Nonlinear Elastic Response Research Articles

A new strategy to describe nonlinear elastic and asymmetric plastic behaviors with one yield surface

This paper proposes a new strategy to describe both nonlinear elastic and asymmetric plastic behaviors (early re-yielding, transient Bauschinger effect, permanent softening, and work-hardening stagnation) only with one yield surface. There have been the popular models, such as Yoshida-Uemori and Quasi-Plastic-Elastic (QPE) models, that have led to remarkable improvements in describing material's behavior and predicting springback. Although the Yoshida-Uemori model describes the asymmetric plastic behavior, it is challenging to follow the nonlinear elastic response. On the other hand, the QPE model does not capture the work-hardening stagnation though it follows the nonlinear elastic behavior. Besides, the above models have multiple surfaces leading to a complex mathematics and, then leading to increased computation time. In this study, a new one surface method incorporates state of strain change or state of energy change as well as the stress state in determining the deformation mode, so that it is possible to keep only one yield surface. In order to capture the work-hardening stagnation, the one surface method traces the equivalent plastic strain, then compares the change in the incremental equivalent plastic strain of the current path to the accumulated one up to the previous path when the loading direction reverses. This way makes the computation time much faster than an existing method which employs an additional surface to capture the work-hardening stagnation. The one surface method has been implemented into a user-defined material (UMAT) subroutine, and validated by comparing it with the experimental results and the results from Yoshida-Uemori and QPE models with cycling loading conditions and U-draw bending test. This work shows that the one surface method can describe both nonlinear elastic and asymmetric plastic behaviors with the reduction of complexity and computation time.

Dynamic acousto-elastic testing of concrete with a coda-wave probe: comparison with standard linear and nonlinear ultrasonic techniques

The use of nonlinear acoustic techniques in solids consists in measuring wave distortion arising from compliant features such as cracks, soft intergrain bonds and dislocations. As such, they provide very powerful nondestructive tools to monitor the onset of damage within materials. In particular, a recent technique called dynamic acousto-elasticity testing (DAET) gives unprecedented details on the nonlinear elastic response of materials (classical and non-classical nonlinear features including hysteresis, transient elastic softening and slow relaxation). Here, we provide a comprehensive set of linear and nonlinear acoustic responses on two prismatic concrete specimens; one intact and one pre-compressed to about 70% of its ultimate strength. The two linear techniques used are Ultrasonic Pulse Velocity (UPV) and Resonance Ultrasound Spectroscopy (RUS), while the nonlinear ones include DAET (fast and slow dynamics) as well as Nonlinear Resonance Ultrasound Spectroscopy (NRUS). In addition, the DAET results correspond to a configuration where the (incoherent) coda portion of the ultrasonic record is used to probe the samples, as opposed to a (coherent) first arrival wave in standard DAET tests. We find that the two visually identical specimens are indistinguishable based on parameters measured by linear techniques (UPV and RUS). On the contrary, the extracted nonlinear parameters from NRUS and DAET are consistent and orders of magnitude greater for the damaged specimen than those for the intact one. This compiled set of linear and nonlinear ultrasonic testing data including the most advanced technique (DAET) provides a benchmark comparison for their use in the field of material characterization.

Compressive elasticity of polydisperse biopolymer gels.

We theoretically predict the nonlinear elastic responses of polydisperse biopolymer gels to uniaxial compression. We analyze the competition between compressive stiffening due to polymer densification by out-going solvent flow and compressive softening due to continuous polymer buckling. We point out that the polydispersity in polymer lengths can result in an intrinsic, equilibrium mode of nonaffine compression: nonuniform strain but with uniform force distribution, which is found to be more energetically preferable than affine deformation. In this case, the gel softens significantly after the onset of polymer buckling at small compression, but as compression increases, densification-induced stiffening becomes important and a modulus plateau should be observed for a large range of strain. We also relate our results to recent compression experiments on collagen gels and fibrin gels.

Vibration damping materials with enhanced loss due to microstructural nonlinearity

One conventional approach to attenuating structure-borne waves or reduce the ringdown time of modes in structural elements is to attach damping layers or patches to vibrating components. Common patch and layer materials are specially formulated polymers or engineered polymeric composites that demonstrate elevated viscoelastic loss factors in the frequency range of interest. Recent research has shown that small volume fractions of negative stiffness inclusions embedded in a lossy material generate effective loss factors that exceed that of the host material. The ability to generate negative stiffness behavior, however, is often the result of nonlinear inclusion material response. Presented here is a multiscale model of a particulate composite material consisting of a nearly incompressible host material containing small-scale heterogeneities with a nonlinear elastic stress-strain response. We investigate the nonlinear dynamic behavior of the heterogeneous medium for small harmonic perturbations about several pre-strain states to demonstrate the influence of the microscale dynamic response on the macroscopically observable mechanical loss. Of primary interest is the energy dissipation capabilities of the composite which can be tuned using inclusion pre-strain. Loss for composites with nonlinear inclusions is compared to conventional composites containing air voids or steel inclusions. [Work supported by ONR.]

Time-domain characterization of the nonlinear viscoelastic memory of rocks with ultrasound waves

The nonlinear elastic response of rocks is known to be caused by internal microstructure, particularly cracks and fluids. In order to quantify this nonlinearity, we describe a new sensitive acoustic method to measure nonlinear viscoelastic properties of rocks directly in the time domain. Our method utilizes co-propagating longitudinal acoustic waves to perturb rocks with dynamically applied strain. Data indicates that rocks have a short period nonlinear viscoelastic memory that is a function of the time history of the loaded strain. We developed a phenomenological model with second and third order nonlinear elastic constants and a memory strength parameter to describe the short period nonlinear viscoelastic memory and the nonlinear elastic behavior of rock. Results of experiments on Crab Orchard sandstone and Lucite samples are presented. We have found both Lucite and Crab Orchard sandstone to be viscoelastic but the sandstone is much more nonlinear. These new observations and methods have significance for quantifying changes in microstructure and pore fluids in rocks and other materials.

Mechanical properties of single-walled carbon nanotubes: a comprehensive molecular dynamics study

There is inconsistent information across the literature related to tensile and compressive mechanical properties of carbon nanotubes (CNTs). This inconsistency arises from different sources such as the technical difficulties associated with testing and measuring the CNTs mechanical properties in the laboratory or the use of different input parameters (e.g. CNT wall thickness or boundary conditions) in numerical simulations. To address this inconsistency, an extensive series of molecular dynamics (MD) simulations is carried out in this study to investigate the influences of major factors including chirality, size, aspect ratio and slenderness ratio of CNTs on their mechanical properties, which have not been examined as extensively in previous studies. Numerical simulations are repeated for different temperatures of 100 K, 300 K and 500 K to investigate the influence of temperature in different practical applications. The agreements between the findings of this study and those from related previous studies are discussed in detail in the paper. Results on compressive behavior show the emergence of secondary and tertiary buckling modes as the slenderness ratio of the CNT increases, especially at higher temperatures. In addition, chiral CNTs are found to buckle at lower stresses and strains. Results on tensile behavior indicate a nonlinear-elastic response for all CNTs examined. Larger-diameter CNTs are found to fail at lower values of tensile stress and strain.

Computation of the homogenized nonlinear elastic response of 2D and 3D auxetic structures based on micropolar continuum models

We develop mechanical and numerical homogenization models for the large strains response of auxetic structures in order to derive their effective elastic response accounting for large changes of the geometry. We construct a strain driven nonlinear scheme for the computation of the stress-strain relation of these repetitive networks over a reference representative unit cell (abbreviated as RUC). Homogenization schemes are developed to evaluate the effective nonlinear mechanical response of periodic networks prone to an auxetic behavior, in both planar and 3D configurations, which are substituted by an effective micropolar continuum at the intermediate mesoscopic level. The couple stress part of the homogenized constitutive law takes into account the impact of the local rotations at the mesoscopic level and allows computing the bending response. This methodology is applied to four planar auxetic periodic lattices (the re-entrant hexagonal honeycomb and alternative honeycomb topologies, including the arrowhead, Milton and hexachiral structures), and to the 3D re-entrant and pyramid-shaped lattices. We have considering successively the in-plane and out-of-plane responses of these structures. The transition to an auxetic behavior is shown to be triggered by the imposed strain over the unit cell boundary. The computed evolutions of Poisson’s ratio versus the imposed stretch traduce an enhanced auxetic response as the stretch is increased. A satisfactory agreement is obtained between the homogenized stress-strain responses and the responses computed numerically by finite element simulations performed over a repeating unit cell.

Pulmonary arterial strain- and remodeling-induced stiffening are differentiated in a chronic model of pulmonary hypertension

Pulmonary hypertension (PH) is a debilitating vascular disease that leads to pulmonary artery (PA) stiffening, which is a predictor of patient mortality. During PH development, PA stiffening adversely affects right ventricular function. PA stiffening has been investigated through the arterial nonlinear elastic response during mechanical testing using a canine PH model. However, only circumferential properties were reported and in the absence of chronic PH-induced PA remodeling. Remodeling can alter arterial nonlinear elastic properties via chronic changes in extracellular matrix (ECM) content and geometry. Here, we used an established constitutive model to demonstrate and differentiate between strain-stiffening, which is due to nonlinear elasticity, and remodeling-induced stiffening, which is due to ECM and geometric changes, in a canine model of chronic thromboembolic PH (CTEPH). To do this, circumferential and axial tissue strips of large extralobar PAs from control and CTEPH tissues were tested in uniaxial tension, and data were fit to a phenomenological constitutive model. Strain-induced stiffening was evident from mechanical testing as nonlinear elasticity in both directions and computationally by a high correlation coefficient between the mechanical data and model (R2=0.89). Remodeling-induced stiffening was evident from a significant increase in the constitutive model stress parameter, which correlated with increased PA collagen content and decreased PA elastin content as measured histologically. The ability to differentiate between strain- and remodeling-induced stiffening in vivo may lead to tailored clinical treatments for PA stiffening in PH patients.

Many-body interactions in soft jammed materials.

In jammed packings of soft frictionless particles such as foams or emulsions, stress is transmitted via a network of mechanical contacts between neighbors. In generic simplified models of such materials, particle interaction energies are assumed to be pairwise additive. We report ab initio simulations of foam microstructures, showing that in general, this fundamental assumption is not justified: the conservation of bubble volumes introduces a many-body coupling between all the contacts of a given particle. It strongly modifies the relation between forces and displacements at individual contacts, in a way that cannot be captured by an effective two-body interaction. We report the impact of this effect on the linear and nonlinear elastic response of ordered bubble packings with coordination numbers ranging from 6 to 12, used as simple model systems, and we present an analytical model without free parameters which is valid as long as bubbles have an approximately spherical shape. It predicts the many-body coupling of particle contact forces, as well as the macroscopic mechanical response. For packing fractions approaching the jamming transition where contact forces go to zero, we derive an asymptotic two-body interaction law. It contains a logarithmic term, yielding a critical scaling that cannot be approximated by a power law.

Non-linear elastic response of layered structures, alternating pentamode lattices and confinement plates

Abstract We study the mechanical response under large elastic strains of pentamode layers confined between stiffening plates, and the potential use of such systems as novel seismic isolation and impact protection devices. We analyze pentamode materials that exhibit three soft deformation modes in the infinitesimal stretching regime that follow by the presence of perfectly hinged connections between the rods. The response of these metamaterials under large elastic strains is characterized by an elastic-stiffening effect in terms of the lateral force-displacement response, which increases in the presence of rigid connections and decreases by increasing the number of layers. Our results lead us to conclude that the analyzed pentamode metamaterials can be effectively employed as novel, performance-based devices for seismic and mechanical vibration protection, by designing the lattice geometry, the stiffness properties of the joints, and the lamination scheme in a suitable manner and as a function of the operating conditions.

Nonlinear coda wave analysis of hysteretic elastic behavior in strongly scattering media

Strongly scattering elastic media, such as consolidated granular materials, respond to ultrasonic pulse excitations with a long response signal with peculiar properties. The portion of the signal at late times, termed coda, is due to multiple scattering. It contains information about the elastic properties of the material, and it has been proven to be very sensitive to small variations in the modulus. Here we propose a technique based on a nonlinear analysis of the coda of a signal, which might be applied to quantify the nonlinear elastic response in consolidated granular media exhibiting a hysteretic elastic behavior. The method proposed allows for an intrinsic definition of the reference signal which is normally needed for applying coda-based methods.

Fast and slow nonlinear elastic response of disparate rocks and the influence of moisture

We study nonlinear elastic phenomena in rocks at the laboratory scale, with the goal of characterizing and understanding observations at crustal scales, for instance, during strong ground motion and earthquake slip processes. A dynamic perturbation of microstrain amplitude in rocks results in a transient elastic softening followed by a log(t)-type relaxation back to the initial unperturbed elastic modulus as soon as the excitation is removed. Here we use Dynamic Acousto-Elastic Testing (DAET) to investigate the relaxation behavior over 7 orders of magnitude in time (from 10-4s to more than 103s). We find that relaxation starts for all samples between 10-3 and 10-2s. Some samples then exhibit a nearly perfect log(t)-relaxation, implying that no characteristic time can be extracted, while some other samples show a preferential relaxation around 0.1s/1s. Such features appear insensitive to the amplitude of the dynamic perturbation and to the moisture content within the sample. The full nonlinear elastic response (fast dynamics) is also extracted at all amplitudes and moisture content. Adsorption of water on the grains strongly increases the elastic softening during the dynamic perturbation and the non-classical nonlinear features, whereas the classical features seem rather unaffected.

Elastic wave dispersion in pre-strained negative stiffness honeycombs

A doubly periodic array of curved beams, known as a negative stiffness (NS) honeycomb, has recently been shown to exhibit tunable impact isolation by exploiting nonlinear mechanical behavior [Correa et al., Rapid Prototyping J., 21 (2), 2015]. The primary benefit of NS honeycombs over standard honeycomb materials lies in their ability to return to their original shape after experiencing an impact. The recoverable nature of their response is a result of the curved beam structure which has a single stable configuration but experiences one or more buckling events when loaded. The complex nonlinear elastic response of NS honeycombs lead to large variations in mechanical stiffness, which makes these materials compelling candidates for tunable control of elastic waves. The present work investigates linear elastic wave propagation of a representative NS honeycomb lattice at varying levels of pre-strain. Eigenfrequency finite element analysis is performed on the unit cell subjected to finite uniaxial deformation. The longitudinal and transverse phase speeds are shown to be anisotropic and highly dependent on uniaxial pre-strain. This is especially true near points of instability, where one observes very different functional dependence of longitudinal and transverse wave motion on pre-strain.

The Inelastic Hierarchy: Multiscale Biomechanics of Weak Bonds

Living tissue is a mechanical chameleon. It may behave as a viscous or elastic material, depending on how fast and strong you pull on it. It also responds plastically to permanent loads, as evidenced by the Tutankhamun-style lobe-stretching ear piercings that are currently regaining popularity. To mechanistically explain and quantitatively predict how such impressive versatility and resilience is accomplished by the molecular composition and architecture of living matter is the holy grail of biomechanics.

Nonlinear elastic response and anharmonic properties of MgO single crystal: First-principles investigation

Knowledge of the nonlinear elastic response and anharmonic properties of MgO single crystal is important for interpreting the seismically observed velocity variations in the Earth’s mantle and constructing reliable mineralogical modes of the Earth’s interior. However, investigations on these properties of it have not been performed sufficiently to date. So the nonlinear elastic response of the MgO is first studied by calculating the second-, third-, and fourth-order elastic constants. Results show that our predications on the high-order elastic constants exhibit excellent agreement with the available experimental ones and other theoretical values, which indicates that the present computational approach is suitable for the calculations of the high-order elastic constants and it can be expanded into other complex systems. Then on the basis of the elastic constants, the related anharmonic properties such as the first-order pressure derivatives of the second- and third-order elastic constants, Grüneisen constants of the long-wavelength acoustic modes, and ultrasonic nonlinear parameters are also investigated. The predications on the anharmonic properties first given in the present work are hoped to serve as a valuable guidance or reference for further related investigations.

Nonlinear elastic response of strong solids: First-principles calculations of the third-order elastic constants of diamond

Accurate theoretical calculations of the nonlinear elastic response of strong solids (e.g., diamond) constitute a fundamental and important scientific need for understanding the response of such materials and for exploring the potential synthesis and design of novel solids. However, without corresponding experimental data, it is difficult to select between predictions from different theoretical methods. Recently the complete set of third-order elastic constants (TOECs) for diamond was determined experimentally, and the validity of various theoretical approaches to calculate the same may now be assessed. We report on the use of density functional theory (DFT) methods to calculate the six third-order elastic constants of diamond. Two different approaches based on homogeneous deformations were used: (1) an energy-strain fitting approach using a prescribed set of deformations, and (2) a longitudinal stress-strain fitting approach using uniaxial compressive strains along the [100], [110], and [111] directions, together with calculated pressure derivatives of the second-order elastic constants. The latter approach provides a direct comparison to the experimental results. The TOECs calculated using the energy-strain approach differ significantly from the measured TOECs. In contrast, calculations using the longitudinal stress-uniaxial strain approach show good agreement with the measured TOECs and match the experimental values significantly better than themore » TOECs reported in previous theoretical studies. Lastly, our results on diamond have demonstrated that, with proper analysis procedures, first-principles calculations can indeed be used to accurately calculate the TOECs of strong solids.« less

Seismic performance of precast concrete rocking walls with buckling restrained braces

The application of a buckling restrained brace (BRB) to improve the energy dissipation of a precast concrete rocking wall (PCRW) was investigated. A series of quasi-static reversed cyclic loading tests was conducted for the BRB, the PCRW subassemblage and the combined PCRW–BRB system. A commonly adopted fibre model was then developed using the Ruaumoko computer program and verified using the experimental results. The results indicate that the stable elasto-plastic BRB supplies vibration energy to the nonlinear elastic responses of the PCRW. This finding confirms that the PCRW–BRB system is a suitable alternative for use in seismic-resistant structures.

Wedge-shaped twins and pseudoelasticity in fcc metallic nanowires under bending

Molecular dynamics simulations were performed to study the deformation mechanisms of 〈110〉-oriented, faceted Cu and Au nanowires under bending along three different crystallographic directions. Independent of the bending direction, the stress field is characterized by a highly nonlinear elastic response, leading to a shift of the neutral fiber away from the central wire axis. The nanowires show ultra-high yield strengths, and the achievable large elastic strains directly influence the dislocation nucleation through the change of the unstable stacking fault energy. In agreement with theory and experiments on face-centered cubic 〈110〉 nanowires under uniaxial load, the tensile part of the wires exhibit deformation twinning, while plastic deformation in the compressed part takes place by slip of perfect dislocations. Independent of the bending direction, wire size, temperature and bending rate, all wires showed the formation of wedge-shaped twins. Upon instantaneous load removal, wires bent in two of the three directions showed spontaneous, pseudoelastic unbending. The findings of this study could be relevant for the design of flexible electronics and mechanical energy storage applications at the nanoscale.

Frequency, pressure, and strain dependence of nonlinear elasticity in Berea Sandstone

AbstractAcoustoelasticity measurements in a sample of room dry Berea sandstone are conducted at various loading frequencies to explore the transition between the quasi‐static ( ) and dynamic (few kilohertz) nonlinear elastic response. We carry out these measurements at multiple confining pressures and perform a multivariate regression analysis to quantify the dependence of the harmonic content on strain amplitude, frequency, and pressure. The modulus softening (equivalent to the harmonic at 0f) increases by a factor 2–3 over 3 orders of magnitude increase in frequency. Harmonics at 2f, 4f, and 6f exhibit similar behaviors. In contrast, the harmonic at 1f appears frequency independent. This result corroborates previous studies showing that the nonlinear elasticity of rocks can be described with a minimum of two physical mechanisms. This study provides quantitative data that describes the rate dependency of nonlinear elasticity. These findings can be used to improve theories relating the macroscopic elastic response to microstructural features.

Response of Honeycombs Subjected to In-Plane Shear

Study on the response of honeycombs subjected to in-plane shear helps establish the constitutive relations for honeycombs and shed light on the mechanics of cellular materials. The present study explores the nonlinear elastic response of honeycombs under in-plane shear by analyzing the large deflection of cell walls in a unit cell. Governing equations are established which relate the macroscopic response of honeycombs to the deflection of cell walls. Solving these equations, the behavior of regular honeycombs under in-plane shear along horizontal (X) and vertical (Y) directions was investigated. It is found that the response of regular honeycombs under in-plane shear depends on the nondimensional shear stress which is a parameter combining the thickness-to-length ratio of cell walls, the Young's modulus of base materials, and macroscopic shear stress. Lateral shrinking is a distinctive characteristic for honeycombs under in-plane shear, which should be taken into account when establishing constitutive relations and performing simple shear experiments. Expressions for predicting the shear strength of honeycombs are formulated in this paper. It is noted that the normalized shear strength of regular honeycombs depends on two ratios: the thickness-to-length ratio of cell walls and the ratio of Young's modulus to yield strength of base materials, and the former has a dominant effect. By comparing honeycombs with cell walls of uniform thickness against honeycombs with vertical cell walls of double thickness, it is found that doubling the thickness of vertical cell walls of honeycombs increases their shear strength along horizontal (X) direction nearly twice, but does not improve the shear strength that much along the vertical (Y) direction.