Macroscopic Stress Research Articles

Simulations of multivariant Si I to Si II phase transformation in polycrystalline silicon with finite-strain scale-free phase-field approach

Scale-free phase-field approach and corresponding finite element method simulations for multivariant martensitic phase transformation from cubic Si I to tetragonal Si II in a polycrystalline aggregate are presented. Important features of the model are large and very anisotropic transformation strain tensor ɛt={0.1753;0.1753;−0.447} and stress-tensor dependent athermal dissipative threshold for transformation, which produce essential challenges for computations. 3D polycrystals with stochastically oriented grains are subjected to uniaxial strain- and stress-controlled loadings under periodic boundary conditions and zero averaged lateral strains. Coupled evolution of discrete martensitic microstructure, volume fractions of martensitic variants and Si II, stress and transformation strain tensors, and texture are presented and analyzed. Macroscopic variables effectively representing multivariant transformational behavior are introduced. Macroscopic stress–strain and transformational behavior for 55 and 910 grains are close. Large transformation strains and grain boundaries lead to huge internal stresses of tens GPa, which affect microstructure evolution and macroscopic behavior. In contrast to a single crystal, the local mechanical instabilities due to phase transformation and negative local tangent modulus are stabilized at the macroscale by arresting/slowing the growth of Si II regions by the grain boundaries. This leads to increasing stress during transformation. The developed methodology can be used for studying similar phase transformations with large transformation strains and for further development by including plastic strain and strain-induced transformations.

Macroscopic and mesoscopic simulation of polystyrene melt in the filling process of injection molding

AbstractIn this article, a fluid flow and heat transfer model which consist of the mass, momentum, and energy conservation equations together with the level set equation and Rolie‐Poly constitution equation is proposed and applied to simulate the rheological behaviors of viscoelastic polystyrene melt in the filling process of injection molding. The macroscopic velocity, pressure, temperature, stress, and mesoscopic molecular conformation of polystyrene melt are shown and analyzed detailedly during the mold filling process. The single‐mode Rolie‐Poly model is employed to describe the stress–strain relationship of polystyrene melt, and the level set method is used to capture the evolution of melt front interface. The governing equations for polymer melt are discretized by the finite volume method on collocated mesh, and solved by the SIMPLE algorithm. Numerical simulations have been implemented based on rectangular cavities with different gate locations, and rectangular and circular inserts. Numerical results indicate that the gate locations, gate numbers, insert size and shape are major factors that affect the rheological behaviors of polystyrene melt in the filling stage of injection molding, and polymer chains experienced higher shear and extension effects in the regions near to mold walls and obstacle. In addition, the length and position of weld lines are discussed in the current study.

Sliding modes of fault activation under constant normal stiffness conditions

Fault activation has been the focus of research community for years. However, the studies of fault activation remain immature, such as the fault activation mode and its major factors under constant normal stiffness (CNS) conditions associated with large thickness of fault surrounding rock mass. In this study, the rock friction experiments were conducted to understand the fault activation modes under the CNS conditions. Two major parameters, i.e. the initial normal stress and loading rate, were considered and calibrated in the tests. To reveal the response mechanism of fault activation, the local strains near the fault plane were recorded, and the macroscopic stresses and displacements were analyzed. The testing results show that the effect of displacement-controlled loading rate is more pronounced under the CNS conditions than that under constant normal load (CNL) conditions. Both the normal and shear stresses drop suddenly when the stick-slip occurs. The decrease and increase of the normal stress are synchronous with the shear stress in the regular stick-slip scenario, but mismatch with the shear stress during the chaotic stick-slip process. The results are helpful for understanding the fault sliding mode and the prediction and prevention of fault slip.

Macroscopic and mesoscopic biomechanical analysis of the bone unit in idiopathic scoliosis

To investigate the effects of postoperative fusion implantation on the mesoscopic biomechanical properties of vertebrae and bone tissue osteogenesis in idiopathic scoliosis, a macroscopic finite element model of the postoperative fusion device was developed, and a mesoscopic model of the bone unit was developed using the Saint Venant sub-model approach. To simulate human physiological conditions, the differences in biomechanical properties between macroscopic cortical bone and mesoscopic bone units under the same boundary conditions were studied, and the effects of fusion implantation on bone tissue growth at the mesoscopic scale were analyzed. The results showed that the stresses in the mesoscopic structure of the lumbar spine increased compared to the macroscopic structure, and the mesoscopic stress in this case is 2.606 to 5.958 times of the macroscopic stress; the stresses in the upper bone unit of the fusion device were greater than those in the lower part; the average stresses in the upper vertebral body end surfaces were ranked in the order of right, left, posterior and anterior; the stresses in the lower vertebral body were ranked in the order of left, posterior, right and anterior; and rotation was the condition with the greatest stress value in the bone unit. It is hypothesized that bone tissue osteogenesis is better on the upper face of the fusion than on the lower face, and that bone tissue growth rate on the upper face is in the order of right, left, posterior, and anterior; while on the lower face, it is in the order of left, posterior, right, and anterior; and that patients' constant rotational movements after surgery is conducive to bone growth. The results of the study may provide a theoretical basis for the design of surgical protocols and optimization of fusion devices for idiopathic scoliosis.

Designing shape-memory-like microstructures in intercalation materials

During the reversible insertion of ions, lattices in intercalation materials undergo structural transformations. These lattice transformations generate misfit strains and volume changes that, in turn, contribute to the structural decay of intercalation materials and limit their reversible cycling. In this paper, we draw on insights from shape-memory alloys, another class of phase transformation materials, that also undergo large lattice transformations but do so with negligible macroscopic volume changes and internal stresses. We develop a theoretical framework to predict structural transformations in intercalation compounds and identify crystallographic design rules necessary for forming shape-memory-like microstructures in intercalation materials. We use our approach to systematically screen open-source structural databases comprising n>5000 pairs of intercalation compounds. We identify candidate compounds, such as LixMn2O4 (Spinel), LixTi2(PO4)3 (NASICON), that approximately satisfy the crystallographic design rules and can be precisely doped to form shape-memory-like microstructures. Throughout, we compare our analytical results with experimental measurements of intercalation compounds. We find a direct correlation between structural transformations, microstructures, and increased capacity retention in these materials. These results, more generally, show that crystallographic designing of intercalation materials could be a novel route to discovering compounds that do not decay with continuous usage.

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.

A physically based elasto-viscoplastic constitutive model for modeling the hot deformation and microstructure evolution of a near α Ti alloy

In this work, a general elasto-viscoplastic constitutive model incorporating a finite strain viscoplastic flow law, the Kocks-Mecking dislocation density evolution, work hardening, stress softening due to dynamic recovery and dynamic recrystallization (DRX), and the Johnson-Mehl-Avarmi-Kolmogorov (JMAK) DRX kinetics was established for modeling the deformation and microstructure evolution of a near-α Ti alloy. A full-implicit stress integration scheme was proposed to implement the constitutive model into the commercial FEM solver ABAQUS/Standard. Isothermal hot compression experiments under various deformation conditions and detailed microstructure characterizations were conducted to study the deformation and DRX behaviors of the alloy, as well as to validate the developed material model. The alloy exhibits a typical discontinuous DRX behavior, macroscopically sensitive to various deformation conditions and microscopically strongly affected by the orientation of prior grains. The simulation results regarding the macroscopic flow stress curves and the local volume fractions of DRX in various regions in the specimen agree well with the experimental results. The proposed material model and the robust numerical implemental scheme can be extended to consider more physical deformation mechanisms and microstructure evolution of advanced structural metals and alloys.

On the strengthening and slip activity of Mg-3Al-1Zn alloy with pre-induced {101¯2} twins

{101¯2} twins were introduced into the magnesium (Mg) plate AZ31 via pre-rolling along its transverse direction. The plates, both with and without the pre-induced {101¯2} twins, were subjected to uniaxial tension along different directions. Using crystal plasticity modeling, we found that the strengthening effect of the pre-induced {101¯2} twins on the macroscopic flow stress primarily arised from the increased slip resistance caused by the boundaries, rather than the orientation hardening due to the twinning reorientation (although the latter did make its contribution in some specific loading directions). Besides, the pre-existing {101¯2} twins were found, by both experiments and simulation, to promote the activity of prismatic 〈a〉 and pyramidal 〈c + a〉 in the parent matrix of the material. Further analysis showed that the enhanced non-basal slip activity is related to the {101¯2} twin boundaries' low micro Hall-Petch slope ratios of non-basal slips to basal slip. With the critical resolved shear stress (CRSS) obtained from crystal plasticity modeling and the orientation data from EBSD, a probability-based slip transfer model was proposed. The model predicts higher slip transfer probabilities and thus lower strain concentration tendencies at {101¯2} twin boundaries than that at grain boundaries, which agrees with the experimental observation that the strain localization was primarily associated with the latter. The present findings are helpful scientifically, in deepening our understanding of how the pre-induced {101¯2} twins affect the strength and slip activity of Mg alloys, and technologically, in guiding the design of the pre-strain protocol of Mg alloys.

Failure of partially saturated frozen soils: A micromechanical analysis

Modelling the mechanical behaviour of partially saturated soils requires special treatment due to interactions among solid, water and air phases. The issue extends also to frozen soils in which ice forms cohesive bridges between soil particles. This study develops a micromechanically based failure criterion for shear strength of partially saturated frozen soil that relates the macroscopic strength of frozen soils to their particle-scale characteristics. Assuming idealized spherical particles, the strength of individual ice bonds is calculated from their geometry and the intrinsic strength of ice. Following a homogeneous stress assumption, a relation between the macroscopic stresses and the interparticle forces is formulated, which eventually leads to a failure criterion at the macroscopic level in terms of parameters such as ice strength, matric suction, void ratio, average particle size, and contact fabric. Other dependencies of the ice strength on parameters such as temperature and strain rate are also naturally captured. The predictions are compared to three different experimental datasets where the model demonstrates good accuracy without any retrospective calibration. Crucially, the failure model establishes a connection between strength of soil at frozen and unfrozen states, which can be used in estimating the strength degradation in frozen soils upon thawing.

Residual Stress Evolution in Low-Alloyed Steel at Three Different Length Scales

Quantitative and qualitative residual stress evolution in low-alloyed steel during heat treatment is investigated on three different length scales for sourgas resistant seamless steel tubes: on the component level, on the level of interdendritic segregation and on precipitate scale. The macroscopic temperature, phase and stress evolution on the component scale result from a continuum model of the heat treatment process. The strain and temperature evolution is transferred to a mesoscopic submodel, which resolves the locally varying chemistry being a result of interdendritic segregation. Within the segregation area and the surrounding matrix precipitates form. They are categorized with respect to their tendency for formation of microscopic residual stresses. After rapid cooling macroscopic stresses up to 700 MPa may form dependent on the cooling procedure. Mesoscopic stresses up to Δ50 MPa form depending on the extent of segregation. Carbides and inclusions occuring in low-alloyed steel are ranked by their tendency for residual stress formation in the iron matrix. This scale bridging study gives an overview of residual stresses, their magnitude and evolution on three different length scales in low-alloyed steels and the results presented can serve as a input for steel design.

Validating texture and lattice strain evolution models via in-situ neutron diffraction and shear tests

Validating micromechanical models under different loadings is challenging due to limited availability of experimental setups that can probe complex stress states. In this work, we perform in situ neutron diffraction experiments during shear loading to study the evolution of crystallographic texture and lattice strains that develop in different grain families. The experiments are conducted on 316 stainless steel samples, having a flat shear sample geometry suitable for testing in uniaxial rigs. The complex stress state in the gauge volume is estimated using an elasto-plastic finite element (FE) simulation, which is validated via experimentally measured gauge strains. The FE simulation reveals the generation of non-negligible in-plane normal stress components together with the in-plane shear stress component. The predicted macroscopic stresses are used as boundary conditions to drive an elasto-viscoplastic fast Fourier transform (EVP-FFT) based crystal plasticity model, which predicts lattice strain evolution. In addition, the evolution of the diffraction intensity is simulated using the Taylor model. The predictions of both models match very well with the experimental results. The experiments reveal that the {220} and {311} grain families have higher lattice strains than the {200} family. This surprising result is explained with the help of simulations.

Validation of a new viscoelastic model for unidirectional polymer matrix composites by analytical and numerical homogenisation

The long-term response of unidirectional, fibres-reinforced, polymer matrix composites is affected by the viscoelastic behaviour of the matrix. Most composite models introduce a different relaxation function for each of the five elastic parameters describing the transversely isotropic response at the scale of the homogenised ply, thus requiring long identification procedures based on multiple experimental tests. A new transversely isotropic model in the small strains regime was recently introduced by the authors, who used a Cartan decomposition in order to derive a new set of elastic parameters, which are more directly related to the behaviour of the underlying constituents. In particular, the deviatoric viscoelastic response of the matrix strongly affects only two of the five newly introduced parameters. In this work, analytical and numerical homogenisation approaches are developed and implemented in order to challenge such hypothesis and to validate the proposed model. In particular, a periodic homogenisation strategy enabling to impose general loading paths in terms of macroscopic strains and/or stresses on a Representative Volume Element (RVE) was developed in Abaqus, and used as a “virtual testing machine” to provide data to compare the transversely isotropic composite model proposed by the authors to the homogenised response of the RVE.

In Situ Nanostress Visualization Method to Reveal the Micromechanical Mechanism of Nanocomposites by Atomic Force Microscopy.

An in situ atomic force microscopy (AFM) nanomechanical technique was used to directly visualize the micromechanical behaviors of polymer nanocomposites during compressive strain. We obtained a stress distribution image of carbon black (CB)-filled rubber at the nanoscale for the first time, and we traced the microscopic deformation behaviors of CB particles. Through this experiment, we directly revealed the microscopic reinforcement mechanisms of rubber composites. We found that CB-filled rubbers exhibited heterogeneous local microscopic deformations, which were related to the dispersion of CB particles in rubber matrices. The local stress distributions of the rubber composites showed heterogeneity, and the stresses were concentrated in the regions near the CB particles during compression. The area of stress concentration gradually expanded with increasing strain and eventually formed a stress network structure. This stress network bore most of the macroscopic stress and was considered the key reinforcement mechanism of CB-filled rubber. The stress transfer process in the rubber matrix was visualized in real space for the first time. Based on the image data from the AFM experiments, we used finite-element method (FEM) simulations to reproduce the microscopic deformation process of CB-filled rubber. The stress distribution images simulated by FEM showed heterogeneity consistent with AFM. In this study, an in situ visualization of material deformation confirmed the predictions of microscopic deformation behavior from previous theories and models; it also provided new insights into the microscopic reinforcement mechanisms of CB-filled rubber composites based on microscopic stress distribution images.

Spatial clustering of microscopic dynamics governs the slip avalanche of sheared granular materials

Establishing quantifiable links between individual-particle dynamics and macroscopic response of granular materials has been a longstanding challenge, with implications in material science, geology and industry. Despite sustained efforts in uncovering generic features in both macroscopic flow and microscopic dynamics, further advance on the subject matter demands quantitative correlations to be established. We propose a 3D convolution neural network (CNN) to quantify the link between microscopic dynamics and macroscopic stress fluctuations, including both stress recharge (stick regime) and stress drop (slip regime). Through the model interpretation, microscopic dynamics is found to demonstrate distinctive spatial patterns in the stick and slip regimes, which root in the result of free volume-induced structural rearrangements and contact network dynamics, respectively. We conclude that the spatial clustering of microscopic dynamics governs the occurrence of slip avalanches and acts as the “fingerprint” of macroscopic stress fluctuation. The data-driven framework developed in this paper can be readily extended to other amorphous solids for building cross-scale relations, paving a new way to understand the complex behavior of amorphous solids.

Multiscale modeling of laminated thin-shell structures with Direct FE[formula omitted

Shell finite element is computationally efficient and thereby favored in large-scale structural analysis. With the application of advanced materials, understanding the relation between micro-heterogeneities and structural behavior is important. To this end, we propose a multiscale method in the current contribution for modeling thin-walled fiber reinforced composite laminates with Mindlin–Reissner shell theory and Direct FE2. By exploiting the length scales in layered structure together with a shell formulation degenerated from continuum solid element, both in-plane and through-thickness periodicity can be assumed. A single-fiber unit cell attached to each Gauss point is employed as a representative volume element (RVE) rather than a through-thickness microstructure. In this way, the scale transitions are naturally dealt with using a first-order homogenization framework, in which the downscaling of transverse shear strain, commonly troublesome for the through-thickness RVE, is straightforward. The other merit inherited from Direct FE2 is that the two-scale problem is formulated in a single finite element analysis wherein the macro and micro DOFs are coupled based on the downscaling kinematics. It avoids the upscaling of the macroscopic stresses and tangents in the conventional nested FE2 scheme, which is challenging in the case of complex microstructural response. The developed method is then verified through several benchmarking examples, including a flat plate and two curve shells. Isotropic materials and composite laminates with homogeneous and heterogeneous RVEs have been examined. Two examples with plasticity and cohesive cracks are further presented to showcase its potential in modeling damage propagation in composite structures.

An innovative experimental energy-based approach for the life prediction of notched details

Fatigue assessment of mechanical components can be very challenging due to the presence of macroscopic defects and notches. However, local approaches as the Stain Energy Density, have shown how the fatigue resistance is strictly related to the local stress field near the notch tip and not on the overall dimensions of the component. It is necessary to know some peculiar material’s parameters to apply those local approaches. A short way to obtain such parameters can be to apply Thermographic Methods, such as the Risitano’s Thermographic Method and the Static Thermographic Method, monitoring the energy release during stepwise fatigue and static tensile tests. In this work, experimental tests were performed on plain and double edge steel notched specimens. The SED method has been applied through finite element models, with the easy application of the volume-free procedure, to evaluate its trend at several stress levels. The analysis of the energy release and of the local averaged strain energy density during a static tensile test may be adopted as a novel approach to relate the local stress state with a macroscopic stress value that introduce in the material the first micro plasticization leading to the failure of the component.

Coupled microplane and micromechanics model for describing the damage and plasticity evolution of quasi-brittle material

Quasi-brittle materials exhibit significant heterogeneity due to their complex mesoscopic components and microdefects, and their macroscopic mechanical behavior depends strongly on crack initiation, propagation and friction-slip. In this study, by coupling microplane and micromechanics, a new constitutive model that can consider the mesoscopic damage and plasticity mechanisms of quasi-brittle materials is proposed. The microplane is used throughout the whole construction process for the model: first, a representative volume element (RVE) is established to characterize the mesoscopic structure of quasi-brittle materials, and the advantages of microplanes are fully exploited to characterize the equivalent cracks‒matrix system. Then, the relationship between macroscopic and microplane strains (stresses) is elucidated. Second, considering the unilateral effect for cracks, the expressions for the free energy potential functions for a microplane and for the RVE are established. Third, for each microplane, the damage criterion characterized by the damage driving force is constructed for the open and closed cracks, and in particular, a new plasticity criterion characterizing the static friction property coupled damage and dilatancy behavior during strain softening deformation is constructed. By considering the characteristics of the microplane to relate macroscopic and mesoscopic stress and strain, an efficient return mapping algorithm based on the Newton‒Raphson method is derived. Finally, the stress‒strain curves and the evolution of damage and plasticity are simulated under conventional triaxial compression, uniaxial tensile, direct shear and true triaxial compression stress paths, and the effectiveness of the model is verified by comparison to the experimental data. To better reflect the inherent correlation between mesoscopic damage and macroscopic mechanical behavior, a microplane configuration that can visually characterize the mesoscopic evolution characteristics of damage is established, and it is shown that the simulated results correspond well to the macroscopic deformation and failure pattern of granite samples under different loading paths.

Resolving intragranular stress fields in plastically deformed titanium using point-focused high-energy diffraction microscopy

The response of a polycrystalline material to a mechanical load depends not only on the response of each individual grain, but also on the interaction with its neighbors. These interactions lead to local, intragranular stress concentrations that often dictate the initiation of plastic deformation and consequently the macroscopic stress–strain behavior. However, very few experimental studies have quantified intragranular stresses across bulk, three-dimensional volumes. In this work, a synchrotron X-ray diffraction technique called point-focused high-energy diffraction microscopy (pf-HEDM) is used to characterize intragranular deformation across a bulk, plastically deformed, polycrystalline titanium specimen. The results reveal the heterogenous stress distributions within individual grains and across grain boundaries, a stress concentration between a low and high Schmid factor grain pair, and a stress gradient near an extension twinning boundary. This work demonstrates the potential for the future use of pf-HEDM for understanding the local deformation associated with networks of grains and informing mesoscale models.Graphical abstract

Implementation and experimental validation of nonlocal damage in a large-strain elasto-viscoplastic FFT-based framework for predicting ductile fracture in 3D polycrystalline materials

Ductile materials, such as metal alloys, can undergo substantial deformation before failure. Additionally, these materials are usually of polycrystalline composition and exhibit strongly anisotropic behavior at small length scales. Previously developed fast Fourier transform (FFT)-based models can model ductile fracture of isotropic materials or the elastic–plastic behavior of anisotropic polycrystalline materials; however, there remains a need to couple both capabilities. This work extends a large-strain FFT-based crystal plasticity model to simulate ductile fracture of polycrystalline materials. A triaxiality-based continuum damage mechanics (CDM) formulation is incorporated into a large-strain elasto-viscoplastic FFT (LS-EVPFFT) framework. The CDM formulation is augmented with an integral-based nonlocal regularization approach that correctly handles gas-phase material necessary to model unconstrained surfaces. To validate the damage-enabled LS-EVPFFT framework, mesoscale copper tensile coupons were machined using microwire electrical discharge machining and experimentally characterized using electron backscatter diffraction. In-situ optical digital image correlation was performed during uniaxial testing to provide a side-by-side comparison of the experimental and computational strain fields and stress–strain responses. The damage-enabled LS-EVPFFT framework can simulate the complete macroscopic stress–strain response of ductile polycrystals to failure. The model reproduces necking behavior that qualitatively agrees with experimental observations. By leveraging the relatively low computational cost of the damage-enabled LS-EVPFFT framework, the framework presented here allows the ductile fracture response of 3D polycrystalline materials to be tractably predicted.

Lung Mechanics: A Review of Solid Mechanical Elasticity in Lung Parenchyma.

The lung is the main organ of the respiratory system. Its purpose is to facilitate gas exchange (breathing). Mechanically, breathing may be described as the cyclic application of stresses acting upon the lung surface. These forces are offset by prominent stress-bearing components of lung tissue. These components result from the mechanical elastic properties of lung parenchyma. Various studies have been dedicated to understanding the macroscopic behaviour of parenchyma. This has been achieved through pressure-volume analysis, numerical methods, the development of constitutive equations or strain-energy functions, finite element methods, image processing and elastography. Constitutive equations can describe the elastic behaviour exhibited by lung parenchyma through the relationship between the macroscopic stress and strain. The research conducted within lung mechanics around the elastic and resistive properties of the lung has allowed scientists to develop new methods and equipment for evaluating and treating pulmonary pathogens. This paper establishes a review of mathematical studies conducted within lung mechanics, centering on the development and implementation of solid mechanics to the understanding of the mechanical properties of the lung. Under the classical theory of elasticity, the lung is said to behave as an isotropic elastic continuum undergoing small deformations. However, the lung has also been known to display heterogeneous anisotropic behaviour associated with large deformations. Therefore, focus is placed on the assumptions and development of the various models, their mechanical influence on lung physiology, and the development of constitutive equations through the classical and non-classical theory of elasticity. Lastly, we also look at lung blast mechanics. No explicit emphasis is placed on lung pathology.