Spring Lattice Research Articles

Microstructural Simulations of Polymer Composites by a Viscoelastic Spring Lattice Model

AbstractAn improved viscoelastic spring lattice model is used to analyze the mechanical properties of polymer composites containing different microstructures, as exemplified by hydroxyl‐terminated polybutadiene‐based solid propellants. A drop‐on‐demand structural model is programmed using the C language to simulate the real solid propellant microstructure. The results show that increasing the particle content has a positive effect on the tensile strength of the propellant, but is detrimental to the ductility. The increase in particle size decreases the maximum tensile strength of the material, reflecting the importance of the dewetting process in the microstructure analysis. Finally, the model accurately predicts that initial defects have a destructive effect on the mechanical properties of the material.

Simulation of mechanical properties of HTPB‐based propellant by a viscoelastic spring lattice model

AbstractAn improved lattice spring model extended by viscoelastic part using the Maxwell model was applied to predict the mechanical properties of HTPB‐based solid propellants. The influence of both strain rate and temperature on the maximum elongation of HTPB‐based propellants was quantitatively analyzed by simulating the uniaxial tensile test, as well as the process of crack development during tensile loads. The results show that the maximum elongation of HTPB‐based propellant decreases as a power function along with the linear increase of strain rate. In contrast, the maximum stress increases as a power function. Increasing the temperature linearly, both the maximum stress and the maximum elongation decrease as a power function.

Spring lattice models in the nonlinear analysis of membrane shells – An applicability study in elasticity, plasticity and damage

Following the accomplishment of fundamental research on- and the development of improved spring models, this work investigates the utility of spring lattice models in representing actual structures. The study explores the applicability of spring lattice models for the analysis of nonlinear shell structures. Of importance the limits of the approximation of defective bar-spring models, on the one hand, and the attempt of a rigorous representation of the continuum by introducing additional spring elements, on the other hand. Preliminary investigations indicate appropriateness for membrane shells, but reveal weakness in modelling shells with bending stiffness by axial bar elements despite three-dimensional cell arrangements. Therefore the focus here is on membrane structures. A direct insight to the employed triangular spring cells is offered by the approach based on the natural description of the continuum suiting the individual geometry. The investigations address specifically the performance of the spring models when applied to problems involving geometrical and material non-linearity. For this purpose the spring models have been tested against the continuum finite element distinctly in elasticity and in plasticity in the context of large deformations. In addition, the effect of damage is simulated by means of a regularized constitutive model of softening plasticity.

Longitudinal elastic wave control by pre-deforming semi-linear materials.

An incremental wave superimposed on a pre-deformed hyper-elastic material perceives an elastic media with the instantaneous modulus of the current material. This offers a new route with a broadband feature to control elastic waves by purposely creating finite deformation field. This study proves that the governing equation of a semi-linear material under a symmetric pre-deformation condition maintains the form invariance for longitudinal wave, so the longitudinal wave control can be made by transformation method without the constraint condition on principle stretches; however, this is not the case for shear waves. Therefore, pre-deforming a semi-linear material provides a potential method for treating longitudinal and shear waves differently. Examples with elastic wave control and band structure shift through pre-deforming a semi-linear material are provided to illustrate this finding. Finally, a one-dimensional spring lattice is proposed to mimic a semi-linear material, and the dispersion relation for longitudinal waves in a sandwich structure with such spring lattice is shown to be invariant during elongation, confirming the result found based on a homogeneous semi-linear material. These results may stimulate researches on designing new hyper-elastic microstructures as well as designing new devices based on pre-deformed hyper-elastic materials.

Compaction dynamics of crunchy granular material

Compaction of brittle porous material leads to a wide variety of densification patterns. Static compaction bands occurs naturally in rocks or bones, and have important consequences in industry for the manufacturing of powder tablets or metallic foams for example. Recently, oscillatory compaction bands have been observed in brittle porous media like snow or cereals. We will discuss the great variety of densification patterns arising during the compaction of puffed rice, including erratic compaction at low velocity, one or several travelling compaction bands at medium velocity and homogeneous compaction at larger velocity. The conditions of existence of each pattern are studied thanks to a numerical spring lattice model undergoing breakage and is mapped to the phase diagram of the patterns based on dimensionless characteristic quantities. This also allows to rationalise the evolution of the compaction behaviour during a single test. Finally, the localisation of compaction bands is linked to the strain rate sensitivity of the material.

Representing stochastic damage evolution in disordered media as a jump Markov process using the fiber bundle model

The damage evolution in quasi-brittle materials is inherently stochastic due to the presence of strong disorder in the form of heterogeneities, voids, and microcracks. The final macroscopic failure is foreshadowed by accumulation of a significant amount of distributed damage that results in precursory events observed as avalanches in experiments and simulations. Simulations on spring lattice models of disordered media have been widely used to understand the collective nature of the quasi-brittle material failure process. In this study, we use the jump Markov process to model stochastic damage evolution, which is informed by the avalanche size distributions for a given material. The jump Markov process is defined based on the probability distributions of the jump sizes, the wait-time between consecutive jumps, and the failure strength. The fiber bundle model is used as an example to obtain the required inputs and test the viability of the proposed approach. The stochasticity and size-dependence of the damage evolution process is inherently captured through the inputs provided for the jump Markov process. The avalanche and strength distributions are used to describe the effect of microscopic information present in the form of disorder, on the macroscopic damage evolution behavior.

Higher‐order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending

AbstractThe quasicontinuum (QC) method is a numerical strategy to reduce the computational cost of direct lattice computations ‐ in this study we achieve a speed up of a factor of 40. It has successfully been applied to (conservative) atomistic lattices in the past, but using a virtualpower statement it was recently shown that QC approaches can also be used for spring and beam lattice models that include dissipation. Recent results have shown that QC approaches for planar beam lattices experiencing in‐plane and out‐of‐plane deformation require higherorder interpolation. Higher‐order QC frameworks are scarce nevertheless. In this contribution, the possibilities of a second‐order and third‐order QC framework are investigated for an elastoplastic spring lattice. The higher‐order QC frameworks are compared to the results of the direct lattice computations and to those of a linear QC scheme. Examples are chosen so that both a macroscale and a microscale quantity influences the results. The two multiscale examples focused on are (i) macroscopically prescribed uniaxial deformation and (ii) macroscopically prescribed pure bending. Furthermore, the examples include an individual inclusion in a large lattice and hence, are concurrent in nature. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stochastic Continuum Damage Mechanics Using Spring Lattice Models

In this study, a planar spring lattice model is used to study the evolution of damage variabledLin disordered media. An elastoplastic softening damage constitutive law is implemented which introduces a cohesive length scale in addition to the disorder-induced one. The cohesive length scale affects the macroscopic response of the lattice with the limiting cases of perfectly brittle and perfectly plastic responses. The cohesive length scale is shown to affect the strength-size scaling such that the strength increases with increasing cohesive length scale for a given size. The formation and interaction of the microcracks is easily captured by the inherent discrete nature of the model and governs the evolution ofdL. The proposed method provides a way to extract a mesoscale dependent damage evolution rule that is linked directly to the microstructural disorder.

Morphological study of elastic-plastic-brittle transitions in disordered media.

We use a spring lattice model with springs following a bilinear elastoplastic-brittle constitutive behavior with spatial disorder in the yield and failure thresholds to study patterns of plasticity and damage evolution. The elastic-perfectly plastic transition is observed to follow percolation scaling with the correlation length critical exponent ν≈1.59, implying the universality class corresponding to the long-range correlated percolation. A quantitative analysis of the plastic strain accumulation reveals a dipolar anisotropy (for antiplane loading) which vanishes with increasing hardening modulus. A parametric study with hardening modulus and ductility controlled through the spring level constitutive response demonstrates a wide spectrum of behaviors with varying degree of coupling between plasticity and damage evolution.

Elastic-Plastic-Brittle Transitions and Avalanches in Disordered Media

A spring lattice model with the ability to simulate elastic-plastic-brittle transitions in a disordered medium is presented. The model is based on bilinear constitutive law defined at the spring level and power-law-type disorder introduced in the yield and failure limits of the springs. The key parameters of the proposed model effectively control the disorder distribution, significantly affecting the stress-strain response, the damage accumulation process, and the fracture surfaces. The model demonstrates a plastic strain avalanche behavior for perfectly plastic as well as hardening materials with a power-law distribution, in agreement with the experiments and related models. The strength of the model is in its generality and ability to interpolate between elastic-plastic hardening and elastic-brittle transitions.

Lattice-particle simulation of stress patterns in a Rwenzori-type rift transfer zone

A new 3D spring lattice computer model has been developed and used to calculate the stress-field in the vicinity of a rift transfer zone. The numerical setup is based on the Rwenzori block, a transfer zone in the Western Branch of the East African Rift Valley. The study has two closely related, yet independent aims: primarily to gain insight into the pattern and the causes of the stress field in the Rwenzori area. The second aim is the evaluation of the model itself, based on a comparison of the model results with local geological structures. The simulations calculate the stress in the brittle part of the crust, at the topographic surface and at a depth of 10 km. The model does therefore not rely on the rate of the continental extension, but only on the amount. The generated stress maps display the maximum and minimum principal normal stress. The stress is calculated for a total extension of 7.5 km, a value based on common estimates for the local rift extension. The local stress field is created by the interplay of the plate driven far-field stress and the mechanics of the rift system. The propagation of the rift system leads thus to stress rotations and to characteristic stress patterns. An important mechanism is lithospheric bending caused by rift flank uplift, which reverses the far field stress in rift vicinity. The stress-field resulting from the simulations explains several previously unexplained structural features encountered in the Rwenzori block.

Lattice models of polycrystalline microstructures: A quantitative approach

This paper addresses the issue of creating a lattice model suitable for design purposes and capable of quantitative estimates of the mechanical properties of a disordered microstructure. The lack of resemblance between idealized lattice models and real materials has limited these models to the realm of qualitative analysis. Two procedures based on the same methodology are presented in the two-dimensional case to achieve the rigorous mapping of the geometrical and the elastic properties of a disordered polycrystalline microstructure into a spring lattice. The theory is validated against finite elements models and literature data of NiAl. The statistical analysis of 900 models provided the effective Young’s modulus and Poisson ratio as function of the lattice size. The lattice models that were created have in average the same Young’s modulus of the real microstructure. However, the Poisson’s ratio could not be matched in the two-dimensional case. The spring constants of the lattices from this technique follow a Gaussian distribution, which intrinsically reflects the mechanical and geometrical disorder of the microscale. The detailed knowledge of the microstructure and the Voronoi tessellation necessary to implement this technique are supplied by modern laboratory equipments and software. As an illustrative example of lattice application, damage simulations of several biaxial loading schemes are briefly reported to show the effectiveness of discrete models towards elastic anisotropy induced by damage and damage localization.

Scaling, localization and anisotropy in fracturing central-force spring lattices with strong disorder

We analyze scaling and localization phenomena in the fracture of a random central-force spring lattice model with strong disorder by means of computer simulation. We investigate the statistical and topological properties of the developing damage pattern and the scaling behaviour of the threshold. Our observations show that from the beginning and up to the point of maximum stress, damage develops in a uniform manner, qualitatively like in a percolating lattice, but numerically different from random percolation. Beyond the maximum-stress point localization and anisotropy come into play, resulting in final crack formation. The fraction of broken bonds at which the lattice fails, as well as the strain corresponding to failure, scale with the lattice size via power laws. The roughness of the final crack scales as a power law of the crack length over three decades of lengthscale.

Elastic medium confined in a column versus the Janssen experiment

We compute the stresses in an elastic medium confined in a vertical column, when the material is at the Coulomb threshold everywhere at the walls. Simulations are performed in 2 dimensions using a spring lattice, and in 3 dimensions, using the Finite Element Method. The results are compared to the Janssen model and to experimental results for a granular material. The necessity to consider elastic anisotropy to render qualitatively the experimental findings is discussed.

Mass spring lattice modeling of the scanning laser source technique

The scanning laser source (SLS) technique is a promising new laser ultrasonic tool for the detection of small surface-breaking defects. The SLS approach is based on monitoring the changes in laser generated ultrasound as a laser source is scanned over a defect. Changes in amplitude and frequency content have been observed for ultrasound generated by the laser over uniform and defective areas. In this paper, the SLS technique is simulated numerically using the mass spring lattice model. Thermoelastic laser generation of ultrasound in an elastic material is modeled using a shear dipole distribution. The spatial and temporal energy distribution profiles of typical pulsed laser sources are used to model the laser source. The amplitude and spectral variations in the laser generated ultrasound as the SLS scans over a large aluminum block containing a small surface-breaking crack are observed. The experimentally observed SLS amplitude and spectral signatures are shown to be captured very well by the model. In addition, the possibility of utilizing the SLS technique to size surface-breaking cracks that are sub-wavelength in depth is explored.

Granular community goes to the desert

The physics of granular media is an ancient science. It is present in practically all fields of applied physical sciences. At a fundamental level, models of hard spheres or disks have been used from the early times of classical Greece to nowadays investigations to characterize the microscopic states of organisation of matter. Conversely, in the last two decades, a large community of physicists has made use of what had been acquired in the microscopic physics world to try to enrich our understanding of the science of “real grains”. In the eighties, as we were testing percolation ideas by mixing conducting and insulating otherwise identical grains in various concentrations [1], we realized then that the way the packings were built had a strong influence on the result of the experiments. Replacing a single conducting grain by an insulating one tells one in particular how many good electrical contacts this grain had, and this varies with filling conditions and packing pressure. The question led us to be acquainted with the British school which since Bernal [2] had studied arrangements of packings in connection with the structure of the liquid state. But probably the most spectacular finding for us was that of a photoelastic experiment by P. Dantu [3] on compressed arrays of parallel aligned photoelastic cylinders which clearly showed that the stresses propagated along continuous chains of cylinders leaving a lot of them practically free of stress due to arching (similar work had been done independently by Josselin de Jong in Holland). This result was not well recognized at this time partly because of a prevalent dogma in mechanics based on homogenization procedures which use phenomenological local constitutive laws to evaluate global properties of heterogeneous - generally periodic structures. Clearly this can not apply in the granular case of a granular assembly; because of the very large distribution of local forces a global geometrical description is needed [4]. Percolation ideas have been applied to the problem taking into account the asymmetry between presence or absence of inter-grain forces when the grains are either pressed together or pulled apart, like a network of fuses or of diodes and unlike a mechanical spring lattice. Such models can account for the fact that the value of the exponent x of the force F deformation D law, F = D x , is much larger than that of an individual contact (given by Hertz law f = d 3/2 ) due to the increase of active mechanical contacts where forces increase. More