Elastic Constitutive Relation Research Articles

In-plane buckling of prismatic funicular arches with shear deformations

The in-plane buckling behavior of funicular arches is investigated numerically in this paper. A finite strain Timoshenko beam-type formulation that incorporates shear deformations is developed for generic funicular arches. The elastic constitutive relationships for the internal beam actions are based on a hyperelastic constitutive model, and the funicular arch equilibrium equations are derived. The problems of a submerged arch under hydrostatic pressure, a parabolic arch under gravity load and a catenary arch loaded by overburden are investigated. Buckling solutions are derived for the parabolic and catenary arch. Subsequent investigation addresses the effects of axial deformation prior to buckling and shear deformation during buckling. An approximate buckling solution is then obtained based on the maximum axial force in the arch. The obtained buckling solutions are compared with the numerical solutions of Dinnik (Stability of arches, 1946) [1] and the finite element package ANSYS. The effects of shear deformation are also evaluated.

A Finite Element Model for the Prediction of Force Reduction of Athletics Tracks

The Force Reduction (FR) impact test, performed by means of an apparatus called artificial athlete, has been chosen by IAAF (International Association of Athletics Federations) as a standard to evaluate the performance of athletics tracks. The test procedure consists in dropping a mass on the sample, recording the evolution of the impact force and taking its maximum value normalized with respect to a reference one. In this work a Finite Element (FE) model of the FR test was developed to investigate the effects of sample thickness and material properties. Two athletics tracks and, for comparison, a sample of natural rubber were considered. Their mechanical behaviour was characterized, extrapolated to the strain rate of interest and modelled using hyperelastic constitutive equations. With data so derived a number of numerical (FE) simulations of the FR test on the three materials with varying thickness were performed. The FR values predicted by the simulations resulted to be in very good agreement with experimental FR data, particularly for thicknesses of practical interest. Finally, the suitability of an alternative model based on a linear elastic constitutive relationship was considered and results were discussed.

Three-dimensional elastic constitutive relations of aligned carbon nanotube architectures

Tailorable anisotropic intrinsic and scale-dependent properties of carbon nanotubes (CNTs) make them attractive elements in next-generation advanced materials. However, in order to model and predict the behavior of CNTs in macroscopic architectures, mechanical constitutive relations must be evaluated. This study presents the full stiffness tensor for aligned CNT-reinforced polymers as a function of the CNT packing (up to ∼20 vol. %), revealing noticeable anisotropy. Finite element models reveal that the usually neglected CNT waviness dictates the degree of anisotropy and packing dependence of the mechanical behavior, rather than any of the usually cited aggregation or polymer interphase mechanisms. Combined with extensive morphology characterization, this work enables the evaluation of structure-property relations for such materials, enabling design of aligned CNT material architectures.

Uniaxial Tension of a Class of Compressible Solids With Plastic Non-Normality

Abstract Motivated by a model that qualitatively captured the response of vertically aligned carbon nanotube (VACNT) pillars in uniaxial compression, we consider the uniaxial tensile response of a class of compressible elastic-viscoplastic solids. In Hutchens et al. [“Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech. Phys. Solids, 59, pp. 2227–2237 (2011), Erratum 60, 1753–1756 (2012)] an elastic viscoplastic constitutive relation with plastic compressibility, plastic non-normality, and a hardening-softening-hardening hardness function was used to model experimentally obtained uniaxial compression data of cylindrical VACNT micropillars. Complex deformation modes were found in uniaxial compression, which include a sequential buckling-like collapse of the type seen in experiments. These complex deformation modes led to the overall stress-strain signature of the pillar not being of the same form as the input material hardness function. A fundamental question that motivates exploring the deformation of this class of materials—both experimentally and theoretically—is how to extract the intrinsic material response from simple tests. In this study we explore the relation between the input material response and the overall stress strain behavior in uniaxial tension using the constitutive framework of Hutchens et al. A simple one-dimensional analysis reveals the types of instability modes to be expected. Dynamic, finite deformation finite element calculations are carried out to explore the dependence of diffuse necking, localized necking, and propagating band deformation modes on characteristics of the hardness function. Attention is devoted to uncovering implications for obtaining intrinsic material properties of complex hierarchical structures; for example, vertically aligned carbon nanotubes (VACNTs), from uniaxial tension experiments.

Full elastic constitutive relation of non-isotropic aligned-CNT/PDMS flexible nanocomposites

The elastic response of vertically aligned-carbon nanotube/polydimethylsiloxane (A-CNT/PDMS) nanocomposites is presented in this study and related to the underlying aligned-CNT morphology. Multiwalled carbon nanotubes (MWCNTs) at 1% Vf are embedded in a flexible substrate of PDMS to create a flexible polymer nanocomposite (PNC). The PNC properties are evaluated using scanning electron microscopy (SEM), differential scanning calorimetry (DSC), dynamic mechanical analysis (DMA) and tensile mechanical tests, and the full linearly elastic constitutive relation is established for such a PNC. The results suggest that the CNTs retain the alignment after wetting and curing of PDMS. PDMS is significantly modified by the reinforcing aligned-CNT fibers, demonstrating non-isotropic (as opposed to the isotropic neat PDMS) elastic properties all different from PDMS (Young's modulus of 0.8 MPa), including an anisotropy ratio of 4.8 and increases in the modulus of A-CNT/PDMS over PDMS by more than 900% and 100%, in the CNT longitudinal and transverse directions, respectively. This study reports the first full constitutive relation that may be useful in modeling PNCs as composites or as elements of hierarchical nanoengineered composites, particularly PDMS-CNT PNCs are envisioned as elements in biomedical devices such as pressure transducers and energy harvesters.

Theoretic Analysis for Composite Elastic Modulus of Concrete-Filled Rectangular Steel Tube Member

Considering the hooping effect between steel tube and core concrete, established a small-deformation elastic analysis theory for the concrete-filled rectangle steel tube (CFRST) member, founded the elastic constitutive relations of CFRST, and accomplished the semi-analytical and semi-numerical solution of the composite elastic modulus. Research shows: composite elastic modulus of CFRST is higher than conversion elastic modulus without considering the hooping effect. Within certain value range of parameters, composite elastic modulus is 1.36% - 15.3% higher than conversion elastic modulus. Further more, as the steel ratio increases, the composite elastic modulus increases significantly, but the sensitivity of the composite elastic modulus to the strength of concrete is small.

Experimental Study on the Compressive Mechanical Performance of Bamboo Strip Plywood under Static Loads

The compressive mechanical performance of bamboo strip plywood was studied in the paper. Based on analysis of the related experiments datum , the principal compressive mechanical performance under static load were expounded perfectly, and a number of mechanical parameters such as compressive strength, compressive elastic modulus and compressive constitutive relations were obtained. It was shown that the mechanical performance of bamboo strip plywood was better than that of wood material such as Northeast larch, the compressive elastic modulus was lower than that of concrete, so that there existed a more great deformation when the bamboo beam structure was loaded, and the constitutive relations of bamboo materials showed that there was excellent ductility with an obvious characteristic of plastic deformation . This research result showed that there was a great significance on studying and application of bamboo bridges in the future.

An approximate spreadsheet integration method for foundation settlements in two-layered medium

Traditional method of calculating foundation settlement in layered soil medium using Boussinesq solution for stresses and using elastic properties of the layers over-estimates the settlements. A simple spreadsheet integration method is proposed here to rationally calculate the elastic settlements of shallow foundations in layered medium. Two layer systems with finite stiffer layers overlying a semi-infinite softer layer are considered. Stresses in the layered system are calculated by combining Boussinesq theory with concept of "equivalent depth" to account for the stress redistribution. In this method, vertical strains at mid-layers are determined using soil elastic constitutive relations. Next, the strains are integrated over the layer depths to determine layer settlements. For validation, soil stresses and foundation settlements of the proposed method are compared to those from finite element analyses. The comparison reveals that the proposed method is reasonably accurate and suitable for design engineers to calculate settlements in layered medium.

Nonlinear homogenization techniques to solve masonry structures problems

The behaviour of masonry material subjected to different in-plane loading combination is studied in this work. The masonry is considered as a periodic composite material composed by a regular distribution of brick and mortar and it is analyzed using a homogenization technique. The mechanical properties of the masonry, as an orthotropic homogeneous material, depend on the geometrical and mechanical properties of the components based on the study of the equilibrium and compatibility of a basic cell. The masonry is a frictional material and its behaviour depends on the loading direction, for these reasons, a unilateral damage model is chosen for the analysis. This model describes the behaviour of brittle materials subjected to tension–compression cyclic loads based on the introduction of two damage variables and it assumes that the damage is due to the beginning and growth of cracks only in the mortar joints. It is considered that the bricks have a linear elastic constitutive relationship. Numerical applications are performed with a nonlinear finite element code in order to test the proposed procedure by comparing the results with those available in the literature and also with experimental data.

Numerical Simulation on Rock Shock Accumulation Damage Effect

Shock accumulation damage failure of rock is often seen in engineering at present. The damage model based on stress wave attenuation is firstly adopted to improve the linear elastic constitutive relationship and load type used in original numerical manifold method program, then the influence of load intensity, action time and rock initial damage on accumulation damage is discussed with it. The simulation results show that: (1) Rock shock accumulation damage shows a nonlinearly rise with increase of load action time; (2) Rock shock accumulation damage shows a non-proportional rise with increase of load intensity, and there will be a load threshold which cannot produce accumulation damage; (3) The initial damage of rock has little influence on accumulation damage of time effect.

Linear elastic constitutive relation for multiphase porous media using microstructure superposition: Freeze–thaw soils

Abstract The constitutive relation derived in our previous work based on the microstructure superposition technique is implemented here for three-phase microstructure configuration to study the mechanical behavior of freeze–thaw soils. Three scenarios were considered for the frozen soils: frozen soil consisting of only two solids, soil skeleton and ice; frozen soil with soil skeleton, ice, and void; and frozen soil consisting soil skeleton, ice and pores saturated with fluid. The frozen soil studied is Alaska frozen soil mainly consisting of clay and silt particles at temperature about −10 °C. The effective elastic constants were calculated for the media under each scenario using two sets of elastic constants of soil skeleton (clay mineral). The modeled results were compared with Hashin–Shtrikman's upper bound solution and the experimentally measured data. In addition to be able to model the mechanical behavior of freeze–thaw soils, the derived constitutive relation, as indicated in the results of this study, could also be used as a tool in determining the microstructure of freeze–thaw soils by measuring the elastic constants of soil skeleton, the elastic properties and unfrozen water content of the media.

An automatic time increment selection scheme for simulation of elasto-viscoplastic consolidation of clayey soils

An automatic time increment selection scheme for numerical analysis of long-term response of geomaterials is presented. The scheme is simple, rational and stable. Governed by a simple empirical expression, it can adaptively adjust the time increments depending on the strain rate-dependent temporal history of the material response. The proposed expression requires only a few parameters whose selection is a trivial task since they have a small effect on accuracy but have a significant effect on computational efficiency. This generalization has been made possible because of the enforcement of certain predefined control criteria to avoid extreme conditions. If any of the control criteria is satisfied, the computation is restarted by going a few time steps back to ensure the smoothness of the computed responses and time increments are again continuously adjusted through the governing equation provided. Performance of the automatic time increment selection scheme is investigated through finite element analyses of the long-term consolidation response of clay under different geotechnical profiles and loading conditions. Both elastic and elasto-viscoplastic constitutive relations are considered, including the consideration of the destructuration effects of geomaterials. Numerical results show that the performance of the automatic time increment selection scheme is reasonably excellent. While offering reasonable accuracy of the numerical solution, it can ensure temporal stability at optimal computational efficiency. In addition to the Euler implicit method, the automatic time increment selection scheme also performs well even when the explicit fourth-order Runge–Kutta method is employed for the integration of time derivatives.

Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

AbstractThis paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

Linear elastic constitutive relation for multiphase porous media using microstructure superposition: Dry media

A constitutive relation for multiphase porous media containing a phase-change constituent is developed based on a periodic representative unit cell, microstructure superposition technique, and homogenization scheme. Multiphase porous mediae, consisting of different solid phases, fluid phases, and phase-change constituents, play an important role in variety engineering applications. A microstructure superposition method, stemmed from linear superposition theory, is devised such that the constitutive relation of the media can be obtained in an analytical form and the derivation is straightforward and tractable. The original microstructure of the media is firstly divided into several simple microstructure configurations. The microscopic stresses and strains are obtained for each individual microstructure configuration based on its own equilibrium and equivalent equations, and complex Fourier series expansion of strains. The microscopic strains are then superposed together to obtain the constitutive relation of the multiphase porous media using homogenization scheme. The derived constitutive relation can be applied to study the mechanical behavior of porous media consisting of two solid phases with one being able to change phase between solid and liquid and one liquid phase. As a particular application of the derived constitutive relation, the mechanical behavior of porous media with a single solid phase under dry condition was studied and the results were consistent with other models and agreed well with available experimental data.

Cosserat model for periodic masonry deduced by nonlinear homogenization

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.

A Nonlinear Transformation Field Procedure for Periodic Masonry Based on an Equivalent Cosserat Medium

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The masonry is considered as a composite material obtained as a regular distribution of blocks connected by horizontal and vertical mortar joints. The macromechanical equivalent Cosserat medium is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for the masonry components. In particular, linear elastic constitutive relationship is considered for the blocks, while nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of the masonry material.

Experimental tests and numerical modeling of reinforced masonry arches

The present paper deals with the experimental campaign and the numerical modeling of reinforced masonry arches strengthened with fiber reinforced plastic (FRP) strips. The mechanical properties of masonry material constituents, clay bricks and mixed mortar, are determined through laboratory tests. The experimental response of unreinforced and reinforced arches is investigated. A nonlinear elastic constitutive law for the masonry material is proposed; in particular, the stress–strain relationship is characterized by no-tensile strength and by a quadratic response with exponential post-peak softening in compression. The FRP reinforcement is modeled as a no-compressive material with uniaxial linear elastic constitutive relationship in tension, until a limit strength response characterized by brittle collapse. A beam finite element is developed and a numerical procedure, based on the Newton–Raphson method, is implemented to solve the nonlinear structural problem. The question of the proper setting of the material parameters to consider in the numerical analyses is addressed. Results obtained using the numerical procedure are compared with the experimental ones in order to assess the ability of the proposed procedure and of the model in predicting the failure mechanisms and the global structural response.

Thermodynamic consistency of third grade finite strain elasticity

The thermodynamic framework of finite strain viscoelasticity with second order weak nonlocality in the deformation gradient is investigated. The application of Liu's procedure leads to a class of third grade elastic materials where the second gradient of the stress appears in the elastic constitutive relation. Finally the dispersion relation of longitudinal plane waves is calculated in isotropic materials.

First-order hyperbolic form of velocity-stress equations for waves in elastic solids with hexagonal symmetry

This paper reports a theoretical framework to analyze wave propagation in elastic solids of hexagonal symmetry. The governing equations include the equations of motions and partial differentiation of elastic constitutive relations with respect to time. The result is a set of nine, first-order, fully-coupled, hyperbolic partial differential equations with velocities and stress components as the unknowns. The equation set is then cast into a vector form with three 9 × 9 coefficient (or Jacobian) matrices. Physics of wave propagation are fully described by the eigen structure of these matrices. In particular, the eigenvalues of the Jacobian matrices are the wave speeds and a part of the left eigenvectors represents the wave polarization. Without invoking the plane wave solution and the Christoffel equation, two- and three-dimensional slowness profiles can be calculated. As an example, slowness profiles of a cadmium sulfide crystal are presented.

Reinvestigation of the effect of plasticity on delamination process of Si/Cu interface in a ductile nano-cantilever

Previous experimental investigations (Hirakata et al. Int J Fract 145:261–271, 2007) have demonstrated that Si/Cu/SiN/Pt/C nano-cantilever is delaminated along the interface between Cu and Si layers when subjected to monotonically bending load, and the measured load-displacement relationship shows a nonlinear behavior. Based on the continuum mechanics model, this study carries out numerical simulations on the crack nucleation and propagation along the Si/Cu interface in order to clarify the effect of plasticity on the fracture behavior of the ductile nano-component. Exponential type of cohesive zone model (CZM) combined with finite element method was adopted to characterize the constitutive relationship of the Si/Cu interface. Two sets of simulations are performed, i.e., Cu layer obeys either linear elastic or Ramberg–Osgood elasto-plastic constitutive relation. The characteristic parameters of interfacial adhesion are extracted through calibration via experimental results. The simulation results indicate that (i) cohesive strength and work of separation are the dominating CZM parameters, and the exponential CZM is suitable for describing the interfacial delamination between the Cu and Si film layers; (ii) the Cu film layer in this nano-cantilever more favorably obeys a linear elastic constitutive relation; (iii) comparing to bulk Cu, nano-scale Cu has a much higher yield stress and hardening rate, which leads to little plastic deformation of the nano-cantilever specimen during the entire delamination process. The numerical predictions are in good agreement with the experimental results, wherein brittle fracture occurred during the Si/Cu interfacial delamination. And the nonlinear load-displacement behavior observed by the tests may be due to the cohesive law of the Si/Cu interface, instead of the plastic deformation of the Cu film layer.