Multi-scale Constitutive Model Research Articles

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

Numerical Simulation of Container Breach and Airborne Release of Solids Due to Mechanical Insults

Abstract Throughout U.S. Department of Energy (DOE) complexes, safety engineers employ the five-factor formula to calculate the source term (ST) that includes parameters of airborne release fraction (ARF), respirable fraction (RF) and damage ratio (DR). Limited experimental data on fragmentation of solids, such as ceramic pellets (i.e., PuO2), and container breach due to mechanical insults (i.e., drop and forklift impact), can be supplemented by modeling and simulation using high fidelity computational tools to estimate these parameters. This paper presents the use of Sandia National Laboratories' SIERRA solid mechanics (SM) finite element code to investigate the behavior of the widely utilized waste container (such as 7A Drum) subject to a range of free fall impact and puncture scenarios. The resulting behavior of the container is assessed, and the estimates are presented for bounding DRs from calculated breach areas for the various accident conditions considered. This paper also describes a novel multiscale constitutive model recently implemented in SIERRA/SM that simulates the fracture of brittle materials such as PuO2 and determines ARF during the fracture process. Comparisons are made between model predictions and simple bench-top experiments.

Multiscale constitutive modeling of additively manufactured Al–Si–Mg alloys based on measured phase stresses and dislocation density

To better understand and predict the mechanical properties of additive manufacturing (AM) Al–Si–Mg alloys, developing a physically-based constitutive model is crucial. Among different models, the dislocation-density-based Kocks-Mecking (K-M) constitutive model has been widely used. Unfortunately, two challenges arise when the K-M model is used for the multiphase Al alloys. Firstly, an accurate K-M model demands the separation of phase stresses. Secondly, a thorough analysis of the K-M model involves the measurement of dislocation density during deformation. In-situ neutron diffraction, a powerful method to measure the phase stress and dislocation density in bulk polycrystalline materials under loading, is employed to investigate the AM AlSi3.5Mg1.5 and AlSi3.5Mg2.5 (wt.%) alloys. Based on the present results and reported data of AlSi10Mg, a multiscale constitutive model is developed for different AM Al–Si–Mg alloys. At the microscale, the evolution of dislocation density in the Al matrix with plastic strain can be well predicted by the K-M model. Meanwhile, the developments of the average stresses in different phases with plastic strain can be well captured by the Voce model. The measured microscopic k2/nc values agree with the theoretical value two of the K-M model quite well. Here, nc is the characteristic factor of the microscale Voce model, while k2 is the coefficient associated with the dynamic recovery process in the microscale K-M model. At the macroscale, the mechanical behavior can also be well reproduced by the K-M model and the Voce model. However, the macroscopic K2/Nc ratio is far away from two, where Nc and K2 are the characteristic factor and coefficient of the macroscale Voce model and K-M model, respectively.

Multiscale modeling of coupling mechanisms in electrically assisted deformation of ultrathin sheets: An example on a nickel-based superalloy

Electrically assisted (EA) forming has ubiquitous merits over room-temperature (RT) forming and thermally aided forming for the fabrication of difficult-to-form microscale products. However, it is difficult to predict the material deformation behavior in the EA microforming process owing to the coupling between the electric current and microstructural size effect. To develop a robust constitutive model that considers the interplay between the electric and grain size effects, RT and EA quasi-static uniaxial tensile tests were performed on ultrathin nickel-based superalloy sheets with a thickness of 0.2 mm and grain sizes ranging from 27.2 to 79.4 μm. The experimental results demonstrated that the Joule heating effect and the normalized flow stress reduction were non-monotonically related to the grain size of the superalloy. The grain size effect in the polycrystalline superalloy was suppressed by the enhanced current density. A multiscale constitutive model that considered multiple strengthening mechanisms was proposed to describe the EA deformation behavior of ultrathin superalloy sheets. The multiscale model was proven to have a desirable predictive ability for the EA drawing force of thin-walled superalloy capillaries. Furthermore, the model revealed that the weakening of the grain size effect with increasing current density in the polycrystalline superalloy was caused by the combined variations in dislocation interaction, shear modulus, and strengthening mechanisms. The abnormal evolution of the surface effect with the current density was captured by the proposed constitutive model, which demonstrates that the electric current can promote the grain size effect in the multicrystalline superalloy.

Micromechanical modeling of work hardening for coupling microstructure evolution, dynamic recovery and recrystallization: Application to high entropy alloys

A dislocation density-based multiscale constitutive model is developed to describe the hot deformation of high entropy alloys. The work hardening process of high entropy alloy is divided into several stages according to the change of hardening rate. As crystalline material, high entropy alloys are regarded as two-phase materials. In the current study, a physical-based constitutive model with three key microstructural state variables, cell mobile dislocation density, cell immobile dislocation density and wall immobile dislocation density, is proposed to simulate the phases deformations. The physical mechanisms including strain hardening, dynamic recovery, and recrystallization are considered in describing the evolution of dislocation densities. It is noted that the competition and interaction among the three mechanisms result in a typical hardening behavior of high entropy alloys. The dislocation density-based constitutive model is embedded in the modified Mori-Tanaka scheme to describe the hardening behaviors of two-phase materials with an elastic phase cell wall and an elastic-inelastic phase cell interior. The model is applied for simulating the hardening curves of CoCrFeMnNiC0.5 high entropy alloy at different temperatures and strain rates. Compared with experimental data, it shows that the developed model can describe hot deformation of high entropy alloys with reasonable accuracy. The macroscopic inelastic strain in the Mori-Tanaka scheme is calculated by a classical plasticity method. The simulation results show that the modified method can give a more accurate prediction compared to the average inelastic strain of all phases.

Experimental investigation and multiscale modeling of reactive powder cement pastes subject to triaxial compressive stresses

This paper presents experimental and numerical studies on the triaxial mechanical behaviors of reactive powder cement (RPC) pastes prepared with an alcohol-based shrink-reducing admixture (ASRA) and ice-replaced mixing procedure. Standard cylindrical samples were cored from three kinds of high strength RPC pastes. For each kind of RPC pastes, a series of triaxial compression tests were performed under different confining pressures. It is shown that the RPC pastes under consideration present a typical quasi-brittle behaviours, including nonlinearity of stress-strain curves, strain hardening/softening, pressure sensitivity, volumetric dilatancy, triaxial strength nonlinearity, etc.For completeness, triaxial cyclic tests with two different confining pressure were conducted to analyse the mechanical damage characteristics of RPC pastes. Based on the experimental investigations and considering the mircostructure evolution of RPC pastes, a two-step homogenization-based multiscale constitutive model was developed. The model allows to take into account the influence of porosity on elastic constants, material damage induced by mesocracking as well as plastic deformation caused by frictional sliding. There is a good agreement between numerical simulations and experimental data.

Multiscale constitutive model with progressive recruitment for nanofibrous scaffolds

Biomedical applications need tailor-made scaffolds that exhibit biomimetic mechanical properties. In this context, electrospinning has emerged as a technique with promising features for their production. However, the electrospun scaffolds mechanical behavior as a function of the microstructure and nanofiber properties is still poorly understood. Besides, multiscale constitutive modeling appears as a powerful design tool, not only able to characterize electrospun structures, but also to determine the fiber properties and scaffold microstructure that would achieve the objective response. With focus in this last aspect, we developed a multiscale constitutive model for nanofibrous structures that takes into account the material constitutive properties, scaffold microstructure, and nanofiber progressive recruitment. A statistical approach of the nanofibers tortuosity with a modified Gaussian distribution was adopted, which allowed for reproducing the scaffolds macroscopic nonlinear mechanical behavior. It was observed that such behavior arises even if the nanofibers response is considered as mechanically linear. Experimental data from pressure vs. diameter inflation tests of electrospun tubular scaffolds was used to validate the model. In addition, the influence of the microstructural parameters upon the macroscopic constitutive behavior was studied. Finally, the model parameters were adjusted to obtain a vascular graft able to reproduce the mechanical response of a target natural tissue. The current study presents a step towards understanding, characterizing, and optimizing the mechanical properties of nanofibrous biomaterials.

Constitutive model for the growth of a gel network regulated by its microscopic mechanics toward a homeostatic state

Growth is one of key issues in understanding how a single fertilized cell can develop into tissues, organs, and a life body of particular shapes to perform diverse functions. Here we provide a multi-scale constitutive modeling framework for the growth of a gel network regulated by its microscopic mechanics. Within the modeling framework, the mechanics of a single constituent polymer chain, chemical potentials of the solution, the interaction between polymer chains and the solution, the regulation of growth by varied physical parameters at the microscopic scale, etc., can all be considered in detail. The theory is then utilized to predict the isotropic growth of a cube or the anisotropic growth of a thin cylindrical shell when subjected to varied mechanical loads. Our analysis indicates how macroscopic shapes and mechanical properties of structures in growth are affected by mechanical loads, as structures try to maintain a homeostatic state at the microscopic scale. It is expected that our theory can be utilized to investigate varied sorts of growth existing at different scales in development.

A multiscale synthesis: characterizing acute cartilage failure under an aggregate tibiofemoral joint loading.

Knee articular cartilage is characterized by a complex mechanical behavior, posing a challenge to develop an efficient and precise model. We argue that the cartilage damage, in general, can be traced to the fibril level as a plastic deformation, defined as micro-defects. To investigate these micro-defects, we have developed a detailed finite element model of the entire healthy tibiofemoral joint (TF) including a multiscale constitutive model which considers the structural hierarchies of the articular cartilage. The net model was simulated under physiological loading conditions to predict joint response under 2000N axial compression and damage initiation under high axial loading (max 7 KN) when the TF joint flexed to 30°. Computed results sufficiently agreed with earlier experimental and numerical studies. Further, initiation and propagation of damage in fibrils were computed at the tibial cartilage located mainly in the superficial and middle layers. Our simulation results also indicated that the stiffer the fibril is (higher cross-link densities), the higher the contact stress required to elicit a fibril yield and the higher the rate of yielding as a function of increased contact stress. To the best of our knowledge, this is the first model that combines macro-continuum joint mechanics and micromechanics at the tissue level. The computational construct presented here serves as a simulation platform to explore the interplay between acute cartilage damage and micromechanics characteristics at the tropocollagen level.

Simplified Formulation of Mechanics of Structure Genome

This paper presents a simplified formulation of the mechanics of a structure genome (MSG). The MSG is a unified theory for multiscale constitutive modeling for all types of composite structures. It is generalized from the previous research works based on the variational asymptotic method (VAM), including the variational asymptotic beam sectional analysis, the variational asymptotic plate and shell analysis, and the variational asymptotic method for unit cell homogenization. In this paper, the original complex formulation of the MSG is simplified and the principle of minimum information loss underpinning the MSG is formalized. The simplified formulation of the MSG is applied to construct the linear elastic Euler–Bernoulli beam model, the Kirchhoff plate model, and three-dimensional Cauchy continuum model. The connections of the MSG to previous VAM-based theories are clearly pointed out.

The effect of fibrillar degradation on the mechanics of articular cartilage: a computational model.

The pathogenesis and pathophysiological underpinnings of cartilage degradation are not well understood. Either mechanically or enzymatically mediated degeneration at the fibril level can lead to acute focal injuries that will, overtime, cause significant cartilage degradation. Understanding the relationship between external loading and the basic molecular structure of cartilage requires establishing a connection between the fibril-level defects and its aggregate effect on cartilage. In this work, we provide a multiscale constitutive model of cartilage to elucidate the effect of two plausible fibril degradation mechanisms on the aggregate tissue: tropocollagen crosslink failure (β) and a generalized surface degradation (δ). Using our model, the mechanics of aggregate tissue shows differed yield stress and post-yield behavior after crosslink failure and surface degradation compared to intact cartilage, and the tissue-level aggregate behaviors are different from the fibrillar behaviors observed in the molecular dynamics simulations. We also compared the effect of fibrillar defects in terms of crosslink failure and surface degradation in different layers of cartilage within the macroscale tissue construct during a simulated nanoindentation test. Although the mechanical properties of cartilage tissue were largely contingent upon the mechanical properties of the fibril, the macroscale mechanics of cartilage tissue showed ~ 10% variation in yield strain (tissue yield strain: ~ 27 to ~ 37%) compared to fibrillar yield strain (fibrillar yield strain: ~ 16 to ~ 26%) for crosslink failure and ~ 7% difference for the surface degradation (yield strain variations at the tissue: ~ 30 to ~ 37% and fibril: ~ 24 to ~ 26%) at the superficial layer. The yield strain was further delayed in middle layers at least up to 30% irrespective of the failure mechanisms. The cartilage tissue appeared to withstand more strain than the fibrils. The degeneration mechanisms of fibril differentially influenced the aggregate mechanics of cartilage, and the deviation may be attributed to fiber-matrix interplay, depth-dependent fiber orientation and fibrillar defects with different degradation mechanisms. The understanding of the aggregate stress-strain behavior of cartilage tissue, cartilage degradation and its underlying biomechanical factors is important for developing engineering approaches and therapeutic interventions for cartilage pathologies.

New micromechanics approaches for the effective properties of multiferroics composites with spring-type imperfect interfaces

This study proposed two new micromechanics models to predict the effective properties of multiferroics composites with spring-type imperfect interfaces. The first model is a full field micromechanics scheme based on the mechanics of structure genome, a multiscale constitutive modeling framework, recently discovered to unify the micromechanics and the structural mechanics. The second model is a mean field micromechanics approach that extends the Mori-Tanaka scheme using the concept of the interior and exterior-point Eshelby tensors to multiferroics composites. The imperfect interface model assumes that the normal components of the fluxes (stress, electric displacement and magnetic flux) are continuous across the interface, whereas the potential fields (displacement, electric potential and magnetic potential) suffer interfacial jumps proportional to the normal components of the fluxes. In contrast to multiferroics composites with perfect contact conditions, the effective properties formulations show the dependence on the size of the reinforcements. Numerical examples of fibrous multiferroics composites are used to demonstrate the robustness and accuracy of the proposed micromechanics theories. The size-dependency of the overall properties shows the importance of imperfect interfaces in predicting the effective properties of multiferroics composites.

Multi-scale constitutive modeling of natural fiber fabric reinforced composites

Natural fiber reinforced structural composites have a hierarchical structure made from stacked fabrics of interwoven yarns formed by twisting together discontinuous natural fibers. We develop multi-scale constitutive models to investigate the mechanical properties of plain woven composites. The multi-scale constitutive modeling is carried out in two steps: the effective micromechanical properties of a micro-scale representative volume element (RVE) of the twisted yarn are calculated using an orientation averaging method, which are subsequently transferred to a meso-scale RVE of the final composite to compute the elastic constants by homogenization over the RVE. 3D finite element models of the meso-scale RVEs of composites are developed to verify the accuracy of the proposed model. The multi-scale modeling results show that the yarn twist angle has significant effects on the elastic properties of the composites. The predicted results from the multi-scale constitutive model show good agreements with that from finite element analysis and experiment.

Numerical study of grain refinement induced by severe shot peening

Three-dimensional finite element models of single shot impact and multiple shot impacts are developed to study grain refinement induced by severe shot peening. A multiscale constitutive model coupling the microscopic dislocation density and the macroscopic plasticity theory is proposed and implemented into the finite element models to characterize the formation and evolution of dislocation cell structure. The random distribution of the indentations under probability control is achieved by parametric modeling to simulate multiple shot impacts with respect to 100% peening coverage. Severe shot peening of AISI 4340 steel under various coverages is simulated and the cell sizes predicted by the finite element models are in good agreement with the experimental data. Parametric analysis of severe shot peening is carried out by using different size shots with the same kinetic energy. The numerical results indicate that: (1) the larger shot causes a larger depth of grain refinement, (2) the smaller shot is beneficial for grain refinement of the peened surface and subsurface, and (3) the shot oblique impact results in a finer dislocation cell structure. Moreover, the interactive effect of multiple shot impacts can effectively improve grain refinement of near surface materials.

Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling

In recent experimental studies a possible damage mechanism of collagenous tissues mainly caused by fatigue was disclosed. In this contribution, a multi-scale constitutive model ranging from the tropocollagen (TC) molecule level up to bundles of collagen fibers is proposed and utilized to predict the elastic and inelastic long-term tissue response. Material failure of collagen fibrils is elucidated by a permanent opening of the triple helical collagen molecule conformation, triggered either by overstretching or reaction kinetics of non-covalent bonds. This kinetics is described within a probabilistic framework of adhesive detachments of molecular linkages providing collagen fiber integrity. Both intramolecular and interfibrillar linkages are considered. The final constitutive equations are validated against recent experimental data available in literature for both uniaxial tension to failure and the evolution of fatigue in subsequent loading cycles. All material parameters of the proposed model have a clear physical interpretation. Statement of significanceIrreversible changes take place at different length scales of soft fibrous tissues under supra-physiological loading and alter their macroscopic mechanical properties. Understanding the evolution of those histologic pathologies under loading and incorporating them into a continuum mechanical framework appears to be crucial in order to predict long-term evolution of various diseases and to support the development of tissue engineering.

Numerical Simulation of Early Age Cracking of Reinforced Concrete Bridge Decks with a Full-3D Multiscale and Multi-Chemo-Physical Integrated Analysis

In November 2011, the Japanese government resolved to build “Revival Roads” in the Tohoku region to accelerate the recovery from the Great East Japan Earthquake of March 2011. Because the Tohoku region experiences such cold and snowy weather in winter, complex degradation from a combination of frost damage, chloride attack from de-icing agents, alkali–silica reaction, cracking and fatigue is anticipated. Thus, to enhance the durability performance of road structures, particularly reinforced concrete (RC) bridge decks, multiple countermeasures are proposed: a low water-to-cement ratio in the mix, mineral admixtures such as ground granulated blast furnace slag and/or fly ash to mitigate the risks of chloride attack and alkali–silica reaction, anticorrosion rebar and 6% entrained air for frost damage. It should be noted here that such high durability specifications may conversely increase the risk of early age cracking caused by temperature and shrinkage due to the large amounts of cement and the use of mineral admixtures. Against this background, this paper presents a numerical simulation of early age deformation and cracking of RC bridge decks with full 3D multiscale and multi-chemo-physical integrated analysis. First, a multiscale constitutive model of solidifying cementitious materials is briefly introduced based on systematic knowledge coupling microscopic thermodynamic phenomena and microscopic structural mechanics. With the aim to assess the early age thermal and shrinkage-induced cracks on real bridge deck, the study began with extensive model validations by applying the multiscale and multi-physical integrated analysis system to small specimens and mock-up RC bridge deck specimens. Then, through the application of the current computational system, factors that affect the generation and propagation of early age thermal and shrinkage-induced cracks are identified via experimental validation and full-scale numerical simulation on real RC slab decks.

Cyclic welding behavior of covalent adaptable network polymers

ABSTRACTSurface welding effect of covalent adaptable network (CAN) polymers enables self‐healing, reprocessing and recycling of thermosets, but little is known about their welding behaviors during repeated welding‐peeling cycles. In this article, we study the cyclic welding effect of an epoxy based thermal‐sensitive CAN. Surface roughness is generated by rubbing the sample on sandpapers with different grid sizes. The welding‐peeling cycles are repeated on the same pair of samples for five times, with roughness amplitude and interfacial fracture energy measured in each cycle. It is shown that the roughness gradually decreases during the repeated welding cycles, especially when a long welding time or high welding pressure is applied. Even though lower roughness amplitude promotes the contact area, the interfacial fracture energy reduces due to the increased BER activation energy after long‐time heating. A multiscale constitutive model is adopted, where we incorporate an explicit expression of interfacial contact area as a function of root‐mean‐square roughness parameter. The model is able to capture the evolving interfacial fracture energy during repeated welding cycles by using the measured roughness parameter, network modulus and BER activation energy. The study provides theoretical basis for the design and applications of CANs involving cyclic welding‐peeling operations. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018, 56, 402–413.

A new method for topology design of electromagnetic antennas in hyperthermia therapy

The topological derivative concept has been proved to be useful in many relevant applications such as topology optimization, inverse problems, image processing, multi-scale constitutive modeling, fracture mechanics and damage evolution modeling. In this work, we develop a new optimization method based on the topological derivative concept applied to the cancer treatment by hyperthermia. Hyperthermia therapy is a non-invasive medical treatment in which body tissue is artificially heated through electromagnetic waves, focusing the heat in cancerous cells undergoing apoptosis. The basic idea, therefore, consists in finding a distribution of heat source generated by electromagnetic antenna aiming to increase the temperature in the region occupied by the tumor, while keeping the temperature in the remainder part of the body. Numerical results are presented illustrating possible application of the proposed methodology to treatment of cancer by hyperthermia.

Arterial mechanics considering the structural and mechanical contributions of ECM constituents

Considering the organization and engagement behavior of different extracellular matrix (ECM) constituents in the medial and adventitial layer of the arterial wall, in this study, we proposed a new constitutive model of ECM mechanics that considers the distinct structural and mechanical contributions of medial elastin, medial collagen, and adventitial collagen, to incorporate the constituent-specific fiber orientation and the sequential fiber engagement in arterial mechanics. Planar biaxial tensile testing method was used to characterize the orthotropic and hyperelastic behavior of porcine thoracic aorta. Fiber distribution functions of medial elastin, medial collagen, and adventitial collagen were incorporated into the constitutive model. Considering the sequential engagement of ECM constituents in arterial mechanics, a recruitment density function was incorporated into the model to capture the delayed engagement of adventitial collagen. A freely jointed chain model was used to capture the mechanical behavior of elastin and collagen at the fiber level. The tissue-level ECM mechanics was obtained by incorporating fiber distribution, engagement, and elastin and collagen content. The multi-scale constitutive model considering the structural and mechanical contributions of the three major ECM constituents allows us to directly incorporate information obtained from quantitative multi-photon imaging and analysis, and biochemical assay for the prediction of tissue-level mechanical response. Moreover, the model shows promises in fitting and predicting with a small set of material parameters, which has physical meanings and can be related to the structure of the ECM.

Kinetics of evolution of radiation induced micro-damage in ductile materials subjected to time-dependent stresses

The present paper aims at predicting evolution of radiation induced damage in the solids subjected to mechanical loads beyond the yield stress. Moreover, the evolution of radiation induced damage is combined with the evolution of mechanically induced damage within the common framework of Continuum Damage Mechanics (CDM). An additive formulation with respect to damage parameters (tensors) has been postulated. Multiscale constitutive model containing strong physical background related to the mechanism of generation of clusters of voids in the irradiated solids has been built. The model is based on the experimental estimation of concentration of lattice defects (interstitials, di-interstitials, interstitial clusters, vacancies, di-vacancies, vacancy clusters) in Al as a function of dpa (displacement per atom), and comprises the relevant kinetics of evolution of radiation induced damage under mechanical loads. Two kinetic laws of damage evolution were taken into account: the Rice & Tracey model and – for comparison – the Gurson model. As an application, estimation of lifetime of a cylindrical shell (coaxial target embedded in a detector of particles) subjected to combination of irradiation and mechanical loads, has been carried out. It is demonstrated that the number of cycles to failure depends strongly on the accumulation of micro-damage due to irradiation. The lifetime of irradiated components has been expressed as a function of two parameters: maximum dpa and axial stress amplitude on cycle.