# DECM: A Discrete Element for Multiscale Modeling of Composite Materials Using the Cell Method

## Abstract

**:**

## 1. Introduction

## 2. DEM, CM, and DECM Approaches to Model the Continuum

## 3. Basic Principles of the Discrete Elements Cell Method (DECM)

#### 3.1. Contact Detection Algorithm

#### 3.2. Direction of Crack Propagation

#### 3.3. Constitutive Assumptions

## 4. DECM for Periodic Composite Continua

#### 4.1. Two-Dimensional Problems

- The forces imposed on the nodes of the left side of the $\left(j+1\right)\text{-}\mathrm{th}$ element, with $1\le j<{n}_{c}$, are half the differences (semi-differences) between the forces already present on the nodes and the forces calculated for the twin nodes on the right side of the $j\text{-}\mathrm{th}$ element,
- The displacements imposed on the nodes of the right side of the $\left(j-1\right)\text{-}\mathrm{th}$ element, with $1<j\le {n}_{c}$, are half the sums (semi-sums) of the displacements already calculated for the nodes and the displacements calculated for the twin nodes on the left side of the $j\text{-}\mathrm{th}$ element.

- The forces imposed on the nodes of the ${n}_{c}$ upper sides of the $\left(i+1\right)\text{-}\mathrm{th}$ row, with $1\le i<{n}_{r}$, are half the differences (semi-differences) between the forces already present on the nodes and the forces calculated for the twin nodes on the ${n}_{c}$ lower sides of the $i\text{-}\mathrm{th}$ row,
- The displacements imposed on the nodes of the ${n}_{c}$ lower sides of the $\left(i-1\right)\text{-}\mathrm{th}$ row, with $1<i\le {n}_{r}$, are half the sums (semi-sums) of the displacements already calculated for the nodes and the displacements calculated for the twin nodes on the ${n}_{c}$ upper sides of the $i\text{-}\mathrm{th}$ row.

#### 4.2. The Effect of the Inclusions for Shear Loads

#### 4.3. The Effect of the Inclusions for Axial Loads

#### 4.3.1. Comparison with the Results of a Previous CM Analysis

- Young’s modulus of both the inclusion and the matrix;
- Poisson’s ratio of both the inclusion and the matrix;
- Length of the base of the discrete element;
- Height of the discrete element (not necessarily equal to the base);
- Shape of the inclusion (no inclusion, round inclusion, polygonal inclusion with a randomly generated shape [107], or straight crack that form a random angle with the $x\text{-}\mathrm{axis}$);
- Radius of the round inclusion, which also serves to generate the polygonal inclusion [107];
- Coordinates of the center of the inclusion;
- Number of subdivisions of the base (to generate the mesh);
- Number of subdivisions of the height (to generate the mesh);
- Number of subdivisions of the circular contour (in the case of round inclusion);
- Number of rows of the array;
- Number of columns of the array;
- Loading condition (concentrated force or distributed load);
- Position of the load (one of the two unconstrained corners or one of the midpoints of the three unconstrained sides, in the case of concentrated forces and one of the three unconstrained sides, in the case of distributed loads);
- Intensity of the load.

- ${d}_{1}=0.5\mathrm{m}$ (stresses calculated along a line at a distance equal to 1/8 of the longest side);
- ${d}_{2}=1\mathrm{m}$ (stresses calculated along a line at a distance equal to 1/4 of the longest side);
- ${d}_{3}=1.71\mathrm{m}$ (stresses calculated along a line tangent to the inclusion, on the constraint side);
- ${d}_{4}=2.11\mathrm{m}$ (stresses calculated along a line that passes through the center of the inclusion);
- ${d}_{5}=2.51\mathrm{m}$ (stresses calculated along a line tangent to the inclusion, on the opposite side of the constraint);
- ${d}_{6}=3\mathrm{m}$ (stresses calculated along a line at a distance equal to 3/4 of the longest side);
- ${d}_{7}=3.5\mathrm{m}$ (stresses calculated along a line at a distance equal to 7/8 of the longest side);
- ${d}_{8}=4\mathrm{m}$ (stresses calculated on the loaded side).

#### 4.3.2. DECM Results for a Periodic Composite Specimen

## 5. Future Developments

## 6. Conclusions

- The DECM does not require the calibration of the stable time step, which is, instead, needed by the DEM to allow the convergence of the numerical solution;
- The DECM does not require a preliminary assessment of the minimum number of contact points to obtain the correct solution, which happens in the case of the DEM with deformable discrete elements to limit the computational cost of the dynamic relaxation technique.

## Funding

## Conflicts of Interest

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**Figure 1.**Isolines of the normal stress ${\sigma}_{y}$ (the $y\text{-}\mathrm{axis}$ runs vertically): (

**a**) With and (

**b**) without the arrangement of the bricks in the foreground (stress values in $\mathrm{N}/{\mathrm{mm}}^{2}$ and linear measurements in mm).

**Figure 2.**The Lok-test: (

**a**) Geometric characteristics of the insert and counter-pressure ring for the pullout test. (

**b**) Shape of the extracted portion in concrete solids. (

**c**) Shape of the modeled domain.

**Figure 3.**CM modeling of temperature variation in the sub-base of a radiant heat floor: Shape of the sub-domains (

**a**) before and (

**b**) after restoring compatibility between the sub-domains.

**Figure 4.**Modeling of radiant heat floors: (

**a**) Detail of a modeled joint, (

**b**) plot of the vertical stresses (positive stresses in warm colors and negative stresses in cool colors), and (

**c**) principal directions of stress.

**Figure 5.**Shape of the domain modeled for the Lok-test simulation: (

**a**) At the beginning of the computation and (

**b**) after several steps of crack propagation.

**Figure 8.**The DECM stabilization procedure to take into account both the compatibility and the equilibrium on the interfaces.

**Figure 9.**Deformed configuration and detail of the identified boundary conditions for the pullout test: (

**a**) On the thickness of the disc and (

**b**) on the rod.

**Figure 12.**Concrete plate in biaxial tensile load: (

**a**) Geometric characteristics and (

**b**) crack trajectories for a given $k$ and $1/k$.

**Figure 14.**Geometry of a two-dimensional domain consisting of one row of three square elements (linear measurements in mm).

**Figure 15.**First iteration from left to right: the $j\text{-}\mathrm{th}$ element generates the forces on the left side of the $\left(j+1\right)\text{-}\mathrm{th}$ element, with $j=1,2$ (Thin line: undeformed configurations. Thick line: deformed configurations. Amplification factor of the displacements: $k=200$ ).

**Figure 16.**First iteration from right to left: the $j\text{-}\mathrm{th}$ element generates the displacements on the right side of the $\left(j-1\right)\text{-}\mathrm{th}$ element, with $j=3,2$ (Thin line: undeformed configurations. Thick line: deformed configurations. Amplification factor of the displacements: $k=200$ ).

**Figure 17.**Stabilization cycle on the elements of the same row: deformed configurations provided by the ${k}_{j}\text{-}\mathrm{th}$ left-to-right-to-left iteration, for $1\le {k}_{j}\le 8$ (Thin line: undeformed configurations. Thick line: deformed configurations. Amplification factor of the displacements: $k=200$ ).

**Figure 18.**Deformed configurations and stress analysis on the constrained sides (the $y\text{-}\mathrm{axis}$ runs vertically) for the first top-to-bottom-to-top stabilization cycle on the rows. (

**a**) In the first iteration from top to bottom, the $i\text{-}\mathrm{th}$ row generates the forces on the upper sides of the $\left(i+1\right)\text{-}\mathrm{th}$ row, with $i=1,2$. (

**b**) In the first iteration from bottom to top, the $i\text{-}\mathrm{th}$ row generates the displacements on the lower sides of the $\left(i-1\right)\text{-}\mathrm{th}$ row, with $i=3,2$ (Thin line: undeformed configurations. Thick line: deformed configurations. Amplification factor of the displacements: $k=200$ ).

**Figure 19.**Deformed configurations of the nine discrete elements and stress analysis on the lower constrained sides (the $y\text{-}\mathrm{axis}$ runs vertically), after 15 top-to-bottom-to-top stabilization cycles on the rows (Thin line: undeformed configurations. Thick line: deformed configurations. Amplification factor of the displacements: $k=200$ ).

**Figure 21.**Stabilization cycle on the rows: deformed configurations provided by the ${k}_{i}\text{-}\mathrm{th}$ top-to-bottom-to-top iteration, for $1\le {k}_{i}\le 12$ (Thin line: undeformed configurations. Thick line: deformed configurations. Amplification factor of the displacements: $k=200$ ).

**Figure 22.**Normal stresses ${\sigma}_{y}$ (the $y\text{-}\mathrm{axis}$ runs vertically) in the discrete elements of a $3\times 3$ array subjected to shear load: (

**a**) Without inclusions and (

**b**) with round inclusions, which are stiffer than the matrix.

**Figure 23.**Normal stresses ${\sigma}_{x}$ (the $x\text{-}\mathrm{axis}$ runs horizontally) in the discrete elements of a $3\times 3$ array subjected to shear load: (

**a**) Without inclusions and (

**b**) with round inclusions, which are stiffer than the matrix.

**Figure 24.**Shear stresses ${\tau}_{xy}$ in the discrete elements of a $3\times 3$ array subjected to shear load: (

**a**) Without inclusions and (

**b**) with round inclusions, which are stiffer than the matrix.

**Figure 25.**Detail of the shear stress ${\tau}_{xy}$ for the discrete element $i=2,j=2$ (colors equally ranged between the extreme values of shear stress for the element).

**Figure 26.**Geometry and loading condition of the elastic cantilever beam with a round inclusion [5].

**Figure 27.**CM static analysis: normal stresses ${\sigma}_{x}$ plotted on the deformed configuration (Thin line: undeformed configuration. Thick line: deformed configuration. Amplification factor of the displacements: $k=500$ ) [5].

**Figure 28.**CM static analysis: shear stresses ${\tau}_{xy}$ plotted on the deformed configuration (Thin line: undeformed configuration. Thick line: deformed configuration. Amplification factor of the displacements: $k=500$ ) [5].

**Figure 29.**DECM static analysis for the $1\times 1$ array generated by the discrete element with the geometric characteristics defined in Table 2. (

**a**) Array geometry and loading conditions. (

**b**) normal stresses ${\sigma}_{y}$ plotted on the deformed configuration (Thin line: undeformed configuration. Thick line: deformed configuration. Amplification factor of the displacements: $k=500$ ).

**Figure 30.**3D plots and isolines of the axial stresses for: (

**a**) The numerical results of the CM model and (

**b**) the numerical results of the DECM model for ${n}_{r}=1$ and ${n}_{c}=1$.

**Figure 31.**Axial stresses calculated at a constant distance from the constraint, along a line that passes through the center of the inclusion for: (

**a**) The numerical results of the CM model and (

**b**) the numerical results of the DECM model for ${n}_{r}=1$ and ${n}_{c}=1$.

**Figure 32.**Geometry and loading condition of the $6\times 4$ array with round inclusions (linear measurements in mm).

**Figure 33.**Elastic modeling of the $6\times 4$ array subjected to uniaxial traction: (

**a**) Deformed configuration (amplification factor of the displacements: $k=400$ ) and (

**b**) normal stresses ${\sigma}_{y}$.

**Figure 34.**Stress fields in the $6\times 4$ array subjected to uniaxial traction. (

**a**) Normal stresses ${\sigma}_{x}$ and (

**b**) shear stresses ${\tau}_{xy}$.

**Figure 36.**Flow chart of a CM code for the analysis of crack propagation in the displacement control.

**Figure 37.**The two-time elements of the CM, represented as elements of a one-dimensional cell-complex.

Symbol | Description | Value |
---|---|---|

$L$ | Base | $4\mathrm{m}$ |

$D$ | Height | $1\mathrm{m}$ |

$R$ | Radius of the round inclusion | $0.4\mathrm{m}$ |

${L}_{1}$ | Distance of the center C from the left side | $2.11\mathrm{m}$ |

${D}_{1}$ | Distance of the center C from the lower side | $0.5\mathrm{m}$ |

${n}_{div}^{L}$ | Number of subdivisions of the base | 32 |

${n}_{div}^{D}$ | Number of subdivisions of the height | 8 |

${n}_{div}^{crf}$ | Number of subdivisions of the circular contour | 80 |

Symbol | Description | Value |
---|---|---|

$L$ | Base | $1\mathrm{m}$ |

$D$ | Height | $4\mathrm{m}$ |

$R$ | Radius of the round inclusion | $0.4\mathrm{m}$ |

${L}_{1}$ | Distance of the center of the inclusion from the lower side | $2.11\mathrm{m}$ |

${D}_{1}$ | Distance of the center of the inclusion from the left side | $0.5\mathrm{m}$ |

${n}_{div}^{L}$ | Number of subdivisions of the base | 8 |

${n}_{div}^{D}$ | Number of subdivisions of the height | 32 |

${n}_{div}^{crf}$ | Number of subdivisions of the circular contour | 80 |

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## Share and Cite

**MDPI and ACS Style**

Ferretti, E.
DECM: A Discrete Element for Multiscale Modeling of Composite Materials Using the Cell Method. *Materials* **2020**, *13*, 880.
https://doi.org/10.3390/ma13040880

**AMA Style**

Ferretti E.
DECM: A Discrete Element for Multiscale Modeling of Composite Materials Using the Cell Method. *Materials*. 2020; 13(4):880.
https://doi.org/10.3390/ma13040880

**Chicago/Turabian Style**

Ferretti, Elena.
2020. "DECM: A Discrete Element for Multiscale Modeling of Composite Materials Using the Cell Method" *Materials* 13, no. 4: 880.
https://doi.org/10.3390/ma13040880