Isotropic Elastic Material Research Articles

Analytical applications and effective properties of a second gradient isotropic elastic material model

Recently, the works by Toupin, Mindlin, Sokolowski and Germain have been developed following two research streams. In the first one, higher-order gradient continuum models were developed based on the Cauchy tetrahedron argument (see, e.g., dell’Isola and Seppecher in Comptes Rendus de l Academie de Sciences 17 Serie IIb: Mecanique, Physique, Chimie, Astronomie 321:303–308, 1995, Meccanica 32:33–52 1997, Zeitschrift fr Angewandte Mathematik und Physik 63(6):1119–1141, 2012). In the second one, the structure of higher-order gradient models is developed with a view to the applications. In particular in the model of linear isotropic solids proposed by Dell’Isola, Sciarra and Vidoli (DSV), the main constitutive equation is obtained for the case of second gradient models. This model introduces in addition to the two well-known Lame’s elastic constants five constitutive constants. The practical applications of this model remain in its infancy since the issue of determining the new moduli it introduces is not yet completely addressed. Also, analytical solutions of simple boundary value problems that can be helpful to grasp some of the physical foundations of this model are missing. This paper aims to address these two issues by providing the analytical solutions for two model problems, a spherical shell subjected to axisymmetric loading conditions and the circular bending of a beam in plane strain, both the beam and the shell obeying the DSV second gradient isotropic elastic model. The solution of the circular bending of a beam has served to grasp some of the physical soundness of the model. A framework based on homogenization under inhomogeneous boundary conditions is also suggested to determine the unknown constitutive constants, which are provided in the particular case of elastic porous heterogeneous materials.

Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory

We investigate recovery of through-the-thickness transverse normal and shear strains and stresses in statically deformed functionally graded (FG) doubly-curved sandwich shell structures and shells of revolution using the generalized zigzag displacement field and the Carrera Unified Formulation (CUF). Three different through-the-thickness distributions of the volume fractions of constituents and two different homogenization techniques are employed to deduce the effective moduli of linear elastic isotropic materials. The system of partial differential equations for different Higher-order Shear Deformation Theories (HSDTs) is numerically solved by using the Generalized Differential Quadrature (GDQ) method. Either the face sheets or the core is assumed to be made of a FGM. The through-the-thickness stress profiles are recovered by integrating along the thickness direction the 3-dimensional (3-D) equilibrium equations written in terms of stresses. The stresses are used to find the strains by using Hooke’s law. The computed displacements and the recovered through-the-thickness stresses and strains are found to compare well with those obtained by analyzing the corresponding 3-D problems with the finite element method and a commercial code. The stresses for the FG structures are found to be in-between those for the homogeneous structures made of the two constituents of the FGM.

Investigation of a crack situated in a thin adhesive layer between two different isotropic materials

A plane strain problem for a crack in a thin adhesive elastic-perfectly plastic layer between two different isotropic elastic materials under the action of remote mechanical loading is considered. First, the problem is solved numerically using the finite element method. The stress distributions at the crack continuations and the J-integral values are found. Further, the analytical investigation of the formulated problem is performed. It is assumed that the pre-fracture zones arise at the crack continuations and the interlayer thickness is negligibly small in comparison with the crack length. Modeling the pre-fracture zones by the crack continuations with unknown constant normal and shear cohesive stresses applied to faces of the crack, the problem of linear relationship is formulated and solved analytically. The unknown pre-fracture zone lengths and cohesive stresses are derived from the condition of stress finiteness at the new crack tips and the yielding criterion of the interlayer material. The method of definition of the normal stress co-directed with the crack faces which is used in the mentioned yielding criterion is discussed. Expressions for the crack opening displacement and for the J-integral are obtained in an analytical form as well. The values of the J-integral obtained by means of the analytical method are in a good agreement with the results of the finite element solution.

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.

Lower bounds of end effects for a nonhomogeneous isotropic linear elastic solid in anti-plane shear

In this paper we give lower bounds for the spatial decay of the solutions for anti-plane shear deformations in the case of isotropic inhomogeneous elastic materials. We first consider the case when the shear modulus only depends on the lateral direction. By means of the logarithmic convexity arguments we obtain the required estimates. Some pictures illustrate our results. We also study the general inhomogeneity. We give some lower bounds whenever shear modulus satisfies several requirements.

Maximizing phononic band gaps in piezocomposite materials by means of topology optimization.

Phononic crystals (PCs) can exhibit phononic band gaps within which sound and vibrations at certain frequencies do not propagate. In fact, PCs with large band gaps are of great interest for many applications, such as transducers, elastic/acoustic filters, noise control, and vibration shields. Previous work in the field concentrated on PCs made of elastic isotropic materials; however, band gaps can be enlarged by using non-isotropic materials, such as piezoelectric materials. Because the main property of PCs is the presence of band gaps, one possible way to design microstructures that have a desired band gap is through topology optimization. Thus in this work, the main objective is to maximize the width of absolute elastic wave band gaps in piezocomposite materials designed by means of topology optimization. For band gap calculation, the finite element analysis is implemented with Bloch-Floquet theory to solve the dynamic behavior of two-dimensional piezocomposite unit cells. Higher order frequency branches are investigated. The results demonstrate that tunable phononic band gaps in piezocomposite materials can be designed by means of the present methodology.

Analytical solution of classical and non-classical boundary value contact problems of thermoelasticity for a rectangular parallelepiped consisting of compressible and incompressible elastic layers and numerical example of the solution of such problems

The paper deals with a static thermoelastic equilibrium of an N-layer rectangular parallelepiped. The layers of the considered body are made of an isotropic homogeneous elastic material. A case when some of the layers consist of incompressible elastic materials, which are also assumed to be isotropic and homogeneous, is considered as well. Boundary conditions of symmetric or antisymmetric continuous extension of the solution are imposed on the lateral facets of the parallelepiped. Between the layers contact conditions of rigid, sliding or other type of contact can be defined. On the upper and lower facets of the parallelepiped, arbitrary boundary conditions are defined. Solution of the stated problems is made analytically using the method of separation of variables. The solution of the problems is reduced to the solution of systems of linear algebraic equations with block diagonal matrices. In the conclusion, a practical example establishing the elastic equilibrium of a three-layer rectangular parallelepiped is given.

Solution of multiple cracks in a finite plate of an elastic isotropic material with the distributed dislocation method

This paper presents a numerical solution to model multiple cracks in a finite plate of an elastic isotropic material. Both the boundaries and the cracks are modeled by distributed dislocations. This method results in a system of singular integral equations with Cauchy kernels which can be solved by Gauss-Chebyshev quadrature method. Four examples are provided to assess the capability of this method.

Non-local theory solution for a plane rectangular crack in a 3D infinite transversely isotropic elastic material under a time-harmonic elastic P-wave

Abstract The non-local theory solution for a plane rectangular crack in a 3D infinite transversely isotropic elastic material is presented under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. In order to overcome the mathematical difficulties, a two-dimensional non-local kernel is used instead of a three-dimensional one for 3D dynamic problem to obtain the stress near the crack edges. With the help of the Fourier transform, the problem is formulated into three pairs of dual integral equations with the jumps of displacement across the crack surfaces as the unknown variables. To solve the dual integral equations, the jumps of displacement across the crack surface are directly expanded as a series of Jacobi polynomials. Numerical examples show how the stress is influenced by the geometric shape of the rectangular crack, the circular frequency of the incident waves and the lattice parameter of the material on the dynamic stress fields near the crack edges. Unlike the classical solutions, the present solution exhibits no stress singularity along the rectangular crack edges, i.e. the dynamic stress field near the rectangular crack edges is finite. Thus, this allows us to use the maximum stress as a fracture criterion.

Finite Element Simulation of Shielding/Intensification Effects of Primary Inclusion Clusters in High Strength Steels Under Fatigue Loading

The change of potency to nucleate cracks in high cycle fatigue (HCF) at a primary nonmetallic inclusion in a martensitic gear steel due to the existence of a neighboring inclusion is computationally investigated using two-and three-dimensional elastoplastic finite element (FE) analyses. Fatigue indicator parameters (FIPs) are computed in the proximity of the inclusion and used to compare crack nucleation potency of various scenarios. The nonlocal average value of the maximum plastic shear strain amplitude is used in computing the FIP. Idealized spherical (cylindrical in 2D) inclusions with homogeneous linear elastic isotropic material properties are considered to be partially debonded, the worst case scenario for HCF crack nucleation as experimentally observed for similar systems (Furuya et al., 2004, “Inclusion-Controlled Fatigue Properties of 1800 Mpa-Class Spring Steels,” Metall. Mater. Trans. A, 35A(12), pp. 3737–3744; Harkegard, 1974, “Experimental Study of the Influence of Inclusions on the Fatigue Properties of Steel,” Eng. Fract. Mech., 6(4), pp. 795–805; Lankford and Kusenberger, 1973, “Initiation of Fatigue Cracks in 4340 Steel,” Metall. Mater. Trans. A, 4(2), pp. 553–559; Laz and Hillberry, 1998, “Fatigue Life Prediction From Inclusion Initiated Cracks,” Int. J. Fatigue, 20(4), pp. 263–270). Inclusion-matrix interfaces are simulated using a frictionless contact penalty algorithm. The fully martensitic steel matrix is modeled as elastic-plastic with pure nonlinear kinematic hardening expressed in a hardening minus dynamic recovery format. FE simulations suggest significant intensification of plastic shear deformation and hence higher FIPs when the inclusion pair is aligned perpendicular to the uniaxial stress direction. Relative to the reference case with no neighboring inclusion, FIPs decrease considerably when the inclusion pair aligns with the applied loading direction. These findings shed light on the anisotropic HCF response of alloys with primary inclusions arranged in clusters by virtue of the fracture of a larger inclusion during deformation processing. Materials design methodologies may also benefit from such cost-efficient parametric studies that explore the relative influence of microstructure attributes on the HCF properties and suggest strategies for improving HCF resistance of alloys.

Hyperlipidemia affects multiscale structure and strength of murine femur

To improve bone strength prediction beyond limitations of assessment founded solely on the bone mineral component, we investigated the effect of hyperlipidemia, present in more than 40% of osteoporotic patients, on multiscale structure of murine bone. Our overarching purpose is to estimate bone strength accurately, to facilitate mitigating fracture morbidity and mortality in patients. Because (i) orientation of collagen type I affects, independently of degree of mineralization, cortical bone׳s micro-structural strength; and, (ii) hyperlipidemia affects collagen orientation and μCT volumetric tissue mineral density (vTMD) in murine cortical bone, we have constructed the first multiscale finite element (mFE), mouse-specific femoral model to study the effect of collagen orientation and vTMD on strength in Ldlr−/−, a mouse model of hyperlipidemia, and its control wild type, on either high fat diet or normal diet. Each µCT scan-based mFE model included either element-specific elastic orthotropic properties calculated from collagen orientation and vTMD (collagen-density model) by experimentally validated formulation, or usual element-specific elastic isotropic material properties dependent on vTMD-only (density-only model). We found that collagen orientation, assessed by circularly polarized light and confocal microscopies, and vTMD, differed among groups and that microindentation results strongly correlate with elastic modulus of collagen-density models (r2=0.85, p=10−5). Collagen-density models yielded (1) larger strains, and therefore lower strength, in simulations of 3-point bending and physiological loading; and (2) higher correlation between mFE-predicted strength and 3-point bending experimental strength, than density-only models. This novel method supports ongoing translational research to achieve the as yet elusive goal of accurate bone strength prediction.

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.

Three-phase anisotropic elliptical inclusions with internal uniform in-plane and anti-plane stresses

We study the stress field of a three-phase composite in which an internal elliptical inclusion is bonded to the surrounding matrix through an interphase layer. The linearly elastic materials occupying both the inclusion and the matrix are generally anisotropic, whereas the interphase layer is made of an isotropic elastic material. The two interfaces of the three-phase composite are confocal ellipses. Two conditions are found that ensure that the internal in-plane and anti-plane stress field is uniform. When these conditions are met, the mean stress within the isotropic interphase layer is also uniform. A real form expression of the internal uniform stress field inside the inclusion is derived. Several examples are presented to demonstrate and validate the obtained results.

Fatigue constrained topology optimization

We present a contribution to a relatively unexplored application of topology optimization: structural topology optimization with fatigue constraints. A probability based high-cycle fatigue analysis is combined with principal stress calculations in order to find the topology with minimum mass that can withstand prescribed variable-amplitude loading conditions for a specific life time. This allows us to generate optimal conceptual designs of structural components where fatigue life is the dimensioning factor. We describe the fatigue analysis and present ideas that make it possible to separate the fatigue analysis from the topology optimization. The number of constraints is kept low as they are applied to stress clusters, which are created such that they give adequate representations of the local stresses. Optimized designs constrained by fatigue and static stresses are shown and a comparison is also made between stress constraints based on the von Mises criterion and the highest tensile principal stresses. The paper is written with focus on structural parts in the avionic industry, but the method applies to any load carrying structure, made of linear elastic isotropic material, subjected to repeated loading conditions.

First strain gradient elasticity solution for nanotube-reinforced matrix problem

In the current paper, a rigorous proof of an important theorem which has been used frequently for derivation of field equations of first gradient elasticity is given for the first time. After that, due to the wide use of nanoparticles as reinforcements to different types of matrices, the nanotube-reinforced matrix problem is investigated in cylindrical coordinates. Then, using the most general model for an isotropic gradient elastic material, the displacement formulation is employed to solve the governing equations of the nanotube-reinforced matrix problem. For this purpose, the generalized perfect interface conditions for the nonhomogeneous representative volume element (RVE) are introduced and used to derive the solution. Numerical results reveal that as the matrix characteristic length parameter becomes larger in comparison to that of nanotube, the difference between the results of the classical theory and the strain gradient theory will increase and classical theory cannot accurately predict the mechanical response of the RVE; In addition, increasing the nanotube’s volume fraction results in reduction of the maximum compressive stress and a rise in the overall stiffness of RVE.

Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ

Abstract This paper investigates the in-plane free vibrations of multi-stepped and multi-damaged parabolic arches, for various boundary conditions. The axial extension, transverse shear deformation and rotatory inertia effects are taken into account. The constitutive equations relating the stress resultants to the corresponding deformation components refer to an isotropic and linear elastic material. Starting from the kinematic hypothesis for the in-plane displacement of the shear-deformable arch, the equations of motion are deduced by using Hamilton’s principle. Natural frequencies and mode shapes are computed using the Generalized Differential Quadrature (GDQ) method. The variable radius of curvature along the axis of the parabolic arch requires, compared to the circular arch, a more complex formulation and numerical implementation of the motion equations as well as the external and internal boundary conditions. Each damage is modelled as a combination of one rotational and two translational elastic springs. A parametric study is performed to illustrate the influence of the damage parameters on the natural frequencies of parabolic arches for different boundary conditions and cross-sections with localizeddamage.Results for the circular arch, derived from the proposed parabolic model with the derivatives of some parameters set to zero, agree well with those published over the past years.

Optimal shapes and stresses of adherent cells on patterned substrates

We investigate a continuum mechanical model for an adherent cell on two dimensional adhesive micropatterned substrates. The cell is modeled as an isotropic and homogeneous elastic material subject to uniform internal contractile stresses. The build-up of tension from cortical actin bundles at the cell periphery is incorporated by introducing an energy cost for bending the cell boundary, resulting in a resistance to changes in the local curvature. Integrin-based adhesions are modeled as harmonic springs that pin the cell to adhesive patches of a predefined geometry. Using Monte Carlo simulations and analytical techniques we investigate the competing effects of bulk contractility and cortical bending rigidity in regulating cell shapes on non-adherent regions. We show that the crossover from convex to concave cell edges is controlled by the interplay between contractile stresses and boundary bending rigidity. In particular, the cell boundary becomes concave beyond a critical value of the contractile stress that is proportional to the cortical bending rigidity. Furthermore, the intracellular stresses are found to be largely concentrated at the concave edge of the cell. The model can be used to generate a cell-shape phase diagram for each specific adhesion geometry.

A Line Inhomogeneity in an Elastic Half Plane Under Anti-Plane Shear Loading

An elastic homogeneous isotropic material with a right line inhomogeneity embedded in the material under Anti-shear is analyzed; the mathematical model of the problem is a boundary value problem formulated using the mellin transform and solved by the Wiener-Hoph Techniques. A closed form solution for displacement is obtained from which the stress intensity factor is calculated. The stress field were found to have square-root singularity at the inner tip. As a result of this, micro-cracking can initiate at the inner tip of the line inhomogeneity in the matrix depending on the applied loads. The outer tip showed no singularly.

Transonic gliding edge dislocations/slip pulse near and on an interface/fault

Three problems are solved in this paper that are related to transonic earthquakes. (1) The shear shock Mach wave emanating from a transonic gliding edge dislocation which impacts an interface. (2) A transonic edge dislocation gliding parallel and near to an interface. (3) The transonic edge dislocation gliding on the interface itself. The dislocation is in uniform or quasi‐uniform motion. The interface separates isotropic elastic material of slightly different properties. The first and last problems are essentially the same. The problems are solved with the building blocks of Mach waves, with logarithmic and arctan distributions of infinitesimal shock waves and with subsonic (with respect to longitudinal wave velocities) glide edge and climb edge dislocations. The need for the smeared infinitesimal shock wave distributions is an interesting feature of the solutions. Distributions of infinitesimal smeared Mach waves will exist wherever physical discontinuities exist near a transonic dislocation. A slip pulse of infinitesimal moving edge dislocations can be self sharpening because of attractive forces between like sign dislocations in certain velocity ranges.

Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity

We study spring-block systems which are equivalent to the P1-finite element methods for the linear elliptic partial differential equation of second order and for the equations of linear elasticity. Each derived spring-block system is consistent with the original partial differential equation, since it is discretized by P1-FEM. Symmetry and positive definiteness of the scalar and tensor-valued spring constants are studied in two dimensions. Under the acuteness condition of the triangular mesh, positive definiteness of the scalar spring constant is obtained. In case of homogeneous linear elasticity, we show the symmetry of the tensor-valued spring constant in the two dimensional case. For isotropic elastic materials, we give a necessary and sufficient condition for the positive definiteness of the tensor-valued spring constant. Consequently, if Poisson's ratio of the elastic material is small enough, like concrete, we can construct a consistent spring-block system with positive definite tensor-valued spring constant.