Homogeneous Linear Elastic Material Research Articles

Cells and Gels: A Comparison of Indentation Behavior

Understanding and modeling the mechanical behavior of biological systems requires high quality experimental data to be acquired and analyzed. Conventional indentation tests assume homogeneous linear elastic material properties, and therefore are of limited use in characterizing inhomogeneous biological tissues that generally exhibit complex nonlinear elastic response. Despite their shortcomings, the Hertzian models commonly applied in the analysis of the indentation of soft materials are accurate for determining the quasi-static elastic properties of many synthetic gels when applied within their restrictions, including small strain deformation.

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.

The comparative body model in material and geometric nonlinear analysis of space R/C frames

PurposeThis paper aims to present a new numerical model for the stability and load‐bearing capacity computation of space reinforced‐concrete (R/C) frame structures. Both material and geometric nonlinearities are taken into account. The R/C cross‐sections are assumed to undergo limited distortion under torsional action.Design/methodology/approachA simple, global discretization using beam‐column finite elements is preferred to a full, global discretization using 3D elements. This is more acceptable from a practical point of view. The composite cross‐section is discretized using 2D elements to apply the fiber decomposition procedure to solve the material and geometrical nonlinear behavior of the cross‐section under biaxial moments and axial forces. A local discretization of each beam element based on the comparative body model (i.e. a prismatic body discretized using brick elements, element by element, during the incremental‐iterative procedure) allows determining the torsional constant of the cross‐section under limited warping. The classical global iterative‐incremental procedure is then used to solve the resulting material and geometric nonlinear problem.FindingsIt has been noticed that, in case of a limited distortion of the cross‐section, the torsional constant of homogeneous (linear elastic) materials is greater than the one obtained from the Saint‐Venant theory. However, due to low‐tensile strength of concrete materials, the torsional constant decreases significantly after an early loading phase, primarily due to the lack of reinforcing flanges.Research limitations/implicationsThe current study does not cover the torsion analysis of R/C cross‐section with stirrups. Besides, the bond‐slip effect between concrete and steel reinforcement is not taken into account, nor is the local buckling of the beam flanges and rebar.Practical implicationsThis new numerical model has been implemented in a computer program for effectively computing the nonlinear stability and load bearing capacity of space R/C frames.Originality/valueThe authors believe that the comparative body model should bring a new approach to the solution of torsion problems with limited distortion of cross‐sections in material and geometric nonlinear analysis of space R/C frames.

Intersonic shear rupture mechanism

According to the theory the rupture speed in solids for mode-I cracks is limited by the Rayleigh speed c R, while mode-II cracks can propagate intersonically. These theoretical predictions and sustaining experiments were made for the idealized condition of a crack propagating along a predetermined weak and straight-line path in a homogeneous linear elastic material. In real materials, however, the mode-I crack speed has never been observed to exceed 0.65 c R. The reason for this is the natural tendency for physical cracks to follow a wavy path and for microbranching, which results in a significant increase in microcrack population and, consequently, in the fracture energy. At the same time, intersonic shear ruptures (mode-II cracks) have been reported for crustal earthquakes. It seems paradoxical because earthquake ruptures are normally associated with high complexity and extreme damage in the rupture zone. The present paper shows that nature has provided special shear rupture mechanisms acting in hard rocks at high confining pressure that minimize the rupture energy, causing the increase in rupture speed. These mechanisms are different for primary ruptures (continuous thin ruptures with uniform structure) and general faults (complex discontinuous systems). The general faults propagate in a jump-like manner, forming a cascade of segments due to an advanced triggering mechanism. The advanced triggering mechanism triples the propagation speed of a general fault compared with primary fractures involved in the fault. The propagation of primary ruptures is facilitated by another mechanism, which is a fan-shaped self-equilibrated mechanism created on the basis of an echelon of rotating blocks representing the intrinsic nature of the shear crack structure. These two mechanisms acting in combination can provide intersonic propagation of general faults.

Application of configurational forces in the context of a kinked crack

AbstractThe application of the configurational force approach in crack problems is often used in order to establish fracture criteria that are adapted to a specific material behaviour. The tangential component of the calculated vectorial quantity that acts at the crack tip is a generalisation of the conventional J‐integral and can be interpreted as the energy release rate when the crack extends in this direction. However, the interpretation of nontangential components in the same way, and hence the interpretation of this vectorial quantity as the crack driving force, is not consistent with established kink criteria in the special case of linear elastic fracture mechanics.As a classical example, an in‐plane loaded crack in a homogeneous isotropic linear elastic material is considered under the small strain assumption. Using the expansion of stress intensity factors at the extended crack tip, nontangential components of the configurational force can be interpreted as sensitivities to crack deflection. This perspective has the potential of generalisation which can be applied to more complex situations in order to study the interplay between mechanical fields in the vicinity of the crack tip and the microstructural influence within the process zone. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatial Estimates for the Constrained Anisotropic Elastic Cylinder

This paper establishes spatial estimates in a prismatic (semi-infinite) cylinder occupied by an anisotropic homogeneous linear elastic material, whose elasticity tensor is strongly elliptic. The cylinder is maintained in equilibrium under zero body force, zero displacement on the lateral boundary and pointwise specified displacement over the base. The other plane end is subject to zero displacement (when the cylinder is finite, say). The limiting case of a semi-infinite cylinder is also considered and zero displacement on the remote end (at large distance) is not assumed in this case. A first approach is developed by considering two mean-square cross-sectional measures of the displacement vector whose spatial evolution with respect to the axial variable is studied by means of a technique based on a second-order differential inequality. Conditions on the elastic constants are derived that show the cross-sectional measures exhibit alternative behaviour and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay. A second approach considers cross-sectional integrals involving the displacement and its gradient and furnishes information upon the spatial evolution, without restricting the range of strongly elliptic elastic constants. Such models are principally based upon a first-order differential inequality as well as on one of second order. The general results are explicitly presented for transversely isotropic materials and graphically illustrated for a cortical bone.

Non-Hertzian Approach to Analyzing Mechanical Properties of Endothelial Cells Probed by Atomic Force Microscopy

Detailed measurements of cell material properties are required for understanding how cells respond to their mechanical environment. Atomic force microscopy (AFM) is an increasingly popular measurement technique that uniquely combines subcellular mechanical testing with high-resolution imaging. However, the standard method of analyzing AFM indentation data is based on a simplified "Hertz" theory that requires unrealistic assumptions about cell indentation experiments. The objective of this study was to utilize an alternative "pointwise modulus" approach, that relaxes several of these assumptions, to examine subcellular mechanics of cultured human aortic endothelial cells (HAECs). Data from indentations in 2- to 5-microm square regions of cytoplasm reveal at least two mechanically distinct populations of cellular material. Indentations colocalized with prominent linear structures in AFM images exhibited depth-dependent variation of the apparent pointwise elastic modulus that was not observed at adjacent locations devoid of such structures. The average pointwise modulus at an arbitrary indentation depth of 200 nm was 5.6+/-3.5 kPa and 1.5+/-0.76 kPa (mean+/-SD, n=7) for these two material populations, respectively. The linear structures in AFM images were identified by fluorescence microscopy as bundles of f-actin, or stress fibers. After treatment with 4 microM cytochalasin B, HAECs behaved like a homogeneous linear elastic material with an apparent modulus of 0.89+/-0.46 kPa. These findings reveal complex mechanical behavior specifically associated with actin stress fibers that is not accurately described using the standard Hertz analysis, and may impact how HAECs interact with their mechanical environment.

Decay and Other Estimates for an Annular Elastic Cylinder in an Axisymmetric State of Stress

This paper considers an annular circular cylinder, consisting of homogeneous isotropic linear elastic material, in a state of axisymmetric (torsionless) stress corresponding to traction-free lateral boundaries and such that the traction on every cross-section is self-equilibrated, body forces beign absent; in the main case considered, one plane end is traction-free, the other being self-equilibrated. A positive-definite cross-sectional measure of stress is defined and shown to be a convex function of the axial coordinate, and, in the main case considered, to be a function of “generalized convexity”. Interences are drawn, including a spatial decay estimate for a semi-infinite cylinder reflecting Saint-Venant's principle. The paper concludes with a discussion of the estimate obtained.

Discussion of “Buckling of Columns with Intermediate Elastic Restraint” by C. M. Wang and I. M. Nazmul

This paper presents the classical flexural buckling solution to an ideal prismatic column with an intermediate lateral elastic restraint and subjected to a concentrated axial load at the top end only. The end conditions are ideal ~clamped, hinged, and free!. The discussor finds this paper useful, particularly for testing a matrix solution proposed by the writer that appears much more general and simpler to program. Consider the stepped column that connects points A and B ~Fig. 1!. This consists of two column segments, AC and CB, with semirigid connections at ends A and B and at the intermediate joint C. It is assumed that ~1! both column segments, AC and CB, are made of homogeneous linear elastic materials with mechanical and geometric properties Et I t At ht and Eb I b Ab hb ~where E5elastic modulus; I5moment of inertia; A5cross-sectional area; and h5span). The subindices t and b indicate the top and bottom column segments, respectively; ~2! the centroidal axis of each segment is a straight line with their centroidal axes lineup; ~3! the column is subject simultaneously to axial loading Pa and transverse load Va at point A, to concentrated loads Pc and Vc , and bending moment Mc at the intermediate joint C; and ~4! the lateral sways of the column are partially inhibited by linear springs Sa and Sc located at A and C, respectively, and by an external rotational spring of magnitude SQ at joint C. The rotational fixity factors at ends A and B and the intermediate joint C ~of column segment AC! are assumed to be r a , r b , and r c , respectively. Further details of this model are given by Aristizabal-Ochoa ~1997b!. A general solution to the second-order analysis and buckling load of the column in Fig. 1 is as follows:

Discussion of “Effect of Base Fixity on Buckling of Heavy Column with End Load” by C. Y. Wang

This paper presents the classical flexural buckling solution to an ideal prismatic column with an intermediate lateral elastic restraint and subjected to a concentrated axial load at the top end only. The end conditions are ideal ~clamped, hinged, and free!. The discussor finds this paper useful, particularly for testing a matrix solution proposed by the writer that appears much more general and simpler to program. Consider the stepped column that connects points A and B ~Fig. 1!. This consists of two column segments, AC and CB, with semirigid connections at ends A and B and at the intermediate joint C. It is assumed that ~1! both column segments, AC and CB, are made of homogeneous linear elastic materials with mechanical and geometric properties Et It At ht and Eb Ib Ab hb ~where E5elastic modulus; I5moment of inertia; A5cross-sectional area; and h5span). The subindices t and b indicate the top and bottom column segments, respectively; ~2! the centroidal axis of each segment is a straight line with their centroidal axes lineup; ~3! the column is subject simultaneously to axial loading Pa and transverse load Va at point A, to concentrated loads Pc and Vc , and bending moment M c at the intermediate joint C; and ~4! the lateral sways of the column are partially inhibited by linear springs Sa and Sc located at A and C, respectively, and by an external rotational spring of magnitude SQ at joint C. The rotational fixity factors at ends A and B and the intermediate joint C ~of column segment AC! are assumed to be ra , rb , and rc , respectively. Further details of this model are given by Aristizabal-Ochoa ~1997b!. A general solution to the second-order analysis and buckling load of the column in Fig. 1 is as follows:

Well-posedness of the Dirichlet problem in non-elliptic isotropic elasticity

The well-posedness of the Dirichlet boundary value problem for an isotropic homogeneous linear elastic material is established for non-elliptic equilibrium systems. Mechanical properties of elastic materials with negative shear modulus are also deduced.

Spatial behaviour in the incompressible linear elastic free cylinder

Exponentially increasing and decreasing estimates are derived in terms of various mean-square cross–sectional measures for the longitudinal alternative growth and decay behaviour in a semi–infinite prismatic cylinder composed of incompressible homogeneous isotropic linear elastic material. The cylinder is in equilibrium under zero body–force, self–equilibrated base load and traction–free lateral boundary. The method relies upon the derivation and integration of a pair of differential inequalities for the cross–sectional energy flux and a function related to its first-order counterpart. Attention is predominantly directed to a qualitative description, although the final section considers quantitative aspects. Numerical evaluation of the estimates, facilitated by the constants involved being dependent upon the cross–sectional geometry, enables easy comparison with the exact solutions known for the circular cylinder.

Continuous dependence on base data in an elastic prismatic cylinder

This paper establishes the continuous dependence of the displacement on the base geometry and load of the decay behaviour in a prismatic semi-infinite cylinder occupied by an anisotropic homogeneous linear elastic material. The cylinder is in equilibrium under zero body-force, zero displacement on the lateral boundary and pointwise specified displacement on the base. A differential inequality is derived which on integration leads to an inequality in terms of the perturbation of the base geometry and differences in the base displacement. Continuous dependence is an immediate consequence.

A symmetric Galerkin boundary/domain element method for finite elastic deformations

The Symmetric Galerkin Boundary Element Method (SGBEM) is reformulated for problems of finite elasticity with hyperelastic material and incompressibility, using fundamental solutions related to a (fictitious) homogeneous isotropic and compressible linear elastic material. The proposed formulation contains, besides the standard boundary integrals, domain integrals which account for the problem's nonlinearities through some (fictitious) initial strain and stress fields required to satisfy appropriate “consistency” equations. The boundary/domain integral equation problem so obtained is shown to admit a stationarity principle (a consequence of the Hu-Washizu one), which covers a number of particular cases, including the compressible case, as well as the fully linear case, either incompressible and compressible (so recovering known results of the literature). The continuum rate problem is also addressed and suitably discretized by boundary and domain elements. The related algebraic solving equation system is shown to exhibit symmetry and some sign definiteness features; it is formally solved in terms of the node values of the displacement gradient, the idrostatic pressure, the initial strain and stress fields, a long with the unknown boundary displacements and tractions. Details regarding the fundamental solutions and the stationarity principle are reported in three appendices at the end of the paper.

On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method are among the most frequently applied tools in the numerical treatment of partial differential equations. However. their properties appear to be complementary: while the boundary element method is appropriate for the most important linear partial differential equations with constant coefficients in bounded or unbounded domains, the finite element method seems to be more appropriate for inhomogeneous or even nonlinear problems, but is somehow restricted to bounded domains. The symmetric coupling of the two methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elements and boundary elements for a model transmission problem in two and three dimensions where we have two domains: a bounded domain with nonlinear, even plastic material behaviour, is surrounded by an unbounded, exterior, domain with isotropic homogeneous linear elastic material. Practically, the coupling is performed such that the boundary element method contributes a macro-element, like a large finite element, within a standard finite element analysis program. Emphasis is on two-dimensional problems where the approach using the Poincare-Steklov operator seems to be impossible at first glance.

Anti-plane shear deformations of anisotropic sandwich structures: End effects

Saint-Venant decay lengths for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The most general anisotropy consistent with a state of anti-plane shear is considered. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement that depends only on the in-plane coordinates. In linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Selfequilibrated loads area class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity and anisotropy significantly affects the practical application of SaintVenant's principle to sandwich structures.

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.

Conservation laws in elasticity. III. Planar linear anisotropic elastostatics

The first order conservation laws for an arbitrary homogeneous linear planar elastic material are completely classified. In all cases, both isotropic and anisotropic, besides the standard Betti reciprocity laws, there are two infinite-dimensional families of conservation laws, each depending on an arbitrary analytic function of two complex variables.

Predicted dynamic response of a composite beam with history-dependent damage

The transverse vibration solution for both uniform and non-uniform (variable cross-section) beams, with homogeneous or non-homogeneous (such as composites) linear elastic material properties has been widely discussed and is well documented. However, spatial variations in material properties due to history-dependent damage have not been previously considered. In this paper, the internal state variable concept is introduced to account for the cumulative stress-induced damage in a simple composite beam structure resulting from a fatigue-type loading of several hundred thousand cycles. Furthermore, the transverse vibration solution based on the history-dependent material is presented for a simple beam.

Free Vibration of a Rectangular Parallelepiped Using the Theory of a Cosserat Point

Free vibration of a rectangular parallelepiped composed of a homogeneous linear elastic isotropic material is studied. The parallelepiped is modeled as an isotropic Cosserat point and simple formulas are developed to predict the lowest frequencies of vibration. Within the context of the theory, extensional and shear vibrations are uncoupled. The lowest extensional frequency predicted by the Cosserat theory is compared with available exact solutions and with predictions of thin rod theory. Finally, by introducing a simple modification of the director inertia coefficient it is shown that the Cosserat predictions of the extensional frequencies are correct.