Stress-free State Research Articles

Multistability of segmented rings by programming natural curvature

Multistable structures have widespread applications in the design of deployable aerospace systems, mechanical metamaterials, flexible electronics, and multimodal soft robotics due to their capability of shape reconfiguration between multiple stable states. Recently, the snap-folding of rings, often in the form of circles or polygons, has shown the capability of inducing diverse stable configurations. The natural curvature of the rod segment (curvature in its stress-free state) plays an important role in the elastic stability of these rings, determining the number and form of their stable configurations during folding. Here, we develop a general theoretical framework for the elastic stability analysis of segmented rings (e.g., polygons) based on an energy variational approach. Combining this framework with finite element simulations, we map out all planar stable configurations of various segmented rings and determine the natural curvature ranges of their multistable states. The theoretical and numerical results are validated through experiments, which demonstrate that a segmented ring with a rectangular cross-section can show up to six distinct planar stable states. The results also reveal that, by rationally designing the segment number and natural curvature of the segmented ring, its one- or multiloop configuration can store more strain energy than a circular ring of the same total length. We envision that the proposed strategy for achieving multistability in the current work will aid in the design of multifunctional, reconfigurable, and deployable structures.

Improved energy method and agglomeration influence of carbon nanotubes on polymer composites

In this paper, the geometric analysis of carbon nanotubes (CNTs) without external loading is carried out by energy method. Based on the theory of molecular mechanics, an improved mechanical model is proposed to predict the energy of armchair carbon nanotubes under stress-free conditions, and the diameter of CNTs is estimated according to the principle of minimum energy. The results show that the diameter obtained by the improved model is larger, but basically consistent with that obtained by conformal mapping. The inversion energy term is added to the modified model, and the inversion energy term related to atomic curvature is characterized by the conization angle. It can be seen from the error that the inversion energy of carbon nanotubes can not be neglected in the stress-free state, especially in the case of small diameter. The agglomeration of nanotubes is one of the important factors, which affects the effective elastic modulus of nanocomposites. Here, a new micro-mechanics model consisting of both agglomeration of CNTs and pure matrix is also presented to analyze its effect on the effective elastic modulus. It is noted from the results that the stiffness of nanocomposites is very sensitive to the CNTs agglomeration.

Residual stresses for a new class of transversely isotropic nonlinear elastic solid

We study the response of a class of transversely elastic bodies, wherein the Green–Saint Venant strain tensor is a function of the second Piola–Kirchhoff stress tensor, when the body is residually stressed. The notion of such non-Cauchy elastic bodies being transversely isotropic is defined in Rajagopal (Mech. Res. Commun. 64, 2015, 38–41), and by a body being residually stressed, we mean the interior of the body is not in a stress-free state although the boundary is free of traction as considered by Coleman and Noll (Arch. Ration. Mech. Anal. 15, 1964, 87–111) and by Hoger (Arch. Ration. Mech. Anal. 88, 1985, 271–289).

In-situ synchrotron X-ray diffraction investigation of martensite decomposition in Laser Powder Bed Fusion (L-PBF) processed Ti–6Al–4V

Laser powder bed fusion (L-PBF) processed Ti–6Al–4V components present two substantial challenges: the presence of tensile residual stress and brittleness of the as-fabricated component due to the formation of the martensite (α′) phase. Industrially, post-heat treatment is performed to improve the quasi-static mechanical properties of the Ti–6Al–4V component through martensite (α′) decomposition and residual stress relaxation. However, detailed insights into the martensite decomposition and stress relaxation mechanisms are still lacking. To address this, in-situ high-energy synchrotron x-ray diffraction (HEXRD) was employed to simultaneously track: (I) martensite (α′) decomposition into a mixture of hexagonal α and body-centered cubic β phases, (II) residual stress relaxation process during heating. Rietveld refinement of 2D diffraction patterns recorded over a temperature range spanning from 25 to 1000°C revealed a three-stage evolution in the α′ unit cell parameters. Unit cell parameter analysis combined with the phase fraction analysis based on the diffraction patterns and the chemical composition profiles predicted by Thermocalc simulation gave detailed insight into the martensite (α′) decomposition process. In Stage 1 (until 375°C), the α′ phase unit cell exhibited linear expansion without indications of any phase transformation or stress relaxation. Upon commencement of stage II at 375°C, an increased expansion rate of the martensite (α′) unit cell was observed, attributed to the enhanced outward diffusion of vanadium (V), succeeded by β phase nucleation at 575°C. Simultaneously, during stage II, residual stress relaxation was observed at both macro and micro scale, culminating in a stress-free state achieved at approximately 750–800°C. Lastly, α-to-β phase transformation was observed in stage 3. The thermal evolution of the hexagonal unit cell dimensions was compared between martensitic and mill-annealed α+β microstructures. This comparison unraveled the link between the initial microstructure and the thermal expansion behavior of the hexagonal α′ and α phase.

Data-driven unlocking volume conservation in single-crystal Ni-rich layered oxides

Our study redefines anisotropic lattice variation through volume conservation (VC) in single-crystal (SC) Ni-rich layered oxides to facilitate long-term cycling stability. Data-driven analysis of the correlation heatmap of the five covariate variables and cycling retention enables precise cut-off voltage modulation for high-energy-density Ni-rich cathodes. We establish the anisotropic variation between a and c lattice constants as critical, leading to interfacial microcracks formation in degrading cycling retention. A balanced use of this variation, verified through computational models, Li1−x[Ni10/12Co1/12Mn1/12]O2 (NCM) and Li1−x[Ni10/12Co1/12Ti1/12]O2 (NCT), creates a volumetric stress free state at the inflection point. Unfortunately, two-scale challenges, namely macroscopic and local structure lattice misfits, which trigger intragranular microcracks, remain unresolved. Formation energy diagrams show biphasic reaction causing macroscopic lattice misfit based on the lattice direction occurring below x = 0.75 in both cathode models. Its effect can be mitigated by doping Ti into the Ni-rich material. Atomistic misfits of in-plane and out-of-plane directions are observed. Furthermore, MO2 and LiO2 slabs anisotropically vary in the complexes for NCM and NCT. Moreover, Ti-doped layered oxide clearly reduces local structural misfits and anisotropy. The proposed VC holds potential as a universal parameter for durable Ni-rich layered oxide fabrication for lithium-ion batteries.

Modeling the Stimulus-Responsive Behaviors of Fiber-Reinforced Soft Materials

Hydrogels can change their size upon swelling. The swelling ratio is the same for all directions in the stress-free state. Dielectric elastomers can reduce their thickness and expand the area upon an electric field. Similarly, the expansion ratio in the plane is also the same for different directions. This isotropic shape change effect limits the function of these soft materials in certain circumstances. To address this issue, recent works have shown that the incorporation of fibers into the polymer matrix can induce an anisotropic response upon external stimulus. In this work, we develop multi-field coupling models for both fiber-reinforced hydrogels and dielectric elastomers. For the former, the change in free energy is caused by the stretching of polymer chains and fibers and the mixing of solvents and polymer networks. The Fickian-type law is adopted for the solvent diffusion. The free energy density for the latter consists of a mechanical part, considering the deformation of both polymer matrix and fibers, and an electric polarization component. Gauss’s law is adopted to obtain the distribution of the electric field. The multi-field models are then implemented for finite element analysis. We consider the stimulus-responsiveness of bilayer strips with an active layer and a passive layer. Without fibers, the strips bend upon the external stimulus. In contrast, the shape changes to helix shapes, which can be further tuned by changing the distribution of fibers. The work provides an efficient design tool for self-folding structures based on stimulus-responsive polymers.

A continuous phase-evolution model for cold and strain-induced crystallization in semi-crystalline polymers

The dynamic crystallization during extreme thermomechanical history under producing or service strongly influences the dimensional stability of the semi-crystalline thermoplastic polymers. To produce high-precision structural components, the proper thermomechanical history needs to be carefully designed to eliminate the crystallization-induced shrinkage. On the contrary, the rapidly developed four-dimensional (4D) printing technology tries to amplify the crystallization-induced shrinkage to directly produce the complex curvatures without supporting materials. However, the crystals formed at different deformation states might have different orientations, leading to an anisotropic residual deformation. Therefore, tracing of all crystals formed with different initial configurations is the key issue to ensure the geometric accuracy of both the high-precision and the deformable components. In this study, we developed a continuous phase-evolution model for both the cold and the strain-induced crystallization in semi-crystalline thermoplastics. The irreversible cold crystallization is analogized as the raindrop falling into the pool, and the reversible strain-induced crystallization is analogized as the nonequilibrium liquid-gas phase transformation. Using the continuous phase-evolution concept, the crystal growth is described by a series of continuously formed crystal phases sequentially added into the initial amorphous medium. Each newly formed crystal phase is in a stress-free state at the formation moment, and therefore the crystallization history coupled with the whole deformation history can be memorized. By introducing the oriented growth tensor representing the stress-free state of all formed crystals, the deformation-history dependent anisotropic crystallization and the corresponding residual deformation can be traced. The developed model is validated by comparing with the experimental data of the dynamic crystallization in amorphous poly l-lactide polymers under thermomechanical loading cycles. Finally, the predictive capability of the model is illustrated by several demonstrations, to show the influences of deformation-history dependent crystallization orientations and the corresponding anisotropic residual deformation.

An inverse fitting strategy to determine the constrained mixture model parameters: application in patient-specific aorta.

The Constrained Mixture Model (CMM) is a novel approach to describe arterial wall mechanics, whose formulation is based on a referential physiological state. The CMM considers the arterial wall as a mixture of load-bearing constituents, each of them with characteristic mass fraction, material properties, and deposition stretch levels from its stress-free state to the in-vivo configuration. Although some reports of this model successfully assess its capabilities, they barely explore experimental approaches to model patient-specific scenarios. In this sense, we propose an iterative fitting procedure of numerical-experimental nature to determine material parameters and deposition stretch values. To this end, the model has been implemented in a finite element framework, and it is calibrated using reported experimental data of descending thoracic aorta. The main results obtained from the proposed procedure consist of a set of material parameters for each constituent. Moreover, a relationship between deposition stretches and residual strain measurements (opening angle and axial stretch) has been numerically proved, establishing a strong consistency between the model and experimental data.

On the Impact of Mechanics on Electrochemistry of Lithium-Ion Battery Anodes

Models exploring electrochemistry-mechanics coupling in liquid electrolyte lithium-ion battery anodes have traditionally incorporated stress impact on thermodynamics, bulk diffusive transport, and fracture, while stress-kinetics coupling is more explored in the context of all solid-state batteries. Here, we showcase the existence of strong link between active particle surface pressure and reaction kinetics affecting performance even in liquid electrolyte systems. Traction-free and immobile particle surface mechanical boundary conditions are used to delineate the varying pressure magnitudes in graphite host during cycling. Both tensile and compressive stresses are generated in traction-free case, while a fixed surface subjects the entire particle to a compression state. Pressure magnitudes are nearly two to three orders of magnitude higher for the latter resulting in significant depression of open circuit potential and improvement of exchange current densities compared to stress-free state. The results demonstrate the need for incorporating stress-kinetics linkage in models and provide a rationale for putting battery electrodes under compression to improve kinetics.

Cable Force Calculation of Cable Hoisting of CFST Arch Bridge Research

To effectively control the stress state and spatial alignment of arch ribs in the cable hoisting construction of a long-span, concrete-filled, steel tube arch bridge and ensure the safety of the structure, it is necessary to calculate and determine the appropriate cable force. Based on the actual project of a double-span, concrete-filled, steel tubular arch bridge, the construction stage of the left span of the bridge from the beginning of construction to the closure is taken as an example. The linear control method of “quiet do not move” is adopted. Based on the principle that the vertical displacement of the front end of the installed segment caused by the self-weight of the new hoisting segment is equal to the vertical displacement of the front end of the previous segment caused by the tension of the new hoisting segment, the tension cable force is calculated by forward iteration. Finally, based on the theory of the stress-free state method, the ideal linear design of the structure was achieved. The results show that after the closure of the bridge, the error range of the cable tension force is −13.33–15.40% on the left bank and −8.37–11.00% on the right bank. The elevation error of the arch rib is −0.003–0.043 m on the left bank and −0.007–0.032 m on the right bank. The overall stress error of the bridge arch is ±7.0 MPa. The error between the theoretical value and the actual value is within the scope of the specification requirements, which meets the specification requirements. After the closure, the arch shape of the bridge meets the smooth requirements.

Metal identity effects in the fracture behavior of coordinatively crosslinked elastomers

Polymers comprising polybutadiene backbones with 2,6-bis(1′-methyl-benzimidazolyl)pyridine (MeBip) sidechains were crosslinked by complexation with two different metal salts, either with copper(II) trifluoromethanosulfonate or with iron(II) trifluoromethanosulfonate. Dynamic mechanical analysis (DMA) and small-angle X-ray scattering (SAXS) data indicate that the crosslinking density and topology of the two materials are the same. The material crosslinked with copper ions, however, exhibits a higher extensibility and fracture energy than the polymer crosslinked with iron. These differences are attributed to differing mechanochemical responses of the metal complexes to applied stress. Computational results further indicate that the copper complexes are more labile, both in the stress-free state as well as upon application of force, and that the “open” complex in which only one MeBip ligand coordinates copper binds fewer counter-ions than the iron-coordinated analog. Both these factors enable easier re-binding of a second MeBip ligand. The computations further suggest that mechanochemically coupled spin-crossover behavior must be considered to fully understand the response of these metal-ligand complexes to mechanical stimuli. The data presented here furthers the facile manipulation of a material's strain response via metal species modulation, and the results offer a way to understand the relationship between bulk and molecular strain response.

Curved Ring Origami: Bistable Elastic Folding for Magic Pattern Reconfigurations

Abstract Ring origami has emerged as a robust strategy for designing foldable and deployable structures due to its impressive packing abilities achieved from snap-folding. In general, polygonal rings with rationally designed geometric parameters can fold into compact three-loop configurations with curved segments which result from the internal bending moment in the folded state. Inspired by the internal bending moment-induced curvature in the folded state, we explore how this curvature can be tuned by introducing initial natural curvature to the segments of the polygonal rings in their deployed stress-free state, and study how this initial curvature affects the folded configurations of the rings. Taking a clue from straight-segmented polygonal rings that fold into overlapping curved loops, we find that this behavior can be reversed by introducing curvature into the ring segments in the stress-free initial state such that the rings fold into a looped straight-line configuration with “zero” area. This strategy realizes extreme packing of the rings. In this work, by a combination of experimental observation, finite element analysis, and theoretical modeling, we systematically study the effect of segment curvature on folding behaviors, folded configurations, and packing abilities of curved ring origami with different geometries. It is anticipated that curved ring origami can open a new avenue for the design of foldable and deployable structures with simple folded configurations and high packing efficiency.

Some models for initially stressed and initially strained structurally anisotropic incompressible materials

In this paper, we determine the constitutive relations for structurally anisotropic materials from a reference with initial strain or initial stress. Using reference independence of strain energy density, we first deduce the relationship of various initial strain measures with material symmetry and determine the resulting structural tensors. We use these structural tensors to derive the necessary invariants for initially strained materials. We also Examine the polyconvexity of the derived invariants and determine constitutive relations for the initially strained Ogden model. Next, we determine the constitutive relation for a combination of initial stress-texture symmetry which may be a more popular approach of recent interest. We deduce four reference-independent constitutive relations for initially stressed transversely isotropic incompressible materials. The first two strain energy functions are linear in the left Cauchy stretch and its cofactor, respectively, in the stress-free state. In the absence of initial stress, the third strain energy function involves all standard invariants for transversely isotropy. To deduce this strain energy in the presence of initial stress, we invert a fourth-order tensor and perform several involved tensor calculations. Finally, we obtain the constitutive relation for an initially stressed transversely isotropic limited chain extensible (LCE) model. To determine the initial stretch in the preferred texture direction in terms of initial stress, we utilize the geometrical interpretation of the Almansi strain measure.

The Biophysical Modelling of the Stress Theory

Selye published a short note about his findings in Nature in 1936. Selye applied his famous theory of stress to everyday life. A stress-free state is equivalent to being dead. In our paper we write about the some directions in stress research and the stress syndromes in clinical medicine. We show that eustress suits to situations when the psychoneuroendocrine stimulation of the organism and its behavior is temperate, maintaining at an optimal level the physical and mental resources and the health status and inducing a positive adaptation to the environment. We give a comparative analysis between the eustress and distress. Psychical stressors are those which come into being independently from the man’s will and destroy – as social factors – psychic and organic components taking place in the response, the organism consequently get tired. During the harmonic life gradually all human psycho-organic components shall get exhausted. Forming a biophysical model is not the task of the biologist or physicist alone, a good model can successfully constructed only by common, collective-work. This is typically the task of the biophysicist and a problem which falls within the competence of this discipline. Model is always an approximation, the user of the model has to take into consideration that he can approach only the absolute truth just through the endless series of relative truths. We use the Le-Chatelier principle and determine the measure of psychical organization.

Phase-field study of the effect of stress field and fission rate on intragranular Xe bubble evolution in U3Si2 nuclear fuel

Due to the superior thermal conductivity and high uranium density, U3Si2 is an excellent candidate for conventional UO2 nuclear fuel and shows great potential application in accident-tolerant fuel (ATF) assembly of light water reactors (LWRs). Currently, the behavior of Xe bubbles with internal or applied stress is rarely investigated, restricting further understanding of swelling in U3Si2. The mesoscopic phase-field method has been developed in this work to study the spatial and temporal Xe bubble evolution in U3Si2. The results show that the bubble density and its average size increase as the fission rate increases. Applied stress accelerates the nucleation and growth of gas bubbles, reshaping the bubbles’ morphology from spherical in a stress-free state into elongated along the applied direction in a stressed state. The gas bubbles in a local dislocation stress field nucleate preferentially at stress-concentrated sites and spread over the whole system in succession, and the bubble coarsening is controlled by the stress overlap of the dislocation pair. The results show a practical phase-field method for Xe bubble evolution study in U3Si2, which can be expanded into swelling behavior investigation in other fuels and lay a solid foundation for the development of ATF assembly.

The stress-free state of human erythrocytes: Data-driven inference of a transferable RBC model

The stress-free state (SFS) of red blood cells (RBCs) is a fundamental reference configuration for the calibration of computational models, yet it remains unknown. Current experimental methods cannot measure the SFS of cells without affecting their mechanical properties, whereas computational postulates are the subject of controversial discussions. Here, we introduce data-driven estimates of the SFS shape and the visco-elastic properties of RBCs. We employ data from single-cell experiments that include measurements of the equilibrium shape of stretched cells and relaxation times of initially stretched RBCs. A hierarchical Bayesian model accounts for these experimental and data heterogeneities. We quantify, for the first time, the SFS of RBCs and use it to introduce a transferable RBC (t-RBC) model. The effectiveness of the proposed model is shown on predictions of unseen experimental conditions during the inference, including the critical stress of transitions between tumbling and tank-treading cells in shear flow. Our findings demonstrate that the proposed t-RBC model provides predictions of blood flows with unprecedented accuracy and quantified uncertainties.

Impact of fractional strain on medium containing spherical cavity in the framework of generalized thermoviscoelastic diffusion

Viscoelastic materials (elastomers, resins, and polycrystalline metals) are of significant importance due to their numerous applications in biology, civil, and mechanical engineering alongwith other scientific disciplines. Viscoelastic materials possess strong relation with temperature and diffusion phenomenon and their mechanical properties exhibit memory dependence. With this view, fractional order strain is considered to study thermoviscoelastic diffusion interactions in the context of hyperbolic two-temperature-three-phase lag model. The investigation is carried out for an infinite, homogeneous, isotropic medium containing a spherical cavity. Initially, the medium is held quiescent. The boundary of the cavity is subjected to continuous concentration and ramp type thermal load in stress-free state. The problem is solved in Laplace transformed domain. The aim of this article is to analyze the behavior of field variables under thermoelastic models with different fractional order strain, ramping, and viscosity parameters. Physical data of Copper material is considered for numerical inversion technique and obtained results are plotted graphically. Here it is revealed that fractional order strain parameters lead to lower perturbations of stress and viscous effects result in higher temperature. One temperature and hyperbolic two temperature theories produce similar solutions. Ramp type thermal loading at the boundary significantly influences the temporal profile.

Shape memory behaviors of 3D printed liquid crystal elastomers

As soft active materials, shape-memory polymers (SMPs) and liquid crystal elastomers (LCEs) have attracted considerable research interest due to their potential applications in various areas. SMPs refer to polymeric materials that can return to their permanent shape in response to external stimuli, such as heat, light, and solvent. In this sense, LCEs can exhibit intrinsic shape-memory behaviors since LCEs can switch between two shapes with temperature change due to the order-disorder transition of liquid crystals. In this work, we fabricate both the polydomain and monodomain nematic LCEs through direct ink writing 3D printing. With increasing the temperature of the substrates, the printed LCEs change from the monodomain state to the polydomain state. For polydomain LCEs, a reversible shape change can occur upon constant loading, while the monodomain can switch the shape with temperature in the stress-free state. This two-way shape-memory behavior is caused by the nematic-isotropic phase transition. We further show that the printed LCEs exhibit a good one-way shape-memory effect due to glass transition. The shape recovery region increases with the programming temperature, which is a typical temperature memory effect. Finally, it is demonstrated that complex shape-memory performance can be designed by combining one-way and two-way shape-memory effects. Specifically, for the monodomain LCEs, with increasing temperature, the programmed shape first recovers, and a second shape change can further occur due to the nematic-isotropic transition.

GEOMETRICAL MODELLING OF CORNEAL STRUCTURE IN NON-PATHOLOGICAL INITIAL CONDITIONS. A COMPARATIVE STUDY

Modelling and numerical simulation have involved a great advance in clinical practice for the prediction of the structural behaviour of soft tissues in certain healthy or pathological scenarios. One of the main features of these simulations is that they can define the geometric parameters of the structure in its initial boundary conditions. In the case of the cornea, there are discrepancies with respect to the initial conditions that depend on whether if it is considered its physiological geometry measured in vivo, or otherwise its geometry in its stress-free state. The choice of one or the other could lead to errors in the simulation model due to the non-linearity of these tissues. In this study, two methodologies were evaluated, the displacement method and the prestress method, to obtain the geometry of non-pathological cornea in its free-of-stress state. Both methodologies allow obtaining the physiological geometry measured in-vivo of the healthy cornea, however, the method of displacements presents a computational architecture that makes it more efficient in numerical simulation, when compared to the pre-stress method. Keywords: 3D Modelling; Computational Modelling; Ophthalmology; Biological Morphology

Explicitly Modeling Stress Softening and Thermal Recovery for Rubber-like Materials

Rubber-like materials exhibit stress softening when subject to loading–unloading cycles, i.e., the Mullins effect. However, this phenomenon can be recovered after annealing the previously stretched sample under a stress-free state. The aim of this paper is to establish a constitutive model with thermodynamic consistency to account for the stress softening and thermal recovery. Towards this goal, (i) an explicit form of Helmholtz free energy can be found such that the restrictions from thermodynamic law can be satisfied; (ii) a compressible, multi-axial strain-energy function considering energy dissipation is proposed by introducing specific invariants; (iii) a unified shape function based on the symmetry property of the test data in a one-dimensional case with stress softening and thermal recovery is provided by introducing a weight variant; (iv) it is proven that the new potential can automatically reduce to the one-dimensional case, i.e., uniaxial tension, equal biaxial, or plane strain; (v) numerical results for model validation are exactly matched with classical experimental data.