scholarly journals Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

2012 ◽  
Vol 10 (1) ◽  
Author(s):  
V. Mantič ◽  
L. Távara ◽  
J.E. Ortiz ◽  
F. París
Keyword(s):  
Fundamental Solution ◽  
Rotational Symmetry ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Real Variable ◽  
Radius Vector ◽  
Linear Elastic ◽  
Plane Normal ◽  
Starting Point ◽  
Recent Developments

<p class="p1">Explicit closed-form real-variable expressions of a fundamental solution and its derivatives for three-dimensional problems in transversely linear elastic isotropic solids are presented. The expressions of the fundamental solution in displacements <span class="s1">U</span><span class="s2">ik </span>and its derivatives, originated by a unit point force, are valid for any combination of material properties and for any orientation of the radius vector between the source and field points. An ex- pression of <span class="s1">U</span><span class="s2">ik </span>in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector is used as starting point. Working from this expression of <span class="s1">U</span><span class="s2">ik</span>, a new approach (based on the application of the rotational symmetry of the material) for deducing the first and second order derivative kernels, <span class="s1">U</span><span class="s2">ik,j </span>and <span class="s1">U</span><span class="s2">ik,jl </span>respectively, has been developed. The expressions of the fundamental solution and its derivatives do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational symmetry axis. The expressions of <span class="s1">U</span><span class="s2">ik</span>, <span class="s1">U</span><span class="s2">ik,j </span>and <span class="s1">U</span><span class="s2">ik,jl </span>are presented in a form suitable for an efficient computational implementation in BEM codes.</p>

scholarly journals Elastic Characterization of Transversely Isotropic Soft Materials by Dynamic Shear and Asymmetric Indentation

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
R. Namani ◽  
Y. Feng ◽  
R. J. Okamoto ◽  
N. Jesuraj ◽  
S. E. Sakiyama-Elbert ◽  
...  
Keyword(s):  
Soft Matter ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Soft Materials ◽  
Fiber Direction ◽  
Dynamic Shear ◽  
Fibrin Gels ◽  
Linear Elastic

The mechanical characterization of soft anisotropic materials is a fundamental challenge because of difficulties in applying mechanical loads to soft matter and the need to combine information from multiple tests. A method to characterize the linear elastic properties of transversely isotropic soft materials is proposed, based on the combination of dynamic shear testing (DST) and asymmetric indentation. The procedure was demonstrated by characterizing a nearly incompressible transversely isotropic soft material. A soft gel with controlled anisotropy was obtained by polymerizing a mixture of fibrinogen and thrombin solutions in a high field magnet (B = 11.7 T); fibrils in the resulting gel were predominantly aligned parallel to the magnetic field. Aligned fibrin gels were subject to dynamic (20–40 Hz) shear deformation in two orthogonal directions. The shear storage modulus was 1.08 ± 0. 42 kPa (mean ± std. dev.) for shear in a plane parallel to the dominant fiber direction, and 0.58 ± 0.21 kPa for shear in the plane of isotropy. Gels were indented by a rectangular tip of a large aspect ratio, aligned either parallel or perpendicular to the normal to the plane of transverse isotropy. Aligned fibrin gels appeared stiffer when indented with the long axis of a rectangular tip perpendicular to the dominant fiber direction. Three-dimensional numerical simulations of asymmetric indentation were used to determine the relationship between direction-dependent differences in indentation stiffness and material parameters. This approach enables the estimation of a complete set of parameters for an incompressible, transversely isotropic, linear elastic material.

scholarly journals Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150272 ◽  
Author(s):  
Longtao Xie ◽  
Chuanzeng Zhang ◽  
Jan Sladek ◽  
Vladimir Sladek
Keyword(s):  
Three Dimensional ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Higher Order ◽  
The Novel ◽  
Elastic Materials ◽  
Linear Elastic ◽  
Second Derivatives ◽  
Analytical Expressions ◽  
Derivatives Of

Novel unified analytical displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions for three-dimensional, generally anisotropic and linear elastic materials are presented in this paper. Adequate integral expressions for the displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions are evaluated analytically by using the Cauchy residue theorem. The resulting explicit displacement fundamental solutions and their first and second derivatives are recast into convenient analytical forms which are valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. The correctness and the accuracy of the novel unified and closed-form three-dimensional anisotropic fundamental solutions are verified by using some available analytical expressions for both transversely isotropic (non-degenerate or partially degenerate) and isotropic (fully degenerate) linear elastic materials.

scholarly journals Three-dimensional fundamental solution in transversely isotropic thermoelastic diffusion material

2012 ◽  
Vol 39 (2) ◽  
pp. 165-184 ◽  
Author(s):  
Rajneesh Kumar ◽  
Vijay Chawla
Keyword(s):  
Fundamental Solution ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Thermoelastic Diffusion ◽  
Diffusion Media ◽  
Stress Components ◽  
Steady State Problem ◽  
Special Case

The aim of the present investigation is to study the fundamental solution for three dimensional problem in transversely isotropic thermoelastic diffusion medium. After applying the dimensionless quantities, two displacement functions are introduced to simplify the basic threedimensional equations of thermoelastic diffusion with transverse isotropy for the steady state problem. Using the operator theory, we have derived the general expression for components of displacement, mass concentration, temperature distribution and stress components. On the basis of general solution, three dimensional fundamental solutions for a point heat source in an infinite thermoelastic diffusion media is obtained by introducing four new harmonic functions. From the present investigation, a special case of interest is also deduced to depict the effect of diffusion.

Reconstruction of Objects from their Projections Using Orthogonal Functions

1973 ◽  
Vol 31 ◽  
pp. 268-269
Author(s):  
Elrnar Zeitler
Keyword(s):  
Unit Area ◽  
Three Dimensional ◽  
One Dimension ◽  
Spatial Density ◽  
Dimensional Representation ◽  
Orthogonal Functions ◽  
Object A ◽  
Plane Normal ◽  
Dimensional Object ◽  
Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

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