Linear Stress-strain Relationship Research Articles

Mechanical behaviors and rupture processes of a typical granitic stratum

Granitic veins (GVs) have a significant influence on the mechanical responses of tunnels excavated in granitic strata. Distinguishing the mechanical properties of host granites (HGs), GVs and vein-granite interfaces (VGIs) is critical. For this, this paper analyzed the mechanical behaviors and rupture processes of typical HG, GV, and VGI samples under uniaxial compression condition. For the rocks studied, although the linear axial stress‒strain relation can be identified and the deformation modulus can be determined, the transverse deformation developed nonlinearly with axial stress. As a result, the instantaneous Poisson's ratio increases continuously and may even exceed 0.5, making it extremely difficult to accurately determine the Poisson's ratio. In addition, the studied GV samples were found to be significantly brittle, indicating that large-scale GVs cannot be ignored when assessing rockburst hazards in granitic strata with brittle GVs. In terms of the rupture process, the HG and GV samples were gradually damaged by the formation of small-scale cracks and then ruptured by large cracks formed from small-scale cracks, whereas the VGI samples ruptured along large cracks with significant energy release. By examining the characteristic stress thresholds of these three granites, it is noted that the crack closure stress σcc exceeds both the crack initiation stress σci and the crack damage stress σcd for the HG and VGI samples. The transverse damage to a tested sample appears to be significantly greater than the axial damage, which is essentially related to the rock grain size and grain size distribution.

P-M INTERACTION DIAGRAMS FOR REINFORCED CONCRETE WALLS RETROFITTED WITH EXTERNALLY BONDED CARBON-FIBER REINFORCED POLYMER

This research develops analytical axial load-bending moment (P-M) interaction diagrams for reinforced concrete (RC) walls retrofitted with carbon fiber reinforced polymer (CFRP). A 12-inch thick, 1-foot strip of RC wall section is considered with varying reinforcement ratios, varying axial compressive loading, and varying number of CFRP layers. The CFRP material is treated as externally bonded onto the tension face of the RC wall to investigate its impact on the flexural capacity of the section. Each P-M interaction diagram was generated considering a discretized compression zone and by satisfying principles of static equilibrium and strain compatibility. An elastic-plastic steel stress-strain relationship is used for Grade 60 reinforcement; a uniaxial nonlinear compressive stress-strain relationship is used for concrete; and a linear stress-strain relationship is used for CFRP in tension. The failure modes considered are steel yielding, concrete crushing, and CFRP debonding. The computed P-M interaction diagrams are normalized for their general use in the retrofit of existing RC walls using CFRP. Results show that as the axial compressive force on the RC wall increases, the less effective CFRP is. CFRP is more effective in sections with beam-like behavior, where the reinforcement ratio tends to be the smallest.

Direct measurement of the effective properties of an additively manufactured titanium octet truss unit cell using high energy X-ray diffraction

Additively manufactured lattice structures offer a high specific strength-to-density ratio in comparison to conventional, fully dense parts. The small feature size and intricate geometry of lattice structures make it challenging to directly characterize their mechanical properties. This study presents direct measurement of the elastic mechanical properties of an additively manufactured titanium alloy octet truss unit cell. Strut stress and strain were measured using high-energy X-ray diffraction during quasi-static compressive loading. The crystallographic stress-strain relationship was converted into the stress-strain relationship of the octet truss lattice unit cell using the known symmetry of the struts. The sample studied herein was best fit by an orthorhombic linear elastic stress-strain relationship. This technique shows promise for studying additively manufactured lattice structures at length scales not previously reported.

From a distance: Shuttleworth revisited.

The Shuttleworth equation: a linear stress-strain relation ubiquitously used in modeling the behavior of soft surfaces. Its validity in the realm of materials subject to large deformation is a topic of current debate. Here, we allow for large deformation by deriving the constitutive behavior of the surface from the general framework of finite kinematics. We distinguish cases of finite and infinitesimal surface relaxation preceding an infinitesimal applied deformation. The Shuttleworth equation identifies as the Cauchy stress measure in the fully linearized setting. We show that both in finite and linearized cases, measured elastic constants depend on the utilized stress measure. In addition, we discuss the physical implications of our results and analyze the impact of surface relaxation on the estimation of surface elastic moduli in the light of two different test cases.

Practical evaluation of high-strength concrete corbels reinforced with GFRP bent bars

Corbels, designed primarily as discontinuity-region structural elements, are often used as components of bridges and buildings to support beams and girders. There has been extensive research on steel-reinforced concrete (RC) corbels; however, there is a noticeable gap in research to investigate the behavior of glass fiber-reinforced polymer (GFRP)-RC corbels. Considering the linear stress–strain relationship up to failure of GFRP, the use of high-strength concrete (HSC) in lieu of the normal-strength concrete (NSC) in corbels would reduce their sizes considerably; yet it will aggravate their brittle nature. In this study, nine full-scale concrete corbels were constructed and tested until failure. One reference specimen was reinforced with steel bars, while the other eight were GFRP-RC corbels. The main test parameters were the type of concrete (HSC or NSC), the shear span-to-depth ratio, and the total reinforcement ratio of the main and horizontal bars. The tests showed that the principal mode of failure was concrete strut crushing for specimens with a shorter shear-span-to-depth ratio. The steel-RC corbels failed by tension tie yielding, which showed large ductility. The Canadian standards for FRP-RC building structures predicted a conservative mean capacity with Ppre/Pexp ratio of 0.59 compared to the American code and Eurocode, which overestimated the capacities with a ratio of 1.18 and 1.25, respectively.

Comment on: “The structural, elastic, electronic and optical properties of MgCu under pressure: A first-principles study”

Rahman et al. [International Journal of Mod. Phys. B 30, 1650199 (2016)] found that MgCu intermetallic compound with CsCl-type (B2) structure becomes unstable under pressure at about 15[Formula: see text]GPa. This prediction has been made using an elastic instability condition for second-order elastic constants given by Born. They determined the elastic constants using Hooke’s law for the linear stress–strain relationship which does not hold good at high pressures. It has been shown here that the values of elastic constants for MgCu at high pressures determined by Rahman et al. are not consistent with the values of bulk modulus computed from the Birch–Murnaghan equation of state (BM-EOS). We have used the Tallon formulation based on the thermodynamics of elastic deformation to estimate elastic constants which yields results for the bulk modulus of MgCu at different pressures in good agreement with those determined from the BM-EOS. It has been found that MgCu remains stable up to about 20 GPa which is significantly higher than the limit of elastic stability predicted by Rahman et al.

A comment on the paper “The structural, elastic, electronic and optical properties of MgCu under pressure: A first-principles study”

Rahman et al. [International Journal of Mod. Phys. B 30, 1650199 (2016)] found that MgCu intermetallic compound with CsCl-type (B2) structure becomes unstable under pressure at about 15[Formula: see text]GPa. This prediction has been made using an elastic instability condition for second-order elastic constants given by Born. They determined the elastic constants using Hooke’s law for the linear stress–strain relationship which does not hold good at high pressures. It has been shown here that the values of elastic constants for MgCu at high pressures determined by Rahman et al. are not consistent with the values of bulk modulus computed from the Birch–Murnaghan equation of state (BM-EOS). We have used the Tallon formulation based on the thermodynamics of elastic deformation to estimate elastic constants which yields results for the bulk modulus of MgCu at different pressures in good agreement with those determined from the BM-EOS. It has been found that MgCu remains stable up to about 20 GPa which is significantly higher than the limit of elastic stability predicted by Rahman et al.

Numerical Modelling of the Compressive and Tensile Behavior of Ultra-High Performance Concrete in Beams

INTRODUCTION Ultra-high performance concrete (UHPC) differs in its structural behavior from conventional concrete upon loading due to its high compressive and tensile strength, stiffness, toughness, and durability. Therefore, UHPC needs an appropriate constitutive model to simulate its mechanical properties in finite element analysis (FEA). METHOD In this study, numerical models were developed to trace the structural behavior of UHPC beams upon loading since beam behavior depends on the constituents' response to compression and tension. New numerical models were formulated to display the stress-strain relationships of UHPC in compression and tension by adopting a new methodology that depended on actual results. The compressive stress-strain relationship consisted of two portions; the ascending one for elastic and strain hardening up to compressive strength and a descending curve for the strain-softening until reaching a strain of 0.0062. RESULT A linear tensile stress-strain relation was applied for the elastic tensile behavior up to tensile strength. Then, a tri-linear relationship was applied for the stiffness degradation and crack propagation upon debonding fibers from the concrete matrix until fracture. These numerical models were used in Abaqus software to simulate the UHPC beam behavior. CONCLUSION The developed models were verified and approved for beams' behavior upon loading in flexure and shear. The results indicated that the models could predict the UHPC beams' response throughout the entire loading range from the beginning until failure. The verification included bear capacity, deflection, crack pattern, and stress distribution.

Uniaxial tensile response and tensile constitutive model of ultra-high performance concrete containing coarse aggregate (CA-UHPC)

To establish the tensile constitutive model of ultra-high performance concrete containing coarse aggregate (CA-UHPC), monotonic and cyclic uniaxial tensile tests for CA-UHPC with fiber volume fractions of 2.5% and 2.0% were conducted. Test results showed that CA-UHPC exhibits approximately linear stress-strain relation up to the tensile strength, and tensile softening response composed of the smeared- and localized-cracking stages, regardless of the tested fiber contents. Based on the monotonic test data, the tensile stress-crack opening model of CA-UHPC was established, and the model was further simplified into tri-linear relation. Based on the cyclic test results, tensile damage evolution laws according to the strain equivalence principle and the energy equivalence principle were developed, respectively. Finally, the proposed tensile constitutive model and the calibrated tensile damage evolution laws were demonstrated to effectively predict the mechanical response of CA-UHPC members under both monotonic tension and cyclic tension through numerical simulations.

Non-Hookean large elastic deformation in bulk crystalline metals

Crystalline metals can have large theoretical elastic strain limits. However, a macroscopic block of conventional crystalline metals practically suffers a very limited elastic deformation of <0.5% with a linear stress–strain relationship obeying Hooke’s law. Here, we report on the experimental observation of a large tensile elastic deformation with an elastic strain of >4.3% in a Cu-based single crystalline alloy at its bulk scale at room temperature. The large macroscopic elastic strain that originates from the reversible lattice strain of a single phase is demonstrated by in situ microstructure and neutron diffraction observations. Furthermore, the elastic reversible deformation, which is nonhysteretic and quasilinear, is associated with a pronounced elastic softening phenomenon. The increase in the stress gives rise to a reduced Young’s modulus, unlike the traditional Hooke’s law behaviour. The experimental discovery of a non-Hookean large elastic deformation offers the potential for the development of bulk crystalline metals as high-performance mechanical springs or for new applications via “elastic strain engineering.”

Aeroelastic method to investigate nonlinear elastic wing structures

Stiffness directions of wing structures are already part of the optimisation in aircraft design. Aircraft like the A350 XWB and the Boeing 787 mainly consist of such composite material, whose stiffness directions can be optimised. To proceed with this stiffness optimisation, the aim of this work is to modify and optimise also the linear stress-strain relation. On that account, the Hooke’s law is exchanged by a multi-linear formulation to analyse any nonlinear elastic structural technology on wing structures. The wing structures, which are used to investigate the nonlinear behaviour, are deduced from a mid-range and a long-range aircraft configuration. These wings are analysed with an extended beam method and coupled with a VLM solution to calculate the aeroelastical loading. The proposed beam method is capable of analysing any multi-linear wing structure technology. A degressive structural behaviour shows up a good potential to reduce the bending moment which is one of the main drivers of the structural weight.

Limitations of the St. Venant–Kirchhoff material model in large strain regimes

The St. Venant–Kirchhoff law is a widely used constitutive relation in engineering applications. However, its limitation to small strains is frequently mentioned in the literature but rarely elaborated on. Therefore, we present a comprehensive study discussing the St. Venant–Kirchhoff law and its shortcomings in large stain regimes, focusing on compressive loading based on one-dimensional stress and strain states. The linear stress–strain relation in combination with the associated Green–Lagrange strain yields a nonphysical structural behavior for large compressive strains, which is reflected in instabilities and nonphysical softening. Common misconceptions concerning the St. Venant–Kirchhoff law and alternative constitutive models are presented and evaluated based on a two-bar snap-through problem.

3D printing interface-modified PDMS/MXene nanocomposites for stretchable conductors

Additive manufacturing has rapidly evolved over recent years with the advent of polymer inks and those inks containing novel nanomaterials. The compatibility of polymer inks with nanomaterial inks remains a great challenge. Simple yet effective methods for interface improvement are highly sought-after to significantly enhance the functional and mechanical properties of printed polymer nanocomposites. In this study, we developed and modified a Ti3C2 MXene ink with a siloxane surfactant to provide compatibility with a polydimethylsiloxane (PDMS) matrix. The rheology of all the inks was investigated with parameters such as complex modulus and viscosity, confirming a self-supporting ink behaviour, whilst Fourier-transform infrared spectroscopy exposed the inks’ reaction mechanisms. The modified MXene nanosheets have displayed strong interactions with PDMS over a wide strain amplitude. An electrical conductivity of 6.14 × 10−2 S cm−1 was recorded for a stretchable nanocomposite conductor containing the modified MXene ink. The nanocomposite revealed a nearly linear stress-strain relationship and a maximum stress of 0.25 MPa. Within 5% strain, the relative resistance change remained below 35% for up to 100 cycles, suggesting high flexibility, conductivity and mechanical resilience. This study creates a pathway for 3D printing conductive polymer/nanomaterial inks for multifunctional applications such as stretchable electronics and sensors.

Machine-learning for turbulence and heat-flux model development: A review of challenges associated with distinct physical phenomena and progress to date

This review paper surveys some of the progress made to date in the use of machine learning (ML) for turbulence and heat transfer modeling. We start by identifying the challenges that various flow phenomena pose to closure modeling. These range from the misalignment between the turbulence stress tensor with the strain rates, that cannot be captured by linear stress–strain relationships, to non-constant turbulence Prandtl numbers, the coupling of multiple closures and deterministic unsteadiness, to name a few. We then introduce several machine learning concepts and frameworks for turbulence stress and heat-flux closure modeling, with a focus on model consistency. Various examples are then provided where applications of ML methods have to some degree succeeded at addressing the identified modeling challenges. We close by outlining some of the remaining challenges, in particular around the generalizability of ML-based models. Overall, further advances in ML techniques and availability of high-quality data sets will see this exciting research direction thrive and promises to lead to improved models with a wider range of applicability.

Rational design of hyperelastic semi-linear material and its application to elastic wave control

Incremental elastic wave in a hyperelastic semi-linear (SL) material can be purposely controlled by pre-deformation with help of hyperelastic transformation (HT) theory. However, this material model, assuming a linear stress-strain relation for large deformation, has neither been realized with naturally occurring materials nor through microstructure design. By balancing the stiffness of axial and rotational springs in a hexagonal mass-spring lattice, we propose a lattice prototype to realize the SL material. The performance of the model is systematically verified from static strain energy to wave property on different finitely deformed configurations. Based on this SL lattice, an elastic wave cloak is designed and validated through numerical simulation. This work provides the first two-dimensional (2D) microstructure model for realizing SL material and paves the way for elastic wave control with HT method.

A Nonlinear Elastic Model for Compressible Aluminum Alloys with Finite Element Implementation.

In this paper, a three-dimensional model of nonlinear elastic material is proposed. The model is formulated in the framework of Green elasticity, which is based on the specific elastic energy potential. Equivalently, this model can be associated to the deformation theory of plasticity. The constitutive relationship, derived from the assumed specific energy, divides the material’s behavior into two stages: the first one starts with an initial almost linear stress–strain relation which, for higher strain, smoothly turns into the second stage of hardening. The proposed relation mimics the experimentally observed response of ductile metals, aluminum alloys in particular. In contrast to the classic deformation theory of plasticity or the plastic flow theory, the presented model can describe metal compressibility in both stages of behavior. The constitutive relationship is non-reversible expressing stress as a function of strain. Special attention is given to the calibration process, in which a one-dimensional analog of the three-dimensional model is used. Various options of calibration based on uniaxial stress test are extensively discussed. A finite element code is written and verified in order to validate the model. Solutions of selected problems, obtained via ABAQUS, confirm the correctness of the model and its usefulness in numerical simulations, especially for buckling.

Dynamic Response and Long-Term Settlement of a Compacted Loess Embankment under Moving Train Loading

There have been many railway construction projects in the loess region of China. Embankment is typically required for rail projects in these regions, since the railway basement is restricted by longitudinal slope requirements. However, there has been little study of the dynamic response of compacted loess embankment under moving train loading. The 2.5D finite element method was adopted to investigate this process and characterize the effects of train speed, height of embankment, and axle weight on the dynamic behavior of subgrade. A rectangular core zone of subgrade was determined, and a prediction model was established to evaluate the long-term settlement of embankment generated by moving train loading. The results showed that embankment height had negligible influence on the variation of dynamic stress. Decays of stress amplitude in both the vertical and horizontal directions slowed with increased train speed. Additionally, the dynamic stress increased linearly with the increase in axle weight due to the linear stress-strain relationship of soil. In the practical speed range (≤ 100 m/s), the dynamic influence depth increased with increasing speed in a range of 3–6 m. A core zone depth of 6 m reflects the effects of moving train loading, with a width of 4 m. For practical conditions (v ≤ 100 m/s), only slight settlement of embankment was observed (≤ 6 mm). However, it is difficult to achieve the same physical parameters used in the experiments (moisture content and compaction degree) in engineering practice. Further work should explore long-term dynamic settlement with relation to the degree of compaction of the embankment.

A novel approach to the complete stress strain curve for plastically damaged concrete under monotonic and cyclic loads

In this paper, new analytical formulations, which can be easily applied to normal and high-strength concretes under monotonic compressive and direct tensile loads, are proposed. Moreover, plastic damage model for concrete under uniaxial cyclic loading is also employed by introducing a simple damage function. The majority of published material models focuses on a certain type of concrete based on their testing results, which can be hardly applied to other types of concrete such as normal or high strength concrete. This paper presents a novel approach that can be applied to different types of concrete and highlights that the differences among testing results may come from the variations of strain at the peak stress. The damage phenomenon of concrete is simplified by a non-linear degradation of elastic modulus function, which in turn creates a linear stress-strain relation under cyclic loading. The damage function can be easily and quickly used to calibrate concrete properties for plastic damage model, which is very useful for industrial applications. Finally, the accuracy and pre-eminence of the proposed damage model are verified through comparison with experimental data and analytical solutions.

Numerical study on rheology of two-dimensional dry foam

We study the mechanical response of two-dimensional aqueous foams subjected to an oscillating shear strain using numerical simulations based on the immersed boundary method. Foams have unique rheological properties ranging from solid-like to fluid-like. They have an elastic property under a small strain with a linear stress-strain relationship. As the strain increases, the foams undergo topological rearrangements with a sudden release of energy and stress. Then the energy-strain and stress-strain curves exhibit hysteresis behaviors under the oscillating shear strain, and the macroscopic response of foams changes from a viscoelastic solid to a viscoelastic fluid. These wide-ranging dynamical responses of foams result from avalanches of topological rearrangements which are concentrated in a single localized region. This is called a shear localization, and we investigate the effect of the topological rearrangements and the shear localization on the rheology of foams.

A partitioned solution approach for the fluid–structure interaction of thin-walled structures and high-Reynolds number flows using RANS and hybrid RANS–LES turbulence models

In this work the partitioned solution approach for the fluid–structure interaction (FSI) of thin-walled structures and high-Reynolds number (Re) flows modeled using Reynolds–Averaged Navier–Stokes (RANS) and hybrid Reynolds–Averaged Navier–Stokes – Large Eddy Simulation (RANS–LES) turbulence models are described. The advanced turbulence modeling is needed to capture very complex fluid phenomena which triggers instabilities of thin–walled structures present in supersonic flow regimes. The finite element (FE) updated Lagrangian formulation (ULF) for the nonlinear elastic solids is used to predict its dynamical behavior. The main contribution addresses to the linear stress–strain relation Laplacian members, which are solved implicitly, on that way decreasing required memory resources and improving solution stability in the same time. The structures of the interest include the vast variety of membranes, curved shells and plates. The instabilities encountering these structures include limit cycle oscillations (LCO), flutter and buckling of the panels. The phenomena appear in everyday engineering practice and a need for the powerful tools to handle such problems is a common goal. Utilization of the unstructured non-regular meshes allows the precise distribution of computational nodes at the physical boundaries of the fluid and solid domains. It is naturally allowing application of the common approach for the fluid–solid interface coupling, as well as classical data interpolation schemes between fluid and solid on the FSI interface. High-Re flows, both 2D (benchmark) and 3D turbulent FSI case are chosen for the validation. Two numerical methods are coupled via a moving boundary treatment, in a staggered way. The proposed coupling method showed a good agreement with the reference test cases. The current FSI framework is developed to serve as a tool for the liquid rocket engine development.