right] \end{aligned}$$

(30)

$$\begin{aligned} \tau ^-&= H_0\left[ -\bar{\sigma }_{min} \right] \left[ \frac{1}{1-a} \left( a \bar{I_1} + \sqrt{3 \bar{J_2}} + \kappa _1 b \langle \bar{\sigma }_{max} \rangle \right) \right] \end{aligned}$$

(31)

$$\begin{aligned} a&= \frac{\left( f^-_{b}/f^- \right) -1 }{2 \left( f^-_{b}/f^- \right) -1} \end{aligned}$$

(32)

$$\begin{aligned} b&= \left( 1 - a \right) f^-/f^+ - \left( 1 + a \right) . \end{aligned}$$

(33)

In the above, $f^+$ and $f^-$ stand for the tensile and compressive uniaxial strengths, respectively, and $f_{b}^-$ for the biaxial compressive strength. $\bar{I_1}$ is the first invariant of the effective stress tensor and $\bar{J_2}$ the second invariant of the deviatoric effective stress tensor. The $\kappa _1$ variable in Eq. (31) was introduced in [287, 288] as a way to control the shape of the compressive damage surface in the shear quadrants, and through this the dilatant behaviour of the material under shear stress states. Its value varies between 0 (i.e. the Drucker-Prager criterion) and 1 (i.e. the criterion proposed by Lubliner et al. [15]). Finally, $\bar{\sigma }_{max}$ and $\bar{\sigma }_{min}$ designate the maximum and minimum principal effective stresses respectively, whereas $H_0$ is the specific Heaviside function ($H_0[x] = 1 \, \text {for}\, x > 0 \, \text {and} \, H_0[x] =0 \, \text {for}\, x \le 0$).

The evolution of tensile and compressive damage is controlled with two damage criteria ($\varPhi ^\pm$), which are defined as

$$\begin{aligned} \varPhi ^\pm (r^\pm ,\tau ^\pm ) = \tau ^\pm - r^\pm \le 0. \end{aligned}$$

(34)

The stress thresholds ($r^\pm$) are internal stress-like variables representing the current damage threshold. Their initial values are equal to the uniaxial tensile and compressive strength at the moment of damage initiation $r_0^\pm = f^\pm$. After damage is triggered, both thresholds become equal to the maximum attained values by the equivalent stresses and can be explicitly computed for a generic time instant t as

$$\begin{aligned} r^\pm _t = \max \left[ r_0^\pm , \max \limits _{i \, \in \, (0,t)} \left( \tau ^\pm _i \right) \right] . \end{aligned}$$

(35)

1.3 Irreversible Strains

The irreversible strains are considered as an internal variable with the following evolution law proposed in [116]

$$\begin{aligned} \Delta \varvec{\varepsilon }^i_{n+1}= \beta \frac{r(\tilde{\bar{\varvec{\sigma }}}_{n+1})-r_n}{r_{n+1}} \varvec{\varepsilon }^e_{{n+1}}. \end{aligned}$$

(36)

where $\tilde{\bar{\varvec{\sigma }}}_{n+1}$ are the trial effective stresses and $r(\tilde{\bar{\varvec{\sigma }}}_{n+1})$ the equivalent stress threshold computed as

$$\begin{aligned} \tilde{\bar{\varvec{\sigma }}}_{n+1}&= (\bar{\varvec{\sigma }}|_{\Delta \varvec{\varepsilon }^i= 0})_{n+1}= \bar{\varvec{\sigma }}_n+ \varvec{C_0}:\Delta \varvec{\varepsilon }_{n+1} \end{aligned}$$

(37)

$$\begin{aligned} r(\tilde{\bar{\varvec{\sigma }}}_{n+1})&= \max \left[ r_n, \tau (\tilde{\bar{\varvec{\sigma }}}_{n+1}) \right] . \end{aligned}$$

(38)

The parameter $\beta = [0,1]$ is used to determine the magnitude of the incremental irreversible strains. For $\beta = 0$ no increment of irreversible strains is considered, while for $\beta = 1$ the total strain increment is irreversible. Numerically, the effect of the irreversible strains is considered through the update of the effective stresses according to the following expressions (see [116, 260])

$$\begin{aligned} \bar{\varvec{\sigma }}_{n+1}&= \lambda \, \tilde{\bar{\varvec{\sigma }}}_{n+1} \end{aligned}$$

(39)

$$\begin{aligned} \lambda&= 1 - \beta \left( 1 - \frac{r_n}{r(\tilde{\bar{\varvec{\sigma }}}_{n+1})} \right) . \end{aligned}$$

(40)

1.4 Damage Variables

The damage variables are defined according to the exponential softening law proposed in [289]

$$\begin{aligned} d^\pm = 1 - \frac{r_0^\pm}{r^\pm} \exp \Bigg \{ 2 H_d^\pm \left( \frac{r_0^\pm - r^\pm}{r_0^\pm} \right) \Bigg \} \qquad r^\pm \ge r_0^\pm. \end{aligned}$$

(41)

Tension and compression evolution laws consider the positive $G_f^+$ and negative $G_f^-$ fracture energies, respectively, as well as the characteristic finite element width $l_{dis}$ through the corresponding discrete softening parameter $H_d^\pm$ ensuring mesh-size independent energy dissipation according to the crack-band theory [5]. For the case of tension this is

$$\begin{aligned} H_d^+ = \frac{l_{dis}}{l_{mat}^+ - l_{dis}} \end{aligned}$$

(42)

For the case of compressive damage, the expression proposed in [116, 260] is used to compute the discrete softening parameter ${H}_d^-$

$$\begin{aligned} H_d^- = \frac{1}{1-\beta } \left( \frac{l_{dis}}{l_{mat}^- - l_{dis}}\right) \end{aligned}$$

(43)

The above definition is consistent with the crack band-width approach yielding objective results for different values of $\beta$ by considering the contributions to the dissipated energy due to the evolutions of the irreversible strains and the compressive damage. In the above, the material characteristic length $l_{mat}^\pm$ for tension and compression is

$$\begin{aligned} l_{mat}^\pm = \frac{2E G_f^\pm }{{(f^\pm )}^2}. \end{aligned}$$

(44)

The current work considers the use of constant strain triangles for which the characteristic finite element length has been considered as $l_{dis}=\sqrt{2A_{fe}}$, with $A_{fe}$ representing the area of the finite element. This is a standard procedure that can be refined to consider the crack direction and the finite element size according to references [110, 290, 291].