Damage propagation in 2d beam lattices: 1. Uncertainty and assumptions

Abstract

The paper studies damage propagation in brittle elastic beam lattices, using the quasistatic approach. The lattice is subjected to a remote tensile loading; the beams in the lattice are bent and stretched. An introduced initial flaw in a stressed lattice causes an overstress of neighboring beams. When one of the overstressed beams fails, it is eliminated from the lattice; then, the process repeats. When several beams are overstressed, one has to choose which beam to eliminate. The paper studies and compares damage propagation under various criteria of the elimination of the overstressed beams. These criteria account for the stress level, randomness of beams properties, and decay of strength due to micro-damage accumulation during the loading history. A numerical study is performed using discrete Fourier transform approach. We compare damage patterns in triangular stretch-dominated and hexagonal bending-dominated lattices. We discuss quantitative characterization of the damage pattern for different criteria. We find that the randomness in the beam stiffness increases fault tolerance, and we outline conditions restricting the most dangerous straight linear crack-like pattern.

Appendix: Numerical scheme

1.1 Periodicity cells

The described scheme of fictitious forces that imitate damage allows for use the representative cell approach [26, 29]. This scheme views the lattice as an assemblage of repetitive cells as it is shown in Fig. 16 right by dashed lines. The representative cell $\Omega ^*$ includes 19 beams rigidly connected at 12 nodal points (see Fig. 16).

Notice that the chosen representative cell is not a minimal possible: An analysis of the triangular lattice may include only three beams as it is done in [18]. However, the chosen here cell allows to easily address both triangular and hexagonal lattices. The cell becomes a hexagon when the stiffness of beams 1, 3, 5, 7, 8, 9, 10, 11, 12, 15, 16, 18, 19 (Fig. 16) vanishes. In addition, such choice of the representative cell enables one to design heterogeneous fault resistant lattices, see [6].

1.2 Lattice

Domain $\Omega$ in Fig. 16, left, is subdivided into $2N_1 \times 2N_2$ cells ($2N_1$ cells the direction $X_1$ and $2N_2$ cells in the direction $X_2$). The Figure shows the case $N_1=N_2=2$. The cells are numbered by a vector index $\mathbf{k }=\{k_1,k_2\}$, where $k_i=-\,N_i,N_i+1,\ldots , N_i-1; i=1,2.$. Nodes and beams within each cell are numbered as it is shown in Fig. 16, right, and the displacement and nodal external forces vectors are labeled as $\mathbf{u }^{(\mathbf{k })}_j,\mathbf{P }^{(\mathbf{k })}_j, j=1,2,\ldots , 12$. The external elongations $\varDelta _1$ and $\varDelta _2$ are applied to the opposite boundaries of $\Omega$.

The nodal equilibrium equations are

$$\begin{aligned} \mathbf{K }\mathbf{u }^{(\mathbf{k })}=\mathbf{P }^{(\mathbf{k })}+\mathbf{F }^{(\mathbf{k })}, \quad \mathbf{k }=\{k_1,k_2\}. \end{aligned}$$

(21)

Here, indices $k_i, i=1,2$ vary: $k_i=-N_i,-N_i+1,\ldots , N_i-1.$ Stiffness matrix $\mathbf{K }$ has $36\times 36$ entries, it corresponds to twelve nodal points in the cell with three degrees of freedom in each node. The lattice is periodic, that is $\mathbf{K }$ is independent of the index $\mathbf{k }$, but the nodes displacements $\mathbf{u }^{\mathbf{k }}$ depend on $\mathbf{k }$ and are different in different cells.

1.3 Forces

Term $\mathbf{F }^{\mathbf{k }}=\{\mathbf{F }^{\mathbf{k }}_i\}$ in the right hand part of (21) represents internal forces that are applied to a cell by the neighboring ones. For the considered representative cell, see Fig. 16, these forces $\mathbf{F }^{\mathbf{k }}=\{\mathbf{F }^{\mathbf{k }}_i\}$ are applied to the boundary nodes 3–12.

Term $\mathbf{P }^{\mathbf{k }}=\{\mathbf{P }^{\mathbf{k }}_i\}$ represents self-equilibrated external forces that mimic damage; they are applied to the nodes at the ends of broken beams. The values of $\mathbf{P }^{\mathbf{k }}$ are determined by an iterative procedure; at each iteration, they are altered so that they compensate internal forces in the loaded lattice computed at the previous iteration. Initially, these fictitious forces are set to be equal to the double of internal forces acting at the beam ends in the undamaged lattice.

In the example illustrated in Fig. 16, these forces are as follows. Here, beam 2 that connects nodes 4 and 7 in (0, 0) cell is removed, and forces $\mathbf{P }_4$ and $\mathbf{P }_7$ are applied to these nodes. Vector $\mathbf{P }^{\mathbf{k }}$ is equal to

$$\begin{aligned} \mathbf{P }^{(k_1, k_2)}=\{0,0,0,\mathbf{P }_4 \delta _{k_1,0}\delta _{k_2,0}, 0,0, \mathbf{P }_7 \delta _{k_1,0}\delta _{k_2,0},0,0,0,0,0\} \end{aligned}$$

(22)

where $\mathbf{P }^{(k_1, k_2)}$ is a block vector, its 12 entries correspond to the generalized forces at the nodes of a cell, $\delta _{ij}$ points to the cell with the damaged beam so that a pair of forces $\mathbf{P }^{(k_1, k_2)}$ is applied to the nodes at the ends of this beam.

1.4 Continuity conditions

The continuity conditions at the interfaces between the neighboring cells are as follows. The forces and displacements satisfy conditions at the interfaces parallel to $X_2$-axis

$$\begin{aligned} \{\mathbf{u }_5,\mathbf{u }_6\}^{(k_1,k_2)}= & {} \{\mathbf{u }_3,\mathbf{u }_4\}^{(k_1+1,k_2)}, \nonumber \\ \{\mathbf{f }_5,\mathbf{f }_6\}^{(k_1,k_2)}= & {} \{\mathbf{f }_3,\mathbf{f }_4\}^{(k_1+1,k_2)}, \end{aligned}$$

(23)

where $k_1=-\,N_1,\ldots , N_1-2;\; k_2=-N_2,\ldots , N_2-1$, and at the interfaces parallel to $X_1$-axis

$$\begin{aligned} \{\mathbf{u }_7,\mathbf{u }_8,\mathbf{u }_9\}^{(k_1,k_2)}= & {} \{\mathbf{u }_{10},\mathbf{u }_{11},\mathbf{u }_{12}\}^{(k_1,k_2+1)}, \nonumber \\ \{\mathbf{f }_7,\mathbf{f }_8,\mathbf{f }_9\}^{(k_1,k_2)}= & {} \{\mathbf{f }_{10},\mathbf{f }_{11},\mathbf{f }_{12}\}^{(k_1,k_2+1)}, \end{aligned}$$

(24)

where $k_1=-\,N_1,\ldots , N_1-1;\; k_2=-\,N_2,\ldots , N_2-2$.

1.5 Boundary conditions for $\Omega$

The boundary conditions for the domain $\Omega$ are formulated for jumps of the generalized displacements $\Delta \mathbf{u }=(\Delta u_1, \Delta u_2, \Delta u_{\theta })$ and forces $\Delta \mathbf{F }=(\Delta F_1, \Delta F_2, \Delta M)$ between the opposite sides of the rectangle, see Fig. 16.

Fig. 16

Left: triangular beam lattice subjected to isotropic tensile deformation. Dashed lines indicate the division into repetitive cells. Right: representative cell with nodes and beam elements numbering

We mention that a similar formulation is employed in [27, 28] for periodic elastic continuum. Uniqueness of the stress state derived from such formulation is addressed in [27].

The jump in the displacement components for the considered loading scheme illustrated in Fig. 16 is $\{\Delta _1,0,0\}$ in the direction $X_1$ and $\{0,\Delta _2,0\}$ in the direction $X_2$.

In the numerical experiments, we choose the size of the domain so that the stresses near its edges are practically insensitive to the presence of flaws, namely $\Omega$ is ($50\times 50$) cells, while the diameter of damaged region is about 4 cells. We set the jump in the boundary forces to zero, meaning that perturbation caused by the missing links do not reach the boundaries. We come to the conditions:

$$\begin{aligned}&\{\mathbf{u }_5,\mathbf{u }_6\}^{(N_1-1,k_2)} - \{\mathbf{u }_3,\mathbf{u }_4\}^{(-N_1,k_2)}= \mathbf{q}_1(\Delta _1),\nonumber \\&\{\mathbf{f }_5,\mathbf{f }_6\}^{(N_1-1,k_2)}- \{\mathbf{f }_3,\mathbf{f }_4\}^{(-N_1,k_2)}= \mathbf{0} \end{aligned}$$

(25)

$$\begin{aligned}&\text{ with } \; k_2=-N_2,..., N_2-1 \nonumber \\&\{\mathbf{u }_7,\mathbf{u }_8,\mathbf{u }_9\}^{(k_1,N_2-1)}- \{\mathbf{u }_{10},\mathbf{u }_{11},\mathbf{u }_{12}\}^{(k_1,-N_2)} = \mathbf{q}_2(\Delta _2),\nonumber \\&\{\mathbf{f }_7,\mathbf{f }_8,\mathbf{f }_9\}^{(k_1,N_2-1)} - \{\mathbf{f }_{10},\mathbf{f }_{11},\mathbf{f }_{12}\}^{(k_1,-N_2)} = \mathbf{0} ,\nonumber \\&\text{ with } \; k_1=-N_1,..., N_1-1; \end{aligned}$$

(26)

where (see Fig. 16 for the numbering of nodes in a cell)

$$\begin{aligned} q_1(\Delta _1) = \{ \Delta _1,0,0, \Delta _1,0,0 \} , \quad q_2(\Delta _2)= \{0,\Delta _2,0,0,\Delta _2,0,0,\Delta _2,0\}. \end{aligned}$$

1.6 Fourier transform

The obtained formulation of the problem for the periodic lattice with non-periodic stress state naturally calls for the application of the discrete Fourier transform. This allows to replace the analysis of the entire lattice by multiple analysis of the single repetitive module in the transform space.

The application the transform

$$\begin{aligned}&f^{F}=\sum _{k_1=-N_1}^{N_1-1} \sum _{k_2=-\,N_2}^{N_2-1}f^{(k_1,k_2)} e^{i(\varphi _{1} k_1+\varphi _2 k_2)}\nonumber \\&\text{ where } \;\;\varphi _{r}=\frac{\pi q_r}{ N_r}\;,\quad q_r=-\,N_r,-N_r+1,\ldots ,N_r-1\;,\;\;r=1,2. \end{aligned}$$

(27)

to the problem (21)–(26) after some manipulation leads to the problem for the representative cell with respect to complex valued displacements and force transforms, as in [28]

$$\begin{aligned}&\mathbf{K }\mathbf{u }^{F}=\mathbf{P }^{F}+\mathbf{F }^{F}, \nonumber \\&\{\mathbf{u }_5,\mathbf{u }_6\}^F- e^{-i\varphi _1} \{\mathbf{u }_3,\mathbf{u }_4\}^F = q_1(\Delta _1^F) \nonumber \\&\{\mathbf{f }_5,\mathbf{f }_6\}^F-e^{-i\varphi _1}\{\mathbf{f }_3,\mathbf{f }_4\}^F= \mathbf{0} \nonumber \\&\{\mathbf{u }_{10},\mathbf{u }_{11},\mathbf{u }_{12}\}^F - e^{-i\varphi _2} \{\mathbf{u }_7,\mathbf{u }_8,\mathbf{u }_9\}^F =q_2(\Delta _2^F) \nonumber \\&\{\mathbf{f }_{10},\mathbf{f }_{11},\mathbf{f }_{12}\}^F - e^{-i\varphi _2} \{\mathbf{f }_7,\mathbf{f }_8,\mathbf{f }_9\}^F = \mathbf{0} \end{aligned}$$

(28)

where $\Delta _j^F =2N_{3-j}\Delta _j\delta _{0,q_{3-j}}e^{\varphi _j(N_j-1)}\;,\quad j=1,2$

After the solving this problem for all $4N_1N_2$ combinations of the parameters $(q_1,q_2)$, one can reveal the generalized displacement in each node of the lattice by the inverse transform formula

$$\begin{aligned} {\mathbf{u }}^{(k_1,k_2)}=\frac{1}{4N_1N_2}\sum _{q_1=-\,N_1}^{N_1-1}\sum _{q_2=-\,N_2}^{N_2-1}\mathbf{u }^F(q_1,q_2) e^{ [-i(\varphi _1 k_1+\varphi _2 k_2)]}\;. \end{aligned}$$

(29)

The employed method may be viewed as representation of arbitrary non-periodic stress state in the entire lattice as sum of different harmonics. This insight helps to understand that cases $q_1=0$ and $q_2=0$ are correspond to simple periodicity in respective directions. In these cases, rigid body motion is possible and certain degrees of freedom are to be fixed in order to eliminate the singularity in the system (28).