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Elastic Solids (elastic + solid)

Distribution by Scientific Domains

Engineering	66%

Selected Abstracts

Conserving Galerkin weak formulations for computational fracture mechanics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2002
Shaofan Li
Abstract In this paper, a notion of invariant Galerkin-variational weak forms is proposed. Two specific invariant variational weak forms, the J-invariant and the L-invariant, are constructed based on the corresponding conservation laws in elasticity, one of which is the conservation of Eshelby's energy-momentum (Eshelby. Philos. Trans. Roy. Soc. 1951; 87: 12; In Solid State Physics, Setitz F, Turnbull D (eds). Academic Press: New York, 1956; 331; Rice, J. Appl. Mech. 1968; 35: 379). It is shown that the finite element solution obtained from the invariant Galerkin weak formulations proposed here can conserve the value of J-integral, or L-integral exactly. In other words, the J and L integrals of the Galerkin finite element solutions are path independent in the discrete sense. It is argued that by using the J-invariant Galerkin weak form to compute near crack-tip field in an elastic solid, one may accurately calculate the crack extension energy release rate and subsequently the stress intensity factors in numerical computations, because the flux of the energy-momentum is conserved in discrete computations. This may provide an alternative means to accurately simulate crack growth and propagation. Copyright © 2002 John Wiley & Sons, Ltd. [source]

An appropriate quadrature rule for the analysis of plane crack problems in the boundary-element method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2001
E. E. Theotokoglou
Abstract An hypersingular integral equation of a three-dimensional elastic solid with an embedded planar crack subjected to a uniform stress field at infinity is derived. The solution of the boundary-integral equation is succeeded taking into consideration an appropriate Gauss quadrature rule for finite part integrals which is suitable for the numerical treatment of any plane crack with a smooth-contour shape and permit the fast convergence for the results. The problem of a circular and of an elliptical crack in an infinite body subjected to a uniform stress field at infinity is confronted; and the stress intensity factors are calculated. Copyright © 2001 John Wiley & Sons, Ltd. [source]

A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2007
C. Miehe
Abstract The paper considers a variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius,Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this variational structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading,release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003
J. L. Ferrin
We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ,. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size ,tends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2-scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2-scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ,a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress,strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Regional, ontogenetic, and sex-related variations in elastic properties of cortical bone in baboon mandibles

AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY, Issue 4 2010
Qian Wang
Abstract Understanding the mechanical features of cortical bone and their changes with growth and adaptation to function plays an important role in our ability to interpret the morphology and evolution of craniofacial skeletons. We assessed the elastic properties of cortical bone of juvenile and adult baboon mandibles using ultrasonic techniques. Results showed that, overall, cortical bone from baboon mandibles could be modeled as an orthotropic elastic solid. There were significant differences in the directions of maximum stiffness, thickness, density, and elastic stiffness among different functional areas, indicating regional adaptations. After maturity, the cortical bone becomes thicker, denser, and stiffer, but less anisotropic. There were differences in elastic properties of the corpus and ramus between male and female mandibles which are not observed in human mandibles. There were correlations between cortical thicknesses and densities, between bone elastic properties and microstructural configuration, and between the directions of maximum stiffness and bone anatomical axes in some areas. The relationships between bone extrinsic and intrinsic properties bring us insights into the integration of form and function in craniofacial skeletons and suggest that we need to consider both macroscopic form, microstructural variation, and the material properties of bone matrix when studying the functional properties and adaptive nature of the craniofacial skeleton in primates. The differences between baboon and human mandibles is at variance to the pattern of differences in crania, suggesting differences in bone adaption to varying skeletal geometries and loading regimes at both phylogenetic and ontogenetic levels. Am J Phys Anthropol, 2010. © 2009 Wiley-Liss, Inc. [source]

STRANDS: Interactive Simulation of Thin Solids using Cosserat Models

COMPUTER GRAPHICS FORUM, Issue 3 2002
Dinesh K. Pai
Strandsare thin elastic solids that are visually well approximated as smooth curves, and yet possess essential physical behaviors characteristic of solid objects such as twisting. Common examples in computer graphics include: sutures, catheters, and tendons in surgical simulation; hairs, ropes, and vegetation in animation. Physical models based on spring meshes or 3D finite elements for such thin solids are either inaccurate or inefficient for interactive simulation. In this paper we show that models based on the Cosserat theory of elastic rods are very well suited for interactive simulation of these objects. The physical model reduces to a system of spatial ordinary differential equations that can be solved efficiently for typical boundary conditions. The model handles the important geometric non-linearity due to large changes in shape. We introduce Cosserat-type physical models, describe efficient numerical methods for interactive simulation of these models, and implementation results. [source]

A new mixed finite element method for poro-elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2008
Maria Tchonkova
Abstract Development of robust numerical solutions for poro-elasticity is an important and timely issue in modern computational geomechanics. Recently, research in this area has seen a surge in activity, not only because of increased interest in coupled problems relevant to the petroleum industry, but also due to emerging applications of poro-elasticity for modelling problems in biomedical engineering and materials science. In this paper, an original mixed least-squares method for solving Biot consolidation problems is developed. The solution is obtained via minimization of a least-squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involves four separate categories of unknowns: displacements, stresses, fluid pressures and velocities. Each of these unknowns is approximated by linear continuous functions. The mathematical formulation is implemented in an original computer program, written from scratch and using object-oriented logic. The performance of the method is tested on one- and two-dimensional classical problems in poro-elasticity. The numerical experiments suggest the same rates of convergence for all four types of variables, when the same interpolation spaces are used. The continuous linear triangles show the same rates of convergence for both compressible and entirely incompressible elastic solids. This mixed formulation results in non-oscillating fluid pressures over entire domain for different moments of time. The method appears to be naturally stable, without any need of additional stabilization terms with mesh-dependent parameters. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Variational approach to the free-discontinuity problem of inverse crack identification

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2008
R. TsotsovaArticle first published online: 17 DEC 200
Abstract This work presents a computational strategy for identification of planar defects (cracks) in homogenous isotropic linear elastic solids. The underlying strategy is a regularizing variational approach based on the diffuse interface model proposed by Ambrosio and Tortorelli. With the help of this model, the sharp interface problem of crack identification is split into two coupled elliptic boundary value problems solved using the finite element method. Numerical examples illustrate the application of the proposed approach for effective reconstruction of the position and the shape of a single crack using only the information collected on the surface of the analyzed body. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
C. Miehe
Abstract The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that ,-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd. [source]

A hypersingular time-domain BEM for 2D dynamic crack analysis in anisotropic solids

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2009
M. Wünsche
Abstract A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment

Biomechanical models for clinical data analysis of pressure-volume relationship in the human eye

ACTA OPHTHALMOLOGICA, Issue 2009
S BAUER
Purpose To develop mechanical models describing the pressure-volume relationship for the human eye and to compare the results obtained with clinical data in order to find out which mechanical characteristics affect this relationship most significantly. Methods The fibrous coat is treated as a connected elastic shell consisting of two spherical segments with different radii and mechanical characteristics. The dependence of the intraocular pressure on the volume is analyzed using three different models in which the sclera and the cornea are modeled (1) by soft shells; (2) by transversally isotropic shells with small tension modules in the transverse direction; and (3) by 3D elastic solids. The models are studied analytically and numerically, the latter using FEM ANSYS software. Results The results are obtained over a wide range of parameters using all three models. Conclusion The models proposed predict a generally nonlinear relationship between the intraocular pressure and volume. The parameters of both the sclera and the cornea affect this relationship, the role of the sclera being more important. In the first approximation the simple soft shell model is in good agreement with the clinical data. [source]