Constitutive equations for linear elasticity

In this chapter the constitutive equations for linear elasticity in which the free energy is a quadratic function, and the stress a linear function, of the small strain tensor are introduced. In general, the fourth-order linear elasticity tensor which relates the stress to the strain has twenty-one independent elastic constants. Most solids, however, exhibit some symmetry, the effect of which is to reduce the number of elastic constants. Forms of the linear elasticity tensor for some anisotropic materials, as well as the widely-used forms for an isotropic material, which has only two independent elastic constants, are discussed. The engineering elastic moduli known as the shear modulus and the bulk modulus, as well as the Young’s modulus and the Poisson’s ratio for isotropic linear elastic materials are introduced. Considerations of temperature changes necessitate the introduction of another tensorial material property called the thermal expansion tensor, which for an isotropic material reduces to a scalar coefficient of thermal expansion.

Structure, thermal expansion and incompressibility of MgSO4·9H2O, its relationship to meridianiite (MgSO4·11H2O) and possible natural occurrences

Since being discovered initially in mixed-cation systems, a method of forming end-member MgSO4·9H2O has been found. We have obtained powder diffraction data from protonated analogues (using X-rays) and deuterated analogues (using neutrons) of this compound over a range of temperatures and pressures. From these data we have determined the crystal structure, including all hydrogen positions, the thermal expansion over the range 9–260 K at ambient pressure, the incompressibility over the range 0–1.1 GPa at 240 K and studied the transitions to other stable and metastable phases. MgSO4·9D2O is monoclinic, space group P21/c, Z = 4, with unit-cell parameters at 9 K, a = 6.72764 (6), b = 11.91154 (9), c = 14.6424 (1) Å, β = 95.2046 (7)° and V = 1168.55 (1) Å3. The structure consists of two symmetry-inequivalent Mg(D2O)6 octahedra on sites of \bar 1 symmetry. These are directly joined by a water–water hydrogen bond to form chains of octahedra parallel with the b axis at a = 0. Three interstitial water molecules bridge the Mg(D2O)6 octahedra to the SO4 2− tetrahedral oxyanion. These tetrahedra sit at a ≃ 0.5 and are linked by two of the three interstitial water molecules in a pentagonal motif to form ribbons parallel with b. The temperature dependences of the lattice parameters from 9 to 260 K have been fitted with a modified Einstein oscillator model, which was used to obtain the coefficients of the thermal expansion tensor. The volume thermal expansion coefficient, αV, is substantially larger than that of either MgSO4·7D2O (epsomite) or MgSO4·11D2O (meridianiite), being ∼ 110 × 10−6 K−1 at 240 K. Fitting to a Murnaghan integrated linear equation of state gave a zero-pressure bulk modulus for MgSO4·9D2O at 240 K, K 0 = 19.5 (3) GPa, with the first pressure derivative of the bulk modulus, K′ = 3.8 (4). The bulk modulus is virtually identical to meridianiite and only ∼ 14% smaller than that of epsomite. Above ∼ 1 GPa at 240 K the bulk modulus begins to decrease with pressure; this elastic softening may indicate a phase transition at a pressure above ∼ 2 GPa. Synthesis of MgSO4·9H2O from cation-pure aqueous solutions requires quench-freezing of small droplets, a situation that may be relevant to spraying of MgSO4-rich cryomagmas into the surface environments of icy satellites in the outer solar system. However, serendipitously, we obtained a mixture of MgSO4·9H2O, mirabilite (Na2SO4·10H2O) and ice by simply leaving a bottle of mid-winter brine from Spotted Lake (Mg/Na ratio = 3), British Columbia, in a domestic freezer for a few hours. This suggests that MgSO4·9H2O can occur naturally – albeit on a transient basis – in certain terrestrial and extraterrestrial environments.

Phonon optics, thermal expansion tensor, thermodynamic and chemical bonding properties of Al4SiC4 and Al4Si2C5: a first-principles study

The creation of stacking fault in Al4SiC4 crystal structure due to a phonon mode (E1, 139.7 cm−1, Raman active) at Γ-point with negative mode-Grüneisen constant (−0.28). (a) 3-D side-view; (b) 2-D side view.

CTEAS: a graphical-user-interface-based program to determine thermal expansion from high-temperature X-ray diffraction

The coefficient of thermal expansion analysis suite (CTEAS) has been developed to calculate and visualize thermal expansion properties of crystalline materials in three dimensions. The software can be used to determine the independent terms of the second-rank thermal expansion tensor usinghklvalues, correspondingdhkllistings and lattice constants obtained from powder X-ray diffraction patterns collected at different temperatures. UsingCTEAS, a researcher can also visualize the anisotropy of this essential material property in three dimensions. In-depth understanding of the thermal expansion of crystalline materials can be a useful tool in understanding the dependence of the thermal properties of materials on temperature when correlated with the crystal structure.