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The original of this paper, Hamid Sharifi and Daniel Larouche “Effect of Grain Shapes of the Microstructure on the Mechanical Properties of the Binary Aluminium Alloy”, had been presented in the INALCO 2013 conference.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

To study the variation of the mechanical behavior of binary aluminum copper alloys with respect to their microstructure, a numerical simulation of their granular structure was carried out. The microstructures are created by a repeated inclusion of some predefined basic grain shapes into a representative volume element until reaching a given volume percentage of the α-phase. Depending on the grain orientations, the coalescence of the grains can be performed. Different granular microstructures are created by using different basic grain shapes. Selecting a suitable set of basic grain shapes, the modeled microstructure exhibits a realistic aluminum alloy microstructure which can be adapted to a particular cooling condition. Our granular models are automatically converted to a finite element model. The effect of grain shapes and sizes on the variation of elastic modulus and plasticity of such a heterogeneous domain was investigated. Our results show that for a given α-phase fraction having different grain shapes and sizes, the elastic moduli and yield stresses are almost the same but the ultimate stress and elongation are more affected. Besides, we realized that the distribution of the θ phases inside the α phases is more important than the grain shape itself.

A longstanding principle states that there is a distinct and identifiable relation between the interior structure of a material and its properties [

Several authors have worked on the development of microstructure models for the simulation of recrystallization and grain growth problems. These models can be divided into two groups [

When an acceptable microstructure geometric with distinguishable solid phases has been produced, the overall mechanical properties of the material can be found. There exist different approaches from different scales that can be used to calculate such properties; some of them are reviewed by Ortiz and Phillips [

In this paper, we investigate the structure-properties relationship for a binary aluminum copper alloy having 4.6 weight percentage (wt%) Cu. A new method for producing discretized microstructure of alloys is presented, which focuses on the generation of a realistic microstructure while avoiding the complex procedures normally required when precipitation and growth of solid phases are considered. The generated microstructures can mimic typical 2D microstructures if suitable basic grain shapes are used. The resulting microstructure is transformed automatically into a finite element mesh. Using the ABAQUS software package, we investigated the variation of the mechanical properties with respect to variation of the microstructure grain shapes and sizes.

Let us consider a tensile test of a solid material using a thin specimen with a uniform thickness, as presented in

Our method is based on the creation of the grains inside this rectangular piece of material, which from now on will be called the representative volume element (RVE). As an example, if we consider the binary aluminum alloy having 4.6% copper, it is clear from the aluminum-copper phase diagram that the face centered cubic (FCC) phase α and the intermetallic phase θ (Al_{2}Cu) can coexist in this alloy. Depending on the cooling conditions, each phase volume percentage can be calculated or determined by image analysis.

The general procedure is as follows: take a RVE completely fill with the θ phase, a basic grain shape is randomly chosen from a predefined basic grain shapes library, its orientation and its size are chosen randomly, and then it is placed in a random position inside the RVE. Some basic shapes will intersect each other inside the RVE and therefore some intersection rules, which depend on the grain orientation, are needed. The basic grains insertion is continued until a given volume fraction of α and θ phases is obtained, the latter being previously determined from a solidification model or from experimental measurements.

The basic grains which are shown in

After the creation of basic grain shapes, they must be projected into the RVE to obtain a granular structure. In the projection procedure, the size, the position and the orientation of each grain inside the RVE are chosen randomly. When a new grain occupies an area already covered by previous grains, a connection criterion is needed to decide whether these grains must be connected together or whether a new grain, with a θ phase joining band separator must be inserted.

Hasson and Goux [_{gb}) as a function of the misorientation of the grains (Δθ) for an aluminum alloy. A simple approximation of this function was given by Mathier _{s/L}, and the formation of a solid-solid grain boundary with energy γ_{gb}. Considering that γ_{gb} < 2γ_{s/L}, then the free energy will be decreased if the solid-solid junction is formed. So it is an attractive situation. Looking at _{gb} = 2γ_{s/L} then the free energy stays the same whether or not the solid-solid junction is formed. This refers to the neutral case. On the other hand, if γ_{gb} > 2γ_{s/L}, then the free energy will be increased if the solid-solid junction is formed. So, it is a repulsive situation. Considering

As a result, if a portion of the inserted grain is already occupied with the previous grain in the domain, the decision about the type of junction between the inserted grain and the old one will depend on the orientation difference between these two grains. If it is an attractive case, the grains must be connected and a single grain is formed. If it is a repulsive case, the old grain must be cut and the new grain is inserted in its position. Between the cut and the new grain, a θ phase joining band (channel) must be inserted to separate these grains. The existence of this phase at the intergranular position is to mimic the late solidification of the liquid phase existing between the grains where the liquid is rich in copper. Here, it is assumed that the liquid is forming a divorced eutectic microconstituent where the θ phase remains as a continuous film at the boundary.

The blue lines are the joining θ phase channels generated by the grain connection procedure. The red squares are the θ phase elements that: (1) came with the basic grains and (2) remained between the grains once the insertion procedure finished. The comparison of this figure with

For microstructures presented in

Each generated grain structure is constructed for a given α to α + θ fraction determined according to the solidification path. One can divide the θ fraction in two parts; the first part contains the θ phase remaining between the grains after the insertion procedure and the θ phase inside the grains, which is already included in the basic grain shape. This subset of θ phase will be designated “pockets”. The second part contains the θ phases which exist in the form of channels between the grains made from a repulsive contact between the grains. This subset of θ phase will be designated “channel”. The volume fraction of these two subsets of θ phase can be written as follow:

Here, _{θ} is the global fraction of θ phase, _{θP} is the fraction of θ phase pockets and _{θC} is the fraction of θ in the channels.

These values must be evaluated before starting the grain generation procedure. For instance, it is known that the volume fraction of channels depends on the cooling rate because of the back-diffusion phenomenon. So, an estimation of channel thicknesses is required to generate a suitable microstructure. At the beginning, the RVE is composed of N by N elements. A fraction _{θP} of these elements will be θ phase at the end of the projection procedure. It is important to mention that channel elements will be added after this step. Since all elements have the same initial size at the projection procedure, one can write at this stage:

The volume of θ phase inside the channels (_{θC}), is given by.

where _{RVE} is the volume of the RVE. Let _{i} be the volume of a channel segment separating two adjacent elements, each belonging to different grains. The volume of channels must be equal to the sum of all these segments so one can write:

Channel segments may have different thicknesses (δ_{i}). If _{elem} is the length of the channel segment separating two adjacent elements, then we have:

For each pair of adjacent grains, a random channel thickness δ_{i} is assumed and is given as:

where _{i} is a random real number between 0.0 and 1.0 and

As a result, one can write:

Now, this equation can be solved for k.

For an as-cast Al-Cu alloy, the copper concentration varies from the center of a dendrite to its boundary because of the microsegregation phenomenon. Levasseur and Larouche [

To classify α phase elements, the list of elements was searched and the elements having a θ phase neighbor were marked. This type of search can be repeated to mark the elements of the other neighboring layers. Different material properties could be assigned to the elements of each layer.

Creation of α phase finite elements are straightforward since each α element of the discrete domain can be presented as an α phase finite element. This is also true for the θ pocket zones as each θ element of the discrete domain can be converted to a θ phase finite element. The sole difficulty is the presentation of the θ phase channels.

The θ phase channels are between two solid elements. The sizes of these channels, using calculations of the previous section, are already known. To create them, parts of the adjacent elements in both sides of the channel are taken and a θ phase finite element is created from these parts. To have a uniform finite elements mesh, half of the θ phase channel thickness is taken from each adjacent α elements. Notice that these elements belong to different grains so they can have different orientations.

The number of generated elements depends on the size and the resolution of RVE. It does not depend on the thickness of the channel since only one θ phase finite element is created between two grains.

Using our grain generation procedure, different grain structures were created and converted in a finite element mesh. The mesh was thereafter introduced in the finite element software package ABAQUS. The Al-4.6% Cu was considered to have 4.4 volume percentage (vol%) of θ phase, among which, approximately 2.5 vol% was present in the channels generated by the connection procedure. The thickness of each channel, between two grains, was chosen randomly as mentioned in the previous section. Different loading and boundary conditions were applied. In these analyses, the α phase elements were considered as an elastoplastic material and the θ phase as an elastic material until its fracture point.

As mentioned in previous sections, the copper weight percentage varies considerably near the θ phase. For an Al-4.6% Cu, the Cu concentration can vary from 2 wt% in the center of the dendrite to 5 wt% near the edge, so the material properties are expected to be not uniform in the α phase. It was therefore decided to give different material properties to the α phase elements depending on whether they were next to a θ phase or not. _{2}Cu phase, the value of Young’s modulus reported by Eshelman and Smith [

A variable displacement boundary condition was applied at the top of RVE. The imposed displacement started from zero and increased linearly until it reached its maximum values before the breakdown of the RVE. Using the

As an example,

The distribution of the Von-Mises stresses at approximately the same imposed strain of

If one calculates the average stress as the sum of nodal reaction forces at the bottom of the RVE over the cross section area and the average strain as the displacement of the top nodes over the height of the RVE, the Young’s modulus can be found. For an elastic plane stress element the relation between stresses and strains are:

where _{xx}=0, and γ_{xy}=0. So, we have:

After the finite element execution, the ε_{yy} at the top of the RVE is known and the σ_{yy} can be evaluated as the sum of the reaction forces at the bottom of the RVE divided by the cross section area, so the Young’s modulus can be calculated and the stress-strain curve can be drawn. Those presented in

Although the microstructures have obvious differences, one can see that they produce similar stress-strain curves from 0 to 190 MPa. So having the same volume fraction of α to θ phases, no matter the grain shapes, we have the same elastic behavior of the microstructure.

All the above models yield approximately at 165 MPa. The hardening parts of the stress-strain curves start to deviate after 190 MPa. Even after 190 MPa, we have some microstructures which behave similarly. The comparison of our simulation results with an as-cast B206 alloy (aluminum 4.6% copper) measurements is presented in

Three groups of microstructures can be identified from the above curves. Group A, which includes microstructures of

Considering

To see the effect of the grain sizes on the mechanical properties of the alloy, microstructures of

In general, we can say that for the same θ and α phase fractions the smaller grain size promotes a higher hardening rate in the plastic regime. The effect of θ phase distribution must not be ignored since a microstructure having larger grains with a lot of θ phases between dendrite arms can present tougher properties. This is why the results of the model presented in

Using a simple numerical procedure, discretized microstructures were generated to mimic a realistic multiphase grain microstructure. A predefined basic grains database was used and a complex grain microstructure can be constructed with these shapes. The time consuming grain growth procedures were not used in this approach, which simplifies the method a lot. Zones of different phases can be formed between the generated grains depending of the connection procedure, the latter assuming the presence of intergranular secondary phases. The grain structure inside a given representative volume element is automatically transformed into a finite element mesh. Using a commercial finite element code, the mechanical properties of a binary aluminum 4.6% copper alloy having the same θ to α phase fraction but with different grain shapes and sizes were calculated. Our results show that even with different grain shapes and sizes, the elastic moduli and yield stresses are almost the same. We observe some difference in elongation, ultimate stresses and the slopes of the hardening part of stress-strain curves which indicates that grain shapes have a measurable influence on the plastic behavior of the alloys. According to these simulations, it seems that the distribution of the θ phase inside the α phase is more important than the grain shape. Note that, different distributions of the θ phase inside the α phase produce different stress fields inside the material.

Although, we have used this method for a binary aluminum alloy, it has the capacity to be used for the generation of other material microstructures. This model is also extendable to a 3D granular model.

The authors declare no conflict of interest.

The original idea of the presented grain generation method and its applications comes from the first author. These ideas are developed and completed with the help of the second author. The financial support was from the second author grants.

A tensile test specimen and a piece of such a specimen.

Some basic grain shapes. (

Basic grain shapes constructed in arrays of 50 × 50 elements. (

Grain boundary energy

(

Generated microstructures using the basic grain shapes a to f of

Generated microstructures, having a volume fraction 95.6% of α phase, using the basic grain shapes b and f of

(

θ phase finite element generation; (

Finite element model of a 1.5 mm × 1.5 mm representative volume element (RVE). It contains 89,401 2D plain stress elements.

Von-Mises stresses inside the RVE of the microstructure of

Von-Mises stresses inside the RVE of the microstructure of

Overall stress-strain curves obtained with the discretized microstructures presented in

Overall stress-strain curves obtained with the discretized microstructures presented in

Mechanical properties of the phases. The α phase properties were taken from the B2117-T4 (Al-2.6 wt% Cu) and the B2219-T87 (Al-6.3 wt% Cu) alloys [_{2}Cu phase, the value of Young’s modulus reported by Eshelman and Smith [

Phase | α | α near θ | θ |
---|---|---|---|

Young’s modulus (GPa) | 70 | 73 | 110 |

Poisson’s ratio | 0.34 | 0.34 | 0.34 |

Yield strength (MPa) | 165 | 395 | ≈1100 |

Ultimate strength (MPa) | 295 | 475 | – |

Elongation (%) | 24 | 10 | – |

Copper (%) | 2.6 | 6.3 | – |

Other Elements (%) | 0.35 | 0.36 | – |

Comparison of properties obtained from different model microstructures with B206 alloy (as cast).

Alloy or Model | Young’s Modulus (GPa) | Yield Stress (MPa) | Ultimate Stress (MPa) | Elongation (%) |
---|---|---|---|---|

B206 as cast [ |
≈72 | 160–172 | 232–255 | 2.8–5 |

72.1 | 163.0 | 222.3 | 1.9 | |

72.0 | 162.8 | 227.6 | 2.6 | |

72.7 | 164.4 | 240.5 | 2.1 | |

72.8 | 164.6 | 234.7 | 1.7 | |

73.1 | 165.2 | 233.0 | 2.4 | |

72.9 | 164.9 | 234.5 | 2.3 |