Linear Elastic Regime Research Articles

A strain-interfaced digital twin solution for corner fatigue crack growth using Bayesian inference

This paper introduces a digital twin solution for corner fatigue crack growth assessment. The digital twin comprises three core features: (1) diagnosis, (2) prognosis and (3) updating. The diagnosis arm performs remote crack size measurement via strain data collected from strategically identified locations. The prognosis component postulates the fatigue life across both linear-elastic and elasto-plastic loading regimes through a fatigue crack growth power law with the cyclic J-integral, ΔJ, as the crack driving force. Uncertainty in power law parameters, however, may result in differences between the prognosis and observed fatigue life. Hence, the digital twin completes the feedback loop via Bayesian updating of the power law parameters, thereby mirroring its physical counterpart closely. An improved estimation of the remaining useful life follows. The proposed digital twin solution validates against three specimens under constant amplitude loading and a single specimen under variable amplitude loading. The successful application of the approach marks a significant step toward operational digital twins within practical settings.

Nonlinear Poisson effect in affine semiflexible polymer networks.

Stretching an elastic material along one axis typically induces contraction along the transverse axes, a phenomenon known as the Poisson effect. From these strains, one can compute the specific volume, which generally either increases or, in the incompressible limit, remains constant as the material is stretched. However, in networks of semiflexible or stiff polymers, which are typically highly compressible yet stiffen significantly when stretched, one instead sees a significant reduction in specific volume under finite strains. This volume reduction is accompanied by increasing alignment of filaments along the strain axis and a nonlinear elastic response, with stiffening of the apparent Young's modulus. For semiflexible networks, in which entropic bending elasticity governs the linear elastic regime, the nonlinear Poisson effect is caused by the nonlinear force-extension relationship of the constituent filaments, which produces a highly asymmetric response of the constituent polymers to stretching and compression. The details of this relationship depend on the geometric and elastic properties of the underlying filaments, which can vary greatly in experimental systems. Here, we provide a comprehensive characterization of the nonlinear Poisson effect in an affine network model and explore the influence of filament properties on essential features of both microscopic and macroscopic response, including strain-driven alignment and volume reduction.

Unified Understanding of Nonlinear Rheology near the Jamming Transition Point.

When slowly sheared, jammed packings respond elastically before yielding. This linear elastic regime becomes progressively narrower as the jamming transition point is approached, and rich nonlinear rheologies such as shear softening and hardening or melting emerge. However, the physical mechanism of these nonlinear rheologies remains elusive. To clarify this, we numerically study jammed packings of athermal frictionless soft particles under quasistatic shear γ. We find the universal scaling behavior for the ratio of the shear stress σ and the pressure P, independent of the preparation protocol of the initial configurations. In particular, we reveal shear softening σ/P∼γ^{1/2} over an unprecedentedly wide range of strain up to the yielding point, which a simple scaling argument can rationalize.

The anti-dogbone: Evaluating and designing optimal tensile specimens for deep learning of constitutive relations

Traditional tensile testing with “dogbone”-shaped specimen (ASTM E8, first standardized in 1924) strives for strain uniformity. Multiple tests with such samples help fit simple constitutive relation parameters on real materials. With the development of deep learning, the concept of employing entirely data-driven constitutive relations to capture more intricate material behavior has arisen. Nevertheless, these methods demand experimental data that are distributed throughout the complete stress–strain configuration space to effectively train the machine learning models. This is particularly crucial for mechanisms like hardening, which are time-dependent and sensitive to loading history. In this work, we investigate the potential to efficiently gather a wider range of experimental data points in the stress–strain configuration space using non-uniform samples and displacement-field mapping, leveraging advancements in computer vision techniques. We developed a metric to quantify stress state diversity in 2D tensile experiments and used it to optimize the shape of the sheet sample. The goal was to increase stress–strain diversity obtained within a single test, particularly in the linear elastic regime. Additional geometric constraints can be introduced on the design features, considering factors such as size and curvature to adapt to the microstructural characteristics of the sample material.

Heterogeneity characterization of commercial structural papers

Abstract Evaluation of physical and mechanical heterogeneity in commercial paperboards is needed to promote their use in structural applications, especially within the field of packaging. Understanding the range of their behaviors is needed to compete with other materials in the current marketplace and expand in others. This work describes the physical and stiffness heterogeneities of twelve commercial materials using tensile tests in the cross-machine direction and several inverse analyses. The effects of grammage, thickness, and apparent density on tensile stiffness were evaluated in both the linear elastic and nonlinear regimes. Thickness and density provided the best explanation for elastic heterogeneous behavior in most of the materials; local grammage was not the best descriptor for any material. The analyses used here were not able to provide a good explanation of the nonlinear behavior, which was attributed to the development of large shear strains within the materials as they neared failure. This work provides a methodology for additional heterogeneous behavior examinations.

Linear elastic diatomic multilattices: Three-dimensional constitutive modeling and solutions of the shift vector equation

Diatomic multilattices are congruences of simple lattices each made out of atoms of two possible chemical species. We here constitutively characterize, in three dimensions, diatomic multilattices for the geometrically and materially linear elastic regime. We give the most generic expression for the energy for ( n + 1 ) diatomic multilattices and characterize explicitly the tensors present in such an expression for all 122 two-color point groups. For the specific case of diatomic 2- and 3-lattices, we delineate how one can solve the shift vector equation in the static case. For cases where the unique solution of the shift vector in terms of the strain tensor is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem.

Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response

This research presents a novel approach to modeling fracture propagation using a discrete lattice element model with embedded strong discontinuities. The focus is on enhancing the linear elastic response within the model followed by propagation of fractures until total failure. To achieve this, a generalized beam lattice element with an embedded strong discontinuity based on the kinematics of a rigid-body spring model is formulated. The linear elastic regime is refined by correcting the stress tensor at nodes within the domain based on the internal forces present in lattice elements, which is achieved by introducing fictitious forces into the standard internal force vectors to predict the right elastic response of the model related to Poisson’s effect. Upon initiation of the first fractures, the procedure for the computation of the fictitious stress tensor is terminated, and the embedded strong discontinuities are activated in the lattice elements for obtaining an objective fracture and failure response. This transition ensures a shift from the elastic phase to the fracture propagation phase, enhancing the predictive capabilities in capturing the full fracture processes.

Transverse-inertia-induced wave dispersion in plate–shell chains with a variable Poisson ratio

One-dimensional shell chain structures can cause stress wave dispersion, and the dispersion ability is mainly affected by the apparent elastic Poisson ratio of the unit cell composing the chain. In this study, a novel plate–shell chain structure with a variable Poisson ratio is proposed by combining bent plates and arc shells (BPAS). The dispersive wave propagation features in the BPAS chains under striker impact are studied theoretically, numerically, and experimentally. An equivalent continuum model of the unit cell under compression in a linear-elastic deformation regime is established, which contains two material parameters, i.e., the apparent elastic modulus and the apparent elastic Poisson ratio. It is found that the unit cell can reach a relatively large Poisson ratio by adjusting its structural parameters. A continuum-based wave propagation model considering transverse inertia is developed, and an explicit analytical solution is obtained to represent the dispersion wave propagation behavior within a linear-elastic deformation regime. The model can reasonably predict the force and velocity evolution features of the chains and is verified by the numerical/experimental tests. The results show that the BPAS chain with a large Poisson ratio exhibits superior wave dispersion performances. The underlying mechanism is that more impact energy is dispersed by the transverse movement of the cells in the BPAS chain. Increasing the aspect ratio of the unit cell can improve the wave dispersion ability of the BPAS chains, achieving a high attenuation of the stress wave and is helpful for impact mitigation.

Predictive methods for initiation of delamination and intra-laminar damage in carbon fibre reinforced polymer laminates subject to impact

AbstractA comparison of the modelling methodologies to capture the damage onset and delamination initiation in Abaqus and LS-Dyna is presented. A quasi-isotropic carbon fibre reinforced polymer laminate is modelled under a low-energy impact scenario. Hashin, Puck and Cuntze criteria are implemented for assessing intra-laminar damage in Abaqus in the linear elastic regime without damage evolution, with Virtual Crack Closure Technique being used for inter-laminar failure. In LS-Dyna, the Chang-Chang criterion is used for the intra-lamina failure with damage evolution, whereas delamination is captured using cohesive zone model and the tiebreak contact algorithm. The implementations carried out by both finite element software result in a modelling work well set to analyse and predict the impact response at the initial stages of delamination and damage within the plies. The composite damage criteria used in both finite element codes overall predict stiffer results when compared with the experimental data, however, remain in close agreement with each other.

Mechanical criticality of fiber networks at a finite temperature.

At zero temperature, spring networks with connectivity below Maxwell's isostatic threshold undergo a mechanical phase transition from a floppy state at small strains to a rigid state for applied shear strain above a critical strain threshold. Disordered networks in the floppy mechanical regime can be stabilized by entropic effects at finite temperature. We develop a scaling theory for this mechanical phase transition at finite temperature, yielding relationships between various scaling exponents. Using Monte Carlo simulations, we verify these scaling relations and identify anomalous entropic elasticity with sublinear T dependence in the linear elastic regime. While our results are consistent with prior studies of phase behavior near the isostatic point, the present work also makes predictions relevant to the broad class of disordered thermal semiflexible polymer networks for which the connectivity generally lies far below the isostatic threshold.

Studying the mechanical behavior of a generic thermoplastic by means of a fast coarse-grained molecular dynamics model

Polymers play an emerging role in modern engineering applications due to their comparatively low cost, low density, and versatile manufacturing. The addition of nano-sized fillers further enhances the polymer’s properties but also induces a strong dependence on the resulting microstructure, particularly the matrix-filler interphase. Since an experimental characterization of this nano-sized interphase is extremely difficult, molecular dynamics (MD) simulations are used to study the effects at such small scales. However, MD’s high computational costs usually limit the scope of a mechanical characterization. Therefore, this study presents the methodology and tools to generate and analyze samples of an efficient generic thermoplastic model. In this first contribution, we focus on the neat polymer and introduce a versatile and numerically stable self-avoiding random walker with adjustable linearity of chain growth. Moreover, we verify our equilibration procedure by preparing samples in liquid and solid state which behave physically sound. Finally, we perform uniaxial tensile tests with a maximum strain of 10 % to evaluate the mechanical properties. In the liquid case, the polymer chains are sufficiently mobile, such that the tensile stresses fluctuate only around zero, while the solid exhibits an almost linear elastic regime followed by a nonlinear part. This contribution forms the basis for a thorough mechanical characterization of polymer nanocomposites which we will address in future studies. The methodology and tools introduced are not limited to our generic polymer, but applicable to many coarse-grained models.

Atomistic simulations of compressive response and deformation mechanisms of body-centered-cubic AlCrFeCoNi high-entropy alloys

Atomistic simulations are performed to study the compressive response and deformation mechanisms of the body-centered-cubic (BCC) AlCrFeCoNi high-entropy alloys (HEAs). Our computations show that the BCC HEA mainly experiences three typical deformation stages, i.e. linear elastic, plastic yielding, and yield flow regimes, and the predicted Young's modulus and yield strength agree well with available experiments. The plastic deformation mechanism of BCC HEA is dominated by the dislocation nucleation and slip accompanied by shear bands but depends little on the phase transformation of the HEAs. Compared with the face-centered-cubic counterpart, the BCC HEA is harder and more brittle due to the less dislocation activities in the alloy. In addition, the dependences of mechanical properties and deformation mechanisms on influential factors including temperature, strain rate, and Cr concentration are also analyzed in detail. These findings may deepen our understanding of the underlying deformation mechanisms in BCC HEAs.

Elasticity and Characteristic Stress Thresholds of Shale under Deep In Situ Geological Conditions.

The mechanical properties of shale are generally influenced by in situ geological conditions. However, the understanding of the effects of in situ geological conditions on the mechanical properties of shale is still immature. To address this problem, this paper provides insight into the elasticity and characteristic stress thresholds (i.e., the crack closure stress σcc, crack initiation stress σci, and crack damage stress σcd) of shales with differently oriented bedding planes under deep in situ geological conditions. To accurately determine the elastic parameters and crack closure and initiation thresholds, a new method-i.e., the bidirectional iterative approximation (BIA) method-which iteratively approaches the upper and lower limit stresses of the linear elastic stress-strain regime, was proposed. Several triaxial compression experiments were performed on Longmaxi shale samples under coupled in situ stress and temperature conditions reflecting depths of 2000 and 4000 m in the study area. The results showed that the peak deviatoric stress (σp) of shale samples with the same bedding plane orientation increases as depth increases from 2000 m to 4000 m. In addition, the elastic modulus of the shale studied is more influenced by bedding plane orientation than by burial depth. However, the Poisson's ratios of the studied shale samples are very similar, indicating that for the studied depth conditions, the Poisson's ratio is not influenced by the geological conditions and bedding plane orientation. For the shale samples with the two typical bedding plane orientations tested (i.e., perpendicular and parallel to the axial loading direction) under 2000 and 4000 m geological conditions, the ratio of crack closure stress to peak deviatoric stress (σcc/σp) ranges from 24.83% to 25.16%, and the ratio of crack initiation stress to peak deviatoric stress (σci/σp) ranges from 34.78% to 38.23%, indicating that the σcc/σp and σci/σp ratios do not change much, and are less affected by the bedding plane orientation and depth conditions studied. Furthermore, as the in situ depth increases from 2000 m to 4000 m, the increase in σcd is significantly greater than that of σcc and σci, indicating that σcd is more sensitive to changes in depth, and that the increase in depth has an obvious inhibitory effect on crack extension. The expected experimental results will provide the background for further constitutive modeling and numerical analysis of the shale gas reservoirs.

Carbon sequestration reservoir at Rock Springs Uplift, Wyoming, USA

The Rock Springs Uplift (RSU) located in southwest Wyoming USA as indicated in Figure 1 is one of several sites across the country to be characterized for suitability for CO2 sequestration. The RSU was identified for several reasons: proximity to a large CO2 emissions point source, the overall geometry of the RSU structure, and geologic formations recognized to have properties required for long term storage of CO2. Original rock cores of the Weber Sandstone formation and the dolomite facies of the Madison Limestone formation from the RSU were prepared into 25-mm diameter rock samples. These samples were vacuum-saturated to 100% saturation with synthetic brine, aged with brine for 800 hours at in-situ temperatures and initial confining pressure of 3.5 MPa, and aged with CO2-rich brine (400 hours first with brine and another 400 hours with compressed CO2-saturated brine) at in-situ temperatures and initial confining pressure of 3.5 MPa. During the aging process, the pore and confining pressure were increased in small steps, typically 0.69 to 1.38 MPa at a time using high precision syringe pumps (Teledyne ISCO 260D, Lincoln, NE, USA) until the test conditions were met without fluid flow. Laboratory hydrostatic and triaxial experiments were performed at in-situ reservoir conditions with the temperatures of 90°C for Weber Sandstone and and 93°C for Madison Limestone under three differential pressures of 6.9 MPa, 34.5 MPa and 55.2 MPa.
 Geomechanical results of the aged rock samples are presented and discussed. Under the triaxial compression, Weber Sandstone exhibited brittle failure strengths while Madison Limestone exhibited a brittle-ductile transition behavior. For the sandstone samples with similar initial porosity, the Young’s moduli increased and Poisson’s ratios decreased as the result of CO2 under the linear elastic regime. No relationship between stress (or strain) data and CO2 in the nonlinear plastic regime was observed. However, the confining pressure has a greater effect than CO2 on geomechanical behaviors. For the limestone samples, the change of elastic constants due to CO2 is more significant than that of sandstone samples. However, no consistent trend was observed on the limestone samples,
 and the effect of CO2 is not the dominant factor influencing the plastic properties of rocks. DH1c with the lowest initial porosity and differential pressure exhibits the highest peak strength, Young’s modulus and Poisson’s ratio, indicating the effect of initial porosity. Considering the effect of CO2 on Mohr failure envelopes, the decreasing trend of cohesion and increasing friction angle was observed in both sandstone and limestone samples. Table 1 summarizes the maximum deviatoric stress, elastic properties, and shear strength parameters of Weber Sandstone and Madison Limestone samples.

Efficiency of the grid analogy method in the analysis of solid slabs: study of a residential floor

Given the importance of structural analysis, slabs, particularly those made of solid concrete, have several models that allow analyzing them and obtaining the necessary efforts for their dimensioning, among them the grate analogy method. Since this is attractive for this purpose, since it has a simple approach without losing efficiency, this work proposes to study, using this approach, the mechanical behavior, in the linear elastic regime, of solid slabs of a residential floor. Thus, the general objective is to obtain the efforts and displacements of the referred pavement with this technique. For that, a computational code was developed in the Scilab programming language. In order to establish a comparison with other methods, the pavement was also analyzed using Kirchhoff's plate flexion theory and using shell finite elements with the aid of the SAP2000 software. It can be proved, through the study, the efficiency of the analysis of slabs through grid analogy. In addition, the importance of the study is verified considering the floor coupled to the beams, as well as taking into account the cracked state of the concrete, ensuring the achievement of more realistic results, which is important for design purposes.

Shape optimization of porous structures by phase-field modeling with strain energy density reduction

Triply periodic minimal surfaces (TPMS) are cell structures defined by mathematical formulas. Due to their special structure and properties, such as a high surface-to-volume ratio and good mechanical properties, these structures are used in a wide range of applications, e.g., for implants or heat exchangers. Given the diverse applicability of these structures, both scientific and industrial interest resides in the tailored shape optimization, where structural integrity is preserved on the one hand, while stiffness is selectively enhanced on the other.Therefore, in the following investigation, three TPMS unit cells are shape-optimized under mechanical loading, in the linear-elastic regime, by minimizing the strain energy density, using the phase-field method. The novelty lies in the shape optimization of TPMS structures with respect to the effective stiffness under mechanical load which is addressed with a phase-field method. It is shown that with the present optimization approach, an increase in the effective Young’s modulus of up to 55% can be achieved for specific TPMS structures.

Elastocaloric effect of shape memory polymers in elastic response regime

Solid-state cooling/heating technology based on the elastocaloric effect is one of the promising alternatives to vapor compression systems. Large elastocaloric temperature modulation is often generated through the non-linear strain-induced structural transition by applying large strain and/or stress to ferroelastic materials. Recently, an unconventional approach to expand the application possibilities of the elastocaloric effect was demonstrated by processing elastocaloric materials into kirigami structures, which was inspired by the art of paper cutting. Using this approach, only a small stretch of processed conventional plastics can locally provide more efficient performance of elastocaloric temperature modulation than that of ferroelastic materials. To further improve such a unique functionality, it is necessary to find plastic or polymeric materials showing large elastocaloric effects in the linear elastic response regime that can be driven by a MPa-order weak stress application, where the non-linear structural transition is irrelevant. In this work, by means of a recently developed measurement technique for the elastocaloric effect based on the lock-in thermography, we found that shape memory polymers (SMPs) show prominent performance for elastocaloric temperature modulation that is larger than conventional plastics. SMPs enable the control of crystallinity by changing the cross-linking agents, melting temperature by changing the degree of polymerization, and orientation of the polymer chain segment by the shape memory effect. By utilizing the unique properties of SMPs, we manipulated their elastocaloric performance. The experimental results reported here will highlight the potential of smart polymers for flexible and durable elastocaloric applications.

A Problem-Independent Machine Learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures

Structural analysis for structures composed of highly heterogeneous materials which often involves the solution of large-scale linear algebraic equations is very time-consuming even within the linear elastic regime. Furthermore, the tremendous computational cost of iterative large-scale finite element analysis also prevents the widespread use of topology optimization as a powerful design tool especially when the desired design resolution is very high. In order to break the bottleneck hindering the efficient solutions of large-scale structural analysis and design optimization problems, in the present article, a general machine learning (ML) enhanced substructure-based framework is proposed. The essential idea is resorting to the classical substructure-based finite element analysis approach and establishing an implicit mapping between the parameters characterizing the material distribution within a substructure and the corresponding condensed stiffness matrix/numerical shape functions through offline trained deep neural networks. In contrast to most of the existing ML enhanced approaches, the proposed framework is truly independent of the forms of structural geometry, boundary condition and external load, and can be applied to solve various boundary value problems governing by the same type of partial differential equation once the offline training is completed. Armed with modern artificial intelligence techniques, the proposed approach in some sense revives the classical substructure approach in finite element analysis. Compared with the traditional paradigm, it can achieve 104–105 times solution efficiency for tested large-scale examples with satisfactory accuracy. The effectiveness of the approach was also validated for the non-adjoint topology optimization problems, i.e., 3D compliant mechanism design. Finally, to demonstrate the proposed approach’s capability in dealing with extremely large-scale three-dimensional problems, a three-dimensional topology optimization problem with about 109 design variables and 3×109 degrees of freedom (Dofs) is solved on a laptop without resorting to any parallel computing techniques.

Alginate-Chitosan Microgel Particles, Water–Oil Interfacial Layers, and Emulsion Stabilization

In this work, alginate-chitosan microgel particles were formed at different pH levels with the aim of using them as viscoelastic interfacial layers, which confer emulsion stability to food systems. The particles’ size and structural characteristics were determined using laser diffraction, confocal laser microscopy (CLSM), and time-domain nuclear magnetic resonance (TD-NMR). The pH affected the microgel characteristics, with larger particles formed at lower pH levels. T2 relaxation measurements with TD-NMR did not reveal differences in the mobility within the particles for the different pH levels, which could have been related to the more or less swollen structure. The rate of adsorption of the particles at the sunflower oil–water interface differed between particles formed at different pH levels, but the equilibrium interfacial tension of all systems was similar. Higher interfacial dilatational viscoelasticity was obtained for the systems at lower pH (3, 4, 5), with G’ reaching 13.6 mN/m (0.1 Hz) at pH 3. The interfacial rheological regime transitioned from a linear elastic regime at lower pH to a linear but more viscoelastic one at higher pH. The thicker, highly elastic interfacial layer at low pH, in combination with the higher charges expected at lower pH, was related to its performance during emulsification and the performance of the emulsion during storage. As revealed by laser diffraction and CLSM, the droplet sizes of emulsions formed at pH 6 and 7 were significantly larger and increased in size during 1 week of storage. CLSM examination of the emulsions revealed bridging flocculation with the higher pH. Nevertheless, all emulsions formed with microgel systems presented macroscopic volumetric stability for periods exceeding 1 week at 25 °C. A potential application of the present systems could be in the formation of stable, low-fat dressings without the addition of any emulsifier, allowing, at the same time, the release of the bioactive compounds for which such particles are known.

Investigation of quasistatic and dynamic mechanical properties of thermoplastic polyurethane magnetorheological elastomers

Thermoplastic magnetorheological elastomers (MRE) are soft magnetic materials with multi stimuli response. As the name suggests, thermoplastic MREs are formed by the suspension of magnetic particles in a thermoplastic elastomer matrix, whose rheological properties can be controlled by temperature. The composite itself can thus be controlled by temperature and magnetic field. In this article, we present the method and results of quasistatic and dynamic mechanical properties of thermoplastic polyurethane MREs (TPU-MREs) based on the observations made on the influence of hardener and plasticizer content from the composition point of view and the directional interdependence of strain and magnetic field on the magnetorheological (MR) effect. Quasistatic tensile and torsion tests followed by dynamic oscillation tests conducted within the linear elastic regime at three different temperatures with and without externally applied magnetic field are performed on the TPU-MRE samples. The tests are conducted with an inhouse developed test setup adapted to a commercial rheometer. The obtained results demonstrate a strong correlation to the change in moduli of the TPU-MRE with respect to change in composition and temperature. From the results obtained, one can conclude that samples with lower hardener content with plasticizer have the highest relative MR effect.