On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—Theory

This work presents an analytical framework for determining the overall constitutive response of elastomers that are reinforced by rigid or compliant fibers, and are subjected to finite deformations. The framework accounts for the evolution of the underlying microstructure, including particle rotation, which results from the finite changes in geometry that are induced by the applied loading. In turn, the evolution of the microstructure can have a significant geometric softening (or hardening) effect on the overall response, leading to the possible development of macroscopic instabilities through loss of strong ellipticity of the homogenized incremental moduli. The theory is based on a recently developed “second-order” homogenization method, which makes use of information on both the first and second moments of the fields in a suitably chosen “linear comparison composite,” and generates fairly explicit estimates—linearizing properly—for the large-deformation effective response of the reinforced elastomers. More specific applications of the results developed in this paper will be presented in Part II.     

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: II—Results

In Part I of this paper, we developed a homogenization-based constitutive model for the effective behavior of isotropic porous elastomers subjected to finite deformations. In this part, we make use of the proposed model to predict the overall response of porous elastomers with (compressible and incompressible) Gent matrix phases under a wide variety of loading conditions and initial values of porosity. The results indicate that the evolution of the underlying microstructure—which results from the finite changes in geometry that are induced by the applied loading—has a significant effect on the overall behavior of porous elastomers. Further, the model is in very good agreement with the exact and numerical results available from the literature for special loading conditions and generally improves on existing models for more general conditions. More specifically, we find that, in spite of the fact that Gent elastomers are strongly elliptic materials, the constitutive models for the porous elastomers are found to lose strong ellipticity at sufficiently large compressive deformations, corresponding to the possible onset of “macroscopic” (shear band-type) instabilities. This capability of the proposed model appears to be unique among theoretical models to date and is in agreement with numerical simulations and physical experience. The resulting elliptic and non-elliptic domains, which serve to define the macroscopic “failure surfaces” of these materials, are presented and discussed in both strain and stress space.     

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: I—Analysis

The purpose of this paper is to provide homogenization-based constitutive models for the overall, finite-deformation response of isotropic porous rubbers with random microstructures. The proposed model is generated by means of the “second-order” homogenization method, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” The constitutive model takes into account the evolution of the size, shape, orientation, and distribution of the underlying pores in the material, resulting from the finite changes in geometry that are induced by the applied loading. This point is key, as the evolution of the microstructure provides geometric softening/stiffening mechanisms that can have a very significant effect on the overall behavior and stability of porous rubbers. In this work, explicit results are generated for porous elastomers with isotropic, (in)compressible, strongly elliptic matrix phases. In spite of the strong ellipticity of the matrix phases, the derived constitutive model may lose strong ellipticity, indicating the possible development of shear/compaction band-type instabilities. The general model developed in this paper will be applied in Part II of this work to a special, but representative, class of isotropic porous elastomers with the objective of exploring the complex interplay between geometric and constitutive softening/stiffening in these materials.     

An atomistic investigation of the kinetics of detwinning

This paper presents a “first principles” atomistic study of the dynamics of detwinning in a shape-memory alloy. In order to describe the macroscopic motion of twin boundaries, the continuum theory of twinning must be provided with a “kinetic relation”, i.e. a relation between the driving force and the propagation speed. This kinetic relation is a macroscopic characterization of the underlying atomistic processes. The goal of the present atomistic study is to provide the continuum theory with this kinetic relation by extracting the essential macroscopic features of the dynamics of the atoms. It also aims to elucidate the mechanism underlying the process of detwinning.The material studied is stoichiometric nickel-manganese, and interatomic interactions are described using three physically motivated Lennard-Jones potentials. The effect of temperature and shear stress on detwinning — specifically on the rate of transformation from one variant of martensite to the other — is examined using molecular dynamics. An explicit formula for this (kinetic) relation is obtained by fitting an analytic expression to the simulation results. The numerical experiments also verify that transverse ledge propagation is the mechanism underlying twin-boundary motion. All calculations are carried out in a two-dimensional setting.     

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II—Application to cylindrical fibers

In Part I of this paper, we presented a general homogenization framework for determining the overall behavior, the evolution of the underlying microstructure, and the possible onset of macroscopic instabilities in fiber-reinforced elastomers subjected to finite deformations. In this work, we make use of this framework to generate specific results for general plane-strain loading of elastomers reinforced with aligned, cylindrical fibers. For the special case of rigid fibers and incompressible behavior for the matrix phase, closed-form, analytical results are obtained. The results suggest that the evolution of the microstructure has a dramatic effect on the effective response of the composite. Furthermore, in spite of the fact that both the matrix and the fibers are assumed to be strongly elliptic, the homogenized behavior is found to lose strong ellipticity at sufficiently large deformations, corresponding to the possible development of macroscopic instabilities [Geymonat, G., Müller, S., Triantafyllidis, N., 1993. Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal. 122, 231-290]. The connection between the evolution of the microstructure and these macroscopic instabilities is put into evidence. In particular, when the reinforced elastomers are loaded in compression along the long, in-plane axis of the fibers, a certain type of “flopping” instability is detected, corresponding to the composite becoming infinitesimally soft to rotation of the fibers.     

Stability of crystalline solids—I: Continuum and atomic lattice considerations

Many crystalline materials exhibit solid-to-solid martensitic phase transformations in response to certain changes in temperature or applied load. These martensitic transformations result from a change in the stability of the material's crystal structure. It is, therefore, desirable to have a detailed understanding of the possible modes through which a crystal structure may become unstable. The current work establishes the connections between three crystalline stability criteria: phonon-stability, homogenized-continuum-stability, and the presently introduced Cauchy-Born-stability criterion. Stability with respect to phonon perturbations, which probe all bounded perturbations of a uniformly deformed specimen under “hard-device” loading (i.e., all around displacement type boundary conditions) is hereby called “constrained material stability”. A more general “material stability” criterion, motivated by considering “soft” loading devices, is also introduced. This criterion considers, in addition to all bounded perturbations, all “quasi-uniform” perturbations (i.e., uniform deformations and internal atomic shifts) of a uniformly deformed specimen, and it is recommend as the relevant crystal stability criterion.     

Cavitation in elastomeric solids: I—A defect-growth theory

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes. This paper introduces a new theory to study the phenomenon of cavitation in soft solids that: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii) applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and (iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as a homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. This problem is then addressed by means of a novel iterated homogenization procedure, which allows to construct solutions for a specific, yet fairly general, class of defects. These include solutions for the change in size of the defects as a function of the applied loading conditions, from which the onset of cavitation — corresponding to the event when the initially infinitesimal defects suddenly grown into finite sizes — can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the defects plays the role of “time” and the applied load plays the role of “space”. When specialized to the case of hydrostatic loading conditions, isotropic solids, and defects that are vacuous and isotropically distributed, the proposed theory recovers the classical result of Ball (1982) for radially symmetric cavitation. The nature and implications of this remarkable connection are discussed in detail.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

Frequency and development of slugs in a horizontal pipe at large liquid flows

The generation of slugs was studied for air–water flow in horizontal 0.0763 m and 0.095 m pipes. The emphasis was on high liquid rates (u_LS ? 0.5 m/s) for which slugs are formed close to the entry and the time intervals between slugs are stochastic. A “fully developed” slug flow is defined as consisting of slugs with different sizes interspersed in a stratified flow with a height slightly larger than the height, h₀, needed for a slug to be stable. Properties of this “fully developed” pattern are discussed. A correlation for the frequency of slugging is suggested, which describes our data as well as the data from other laboratories for a wide range of conditions. The possibility is explored that there is a further increase of slug length beyond the “fully developed” condition because slugs slowly overtake one another.     

One, no one, and one hundred thousand crack propagation laws: A generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth

Barenblatt and Botvina with elegant dimensional analysis arguments have elucidated that Paris’ power-law is a weak form of scaling, so that the Paris’ parameters C and m should not be taken as material constants. On the contrary, they are expected to depend on all the dimensionless parameters of the problem, and are really “constants” only within some specific ranges of all these. In the present paper, the dimensional analysis approach by Barenblatt and Botvina is generalized to explore the functional dependencies of m and C on more dimensionless parameters than the original Barenblatt and Botvina, and experimental results are interpreted for a wider range of materials including both metals and concrete. In particular, we find that the size-scale dependencies of m and C and the resulting correlation between C and m are quite different for metals and for quasi-brittle materials, as it is already suggested from the fact the fatigue crack propagation processes lead to m=2-5 in metals and m=10-50 in quasi-brittle materials. Therefore, according to the concepts of complete and incomplete self-similarities, the experimentally observed breakdowns of the classical Paris’ law are discussed and interpreted within a unified theoretical framework. Finally, we show that most attempts to address the deviations from the Paris’ law or the empirical correlations between the constants can be explained with this approach. We also suggest that “incomplete similarity” corresponds to the difficulties encountered so far by the “damage tolerant” approach which, after nearly 50 years since the introduction of Paris’ law, is still not a reliable calculation of damage, as Paris himself admits in a recent review.     

On the possibility of piezoelectric nanocomposites without using piezoelectric materials

In this work, predicated on nanoscale size-effects, we explore the tantalizing possibility of creating apparently piezoelectric composites without using piezoelectric constituent materials. In a piezoelectric material an applied uniform strain can induce an electric polarization (or vice-versa). Crystallographic considerations restrict this technologically important property to non-centrosymmetric systems. Non-uniform strain can break the inversion symmetry and induce polarization even in non-piezoelectric dielectrics. The key concept is that all dielectrics (including non-piezoelectric ones) exhibit the aforementioned coupling between strain gradient and polarization—an experimentally verified phenomenon known in some circles as the flexoelectric effect. This flexoelectric coupling, however, is generally very small and evades experimental detection unless very large strain gradients (or conversely polarization gradients) are present. Based on a field theoretic framework and the associated Greens function solutions developed in prior work, we quantitatively demonstrate the possibility of “designing piezoelectricity,” i.e. we exploit the large strain gradients present in the interior of composites containing nanoscale inhomogeneities to achieve an overall non-zero polarization even under an uniformly applied stress. We prove that the aforementioned effect may be realized only if both the shapes and distributions of the inhomogeneities are non-centrosymmetric. Our un-optimized quantitative results, based on limited material data and restrictive assumptions on inhomogeneity shape and distribution, indicate that apparent piezoelectric behavior close to 10% of Quartz may be achievable for inhomogeneity sizes in the 4 nm range. In future works, it is not unreasonable to expect enhanced performance based on optimization of shape, topology and appropriate material selection.     

Thermo-elastic dynamic instability (TEDI) in frictional sliding of two elastic half-spaces

In some simplified 1D models, we recently studied the coupling of TEI (thermoelastic instability) and DI (dynamic instability), finding that thermal effects can render unstable the otherwise neutrally stable natural elastodynamic modes of the system, giving rise to a new family of instability which we called TEDI.Here, we study the general case of two sliding elastic half-planes, finding again a relatively weak coupling between thermal and dynamic effects, and the general family of instability TEDI class is found to modify both the otherwise separated TEI and DI classes. The growth factor, the phase velocity and the migrating speeds of the perturbations are wavelength-dependent, and it is difficult to give a complete picture given the high number of materials’ parameters, and the dependence on speed, friction coefficient, and the underlying uniform pressure. However, a set of results are given for “large” and “small” mismatch of shear wave speeds in the materials, and as a function of (i) friction coefficient; (ii) sliding speed V₀; (iii) wavenumber parameter γ. In the case of small mismatch, generalized Rayleigh waves exists already under frictionless conditions, the critical f for instability is zero. DI dominates over TEI typically for large wavenumbers, where the growth factors increase without limit and hence become eventually meaningless, requiring regularizations for example with rate-state dependent friction laws. TEI growth factors vice versa have a maximum at a certain wavenumber and therefore are always well posed. Larger coupling effects are noticed for two materials with large mismatch, but significantly only for sliding speeds comparable with the wave speed. In general, TEI growth factors increase with speed, whereas DI growth factors increase with speed for similar materials and decrease when the mismatch between materials is large.     

Onset of macroscopic instabilities in fiber-reinforced elastomers at finite strain

In this paper, we investigate theoretically the possible development of instabilities in fiber-reinforced elastomers (and other soft materials) when they are subjected to finite-strain loading conditions. We focus on the physically relevant class of “macroscopic” instabilities, i.e., instabilities with wavelengths that are much larger than the characteristic size of the underlying microstructure. To this end, we make use of recently developed homogenization estimates, together with a fundamental result of Geymonat, Müller and Triantafyllidis linking the development of these instabilities to the loss of strong ellipticity of the homogenized constitutive relations. For the important class of material systems with very stiff fibers and random microstructures, we derive a closed-form formula for the critical macroscopic deformation at which instabilities may develop under general loading conditions, and we show that this critical deformation is quite sensitive to the loading orientation relative to the fiber direction. The result is also confronted with classical estimates (including those of Rosen) for laminates, which have commonly been used as two-dimensional (2-D) approximations for actual fiber-reinforced composites. We find that while predictions based on laminate models are qualitatively correct for certain loadings, they can be significantly off for other more general 3-D loadings. Finally, we provide a parametric analysis of the effects of the matrix and fiber properties and of the fiber volume fraction on the onset of instabilities for various loading conditions.     

Optimization of structural topology in the high-porosity regime

We propose a new approach to topology optimization, based on the use of “single-scale laminates” as structural components. The method is well-founded, because in the high porosity limit these structures achieve maximal stiffness and minimal weight. The method is useful, because the Hooke's law of a single-scale laminate has a simple, explicit formula which scales linearly with weight. And it is interesting, because the selection of relatively simple, manufacturable designs can be addressed using linear or quadratic programming. Our contributions are two-fold: (a) we establish the foundation of this approach, by defining single-scale laminates and giving self-contained proofs of their optimality in the high-porosity limit; and (b) we explore two numerical applications—minimizing weight with a constraint on the Hooke's law, and imposing continuity on a spatially varying microstructure.     

A generalized Paris’ law for fatigue crack growth

An extension of the celebrated Paris law for crack propagation is given to take into account some of the deviations from the power-law regime in a simple manner using the Wöhler SN curve of the material, suggesting a more general “unified law”. In particular, using recent proposals by the first author, the stress intensity factor K(a) is replaced with a suitable mean over a material/structural parameter length scale Δa, the “fracture quantum”. In practice, for a Griffith crack, this is seen to correspond to increasing the effective crack length of Δa, similarly to the Dugdale strip-yield models. However, instead of including explicitly information on cyclic plastic yield, short-crack behavior, crack closure, and all other detailed information needed to eventually explain the SN curve of the material, we include directly the SN curve constants as material property. The idea comes as a natural extension of the recent successful proposals by the first author to the static failure and to the infinite life envelopes. Here, we suggest a dependence of this fracture “quantum” on the applied stress range level such that the correct convergence towards the Wöhler-like regime is obtained. Hence, the final law includes both Wöhler's and Paris’ material constants, and can be seen as either a generalized Wöhler's SN curve law in the presence of a crack or a generalized Paris’ law for cracks of any size.     

Modelling of laminated microstructures in stress-induced martensitic transformations

This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro-macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β₁ (austenite) to orthorhombic γ₁′ (martensite) phase transformation in a single crystal of Cu-Al-Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure.     

Homogenization-based constitutive models for magnetorheological elastomers at finite strain

This paper proposes a new homogenization framework for magnetoelastic composites accounting for the effect of magnetic dipole interactions, as well as finite strains. In addition, it provides an application for magnetorheological elastomers via a “partial decoupling” approximation splitting the magnetoelastic energy into a purely mechanical component, together with a magnetostatic component evaluated in the deformed configuration of the composite, as estimated by means of the purely mechanical solution of the problem. It is argued that the resulting constitutive model for the material, which can account for the initial volume fraction, average shape, orientation and distribution of the magnetically anisotropic, non-spherical particles, should be quite accurate at least for perfectly aligned magnetic and mechanical loadings. The theory predicts the existence of certain “extra” stresses—arising in the composite beyond the purely mechanical and magnetic (Maxwell) stresses—which can be directly linked to deformation-induced changes in the microstructure. For the special case of isotropic distributions of magnetically isotropic, spherical particles, the extra stresses are due to changes in the particle two-point distribution function with the deformation, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles. The theory is capable of handling the strongly nonlinear effects associated with finite strains and magnetic saturation of the particles at sufficiently high deformations and magnetic fields, respectively.     

Assessing continuum postulates in simulations of granular flow

Continuum mechanics relies on the fundamental notion of a mesoscopic volume “element” in which properties averaged over discrete particles obey deterministic relationships. Recent work on granular materials suggests that a continuum law may be inapplicable, revealing inhomogeneities at the particle level, such as force chains and slow cage breaking. Here, we analyze large-scale three-dimensional discrete-element method (DEM) simulations of different granular flows and show that an approximate “granular element” defined at the scale of observed dynamical correlations (roughly three to five particle diameters) has a reasonable continuum interpretation. By viewing all the simulations as an ensemble of granular elements which deform and move with the flow, we can track material evolution at a local level. Our results confirm some of the hypotheses of classical plasticity theory while contradicting others and suggest a subtle physical picture of granular failure, combining liquid-like dependence on deformation rate and solid-like dependence on strain. Our computational methods and results can be used to guide the development of more realistic continuum models, based on observed local relationships between average variables.