Stress-free State Research Articles

Modeling thermal recovery of the Mullins effect

Many elastomeric materials exhibit the Mullins effect, or stretch-induced stress softening. However, early experiments show that the Mullins effect may be recovered after annealing previously stretched material in a stress-free state. And that both the rate and degree of recovery are affected by the annealing temperature. In this work, we develop a thermo-mechanically-coupled large-deformation constitutive model that quantitatively captures thermal recovery of the Mullins effect. The model is calibrated to experiments from the literature, and is numerically implemented by writing a user material subroutine for the finite element program Abaqus/Standard. Lastly, our simulation results suggest that unanticipated behavior due to recovery of the Mullins effect is possible.

A physics-based micromechanical model for electroactive viscoelastic polymers

Electroactive polymers with time-dependent behavior are considered in the present paper by way of a new physics-based micromechanical model; such viscoelastic response is described by the internal evolution of the polymer network, providing a new viewpoint on the stress relaxation occurring in elastomers. The main peculiarity of such internally rearranging materials is their capacity to locally reset their reference stress-free state, leading to a mechanical behavior that relaxes out (eases off) an induced stress state and that can thus be assimilated to a sort of internal self-healing process. Such high deformability and recoverability displayed by dynamically cross-linked polymers can be conveniently exploited when they are coupled in electromechanical problems; the deformation induced by an electric field can be easily tuned by the intensity of the electric field itself and the obtained shape can be maintained without any electric influence once the material microstructure has rearranged after a sufficient curing time. In the present paper, both features of the polymeric material, that is, internal remodeling and electromechanical coupled response, are considered and a theoretical framework is established to simulate representative boundary value problems.

Sensitivity of Arterial Hyperelastic Models to Uncertainties in Stress-Free Measurements.

Despite major advances made in modeling vascular tissue biomechanics, the predictive power of constitutive models is still limited by uncertainty of the input data. Specifically, key measurements, like the geometry of the stress-free (SF) state, involve a definite, sometimes non-negligible, degree of uncertainty. Here, we introduce a new approach for sensitivity analysis of vascular hyperelastic constitutive models to uncertainty in SF measurements. We have considered two vascular hyperelastic models: the phenomenological Fung model and the structure-motivated Holzapfel-Gasser-Ogden (HGO) model. Our results indicate up to 160% errors in the identified constitutive parameters for a 5% measurement uncertainty in the SF data. Relative margins of errors of up to 30% in the luminal pressure, 36% in the axial force, and over 200% in the stress predictions were recorded for 10% uncertainties. These findings are relevant to the large body of studies involving experimentally based modeling and analysis of vascular tissues. The impact of uncertainties on calibrated constitutive parameters is significant in context of studies that use constitutive parameters to draw conclusions about the underlying microstructure of vascular tissues, their growth and remodeling processes, and aging and disease states. The propagation of uncertainties into the predictions of biophysical parameters, e.g., force, luminal pressure, and wall stresses, is of practical importance in the design and execution of clinical devices and interventions. Furthermore, insights provided by the present findings may lead to more robust parameters identification techniques, and serve as selection criteria in the trade-off between model complexity and sensitivity.

Modified multiplicative decomposition model for tissue growth: Beyond the initial stress-free state

The multiplicative decomposition model is widely employed for predicting residual stresses and morphologies of biological tissues due to growth. However, it relies on the assumption that the tissue is initially in a stress-free state, which conflicts with the observations that any growth state of a biological tissue is under a significant level of residual stresses that helps to maintain its ideal mechanical conditions. Here, we propose a modified multiplicative decomposition model in which the initial state (or reference configuration) of a biological tissue is endowed with a residual stress instead of being stress-free.Releasing theoretically the initial residual stress, the initially stressed state is first transmitted into a virtual stress-free state, thus resulting in an initial elastic deformation. The initial virtual stress-free state subsequently grows to another counterpart with a growth deformation, and the latter is further integrated into its natural configuration of a real tissue with an excessive elastic deformation that ensures tissue compatibility. With this decomposition, the total deformation arising during growth may be expressed as the product of elastic deformation, growth deformation and initial elastic deformation, while the corresponding free energy density should depend on the initial residual stress and the total deformation. Three key issues including the explicit expression of the free energy density, the predetermination of the initial elastic deformation, and the initial residual stress are addressed.Finally, we consider a tubular organ as a representative example to demonstrate the effects of the proposed initial residual stress on stress distribution and on shape formation through an incremental stability analysis. Our results suggest that the initial residual stress exerts a major influence on the growth stress and the morphology of biological tissues. The model bridges the gap between any two growth states of a biological tissue that is endowed with a certain level of residual stresses.

Periodic crack system in a layered elastic wedge

ABSTRACTThree-dimensional periodic mode I crack problems for an elastic layered wedge are reduced to an integro-differential equation principal part of which is the same as in the classical Griffith problem for one crack in an elastic full space. The infinite system of elliptic cracks, with equal spaces between neighbouring cracks, is situated in the middle of the wedge parallel to its edge. The faces of the wedge with cracks are layered, with sliding support, by identical elastic incompressible wedge-shaped layers so that the system of three wedges have the same edge (for example, rubber–metal–rubber system). Three types of boundary conditions on the outer layered wedge faces are considered (sliding or rigid support, stress-free state). The regular asymptotic method is used to solve the problems effective for fairly big spaces between the cracks as well as for big distances between the cracks and the edge of the wedge. The stress intensity factor (SIF) is calculated for different geometric parameters. It is shown that the solution, including the SIF, for close cracks does not practically depend on the type of the outer boundary conditions.

A non-destructive approach to identify the stress-free state and material properties of arteries

Residual strains and stresses in arteries play a significant role in homogenizing the stress field in the arterial wall and make arteries more compliant. The stress-free state of the artery needs to be identified in order to determine the residual stresses or strains in the unloaded and loaded states of the artery. The stress-free state of an artery can be characterized by two geometric parameters of the open sector achieved after a short artery segment is cut open radially, i.e., the inner radius and the opening angle. A traditional method to determine the stress-free state of arteries is to cut a short artery segment radially and measure the inner radius and the opening angle of the open sector, which is destructive by nature. In this paper, we propose a novel non-destructive method to identify the stress-free state of arteries, i.e., its inner radius and the opening angle, without the need to make the physical cut to the artery. The essence of this method is an inverse problem to identify the inner radius and the opening angle of the stress-free state by applying optimization algorithms based on a series of loaded states of the artery under different intraluminal pressures. Numerical simulations were conducted to generate a series of deformed states of an artery from which the inner radius and the opening angle of the stress-free state were identified thereafter. In addition, the effect of experimental error was investigated, and a method to modify the experimental data with error based on the pressure-radius relation of the artery was proposed. Furthermore, a non-destructive method was proposed to identify the material parameters in Fung’s constitutive model by applying the notion of average stretch ratios and stresses, from which the stress-free state can be further identified.

Comparison of in vivo vs. ex situ obtained material properties of sheep common carotid artery

Patient-specific biomechanical modelling can improve preoperative surgical planning. This requires patient-specific geometry as well as patient-specific material properties as input. The latter are, however, still quite challenging to estimate in vivo.This study focuses on the estimation of the mechanical properties of the arterial wall. Firstly, in vivo pressure, diameter and thickness of the arterial wall were acquired for sheep common carotid arteries. Next, the animals were sacrificed and the tissue was stored for mechanical testing. Planar biaxial tests were performed to obtain experimental stress-stretch curves. Finally, parameters for the hyperelastic Mooney–Rivlin and Gasser–Ogden–Holzapfel (GOH) material model were estimated based on the in vivo obtained pressure-diameter data as well as on the ex situ experimental stress-stretch curves.Both material models were able to capture the in vivo behaviour of the tissue. However, in the ex situ case only the GOH model provided satisfactory results. When comparing different fitting approaches, in vivo vs. ex situ, each of them showed its own advantages and disadvantages. The in vivo approach estimates the properties of the tissue in its physiological state while the ex situ approach allows to apply different loadings to properly capture the anisotropy of the tissue. Both of them could be further enhanced by improving the estimation of the stress-free state, i.e. by adding residual circumferential stresses in vivo and by accounting for the flattening effect of the tested samples ex vivo.• Competing interests: none declared• Word count: 4716

Effect of surface elasticity on extensional and torsional stiffnesses of isotropic circular nanorods

We present a continuum formulation to obtain the effects of surface residual stress and surface elastic constants on extensional and torsional stiffnesses of isotropic circular nanorods. Analytical expressions of axial force, twisting moment, and extensional and torsional stiffnesses are obtained. Unlike the case of rectangular nanorods, we show that the stiffnesses of circular nanorods also depend on surface residual stress components. This is attributed to non-zero surface curvature inherent in circular nanorods. We further normalize these expressions and analyze their asymptotic limits in the limit of the nanorod’s radius approaching both zero and infinity, corresponding to surface-dominated and bulk-dominated regimes, respectively. Finally, we use the recently proposed helical Cauchy–Born rule and perform molecular statics calculations to obtain axial force, twisting moment, and stiffnesses of the tungsten nanorod. The tungsten material is selected since its bulk crystal exhibits isotropy in the stress-free state. The results from molecular statics calculations are shown to match the derived continuum formulas accurately.

Comparison between axial residual stresses measured by Raman spectroscopy and X-ray diffraction in SiC fiber reinforced titanium matrix composite

Accurate measurement and analysis of residual stress state in the SiCf/Ti composites are crucial to optimizing their fabrication process and to understanding their failure mode, but they are still a challenge. In this work, SiCf/C/Ti17 composites with~48% fiber volume fraction, consisting of W-core SiC fibers (~100 m in diameter), turbostratic C coating (~2.5 m in thickness) and Ti17 matrix, are prepared by consolidating precursor wires fabricated by matrix-coated fiber method through hot isostatic pressing at 920℃/120 MPa/2 h; these samples are used for measuring their stresses. It is noted that turbostratic C coating, a necessary diffusion barrier layer between SiC fiber and Ti17 alloy matrix, bears the residual stress caused by the mismatch of thermal expansion coefficients between fiber and matrix during consolidation. It is found that the graphene planes are almost parallel to the axial direction of SiC fibers in the turbostratic C coating revealed by high magnification transmission electron microscope, and thus G peak position of C coating would be sensitive to stress state. Accordingly, micro-Raman spectroscopy is first used to measure the G peak positions of C coating under stress and stress-free state in the SiCf/C/Ti17 composite, respectively. Based on the position shift of G band caused by residual stress, the axial residual compressive stress of SiC fiber in SiCf/C/Ti17 composite is calculated to be~705.0 MPa. For comparison, X-ray diffraction method is also adopted to measure the interplanar spacing values of the Ti17 alloy matrix in different directions to obtain the spatial strains. During measurement, -Ti (213) high-angle diffraction peak is chosen to reduce test error, and then the different interplanar spacing values of -Ti (213) are obtained by varying the values of in three different directions at =0, 45 and 90. As three-axis-stress model is employed, the residual tensile stress of Ti17 alloy matrix in the axial direction of SiCf/C/Ti17 composite is~701.3 MPa, which is transformed through linear elastic theory into the residual compressive stress of SiC fiber of~759.4 MPa. The similar results confirm that it is reliable to characterize the residual stress in the SiCf/C/Ti17 composite with high-texture turbostratic carbon by both the Raman spectroscopy and the X-ray diffraction method.

A nonlinear poroelastic theory of solid tumors with glycosaminoglycan swelling

Mechanics plays a crucial role in the growth, development, and therapeutics of tumors. In this paper, a nonlinear poroelastic theory is established to describe the mechanical behaviors of solid tumors. The free-swollen state of a tumor is chosen as the reference state, which enables us to avoid pursuing a dry and stress-free state that is hard to achieve for living tissues. Our results reveal that the compression resistance of a tumor is primarily attributed to glycosaminoglycan (GAG) swelling, and the compactness of cell aggregates is found to affect tumor consolidation. Over-expressed GAGs and dense cell aggregates can stiffen the tumor, a remodeling mechanism that makes the tumor with higher elastic modulus than its surrounding host tissues. Glycosaminoglycan chains also influence the transient mechanical response of the tumor by modulating the tissue permeability. The theoretical results show good agreement with relevant experimental observations. This study may not only deepen our understanding of tumorigenesis but also provide cues for developing novel anticancer strategies.

On the importance of retaining stresses and strains in repositioning computational biomechanical models of the cervical spine.

Human body models are created in a specific posture and often repositioned and analyzed without retaining stresses that result from repositioning. For example, repositioning a human neck model within the physiological range of motion to a head-turned posture prior to an impact results in initial stresses within the tissues distracted from their neutral position. The aim of this study was to investigate the effect of repositioning on the subsequent kinetics, kinematics, and failure modes, of a lower cervical spine motion segment, to support future research at the full neck level. Repositioning was investigated for 3 modes (tension, flexion, and extension) and 3 load cases. The model was repositioned and loaded to failure in one continuous load history (case 1), or repositioned then restarted with retained stresses and loaded to failure (case 2). In case 3, the model was repositioned and then restarted in a stress-free state, representing current repositioning methods. Not retaining the repositioning stresses and strains resulted in different kinetics, kinematics, or failure modes, depending on the mode of loading. For the motion segment model, the differences were associated with the intervertebral disc fiber reorientation and load distribution, because the disc underwent the largest deformation during repositioning. This study demonstrated that repositioning led to altered response and tissue failure, which is critical for computational models intended to predict injury at the tissue level. It is recommended that stresses and strains be included and retained for subsequent analysis when repositioning a human computational neck model.

Local-stress-induced thermal conductivity anisotropy analysis using non-destructive photo-thermo-mechanical lock-in thermography (PTM-LIT) imaging

When tensile loading is exerted on a sample, internal stress will be generated to counterbalance the external force. The accumulation of local stress results in the change of thermal properties and the sample becomes locally anisotropic, wherein lies the significance of correlation between mechanical properties and thermal parameters. In this research, anisotropic thermal conductivity was considered as the direct result of tensile loading and was interpreted by a theoretical three-dimensional anisotropic diffusion-wave model. Photo-thermo-mechanical lock-in thermography (PTM-LIT) was introduced and used with a focused laser beam and a mid-infrared camera viewing an aluminum alloy sample fixed on a dynamic home-made tensile rig. A numerical two-dimensional Fourier transform was used to compute the thermal-wave field and the effects of several important factors were investigated. Both theoretical and experimental images were analyzed with an isothermal conductivity anisotropy contour fitting approach. It was demonstrated that PTM-LIT can qualitatively and quantitatively reveal otherwise hard-to- measure mechanical property behavior of materials subjected to tensile loading from the stress-free state to its ultimate level of mechanical strength before fracture.

Correlative plasma-surface model for metastable Cr-Al-N: Frenkel pair formation and influence of the stress state on the elastic properties

Correlatively employing density functional theory and experiments congregated around high power pulsed magnetron sputtering, a plasma-surface model for metastable Cr0.8Al0.2N (space group Fm3¯m) is developed. This plasma-surface model relates plasma energetics with film composition, crystal structure, mass density, stress state, and elastic properties. It is predicted that N Frenkel pairs form during Cr0.8Al0.2N growth due to high-energy ion irradiation, yielding a mass density of 5.69 g cm−3 at room temperature and Young's modulus of 358–130 GPa in the temperature range of 50–700 K for the stress-free state and about 150 GPa larger values for the compressive stress of 4 GPa. Our measurements are consistent with the quantum mechanical predictions within 5% for the mass density and 3% for Young's modulus. The hypothesis of a stress-induced Young's modulus change may at least in part explain the spread in the reported elasticity data ranging from 250 to 420 GPa.

On the shape memory of red blood cells

Red blood cells (RBCs) undergo remarkably large deformations when subjected to external forces but return to their biconcave discoid resting shape as the forces are withdrawn. In many experiments, such as when RBCs are subjected to a shear flow and undergo the tank-treading motion, the membrane elements are also displaced from their original (resting) locations along the cell surface with respect to the cell axis, in addition to the cell being deformed. A shape memory is said to exist if after the flow is stopped the RBC regains its biconcave shape and the membrane elements also return to their original locations. The shape memory of RBCs was demonstrated by Fischer [“Shape memory of human red blood cells,” Biophys. J. 86, 3304–3313 (2004)] using shear flow go-and-stop experiments. Optical tweezer and micropipette based stretch-relaxation experiments do not reveal the complete shape memory because while the RBC may be deformed, the membrane elements are not significantly displaced from their original locations with respect to the cell axis. Here we present the first three-dimensional computational study predicting the complete shape memory of RBCs using shear flow go-and-stop simulations. The influence of different parameters, namely, membrane shear elasticity and bending rigidity, membrane viscosity, cytoplasmic and suspending fluid viscosity, as well as different stress-free states of the RBC is studied. For all cases, the RBCs always exhibit shape memory. The complete recovery of the RBC in shear flow go-and-stop simulations occurs over a time that is orders of magnitude longer than that for optical tweezer and micropipette based relaxations. The response is also observed to be more complex and composed of widely disparate time scales as opposed to only one time scale that characterizes the optical tweezer and micropipette based relaxations. We observe that the recovery occurs in three phases: a rapid compression of the RBC immediately after the flow is stopped, followed by a slow recovery to the biconcave shape combined with membrane rotation, and a final rotational return of the membrane elements back to their original locations. A fast time scale on the order of a few hundred milliseconds characterizes the initial compression phase while a slow time scale on the order of tens of seconds is associated with the rotational phase. We observe that the response is strongly dependent on the stress-free state of the cells, that is, the relaxation time decreases significantly and the mode of recovery changes from rotation-driven to deformation-driven as the stress-free state becomes more non-spherical. We show that while membrane shear elasticity and non-spherical stress-free shape are necessary and sufficient for the membrane elements to return to their original locations, bending rigidity is needed for the “global” recovery of the biconcave shape. We also perform a novel relaxation simulation in which the cell axis of revolution is not aligned with the shear plane and show that the shape memory is exhibited even when the membrane elements are displaced normal to the imposed flow direction. The results presented here could motivate new experiments to determine the exact stress-free state of the RBC and also to clearly identify different tank-treading modes.

Effect of Intrinsic Twist and Orthotropy on Extension–Twist–Inflation Coupling in Compressible Circular Tubes

We present effects of intrinsic twist and material orthotropy on extension–twist–inflation coupling in circular tubes about their stress-free state. Simple analytical expressions for coupling stiffnesses corresponding to extension–twist, twist–inflation and extension–inflation couplings are obtained. We show that the sign of the extension–twist coupling stiffness, which governs initial overwinding/unwinding in tubes during their extension, is not just dependent on the tube’s intrinsic twist but also on two other parameters: ratio of the Young’s moduli in the lateral surface of orthotropic tube and the excess of the Poisson’s ratio from an isotropy condition. By tuning these two parameters, one can generate the counter-intuitive overwinding as reported earlier in the case of DNA. Similarly, we show that even with positive Poisson’s ratio, an intrinsically twisted tube could inflate on being stretched. We also present a scheme to obtain all the relevant stiffnesses of chiral single-walled carbon nanotubes from a “one-atom unit cell” calculation. These stiffnesses, when plotted vs. the nanotube’s chirality, exhibit interesting periodicity which has its origin in the 6-fold symmetry of graphene. This trend is captured in the continuum model for a nanotube when it is assumed to be comprised of three families of parallel fibers on its lateral surface.

Microneedle growth method as an innovative approach for growing freestanding single crystal diamond substrate: Detailed study on the growth scheme of continuous diamond layers on diamond microneedles

A detailed study of diamond growth on diamond microneedles was conducted using micro-Raman spectroscopy of the microneedle and coalescence regions to approach the production of freestanding diamond substrate by heteroepitaxy with the microneedle growth method. The high-density non-diamond-phase carbon is contaminated in the initial stage of overgrowth of diamond on diamond microneedles, but this completely disappears through the quick recovery of the crystallinity of the overgrown diamond layers during the coalescence with the lateral direction growth. We also point out the possibility that a strong driving force is applied to the dislocations generated at the regrowth point and at the coalescence front to enhance the mutual annihilation of dislocations. In addition, we reveal that the stress state changes from compressive stress in the initial diamond layers to a nearly stress-free state in the bulk layers on the microneedles through a momentary tensile stress state at the regrowth point at the tip of the microneedle. Overall results indicate the strong feasibility of producing freestanding, stress-free, single-crystal diamond substrate by heteroepitaxy.

A simple statistical approach to model the time-dependent response of polymers with reversible cross-links.

A new class of polymers characterized by dynamic cross-links is analyzed from a mechanical point of view. A thermodynamically consistent model is developed within the Lagrangian framework for polymers that can rearrange their internal cross-links. Such a class of polymers has the capability to reset their internal microstructure and the microscopic remodeling mechanism leads to a behavior similar to that of an elastic fluid. These materials can potentially be used in several fields, such as in biomechanics, smart materials, morphing materials to cite e few. However, a comprehensive understanding is necessary before we can predict their behavior and perform material design for advanced technologies. The proposed formulation-following a statistical approach adapted from classical rubber elasticitye is based on the evolution of the molecular chains' end-to-end distance distribution function. This distribution is allowed here to evolve with time, starting from an initial stress-free state and depending on the deformation history and the cross-link attachment/detachment kinetics. Some simple examples are finally presented and discussed to illustrate the capability and generality of the developed approach.

Growth-induced compatible strains

We studied the time evolution problem driven by growth for a non-Euclidean ball which at its initial state is equipped with a non-compatible distortion field. The problem is set within the framework of non-linear elasticity with large growing distortions. No bulk accretive forces are considered, and growth is only driven by the stress state. We showed that, when stress-driven growth is considered, distortions can evolve along different trajectories which share a common attracting manifold; moreover, they eventually attain a steady and compatible form, to which there corresponds a stress-free state of the ball.

Influence of Copper Pumping on Integrity and Stress of Through-Silicon Vias

The influence of copper pumping on through-silicon vias (TSVs) under thermal loading was investigated using an in situ experimental technique. The morphology of the TSV and its evolution with temperature were observed by scanning electric microscopy, and the process of copper pumping and its damage to the integrity of the TSV were recorded. Some new failure modes were found. The maximum height of pumped copper was as high as $5~\mu \text{m}$ for a single thermal cycle and 25 $\mu \text{m}$ for multiple cycles. The thermal stress of the TSV and its evolution with temperature were then evaluated using a microinfrared photoelastic system. Full-field and real-time measurements were carried out. It was found that copper pumping at high temperature served to release stress and that this stress-free state of the TSV could be maintained. After cooling or annealing, however, the TSV stress again increased.

Finite element simulation of stresses in a plasma-sprayed thermal barrier coating with an irregular top-coat/bond-coat interface

A three-dimensional finite element model of a conventional plasma-sprayed thermal barrier coating was set up with an irregular top-coat/bond-coat interface that was modelled based on the waviness data of a plasma-sprayed coating surface. The residual stresses upon cooling from a high-temperature stress-free state down to a room-temperature stressed state were studied as a function of thickness of the thermally grown oxide (TGO) layer taking into account the plasticity of the bond coat. The calculations indicate that the delamination cracks at the TGO/bond-coat are more likely to appear with progressive oxidation, and, conversely, the initiation of cracks in the top coat at the interfacial peaks in an intact thermal barrier coating is less probable because the tensile stresses in the top coat initially decrease and the tensile regions gradually become smaller and localized due to the observed stress conversion. Based on the obtained results and experimental evidence from the literature, a role of the interfacial topography in the failure sequence is discussed for oxidized and thermally-cycled coatings.