Procedures for construction of anisotropic elastic-plastic property closures for face-centered cubic polycrystals using first-order bounding relations

Microstructure-sensitive design (MSD) is a novel mathematical framework that facilitates a rigorous consideration of the material microstructure as a continuous design variable in the engineering design enterprise [Adams, B.L., Henrie, A., Henrie, B., Lyon, M., Kalidindi, S.R., Garmestani, H., 2001. Microstructure-sensitive design of a compliant beam. J. Mech. Phys. Solids 49(8), 1639-1663; Adams, B.L., Lyon, M., Henrie, B., 2004. Microstructures by design: linear problems in elastic-plastic design. Int. J. Plasticity 20(8-9), 1577-1602; Kalidindi, S.R., Houskamp, J.R., Lyons, M., Adams, B.L., 2004. Microstructure sensitive design of an orthotropic plate subjected to tensile load. Int. J. Plasticity 20(8-9), 1561-1575]. MSD employs spectral representations of the local state distribution functions in describing the microstructure quantitatively, and these in turn enable development of invertible linkages between microstructure and effective properties using established homogenization (composite) theories. As a natural extension of the recent publications in MSD, we provide in this paper a detailed account of the methods that can be readily used by mechanical designers to construct first-order elastic-plastic property closures. The main focus in this paper is on the crystallographic texture (also called Orientation Distribution Function or ODF) as the main microstructural parameter controlling the elastic and yield properties of cubic (fcc and bcc) polycrystalline metals. The following specific advances are described in this paper: (i) derivation of rigorous first-order bounds for the off-diagonal terms of the effective elastic stiffness tensor and their incorporation in the MSD framework, (ii) delineation of the union of the property closures corresponding to both the upper and lower bound theories resulting in comprehensive first-order closures, (iii) development of generalized and readily usable expressions for effective anisotropic elastic-plastic properties that could be applied to all cubic polycrystals, and (iv) identification of the locations of readily available or easily processable ODFs (e.g. textures that are produced by rolling, drawing, etc.) on the property closures. It is anticipated that the advances communicated in this paper will make the mathematical framework of MSD highly accessible to the mechanical designers.     

复合材料焊接线出现裂缝的平面弹性基本问题

本文用复变方法讨论了复合材料任意形状焊接线上出现若干条裂缝时的平面弹性第一和第二基本问题，把寻求复应力函数的问题分别归结为求解某种正则型奇异积分方程和正则型奇异积分方程组，并证明了其解存在且唯一。     

Ab initio calculations of elastic constants and thermodynamic properties of Li2O for high temperatures and pressures

The elastic constants and thermodynamic properties of Li₂O for high temperatures and pressures are calculated by the ab initio unrestricted Hartree-Fock (HF) linear combination of atomic orbital (LCAO) periodic approach. The lattice constant, elastic constants, Debye temperature, and thermal expansion coefficient obtained are in good agreement with the available experimental data and other theoretical results. It is found that at zero pressure the elastic constants C₁₁, C₁₂ and C₄₄, bulk modulus B and Debye temperature Θ_D decrease monotonically over the wide range of temperatures from 0 to 1100 K. When the temperature , C₁₂ approaches zero, consistently with the transition temperature 1200 K. However, with increasing pressure, they all increase monotonically and the anisotropy will weaken.     

More on shear modulus collapse of lattices at high pressure

In his recent paper, Shear modulus collapse of lattices at high pressure, J. Phys. Cond. Matt. 16 (2004) L125, V.V. Kechin claims that the zero temperature shear modulus of a metallic solid vanishes at a high critical pressure, and the critical pressures for this shear modulus collapse lie in the range 0-250 Mbar for elemental metals. Here we demonstrate that Kechin's arguments contain an erroneous assumption, and therefore, do not prove that all metals become mechanically unstable at high pressures. Ab initio calculations and experimental results on a number of solids are analyzed to confirm our conclusion.     

Influence of age and sex on aortic distensibility assessed by MRI in healthy subjects

Magnetic resonance imaging (MRI) is particularly well adapted to the evaluation of aortic distensibility. The calculation of this parameter, based on the change in vessel cross-sectional area per unit change in blood pressure, requires precise delineation of the aortic wall on a series of cine-MR images. Firstly, the study consisted in validating a new automatic method to assess aortic elasticity. Secondly, aortic distensibility was studied for the ascending and descending thoracic aortas in 26 healthy subjects. Two homogeneous groups were available to evaluate the influence of sex and age (with an age limit value of 35 years). The automatic postprocessing method proved to be robust and reliable enough to automatically determine aortic distensibility, even on artefacted images. In the 26 healthy volunteers, a marked decrease in distensibility appears with age, although this decrease is only significant for the ascending aorta (8.97±2.69 10⁻³ mmHg⁻¹ vs. 5.97±2.02 10⁻³ mmHg⁻¹). Women have a higher aortic distensibility than men but only significantly at the level of the descending aorta (7.20±1.61 10⁻³ mmHg⁻¹ vs. 5.05±2.40 10⁻³ mmHg⁻¹). Through our automatic contouring method, the aortic distensibility from routine cine-MRI has been studied on a healthy subject population providing reference values of aortic stiffness. The aortic distensibility calculation shows that age and sex are causes of aortic stiffness variations in healthy subjects.     

Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.     

Self-similar Solutions of Nonlinear Elasticity and Poroelasticity Equations

Recent advances in nonlinear wave propagation in elastic and porous elastic (poro-elastic) material have presented new nonlinear evolutionary equations. The derivation of these equations in three-dimensional space is based on the semilinear Biot theory. The nonlinear elastodynamic equations are derived form the more general model of poro-elastodynamic using consistency arguments. For simplicity, we discuss and carry out the analysis for the nonlinear elastic model. It is found in this article that the methods of symmetry groups and self-similar solutions can furnish solutions to the nonlinear elastodynamic wave equation. It is also found that these models lead to shock wave development in finite time. Necessary conditions for the existence of the solution are given and well-posedness of the Cauchy problem is discussed.     

Effect of B2O3 on the structure and properties of tungsten–tellurite glasses

The elastic properties and Debye temperatures of xB₂O₃–70TeO₂–(30–x)WO₃, (0 ≤ x ≤ 30 mol%) glasses have been investigated using sound velocity measurements at 4 MHz. Ultrasonic and thermal parameters, combined with the results of IR spectroscopic analyses, were employed to explore the effect of B₂O₃ on the structure of tungsten–tellurite glasses. According to IR analysis, there is competition between WO₆ and TeO₄ units to form BO₄ units, and the vibrations of the tellurite structural units are shifted towards lower wavenumbers on the formation of non-bridging oxygens. It is assumed that B₂O₃ acts as a modifier by decreasing the glass-transition temperature T _g and increasing both the thermal stability and glass formation range of the tellurite glasses. The change in density and molar volume with B₂O₃ content reveals that the borate units are less dense than the tellurite structural units. The observed compositional dependence of elastic moduli is interpreted in terms of the effect of B₂O₃ on the coordination number of the tellurite units. A good correlation was observed between experimentally determined elastic moduli and those computed with the Makishima–Mackenzie model.     

Static analysis of rectangular nano-plate using three-dimensional theory of elasticity

Analytical solution for bending of a simply supported rectangular graphene sheets based on three dimensional theory of elasticity, is studied employing non-local continuum mechanics. By applying the Fourier series solution to the both displacement and stress field along the in-plane rectangular coordinates direction, and to the governing equation and constitutive relations, the three-dimensional governing equations in term of displacement components can be obtained. Closed form solution for the bending behavior of nano-plate is obtained by exerting the surface tractions on the state equations. To validate the accuracy and convergence of the present approach, numerical results are presented and compared with the results available in the open literature. Effect of non-local parameter, aspect ratio, thickness to length ratio and half wave numbers in the bending behavior of plate are examined. Furthermore, these results may also serve as benchmark to further results into the two-dimensional plate theories.     

Ultrasound simulation of real-time temperature estimation during radiofrequency ablation using finite element models

Radiofrequency ablation is the most common minimally invasive therapy used in the United States to treat hepatocellular carcinoma and liver metastases. The ability to perform real-time temperature imaging while a patient is undergoing ablation therapy may help reduce the high recurrence rates following ablation therapy. Ultrasound echo signals undergo time shifts with increasing temperature due to sound speed and thermal expansion, which are tracked using both 1D cross correlation and 2D block matching based speckle tracking methods. In this paper, we present a quantitative evaluation of the accuracy and precision of temperature estimation using the above algorithms on both simulated and experimental data.A finite element analysis simulation of radiofrequency ablation of hepatic tissue was developed. Finite element analysis provides a method to obtain the exact temperature distribution along with a mapping of the tissue displacement due to thermal expansion. These local displacement maps were combined with the displacement due to speed of sound changes and utilized to generate ultrasound radiofrequency frames at specified time increments over the entire ablation procedure. These echo signals provide an ideal test-bed to evaluate the performance of both speckle tracking methods, since the estimated temperature results can be compared directly to the exact finite element solution. Our results indicate that the 1D cross-correlation (CC) method underestimates the cumulative displacement by 0.20 mm, while the underestimation with 2D block matching (BM) is about 0.14 mm after 360 s of ablation. The 1D method also overestimates the size of the ablated region by 5.4% when compared to 2.4% with the 2D method after 720 s of ablation. Hence 2D block matching provides better tracking of temperature variations when compared to the 1D cross-correlation method over the entire duration of the ablation procedure. In addition, results obtained using 1D cross-correlation diverge from the ideal finite element results after 7 min of ablation and for temperatures greater than 65 °C.In a similar manner, experimental results presented using a tissue-mimicking phantom also demonstrate that the maximum percent difference with 2D block matching was 5%, when compared to 31% with the 1D method over the 700 s heating duration on the phantom.