Incremental Stiffness Research Articles

Effects of prestress in the coating of an elastic disk

An elastic disk is coated with an elastic rod, uniformly prestressed with a tensile or compressive axial force. The prestress state is assumed to be induced by three different models of external radial load or by ‘shrink-fit’ forcing the coating onto the disk. The prestressed coating/disk system, when loaded with an additional and arbitrary incremental external load, experiences incremental displacement, strain, and stress, which are solved via complex potentials. The analysis incorporates models for both perfect and imperfect bonding at the coating/disk interface. The derived solution highlights the significant influence not only of the prestress but also of the method employed to generate it. These two factors lead, in different ways, to a loss or an increase in incremental stiffness for compressive or tensile prestress. The first bifurcation load of the structure (which differs for different prestress generations) is determined in a perturbative way. The results emphasize the importance of modelling the load and may find applications in flexible electronics and robot arms subject to pressure or uniformly-distributed radial forces.

Theoretical and Simulation Finite Element Modal Analysis of Rotating Cantilever Beam

The finite element method is used to carry out modal analysis of rotating cantilever beam. The virtual work method is then used to derive the stiffness matrix of a rotating beam element. The stiffness matrix of a rotating beam element is simply seen to be sum of the stiffness matrix of the non-rotating beam and an incremental stiffness matrix induced by rotation. This work presents a novel generalized incremental stiffness matrix that takes care of any element at any distance from the rotation axis. The established rotational stiffness matrix is then used together with consistent mass matrix (not affected by rotation) to carry out modal analysis of rotating cantilever beam. Theoretical computations were validated by simulations from ANSYS. It is seen that the contribution to modal frequencies and shapes of the blade of rotation via incremental stiffness matrix is very marginal at low rotational speeds. For exampleit is seen for a numerical case study that five-element model computes slight increase in fundamental natural frequency due to rotation relative to that of the non-rotating blade as 0.000195% for rotational speed of 300rpm. It is also seen that ten times increase in speed leads to about hundred times increase in rotational contribution meaning that it becomes more imperative to model rotation as speed of the blade increases. Points are also made regarding application to avoiding resonance of rotating cantilever beam.

Self-consistent approximations for the tangential-displacement correction to the incremental indentation stiffness

An axisymmetric problem of unilateral frictionless contact between a paraboloidal or conical indenter and a transversely isotropic elastic half-space is considered in the refined formulation by accounting for the tangential (radial) displacements of the surface points of the elastic body. Using the idea of self-consistent approximation, a system of two coupled relations for the main contact variables (contact force, indenter displacement, and contact radius) is derived. The concept of the true contact radius is discussed, and it has been argued that the incremental stiffness relation should be expressed in its terms. Correction factors for the force–displacement relation and the incremental indentation stiffness (as a function of the true contact radius) are evaluated in explicit form.

The stress–strain behavior of refractory microcracked aluminum titanate: The effect of zigzag microcracks and its modeling

AbstractThe stress–strain behavior of ceramics, such as aluminum titanate, has certain features that are unusual for brittle materials—in particular, a substantial nonlinearity under uniaxial tension, and load–unload hysteresis caused by the sharp increase of the incremental stiffness at the beginning of unloading. These features are observed experimentally and are attributed to microcracking. Here we compare different degrees of stress–strain nonlinearity of aluminum titanate materials and quantitatively model them. We use advanced mechanical testing to observe the mechanical response at room and high temperature; electron microscopy, and X‐ray refraction radiography to observe the microstructural changes. Experiments show that two types of microcracks can be distinguished: (i) microcracks induced by cooling from the sintering temperature (due to heterogeneity and anisotropy of thermal expansion), with typical sizes of the order of grain size, and (ii) much larger microcracks generated by the mechanical loading. The two microcrack types produce different effects on the stress–strain curves. Such microcracks and the features of the stress–strain behavior depend on the density of the cooling‐induced microcracks and on the distribution of grain sizes. They are modeled analytically and numerically.

Post-buckling of variable stiffness curvilinear fibre-reinforced general lay-up composite beams by sinusoidal shear flexible theory

The mechanical post-buckling behaviour of variable stiffness layered composite beams reinforced by curvilinear fibres subjected to compressive loads is investigated here using a sine function-based shear flexible beam model. The neutral axis stretching force stemming from the axial movement restraints is accounted for through von Karman’s assumption based geometrical nonlinearity. Furthermore, the modified beam constitutive equation arising from the consideration of the Poisson’s effect is introduced in the formulation for the laminated beam analysis with general lay-up or ply sequences. The governing equations incorporating the incremental stiffness matrices are formed through the minimization of total potential energy principle and are solved by numerical method. The solutions for the developed governing equations are evaluated iteratively based on eigenvalue analysis and the characteristics of post-buckling of laminated beams can be inferred through the relationship between the beam deflection level and post-buckling axial load. An in-depth analysis selecting many design parameters for instance lamina’s curvilinear fibre angles, beam slenderness ratio, lay-up and edge conditions, load type and so on is dealt with in bringing out the behaviour of variable stiffness laminated beam in linear and post-buckling regions. Also, the influence of flexible end supports by restraining elastically against ends rotation is studied on the beam elastic stability behaviour.

Nonlinear thermo-elastic stability of variable stiffness curvilinear fibres based layered composite beams by shear deformable trigonometric beam model coupled with modified constitutive equations

Nonlinear thermo-elastic buckling characteristics of composite variable stiffness beam with layers making use of curvilinear fibres under thermal environment is attempted here. The model is based on a shear deformable theory introducing trigonometric function, and considering von Kármán’s assumptions based geometrical nonlinear effect. The beam constitutive equation is modified according to the stress-free situation in the width direction of beam-Poisson’s effect in the formulation for predicting the behaviour of general lay-up composite beams. By the principle of minimum total potential energy, the governing equations in terms of incremental stiffness matrices are formed introducing general beam finite element. The global equilibrium equations formulated are solved for envisaging the post-buckling path through eigenvalue analysis iteratively, thus establishing the relationship of thermal temperature against moderate amplitude level of beam deflection. A systematic parametric analysis considering different lamina properties such as curvilinear fibre path angles and including lay-up sequences, thermal expansion coefficient, mixed laminate combining straight and curvilinear fibres-based layers is carried out on thermo-structural stability of curvilinear fibre-based beams. Also, the influence of geometric factors, flexible beam end support, and variation in thermal profile, etc. over the stability behaviour of beam is examined.

On band gap and damping of metamaterials involving negative-stiffness elements

This study proposes a viscoelastic metamaterial containing negative-stiffness elements, which exhibits both a damping enhancement effect and a tunable ultra-low frequency band gap. A dynamic model involving static large predeformation of the viscoelastic metamaterial is established based on the updated Lagrangian scheme and solved by finite element method. Numerical results show that the presence of negative-stiffness elements can reduce the frequency of the band gap and increase its damping performance by two orders of magnitude. The mechanism roots in the fact that the negative stiffness elements enable the viscoelastic metamaterials with extremely low local incremental stiffness, resulting in low-frequency local resonance and damping enhancement. This study facilitates the design of damping composites with both high stiffness and damping capacity, which are widely required in vibration and noise control.

Mechanotransduction-induced glycolysis epigenetically regulates a CXCL1-dominant angiocrine signaling program in liver sinusoidal endothelial cells invitro and invivo.

Liver sinusoidal endothelial cells (LSECs) are ideally situated to sense stiffness and generate angiocrine programs that potentially regulate liver fibrosis and portal hypertension. We explored how specific focal adhesion (FA) proteins parlay LSEC mechanotransduction into stiffness-induced angiocrine signaling invitro and invivo. Primary human and murine LSECs were placed on gels with incremental stiffness (0.2 kPa vs. 32 kPa). Cell response was studied by FA isolation, actin polymerization assay, RNA-sequencing and electron microscopy. Glycolysis was assessed using radioactive tracers. Epigenetic regulation of stiffness-induced genes was analyzed by chromatin-immunoprecipitation (ChIP) analysis of histone activation marks, ChIP sequencing and circularized chromosome conformation capture (4C). Mice with LSEC-selective deletion of glycolytic enzymes (Hk2fl/fl/Cdh5cre-ERT2) or treatment with the glycolysis inhibitor 3PO were studied in portal hypertension (partial ligation of the inferior vena cava, pIVCL) and early liver fibrosis (CCl4) models. Glycolytic enzymes, particularly phosphofructokinase 1 isoform P (PFKP), are enriched in isolated FAs from LSECs on gels with incremental stiffness. Stiffness resulted in PFKP recruitment to FAs, which paralleled an increase in glycolysis. Glycolysis was associated with expansion of actin dynamics and was attenuated by inhibition of integrin β1. Inhibition of glycolysis attenuated a stiffness-induced CXCL1-dominant angiocrine program. Mechanistically, glycolysis promoted CXCL1 expression through nuclear pore changes and increases in NF-kB translocation. Biochemically, this CXCL1 expression was mediated through spatial re-organization of nuclear chromatin resulting in formation of super-enhancers, histone acetylation and NF-kB interaction with the CXCL1 promoter. Hk2fl/fl/Cdh5cre-ERT2 mice showed attenuated neutrophil infiltration and portal hypertension after pIVCL. 3PO treatment attenuated liver fibrosis in a CCl4 model. Glycolytic enzymes are involved in stiffness-induced angiocrine signaling in LSECs and represent druggable targets in early liver disease. Treatment options for liver fibrosis and portal hypertension still represent an unmet need. Herein, we uncovered a novel role for glycolytic enzymes in promoting stiffness-induced angiocrine signaling, which resulted in inflammation, fibrosis and portal hypertension. This work has revealed new targets that could be used in the prevention and treatment of liver fibrosis and portal hypertension.

Upper and lower bounds to the overall incremental stiffness of hyperelastic composites

Bounds to the overall stiffness of a composite are well-known within the classical theory of elasticity. They are based on the positive-definiteness of the local stiffness. A transfer to a prestressed state is not trivial. We may study the incremental stiffness that connects the nominal stress rate with the velocity gradient. But when there are mainly compressive stresses, then positive-definiteness can only be secured if this stiffness is replaced by a pseudo-stiffness. Its existence is equivalent to a strengthened form of uniform infinitesimal polyconvexity and is independent of the geometry. The same is the case with the crude Voigt and Reuss bounds. More refined kinematic or dynamic approximations do, of course, depend on the geometry. This is demonstrated with the unidirectional reinforcement of a matrix.

Analysis of Elastic–Plastic Problems Using the Improved Interpolating Complex Variable Element Free Galerkin Method

A numerical model for the two-dimensional nonlinear elastic–plastic problem is proposed based on the improved interpolating complex variable element free Galerkin (IICVEFG) method and the incremental tangent stiffness matrix method. The viability of the proposed model is verified through three elastic–plastic examples. The numerical analyses show that the IICVEFG method has good convergence. The solutions using the IICVEFG method are consistent with the solutions obtained from the finite element method using the ABAQUS program. Moreover, the IICVEFG method shows greater computing precision and efficiency than the non-interpolating meshless methods.

Dynamic buckling of classical/non-classical curved beams by nonlocal nonlinear finite element accounting for size dependent effect and using higher-order shear flexible model

This paper investigates the dynamic snap-through buckling of classical and non-classical curved beams subjected to a suddenly applied step load. The small scale effect prevalent in non-classical beams, viz, micro and nanobeam, is modeled using the nonlocal elasticity approach. The formulation accounts for moderately large deflection and rotation. The governing equilibrium equations are derived using the dynamic version of the principle of virtual work and are subsequently simplified in terms of the generalized displacements for the development of a nonlocal nonlinear finite element model. The spatial domain comprises of 3-noded higher-order curved beam elements based on shear flexible theory associated with sine function. The nonlinear governing equations are solved using the incremental stiffness matrices and by adopting direct time integration method. The critical dynamic buckling load is identified by the smallest load at which there is a sudden rise in the amplitude of vibration. The efficacy of model here is compared against the available analytical studies for the local and nonlocal beams. A detailed study is made to highlight the effects of the geometric parameter, initial condition, nonlocal parameter, load duration, and boundary conditions on the dynamic stability of both classical and non-classical curved beams. The nature and degree of participation of various eigen modes accountable for the dynamic snap-through behavior are examined a posteriori using the modal expansion approach. Some interesting observations made here are valuable for the optimal design of such structural members against fatigue and instability.

Seismic Response of Multistory Buildings with Double Telescoping Self-Centering Energy-Dissipative Brace (DT-SCED)

Telescopic Self-Centering braces are one of the very successful examples of Self-Centering braces which perform well in seismic loading. In this study, a new example of Telescopic Self-Centering brace is introduced, which has superior features over other telescopic braces. These include: high axial load capacity, use of shorter cables in brace construction, simplicity of construction, use of separate cables for compressive and traction modes, less fatigue in cyclic loads and, allowing for more dynamic loading cycles. In this paper, a sample was designed with an axial force capacity of 300kN. Modeling of behavior (DT-SCED) was accurately expressed using numerical relationships. nonlinear incremental stiffness analysis method was also used to calculate the hysteresis brace behavior. The cyclic load test was applied to this brace and the result showed complete Self-Centering behavior. Then, a sample building with the double telescoping Self-Centering Energy-Dissipative Brace (DT-SCED) was subjected Single direction pushover analyses and the results are compared with the sample buildings with the Self-Centering Energy-Dissipative Bracing (SCED) and the Telescoping Self-Centering Energy-Dissipative Bracing (T-SCED). The results of the analysis and comparing with other samples confirm the seismic superiority of performance of the DT-SCED brace over other samples.

Coupled nonlinear and time-dependent analysis for long span cable-stayed bridges

This paper proposes a new cable element and a novel formation procedure of unbalanced loads for cable-stayed bridges. The proposed method couples the effects of geometric nonlinearity and concrete creep/shrinkage. An improved catenary cable element with arbitrary rigid arms is introduced to reduce the degree of freedoms in simulations. An explicit expression of tangent stiffness of the new cable element is derived. Then, an incremental stiffness equation under co-rotational coordinate system is established by introducing a recurrence equation of creep loads. Following this, a new formation method is proposed to evaluate the unbalanced loads on the current deformed equilibrium configuration. This method is illustrated and validated by two actual long span cable-stayed bridges in Yangtze River. The results indicate that the coupled algorithm can significantly enhance the computational efficiency and accuracy for the bridge analysis during construction.

3D composite metamaterial with magnetic inclusions exhibiting negative stiffness and auxetic behaviour

In this work, through the use of a theoretical model and experimental testing, we investigate a novel three-dimensional mechanical metamaterial composed of structural units consisting of two arrowhead-like elements with magnetic inclusions. We show that such system has a capability to exhibit negative Poisson's ratio and negative incremental stiffness at the same time. We also show that both the type of the stiffness exhibited by the system and its magnitude can be solely controlled via the orientation of magnetic inclusions within the structure and the magnitude of the magnetic moment associated with magnetic inclusions respectively. More specifically, for a particular example of magnetic inclusions used in the experiment, the stiffness could assume an arbitrary value from the range approximately between −5 kN/m and 5 kN/m depending on the stage of the mechanical deformation and aforementioned factors. This range can be also conveniently adjusted depending on the size and properties of magnetic inclusions. This in turn allows to adjust mechanical response of the considered structure in accordance to a particular application without a need of reconstructing the system which would not normally be possible for conventional nonmagnetic systems attributing negative stiffness to their elastic properties.

Linear‐frictional contact model for 3D discrete element simulations of granular systems

SummaryThe linear‐frictional contact model is the most commonly used contact mechanism for discrete element (DEM) simulations of granular materials. Linear springs with a frictional slider are used for modeling interactions in directions normal and tangential to the contact surface. Although the model is simple in two dimensions, its implementation in 3D faces certain subtle challenges, and the particle interactions that occur within a single time step require careful modeling with a robust algorithm. The paper details a three‐dimensional algorithm that accounts for the changing direction of the tangential force within a time step, the transition from elastic to slip behavior within a time step, possible contact sliding during only part of a time step, and twirling and rotation of the tangential force during a time step. Without three of these adjustments, errors are introduced in the incremental stiffness of an assembly. Without the fourth adjustment, the resulting stress tensor is not only incorrect but it is also no longer a tensor. The algorithm also computes the work increments during a time step, both elastic and dissipative.

Instability-Triggered Triply Negative Mechanical Metamaterial

We report a three-dimensional mechanical metamaterial that simultaneously possesses negative stiffness, negative bulk modulus, and negative Poisson's ratio. This metamaterial is a periodic arrangement of binder-shell elements. Under compression, the spherical shells dent inwards which cause the material to contract in the lateral directions. At larger deformations, snap through instability occurs and the material exhibits negative incremental stiffness. Interestingly, both incremental stiffness and incremental Poisson's ratio approach negative infinity (under displacement control) when snap-back is observed. We further showed that the multi-negative index metamaterial satisfies the strong ellipticity condition, therefore, a block of the metamaterial with many unit cells is stable under displacement constraint.

Active Adjustment of Surface Accuracy for a Large Cable-Net Structure by Shape Memory Alloy.

The high surface accuracy design of a cable-net antenna structure under the disturbance of the extremely harsh space environment requires the antenna to have good in-orbit adjustment ability for surface accuracy. A shape memory cable-net (SMC) structure is proposed in this paper and believed to be able to improve the in-orbit surface accuracy of the cable-net antenna. Firstly, the incremental stiffness equation of a one-dimensional bar element of the shape memory alloy (SMA) to express the relationship between the force, temperature and deformation was effectively constructed. Secondly, the finite element model of the SMC antenna structure incorporated the incremental stiffness equation of a SMA was established. Thirdly, a shape active adjustment procedure of surface accuracy based on the optimization method was presented. Finally, a numerical example of the shape memory cable net structure applied to the parabolic reflectors of space antennas was analyzed.

An Equivalent Circuit Model for Rectangular Waveguide Performance Analysis Considering Rough Flanges’ Contact

In order to investigate the rough flanges’ impact on the electrical performance of rectangular waveguides in the high-power-microwave and high-frequency-microwave communications, a new equivalent circuit model for the contact problem in waveguide flanges is established. Different from the present method that expresses the contact resistance with the contact radius, this method expresses the contact resistance with the incremental stiffness. The equivalent circuit of the waveguide connection is deduced by combining the mechanical contact model with the contact resistance model. Both metal–insulator–metal contacts and metal–metal contacts are considered. The relationship between electrical characteristics and mechanical characteristics is finally established with the transmission line theory of waveguides. The experimental results are given to demonstrate the proposed methodology.

Numerical simulation of the effect of grain fragmentation on the evolution of microstructure quantities

In this paper the effect of grain fragmentation of a cohesionless granular material on the change of microstructure quantities is investigated using a micropolar continuum model. To this end the change of the grain size distribution is related in a simplified manner to a reduction of the mean grain diameter, which enters the constitutive equation as the internal length. The additional densification of the grain skeleton is modelled by a reduction of the incremental stiffness, which is related to the so-called solid hardness defined in the sense of a continuum description. The reduction of the mean grain diameter and the solid hardness caused by grain fragmentation is described by corresponding evolution equations, which depend on an increasing mean pressure and an increasing deviatoric stress. The focus of the numerical investigations is on the evolution of microstructure quantities like the void ratio, mean grain diameter and couple stresses under monotonic compression. It is shown that for an initially inhomogeneous distribution of the void ratio polar quantities like microrotations and couple stresses can develop even outside of shear bands. With increasing compression the polar quantities can increase, while the range of fluctuation of the void ratio decreases.

Measurement of higher-order stress-strain effects in granular materials undergoing non-uniform deformation

Discrete element (DEM) simulations demonstrate that granular materials are non-simple, meaning that the incremental stiffness of a granular assembly depends on the gradients of the strain increment as well as on the strain increment itself. In quasi-static simulations, two-dimensional granular assemblies were stiffer when the imposed deformation was non-uniform than for uniform deformation. The contacts between particles were modeled as linear–frictional contacts with no contact moments. The results are interpreted in the context of a higher-order micro-polar continuum, which admits the possibility of higher-order stress and couple-stress. Although the behavior was non-simple, no evidence was found for a couple-stress or an associated stiffness. The experimental results apply consistently to three particle shapes (circles, ovals, and a non-convex cluster shape), to assemblies of three sizes (ranging from 250 to 4000 particles), and at pre-peak and post-peak strains.