Tracking multi-directional intersecting cracks in numerical modelling of masonry shear walls under cyclic loading

Abstract

In-plane cyclic loading of masonry walls induces a complex failure pattern composed of multiple diagonal shear cracks, as well as flexural cracks. The realistic modelling of such induced localized cracking necessitates the use of costly direct numerical simulations with detailed information on both the properties and geometry of masonry components. On the contrary, computationally efficient macro-models using standard smeared-crack approaches often result in a poor representation of fracture in the simulated material, not properly localized and biased by the finite element mesh orientation. This work proposes a possible remedy to these drawbacks of macro-models through the use of a crack-tracking algorithm. The macro-modelling approach results in an affordable computational cost, while the tracking algorithm aids the mesh-bias independent and localized representation of cracking. A novel methodology is presented that allows the simulation of intersecting and multi-directional cracks using tracking algorithms. This development extends the use of localized crack approaches using tracking algorithms to a wider field of applications exhibiting multiple, arbitrary and interacting cracking. The paper presents also a novel formulation including into an orthotropic damage model the description of irreversible deformations under shear loading. The proposed approach is calibrated through the comparison with an experimental test on a masonry shear wall against in-plane cyclic loading.

Appendix

The rate of the mechanical dissipation $\gamma$ of the continuum damage model presented in Sect. 2 is [33]

$$\begin{aligned} {} \dot{\gamma } = - \frac{\partial \psi }{\partial \varvec{\epsilon }^i}:\dot{\varvec{\epsilon }}^i - \frac{\partial \psi }{\partial d^+}\dot{d}^+ - \frac{\partial \psi }{\partial d^-}\dot{d}^- = \dot{\gamma }^i + \dot{\gamma }^d \ge 0 \end{aligned}$$

(28)

where $\psi$ is the Helmholtz free energy and $\psi _0^\pm$ elastic free energies which have the following form [33]

$$\begin{aligned}&\psi = (1-d^+) \, \psi _0^+ +(1-d^-) \, \psi _0^- \end{aligned}$$

(29)

$$\begin{aligned}&\psi _0^+ = \frac{1}{2}\, \bar{{\varvec{\sigma }}}^+: \, \varvec{C_0}^{-1} :\, \bar{{\varvec{\sigma }}}\end{aligned}$$

(30)

$$\begin{aligned}&\psi _0^- = \frac{1}{2}\, \bar{{\varvec{\sigma }}}^-: \, \varvec{C_0}^{-1} :\, \bar{{\varvec{\sigma }}}\end{aligned}$$

(31)

It is visible from Eq. (28) that the evolutions of both damage $\dot{\gamma }^d$ and irreversible strains $\dot{\gamma }^i$ contribute to the total dissipation energy of the solid, with each part being

$$\begin{aligned} \dot{\gamma }^i&= - \frac{\partial \psi }{\partial \varvec{\epsilon }^i}:\dot{\varvec{\epsilon }}^i \ge 0 \end{aligned}$$

(32)

$$\begin{aligned} \dot{\gamma }^d&= -\frac{\partial \psi }{\partial d^+}\, \dot{d}^+ - \frac{\partial \psi }{\partial d^-}\, \dot{d}^- \ge 0 \end{aligned}$$

(33)

The total dissipated energy per unit volume $g_f$ is obtained as

$$\begin{aligned} g_f = \int _{0}^{t} \dot{\gamma }^i \, dt + \int _{0}^{t} \dot{\gamma }^d \, dt \end{aligned}$$

(34)

The discrete softening parameter can be defined similarly to [44] considering an ideal uniaxial 1D compressive experiment, with a monotonic increment of the compressive strain (denoted hereafter as $\epsilon ^e$) from an initial unstressed state to full degradation. During the loading, and considering Eqs. (5) and (9) the stress threshold will be

$$\begin{aligned} r^- = E \, \epsilon ^e \end{aligned}$$

(35)

In such case the dissipation due to the damage evolution is (see [44])

$$\begin{aligned} \gamma ^d&= \int _{0}^{t} \dot{\gamma }^d \, dt \end{aligned}$$

(36)

$$\begin{aligned}&= \frac{1}{2E} \int _{r0}^{t} (r^-)^2 \, d' \, dr \end{aligned}$$

(37)

$$\begin{aligned}&= \left( 1 + \frac{1}{H_d^-}\right) \frac{(f^-)^2}{2E} \end{aligned}$$

(38)

The contribution of the irreversible strains to the total dissipated energy can be computed considering

$$\begin{aligned} \dot{\epsilon }^i&= \beta \, \frac{E \dot{\epsilon }}{E \, \epsilon ^e} \epsilon ^e \end{aligned}$$

(39a)

$$\begin{aligned}&= \beta \, \dot{\epsilon } \end{aligned}$$

(39b)

$$\begin{aligned}&= \frac{\beta }{(1-\beta )} \dot{\epsilon }^e \end{aligned}$$

(39c)

and

$$\begin{aligned} - \frac{\partial \psi }{\partial \epsilon ^i} =(1-d^-) \, r^- \end{aligned}$$

(40)

Using the above Eqs. (39c) and (40), the dissipated energy due to the evolution of the irreversible strains is

$$\begin{aligned} \gamma ^i&= \int _{r0}^{t} (1-d^-) \,r^- \, \frac{\beta }{(1-\beta )}\, \dot{\epsilon }^e dt \end{aligned}$$

(41)

$$\begin{aligned}&= \int _{r0}^{r} (1-d^-) \, r^- \, \frac{\beta }{(1-\beta )} \, \frac{1}{E} \, dr \end{aligned}$$

(42)

$$\begin{aligned}&= \frac{\beta }{1 - \beta } \frac{(f^-)^2}{E}\frac{1}{2H_d^-} \end{aligned}$$

(43)

By virtue of Eqs. (36) as well as (41) and considering that the total dissipation should be equal to $Gf^-$, the updated softening modulus is derived as

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

(44)

Note that for the limit case of $\beta =1$, equation (39b) limits to $\dot{\epsilon }^i=\dot{\epsilon }$ and consequently $\dot{\epsilon }^e= 0$. This results in $\Delta r = 0$, which means that there is no damage evolution and hence no softening. In that case, energy is dissipated only due to the evolution of the irreversible strains, which using equations (32), (35) and (40) will be

$$\begin{aligned} \gamma ^i&= E \, \epsilon _0^e \, \int _{0}^{t} \dot{\epsilon } \, dt \end{aligned}$$

(45a)

$$\begin{aligned}&= E \, \epsilon _0^e \int _{\epsilon _0}^{e} d\epsilon \end{aligned}$$

(45b)

$$\begin{aligned}&= E \, \epsilon _0^e \,[\epsilon - \epsilon _0^e] \end{aligned}$$

(45c)

with $\epsilon _0^e$ being the elastic strain at the peak strength. According to the above, the dissipation will keep increasing linearly with the increase of the strains, resembling the behaviour of a perfectly plastic material.