Response Of Viscoelastic Materials Research Articles

Extracting viscoelastic material parameters using an atomic force microscope and static force spectroscopy.

Atomic force microscopy (AFM) techniques have provided and continue to provide increasingly important insights into surface morphology, mechanics, and other critical material characteristics at the nanoscale. One attractive implementation involves extracting meaningful material properties, which demands physically accurate models specifically designed for AFM experimentation and simulation. The AFM community has pursued the precise quantification and extraction of rate-dependent material properties, in particular, for a significant period of time, attempting to describe the standard viscoelastic response of materials. AFM static force spectroscopy (SFS) is one approach commonly used in pursuit of this goal. It is capable of acquiring rich temporal insight into the behavior of a sample. During AFM-SFS experiments the cantilever base approaches samples with a nearly constant velocity, which is manipulated to investigate different timescales of the mechanical response. This manuscript seeks to build upon our previous work and presents an approach to extracting useful linear viscoelastic information from AFM-SFS experiments. In addition, the basis for selecting and restricting the model parameters for fitting is discussed from the perspective of applying this technique on a practical level. This work begins with a guided discussion that develops a fit function from fundamental laws, continues with conditioning a raw SFS experimental dataset, and concludes with the fit and prediction of viscoelastic response parameters such as storage modulus, loss modulus, loss angle, and compliance. These steps constitute a complete guide to leveraging AFM-SFS data to estimate key material parameters, with a series of detailed insights into both the methodology and supporting analytical choices.

Mechanical characterization of PVA hydrogels' rate-dependent response using multi-axial loading.

The time-dependent properties of rubber-like synthesized and biological materials are crucial for their applications. Currently, this behavior is mainly measured using axial tensile test, compression test, or indentation. Limited studies performed on using multi-axial loading measurements of time-dependent material behavior exist in the literature. Therefore, the aim of this study is to investigate the viscoelastic response of rubber-like materials under multi-axial loading using cavity expansion and relaxation tests. The tests were performed on PVA hydrogel specimens. Three hyperelasitc models and one term Prony series were used to characterize the viscoelastic response of the hydrogels. Finite element (FE) simulations were performed to verify the validity of the calibrated material coefficients by reproducing the experimental results. The excellent agreement between the experimental, analytical and numerical data proves the capability of the cavity expansion technique to measure the time-dependent behavior of viscoelastic materials.

Viscoelasticity with limiting strain

<p style='text-indent:20px;'>A self-contained review is given for the development and current state of implicit constitutive modelling of viscoelastic response of materials in the context of strain-limiting theory.

Finite element simulation of restrained shrinkage cracking of cementitious materials: Considering moisture diffusion, aging viscoelasticity, aleatory uncertainty, and the effects of soft/stiff inclusions

Simulation of restrained ring shrinkage and cracking of cementitious materials in a multiphysics simulation framework (MOOSE) is discussed in this paper. The 3D numerical model analyzes residual stress development and crack initiation/propagation in cement pastes by applying an eigenstrain which varies over the depth of the specimen based on the relative humidity of the pores as moisture diffuses from the drying surface. The numerical modeling framework explicitly considers: (i) moisture diffusion that generates differential shrinkage along the depth of the specimen (ii) viscoelastic response of aging cementitious materials through a rate-type creep law based on generalized Maxwell model, (iii) isotropic damage model with Rankine′s criterion determining the failure initiation, and (iv) aleatory uncertainty-based distribution of tensile strengths of individual finite elements to account for statistical variability and associated microstructural size effects. The model is implemented for cement pastes containing compliant/stiff inclusions subjected to variable drying conditions. The numerical model can be used to compare the cracking propensity of different cementitious mixtures.

An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

Computational biomechanics plays an important role in biomedical engineering: using modeling to understand pathophysiology, treatment and device design. While experimental evidence indicates that the mechanical response of most tissues is viscoelastic, current biomechanical models in the computational community often assume hyperelastic material models. Fractional viscoelastic constitutive models have been successfully used in literature to capture viscoelastic material response; however, the translation of these models into computational platforms remains limited. Many experimentally derived viscoelastic constitutive models are not suitable for three-dimensional simulations. Furthermore, the use of fractional derivatives can be computationally prohibitive, with a number of current numerical approximations having a computational cost that is O(NT2) and a storage cost that is O(NT) (NT denotes the number of time steps). In this paper, we present a novel numerical approximation to the Caputo derivative which exploits a recurrence relation similar to those used to discretize classic temporal derivatives, giving a computational cost that is O(NT) and a storage cost that is fixed over time. The approximation is optimized for numerical applications, and an error estimate is presented to demonstrate the efficacy of the method. The method, integrated into a finite element solid mechanics framework, is shown to be unconditionally stable in the linear viscoelastic case. It was then integrated into a computational biomechanical framework, with several numerical examples verifying the accuracy and computational efficiency of the method, including in an analytic test, in an analytic fractional differential equation, as well as in a computational biomechanical model problem.

Prestress Effect on the Thermomechanical Response of Viscoelastic Plate Under Harmonic Loading

A statement of the coupled thermomechanical problem on forced resonant vibrations and dissipative heating of hinged viscoelastic elastomeric plate is given with account of prestresses present in the plate. It is assumed the prestress is generated as a result of the manufacturing process or preliminary plate service. The problem statement is based on the standard Kirchhoff-Love hypotheses and concept of complex moduli that are used to describe the viscoelastic material response to harmonic loading. Under these circumstances, the prestress manifests itself as a membrane forces applied in the plane of the rectangular plate. Therefore, the problem of in-plane stress state and problem of forced transverse vibration of the plate can be solved separately. Both steady-state and transient thermal response is investigated. Influence of the prestress is studied in details. Dissipative heating temperature histories are calculated for the variety of the prestress and loading parameters. Temperature criterion is adopted to determine the critical state. The data obtained are used for the plate fatigue life prediction as well as for the investigation of prestress effect on the plate response. The reliability of the values of frequencies on the several lowest resonances was checked. For the most energy-intensive first mode of transverse vibrations, the influence of the preliminary tensile stress state, as well as the amplitude of the transverse distributed load on the amplitude–frequency characteristics and temperature evolution was studied.

The viscoelastic behaviour of material phases in fired clay identified by means of grid nanoindentation

A systematic experimental identification approach for fired clay, the base material of bricks and clay block masonry, is essential to optimize the mechanical and thermal properties and, in this way, extending the field of application of this sustainable and high-performant building material. For this reason, a comprehensive nanoindentation campaign was performed on two clays, one of them carbonate bearing, each fired at 880 °C and 1100 °C, in extrusion direction x and normal to it. A carefully defined load function, including a short preloading step, a holding phase of 20 s, and a very fast unloading step, allowed for (i) the decomposition of plastic and viscoelastic material response, (ii) the identification of the time-dependent behaviour as well as (iii) a neat determination of elastic moduli of different material phases. A well-developed sample preparation protocol, resulting in a very low surface roughness and, thus, allowing for very small indentation forces, together with the information of scanning electron microscopy images enabled the assignment of indentation responses to single material phases. An orientation dependency of the stiffness response for the matrix of both clays fired at 880 °C was identified, which, however, disappears at a firing temperature of 1100 °C. Furthermore, the indentation results revealed that every material phase within the investigated fired clay samples, even the quartz phase, has a distinct time-dependent behaviour, which could be sufficiently captured within the second loading phase of 20 s and described very well with three different rheological models. Finally, through the analysis of representative area elements of eight different fired clay samples, comprehensive 2D information about the elastic as well as viscous behaviour is given within this work, representing a basis and input for the development of micromechanical models for fired clay material.

Forming of Dynamic Microstructure of Flexible Polymer.

This study focuses on the forming of dynamic microstructure of flexible polymer. The dynamic tensile control microstructure of the polymer mold, along with the gasbag, was used to exert pressure to achieve forming. This study simulated the dynamic control of the flexible mold, and proposed four mechanical models of material viscoelastic response for modeling and evaluation. MATLAB software was used to calculate the imprint prediction calculation theory construction according to the imprint result of curved surface and asymmetric imprint forming. This study designed and developed a gasbag-assisted dynamic forming system, and tested the proposed system for verification. The test results showed that the mechanical stability, curved surface, and asymmetric imprint prediction calculation of the mechanical model of the viscoelastic response of flexible mold material, as proposed in this study, can display the geometric features of the imprinted microstructure. The dynamic mold microstructure control process can accurately transfer a bifacial microstructure and construct the confidence interval for transfer printing forming.

Evaluation of viscoelastic characteristics of PTFE-Based materials

Abstract PTFE-based materials are widely used in areas of tribology, particularly in seal and bearing applications because of their outstanding self-lubricating properties. This paper concentrates on the viscoelastic response of PTFE-based materials. Using a dynamic mechanical analyzer (DMA), the storage modulus, loss modulus and tan δ, are measured and discussed in the frequency domain. The relaxation modulus and creep compliances are also measured and studied in the time domain. Finally, the materials’ compositions are studied using Energy Dispersive X-ray Spectrometer for composites measurement. The experimental data is then modeled with best fit curves using the Prony series and compared to the experimental data.

Microdrop impact on soft substrates at low Weber numbers

In this work, we experimentally investigate the impact of microdrops on diverse soft substrates at low Weber numbers. We show that the dynamic behaviors of the impinging microdrops are very different from that of macroscopic droplets reported in the literature. Whereas various impact phenomena have been identified for millimeter-sized water droplets in previous studies, we only observed regular deposition for droplets of a few tens of micrometers in the experiment. In contrast to the significant slowdown of the receding millimeter-sized droplets, no effects of substrate stiffness on the impact process have been identified at microscale. These different dynamics are explained by the different dynamic response of soft viscoelastic materials during droplet impact over different timescales. Finally, the analysis of the post-impact droplet oscillation with the theory for damped harmonic oscillators reveals that the damping coefficient is also size-dependent while the spring constant is not. A simple scaling argument was proposed to understand this finding.

Rationally Derived Three-Parameter Models for Elastomeric Suspension Bushings: Theory and Experiment

Abstract Linear, causal, and time-invariant viscoelastic material response models are of practical interest in many areas, including our interest in automotive suspension bushings. Empirical fits to experimental data may involve several fitted parameters which, if chosen arbitrarily, violate theoretical restrictions derived from causality, linearity, and time invariance. Here, we retain a correction term in an asymptotic expansion for Bode’s representation of the famous Kramers-Kronig relations. We then derive two new mathematical forms that both satisfy theoretical restrictions and fit experimental data well. Both forms proposed in this article have three free parameters each. The first form has logarithms and power law terms and is valid if the index in the power law is small. The second form has only logarithmic terms and satisfies restrictions exactly. Both forms successfully fit the test data for four different automotive suspension bushings in a frequency range of 1–30 Hz. The first form, with the power law, is more complicated but fits the data slightly better. Because the frequency response involves both a real and imaginary part, simultaneously fitting both parts well with merely three parameters validates the approximations proposed in this article.

Mathematical modeling of viscoelastic material under impact load

A linear physics-based model is developed to investigate the one-dimensional impact on a viscoelastic material. A generalized model with three Maxwell elements is considered to describe the viscoelastic behavior. An analytical method based on Laplace transformation is used to solve the impact problem. To have a comprehensive understanding of viscoelastic material response, drop-weight impact is also considered in this study. For both linear impact and drop-weight cases, a maximum reduction of 15% of the impact force as well as 32% higher energy absorption can be achieved with the increase in tan δ from 0.01 to 0.8 of viscoelastic material. In addition, for linear impact, impact force decreases by 20% when tan δ = 1. With the increase in tan δ, storage modulus decreases by around 57% for maintaining a predetermined deformation. Moreover, for almost constant maximum displacement, materials with a higher storage modulus absorb more impact energy and experience higher impact force as compared to materials with a lower storage modulus.

On the inference of viscoelastic constants from stress relaxation experiments

Several constitutive theories have been proposed in the literature to model the viscoelastic response of materials, including widely used rheological constitutive models. These models are characterized by certain parameters (“time constants”) that define the time scales over which the material relaxes. These parameters are primarily obtained from stress relaxation experiments using curve-fitting techniques. However, the question of how best to estimate these time constants remains open. As a step towards answering this question, we propose an optimal experimental design approach based on ideas from information geometry, namely Fisher information and Kullback–Leibler divergence. The material is modeled as a spring element in parallel with multiple Maxwell elements and described using a one- or two-term Prony series. Treating the time constants as unknowns, we develop expressions for the Fisher information and Kullback–Leibler divergence that allow us to maximize information gain from experimental data. Based on the results of this study, we propose that the largest time constant estimated from a stress relaxation experiment for a linear viscoelastic material should be at most one-fifth of the total time of the experiment in order to maximize information gain. Our results also provide confirmation that the equilibrium modulus of the material cannot be reliably determined from curve-fitting to data from a stress relaxation experiment.

The development of the formalism of movable cellular automata for modeling the nonlinear mechanical behavior of viscoelastic materials

The paper is devoted to the development of the formalism of the computational method of discrete elements (DEM) for describing the mechanical behavior of consolidated viscoelastic materials. We considered an advanced implementation of DEM, namely, the method of movable cellular automata (MCA). A feature of this implementation of DEM is the use of a generalized many-body formulation of the relations for the forces of element-element interaction. 3D numerical models of viscoelastic material with a spectrum of relaxation times (Kelvin and Maxwell models, the standard model of elastomers, and others) were developed within the formalism of MCA. The correctness of the developed discrete element formalism and its applicability for modeling the processes of deformation and fracture of viscoelastic materials under dynamic loading are shown using the standard model of elastomers as an example. The relevance of the results is determined by the prospects for the further development of DEM and its application to study and predict the mechanical response of viscoelastic materials of various nature under dynamic loading including contact problems.

Low-velocity impact response of viscoelastic material filled FG honeycomb reinforced laminate plate in hygrothermal environments

In the present study, the dynamic response of functionally gradient (FG) honeycomb reinforced and viscoelastic material (VEM) embedded composite laminate plates with low-speed impact in the hygrothermal environment is investigated. The equivalent effective mechanical properties of the two-phase macroscopic constructed layers are determined according to Hill's generalized self-consistent model. The Reddy's high-order shear deformation theory (HSDT) is employed to evaluate the in-plane displacement variables. By combining the extended Hamilton's principle with the finite element method and adopting the Rayleigh's proportional damping, the global mass, damping and stiffness matrices are determined first. Then, the central displacement of the FG honeycomb reinforced laminate plate is obtained with the aid of the Newmark's method. Next, several key factors affecting the central displacement in the whole impact duration are analyzed and compared, such as the hygrothermal effects, the radius of the impactor and the functionally gradient factor. Moreover, the finite element method (FEM) is applied to obtain the mode parameters and to verify the convergence of our analysis. Finally, the LS-DYNA explicit solution is employed to validate the present results.

Fracture of viscoelastic solids modeled with a modified phase field method

Fracture of viscoelastic solids plays an important role in many applications but it is not yet well understood. In addition to the time and rate dependent response of viscoelastic materials, fracture of these solids is governed by nonlinear processes at the fracture process zone and could also be accelerated by viscous energy dissipation. To this end, we propose a new phase field formulation in which fracture of viscoelastic solids is driven by both elastic and viscous components of the energy. The formulation requires a single additional parameter to quantify the portion of the viscous energy that contributes to fracture and is shown to be thermodynamically consistent. Viscoelastic material behavior, in the form of a Generalized Maxwell model, is obtained through a standard Prony-series type expansion. Fracture driven by viscous dissipation is studied on several important benchmark problems, including (i) a bar under creep, relaxation, strain rate and cyclic loadings, and (ii) two 3-point asphalt-beam bending problems that lead to crack propagation under mode I and mixed mode conditions. It is shown that at low strain rates viscous dissipation accelerates the fracture growth rate but essentially does not affect the crack path, while at high rates the effect of viscous dissipation is minor.

Micro-computed tomography based experimental investigation of micro- and macro-mechanical response of particulate composites with void growth

ABSTRACTIn this work, we develop an image-based material testing approach using micro-computed tomography to understand the influence of microstructure and local damage phenomenon on the effective mechanical response of rubber-glass bead composites. Furthermore, a nondestructive, three-dimensional image-based analysis protocol which provides high fidelity of sample testing and data assessment has been established. An investigation was performed on various compositions of silicone rubber reinforced with silica particles. In situ compression experiments were used to study how the microscale damage (void creation from debonding) develops and evolves in the context of four primary studies: (i) effect of particle volume fraction, (ii) effect of particle diameter, (iii) local damage phenomena and its evolution (incremental loading/unloading), and (iv) effect of surface treatments on bonding characteristics. A detailed statistical analysis of the evolution of structural features through robust image processing strategies at various stages of loading was conducted. The rich data analysis collected from the experimental studies offers an understanding of the complex phenomena attributing to the material's macro and microscopic response to loading. Altogether, this framework results in the development of microstructure-statistics-property relations. Furthermore, the mechanical and morphological response of non-linear viscoelastic materials subjected to uniaxial compression is investigated.

The viscoelastic response of soft material: A theoretical and experimental study based on barreling deformation

The viscoelastic properties of polydimethylsiloxane (PDMS) measured are demonstrated by the uniaxial indentation test. Based on the barreling deformation, the viscoelastic models are established and the factors influencing the viscoelastic properties are discussed. These results indicate that the deformation response is greatly associated with load conditions and the PDMS materials exhibited more elastic properties at large load conditions as compared to small load conditions. It was reasonable to use the stress at the middle-section zone to investigate the viscoelastic behavior of the soft materials. The number and type of these elements in viscoelastic models affected the soft materials characteristics. These results would be helpful to further understand the viscoelastic properties and characterization of soft tissues.

A Straightforward Procedure to Characterize Nonlinear Viscoelastic Response of Asphalt Concrete at High Temperatures

This paper proposes a straightforward procedure to characterize the nonlinear viscoelastic response of asphalt concrete materials. Furthermore, a model is proposed to estimate the nonlinear viscoelastic parameters as a function of the triaxiality ratio, which accounts for both confinement and deviatoric stress levels. The simplified procedure allows for easy characterization of linear viscoelastic (LVE) and nonlinear viscoelastic (NVE) responses. First, Schapery’s nonlinear viscoelastic model is used to represent the viscoelastic behavior. Dynamic modulus tests are performed to calibrate LVE properties. Repeated creep-recovery tests at variable deviatoric stress levels (RCRT-VS) were designed and conducted to calibrate the nonlinear viscoelastic properties of four types of mixtures used in the Federal Aviation Administration’s National Airport Pavement and Materials Research Center test sections. The RCRT-VS were conducted at 55°C, 140 kPa initial confinement pressure, and wide range of deviatoric stress levels; mimicking the stress levels induced in a pavement structure under traffic. Once calibrated, the model was validated by comparing the model predictions and experimental measurements at different deviatoric stress levels. The predictions indicate that the proposed method is capable of characterizing NVE response of asphalt concrete materials.

Analysis of Cone Wave Reflection in Finite-Size Elastic Membranes and Extension of the Ballistic Impact Problem From Elastic to Viscoelastic Membranes

The transverse ballistic impact on a two-dimensional (2D) membrane causes a truncated deformation cone to develop in the wake of tensile implosion waves. Here, the cone wave reflected from the finite boundaries of the elastic membrane has been studied analytically. A first-order linear nonhomogeneous differential equation for the ratio of the reflected cone wave front velocity to the speed of tensile waves is derived, which is further used to calculate the traveling time taken by the reflected cone wave to reach to the projectile surface. Since the reflected wave starts when the membrane is already in a deformed configuration, the speed of the reflected cone wave is a function of radius r in the cylindrical coordinates as opposed to almost constant speed of the incoming cone wave studied in the literature. The analytical results are validated with molecular dynamics (MD) simulations of the ballistic impact of projectiles onto a single layer of coarse-grained (CG) graphene. In the second part of the paper, we analyze the membrane impact problem for linear isotropic viscoelastic materials and find that the tensile wave speed for stresses and displacements is the same as that obtained in the case of a linear isotropic elastic material. We also show that only under special conditions, self-similar solutions for the cone wave are possible in viscoelastic materials modeled by Maxwell, Kelvin–Voigt, or a combination of similar models. Our findings lay some grounds on which further studies on the ballistic response of viscoelastic materials can be performed.