Nonlinear Elasticity Research Articles

Tricritical state and quasi-periodicity triggered by the non-linear elasticity in an Upper Convected Maxwell fluid confined between two co-oscillating cylinders about zero-mean

The effects of harmonically co-oscillating the inner and outer cylinders about zero mean rotation in a Taylor–Couette flow are examined numerically using Floquet theory, for the case where the fluid confined between the cylinders obeys the upper convected Maxwell model. Although stability diagrams and mode competition involved in the system were clearly elucidated recently by Hayani Choujaa et al. (2021) in weakly elastic fluids, attention is focused, in this paper, on the dynamic of the system at higher elasticity with emphasis on the nature of the primary bifurcation. In this framework, we are dealing with pure inertio-elastic parametric resonant instabilities where the elastic and inertial mechanisms are considered of the same order of magnitude. It turns out, on the one hand, that the fluid elasticity gives rise, at the onset of instability, to the appearance of a family of new harmonic modes having different axial wavelengths and breaking the spatio-temporal symmetry of the base flow: invariance in the axial direction generating the O(2) symmetry group and a half-period-reflection symmetry in the azimuthal direction generating a spatio-temporal Z2 symmetry group. On the other hand, new quasi-periodic flow emerging in the high frequency limit and other interesting bifurcation phenomena including bi and tricritical states are also among the features induced by the fluid elasticity. Lastly, and in comparison with the Newtonian configuration of this system, the fluid elasticity leads to a total suppression of the non-reversing flow besides emergence of instabilities with lower wavelengths. Such a comparison provides insights into the dynamics of elastic hoop stresses in altering the flow reversal in modulated Taylor–Couette flow.

Mesoscopic irreversible thermodynamics of aging kinetics of alpha polypeptides [DNA] under various constraints: Special reference to the simple spring mechanics

The mesoscopic irreversible thermodynamic treatment of α-polypeptides and the helical polynucleotides (DNA) furnishes two sets of analytical expressions, which allow us not only to analyze the reversible force–extension experiments performed by atomic force microscopy (AFM) but also to predict the irreversible “aging” kinetics of the single-stranded and double-stranded polynucleotides (ssDNA and dsDNA) helical conformations exposed to aqueous solutions and applied static stress systems under the various constraints. The present physicochemical cage model emphasizes the fact that the global Helmholtz free energy of the helical conformation acts not only under the stored “intrinsic” unusual torsional and bending elastic energies inherited by the unfolded helical structure of the amino-acid (peptides) or the nucleic-acid (nucleotide) backbone but also reveals the importance of the interfacial Helmholtz free energy density associated with the interaction of the side-wall branches within the surrounding aqueous solutions. The analytical expression obtained for the unfolding force vs extension (FE) shows a strong non-linear elasticity behavior under the twist angle constraint when the interfacial Helmholtz energy term is incorporating into the scenario. This behavior is in excellent quantitative agreement with the AFM test results obtained by Idiris et al. (2000) on the poly-L-glutamic acid [Glu(n)-Cys] exposed to aqueous solutions, which show that acidity increases the degrees of helicity.

Numerical and experimental investigations on quasistatic pulse generation of ultrasonic guided waves in fiber reinforced composite pipes

The quasistatic pulse (QSP) generation of ultrasonic guided waves in composite pipes can exhibit many features that are useful for early-stage material characterization and structural health monitoring. The intrinsic relationship between the QSP generation and the weak elastic nonlinearity of solids is complex. It has yet promised for developing the nonlinear ultrasonic guided wave technique that combines its advantages of high sensitivity to microdamage and low ultrasonic attenuation in composite materials. This study presents a systematic investigation of the QSP generation in fiber reinforced composite pipes. Using a three-dimensional finite element simulation with a nonlinear material model, the temporal waveform, mode conversion effect, cumulative effect, generation efficiency, and duration effect of QSP generation are revealed. The nonlinear QSP signals in laminated composite pipes are confirmed as the fastest wave mode that only has axial displacement. The shape of QSP depends on the group velocity difference between the QSP and the primary wave. The magnitude of QSP is related to the excitation level, frequency, and tone-burst duration of the primary wave. Experiments are conducted using carbon fiber reinforced composite pipes and the signals are measured using a laser vibrometer scanning system for verifying the QSP generation. The experimental results are consistent with the numerical findings. A relative nonlinear acoustic parameter based on the QSP generation is proposed and used to evaluate the early-stage thermal fatigue damage in the pipes. The measured nonlinear acoustic parameter demonstrates high sensitivity to the damage, indicating considerable potential for industrial applications.

Nonlinear topology optimization on thin shells using a reduced-order elastic shell model

We present a novel numerical algorithm to perform nonlinear topology optimization on elastic thin shells. The main component of our method is a differentiable thin-shell simulator based on discrete differential geometry (DDG) discretization and the projected Newton method to solve geometrically nonlinear elasticity and its derivatives on a triangle mesh. We build a density-based topology optimization algorithm, enhanced by a density filter and a Heaviside projection scheme, to emerge and optimize topologically complex shell structures on curved surfaces. We validate our approach using standard test cases for nonlinear topology optimization and demonstrate the efficacy of our method by tackling highly nonlinear topology optimization problems by producing complex and high-resolution shell structural designs under various load conditions.

Quasiconvexity in a model of fiber-reinforced solids based on Cosserat elasticity theory

The quasiconvexity inequality associated with energy minimizers is derived in the context of a nonlinear Cosserat elasticity theory for fiber-reinforced elastic solids in which the intrinsic flexural and torsional elasticities of the fibers are taken into account explicitly. The derivation accounts for non-standard kinematic constraints, associated with the materiality of the embedded fibers, connecting the deformation gradient and the Cosserat rotation field.

Solution of contact pressure between robot edge with soft covering and human body surface considering nonlinear elasticity

Solution of contact pressure between robot edge with soft covering and human body surface considering nonlinear elasticity

Necking of thin-walled cylinders via bifurcation of incompressible nonlinear elastic solids.

Necking localization under quasi-static uniaxial tension is experimentally observed in ductile thin-walled cylindrical tubes, made of soft polypropylene. Necking nucleates at multiple locations along the tube and spreads throughout, involving the occurrence of higher-order modes, evidencing trefoil and fourth-foiled (but rarely even fifth-foiled) shaped cross-sections. No evidence of such a complicated necking occurrence and growth was found in other ductile materials for thin-walled cylinders under quasi-static loading. With the aim of modelling this phenomenon, as well as all other possible bifurcations, a two-dimensional formulation is introduced, in which only the mean surface of the tube is considered, paralleling the celebrated Flügge 's treatment of axially-compressed cylindrical shells. This treatment is extended to include tension and a broad class of nonlinear-hyperelastic constitutive law for the material, which is also assumed to be incompressible. The theoretical framework leads to a number of new results, not only for tensile axial force (where necking is modelled and, as a particular case, the classic Considère formula is shown to represent the limit of very thin tubes), but also for compressive force, providing closed-form formulae for wrinkling (showing that a direct application of the Flügge equation can be incorrect) and for Euler buckling. It is shown that the J2-deformation theory of plasticity (the simplest constitutive assumption to mimic through nonlinear elasticity the plastic branch of a material) captures multiple necking and occurrence of higher-order modes, so that experiments are explained. The presented results are important for several applications, ranging from aerospace and automotive engineering to the vascular mechanobiology, where a thin-walled tube (for instance an artery, or a catheter, or a stent) may become unstable not only in compression, but also in tension.

Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

<abstract><p>An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.</p></abstract>

Generalized Beam Theory for the static and dynamic response of a bridge deck with Structural Health Monitoring purposes

In recent years Structural Health Monitoring (SHM) has been assuming a fundamental role for existing bridges and viaducts, primarily for the aged ones, making compelling the need to resort to adequate techniques that enable the user to correctly process and interpret the amount of produced data. In the present work, with the purpose of defining SHM strategies for viaducts, a mechanical model of an existing bridge deck is realized on the basis of the Generalized Beam Theory (GBT). With the aim of catching the static and dynamic behavior of the structure up to prescribed tolerances, the model takes into account, for instance, the possible in-plane and out-of-plane distortion of the cross-sections and, at the same time, turns out to be easy to use and adapt, according to the level of detail one wants to achieve. Possible implementation of geometric and elastic nonlinearities is discussed as well. Moreover, data gained through static loading tests on the bridge, almost up to the elastic limit of the structure, and dynamic tests in operational conditions, processed in the framework of the Operational Modal Analysis (OMA), are used as benchmark for the calibration of the parameters of the mechanical model. A convergence analysis is then performed, so as to define the minimal number of degrees of freedom required to consistently simulate the mechanical behavior of the structure. Discussion on the effects of either point or distributed support conditions is proposed, with consequences in the implementation of the boundary conditions of the mechanical model.

The Significance of Cross-Sectional Shape Accuracy and Non-Linear Elasticity on the Numerical Modelling of Cerebral Veins under Tensile Loading.

Traumatic brain injury (TBI) is a serious global health issue, leading to serious disabilities. One type of TBI is acute subdural haematoma (ASDH), which occurs when a bridging vein ruptures. Many numerical models of these structures, mainly based on the finite element method, have been developed. However, most rely on linear elasticity (without validation) and others on simplifications at the geometrical level. An example of the latter is the assumption of a regular cylinder with a constant radius, or the geometry of the vein acquired from medical images. Unfortunately, these do not replicate the real conditions of a mechanical tensile test. In this work, the main goal is to evaluate the influence of the vein's geometry in its mechanical behaviour under tensile loading, simulating the real conditions of experimental tests. The second goal is to implement a hyperelastic model of the bridging veins where it would be possible to observe its non-linear elastic behaviour. The results of the developed finite element models were compared to experimental data available in the literature and other models. It was possible to conclude that the geometry of the vein structure influences the tensile stress-strain curve, which means that flattened specimens should be modelled when validating constitutive models for bridging veins. Additionally, the implementation of hyperelastic material models has been verified, highlighting the potential application of the Marlow and reduced polynomial (of fourth and sixth orders) constitutive models.

The Non-Linear Elasticity of Unidirectional Continuous Carbon Fibre-Reinforced Composites and of Carbon Fibres.

A database of non-linear elastic parameters in axial tension and compression is provided for continuous carbon fibre polymer composites and carbon fibres of different stiffnesses. Composite laminates manufactured by conventional or automated processes are tested in bending, and parameters are extracted for strains of less than 0.5%. While fibre composites with fibres of standard and intermediate moduli exhibit a stiffening of ∼15 GPa/% (of strain) and a softening of ∼20 GPa/%, those with high-modulus carbon fibres exhibit much higher values of ∼50 GPa/% for both. This database is useful for designing composite structures in a stiffness-based design and for correlating the processing of carbon fibres with their nanostructure and induced properties. The latter is discussed in terms of reorientation of crystallites of graphene sheets vis-à-vis the carbon fibre axis during loading.

A novel pulse-echo piezoelectric transducer for detecting quasi-static component induced by an ultrasonic longitudinal wave

For effectively detecting the quasi-static component (QSC) of an ultrasonic longitudinal wave, which is closely related to the elastic nonlinearity of material, we proposed a novel pulse-echo piezoelectric transducer consisting of a high-frequency piezoelectric wafer, a frequency selective isolation layer, a low-frequency piezoelectric wafer, and an acoustic backing. The high-frequency wafer generates the primary longitudinal wave (PLW) tone burst, while the high- and low-frequency wafers receive the pulse echo containing both the PLW component and QSC, respectively. We analyze the pulse-echo formation of the high-frequency PLW tone burst in a specimen, and conduct numerical simulations and experiments to validate the effectiveness of the proposed transducer. The results demonstrate that the low-frequency receiver is more efficient at detecting the QSC, even though the high-frequency wafer can also receive the echo of the QSC pulse. Specifically, the QSC pulse can still be detected by the low-frequency receiver when the high-frequency PLW tone burst is completely attenuated. The novel pulse-echo transducer proposed in this paper expands the design perspectives for transducers used in ultrasonic non-destructive testing.

Eringen’s model via linearization of nonlocal hyperelasticity

We consider Riesz’ fractional gradient and a truncated version of it. The equations of nonlocal nonlinear elasticity based on those gradients are known. We perform a formal linearization and arrive at the equations of linear elasticity based on those nonlocal operators. We prove the existence of solutions of the linear equations, notably, by a nonlocal version of Korn’s inequality. Finally, we show that the linearizations obtained are particular cases of Eringen’s model with singular kernels.

Unravelling anisotropic nonlinear shear elasticity in muscles: Towards a non-invasive assessment of stress in living organisms

Acoustoelasticity theory describes propagation of shear waves in uniaxially stressed medium and allows the retrieval of nonlinear elastic coefficients of tissues. In transverse isotropic medium such as muscles the theory leads to 9 different configurations of propagating shear waves (stress axis vs. fibers axis vs. shear wave polarization axis vs. shear wave propagation axis). In this work we propose to use 4 configurations to quantify these nonlinear parameters ex vivo and in vivo. Ex vivo experiments combining ultrasound shear wave elastography and mechanical testing were conducted on iliopsoas pig muscles to quantify three third-order nonlinear coefficients A, H and K that are possibly linked to the architectural structure of muscles. In vivo experiments were performed with human volunteers on biceps brachii during a stretching exercise on an ergometer. A combination of the third order nonlinear elastic parameters was assessed. The knowledge of this nonlinear elastic parameters paves the way to quantify in vivo the local forces produced by muscle during exercise, contraction or movements.

Entropic pressure on fluctuating solid membranes

Biological and crystalline membranes exhibit noticeable fluctuations at room temperature due to their low bending stiffness. These fluctuations have a significant impact on their overall mechanical behavior and interactions with external objects. When two membranes come into proximity, they mutually suppress each other’s fluctuations, leading to a repulsive force that plays a pivotal role in the mechanical behavior of these membranes. From the mechanics point of view, crystalline membranes are modeled as solid membranes with inherent shear resistance, whereas biological membranes are commonly described as fluidic entities without shear resistance. Under this premise, the entropic force between two fluctuating biological membranes is proposed to scale as p∝1/d3, where d is the intermembrane distance. Yet, there are numerous instances where these membranes display shear resistance and behave akin to solid membranes. In this paper, we develop a statistical mechanics model within nonlinear elasticity to study the entropic force acting on a confined, fluctuating solid membrane. We demonstrate that, due to the nonlinear elasticity of solid membranes, the entropic force scales differently compared to that of fluid membranes. Our predictions align well with the results obtained from molecular dynamics simulations involving graphene, a representative of a solid membrane, confined between two rigid walls.

On Solving Nonlinear Elasticity Problems Using a Boundary-Elements-Based Solution Method

The attractiveness of the boundary element method—the reduction in the problem dimension by one—is lost when solving nonlinear solid mechanics problems. The point collocation method applied to strong-form differential equations is appealing because it is easy to implement. The method becomes inaccurate in the presence of traction boundary conditions, which are inevitable in solid mechanics. A judicious combination of the point collocation and the boundary integral formulation of Navier’s equation allows a pure boundary element method to be obtained for the solution of nonlinear elasticity problems. The potential of the approach is investigated in some simple examples considering isotropic and anisotropic material models in the total Lagrangian framework.

A new example for the Lavrentiev phenomenon in nonlinear elasticity

We present a new example for the Lavrentiev phenomenon in context of nonlinear elasticity, caused by an interplay of the elastic energy’s resistance to infinite compression and the Ciarlet–Nečas condition, a constraint preventing global interpenetration of matter on sets of full measure.

Computational Design of Flexible Planar Microstructures

Mechanical metamaterials enable customizing the elastic properties of physical objects by altering their fine-scale structure. A broad gamut of effective material properties can be produced even from a single fabrication material by optimizing the geometry of a periodic microstructure tiling. Past work has extensively studied the capabilities of microstructures in the small-displacement regime, where periodic homogenization of linear elasticity yields computationally efficient optimal design algorithms. However, many applications involve flexible structures undergoing large deformations for which the accuracy of linear elasticity rapidly deteriorates due to geometric nonlinearities. Design of microstructures at finite strains involves a massive increase in computation and is much less explored; no computational tool yet exists to design metamaterials emulating target hyperelastic laws over finite regions of strain space. We make an initial step in this direction, developing algorithms to accelerate homogenization and metamaterial design for nonlinear elasticity and building a complete framework for the optimal design of planar metamaterials. Our nonlinear homogenization method works by efficiently constructing an accurate interpolant of a microstructure's deformation over a finite space of macroscopic strains likely to be endured by the metamaterial. From this interpolant, the homogenized energy density, stress, and tangent elasticity tensor describing the microstructure's effective properties can be inexpensively computed at any strain. Our design tool then fits the effective material properties to a target constitutive law over a region of strain space using a parametric shape optimization approach, producing a directly manufacturable geometry. We systematically test our framework by designing a catalog of materials fitting isotropic Hooke's laws as closely as possible. We demonstrate significantly improved accuracy over traditional linear metamaterial design techniques by fabricating and testing physical prototypes.

Influence of Temperature and Atmospheric Pressure on Nonlinear Elasticity of Rock

We conducted a 28-day long period co-propagating longitudinal acoustic wave experiment to monitor the change of nonlinear elastic modulus of rock. During the monitoring period, we monitored the atmospheric pressure and temperature of the environment. We analyzed the relationship between the change of nonlinear elastic modulus of rock and atmospheric pressure and temperature. We found that the change of them appears high consistency. But the change of ΔM\\/E is not sensitive to small amplitude change of atmospheric pressure. And there is a certain delay with the influence of temperature to the ΔM\\/E.

Coupling diffusion and finite deformation in phase transformation materials

We present a multiscale theoretical framework to investigate the interplay between diffusion and finite lattice deformation in phase transformation materials. In this framework, we use the Cauchy–Born Rule and the Principle of Virtual Power to derive a thermodynamically consistent theory coupling the diffusion of a guest species (Cahn–Hilliard type) with the finite deformation of host lattices (nonlinear gradient elasticity). We adapt this theory to intercalation materials – specifically Li1−2Mn2O4 – to investigate the delicate interplay between Li-diffusion and the cubic-to-tetragonal deformation of lattices. Our computations reveal fundamental insights into the microstructural evolution pathways under dynamic discharge conditions, and provide quantitative insights into the nucleation and growth of twinned microstructures during intercalation. Additionally, our results identify regions of stress concentrations (e.g., at phase boundaries, particle surfaces) that arise from lattice misfit and accumulate in the electrode with repeated cycling. These findings suggest a potential mechanism for structural decay in Li2Mn2O4. More generally, we establish a theoretical framework that can be used to investigate microstructural evolution pathways, across multiple length scales, in first-order phase transformation materials.