Effective Stiffness Research Articles

Can wormlike micelle stiffness be estimated from zero‐shear viscosity? Experimental investigation with a model system and specific salt interactions

AbstractThe microstructure of wormlike micelles (WLM) directly affects the rheological properties of their solutions. Investigating the structure–property relationships of WLM has long been a popular topic for researchers who have developed theoretical and empirical models to describe their viscoelastic behavior. All these models rely on a collection of characteristic “length” parameters that are often difficult to estimate using relatively simple rheological tests. In particular, the micelle stiffness, as described by the persistence length, can be difficult to measure experimentally, while being very impactful on the bulk rheology of WLM solutions. Here, an array of inorganic salts (NaCl, LiCl, MgCl2, NaBr, NaI, and Na2SO4) have been used to induce wormlike micelle formation in an aqueous solution of the surfactant sodium lauryl ether sulfate. Ion dissociation/association with the surfactant head groups and the hydrogen bond network of the water alters the effective stiffness of the micelles, allowing for an estimation of the stiffness using three different methods: (1) small angle neutron scattering (SANS) measurements, (2) oscillatory rheological measurements, and (3) steady rheological measurements with a thermodynamic packing parameter model. Each of these methods are then compared and shown to be consistent with each other for the micelle solutions tested. The consistency of the results across all the measurements suggests that the approach used in this study, which estimates micelle parameters using steady shear rheology and a thermodynamic model, could provide a simpler and more accessible method for estimating micelle parameters in a wide range of surfactant systems.

Effective Stiffness and Seismic Response Modification Models Recommended for Cantilever Circular Columns of RC Bridges, First Part: Serviceability Limit State

This paper’s main aim was to explain the process of characterising the structural over-strength factor (R), seismic behaviour factor (Q), and the effective elastic stiffness, Keff, of cantilever-reinforced concrete (RC) urban bridge columns with solid circular cross-sections for use in seismic design under the Serviceability Limit State (SLS). Similarly, mathematical models have been proposed to determine the average values of effective stiffness and seismic response modification factors suggested for cantilever-reinforced concrete bridge columns at SLS. This is because multiple design codes stipulate that cantilever RC bridge columns must meet the SLS requirements. Therefore, to comply, the lateral displacement ductility demand must not exceed unity after a moderate or small earthquake. While the behaviour of the materials remains in the elastic range, this performance criterion can be conservative. If the materials undergo small deformations, the slight damage can be quickly repaired to meet the SLS.

Performance and strength analysis of shear walls with vertical steel encased profiles under tensile-shear load

This study investigates the behavior of composite steel reinforced concrete (SRC) walls under tension-shear (TS) coupled stress state during earthquakes. Quasi-static tests on five SRC shear walls with a shear span ratio of 1.42 were conducted. The experimental results demonstrated that the SRC shear wall retained its high shear capacity and good ductility in shear failure mode. The stiffness analysis revealed that the concrete provided most axial stiffness before cracking but gradually withdrew from the work as the cracks developed and extended rapidly in the tension loading phase. And the effective lateral stiffness was approximately 10%–15% of the elastic lateral stiffness. Parametric analysis was conducted on the shear capacity, including the axial tensile ratio, steel profiles content, concrete strength grade, and horizontal distributed bar diameter. Additionally, the predictive formulas for the shear capacity in the codes of various countries were compared. It was found that the European and American code inadequately predict the shear capacity of specimens, and the Chinese code has higher safety. These findings can provide references for the practical engineering applications of SRC shear walls in earthquake-prone regions.

Effective Stiffness and Seismic Response Modification Models Recommended for Cantilever Circular Columns of RC Bridges, Second Part: Collapse Prevention Limit State

This paper aims to assess the structural overstrength (R), seismic behaviour factor (Q), and effective inertia factor (keff), of which the effective elastic stiffness (Keff) is a function, i.e., Keff (keff), of reinforced concrete (RC) cantilever columns with solid circular sections in urban bridges. The R and Q factors proposed in the design codes have been based mainly on an empirical basis. However, some studies suggest that, in the case of RC bridge columns, both factors are determined by a range of geometric and structural parameters. These results indicate that equations incorporating the main geometric and structural parameters of the columns, rather than the use of fixed values, are the appropriate method for estimating the values of R, Q and keff for the columns. Therefore, the results confirm the relation between R, Q and keff with the mechanical properties of the concrete, the aspect ratio length–depth, the axial load–strength ratio of a column, as well as its longitudinal reinforcement steel ratio. Finally, the research proposes models for estimating R, Q and keff that are recommended for the collapse prevention limit state of cantilevered circular columns of RC bridges.

Porosity-dependent stability analysis of bio-inspired cellular nanocomposite shells

This paper presents an innovative numerical model to investigate buckling behaviour of bio-inspired continuously graded porous (CGP) nanocomposite cylindrical shells. It is postulated that the shell subjected to combined lateral pressure and axial compressive load is constructed from metal foams with closed-cell structures that possess graded internal pores, which exhibit three types of continuously graded porosity profiles based on a power-law distribution. A scaling relation for the effective Young’s modulus of the cellular structure determined by a variational finite element method (FEM) is used. The effective constitutive law of an elastic isotropic metal matrix containing distributed elastic carbon-nanotubes (CNTs) is estimated in consideration of the impact of CNTs agglomeration using a continuum model based on the Eshelby–Mori–Tanaka (EMT) approach. In contrast to conventional approaches, the study employs Euler–Bernoulli beams to model stiffeners within the CGP shells. This choice allows for a more realistic representation of stiffener effects, as opposed to the prevalent approach of uniform smearing across the shell’s surface. The equilibrium equations of the CGP shell, based on the Reddy higher-order shell theory (RHST), is obtained through the application of the Euler equation. Subsequently, the equations for stability are obtained through the utilization of the variational method. This study emphasizes the effects of geometrical parameters, porosity variability, and distribution of CNTs on the buckling performance of the CGP shells. The intricate interplay between CNTs and porosity distributions critically influences the stability behaviour of CGP shells. CNTs arrangement remarkably impacts buckling behaviour at higher length-to-mean radius ratios, while symmetric porosity near the mid-surface significantly enhances stiffness. These findings provide valuable insights for designing closed-cell cellular stiffened shells with optimal porosity to enhance stability.

The elasticity-conductivity connection for materials with cracks of arbitrary shapes and orientation distributions

The cross-property connection between the effective elastic and conductive properties of a solid with cracks has been first given by Bristow (1960) for randomly oriented cracks, assuming that cracks have circular (penny) shapes. However, cracks in materials typically have “irregular” shapes and this factor, generally, has strong effect on the said properties (the strong stiffening effect produced by contacts between crack faces is an example). In addition, shapes of 3D cracks (for example, the statistics of the mentioned contacts) may be largely unknown. These factors make it difficult to estimate the effect of cracks on the elastic properties. We show that flat cracks of arbitrary shapes and orientation distributions produce similar effects on the elastic and conductive properties, so that changes of stiffnesses can be monitored by changes in electric conductivities. Extension to transversely isotropic materials is given. Possible extensions to non-flat crack shapes are discussed.

Nanoindentation response of small-volume piezoelectric structures and multi-layered composites: modeling the effect of surrounding materials

With piezoelectric small-volume composites gaining importance in smart device applications and nanoindentation being recognized as a versatile method for assessing the properties of layer materials, the present study is focused on the indentation response of the small-volume piezoelectric structures multi-layered composites. In particular, the effects of the nature of the substrate and surrounding materials, on the indentation response of piezoelectric nanocomposites, such as nanoislands, nanowires, and multi-layered composites are investigated. By developing three-dimensional finite element modeling, the complex interaction between the fundamental elastic, piezoelectric and dielectric properties of the piezoelectric materials and the elastic, plastic and electrically conducting or insulating properties of the surrounding materials, on the indentation response of the layered composites is analyzed. It is found that: (i) a substrate material that is elastically stiffer enhances the mechanical indentation stiffness and the electric indentation stiffness while plastic deformation in the substrate causes a reduction in the mechanical and electrical indentation stiffness; (ii) the effective piezoelectric and mechanical indentation stiffnesses of piezoelectric multi-layered composites are bounded by the corresponding characteristics of the bulk material counterparts from which the individual layers are constructed; (iii) electrically conducting surrounding materials produce a softening effect while insulating materials enhance the electrical indentation stiffness resulting in more charges being accumulated during the indentation process.

The structural behaviour of moulded gratings made of glass fibre reinforced plastics

Moulded GFRP gratings provide several advantages over conventional metallic gratings and have steadily increased in popularity, even though their use within the building sector is often accompanied by costly approvals. Those are frequently required due to a lack of adequate design rules. To push the development of such rules, a comprehensive study on the load bearing behaviour of GFRP gratings is conducted and briefly presented herein.Two failure types, shear and bending failure, are identified in three-point bending tests and further studied by numerical analyses. Subroutines are developed and integrated into a commercial FE software to account for nonlinear material behaviour related to microdamage and macroscopic fracture. Thus, relevant stress and damage states at the scale of a lamina are clearly identified. Based on these investigations, a mutual interaction of bending and shear stresses on the load-bearing capacity can be excluded.Additional tests are carried out on full-size grating panels and recalculated with a structural frame analysis program. The research suggests that a warping restraint shall be assumed in the crossings between orthogonal bearing bars which significantly increases the effective torsional stiffness. It can also be shown that torsion-induced shear stresses do not contribute to the development of shear failure.

Geometrically nonlinear analysis of compliant mechanisms using a dynamic beam constraint model (DBCM)

Compliant mechanisms with intermediate-deformation ranges are increasingly applied in large-stroke precision manipulators and other actuators. The nonlinearities of axially-loaded stiffening and kinematics-arching effects (P-Δ effects) have a pronounced influence on the kinetostatic and dynamic behaviors of such a class of compliant mechanisms. A dynamic beam constraint model (DBCM) with the pseudo-static characteristic is reported for the nonlinear kinetostatics and large-amplitude vibration analyses of intermediate-deformation compliant mechanisms. Firstly, the DBCM of flexure beams is derived in a closed-form equation on the frequency domain, including the P-Δ effects. A matrix-based modeling methodology with a step-by-step iteration procedure is then introduced to calculate the nonlinear performance of general planar compliant mechanisms with serial-parallel configurations avoiding inner force analysis and kinematics calculation. Except for the static constraint behaviors in the BCM, the amplitude-dependent resonance frequency is captured with the DBCM. Theoretical, numerical and experimental comparisons for a double parallel guiding flexure pivot and Scott-Russell compliant amplifying mechanism verify the feasibility of the DBCM. In summary, the DBCM enables both the nonlinear kinetostatics and large-amplitude vibration analyses of moderately large-stroke compliant mechanisms in a straightforward way.

Monopile Foundation Stiffness Estimation of an Instrumented Offshore Wind Turbine through Model Updating

Rapid development of offshore wind foundation models has resulted in a large number of built structures with generally underestimated foundation stiffness properties and a need to update and validate both the individual structural models and the underlying foundation design frameworks. This paper outlines a structural health monitoring approach, based on the combination of output only structural health monitoring methods and model updating, to estimate foundation stiffness parameters using field monitored data. Field monitoring data from an offshore wind turbine under idling conditions, over a large monitoring period, are presented and operational modal analysis is applied to estimate the modal parameters. Those are compared to modal properties predicted by finite element models, employing either old (API/DNVGL) or new (PISA) foundation design properties, which are calibrated using geotechnical site investigation data. A new approach to interpret seabed level statically equivalent foundation stiffness, in terms of effective lateral and rotational stiffness against load eccentricity, is presented. Seabed level statically equivalent foundation properties are updated by comparison against the observed modal behaviour and the optimised foundation parameters are presented, demonstrating a close match to the predictions of the PISA method.

Numerical modelling of DMLS Ti6Al4V(ELI) polygon structures

Numerical modelling is particularly advantageous for analysing structures with complex behaviour. It is used to predict the mechanical properties of structures. Analytical modelling, on the contrary, has limited capacity for predicting the behaviour, particularly of structures, because it is based on mathematical equations that do not always exactly represent the geometry of the model. In such cases, numerical modelling is used for predicting structural bending, axial deformation, and buckling behaviour. This study documents numerical modelling of different types of polygon structures. To reduce computation costs, planar and extruded Ti6Al4V(ELI) hexagonal shell structures were used to predict stresses in the out-of-plane and in-plane directions. This was followed by numerical modelling of different types of planar polygon structures to predict their load-bearing capacity and stiffness. Thereafter, the hexagonal polygon was subjected to out-of-plane and in-plane uniaxial compression loads. This was done to compare the bending and buckling behaviour of finite element (FE) models to analytical models. The numerical and analytical results were then compared to determine how the ratio (t/L) of the wall thickness (t) and length of the polygon members (L) influenced the effective stiffness of the hexagonal polygon. The triangular polygon was seen to have the greatest load-bearing capacity and stiffness of all polygons that were modelled. The hexagonal model was observed to generate deformations due to compression, similar to those reported in literature. The critical buckling loads for the analytical honeycomb (HC) models were found to be below the yield stress for (1-, 1.125-, and 1.25-mm wall thicknesses) and above the yield stress for all FE HC models, respectively. The effective stiffness of the HC models were observed to increase with the increasing (t/L) ratio, for both the numerical and analytical models.

Experimental study of innovative bolted shear connectors in demountable cold-formed steel–concrete composite beams

Although cold-formed steel (CFS) members have been widely used in structures, their application is limited due to the buckling and instability of these members. To enhance their strength and stiffness, they are often combined with other materials like concrete or timber. One of the main challenges in composite structures is ensuring proper connection between different components to facilitate safe shear load transfer. This is achieved through various types of shear connectors. Bolted shear connections are preferred over headed shear studs because they offer greater strength and can be easily dismantled. Headed shear studs welded onto a steel section are unsuitable for CFS members due to their low thickness.In this paper, we propose a novel demountable composite connection specifically designed for prefabricated CFS structures. The behavior of this connection was evaluated through fifteen push-out test specimens, with key variables including CFS section thickness and bolt diameter/strength. The ultimate load, stiffness, ductility, and failure modes of all specimens were determined. The results revealed that specimens with a higher thickness of CFS members experienced bolt fracture, exhibiting superior effective stiffness, strength, and ductility. Conversely, specimens with a lower thickness of CFS members failed due to distortional buckling of the CFS members. The test results underscore the importance of carefully designing and providing a proportionate CFS section thickness to ensure shear connectors fracture before the CFS members. This significantly improves the shear strength capacity of the composite connection with CFS sections and precast concrete slabs. Furthermore, the deconstructability of the connection at the end of its service life was investigated. The steel–concrete composite connection with precast concrete slabs and demountable bolted shear connections was also modeled using a 3D finite element method. The numerical model’s accuracy in simulating the observed structural behavior was confirmed against experimental data. Finally, appropriate and straightforward formulations for predicting the ultimate load of the CFS-concrete composite connection were proposed. It was concluded that the predictions of ultimate shear load capacity values by EC4, AISC, and AISI provisions for this composite connection are significantly less accurate.

Bayesian optimisation of hexagonal honeycomb metamaterial

Periodic mechanical metamaterials, such as hexagonal honeycombs, have traditionally been designed with uniform cell walls to simplify manufacturing and modelling. However, recent research has suggested that varying strut thickness within the lattice could improve its mechanical properties. To fully explore this design space, we developed a computational framework that leverages Bayesian optimisation to identify configurations with increased uniaxial effective elastic stiffness and plastic or buckling strength. The best topologies found, representative of relative densities with distinct failure modes, were additively manufactured and tested, resulting in a 54% increase in stiffness without compromising the buckling strength for slender architectures, and a 63% increase in elastic modulus and a 88% increase in plastic strength for higher volume fractions. Our results demonstrate the potential of Bayesian optimisation and solid material redistribution to enhance the performance of mechanical metamaterials.

Graph neural network for predicting the effective properties of polycrystalline materials: A comprehensive analysis

We develop a polycrystal graph neural network (PGNN) model for predicting the effective properties of polycrystalline materials, using the Li7La3Zr2O12 ceramic as an example. A large-scale dataset with >5000 different three-dimensional polycrystalline microstructures of finite-width grain boundary is generated by Voronoi tessellation and processing of the electron backscatter diffraction images. The effective ion conductivities and elastic stiffness coefficients of these microstructures are calculated by high-throughput physics-based simulations. The optimized PGNN model achieves a low error of <1.4% in predicting all three diagonal components of the effective Li-ion conductivity matrix, outperforming a linear regression model and two baseline convolutional neural network models. Sequential forward selection method is used to quantify the relative importance of selecting individual grain (boundary) features to improving the property prediction accuracy, through which both the critical and unwanted node (edge) feature can be determined. The extrapolation performance of the trained PGNN model is also investigated. The transfer learning performance is evaluated by using the PGNN model pretrained for predicting conductivities to predict the elastic properties of the same set of microstructures.:

Concurrent topology optimization of parts and their supports for additive manufacturing with thermal deformation constraint

A typical manufacturing failure in powder bed additive manufacturing (AM) is caused by the collision between the already printed structure and the powder recoater due to large thermal deformation. This paper presents a density-based topology optimization (TO) approach to concurrently design parts and their supports that are not susceptible to such a manufacturing failure. The thermal deformation during the layer-by-layer AM process is predicted based on an inherent strain method, and the top surfaces' upward deformations are constrained below the powder layer thickness to avoid recoater collision. To increase the design freedom of TO, the parts and their supports are composed of lattice unit cells with different volume fractions, whose effective stiffness matrixes are predicted by the numerical homogenization and surrogate models. Besides, an overhang constraint and a two-field-based length scale control formulation are also utilized, ensuring overall manufacturability. With the proposed approach, lightweight structures with optimized mechanical properties and controllable manufacturing qualities can be obtained. Numerical examples are given to demonstrate the effectiveness of the proposed approach.

Fatigue Reliability Analysis of Composite Material Considering the Growth of Effective Stress and Critical Stiffness

Fatigue damage accumulation will not only cause the degradation of material performance but also lead to the growth of effective stress and critical stiffness. However, the existing fatigue reliability models usually ignore the effective stress growth and its influence on the critical stiffness of a composite material. This study considers the combined effects of performance degradation and effective stress growth, and a pair of fatigue reliability models for a composite material are presented. Firstly, the fatigue damage in a composite material is quantified by its performance degradation, and the fitting accuracy of several typical fatigue damage models is compared. Subsequently, the uncertainties of initial strength and initial stiffness are considered, and a pair of probabilistic models of residual strength and residual stiffness are proposed. The performance degradation data of Gr/PEEK [0/45/90/−45]2S laminates are utilized to verify the proposed probabilistic models. Finally, the effective stress growth mechanism and its influence on the failure threshold are elaborated, and a pair of fatigue reliability models for composite materials are developed. Moreover, the differences between the strength-based and stiffness-based reliability analysis results of composite materials are compared and discussed.

Performance of special RC shear walls under lateral cyclic and axial loads

Reinforced concrete (RC) shear walls are common structural elements, although their cyclic behavior under different combinations of axial and lateral loading requires further study. The boundaries of flexural-dominated walls experience the greatest stress and strain; thus, rebar layouts are vital in this area and have significant effects on the cyclic behavior of a wall. In the current study, three full-scale walls were designed and constructed using the ACI318-19 special structural wall criteria with boundary elements placed at each end of the wall. Cyclic testing was performed on RC shear walls for different lateral and axial load combinations. These large-scale tests showed how axial loads affect the seismic performance of special RC shear walls. The seismic parameters of the force–displacement hysteresis curves, concrete strain, rebar strain, dissipated energy and stiffness are presented and discussed. The experimental findings demonstrate the effect of axial loading on the wall deformation capacity, strength capacity and mode of failure. Experimental backbone curves for numerical models of walls as recommended by ASCE41-17 revealed that the ASCE model overestimates the effective stiffness of walls with axial loads of less than 10% fcAg, A four-line model is proposed to estimate the force–deformation relationship. It is demonstrated that the effective stiffness coefficient should be a function of the axial load ratio.

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.

Prediction of the additional compressive strength due to PBO-FRCM-confinement of brick-masonry depending on the stiffening effect caused by different discontinuous wrappings: new design-oriented perspective

Fabric Reinforced Cementitious Mortar (FRCM) is an innovative composite system able to strength or rehabilitate masonry members with an efficacy, which is affected by the mechanical and geometrical properties of its constituents: matrix and fabric. Since one of the most appreciated benefits of FRCM is the breathability, the discontinuous application is commonly not considered in the retrofitting. On the other side, the thickness of the matrix may be considerable (ranging between 10 and 30 mm) implying an over-weight and over-stiffness which may influence the seismic response of the structure in a detrimental manner; especially when dealing with column confinement. Consequently, the experimental evidence and the proper design for the partially wrapping with FRCMs is currently lacking.For this reason, the present research reports on an experimental and analytical investigation about PBO (polybenzoxazoles) FRCM-confinement of clay-brick based masonry columns by varying the confinement scheme. The aim is to evaluate the effectiveness of the confinement in terms of axial bearing load capacity and ductility and, at the same time, focusing on the parameters related to the confining scheme showing how these affect the results. Two columns were tested unconfined as lower-bound reference; six columns (i.e. three couples) were partially confined respectively with 3, 4 and 5 wraps having the same width and lastly, a further couple was fully-jacketed as upper-bound reference. The findings evidenced that an appreciable gain in axial strength and deformability was ensured. On the other side, the axial stiffness increased proportionally to the number of confining wraps.Thus, a new analytical model was herein proposed considering the geometry of the confining scheme, as well as the stiffening effect related to it. In particular, a novel effectiveness coefficient (namely KE) is evaluated affecting the theoretical confining pressure based on the axial stiffness of the FRCM confined masonry. The comparison with the own and other available experimental data confirmed the reliability of the proposal which wants to offer a new perspective for the design of the FRCM partial confinement of masonry that needs further investigations to be confirmed by including a variety of masonry and FRCM typologies.

Effective Young’s modulus of Bézier-based honeycombs: Semi-analytical modeling and the role of design parameters and curvature

Metamaterials and lightweight structures are widely used in various engineering applications and additive manufacturing technologies have enhanced their potential. Curved elements have been increasingly used in the design of planar mechanical metamaterials or honeycombs. A honeycomb cellular structure topology with cell walls parametrized using cubic Bézier curves is proposed and analyzed here. The effective Young’s modulus of the Bézier-based honeycomb is characterized semi-analytically using Castigliano’s second theorem. Semi-analytical estimations of the effective Young’s modulus were contrasted to those obtained with numerical finite element models and measured on additively manufactured samples. By considering the relative influence of bending, axial, and shear loads, the deformation mechanism of the honeycomb can be predicted and explained in terms of the load orientation, which depends on the definition of the Bézier curve. A large region achievable by tailoring the design parameters of the curve is presented in the so-called Ashby maps and compared to other existing topologies. The methodology presented here enables designers to propose topologies with a customized effective stiffness, expanding the design space for honeycombs in various engineering applications.