# Ultimate Mechanical Properties of Forsterite

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}SiO

_{4}forsterite, have been calculated using first-principles calculations and generalized gradient approximation under tensile and shear loading. The ideal tensile strengths (ITS) and ideal shear strengths (ISS) are computed by applying homogeneous strain increments along high-symmetry directions ([100], [010], and [001]) and low index shear plane ((100), (010), and (001)) of the orthorhombic lattice. We show that the ultimate mechanical properties of forsterite are highly anisotropic, with ITS ranging from 12.1 GPa along [010] to 29.3 GPa along [100], and ISS ranging from 5.6 GPa for simple shear deformation along (100) to 11.5 GPa for shear along (010).

## 1. Introduction

_{2}SiO

_{4}is a mineral of prominent importance since it is a major component of the diffuse interstellar medium and of protoplanetary disks around young stars [1]. Olivine dust in the interstellar medium appears to be almost entirely amorphous, whereas the spectra of protoplanetary disks also show evidence of crystallinity. In the solar system, olivine is found in comets [2], chondritic and nonchondritic meteorites [3,4], and in the mantle of terrestrial planets. On Earth, olivine is the main constituent of the upper mantle [5] and its transformation under pressure to wadsleyite and ringwoodite is the main cause of the observed seismic discontinuities at 410 and 520 km depths. Olivine glass is very difficult to obtain from the melt and requires extreme cooling rates [6]. The first report of olivine glass in 1977 by [7] was related to shock experiments of a single crystal of San Carlos olivine. A few years later, [8] reported evidence of fayalite olivine glass formed after heating in a diamond anvil cell. In 1990, [9,10] presented evidence of amorphization of fayalite pressurized above 39 GPa and 35 GPa respectively. Occurrence of pressure induced amorphization of Mg-rich olivines was further documented by [11,12,13]. Although the role of pressure was generally put forward as the cause for amorphization, the influence of non-hydrostatic stresses was highlighted by [12,14]. Pressure-induced amorphization is usually described as a kinetically preferred transformation resulting from frustration in reaching the high-pressure equilibrium crystalline state. This transformation questions the mechanical stability of crystalline solids.

_{2}SiO

_{4}forsterite. The ideal tensile strengths (ITS) and ideal shear strengths (ISS) are computed from a first-principles method along high-symmetry directions [100], [010], and [001] and for homogenous shear of (100), (010), and (001) planes (here given with respect to the Pbnm space group of forsterite).

## 2. Materials and Methods

_{2}SiO

_{4}unit cell has been fully relaxed, tensile or shear tests are performed by applying incremental homogeneous strain, i.e., atomic layers of the crystal are uniformly displaced along the tensile or shear direction (Figure 1). At each deformation state, a relaxation of both the cell shape and the atomic positions is performed until all the components of the stress tensor are brought to zero, except for one corresponding to the applied stress condition. In practice, we verify that structural relaxation allows for residual stresses of the order of a few MPa at the most.

## 3. Results

#### 3.1. Ground State Properties

_{2}SiO

_{4}, we optimized the equilibrium structure. A unit cell has been built and relaxed for the Pbnm configuration, giving rise to the equilibrium lattice parameters a, b, and c. The results are displayed in Table 1, where they are compared with available data (both theoretical and experimental). It is shown that the calculations predict the correct Mg

_{2}SiO

_{4}ground state structure.

#### 3.2. Ideal Strength in Tension and in Shear

#### 3.2.1. Tensile Tests

#### 3.2.2. Shear Tests

_{ii}(with i = 4, 5, or 6 in Voigt notations). In Figure 4, the results are presented in three groups, each corresponding to a pure shear deformation test associated with the two related simple shear experiments. The three groups naturally emerge from Figure 3. A first set, involving [100](001), [001](100) simple shear deformation and pure shear in (010), corresponds to the highest energy curves. On the opposite side, applying [010](001) or [001](010) simple shear deformation or pure shear in (100) corresponds to the smallest energy increase. In between, the last set of experiments corresponds to simple shear [100](010) and [010](100), and pure shear in (001).

_{44}= 58.7 GPa, C

_{55}= 73.7 GPa and C

_{66}= 73 GPa in agreement with the elastic properties of olivine found experimentally [31,32] or numerically [26]. Moreover, in each group, the pure shear stress–strain curve lies in between the simple shear stress–strain curves until the instability is reached. At the ISS, the stress differences between simple or pure shear tests are within a few GPa. Critical strains and ISS are summarized in Table 1.

## 4. Discussion and Conclusions

^{1}and Mg

^{2}. It appears that the stress maximum corresponds to a divergence of the Mg

^{2}–O bond. The authors of [33] show that the strength of metallic bond in oxides correlate with their lengths. Above 2.5 Å, the Pauli strength decreases drastically, and the bond loses its strength. This is what occurs at the inflection point of the energy curve when forsterite is strained along [100]. Loading forsterite in tension along [010] and [100] leads to different behaviors. The case of tension along [010] is interesting since after the maximum, the stress first decreases before stabilizing and progressively stiffening up to an engineering strain of 0.4. Again, the origin of this behavior is found in the bond distances as shown in Figure 5b. After a first increase from 2.2 to ca. 2.35 Å, the Mg

^{2}–O bond length decreases to recover its original value in the strain interval 0.2–0.3 before increasing again. On both Figure 5a,b, one can see that the SiO

_{4}tetrahedra are not affected by the loading since the Si–O distances remain almost constant. This is a general observation for all solicitations investigated here which illustrates the stiffness of the ionocovalent Si–O bond.

_{3}N

_{4}) are around 0.2. Our results on forsterite are consistent with this general pattern since the resistance of the structure depends on the Mg–O bonds mostly.

^{2}shows a bond divergence (with O

^{9}). A second similar feature (bond breakage between Mg

^{2}and O

^{1}) is responsible for the instability. However, in parallel, several Mg–O bonds show their distances decrease significantly (below 2Å) which result in the significant stiffening observed. Such features, with new bond formation or reorganization under large strain, has also been reported in various crystalline solids (for instance in cementite Fe

_{3}C) leading to a strong strain-stiffening effect [36] as observed here in forsterite.

_{0.9}, Fe

_{0.1})

_{2}SiO

_{4}olivine transforms into wadsleyite at ca. 13 GPa and then to ringwoodite at ca. 18 GPa. At room temperature, these reconstructive phase transformations are kinetically hindered and the olivine structure can be further compressed until it collapses to an amorphous phase above ca. 40 GPa [9,14]. The onset of pressure induced amorphization varies depending on the composition, the type of loading (static, dynamic), but also, although this is less constrained, on non-hydrostaticity as pointed out by [12] and [37]. Here, we characterize the limit of mechanical stability of forsterite without confining pressure and under tensile and shear loading. We show that the onset of instability can be as low as 5–6 GPa for some shear conditions (Table 2). Such deviatoric stress conditions can be reached in nanoindentation, which can significantly facilitate high pressure phase transitions and lower the pressure threshold. Evidence for amorphization has been reported under contact loading in silicon [38] and in boron carbide [39]. Nanoindentation has been performed recently on olivine by [40,41], however, no microstructural investigation was conducted to show a possible amorphization. Such characterizations should provide a test for our theoretical predictions.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Illustration of the various loading conditions applied in this study (

**a**) tensile deformation along [001] where ε

_{zz}= ε, (

**b**) simple shear [001](100) where ε

_{xz}= ε/2 and ε

_{zx}= 0 , or (

**c**) pure shear deformation in (001), where ε

_{xy}= ε

_{yx}= ε/2. The colored image represents a deformed state compared with the undeformed, reference structure in grey. Mg is in orange, Si in blue, O in red.

**Figure 2.**(

**a**) Evolution of the strain energy as a function of the engineering strain; (

**b**) Stress as a function of the engineering strain. The tensile directions are [100] (black empty square), [001] (red empty circle), and [010] (green empty triangle). The instability occurs at the inflection point of the strain energy–strain curves, or the maximum of the stress–strain curves. In (b), the lines correspond to the Cauchy stress evaluations from the derivative of the energy curves shown in (a).

**Figure 3.**Variation of the total energy as a function of the engineering strain calculated using a unit cell of forsterite (

**a**) under pure shear deformation (

**b**) under simple shear deformation, for the [100](001) (square), [100](010) (circle), [010](001) (up triangle), [010](100) (down triangle), [001](100) (diamond) and [001](010) (left triangle) shear deformations.

**Figure 4.**Stress as a function of the engineering strain calculated using a unit cell of forsterite under shear deformation for (

**a**) simple shear along [100](001) and [001](100) and pure shear in (010), (

**b**) simple shear along [010](001) and [001](010) and pure shear in (100), and (

**c**) simple shear along [100](010) and [010](100) and pure shear in (001). Arrows in (b) mark the occurrence of Mg–O bonds breaking as described in the discussion section.

**Figure 5.**Typical bond lengths evolution as a function of strain for tensile deformation along (

**a**) [100] and (

**b**) [010]. Mainly because of the ionocovalency of the bond, Si–O bonds are rather unaffected by strain, whereas one may notice the differential behavior between Mg

^{1}and Mg

^{2}sites.

**Figure 6.**Mg–O bond lengths for both (

**a**) Mg

^{1}and (

**b**) Mg

^{2}sites, as a function of the applied strain for [010](001) simple shear deformation. In (b), the two arrows highlight the breaking of Mg–O bonds at, respectively, first inflexion of the stress–strain curve and the instability as quoted in Figure 4b. The oxygen numbering refers to the labeled atoms shown in (

**c**) and (

**d**) within the unit cell of forsterite.

a (Å) | b (Å) | c (Å) | V (Å^{3}) | |
---|---|---|---|---|

This study | 4.79 | 10.27 | 6.03 | 296.63 |

Calculated GGA [24] | 4.79 | 10.28 | 6.02 | 296.43 |

Calculated GGA [26] | 4.79 | 10.28 | 6.04 | 297.68 |

Calculated GGA [27] | 4.71 | 10.15 | 5.96 | 284.92 |

Calculated LDA [28] | 4.64 | 9.99 | 6.07 | 281.67 |

Experimental [29] | 4.75 | 10.19 | 5.98 | 289.58 |

**Table 2.**Ideal stresses (and associated engineering strains) determined in this study under tensile, pure and simple shear loading. For tensile and simple shear tests, we report also the Young’s modulus and Poisson ratio. The normalized stresses are the ideal stresses divided by the elastic modulus (Young’s modulus in tension and shear modulus in shear).

Tensile Tests | [100] | [010] | [001] | |||
---|---|---|---|---|---|---|

ITS (GPa) | 29.3 | 12.1 | 15.9 | |||

Corresponding strain (%) | 13.0 | 11.5 | 16 | |||

Young’s modulus (GPa) | 274.4 | 153.2 | 170.9 | |||

Normalized stress | 0.10 | 0.08 | 0.09 | |||

Pure shear tests | (100) | (010) | (001) | |||

ISS (GPa) | 5.6 | 11.8 | 8.7 | |||

Corresponding strain (%) | 18.5 | 26.5 | 18.5 | |||

Simple shear tests | [010](001) | [001](010) | [100](001) | [001](100) | [010](100) | [100](010) |

ISS (GPa) | 6.2 | 5.3 | 13.4 | 11.2 | 9.0 | 8.5 |

Corresponding strain (%) | 20 | 18 | 29.5 | 26 | 20 | 18 |

Shear modulus (GPa) | 58.7 (i.e., C _{44}) | 58.7 (i.e., C _{44}) | 73.7 (i.e., C _{55}) | 73.7 (i.e., C _{55}) | 73.0 (i.e., C _{66}) | 73.0 (i.e., C _{66}) |

Normalized stress | 0.10 | 0.09 | 0.18 | 0.15 | 0.12 | 0.11 |

Tensile Tests | [100] | [010] | [001] |
---|---|---|---|

υ_{[100]} | - | 0.13 | 0.14 |

υ_{[010]} | 0.23 | - | 0.29 |

υ_{[001]} | 0.20 | 0.24 | - |

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**MDPI and ACS Style**

Gouriet, K.; Carrez, P.; Cordier, P.
Ultimate Mechanical Properties of Forsterite. *Minerals* **2019**, *9*, 787.
https://doi.org/10.3390/min9120787

**AMA Style**

Gouriet K, Carrez P, Cordier P.
Ultimate Mechanical Properties of Forsterite. *Minerals*. 2019; 9(12):787.
https://doi.org/10.3390/min9120787

**Chicago/Turabian Style**

Gouriet, Karine, Philippe Carrez, and Patrick Cordier.
2019. "Ultimate Mechanical Properties of Forsterite" *Minerals* 9, no. 12: 787.
https://doi.org/10.3390/min9120787