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The pressure dependence of thermal expansivity affects mineral density at pressure and is an extrapolator for calculating self-compression adiabats of a self-gravitating body. I review different models for the pressure dependence of expansivity and how to decide which performs best. A finite strain model, proposed here, performs better when used to calculate adiabatic temperature lapses in both the solid silicate and liquid metal parts of a planet than either an ad-hoc exponential dependence on pressure or a commonly used mineral physics model. Choosing a particular thermal expansivity pressure dependence leads to significantly different temperatures in planetary interiors, and to inferred subsolidus properties related to homologous melting temperature. In particular, thermal expansivity in liquid metal in planetary cores at pressures comparable to Earth's core is significantly affected. The universality of the parameterization provides a simple way to model rocky planet interiors in our solar system and exoplanet interiors.

Microstructural, finite strain and vorticity analyses of quartz-rich mylonites were used in order to investigate kinematics of rock flow and deformation temperature in the Sirjan thrust sheet exposed in a structural window within the Sanandaj–Sirjan High Pressure – Low Temperature (HP–LT) metamorphic belt that forms part of the hinterland of the Zagros orogenic belt of Iran. A dominant top-to-the-SW sense of shear in the study area is indicated by several shear sense indicators such as asymmetric boudins, rotated porphyroclasts, mica fish and S/C fabrics. Quantitative analyses reveal approximately plane strain deformation conditions with Rxz values ranging from 2.5 to 4.3 and increasing towards the Sirjan thrust. Opening angles of quartz c-axis fabrics and recrystallization regimes suggest deformation temperatures vary from 430 to 625 ± 50°C in the hanging wall rocks. Oblique grain shape and quartz c-axis fabrics were used to estimate the degree of non-coaxiality during deformation. The obtained vorticity profile indicates a down-section increase in kinematic vorticity number (Wm) from 0.6 to 0.89. This range of vorticity numbers confirms contributions of both simple (41–68 %) and pure shear (32–59 %) deformation components. The structural characteristics of the study area ultimately were controlled by oblique motion of the Afro-Arabian plate relative to the Iranian plate.

Geometrically exact beam theory is extended to allow distortion of the cross section. We present an appropriate set of cross-section basis functions and provide physical insight to the cross-sectional distortion from linear elastostatics. The beam formulation in terms of material (back-rotated) beam internal force resultants and work-conjugate kinematic quantities emerges naturally from the material description of virtual work of constrained finite elasticity. The inclusion of cross-sectional deformation allows straightforward application of three-dimensional constitutive laws in the beam formulation. Beam counterparts of applied loads are expressed in terms of the original three-dimensional data. Special attention is paid to the treatment of the applied stress, keeping in mind applications such as hydrogel actuators under environmental stimuli or devices made of electroactive polymers. Numerical comparisons show the ability of the beam model to reproduce finite elasticity results with good efficiency.

Eulerian finite strain of an elastically isotropic body is defined using the expansion of squared length and the post-compression state as reference. The key to deriving second-, third- and fourth-order Birch–Murnaghan equations-of-state (EOSs) is not requiring a differential to describe the dimensions of a body owing to isotropic, uniform, and finite change in length and, therefore, volume. Truncation of higher orders of finite strain to express the Helmholtz free energy is not equal to ignoring higher-order pressure derivatives of the bulk modulus as zero. To better understand the Eulerian scheme, finite strain is defined by taking the pre-compressed state as the reference and EOSs are derived in both the Lagrangian and Eulerian schemes. In the Lagrangian scheme, pressure increases less significantly upon compression than the Eulerian scheme. Different Eulerian strains are defined by expansion of linear and cubed length and the first- and third-power Eulerian EOSs are derived in these schemes. Fitting analysis of pressure-scale-free data using these equations indicates that the Lagrangian scheme is inappropriate to describe P-V-T relations of MgO, whereas three Eulerian EOSs including the Birch–Murnaghan EOS have equivalent significance.

This work addresses an efficient low order 3D virtual element method for elastic–plastic solids undergoing large deformations. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The successful application of the method to non-linear problems such as finite strain elasticity and plasticity in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. This work is concerned with the extensions of the virtual element method to problems of 3D finite strain plasticity. Low-order formulations for problems in three dimensions, with elements being arbitrary shaped polyhedra, are considered. The formulation is based on minimization of a pseudo energy expression, with a generalization of a stabilization techniques, introduced for two dimensional polygons, to the three-dimensional domain. The resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency, accuracy and convergence properties. For comparison purposes, results of the standard finite element method are also demonstrated.

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a two-dimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

This paper proposes a thermodynamically consistent phase-field damage model for viscoelastic materials following the strategy developed by Boldrini et al. (Methods Appl Mech Eng 312:395–427, 2016). Suitable free-energy and pseudo-potentials of dissipation are developed to build a model leading to a stress-strain relation, under the assumption of finite strain, in terms of fractional derivatives. A novel degradation function, which properly couples stress response and damage evolution for viscoelastic materials, is proposed. We obtain a set of differential equations that accounts for the evolution of motion, damage, and temperature. In the present work, for simplicity, this model is numerically solved for isothermal cases by using a semi-implicit/explicit scheme. Several numerical tests, including fitting with experimental data, show that the developed model accounts appropriately for damage in viscoelastic materials for small and finite strains. Non-isothermal numerical simulations will be considered in future works.

Even when entirely unloaded, biological structures are not stress-free, as shown by Y.C. Fung׳s seminal opening angle experiment on arteries and the left ventricle. As a result of this prestrain, subject-specific geometries extracted from medical imaging do not represent an unloaded reference configuration necessary for mechanical analysis, even if the structure is externally unloaded. Here we propose a new computational method to create physiological residual stress fields in subject-specific left ventricular geometries using the continuum theory of fictitious configurations combined with a fixed-point iteration. We also reproduced the opening angle experiment on four swine models, to characterize the range of normal opening angle values. The proposed method generates residual stress fields which can reliably reproduce the range of opening angles between 8.7±1.8 and 16.6±13.7 as measured experimentally. We demonstrate that including the effects of prestrain reduces the left ventricular stiffness by up to 40%, thus facilitating the ventricular filling, which has a significant impact on cardiac function. This method can improve the fidelity of subject-specific models to improve our understanding of cardiac diseases and to optimize treatment options.