Macroscopic Stress-strain Curves Research Articles

A hybrid algorithm of particle swarm optimization and finite element method to identify local mesoscopic damage of concrete-like materials

Meso-scale material modeling and parameter identification are still unsettled issues in the current multi-scale methods for the field of computational mechanics. In this investigation, a hybrid algorithm of particle swarm optimization and finite element method coupled with continuum damage mechanics is proposed to forecast local mesoscopic elasticity modulus and damage of concrete-like materials based on the macroscopic stress-strain relationship data curve. The algorithm establishes a bridge between mesoscopic mechanical analysis and macroscopic mechanical analysis, which can also guarantee the consistency of analysis in different scales. A representative numerical case is conducted to support the algorithm. Additionally, the effect of different divisions of representative volume element is analyzed based on the algorithm, and the recommended size of representative volume element is 5 mm. The prediction results match well with the corresponding macroscopic experimental results and mesoscopic experimental results based on the chosen recommended size of representative volume element. The effectiveness of the algorithm is verified, which can support an effective numerical tool to optimize and identify mesoscopic mechanical parameters and damage for multi-scale material modeling and analysis.

The microscopic failure mechanism and energy evolution law of foamed concrete under uniaxial compression

This paper uses the macroscopic stress–strain curve of foamed concrete under uniaxial compression as a baseline reference, combining the micro-failure mechanisms of the internal structure and the laws of energy evolution to explain the macroscopic mechanical behavior. This study proposes the mechanisms behind pore fracture and recombination, as well as the mechanical interactions within thin-walled structures. The fractured structure swiftly fills and restores the framework, while the reticulated thin-walled structure adjusts transmission pathways for forces, inducing dormant elastic strain energy and thereby prolonging the interval of residual strain. Coupled with macroscopic laws dictating energy evolution, this paper elaborates on the influence of microscopic structures on stress–strain characteristics. The capacity of foam concrete to proficiently absorb and dissipate external energy, thereby extending the duration of the strain process, and its potential as a buffering material are substantiated.

Application of a nanoindentation-based approach for parameter identification to a crystal plasticity model for bcc metals

The application of an inverse method for determining the parameters of a crystal plasticity constitutive law of a body-centered-cubic (BCC) single phase material is presented. Nanoindentation is used as the primary experimental input. An objective function, based on the deviation between the experimentally measured imprint and the simulated one, is minimized by a differential evolution algorithm to obtain the best fitting crystal plasticity parameters. To aid the identification procedure additional experimental data is used: the upper bounds and the ratios of the critical resolved shear stresses of the three slip plane families in BCC are estimated from micropillar compression experiments and used as a constraint in the optimization. The effect of the imposed constraints and the chosen strategy for mapping experimental to simulated displacements is presented and discussed. The validation of the method is done in the macroscopic regime by comparing an experimental tensile test with a simulated one using the obtained crystal plasticity parameters. Accurate results are achieved from two different indents. Therefore, the method is a promising path for determining crystal plasticity parameters in the case where a direct fitting from a macroscopic stress–strain curve is not possible, i.e. in the case of multi-phase materials.

Microplasticity and macroplasticity behavior of additively manufactured Al-Mg-Sc-Zr alloys: In-situ experiment and modeling

Understanding and controlling the performance of additively manufactured aluminum alloys containing scandium (Sc) and zirconium (Zr) elements heavily relies on knowledge of their microplasticity and macroplasticity behavior. However, this aspect has received very little attention. In this investigation, we examined the microplasticity and macroplasticity behavior of additively manufactured Al-Mg-Sc-Zr alloys before and after aging, using in-situ synchrotron X-ray diffraction and full-field crystal plasticity modeling. Our study provides a quantitative assessment of the transitions from elasticity to microplasticity and then to macroplasticity and analyzes the development of the initial microstructure, particularly the dislocations. We constructed crystal plasticity fast-Fourier-transform models based on dislocation densities. The predicted evolutions of macroscopic stress-strain curves, lattice strains, and dislocation densities agree with in-situ measurements. The present findings provide deep insights into controlling the performance of AM Al-Mg-Sc-Zr alloys. Besides, the micromechanical model developed in this investigation paves the way for predicting the microplasticity and macroplasticity behavior of various metallic materials.

Time-Resolved Extensional Rheo-NMR Spectroscopy for Investigating Polymer Nanocomposites under Deformation

Understanding the microstructure change of polymer nanocomposites (PNCs) under elongation deformation at the molecular level is the key to coupling structure-property relationships of PNCs. In this study, we developed our recently proposed in situ extensional rheology NMR device, Rheo-spin NMR, which can simultaneously obtain both the macroscopic stress-strain curves and the microscopic molecular information with the total sample weight of ∼6 mg. This enables us to conduct a detailed investigation of the evolution of the interfacial layer and polymer matrix in nonlinear elongational strain softening behaviors. A quantitative method is established for in situ analysis of (1) the fraction of the interfacial layer and (2) the network strand orientation distribution of the polymer matrix based on the molecular stress function model under active deformation. The results show that for the current highly filled silicone nanocomposite system, the influence of the interfacial layer fraction on mechanical property change during small amplitude deformation is quite minor, while the main role is reflected in rubber network strand reorientation. The Rheo-spin NMR device and the established analysis method are expected to facilitate the understanding of the reinforcement mechanism of PNC, which can be further applied to understand the deformation mechanism of other systems, i.e., glassy and semicrystalline polymers and the vascular tissues.

A microstructure-based multiscale approach to predict the formability of multiphase steels

A multiscale approach was proposed to predict the macroscopic forming limit of Q&P980 steel. The microstructure-based simulation was implemented in the microscopic simulation to obtain the macroscopic stress-strain curve. The representative volume element (RVE) was established using the data extracted from the characterised results of electron back-scattered diffraction (EBSD). The microscopic constitutive model of each constituent phase was determined based on a new method. The acquired macroscopic stress-strain curve was extrapolated using the modified Power-Law (MPL) strain hardening model. In the macroscopic simulation, the Nakajima and Marciniak-Kuczynski (M-K) simulations were combined to obtain the macroscopic forming limit only based on the macroscopic stress-strain curve. The microscopic and macroscopic simulations show good agreement with the experiment results, verifying the multiscale approach's accuracy. This study provides a solution for the direct connection between the microstructure and macroscopic forming limit.

Experimental investigation of microscale mechanisms during compressive loading of paperboard

Compression of paperboard is a common procedure during industrial package forming and better knowledge of the material response is needed to avoid defective packages and waste. To go beyond current modelling approaches, experimental identification of mechanisms underlying the macroscopic stress–strain responses is needed. In this study, in-situ uniaxial compression of paperboard is studied through synchrotron tomography at high spatiotemporal resolutions. Both the microstructural evolution of the fibre network and the actual boundary conditions of the loading were quantified and analysed. At the microscale, the loading equipment plates were not perfectly flat resulting in an increasing sample-equipment contact area with loading. This is, however, shown to only have a small effect on the form of the macroscopic stress–strain curves. The evolution of 3D strain fields showed that strain accumulated close to the sample surfaces in the early part of the compression process, whereafter the main deformation zone shifted to the out-of-plane centre. Both fibre walls and pore volumes were observed to decrease during loading (and recover partly after unloading). Regarding the pore volume, the main reduction mechanism was seen to be closure of layers between fibres. Even if the total pore volume reduction was seen to be the dominant deformation mechanism in a second stage of compression, the volumetric change of fibre walls was non-negligible. Fibre wall compression is not commonly considered in theoretical treatments of paperboard compression, but this work suggests that the stored elastic energy could be a driver for the elastic recovery of the fibre network during unloading.

Time courses of strains that induce necking and fracturing in high-density polyethylene

High-density polyethylene is widely used in pressure pipe applications, but its necking and pre-cut effects are still poorly understood. Herein, we analyze the spatial distributions of the time courses of strains to highlight the strain field evolution to necking and the effect of pre-cutting on the strain field evolution in a high-density polyethylene material deformed under tension. Digital image correlation was used to measure the strain fields on two perpendicular surfaces of a specimen. Necking and its propagation in the tension direction dominate the failure behavior of an intact specimen. However, in the pre-cut specimen, crack propagation prevents neck propagation in the tension direction. Energy release outside the crack zone is observed as a decrease in strain at approximately the failure time. This leads to a macroscopic stress-strain curve that deviates from that of the intact specimen. These findings provide novel insights that are significant in the theoretical modelling and simulations of advanced polymeric materials and structures.

Yielding, shear banding, and brittle failure of amorphous materials

Widespread processes in nature and technology are governed by the dynamical transition whereby a material in an initially solid-like state then yields plastically. Major unresolved questions concern whether any material will yield smoothly and gradually (ductile behaviour) or fail abruptly and catastrophically (brittle behaviour); the roles of sample annealing, disorder and shear band formation in the onset of yielding and failure; and, most importantly from a practical viewpoint, whether any impending catastrophic failure can be anticipated before it happens. We address these questions by studying the yielding of slowly sheared athermal amorphous materials, within a minimal mesoscopic lattice elastoplastic model. Our contributions are fourfold. First, we elucidate whether yielding will be ductile or brittle, for any given level of sample annealing. Second, we show that yielding comprises two distinct stages: a pre-failure stage, in which small levels of strain heterogeneity slowly accumulate, followed by a catastrophic brittle failure event, in which a crack quickly propagates across the sample via a cooperating line of plastic events. Third, we provide an expression for the slowly growing level of strain heterogeneity in the pre-failure stage, expressed in terms of the macroscopic stress-strain curve and the sample size, and in excellent agreement with our simulation results. Fourth, we elucidate the basic mechanism via which a crack then nucleates and provide an approximate expression for the probability distribution of shear strains at which failure occurs, as determined by the disorder inherent in the sample, expressed in terms of a single annealing parameter, and the system size.

Damage-induced failure analysis of additively manufactured lattice materials under uniaxial and multiaxial tension

Mechanical behavior of additively manufactured lattice materials has been mainly investigated under uniaxial compression, while their performance under uniaxial and multiaxial tension are yet to be understood. To address this gap, a generic elastoplastic homogenization scheme with continuum damage model is developed, and three different lattice materials, namely cubic, modified face-center cubic and body-center cubic, are analyzed under uniaxial, biaxial and triaxial tension. The influence of micro-architecture on the material’s failure behavior as well as its macroscopic mechanical performance is thoroughly discussed. For validation, a set of uniaxial tensile experiments are conducted on functionally graded cubic lattice samples that are additively manufactured using Electron Beam Melting (EBM) process. Digital image correlation technique is employed to obtain the macroscopic stress–strain curves, and manufacturing imperfections are inspected using light omitting microscopy. It turns out that the behavior of as-built samples could substantially differ from numerical predictions. Thus, a defect-informed numerical model is employed to accommodate the effect of imperfections. The outcome is in a very good agreement with experimental data, indicating that with proper input data, the developed scheme can accurately predict the mechanical and failure behavior of a given lattice material.

Mesoscopic damage mechanism and a constitutive model of shale using in-situ X-ray CT device

With the continuous development of shale gas engineering in China, an accurate and adequate understanding of shale damage mechanism has become a critical scientific problem addressed by rock mechanics. So far, many researchers have studied the fracturing process and mechanism of shale in multi-scales. The mesoscopic structural characteristics of shale during loading can be obtained using the in-situ X-ray computed tomography (CT) technology. Some scholars have observed the structure evolution of shale under in-situ uniaxial compression conditions, but the quantitative investigation of damage evolution and constitutive model of shale still requires further studies. Given this, the in-situ micro-CT device was adopted to study the macroscopic mechanical response and the mesoscopic structural evolution of Longmaxi shale under uniaxial compression conditions. The evolution law of the damage variable with the axial strain was analysed according to the characterisation of the damage variable based on the CT image grey variance. In combination with the macroscopic stress-strain curve shape, evolution stage of the mesoscopic void, evolution of the damage variable, mesoscopic mechanism of the macroscopic mechanical response was investigated, and a damage constitutive model of shale under the uniaxial compression test was proposed. The conclusion of this study can provide a foundation for further research into the multi-scale failure mechanism of shale and exert theoretical guidance significance on shale gas fracturing engineering practice.

A directional microcrack damage theory for brittle solids based on continuous hyperplasticity

We present a novel microcrack-damage theory for brittle solids under compression. Instead of using internal variables like zeroth, second or fourth rank damage tensors, the state of material damage is represented by an internal function that encapsulates the information of direction, density and size of microcracks. Just like other internal variables, the evolution of this state function must obey the second law of thermodynamics for arbitrary loading paths. This is done by casting the model in the framework of continuous hyperplasticity and enforcing a non-negative dissipation rate functional. The proposed framework offers predictions on the continuous evolution of microcrack density and the induced material anisotropy along with the macroscopic stress-strain curves. The use of continuous damage function grants the model significantly enhanced resolution in characterizing the direction-dependent response of cracked solids compared to classical models that are based on damage tensors. Two scenarios are considered in developing the theory, one assumes frictionless cracks and the other incorporates friction between crack surfaces. The results highlight that inelasticity, pressure dependence, and loading-unloading hysteresis exhibited by brittle solids are natural consequences of frictional microcracks. The proposed theory offers a generic and versatile framework to upscale micromechanical processes operating at individual crack scale to explain the macroscopic behavior of cracked solids.

A meso-scale size effect study of concrete tensile strength considering parameters of random fields

This study analyses size effects of concrete under uniaxial tension by Monte Carlo simulations, where heterogeneous strength at meso-scale is modelled by Weibull random fields with statistical parameters including correlation length and variance. For a given sample size and different random field parameters, a sufficient number of random field realisations are simulated to obtain statistical information from macroscopic stress-strain curves, while the complex meso-crack initiation and propagation is captured by the phase-field regularized cohesive zone model (PF-CZM). The effects of sample size and material heterogeneity on macroscopic tensile strength are analysed, and the quasi-brittle transition between plasticity and linear elastic fracture mechanics (LEFM) is well simulated using the nonlocal PF-CZM. It is also found that both the correlation length and the variance affect the trend of size effect in varying degrees: larger correlation length and higher variance with higher heterogeneity lead to more dispersed responses that approach the LEFM descending line. A modified law in three-dimensional parametric space is proposed by data regression for effective assessment of size effect and structural reliability.

Influence of Crystal Plasticity Parameters on the Strain Hardening Behavior of Polycrystals

The effective mechanical properties of a polycrystal depend directly on the single-crystal properties of each grain and its crystallographic orientation with respect to the load axis. While the micromechanical approach has been used quite extensively to study the influence of grain shape and crystallographic texture on the resulting mechanical behavior of a polycrystal, the influence of the crystal plasticity parameters, which describe the constitutive behavior of the single crystal, requires to be investigated systemically because, typically, these parameters are fitted to describe a given material behavior. In the current research, this gap is filled by systemically studying the effect of changes in crystal plasticity parameters on the effective mechanical properties of polycrystals. The numerical model employed here consists of a representative volume element of 100 grains, and the material properties are described by using a non-local crystal plasticity model. A proper homogenization technique was used to homogenize the micromechanical results to an effective macroscopic material response. The equivalent stress versus equivalent plastic strain curve was obtained numerically by introducing the Voce-type hardening law, mimicking the material behavior in uniaxial tensile tests. The four parameters of the Voce-type hardening law were fitted to the macroscopic stress-strain curves, and the correlation between the crystal plasticity parameters and the Voce parameters has been studied, which is an efficient way to study the influence of microscopic material descriptions on the macroscopic behavior of polycrystals.

Determination and analysis of the constitutive parameters of temperature-dependent dislocation-density-based crystal plasticity models

Physics-based crystal plasticity models rely on certain statistical assumptions about the collective behavior of dislocation populations on one slip system and their interactions with the dislocations on the other slip systems. One main advantage of using such physics-based constitutive dislocation models in crystal plasticity kinematic frameworks is their suitability for predicting the mechanical behavior of polycrystals over a wide range of deformation temperatures and strain rates with the same physics-based parameter set. In this study, the ability of a widely used temperature-dependent dislocation-density-based crystal plasticity formulation to reproduce experimental results, with a main focus on the yield stress behavior, is investigated. First, the material parameters are identified from experimental macroscopic stress–strain curves using a computationally efficient optimization methodology that uses a genetic algorithm along with the response surface methodology. For this purpose, a systematic set of compression tests on interstitial free (IF) steel samples is performed at various temperatures and strain rates. Next, the influence of the individual parameters on the observed behavior is analyzed. Based on mutual interactions between various parameters, the ability to find a unique parameter set is discussed. This allows identifying shortcomings of the constitutive law and sketch ideas for possible improvements. Particular attention is directed toward identifying possibly redundant material parameters, narrowing the acceptable range of material parameters based on physical criteria, and modifying the crystal plasticity formulation numerically for high-temperature use.

Fiber Rearrangement and Matrix Compression in Soft Tissues: Multiscale Hypoelasticity and Application to Tendon.

It is widely accepted that the nonlinear macroscopic mechanical behavior of soft tissue is governed by fiber straightening and re-orientation. Here, we provide a quantitative assessment of this phenomenon, by means of a continuum micromechanics approach. Given the negligibly small bending stiffness of crimped fibers, the latter are represented through a number of hypoelastic straight fiber phases with different orientations, being embedded into a hypoelastic matrix phase. The corresponding representative volume element (RVE) hosting these phases is subjected to “macroscopic” strain rates, which are downscaled to fiber and matrix strain rates on the one hand, and to fiber spins on the other hand. This gives quantitative access to the fiber decrimping (or straightening) phenomenon under non-affine conditions, i.e. in the case where the fiber orientations cannot be simply linked to the macroscopic strain state. In the case of tendinous tissue, such an RVE relates to the fascicle material with 50 μm characteristic length, made up of crimped collagen bundles and a gel-type matrix in-between. The fascicles themselves act as parallel fibers in a similar matrix at the scale of a tissue-related RVE with 500 μm characteristic length. As evidenced by a sensitivity analysis and confirmed by various mechanical tests, it is the initial crimping angle which drives both the degree of straightening and the shape of the macroscopic stress-strain curve, while the final linear portion of this curve depends almost exclusively on the collagen bundle elasticity. Our model also reveals the mechanical cooperation of the tissue’s key microstructural components: while the fibers carry tensile forces, the matrices undergo hydrostatic pressure.

Micromechanical investigation of the effect of the crystal orientation on the local deformation path and ductile void nucleation in dual-phase steels

The effect of the crystal orientation on the local stress–strain path and ductile damage initiation in dual-phase steels was characterized through an integrated experimental–numerical study. Interrupted tensile tests were carried out on two different steels with varying microstructure to clarify the damage initiation mechanisms, their locations, sequences and proportions. It was found that damage initiated early and mainly from ferrite/martensite decohesion and martensite/martensite cracking with different ratio in the two materials. Finite element simulations were carried out on semi-synthetic microstructures generated from optical micrographs of specimen surfaces, using either an isotropic elasto-plastic model or a phenomenological crystal plasticity model calibrated from macroscopic stress–strain curves. The accuracy of the simulations was assessed through a comparison of the equivalent plastic strain fields estimated experimentally by digital image correlation and predicted numerically. It was found that both models predicted similar plastic strain distribution but slightly different stress triaxiality and Lode angle fields which highlighted the effect of the crystal orientation. However, in the vicinity of strain concentration regions where voids nucleated, the two fields were relatively close suggesting that the phase distribution and strength contrast had a predominant effect on these fields. The local stress–strain paths at void sites were extracted and were observed to significantly differ from that predicted by macroscopic simulations, especially in the case of ferrite damage which highlighted the importance of considering microstructural effects. A general ductile damage model dependent on the equivalent plastic strain, the stress triaxiality and the Lode angle was calibrated based on the stress–strain path history of void sites, taking into account experimental uncertainties. Similar fracture loci were predicted by the isotropic and the crystal plasticity model. Estimated damage initiation parameters suggested that martensite fracture was insensitive to the stress triaxiality and the Lode angle, while a certain sensitivity against the stress triaxiality was observed for ferrite/martensite decohesion.

A multiscale four-layer finite element model to predict the effects of collagen fibers on skin behavior under tension.

Human skin is a complex multilayered multiscale material that exhibits nonlinear and anisotropic mechanical behavior. It has been reported that its macroscopic behavior in terms of progression of wrinkles induced by aging is strongly dependent on its microscopic composition in terms of collagen fibers in the dermis layer. In the present work, a multiscale four-layer 2D finite element model of the skin was developed and implemented in Matlab code. The focus here was to investigate the effects of dermal collagen on the macroscopic mechanical behavior of the skin. The skin was modeled by a continuum model composed of four layers: the Stratum Corneum, the epidermis, the dermis, and the hypodermis. The geometry of the different layers of the skin was represented in a 2D model with their respective thicknesses and material properties taken from literature data. The macroscopic behavior of the dermis was modeled with a nonlinear multiscale approach based on a multiscale elastic model of collagen structure going from cross-linked molecules to the collagen fiber, combined with a Mori-Tanaka homogenization scheme. The model includes the nonlinear elasticity of the collagen fiber density, the fiber radius, the undulation, and the fiber orientation. An axial tension was applied incrementally to the lateral surfaces of the skin model. A parametric study was performed in order to investigate the effect of the collagen constituents on the macroscopic skin mechanical behavior in terms of the predicted macroscopic stress-strain curve of the skin. The results of the FE computations under uniaxial tension showed that the different layers undergo different strains, leading to a difference in the transversal deformation at the top surface. In addition, the parametric study revealed a strong correlation between macroscopic skin elasticity and its collagen structure.

A surface-mesh gradation tool for generating gradated tetrahedral meshes of microstructures with defects

A framework for generating tetrahedral meshes of polycrystalline microstructures with optimal element-size distribution, as determined by the underlying defects (e.g., cracks and voids), is proposed. The framework utilizes a conformal surface mesh of each grain boundary and a point-cloud file to describe, respectively, the polycrystal and its underlying defects. The input meshes are first preprocessed to remove undesired mesh artifacts. To create a suitable element-size distribution informed by the defect structure, elements near the defects are locally refined via edge splitting, while elements far from the defects are coarsened by edge collapsing. The quality of the gradated mesh is improved via edge swapping. The presented framework is robust and flexible, and its ability to selectively refine and coarsen a given mesh is demonstrated by three proof-of-concept models: (1) a polycrystalline open-cell foam with intra-ligament voids, (2) an experimentally measured microstructure with an observed fatigue crack, and (3) an additively manufactured polycrystalline microstructure with an evolving crack. The framework is shown to generate gradated meshes that are converged with respect to both global (e.g., macroscopic stress–strain curve) and local (e.g., crack-front kink angles) metrics, with a significant reduction in both element count and computation time and a minimal reduction in element quality as compared to a uniform mesh. The gradated meshes generated using this framework also provide similar simulation results as produced by their highly refined and uniform mesh counterparts, with the advantage of being more computationally efficient.

Effective elastic and fracture properties of regular and irregular masonry from nonlinear homogenization

Prediction of effective elastic and strength parameters of both regular and irregular masonry walls from homogenization is presented. To that end, the widely accepted first order homogenization method is adopted to provide the homogenized elastic stiffnesses or compliances as well as macroscopic parameters of the selected nonlinear constitutive models. These include the tensile and compressive strength and fracture energies of a generally orthotropic material extracted from the computationally derived macroscopic stress strain curves. In this regard, two types of boundary/loading conditions resulting from the strain-based and mix type formulation of the homogenization problem are examined. The response provided by an orthotropic damage model, expected to describe the behavior of the homogenized structure on macroscale, is compared to that derived via a classical isotropic scalar damage model. It reveals that strong constraints of the orthotropic damage model offer results inapplicable for estimating the macroscopic fracture properties thus promoting the application of a simple isotropic damage model when solving the homogenization problem. The results also show that the mixed boundary conditions, allowing us to represent a pure tension/compression loading mode while being capable of tracking the softening branch of the stress–strain curve, deliver the response comparable to that of a purely strain-based formulation.