CAPITULO 2: MARCO TEÓRICO
2.1. Antecedentes de la Investigación
2.1.2. Antecedentes Nacionales
3.2 Geometric structures
4
This chapter focus on the two-phase interpenetrating composites of which consist two
5
homogenous and isotropic materials. Three different kind of composites are considered.
6
The microstructure of the composites is supposed to be perfectly periodic, containing a
7
large number of identical cubic cells. In FE analysis where time and memory consuming
8
are sensitive, it is meaningful to describe the IPCs by a much smaller model, which is
9
still large enough to include all the features and characteristics of the structure. The
10
representative model is small enough on one hand for short calculation time and large
11
enough on the other hand for fully describe the material. This smaller model is a
12
representative volume element. In this case, the periodic cells are perfect candidates as
13
RVEs. The different selections on the same structure are shown in Figure 3-1 as a
cross-14
sectional view of the material structure. The cyan colour of material represents for the
15
matrix while the magenta colour of material represents for the reinforcements. The In
16
each cubic cell, the reinforced open foam structure is assumed to be reticular cylindrical
17
struts latticed in three different patterns: (I) cross-cubic, (II) cross-cubic with space
18
diagonals, (II) tetrakaidekahedral, while the matrix fills the RVE space other than the
19
reinforcements, as shown in Figure 3-3. The method of existence and size determination
20
of an RVE has been discussed during the research of particle model [179,180].
21
Figure 3-1. RVE1 selection of simple cubic lattice IPC.
The fibre network is constructed in ANSYS and added together with the conjunction
1
spherical reinforce components. And then RVEs are constructed by Boolean operations
2
as Figure 3-2 shows. Firstly, the fibre in an RVE is obtained by intersecting the fibre
3
network from a solid cube in correct dimension and place. Then the matrix in the RVE
4
is obtained by subtracting the fibre in the RVE from the same solid cube. Finally, these
5
two entities are put together to form a complete RVE. The interface of the fibres and
6
matrix is bonded together.
7
RVE 1
RVE 2
Figure 3-2. Boolean operation to build an RVE of regular IPC.
Figure 3-3. The geometrical structures of three different types of self-connected reinforcement composites: (a) Type I, (b) Type II, (c) Type III.
All the three cubic RVEs have an edge length L, and the diameter of the open-foam
1
cylinders strut is d. By changing d, the different set of volume fractions can be obtained.
2
The two phase of the interpenetrating composites RVEs are made from two isotropic
3
materials. 𝐸𝑚, 𝐸𝑓, 𝜈𝑚, 𝜈𝑓 represent the Young’s moduli and Poisson’s ratio of the matrix
4
and fibre reinforcement separately. The volume fractions of the reinforcement and
5
(a) (b) (c)
matrix are denoted as 𝑉𝐹𝑓 and 𝑉𝐹𝑚. The intersections of the rods are reinforced by
1
spherical connections to simulation the crossing of micro lattice structures.
2
In order to predict the elastic performance of the composites by tracking how Young’s
3
Modulus of the composite 𝐸𝑐 is affected by different constituent material properties and
4
Table 3-1. Young’s moduli and Poisson’s ratios of constituent materials 8
of the fibre is considered from 5% to the upper close of the geometrical limit of each
10
As the structure is complex especially in the cross parts of the fibres, it is difficult to
17
mesh the whole RVE by rectangular prism or cuboid elements. Tetrahedral element are
18
a reasonable choice for this kind of structures. The RVEs are meshed by 3D 10-node
19
tetrahedral structural solid element SOLID187 in ANSYS. As it is proved that the
20
solving results of these simple regular models is not sensitive to the element sizes as
1
long as the models can be solved without any shape elements in ANSYS, the size of the
2
element es is set as a function of the fibre volume fraction as to ensure a balance of
3
efficiency and calculation time. A piecewise function is created to describe how to
4
determine the size of the element es according to the fibre volume fraction. At a given
5
fibre volume fraction the fibre diameter can be calculated. If the corresponding fibre
6
diameter to the fibre volume fraction is denoted in specific notation as 𝑑𝑓 = 𝑑𝑓𝑒× 10𝑘.
7
In specific notation 1 ≤ |𝑑𝑓𝑒| < 10, es is specified as
8
𝑒𝑠 = {𝑓𝑙𝑜𝑜𝑟(𝑑𝑓𝑒) × 10𝑘, 𝑑𝑓𝑒 < 5
0.5, 𝑑𝑓𝑒≥ 5 (3. 4)
9
The function floor is defined to round 𝑑𝑓𝑒 toward negative infinity. This can give an
10
element size automatically with a given fibre volume fraction, thus accelerate the
11
solving procedure of FE analysis. After mesh, the nodes on the interfaces of the fibres
12
and matrix are merged as one single nodes. Besides, the nodes of each face of the RVEs
13
are kept exactly the same in both number and positions for periodicity, as Figure 3-4
14
shows.
15
Figure 3-4. The two facets of an RVE with the exactly the same mesh pattern.
representative volume elements, that means a single RVE can form the full material
5
structure by periodically replicate itself. Thus, a matching boundary condition, namely,
6
the periodic boundary conditions need to be applied to constraint the deformation
7
behaviour of the RVE model. It has been suggested that periodic boundary conditions
8
are more suitable than mixed boundary conditions and prescribed displacement
9
boundary conditions for a periodic RVE [109,157,181]. A unified periodical boundary
10
conditions for representative volume elements of composites was presented to predict
11
the rhombohedral RVE models’ elastic moduli for both unidirectional laminate and
12
angle-ply laminates [182]. Any correctly defined RVE with different shapes can obtain
13
the same mechanical properties with correct periodic boundary conditions. This has
14
been proved by Xia’s work [183] with two types of different RVEs of the same structure,
15