On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

Strain and damage interactions during tearing of a ductile Al-alloy with high work hardening are assessed in situ and in 3D combining two recently developed experimental techniques, namely, synchrotron laminography and digital volume correlation. Digital volume correlation consists of registering 3D laminography images. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root of a thin Compact Tension-like specimen, it is found that parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. These bands have an intermittent activity but are stable in space. Even at late stages of deformation strained bands can stop their activity highlighting the importance of plasticity on the failure process rather than damage softening. One void is followed over the loading history and seen to grow and orient along the slant strained band at very late stages of deformation. Void growth and strain are quantified. Gurson–Tvergaard–Needleman-type simulations using damage nucleation for shear, which is based on the Lode parameter, are performed and capture slant fracture but not the initial strain fields and in particular the experimentally found slant bands. The band formation and strain distribution inside and outside the bands are discussed further using plane strain simulations accounting for plastic material heterogeneity in soft zones.     

Computational micromechanics of dynamic compressive loading of a brittle polycrystalline material using a distribution of grain boundary properties

A two-dimensional finite element model is used to investigate compressive loading of a brittle ceramic. Intergranular cracking in the microstructure is captured explicitly by using a distribution of cohesive interfaces. The addition of confining stress increases the maximum strength and if high enough, can allow the effective material response to reach large strains before failure. Increasing the friction at the grain boundaries also increases the maximum strength until saturation of the strength is approached. Above a transitional strain rate, increasing the rate-of-deformation also increases the strength and as the strain rate increases, fragment sizes of the damaged specimen decrease. The effects of flaws within the specimen were investigated using a random distribution at various initial flaw densities. The model is able to capture an effective modulus change and degradation of strength as the initial flaw density increases. Effects of confinement, friction, and spatial distribution of flaws seem to depend on the crack coalescence and dilatation of the specimen, while strain-rate effects are result of inertial resistance to motion.     

A Binary Model of textile composites: III high failure strain and work of fracture in 3D weaves

Prior experiments have revealed exceptionally high values of the work of fracture (0.4-) in carbon/epoxy 3D interlock woven composites. Detailed destructive examination of specimens suggested that much of the work of fracture arose when the specimens were strained well beyond the failure of individual tows yet still carried loads . A mechanism of lockup amongst broken tows sliding across the final tensile fracture surface was suggested as the means by which high loads could still be transferred after tow failure. In this paper, the roles of weave architecture and the distribution of flaws in the mechanics of tow lockup are investigated by Monte Carlo simulations using the so-called Binary Model. The Binary Model was introduced previously as a finite element formulation specialised to the problem of simulating relatively large, three-dimensional segments of textile composites, without any assumption of periodicity or other symmetry, while preserving the architecture and topology of the tow arrangement. The simulations succeed in reproducing all qualitative aspects of measured stress-strain curves. They reveal that lockup can indeed account for high loads being sustained beyond tow failure, provided flaws in tows have certain spatial distributions. The importance of the interlock architecture in enhancing friction by holding asperities on sliding fibre tows into firm contact is highlighted.     

Computational mechanics modelling of cell-substrate contact during cyclic substrate deformation

Experimental testing carried out on various adherent cell types cultured on deformable substrates reveals specific patterns of cell reorientation in response to cyclic stretching of the substrate. In Wang et al. (2001. Specificity of endothelial cell reorientation in response to cyclic mechanical stretching. J. Biomech. 34, 1563), a number of substrate deformation modes were considered: in cases where lateral deformation of the substrate was prohibited (uniaxial case) cells were found to elongate perpendicular to the stretch direction, whereas in cases where the substrate was laterally unrestrained (biaxial case) cells were found to elongate at an angle to the stretch direction. The alignment directions in both cases corresponded to directions of minimum substrate strain. However, the mechanisms underlying such behaviour are not apparent from such in-vitro testing and consequently are not well understood. In this study finite element models are developed in order to investigate the role of cell viscoelasticity in cell debonding and cell realignment under conditions of cyclic substrate stretching using cohesive zone formulations to simulate cell-substrate interfacial behaviour. The characteristic length scale used in such models is based on the length of the receptor-ligand bonds at the cell-substrate interface. Two-dimensional simulations reveal that permanent debonding at the cell-substrate interface occurs due to the accumulation of strain concentrations in the cell. Inclusion of a nucleus in two-dimensional models is shown to have little effect on debonding while discrete cell-substrate contact at focal adhesion sites results in a completion of debonding in fewer cycles. Three-dimensional cohesive zone models are developed in order to compute changes in cell-substrate contact under the aforementioned uniaxial and biaxial modes of substrate deformation. Results reveal that, due to the accumulation of tensile and compressive strains in the cell under cyclic deformation, definite patterns of cell-substrate contact area evolution are computed. With continued cycling, equilibrium contact area profiles with definite orientations are established. These orientations are found to be coincidental with the preferential cell orientation directions seen in the experiments. As no changes in cell morphology are predicted by the models it is concluded that permanent breaking of cell-substrate bonds constitutes the first stage in the process of cell alignment under such mechanical loading.     

Micromechanics of plastic deformation and phase transformation in a three-phase TRIP-assisted advanced high strength steel: Experiments and modeling

The micromechanics of plastic deformation and phase transformation in a three-phase advanced high strength steel are analyzed both experimentally and by microstructure-based simulations. The steel examined is a three-phase (ferrite, martensite and retained austenite) quenched and partitioned sheet steel with a tensile strength of ~980 MPa. The macroscopic flow behavior and the volume fraction of martensite resulting from the austenite–martensite transformation during deformation were measured. In addition, micropillar compression specimens were extracted from the individual ferrite grains and the martensite particles, and using a flat-punch nanoindenter, stress–strain curves were obtained. Finite element simulations idealize the microstructure as a composite that contains ferrite, martensite and retained austenite. All three phases are discretely modeled using appropriate crystal plasticity based constitutive relations. Material parameters for ferrite and martensite are determined by fitting numerical predictions to the micropillar data. The constitutive relation for retained austenite takes into account contributions to the strain rate from the austenite–martensite transformation, as well as slip in both the untransformed austenite and product martensite. Parameters for the retained austenite are then determined by fitting the predicted flow stress and transformed austenite volume fraction in a 3D microstructure to experimental measurements. Simulations are used to probe the role of the retained austenite in controlling the strain hardening behavior as well as internal stress and strain distributions in the microstructure.     

The strength limit in a bio-inspired metallic nanocomposite

Large-scale molecular dynamics simulations are performed to investigate the plastic deformation behavior of a bio-inspired metallic nanocomposite which consists of hard nanosized Ni platelets embedded in a soft Al matrix. The investigation is restricted to an idealized nanocomposite structure with regular platelet distributions in a quasi-two-dimensional geometry under quasi-static loading conditions. This restriction enables us to study size dependent material properties over a wide range of length scales with a fully atomistic resolution of the material and thus without any a priori assumptions of the deformation processes. The simulation results are analyzed with respect to the prevailing deformation mechanisms and their influence on the mechanical properties of the nanocomposite with various geometrical variations. It is found that interfacial sliding contributes significantly to the plastic deformation despite a strong bonding across the interface. Critical for the strength of the nanocomposite is the geometric confinement of dislocation processes in the plastic phase, which strongly depends on the length scale and the morphology of the nanostructure. However, for the smallest structural scales, the softening caused by interfacial sliding prevails, giving rise to a maximum strength.     

Model identification and FE simulations: Effect of different yield loci and hardening laws in sheet forming

The bi-axial experimental equipment [Flores, P., Rondia, E., Habraken, A.M., 2005a. Development of an experimental equipment for the identification of constitutive laws (Special Issue). International Journal of Forming Processes] developed by Flores enables to perform Bauschinger shear tests and successive or simultaneous simple shear tests and plane strain tests. Flores investigates the material behavior with the help of classical tensile tests and the ones performed in his bi-axial machine in order to identify the yield locus and the hardening model. With tests performed on one steel grade, the methods applied to identify classical yield surfaces such as [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic materials. Proceedings of the Royal Society of London A 193, 281–297; Hosford, W.F., 1979. On yield loci of anisotropic cubic metals. In: Proceedings of the 7th North American Metalworking Conf. (NMRC), SME, Dearborn, MI, pp. 191–197] ones as well as isotropic Swift type hardening, kinematic Armstrong–Frederick or Teodosiu and Hu hardening models are explained. Comparison with the Taylor–Bishop–Hill yield locus is also provided. The effect of both yield locus and hardening model choices is presented for two applications: plane strain tensile test and Single Point Incremental Forming (SPIF).     

A general hyperelastic model for incompressible fiber-reinforced elastomers

This work presents a new constitutive model for the effective response of fiber-reinforced elastomers at finite strains. The matrix and fiber phases are assumed to be incompressible, isotropic, hyperelastic solids. Furthermore, the fibers are taken to be perfectly aligned and distributed randomly and isotropically in the transverse plane, leading to overall transversely isotropic behavior for the composite. The model is derived by means of the “second-order” homogenization theory, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” Compared to other constitutive models that have been proposed thus far for this class of materials, the present model has the distinguishing feature that it allows consideration of behaviors for the constituent phases that are more general than Neo-Hookean, while still being able to account directly for the shape, orientation, and distribution of the fibers. In addition, the proposed model has the merit that it recovers a known exact solution for the special case of incompressible Neo-Hookean phases, as well as some other known exact solutions for more general constituents under special loading conditions.     

Actuation of the Kagome Double-Layer Grid. Part 2: Effect of imperfections on the measured and predicted actuation stiffness

The actuation stiffness of a set of steel Kagome Double-Layer Grid (KDLG) structures with brazed joints is measured experimentally and compared with predictions by the finite element method. The predicted actuation stiffnesses for the perfect KDLGs much exceed the measured values, and it is argued that the low values of observed actuation stiffness are due to the presence of geometric imperfections introduced during manufacture. In order to assess the significance of geometric defects upon actuation stiffness, finite element calculations are performed on structures with a stochastic dispersion in nodal position from the perfectly periodic arrangement, and on structures with wavy bars. It is found that bar waviness has the dominant effect upon the actuation stiffness. The predicted actuation stiffness for the imperfect structures are in satisfactory agreement with the measured values assuming the same level of imperfection between theory and experiment.     

Discrete contact mechanics of a fibrillar surface with backing layer interactions

The contact mechanics of a fibrillar micro-fabricated surface structure made of poly(dimethyl siloxane) (PDMS) is studied. The attachment and detachment of individual fibrils to and from a spherical indenter upon approach and retraction are detected as jumps in force and stiffness. A quantitative model describes the stiffness values by taking into account the deformation of the fibrils and the backing layer. The results emphasize the importance of long-range interactions in the contact mechanics of elastic materials and confirm some of the important concepts underlying the development of fibrillar adhesive materials.     

Predicting the Hall–Petch effect in fcc metals using non-local crystal plasticity

Many conventional continuum approaches to solid mechanics do not address the size sensitivity of deformation to microstructural features like grain boundaries, and are therefore unable to capture much of the experimentally observed behavior of polycrystal deformation. We propose a non-local crystal plasticity model, in which the geometrically necessary dislocation (GND) density is calculated using a non-local integral approach. The model is based on augmented F^eF^p kinematics, which account for the initial microstructure (primarily grain boundaries) present in the polycrystal. With the augmented kinematics, the initial GND and the evolving GND state are determined in a consistent manner. The expanded kinematics and the non-local crystal plasticity model are used to simulate the tensile behavior in copper polycrystals with different grain sizes ranging from 14 μm to 244 μm. The simulation results show a grain size dependence on the polycrystal’s yield strength, which are in good agreement with the experimental data.     

Modelling the torsion of thin metal wires by distortion gradient plasticity

Under small strains and rotations, we apply a phenomenological higher-order theory of distortion gradient plasticity to the torsion problem, here assumed as a paradigmatic benchmark of small-scale plasticity. Peculiar of the studied theory, proposed about ten years ago by Morton E. Gurtin, is the constitutive inclusion of the plastic spin, affecting both the free energy and the dissipation. In particular, the part of the free energy, called the defect energy, which accounts for Geometrically Necessary Dislocations, is a function of Nye's dislocation density tensor, dependent on the plastic distortion, including the plastic spin. For the specific torsion problem, we implement this distortion gradient plasticity theory into a Finite Element (FE) code characterised by implicit (Backward Euler) time integration, numerically robust and accurate for both viscoplastic and rate-independent material responses. We show that, contrariwise to other higher-order theories of strain gradient plasticity (neglecting the plastic spin), the distortion gradient plasticity can predict some strengthening even if a quadratic defect energy is chosen. On the basis of the results of many FE analyses, concerned with (i) cyclic loading, (ii) switch in the higher-order boundary conditions during monotonic plastic loading, (iii) the use of non-quadratic defect energies, and (iv) the prediction of experimental data, we mainly show that (a) including the plastic spin contribution in a gradient plasticity theory is highly recommendable to model small-scale plasticity, (b) less-than-quadratic defect energies may help in describing the experimental results, but they may lead to anomalous cyclic behaviour, and (c) dissipative (unrecoverable) higher-order finite stresses are responsible for an unexpected mechanical response under non-proportional loading.     

Models for stiffening in cross-linked biopolymer networks: A comparative study

In a recent publication, we studied the mechanical stiffening behavior in two-dimensional (2D) cross-linked networks of semiflexible biopolymer filaments under simple shear [Onck, P.R., Koeman, T., Van Dillen, T., Van der Giessen, E., 2005. Alternative explanation of stiffening in cross-linked semiflexible networks. Phys. Rev. Lett. 95, 178102]. These simulations make use of a geometrically nonlinear finite-element technique, taking into account the discreteness of the biopolymer network. As an alternative to the prevalent view, these computations relate the stiffening to nonaffine network reorientations. However, this discrete-network model neglects any interaction of the filaments with the surrounding fluid, which is the origin of entropic stiffening in single filaments and in biopolymer networks, according to MacKintosh et al. For this reason, this article is devoted to a thorough study of the difference between both approaches on the 2D single-filament level. In addition, we investigate the deviation from affine deformation behavior, by comparing the discrete calculations with an affine-network model.     

The correlation of the indentation size effect measured with indenters of various shapes

Experimental results are presented which show that the indentation size effect for pyramidal and spherical indenters can be correlated. For a pyramidal indenter, the hardness measured in crystalline materials usually increases with decreasing depth of penetration, which is known as the indentation size effect. Spherical indentation also shows an indentation size effect. However, for a spherical indenter, hardness is not affected by depth, but increases with decreasing sphere radius. The correlation for pyramidal and spherical indenter shapes is based on geometrically necessary dislocations and work-hardening. The Nix and Gao indentation size effect model (J. Mech. Phys. Solids 46 (1998) 411) for conical indenters is extended to indenters of various shapes and compared to the experimental results.     

Understanding the influence of structural hierarchy and its coupling with chemical environment on the strength of idealized tropocollagen–hydroxyapatite biomaterials

Hard biomaterials such as bone, dentin, and nacre have primarily an organic phase (e.g. tropocollagen (TC)) and a mineral phase (e.g. hydroxyapatite (HAP) or aragonite) arranged in a staggered arrangement at the nanoscopic length scale. Interfacial interactions between the organic phase and the mineral phase as well as the structural effects arising due to the staggered arrangement significantly affect the strength of such biomaterials. The effect of such factors is intricately intertwined with the chemical environment of such materials. In the present investigation, an idealized TC–HAP composite system under tensile loading is analyzed using explicit three-dimensional (3-D) molecular dynamics (MD) simulations to develop an understanding of these factors. The material system is analyzed in three different environments: (1) in the absence of water molecules (non-hydrated), (2) in the presence of water molecules (hydrated), and (3) in the presence of water molecules with calcium ions (ionized water). The analyses focus on understanding the correlations among factors such as the structural arrangement, the peak stress during deformation, Young's modulus, the peak interfacial strength, and the length scale of the localization of peak stress during deformation. Analyses show that maximizing the contact area between the TC and HAP phases results in higher interfacial strength as well as higher fracture strength. Due to the staggered arrangement, the orientation of HAP crystals has insignificant effect on the biomaterial strength. Analyses based on strength scaling as a function of structural hierarchy level reveal that while peak strength follows a multiscaling relation, the fracture strength does not. The peak strain for failure was found to be independent of the changes in levels of structural hierarchy. Overall, the analyses, being limited in size due to the computational time constraint, point out important correlations between the mechanical strength and chemically influenced structural hierarchy of biomaterials.     

Coupled continuum and discrete analysis of random heterogeneous materials: Elasticity and fracture

Recent work has suggested that the heterogeneous distribution of mechanical properties in natural and synthetic materials induces a toughening mechanism that leads to a more robust structural response in the presence of cracks, defects or other types of flaws. Motivated by this, we model an elastic solid with a Young′s modulus distribution described by a Gaussian process. We study the pristine system using both a continuum and a discrete model to establish a link between the microscale and the macroscale in the presence of disorder. Furthermore, we analyze a flawed discrete particle system and investigate the influence of heterogeneity on the fracture mechanical properties of the solid. We vary the variability and correlation length of the Gaussian process, thereby gaining fundamental insights into the effect of heterogeneity and the essential length scales of heterogeneity critical to enhanced fracture properties. As previously shown for composites with complex hierarchical architectures, we find that materials with disordered elastic fields toughen by a ‘distribution-of-weakness’ mechanism inducing crack arrest and stress delocalization. In our systems, the toughness modulus can increase by up to 30% due to an increase in variability in the elastic field. Our work presents a foundation for stochastic modeling in a particle-based micromechanical environment that can find broad applications within natural and synthetic materials.     

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.     

FE analysis of stress and strain fields in finite random composite bodies

In this paper the mechanical behaviour of finite random heterogeneous bodies is considered. The analysis of non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. Microstructures can be periodic but generically they are random. In the first case, an exact calculation can be performed but in the second case recourse has to be made either to simulation or to some scheme of approximation. One such scheme is based on a stochastic variational principle. The novelty of the present work is that a stochastic variational principle is projected directly onto a finite-element basis so that all subsequent analysis is performed within a finite-element framework. The proposed formulation provides expressions for the local stress and strain fields in any realization of the medium, from which expressions for statistically-averaged quantities can be derived. Then an approximation of Hashin-Shtrikman type is developed, which generates a FE-based numerical procedure able to take account of interactions between random inclusions and boundary layer effects in finite composite structures. Finally, two examples are presented, namely a cylinder with square cross-section subjected to mixed boundary conditions of different types on different faces and a rectangular body containing a centre crack. The results show that in the vicinity of the boundary or close to the crack tip, the strain and the stress in the matrix and in the inclusions differ considerably from those obtained by the formal application of conventional homogenization.     

The fracture toughness of planar lattices: Imperfection sensitivity

The imperfection sensitivity of in-plane modulus and fracture toughness is explored for five morphologies of 2D lattice: the isotropic triangular, hexagonal and Kagome lattices, and the orthotropic 0/90^° and ±45^° square lattices. The elastic lattices fail when the maximum local tensile stress at any point attains the tensile strength of the solid. The assumed imperfection comprises a random dispersion of the joint position from that of the perfect lattice. Finite element simulations reveal that the knockdown in stiffness and toughness are sensitive to the type of lattice: the Kagome and square lattices are the most imperfection sensitive. Analytical models are developed for the dependence of modes I and II fracture toughness of the 0/90^° and ±45^° lattices upon relative density. These models explain why the mode II fracture toughness of the 0/90^° lattice has an unusual functional dependence upon relative density.