Nonlinear Elastic Deformation Research Articles

Pressure effect on wave propagation in saturated porous rocks with aligned fractures

Rock fracturing is widespread in the upper crust of the earth. Nearly all in-situ rocks are subject to subsurface pressures. Understanding the influence of effective pressure (difference between confining stress and pore pressure) on the elastic properties of fractured rocks is crucial for estimating in-situ seismic properties. Wave-Induced Fluid Flow (WIFF) and fracture Elastic Scattering (ES) attenuation mechanisms have been widely studied. However, the effects of effective pressure on WIFF and ES mechanisms in fractured rocks are relatively unexplored. To investigate the pressure influence on WIFF and ES in P and SV wave propagation, we developed a pressure-dependent dynamic model to incorporate Multi-shaped Microcracks' Squirt Flow (MMSF), Fracture-Background WIFF (FB-WIFF), Biot Flow, and ES mechanisms with non-linear elastic and hyper-elastic deformation stages. The results show that both the WIFF and ES mechanisms are affected by effective pressure, except for the ES mechanism of normal incident SV waves. The WIFF is more susceptible to effective pressure than the ES mechanism. In addition, the closure of microcracks (soft pores) during effective pressure loading continuously decouples the MMSF mechanism from the FB-WIFF, Biot Flow, and ES mechanisms, causing the increase in the velocity of the P- and SV- waves and the variations in the attenuation processes. These wave propagation characteristics help to understand the hydraulic properties of in-situ fractured rocks. By comparing model predictions with ultrasonic laboratory velocity data under varying effective pressure loading, we validated our model.

Hyperelastic model for nonlinear elastic deformations of graphene-based polymer nanocomposites

Graphene-based polymer nanocomposites (PNCs) are increasingly important in engineering applications involving large deformations. However, the nonlinear behavior of these materials has not been thoroughly studied. Current models do not address the specific nonlinear effects of graphene nanofillers under large strains, lack sufficient comparison with experimental data, and primarily focus on uniaxial behavior without exploring biaxial responses, which are relevant in technological applications. This study investigates PNCs composed of silicone elastomer and graphene nanoplatelets (GNPs). We present experimental tests conducted in both simple tension and biaxial inflation on circular membranes. A homogenized hyperelastic model is developed, incorporating distinct contributions from the matrix and the nanofiller. Specifically, we introduce a novel strain energy function for the nanofiller contribution, tailored to reproduce the observed experimental behavior. The model accurately predicts the nonlinear elastic response of the studied PNCs across varying contents of GNPs. The proposed strain energy function is implemented in MATLAB to obtain an exact numerical solution for the inflation of circular PNC membranes. Finally, to demonstrate its broader applicability, the hyperelastic model is applied to additional experimental data from other PNCs found in the literature. This model contributes to establishing a robust framework for the effective use of PNCs.

Nonlinear Isogeometric Analyses of Instabilities in Thin Elastic Shells Exhibiting Combined Bending and Stretching

Abstract Thin structures like shells are highly susceptible to mechanical instabilities. They can undergo large nonlinear elastic deformations while exhibiting combined bending and stretching modes of deformation. Here, we propose a new displacement-based finite element (FE) formulation based on the isogeometric discretization of Koiter’s nonlinear shell theory and use of the method of dynamic relaxation (DR) to solve equilibrium configurations of problems involving fine-scale and hierarchical wrinkling and buckling. No imperfection seeding is required to trigger instabilities. The use of the NURBS basis provides a rotation-free, conforming, higher-order spatial continuity such that curvatures and membrane strains can be computed directly from interpolating the position vectors of the control points using spatial finite differences. The pseudo-dissipative FE dynamical system is updated through explicit time integration and made scalable for parallel computing using a message passing interface (MPI). The proposed FE method is successfully benchmarked against several numerical and experimental results.

Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

Nonlinear Deformation of Cylinders from Materials with Different Behavior in Tension and Compression

A new numerical-analytical method for solving physically nonlinear deformation problems of axisymmetrically loaded cylinders made of materials with different behavior in tension and compression has been developed. To linearize the problem, the uninterrupted parameter continuation method was used. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible displacement rates, is constructed. To find the main unknowns of the problem of physically nonlinear cylinder deformation, the Cauchy problem for the system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta-Merson method with automatic step selection. The initial conditions were established by solving the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson’s scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method. The test problem for the nonlinear elastic deformation of a thin cylindrical shell is solved. Coincidence of the spatial solution with the shell solution was obtained. Physically nonlinear deformation of a thick-walled cylinder was studied. It is shown that failure to take into account the different behavior of the material under tension and compression leads to significant errors in the calculations of stress-strain state parameters.

A nonlinear continuum framework for constitutive modeling of active polymer gels

Chemo-mechanical transduction is one of the key mechanisms that has formed the basis for designing bio-inspired self-driven synthetic systems from soft materials. Polymer hydrogels that use Belousov–Zhabotinsky (BZ) reaction are a unique class of dynamical reaction–diffusion (RD) systems that can continuously transduce internal chemical energy, from the reaction, to produce sustained mechanical work. In particular, BZ gels represent a complex nonlinear chemo-mechanical system, wherein, the autocatalytic oscillatory BZ reaction drives the rhythmic mechanical deformations through polymer-solvent inter–diffusion. The objective of our work is to develop a standardized finite element (FE) framework for chemically driven active hydrogels that captures nonlinear elastic deformations with limited chain extensibility. The distinguishing feature of our approach is that, unlike other approaches, it combines reaction kinetics, solvent transport, elastodynamics of the polymeric network, and polymer-solvent friction under a unified FE framework. Moreover, we adapt our approach to a specific case of BZ gels and capture their swelling-deswelling characteristics. We first implement our FE framework in MATLAB that subsequently, forms the basis for constructing a three-dimensional user element subroutine (3D-UEL) in ABAQUS. Ultimately, through our simulations, we are able to capture all the essential features of BZ gels that includes chemically driven mechanical deformations. In addition, we also demonstrate that our 3D-UEL efficiently captures the chemo-mechanical response of “stent-shaped” BZ gels–a non-standard 3D geometry. In essence, our FE approach not only allows us to simulate BZ gels but also provides a template for other active, dynamical, RD-based systems, driven by chemo-mechanical transduction, irrespective of internal or external mechanisms.

Delayed fracture caused by time-dependent damage in PDMS

An experimental and theoretical study of delayed fracture of polydimethlsiloxane (PDMS) is presented. Previous works have demonstrated that delayed fracture in single edge notch specimens is caused by time dependent damage due to chain scission. Here we study the interactions between damage and the elastic field using different specimens and crack geometries with blunt and sharp cracks. Our experiments show that initial toughness is not well defined, as stable slow crack growth can occur over a range of applied loads. Our experiments demonstrate that there is a universal relation between crack growth rate and applied energy release rate. A model coupling the nonlinear elastic deformation and rate dependent bond scission is proposed and is in good agreement with experimental data.

Research on Dynamic Deformation Laws of Super High-Rise Buildings and Visualization Based on GB-RAR and LiDAR Technology

It is well-known that structures composed of super high-rise buildings accumulate damages gradually due to ultra-long loads, material aging, and component defects. Thus, the bearing capacity of the structures can be significantly decreased. In addition, these effects may cause inestimable life and property losses upon strong winds, earthquakes, and other heavy loads. Hence, it is necessary to develop real-time health monitoring methods for super high-rise buildings to deeply understand the running state during operation, timely discover potential safety potentials, and to provide reference data for reinforcement design. Along these lines, in this work, the built super high-rise buildings (Yunding Building) and super high-rise buildings (the Main Tower of the Shandong International Financial Center), under construction, were selected as the research objects. The overall dynamic deformation laws of super high-rise buildings were monitored by using ground-based real aperture radar (GB-RAR) technology for its advantages in non-contact measurement, remote monitoring, and real-time display of observation results. Denoising of the observation data was also carried out based on wavelet analysis. The visualization of the space state of the Yunding Building was realized based on handheld LiDAR technology. From the acquired results, it was demonstrated that the measuring accuracy of GB-RAR could reach the submillimeter level, while the noises under a natural state of wavelet analysis were eliminated well. The maximum deformation values of the Yunding Building and the Main Tower of Shandong International Financial Center under their natural state were 9.63 mm and 16.46 mm, respectively. Under sudden wind loads, the maximum deformation of the Yunding Building could be as high as 895.79 mm. The overall motion state switched between an S-shaped pattern, hyperbolic-type, and oblique line, presented the characteristics of nonlinear elastic deformation.

Statistical field theory for nonlinear elasticity of polymer networks with excluded volume interactions.

Polymer networks formed by cross linking flexible polymer chains are ubiquitous in many natural and synthetic soft-matter systems. Current micromechanics models generally do not account for excluded volume interactions except, for instance, through imposing a phenomenological incompressibility constraint at the continuum scale. This work aims to examine the role of excluded volume interactions on the mechanical response. The approach is based on the framework of the self-consistent statistical field theory of polymers, which provides an efficient mesoscale approach that enables the accounting of excluded volume effects without the expense of large-scale molecular modeling. A mesoscale representative volume element is populated with multiple interacting chains, and the macroscale nonlinear elastic deformation is imposed by mapping the end-to-end vectors of the chains by this deformation. In the absence of excluded volume interactions, it recovers the closed-form results of the classical theory of rubber elasticity. With excluded volume interactions, the model is solved numerically in three dimensions using a finite element method to obtain the energy, stresses, and linearized moduli under imposed macroscale deformation. Highlights of the numerical study include: (i) the linearized Poisson's ratio is very close to the incompressible limit without a phenomenological imposition of incompressibility; (ii) despite the harmonic Gaussian chain as a starting point, there is an emergent strain-softening and strain-stiffening response that is characteristic of real polymer networks, driven by the interplay between the entropy and the excluded volume interactions; and (iii) the emergence of a deformation-sensitive localization instability at large excluded volumes.

A numerical two-scale approach for nonlinear hyperelastic beams and beam networks

We introduce a numerical framework for modeling hyperelastic slender trusses, oftentimes used as elementary building blocks in architected materials, which accounts for both the geometric nonlinearity inherent in thin structures and the nonlinear constitutive behavior of the base material. Akin to the FE2 method in homogenization, our approach is based on a formal, two-scale expansion. We decompose the three-dimensional (3D) description of a slender structure into a macroscale problem solving for the deformation of the beam center-line (based on an effective strain energy density, which depends on the stretching, bending, and torsional strains of the beam’s center-line) and a series of two-dimensional (2D) microscale boundary value problems (defined over the beam cross-section). We solve a series of 2D problems (covering a range of macroscopic strain combinations) for a given cross-sectional geometry and material distribution in an offline, pre-processing step. Using this pre-computed energy landscape, we solve the macroscopic boundary value problem within a geometrically exact, nonlinear discrete beam framework. We demonstrate the accuracy of this approach through a set of benchmark problems highlighting the nonlinear effects of the cross-sectional geometry and constitutive material, including material heterogeneity and pre-strains. We further illustrate how this technique is applied to truss-based architected materials consisting of networks of slender struts, which are typically made of polymeric base materials and thus allow for large nonlinear elastic deformation.

Nonlinear elastic behaviors and deformation mechanisms of nano-structured crosslinked biopolymer networks

Cells always undergo large deformation in response to external stimulations or internal stresses in many cell functions (e.g., cell migration, cell division and cell growth). When undergoing large deformation, crosslinked actin filament networks (CAFNs) always show strong nonlinear elasticity to maintain the cell shape and integrity, known as strain stiffening, which plays a crucial role in many cell functions. To investigate the nonlinear elastic behaviors of CAFNs, a three-dimensional representative volume element model is used to perform finite element method simulations. Simulation results show that actin filament volume fraction, crosslinking density and components’ Young’s moduli show significant influences on the nonlinear elastic behaviors of CAFNs. In addition, the shear stress–strain curves of CAFNs highly depend on the bending stiffness and tensile stiffness of filamins as well as the bending stiffness of actin filaments, however, they are almost insensitive to the tensile stiffness of actin filaments. The present work not only sheds light on the nonlinear elastic behaviors of CAFNs but also provides a valuable reference for developing advanced artificial composite structures that can be used as semi-flexible biomedical scaffolds and wearable electronics.

Chiral Mechanical Metamaterials for Tunable Optical Transmittance

AbstractFlexible metamaterials have been increasingly harnessed to create functionality through their tunable and unconventional response. Herein, chiral unit cells based on Archimedean spirals are employed to transform a linear displacement into twisting. First, the effect of geometry on such extension‐twisting coupling is investigated. This unravels a wide range of highly nonlinear behaviors that can be programmed. Additionally, it is demonstrated that by combining the spirals with polarizing films one can create mechanical pixels capable of modulating the transmission of light through deformation. Guided by experiments and numerical analyses, pixels are arranged in 2D arrays to realize black and white and color displays, which reveal distinct images at different states of deformation. As such, the study puts forward a methodology for the design of an emerging class of flexible devices that can convert nonlinear elastic deformation to tunable optical transmittance.

Characterizing the biaxial compressive deformation behavior of epoxy polymer through cruciform experiment and finite element analysis

The multiaxial deformation behaviors of amorphous epoxy polymers were characterized using planar biaxial compression tests and finite element (FE) analyses of cruciform specimens. The multiaxial limit of elastic deformation was investigated considering buckling instability of the planar biaxial compression tests. An offset-based yield criterion was proposed to determine the initial yield surface considering the nonlinear elastic deformation characteristics observed in the deformation tests. An FE analysis confirmed using the results of the tests revealed that the proposed method captured the initial multiaxial yielding in the central gauge of the specimen, demonstrating that existing pressure-dependent classical yield functions, such as the paraboloidal and conical yield functions, cannot appropriately describe the initial yield surface of the epoxy polymer. The yield function previously customized from molecular dynamics (MD) data exhibited a relatively good agreement with the initial yield surface. The determined shape of the initial yield surface and its agreement with the MD data–driven yield function were not consistently maintained during subsequent yielding. The results indicate that a new customized yield function properly describing the broader initial yield surface than those of the classical yield functions and its shape transition to plastic behavior is necessary to accurately reflect the entire multiaxial deformation behavior.

Physico-mechanical properties and brittle to plastic transition of a thermally cracked marble under uniaxial tension

Thermal stress induces severe changes in rock microstructures, resulting in the variations of physical and mechanical properties. To study thermal effects on rock physical properties and tensile behaviors, monotonic and cyclic tension tests were conducted on marble samples heated at different temperatures. The results show that marble samples are weakened by thermal damage, showing a decrease in tensile strength and P-wave velocity, and an increase in porosity. The changing trends of Young's modulus and failure strain with temperature appear to be scattering and complicated probably depending on the deformation types (i.e., nonlinear elastic deformation and plastic deformation). Nonlinearity is a fundamental feature of tensile stress versus strain curves, which is enhanced when the sample is heated at higher temperature. Even under low tensile stress levels, permanent damages are always observed, highlighting a hysteresis in the stress-strain curves obtained from cyclic tension tests. A transition from the nonlinear elastic deformation to the plastic deformation occurs at approximately 300 - 400°C. A crack density based constitutive model is introduced for the nonlinear elastic deformation (25 - 300°C). A modified Voce exponential function is recommended to capture the tensile stress and strain relationships for the plastic deformation (400 - 700°C).

Plastic deformation behavior of bending aluminum fiber porous material under compressive load

The 6061 aluminum alloy wires with diameters of 0.15 mm and 0.20 mm were cut into different structural sizes of 5 × 10 mm and 10 × 15 mm by a spring machine to bend the aluminum alloy fibers, one is 5 mm short side, 10 mm long side, and 1 mm arc radius, and the other is 10 mm short side, 15 mm long side, and 1 mm arc radius. Special cylindrical molds are used to prepare folding sheets with different porosities using vacuum hot pressing sintering technology. Bent aluminum fiber porous material. The gas displacement method was used to detect the pore size and pore size distribution, and the Plastic Deformation Behavior were analyzed by quasi-static uniaxial compression test. Research shows that the material presents a three-stage stress-strain curve, from the initial nonlinear elastic deformation stage to the pseudo-platform stage and the densification stage, the wire diameter and structure of the material The smaller the size, the more conducive to the densification of the silk skeleton, and the smaller the average pore size formed, the higher the compressive strength.

Nonlinear FE-analysis and testing of light-frame timber shear walls subjected to cyclic loading

Light-frame timber shear walls have been used as load-bearing elements in buildings for several decades. To predict the performance of such structural elements under loading, numerous analytical and numerical models have been developed. However, little focus has been on the prediction of the plastic damage behaviour and unloading of the walls. In this paper, a parametric Finite Element (FE) model is further developed by introducing elasto-plastic connectors to simulate the mechanical behaviour of the sheathing-to-framing connections. To verify the accuracy of the elasto-plastic model, full-size walls were tested and compared with results from simulations. The numerical results, from a few loading cycles, indicate that the model achieves reasonable accuracy in predicting both the nonlinear elastic and plastic deformations. Both experimental and simulation results demonstrate the importance of opening locations relating to the external racking force. The results also indicate that for a double-layer wall, its racking strength can be achieved by summation of the separate contribution from each layer. Furthermore, the internal layer was observed to contribute significantly less than the external layer since its nail pattern was based on the sheathing pattern of the external layer.

A Scattering Wavelet Network-based Approach to Fingerprint Classification

In a large-scale automatic fingerprint identification system (AFIS), fingerprint classification is an essential indexing step to reduce the search time in a large database for accurate matching. Fingerprint classification is still a challenging machine learning problem due to large intra-class and small inter-class variability. Nonlinear elastic deformation is one of the main sources of intra-class variability, which occurs due to the non-uniform pressure applied during fingerprint acquisition and the elastic nature of the fingerprint itself. This paper proposes a novel approach to fingerprint classification based on a scattering wavelet network to extract translation and small deformation invariant local features. The resulting sparse invariant feature vectors are used as input to a simple generative PCA affine classifier for the classification. The experiments evaluated with two different protocols on a benchmark NIST SD-4 database show the effectiveness and robustness of the proposed fingerprint classification algorithm in terms of classification accuracy.

A Novel Nonlinear Elasticity Approach for Analysis of Nonlinear and Hyperelastic Structures

There are many materials that the linear elasticity theory cannot predict their mechanical behavior against applied loads. This research proposes a comprehensive theoretical method to obtain the mechanical response of hyperelastic models (with nonlinear elastic deformations) such as polymers and rubbers. They are vital in the design phase of complicated engineering structures like engine mounts and structural bearings in aerospace and automotive industries. The presented theory is implemented in detail and has no limitations in analyzing geometrically and physically nonlinear materials. As a test case, the governing equations of a sheet made of nonlinear elastic material are derived within the framework of this new approach. The derived governing equations are completely nonlinear in all major directions. Consequently, results can be one of the most accurate mathematical simulation results for nonlinear elastic material structures. Then, the obtained equations are solved using a meshless solution method named the semi-analytical polynomial method. The stress and deformation results of the structure under uniform and non-uniform transverse external loadings are obtained for different types of boundary conditions, loading, and material properties. Consequently, this research can be widely used as a suitable essential reference for researchers studying the mechanical behavior of nonlinear elastic structures.

Analysis of electromechanical systems based on the absolute nodal coordinate formulation

The absolute nodal coordinate formulation (ANCF) approach has been successfully used to analyze bodies undergoing large deformations in multibody dynamics applications. In this study, the ANCF is extended to the analysis of coupled electromechanical systems. To this end, the electrostatic equations are solved by means of conventional plane finite elements, and the ANCF is used to describe the geometrically nonlinear elastic deformation of a thin beam. Bidirectional coupling between electrostatic and elastic domains was introduced using an iterative staggering algorithm. The results illustrate that the ANCF approach can be applied to electromechanical problems when objects are discretized using beam and plate elements. Two numerical examples of microbeams subject to an electrostatic field are used to validate the proposed solution strategy and to reveal characteristic features of fully coupled electromechanical solutions accounting for finite strain theory.

A Multiple Site Type Nucleation Model and Its Application to the Probabilistic Strength of Pd Nanowires

Pristine specimens yield plastically under high loads by nucleating dislocations. Since dislocation nucleation is a thermally activated process, the so-called nucleation-controlled plasticity is probabilistic rather than deterministic, and the distribution of the yield strengths depends on the activation parameters to nucleate. In this work, we develop a model to predict the strength distribution in nucleation-controlled plasticity when there are multiple nucleation site types. We then apply the model to molecular dynamics (MD) simulations of Pd nanowires under tension. We found that in Pd nanowires with a rhombic cross-section, nucleation starts from the edges, either with the acute or the obtuse cross-section angles, with a probability that is temperature-dependent. We show that the distribution of the nucleation strain is approximately normal for tensile loading at a constant strain rate. We apply the proposed model and extract the activation parameters for site types from both site types. With additional nudged elastic bands simulations, we propose that the activation entropy, in this case, has a negligible contribution. Additionally, the free-energy barriers obey a power-law with strain, with different exponents, which corresponds to the non-linear elastic deformation of the nanowires. This multiple site type nucleation model is not subjected only to two site types and can be extended to a more complex scenario like specimen with rough surfaces which has a distribution of nucleation sites with different conditions to nucleate dislocations.