Micro-inertia Effects Research Articles

Strain-rate dependence for Ni/Al hybrid foams

Shock absorption often needs stiff but lightweight materials that exhibit a large kinetic energy absorption capability. Open-cell metal foams are artificial structures, which due to their plateau stress, including a strong hysteresis, can in principle absorb large amounts of energy. However, their plateau stress is too low for many applications. In this study, we use highly novel and promising Ni/Al hybrid foams which consist of standard, open-cell aluminium foams, where nanocrystalline nickel is deposited by electrodeposition as coating on the strut surface. The mechanical behaviour of cellular materials, including their behaviour under higher strain-rates, is governed by their microstructure due to the properties of the strut material, pore/strut geometry and mass distribution over the struts. Micro-inertia effects are strongly related to the microstructure. For a conclusive model, the exact real microstructure is needed. In this study a micro-focus computer tomography (μCT) system has been used for the analysis of the microstructure of the foam samples and for the development of a microstructural Finite Element (micro-FE) mesh. The microstructural FE models have been used to model the mechanical behaviour of the Ni/Al hybrid foams under dynamic loading conditions. The simulations are validated by quasi-static compression tests and dynamic split Hopkinson pressure bar tests.

Torsional vibrations of a column of fine-grained material: A gradient elastic approach

The gradient theory of elasticity with damping is successfully employed to explain the experimentally observed shift in resonance frequencies during forced harmonic torsional vibration tests of columns made of fine-grained material from their theoretically computed values on the basis of the classical theory of elasticity with damping. To this end, the governing equation of torsional vibrations of a column with circular cross-section is derived both by the lattice theory and the continuum gradient elasticity theory with damping, with consideration of micro-stiffness and micro-inertia effects. Both cases of a column with two rotating masses attached at its top and bottom, and of a column fixed at its base carrying a rotating mass at its free top, are considered. The presence of both micro-stiffness and micro-inertia effects helps to explain the observed natural frequency shift to the left or to the right of the classical values depending on the nature of interparticle forces (repulsive or attractive) due to particle charge. A method for using resonance column tests to determine not only the shear modulus but also the micro-stiffness and micro-inertia coefficients of gradient elasticity for fine-grained materials is proposed.

Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory

This paper presents a new model for the free transverse vibrations of an Euler–Bernoulli beam using the couple stress theory of elasticity with micro-structure. Introducing the kinematic variables, the strain and kinetic energy expressions (involving micro-inertia effect) have been obtained and the Hamilton principle has been used to derive the governing equations and the related boundary conditions of the free vibrations of fixed–fixed and simply supported beams. A numerical solution has been used to study the natural frequencies, mode shapes and free vibrations of the beams. A comparative result has shown that the bending rigidity predicted by the couple stress, is closer to the experiment result than that predicted by the modified couple stress theory. The results have shown that the bending rigidity of the beams depends on the ratio of the length scale to the beam thickness, whereas the micro-inertia term depends on the ratio of the length scale to the beam length.

Generalized Thermoelastic Models for Linear Elastic Materials with Micro-Structure Part I: Enhanced Green–Lindsay Model

In this article, we derive constitutive thermoelastic models for linear elastic materials with micro-structure. The elastic behavior is assumed to be consistent with Mindlins’ Form II gradient elasticity theory, whereas for the thermal behavior the generalization of Clausius-Duhem inequality, proposed by Green and Laws, is adopted. The resulting model is actually a generalization of the thermoelastic theory of Green and Lindsay for linear elastic materials with micro-structure, taking into account micro-inertia effects, as well. It is demonstrated that classical thermoelasticity models are retrieved from the present general formulation, when some of the model constants are set to zero. Finally, the uniqueness of solution for the general case of anisotropic materials is proved.

Generalized Thermoelastic Models for Linear Elastic Materials with Micro-Structure Part II: Enhanced Lord–Shulman Model

In this article, we derive constitutive thermoelastic models for linear elastic materials with micro-structure. The elastic behavior is assumed to be consistent with Mindlin Form II, whereas for the thermal behavior, the generalization of Fourier–Duhamel law proposed by Maxwell-Vernotte-Cattaneo is adopted. The resulting model is actually a generalization of the thermoelastic theory of Lord and Shulman, suitable for linear elastic materials with micro-structure taking into account micro-inertia effects as well. Uniqueness of the solution for the general case of anisotropic materials is proved. An application example is analyzed by means of the Finite Element method and comparisons are made with the generalized Green–Lindsay model derived in “Generalized Thermoelastic Models for Linear Elastic Materials with Micro-Structure Part I: Enhanced Green–Lindsay Model,” also in this issue.

Constitutive behavior of porous ductile materials accounting for micro-inertia and void shape

The goal of this paper is to characterize the mechanical behavior of porous materials by taking into account the void shape and micro-inertia effects. A representative volume element (RVE) defined by two confocal prolate spheroids is used to describe the porous material. The matrix behavior is assumed to be rigid and non linear viscous. Based on the work of Molinari and Mercier (2001), the macroscopic stress is decomposed into static and dynamic parts. In the present work the static contribution is described by the Gologanu et al. model (1993). The dynamic stress is obtained by choosing the trial velocity field proposed by Gologanu et al. (1993). With the proposed modeling a link is established between the macroscopic dynamic stress, on the one hand and, on the other hand, the macroscopic strain rate tensor and its time derivative. To validate our model, finite element simulations have been performed. Two shapes of void (spherical and prolate with an aspect ratio of 5) and two volume fraction of voids (0.001 or 0.1) are considered. The influence of micro-inertia on the macroscopic flow stress surface is analyzed and it is shown that the flow surface obtained by the analytical approach is in good agreement with finite element computations.

A high order homogenization model for transient dynamics of heterogeneous media including micro-inertia effects

This manuscript presents a multi-dimensional, high order homogenization model for elastic composite materials subjected to dynamic loading conditions. The proposed model is derived based on the asymptotic homogenization method with multiple spatial scales. The high order homogenization model permits implementation using the standard finite elements and shape functions since it does not involve higher order terms typically present in dispersion models. The high order homogenization model can accurately capture wave dispersion in the presence of non-uniform density and non-uniform moduli within the material microstructure. Employing the Hybrid Laplace Transform/Finite Element Method, both displacement and traction boundary conditions for the macroscopic problem have been implemented. Finite element formulations for the first and second order influence functions, and the macroscale initial boundary value problem are presented. The performance of the model is verified by comparing model predictions to the local homogenization and the direct numerical simulations. The high order homogenization model is shown to predict the wave dispersion with very reasonable accuracy and cost. The proposed model can also capture the phononic bands – frequency bands within which the micro-inertia effects completely block wave propagation.

Modelling 2D wave motion in microstructured solids

The Mindlin-type model is used for describing the deformation waves in microstructured solids. The 2D evolution equation (one-wave equation) is derived based on hierarchical governing equations by using the perturbation method. This equation is of the Zabolotskaya–Khokhlov-type and is integrated numerically under localized initial conditions (related to appropriate boundary value problems) by the FFT-based pseudospectral method. Analysis of results demonstrates that the derived evolution equation is able to grasp essential nonlinear effects of microinertia and elasticity of microstructure. Such an equation can be used in the NDT of microstructured materials in order to model wave beams generated by ultrasonic transducers.

New hybrid foam materials for impact protection

The present paper investigates the dynamic compression and impact behaviour of a new class of open-cell Ni/Al-hybrid metal foams (nickel strengthened aluminium foams) in comparison to pure aluminium foams. In order to determine the characteristics of the pure aluminium and Ni/Al-hybrid foams, dynamic compression tests using a classical split Hopkinson pressure bar have been performed at strain rates up to 5000 s−1. Whereas the pure aluminium foams show only slight strain rate sensitivity, the hybrid foams are highly strain rate sensitive. The stress strain characteristics show a change in the deformation behaviour of the foams from bending dominated failure under quasi-static compression to a failure mode induced by microinertia effects under dynamic loading with additional stretching. As a fact of this additional stretching the hybrid foams are able to dissipate more energy under dynamic loading.Further ballistic impact tests have been performed on foam sandwich panels at impact velocities of 300 m/s. The hybrid foams show a significant improved ballistic protection performance especially in the case of elastomeric fillings. The paper outlines the possible application of Ni/Al-hybrid foams as crash absorber, security panels in case of blast and ballistic impact.

Performance of closed-cell aluminum foams subjected to impact loading

An experimental study has been conducted to examine the impact behavior, deformation mode and energy absorption ability during axial crushing of aluminum foams. Aluminum foams with two different components, manufactured by molten body transitional foaming process, were used in the experiment. Impacts were imparted with an Instron 9250HV impact system and with the impact energy of 150J. The results showed that the dynamic impact contains two steps: initial compression and gradual crushing. In the displacement–load curve, the enhancement of peak load was due to the micro-inertia effect. Two deformation modes existed in specimens as a result of wave propagation and reflection. During impact, the increasing foam density resulted in a higher value of specific energy absorption. The energy absorption ability of 6063 aluminum alloys was superior to that of the pure aluminum foam.

Constitutive modeling of rate dependence and microinertia effects in porous-plastic materials with multi-sized voids (MSVs)

Micro-voids of varying sizes exist in most metals and alloys. Both experiments and numerical studies have demonstrated the critical influence of initial void sizes on void growth. The classical Gurson–Tvergaard–Needleman model summarizes the influence of voids with a single parameter, namely the void-volume fraction, excluding any possible effects of the void-size distribution. We extend our newly proposed model including the multi-sized void (MSV) effect and the void-interaction effect for the capability of working for both moderate and high loading rate cases, where either rate dependence or microinertia becomes considerable or even dominant. Parametric studies show that the MSV-related competitive mechanism among void growth leads to the dependence of the void growth rate on void size, which directly influences the void's contribution to the total energy composition. We finally show that the stress–strain constitutive behavior is also affected by this MSV-related competitive mechanism. The stabilizing effect due to rate sensitivity and microinertia is emphasized.

Wave propagation in 3-D poroelastic media including gradient effects

In the framework of the theory of mixtures, the governing equations of motion of a fluid-saturated poroelastic medium including microstructural (for both the solid and the fluid) and micro-inertia (for the solid) effects are derived. This is accomplished by appropriately combining the conservation of mass and linear momentum equations with the constitutive equations for both the solid and the fluid constituents. The solid is assumed to be gradient elastic, that is, its stress tensor depends on the strain and the second gradient of strain tensor. The fluid is assumed to have an analogous behavior, that is, its stress tensor depends on the pressure and the second gradient of pressure. A micro-inertia term in the form of the second gradient of the acceleration of the solid is also included in the equations of motion. The equations of motion in three dimensions are seven equations with seven unknowns, the six displacement components for the solid and the fluid and the pore-fluid pressure. Because of the microstructural effects, the order of these equations is two degrees higher than in the classical case. Application of the divergence and the rot operations on these equations enable one to study the propagation of plane harmonic waves in the infinitely extended medium separately in the form of dilatational and rotational dispersive waves. The effects of the microstructure and the micro-inertia on the dispersion curves are determined and discussed.

Effects of microscale inertia on dynamic ductile crack growth

The aim of this paper is to investigate the role of microscale inertia in dynamic ductile crack growth. A constitutive model for porous solids that accounts for dynamic effects due to void growth is proposed. The model has been implemented in a finite element code and simulations of crack growth in a notched bar and in an edge cracked specimen have been performed. Results are compared to predictions obtained via the Gurson–Tvergaard–Needleman (GTN) model where micro-inertia effects are not accounted for. It is found that microscale inertia has a significant influence on the crack growth. In particular, it is shown that micro-inertia plays an important role during the strain localisation process by impeding void growth. Therefore, the resulting damage accumulation occurs in a more progressive manner. For this reason, simulations based on the proposed modelling exhibit much less mesh sensitivity than those based on the viscoplastic GTN model. Microscale inertia is also found to lead to lower crack speeds. Effects of micro-inertia on fracture toughness are evaluated.

A micromechanical constitutive model for dynamic damage and fracture of ductile materials

This paper proposes a detailed theoretical analysis of the development of dynamic damage in plate impact experiments for the case of high-purity tantalum. Our micro-mechanical model of damage is based on physical mechanisms (void nucleation and growth). The model is aimed to be general enough to be applied to a variety of ductile materials subjected to high tensile pressure loading. In this respect, the work of Czarnota et al. (J Mech Phys Solids 56:1624–1650, 2008) has been extended by introducing the concept of nucleation law and by entering a nonlinear formulation of the elastic response based on the Mie-Gruneisen equation of state. This later aspect allows us to consider high impact velocities. All model parameters are directly assessed by experimental measurements to the exception of the nucleation law which is characterized by the way of an inverse identification method using three free-surface velocity profiles (at low, intermediate and high impact velocities). It is shown that the nucleation law can be consistently determined in the range of operating pressures. The nucleation law being identified, the development of internal damage happens to be a natural outcome of the modelling. The model is applied to predict damage development and free-surface velocity profiles for various test conditions. The variety and the quality of results support the physical basis (in particular micro-inertia effects) upon which the proposed model of dynamic damage is based.

Dynamic finite element analysis of a gradient elastic bar with micro-inertia

We study the dynamic behavior of a first strain gradient elastic bar with micro-inertia by means of the finite element method. The partial differential equation describing the motion of the bar expressed in terms of displacement is of fourth order with respect to the spatial variable, therefore, for the standard Galerkin formulation, Hermite elements are required. Consistent mass matrices are employed. The results are validated by comparison with some special cases where the exact solutions are derivable. Dispersion relations for the longitudinal waves are also derived. The effect of micro-inertia in the dynamic response of the bar is analyzed and comparisons are made with the classical elastic case. It is found that the micro-inertia parameter considerably affects the dynamic response of the bar and the dispersion characteristics of longitudinal waves.

Effect of microscopic deformation mechanisms on the dynamic response of soft cellular materials

Cellular materials show progressive or uniform collapse during impact loading depending on their microstructural and material properties. It is generally agreed that a complex interplay among microinertia, microbuckling and microbending of the cell walls of these materials plays an important role in determining their macroscopic stress–strain response. However, an evaluation of the dependency of the overall deformation behavior on these parameters requires sophisticated modeling approach due to extremely fast and complex wave propagation events occurring during dynamic deformation. We have developed a transient finite element based computational framework that can examine the contribution of each of these effects on the deformation history of this class of materials. An in-depth parametric study for different loading, microstuctural and material parameters has been undertaken in this study. Our significant finding is that at high strain rate, shorter pulse rise times lead to higher microinertial stress enhancement due to an increase in apparent microbuckling strength. A variation of cell size shows insignificant effect of microinertia and microbuckling at initial stage but localization can be found at later stage of deformation due to increasing microbuckling and microbending activities. Deformation localization occurs in lower Young’s modulus specimens due to lower buckling and bending strength of the cell walls. A significant inertial stress enhancement can be noticed in the specimens with higher bulk density of the constituent material leading to increased microbuckling activities resulting in localized collapse at the impact end.

On one-dimensional solitary waves in microstructured solids

For describing the longitudinal deformation waves in microstructured solids a Mindlin-type model is used. The embedding of a microstructure in an elastic material is reflected in an inherent length scale causing dispersion of propagating waves. Nonlinear effects, if taken into account, will counteract dispersion. A suitable balance between nonlinearity and dispersion may permit the propagation of solitary waves. Following previous work by Engelbrecht and others the nonlinear hierarchical model is derived in a one-dimensional setting which corresponds to a two-wave equation. The evolution equation as a simplified model, representing a one-wave equation, is able to grasp the essential effects of microinertia and elasticity of the microstructure. It is shown that the nonlinearity in microscale leads to an asymmetry of the wave profile. The nonlinear evolution equation as an extended Korteweg–de Vries equation is solved approximately by a series expansion in a small parameter representing the micro-nonlinearity. Already the first approximation indicates the asymmetry of the solitary waves. It is shown that solitary waves will propagate only if the micro-nonlinearity does not exceed some upper bound. For the limiting case, an analytical solution of the extended Korteweg–de Vries equation can be provided and used as a reference for the approximate solutions.

A Finite Element Model for Calculating the Stresses in Bars with Microstructure Loaded by Ultra-Short Laser Pulses

A finite element solution for the thermal stress field, generated in elastic bars with micro-structure is presented in this paper. The heat transfer phenomenon is governed by the non-Fourier law of Maxwell–Vernotte–Cattanneo. Heat losses due to free convection are supposed to occur along the bar. The micro-structure is taken into account in the elastic model by adopting the first strain gradient model. As a first approximation, the effects of micro-inertia are ignored. Furthermore, the thermal and mechanical fields are assumed uncoupled, so that the energy equation is independent of any strain rates. As a model problem, we consider a micro-bar, mechanically fixed and thermally insulated at both ends, stimulated by an ultra-short laser pulse. For the spatial discretization of the heat equation, we use quadratic C 0-continuous Lagrange elements, while for the corresponding gradient elasticity equations, Hermite C 1-continuous elements are employed. The semi-discrete system of equations is integrated in time using the implicit Newmark method. However, for this class of wave propagation problems, the explicit Newmark or equivalently, the central difference method is the most suitable. Nonetheless, some sort of numerical smoothing procedures are needed to avoid the numerical oscillations which are not physical.

On modelling wave motion in microstructured solids

The Mindlin-type model is used for describing the longitudinal deformation waves in microstructured solids. The evolution equation (one-wave equation) is derived for the hierarchical governing equation (two-wave equation) in the nonlinear case using the asymptotic (reductive perturbation) method. The evolution equation is integrated numerically under harmonic as well as localized initial conditions making use of the pseudospectral method. Analysis of the results demonstrates that the derived evolution equation is able to grasp essential effects of microinertia and elasticity of a microstructure. The influence of these effects can result in the emergence of asymmetric solitary waves.

A variational constitutive model for soft biological tissues

In this paper, a fully variational constitutive model of soft biological tissues is formulated in the finite strain regime. The model includes Ogden-type hyperelasticity, finite viscosity, deviatoric and volumetric plasticity, rate and microinertia effects. Variational updates are obtained via time discretization and pre-minimization of a suitable objective function with respect to internal variables. Genetic algorithms are used for model parameter identification due to their suitability for non-convex, high dimensional optimization problems. The material behavior predicted by the model is compared to available tests on swine and human brain tissue. The ability of the model to predict a wide range of experimentally observed behavior, including hysteresis, cyclic softening, rate effects, and plastic deformation is demonstrated.