Effective Shear Modulus Research Articles

Static and fatigue bending behavior of pultruded GFRP sandwich panels with through-thickness fiber insertions

This paper presents the findings of a research program that was undertaken to evaluate the static and fatigue characteristics of an innovative 3-D glass fiber reinforced polymer (GFRP) sandwich panel proposed for civil infrastructure and transportation applications. The research consists of analytical modeling verified by experimental results. A rational analytical model is presented and used to evaluate the effective elastic modulus, shear modulus and degree of composite interaction of the panels to resist one-way bending. The experimental program was conducted in two phases to study the static and fatigue behavior of the panels. In the first phase a total of 730 sandwich beams were tested to evaluate the effect of different parameters on the fundamental behavior of the panel. The parameters considered include the pattern and density of through-thickness fiber insertions, the overall thickness of the panels, and the number of FRP plies in the face skins. The study indicates that the shear behavior and degree of composite interaction of the panels is sensitive to the configuration of the panel core. The second phase of the experimental program included testing of 24 additional sandwich panels to evaluate the fatigue behavior. The results of the experimental program indicate that the panels with stiffer cores generally exhibited a higher degree of degradation than panels with more flexible cores. The findings of this study indicate that the proposed panels represent a versatile construction system which can be configured to achieve the specific design demands for civil engineering infrastructure applications.

Seismoelectric response of heavy oil reservoirs: theory and numerical modelling

We consider a linear poroelastic material filled with a linear viscoelastic solvent like wet heavy oils (oil as opposed to water or brines is wetting the surfaces of the pores). We extend the electrokinetic theory in the frequency domain accounting for the relaxation effects associated with resonance of the viscoelastic fluid. The fluid is described by a generalized Maxwell rheology with a distribution of relaxation times given by a Cole–Cole distribution. We use the assumption that the charges of the diffuse layer are uniformly distributed in the pore space (Donnan model). The macroscopic constitutive equations of transport for the seepage velocity and the current density have the form of coupled Darcy and Ohm equations with frequency-dependent material properties. These equations are combined with an extended Frenkel–Biot model describing the deformation of the poroelastic material filled with the viscoelastic fluid. In the mechanical constitutive equations, the effective shear modulus is frequency dependent. An amplification of the seismoelectric conversion is expected in the frequency band where resonance of the generalized Maxwell fluid occurs. The seismic and seismoelectric equations are modelled using a finite element code with PML boundary conditions. We found that the DC-value of the streaming potential coupling coefficient is also very high. These results have applications regarding the development of new non-intrusive methods to characterize shallow heavy oil reservoirs in tar sands and DNAPL contaminant plumes in shallow aquifers.

Novel approach for measuring the effective shear modulus of porous materials

A new approach is proposed for the experimental study of the effective shear modulus of porous elastic materials using the uniaxial tension test. The idea is to measure strains at a few points surrounding a cluster of holes that represents the structure of the material. The representative cluster is placed in the material with the same elastic properties as those of the matrix. The measured strains lead to the properties of the equivalent circular inhomogeneity, which would produce the same elastic fields as from the cluster. An aluminum plate containing a cluster of seven circular or hexagonal holes was used. The effective shear modulus obtained from the strain data was compared with theoretical predictions and various bounds, and it was shown that the laboratory estimated values are quite accurate. The experimental technique can be used for the determination of the effective Poisson’s ratio of porous media and/or cellular solids if more detailed strain data are obtained.

Fiber Network Elasticity as Function of Crosslinker Density

Crosslinkers determine the architecture of polymer networks and thus are of great importance for the resulting mechanical properties. A simple morphological model is proposed for the investigation of the linear elastic response of 3D fiber networks to randomly disconnecting network nodes. Isotropic ordered and disordered, 4-coordinated networks are modeled as homogeneous bodies in the shape of a network with a given volume fraction with locally isotropic elastic moduli. The 4-coordinated nodes are randomly split into two locally unconnected fibers representing a morphology change of the network at constant volume fraction. The effective shear modulus is studied using a voxel-based finite element method. Our results show a strong, continuous decrease of the shear modulus with decreasing number of nodes in the network without a percolation transition. The morphology of the networks is characterized by the Euler number that linearly depends on the fraction of split nodes, and that is easily extracted from 3D confocal microscopy data. Associating all network nodes with fiber junctions connected by a crosslinking molecule, this approach is a first order model for elasticity of biological networks with varying crosslinker density.

Elastic and Morphological Properties of Porous Biomaterials

The relationship between effective elastic moduli and morphological properties of microstructured or porous biomaterials including bone, wood, biomineralised skeletons of crustaceans, biopolymer networks and cubic lipid mesophases remains an open question. We compute effective elastic moduli and morphological properties of ordered porous media models based on triply-periodic minimal and constant mean-curvature surfaces of cubic symmetry.Bulk and shear moduli are computed using voxel-based finite-element method considering the solid fraction to be a homogeneous linear elastic solid. For structures with varying volume fraction of the solid fraction, the effective bulk and shear moduli of the microstructures can be related to the porosity by a power law with fractional exponent. For fixed volume fraction of 50%, we find that within classes of geometrically similar structures the effective bulk modulus decreases with increasing heterogeneity of the domain thickness of the solid fraction which is quantified by using euclidean distance maps and percolation critical radii. On the other hand, we find significant differences between the elastic moduli of topologically distinct classes of structures. In particular, a porous medium where the solid fraction comprises a thick warped sheet separating two hollow labyrinthine network domains has larger bulk modulus than a medium where both the solid and the void fraction are represented by congruent labyrinthine domains.

Response of prestretched nematic elastomers to external fields

We investigate the response of prestretched nematic side-chain liquid single-crystal elastomers to superimposed external shear, electric, and magnetic fields of small amplitude. The prestretching direction is oriented perpendicular to the initial nematic director orientation, which enforces director reorientation. Furthermore, the shear plane contains the direction of prestretch. In this case, we obtain a strongly decreased effective shear modulus in the vicinity of the onset and the completion of the enforced director rotation. For the same regions, we find that it becomes comparatively easy to reorient the director by external electric and magnetic fields. These results were derived using conventional elasticity theory and its coupling to relative director-network rotations.

Analytical Approach to Recovering Bone Porosity From Effective Complex Shear Modulus

This work deals with the study of the analytical relations between porosity of cancellous bone and its mechanical properties. The Stieltjes representation of the effective shear complex modulus of cancellous bone is exploited to recover porosity. The microstructural information is contained in the spectral measure in this analytical representation. The spectral function can be recovered from the effective measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed. Regularized algorithm is derived to ensure stability of the results. The proposed method does not use any specific assumptions about the microgeometry of bone. The approach does not rely on correlation analysis, it uses analytical relationships. For validation purposes, complex shear modulus over a range of frequencies was calculated by the finite element method using micro-computed tomography (micro-CT) images of human cancellous bone. The calculated values were used in numerical algorithm to recover bone porosity. At the microlevel, bone was modeled as a heterogeneous medium composed of trabeculae tissue and bone marrow treated as transversely isotropic elastic and isotropic viscoelastic materials, respectively. Recovered porosity values are in excellent agreement with true porosity found from the corresponding micro-CT images.

Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres

This paper presents a closed-form expression for the homogenized longitudinal shear moduli of a linear elastic composite material reinforced by long, parallel, radially-graded circular fibres with a periodic arrangement. An imperfect linear elastic fibre-matrix interface is allowed. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the fibre domain is solved in closed form by applying the theory of hypergeometric functions, for new wide classes of grading profiles defined in terms of special functions. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. A parametric analysis investigating the influence of microstructural and material features on the effective moduli is presented. The feasibility of mitigating the shear stress concentration in the composite by tuning the fibre grading profile is shown.

Negative refraction, surface modes, and superlensing effect via homogenization near resonances for a finite array of split-ring resonators

We present a theoretical and numerical analysis of liquid surface waves (LSWs) localized at the boundary of a phononic crystal consisting of split-ring resonators (SRRs). We first derive the homogenized parameters of the fluid-filled structure using a three-scale asymptotic expansion in the linearized Navier-Stokes equations. In the limit when the wavelength of the LSW is much larger than the typical heterogeneity size of the phononic crystal, we show that it behaves as an artificial fluid with an anisotropic effective shear modulus and a dispersive effective-mass density. We then analyze dispersion diagrams associated with LSW propagating within an infinite array of SRR, for which eigensolutions are sought in the form of Floquet-Bloch waves. The main emphasis is given to the study of localized modes within such a periodic fluid-filled structure and to the control of low-frequency stop bands associated with resonances of SRRs. Considering a macrocell, we are able to compute the dispersion of LSW supported by a semi-infinite phononic crystal of SRRs. We find that the dispersion of this evanescent mode nearly sits within the first stop band of the doubly periodic structure. We further discover that it is linked to the frequency at which the effective-mass density of the homogenized phononic crystal becomes negative. We demonstrate that this surface mode displays the hallmarks of all-angle negative refraction and it leads to a superlensing effect. Last, we note that our homogenization results for the velocity potential can be applied mutatis mutandis to designs of electromagnetic and acoustic superlenses for transverse electric waves propagating in arrays of infinite conducting SRRs and antiplane shear waves in arrays of cracks shaped as SRRs.

Three-dimensional models of elastostatic deformation in heterogeneous media, with applications to the Eastern California Shear Zone

SUMMARY We present a semi-analytic iterative procedure for evaluating the 3-D deformation due to faults in an arbitrarily heterogeneous elastic half-space. Spatially variable elastic properties are modelled with equivalent body forces and equivalent surface traction in a ‘homogenized’ elastic medium. The displacement field is obtained in the Fourier domain using a semi-analytic Green function. We apply this model to investigate the response of 3-D compliant zones (CZ) around major crustal faults to coseismic stressing by nearby earthquakes. We constrain the two elastic moduli, as well as the geometry of the fault zones by comparing the model predictions to Synthetic Aperture Radar inferferometric (InSAR) data. Our results confirm that the CZ models for the Rodman, Calico and Pinto Mountain faults in the Eastern California Shear Zone (ECSZ) can explain the coseismic InSAR data from both the Landers and the Hector Mine earthquakes. For the Pinto Mountain fault zone, InSAR data suggest a 50 per cent reduction in effective shear modulus and no significant change in Poisson's ratio compared to the ambient crust. The large wavelength of coseismic line-of-sight displacements around the Pinto Mountain fault requires a fairly wide (∼1.9 km) CZ extending to a depth of at least 9 km. Best fit for the Calico CZ, north of Galway Dry Lake, is obtained for a 4 km deep structure, with a 60 per cent reduction in shear modulus, with no change in Poisson's ratio. We find that the required effective rigidity of the Calico fault zone south of Galway Dry Lake is not as low as that of the northern segment, suggesting along-strike variations of effective elastic moduli within the same fault zone. The ECSZ InSAR data is best explained by CZ models with reduction in both shear and bulk moduli. These observations suggest pervasive and widespread damage around active crustal faults.

An analytical model to study the effective stiffness of the composites with periodically distributed sphere particles

Abstract In order to analytically study the overall elastic stiffness of the composite containing periodically dispersed sphere particles, a new micro-mechanics model is developed in this paper. Three kinds of typical particle packing arrangements in the form of simple cubic lattice, body-centered cubic lattice and face-centered cubic lattice are considered and compared. The special characteristics of regular distribution are fully considered by incorporating the necessary geometrical symmetry conditions into strain Green’s function. It is found that particle arrangement obviously affects the macroscopic elastic response of such the kind of composite. Moreover, most of the predictions by the present model are in good agreement with the FEM computations. The effective Young’s modulus of BCC composite the effective shear modulus of SC composite are not in the range of the Hashin–Shtrikman bounds. The present model is also useful to verify some other numerical results mainly obtained by the unit-cell model, for instance, damage variables, matrix plasticity, etc.

On the effective electroelastic properties of microcracked generally anisotropic solids

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration.

Modelling of ER squeeze films at low amplitude oscillations

An experimental investigation of electrorheological squeeze film dynamics is presented for constant applied voltage and low strain amplitude. Both broadband random and sinusoidal motion are examined to explain complex film dynamics. Spectral results indicate a primarily elastic response with slip at the plate boundaries. By examining the evolution of an effective shear modulus over time, sinusoidal results show that slip at the boundaries is due to a solvent layer which may be modelled as a separate variable thickness squeeze film.

Analytic model of elastic metamaterials with local resonances

A unified analytic model for effective mass density, effective bulk modulus, and effective shear modulus is presented for elastic metamaterials composed of coated spheres embedded in a host matrix. The effective material properties are derived directly from the averages of local momentum, stress, and strain defined in a single doubly coated sphere. It is shown that the effective material parameters predicted by the proposed model are in excellent agreements with the coherent-potential approximation results at low filling fractions where the anisotropy of periodic structures can be neglected for elastic waves. The advantage of the proposed method is that it can reveal clearly the physical mechanism for negative effective material parameters induced by the resonant effect. It is found that negative effective mass density is induced by negative total momentum of the composite for a positive momentum excitation. Negative effective bulk modulus appears for composites with an increasing (decreasing) total volume under a compressive (tensile) stress. Negative effective shear modulus describes composites with axisymmetric deformation under an opposite axisymmetric loading. Numerical examples are also given to illustrate these mechanisms. These findings may be useful in design of elastic metamaterials.

INVESTIGATION OF ELASTIC INVERSION ATTRIBUTES USING THE EXPANSIBLE CLAY MODEL FOR WATER SATURATION

Quantitative X‐ray diffraction has been used to characterize water saturation levels in complex shaly sand reservoirs (i.e. shaly sands with infrequent carbonates and minor proportions of iron‐rich minerals such as pyrite and siderite). The results led to the design of a total expansible clay model for water saturation, which is similar in form to the Dual Water model except that the excess effect of the clay minerals has been accounted for by a volume‐conductivity relationship, rather than one of the usual volume‐porosity translations, effectively reducing the uncertainties in estimating water saturation. Given the ambiguities associated with predicting these petrophysical properties from data on rock properties, such as mineralogy, an investigation of the relationship of estimated water saturation based on the total expansible clay model to independently determined rock properties was undertaken using well log inversion and forward modelling techniques. The results show that there is consistency in the relationship between water saturation estimates made from the total expansible clay model and known elastic parameters such as primary and shear‐wave sonic velocity (Vp, Vs), bulk density (ρb) and impedance (I), when the Raymer‐Gardner‐Hunt model is used. Use of the Raymer‐Gardner‐Hunt model to reconstruct the required rock‐physics relationship avoids the classic limitation of the more advanced Gassman model, which assumes that the dry shear modulus is equivalent to the wet shear modulus (μdry=μwet).The present work raises further questions on the application of the Voigt‐Reuss‐Hill (VRH) limits, or the Hashin Shtrikman bounds for averaging the effective shear modulus of the dry matrix in complex shaly sand reservoirs, where a two‐mineral matrix is normally assumed. The study shows the inapplicability of the VRH or Hashin‐Shtrikman averaging techniques but provides a minor adjustment to the averaging that solves the problems faced in reconstructing the relationships between directly measured elastic properties and derived petrophysical properties for this type of reservoir rock.

A numerical upscaling procedure to estimate effective plane wave and shear moduli in heterogeneous fluid-saturated poroelastic media

An important loss effect in heterogeneous poroelastic Biot media is the dissipation mechanism due to wave-induced fluid flow caused by mesoscopic scale heterogeneities, which are larger than the pore size but much smaller than the predominant wavelengths of the fast compressional and shear waves. These heterogeneities can be due to local variations in lithological properties or to patches of immiscible fluids. For example, a fast compressional wave traveling across a porous rock saturated with water and patches of gas induces a smaller fluid-pressure in the gas patches than in the water-saturated parts of the material. This in turn generates fluid flow and slow Biot waves which diffuse away from the gas–water interfaces generating significant energy losses and velocity dispersion. To perform numerical simulations using Biot’s equations of motion, it would be necessary to employ extremely fine meshes to properly represent these mesoscopic heterogeneities and their attenuation effects on the fast waves. An alternative approach to model wave propagation in these type of Biot media is to employ a numerical upscaling procedure to determine effective complex P-wave and shear moduli defining locally a viscoelastic medium having in the average the same properties than the original Biot medium. In this work the complex P-wave and shear moduli in heterogeneous fluid-saturated porous media are obtained using numerical gedanken experiments in a Monte Carlo fashion. The experiments are defined as local boundary value problems on a reference representative volume of bulk material containing stochastic heterogeneities characterized by their statistical properties. These boundary value problems represent compressibility and shear tests needed to determine these moduli for a given realization. The average and variance of the phase velocities and quality factors associated with these moduli are obtained by averaging over realizations of the stochastic parameters. The Monte Carlo realizations were stopped when the variance of the computed quantities stabilized at an almost constant value. The approximate solution of the local boundary value problems was obtained using a Galerkin finite element procedure, and the method was validated by reproducing known solutions in the case of periodic layered media. For the spatial discretization, standard bilinear finite element spaces are employed for the solid phase, while for the fluid phase the vector part of the Raviart–Thomas–Nedelec mixed finite element space of order zero was used. Results on the uniqueness of the continuous and discrete problems as well as optimal a priori error estimates for the Galerkin finite element procedure are derived. Numerical experiments showing the implementation of the procedure to estimate the average and variance of the fast compressional and shear phase velocities and inverse quality factors in these kind of highly heterogeneous fluid-saturated porous media are presented.

Exact solution for orthotropic materials weakened by doubly periodic cracks of unequal size under antiplane shear

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.

Micromechanical Modeling of Honeycomb Structures Based on a Modified Couple Stress Theory

A micromechanics model is developed for hexagonal honeycomb structures by using a modified couple stress theory. The formulation is based on structural analysis of repeating units and a homogenization procedure. Both bending and axial deformations of cell walls are considered here, unlike earlier models. The closed-form expressions for five effective in-plane elastic constants (including two Young's moduli, one shear modulus and two Poisson's ratios) of a honeycomb structure are derived in terms of the Young's modulus of the cell wall material and the slenderness (length-to-thickness) ratio of the cell wall. The formulas for two length scale parameters are obtained in terms of the slenderness ratio and thickness of the cell wall, thereby providing a direct measure of the microstructure of the honeycomb. The constitutive relations for the two-dimensional couple stress continuum equivalently representing the discrete honeycomb structure are provided using the five effective elastic constants and two length scale parameters. The numerical results reveal that for honeycombs with small slenderness ratios the effective Young's moduli and shear modulus are strongly dependent on the slenderness ratio. For low-density honeycombs with large slenderness ratios, the effective elastic constants predicted by the current model agree well with those by an existing model that considers only bending deformations.

On estimates for the effective shear modulus of cubic crystal aggregates

AbstractTo avoid time–consuming computations, several analytical approaches can be used to estimate the elastic properties of polycrystalline aggregates. In this paper we compute some well–known bounds and estimates for the effective shear modulus of aggregates of cubic crystals and compare them with results from finite element simulations using a representative volume element (RVE). It is shown that among the evaluated approaches, the singular approximation results in the best agreement with the RVE simulations for the examined materials. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Effective elastic moduli of three-phase composites with randomly located and interacting spherical particles of distinct properties

A micromechanical analytical framework is presented to predict effective elastic moduli of three-phase composites containing many randomly dispersed and pairwisely interacting spherical particles. Specifically, the two inhomogeneity phases feature distinct elastic properties. A higher-order structure is proposed based on the probabilistic spatial distribution of spherical particles, the pairwise particle interactions, and the ensemble-volume homogenization method. Two non-equivalent formulations are considered in detail to derive effective elastic moduli with heterogeneous inclusions. As a special case, the effective shear modulus for an incompressible matrix containing randomly dispersed and identical rigid spheres is derived. It is demonstrated that a significant improvement in the singular problem and accuracy is achieved by employing the proposed methodology. Comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered. Moreover, numerical examples are implemented to illustrate the potential of the present method.