Rate-dependent Mechanical Behavior Research Articles

3D printing of TPMS microlattice: indentation characterisation and numerical analysis

ABSTRACT Mechanical metamaterials are emerging as an important tool in many engineering applications, where microarchitectured lattices have advantages of high strength-to-weight ratios, energy absorption capabilities and versatility in designed mechanical behaviours. Whilst a variety of microarchitectured lattices have been previously investigated for their mechanical properties, they have been mostly constrained to strut and plate-based designs and subjected to additional post-processing to modify their functionality. In this work, we demonstrate a novel approach utilising the two-photon polymerisation (2PP) method to generate complex, defect-free microarchitectured lattices. Unlike traditional methods, our technique allows for the direct fabrication of shell-based triply periodic minimal surfaces (TPMS) without requiring any additional post-processing. We utilise gyroid and diamond TPMS unit cells to perform uniaxial micro-compression testing to characterise their mechanical properties. In doing so, we gain insights into the mechanical behaviours of microlattices, with the finding that diamond cells possess superior stiffness and energy absorption potential compared to gyroid cells. To unravel the size-dependency property alterations at this scale, we further analyse a variety of bending beam samples and observed a strain rate-dependent mechanical behaviour attributable to the material's viscoelastic nature. We employ finite element analysis to enhance our understanding of the deformation mechanisms inherent in TPMS topologies, finding good agreement with measured mechanical properties. This work establishes a comprehensive framework that spans from design and geometry characterisation to robust microstructure testing using indentation, combined with comprehensive numerical simulation.

On the strain rate-dependent mechanical behavior of PE separator for lithium-ion batteries

The separator is a critical component for ensuring electrochemical cycling performance and preventing internal short circuits in lithium-ion batteries. For the collision safety of lithium-ion batteries, understanding the rate-dependent mechanical behavior of the separator is essential for battery impact modeling and safety prediction. This study conducts a comprehensive experimental investigation into the strain rate–dependent tensile/compressive behavior and failure mechanism of the polyethylene (PE) separator under quasi-static and dynamic conditions. The combination of deformation images recorded by cameras and post-mortem characterization using SEM was employed to clarify the rate-dependent deformation and fracture mechanism of the separator under both tensile and compressive loading. The experimental results demonstrate a significant strain rate effect on the tensile/compressive mechanical properties and damage/failure behavior of the separator. Furthermore, the effect of the strain rate on the mechanical properties, including the tensile strength, tensile fracture strain, tensile elastic modulus, compressive modulus, yield stress and yield strain of separator, was analyzed and discussed. A significant strain rate-dependent tensile damage and fracture behavior of the separator was observed, where the fracture site exhibited an obvious phase transition and skeletal lamella fracture under extremely high strain rate tensile loading. The separator underwent severe damage under dynamic compressive conditions. The results of this study provide an important basis for the establishment of rate-dependent safety criterion and short circuit prediction of lithium-ion batteries under impact loading, and shed light on understanding separator failure-induced short circuit issues in battery collision safety scenarios.

A model for capturing the rate-dependent mechanical behaviour of liquid crystal elastomers

A modelling framework is adopted and adapted here to model the rate-dependent deformation behaviour of liquid crystal elastomers (LCEs). On using an isotropic hyperelastic strain energy function WI1,I2 of binomial form, previously employed for capturing the (quasi-static) deformation of polydomain LCEs, an extension is presented for application to the rate-dependent behaviour of LCEs by incorporating the deformation rate into the model parameters. That is, the model parameters of the core hyperelastic strain energy function W are considered to evolve with, i.e., be a function of, the deformation rate. A specific measure of deformation rate ɛ̇ is utilised, and a particular functional dependency of the hyperelastic model parameters on ɛ̇, of a skewed exponential form, is devised and presented. The ensuing rate-dependent model Wr is then applied to a range of extant experimental datasets, encompassing various LCE specimens and across different deformation rates, under uniaxial loading and including both tension and compression deformations. The model is shown to capture and predict the data favourably. Given the simplic of devising and implementing the presented modelling framework, the flexibility it provides for incorporating different choices of strain energy W function as well as the functional form(s) of the dependency of the model parameters on the deformation rate, and the accuracy of the modelling results, the application of this modelling approach to capturing the rate-dependent deformation behaviour of LCEs is hereby presented.

Mechanical behavior of full-thickness burn human skin is rate-independent

Skin tissue is recognized to exhibit rate-dependent mechanical behavior under various loading conditions. Here, we report that the full-thickness burn human skin exhibits rate-independent behavior under uniaxial tensile loading conditions. Mechanical properties, namely, ultimate tensile stress, ultimate tensile strain, and toughness, and parameters of Veronda–Westmann hyperelastic material law were assessed via uniaxial tensile tests. Univariate hypothesis testing yielded no significant difference (p > 0.01) in the distributions of these properties for skin samples loaded at three different rates of 0.3 mm/s, 2 mm/s, and 8 mm/s. Multivariate multiclass classification, employing a logistic regression model, failed to effectively discriminate samples loaded at the aforementioned rates, with a classification accuracy of only 40%. The median values for ultimate tensile stress, ultimate tensile strain, and toughness are computed as 1.73 MPa, 1.69, and 1.38 MPa, respectively. The findings of this study hold considerable significance for the refinement of burn care training protocols and treatment planning, shedding new light on the unique, rate-independent behavior of burn skin.

Rate-dependent energy dissipation of graded viscoelastic structures fabricated by grayscale vat photopolymerization

A major benefit of additive manufacturing technologies is precise control over structural topologies and material properties, which allows to tailor, for instance, energy absorption and dissipation. While vat photopolymerization is generally restricted to a single material, grayscale masked stereolithography (gMSLA) allows to customize material behavior by grading the light intensity within a structure. This study investigates the impact and opportunities of grayscale grading strategies on the rate-dependent mechanical behavior of structures fabricated by gMSLA. Considering the viscoelastic nature of polymers, rate-dependent energy dissipation is explored, introducing a parametric linear viscoelastic constitutive model for varying grayscales. The investigation includes the comprehensive characterization of mechanical properties, numerical finite element simulation, validation through experimental procedures, and exploration of dissipation energy under different strain rates. In this way, a rational function successfully determines the critical strain rate at which the maximum dissipation occurs. Overall, the research offers a comprehensive investigation of the mechanical dissipation behavior of graded 3D printed structures, laying the foundation for further studies and advancements aimed at optimizing these structures for enhanced energy absorption capabilities.

Modelling the rate-dependent mechanical behaviour of the brain tissue

A new modelling approach is employed in this work for application to the rate-dependent mechanical behaviour of the brain tissue, as an incompressible isotropic material. Extant datasets encompassing single- and multi-mode compression, tension and simple shear deformation(s) are considered, across a wide range of deformation rates from quasi-static to rates akin to blast loading conditions, in the order of 1000 s−1 . With a simple functional form and a reduced number of parameters, the model is shown to capture the considered rate-dependent behaviours favourably, including in both single- and multi-mode deformation fits, and over all range of deformation rates. The provided modelling results here are obtained from either first fitting the model to the quasi-static data, or/and predicting the behaviour at a different rate than those used for calibrating the model parameters. Given its simplicity, versatility, predictive capability and accuracy, the application of the utilised modelling framework in this work to the rate-dependent mechanical behaviour of the brain tissue is proposed.

A critical relook of Haasen plots to estimate activation volume in alloys

Haasen plots are well known for quantifying rate-dependent mechanical behaviour. The Haasen plot provides information regarding the activation volume and strain rate sensitivity of plastic deformation. When extending the application of Haasen plots from pure metal to alloys, the conjecture in traditional analysis is inadequate. The present work critically analyses the analytical formulation of a generalized Haasen plot for both pure metals and alloys.

Dynamic enhancement induced by interface for additively manufactured continuous carbon fiber reinforced composites

Additive manufacturing (AM) has become one advanced manufacturing method for continuous carbon fiber reinforced polymer (CCFRP) composites, which is promising to work as aerospace materials. To promote their efficient design and extensive application, series of tensile and fiber-pullout tests were implemented to investigate the rate-dependent mechanical behavior and fracture mechanism. Both tensile strength and fracture strain significantly increase with the increase of strain rate in the studied range of 0.0001–1600 s−1. Fiber-pullout test also confirms that the interfacial shear strength of composites increases with the increasing loading rate. Dynamic enhancement is analyzed quantitatively, and it is mainly contributed by the rate-dependent behavior of fiber-matrix interface. Fracture analysis indicates the stronger interfacial bonding under dynamic loading, which is attributed to the impeded interfacial crack propagation by high strain rate. A constitutive model is built to predict the rate-dependent strength of CCFRP composites, and it fits well with the experimental results.

A numerical multi-scale method for analyzing the rate-dependent and inelastic response of short fiber reinforced polymers: Modeling framework and experimental validation

This research presents a numerical multi-scale approach that efficiently addresses the inelastic and time-dependent mechanical response of short fiber reinforced polymers (SFRPs) under monotonic loading conditions by linking the mechanical analysis from microscale analysis to a continuum model. To do so, first, the mechanical performance of a recently suggested unit cell, considering the intrinsic mechanical characteristics of both fiber and matrix, is studied to address the inelastic and rate-dependent mechanical behavior of completely aligned SFRPs. Then, the evaluated mechanical response is linked to the Hill's plasticity and two-layer viscoplastic (TLVP) models to represent the anisotropic mechanical response of SFRPs. Furthermore, an easy-to-use multi-step homogenization process is considered to numerically incorporate the influence of fiber misalignments. Finally, the suggested multi-scale technique is thoroughly validated at different strain rates, by using experimental observations of short fiber composites with high volume fraction and direct FE simulations of RVEs with complex microstructures.

Effect of inelastic deformation on strain rate-dependent mechanical behaviour of human cortical bone

In this study, the role of inelastic deformation of bone on its strain rate-dependent mechanical behaviour was investigated. For this, human cortical bone samples were cyclically loaded to accumulate inelastic strain and subsequently, mechanical response was investigated under compressive loading at different strain rates. The strain rate behaviour of fatigued samples was compared with non-loaded control samples. Furthermore, cyclic loading-induced microdamage was quantified through histological analysis. The compression test results show that the strength of fatigue-loaded bone reduced significantly at low strain rates but not at high strain rates. The difference in microcrack density was not significant between fatigued and control groups. The results indicate that the mechanism of load transfer varies between low strain rate and high strain rate regimes. The inelastic deformation mechanisms are more prominent at low strain rates but not at high strain rates. This study shed light on the role of inelastic deformation on the rate-dependent behaviour of cortical bone.

Modeling the biaxial, rate-dependent response of filament-wound FRP tubes

This work studies the rate-dependent mechanical behavior of filament-wound fiber-reinforced polymer (FRP) composite pipes. Commercially available tubes with a filament winding angle of ±55° were tested under cyclic axial compression for four loading rates. Stress relaxation under constant strain was observed as well as a dependence of stress on the strain rate. A novel modeling methodology is presented to capture the nonlinear cyclic response, including the viscoelastic behavior of the epoxy matrix and the interaction of axial and hoop strains. This is accomplished by defining an element configuration with separate elements for the epoxy matrix and the glass fibers. The nonlinear and viscoelastic behavior is incorporated using the generalized Maxwell model. A machine learning (ML) calibration framework is adapted for this study and used to calibrate the nonlinear and viscoelastic properties for the analytical model using a convolutional neural network (CNN). The CNN is trained to identify and understand the interdependencies among the model parameters. The calibrated model parameters are used to simulate the experimentally measured response of the FRP tubes and were found to be applicable across the range of strain rates. The proposed modeling methodology accurately predicted the axial stress and hoop strain time histories as well as the rate-dependent stress relaxation during constant axial strains. The accuracy capturing the measured stress-strain responses demonstrated the synthetic dataset was adequate for training the CNN without requiring additional experimental data.

Phase-field modelling and analysis of rate-dependent fracture phenomena at finite deformation

AbstractFracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack driving force has to be accounted for in an appropriate manner. In addition, the resistance against crack propagation can depend on rate of deformation. In this contribution, an energetic phase-field model of rate-dependent fracture at finite deformation is presented. For the deformation of the bulk material, a formulation of finite viscoelasticity is adopted with strain energy densities of Ogden type assumed. The unified formulation allows to study different expressions for the fracture driving force. Furthermore, a possibly rate-dependent toughness is incorporated. The model is calibrated using experimental results from the literature for an elastomer and predictions are qualitatively and quantitatively validated against experimental data. Predictive capabilities of the model are studied for monotonic loads as well as creep fracture. Symmetrical and asymmetrical crack patterns are discussed and the influence of a dissipative fracture driving force contribution is analysed. It is shown that, different from ductile fracture of metals, such a driving force is not required for an adequate simulation of experimentally observable crack paths and is not favourable for the description of failure in viscoelastic rubbery polymers. Furthermore, the influence of a rate-dependent toughness is discussed by means of a numerical study. From a phenomenological point of view, it is demonstrated that rate-dependency of resistance against crack propagation can be an essential ingredient for the model when specific effects such as rate-dependent brittle-to-ductile transitions shall be described.

Effect of Pore Pressure on Strain Rate-Dependency of Coal

Viscoelastic strain rate-dependent behaviour of coal is critical in several subsurface engineering applications especially coal seams gas production. Such rate dependency is controlled by the interaction between coal bulk and gas sorption (a sorbing gas) or gas pressure (a non-sorbing gas). Despite the research conducted to date, the gas pressure effect (non-sorbing) on the viscous behaviour of sediments in particular coal remains unexplored. We, therefore, investigate the strain rate-dependent mechanical behaviour of coal under isotropic loading to specifically explore the effect of gas pressure (Helium) on its rate dependency eliminating the sorption effect. We perform a set of triaxial experiments on coal specimens at dry and pressurised gas (Helium) conditions under different strain rates under isotropic loading. The experimental results show that all coal specimens have viscoelastic strain rate dependency at a dry condition where viscous effect increases with strain rate. As a result, the bulk modulus of the specimens increases with the increase in strain rates. This strain rate dependency response, however, reduces with an increase in pore pressure and vanishes at a certain pore pressure under the same effective stress to that of dry specimens. We further employ X-ray micro-Computed Tomography (XRCT) to 3D scan a coal specimen saturated with Krypton gas undergoing different loading rates to shed light on the micro-mechanisms of gas pressure effect on specimens’ rate dependency. The XRCT results show that gas can be trapped in small-scale fractures and pores during the loading process leading to a localised undrained response that can stiffen the specimen and reduce its ability to show viscous rate dependency. The obtained results are significant in optimizing coal seam gas production and coal seam gas drainage applications.

A New Dissipation Function to Model the Rate-Dependent Mechanical Behavior of Semilunar Valve Leaflets.

A new dissipation function Wv is devised and presented to capture the rate-dependent mechanical behavior of the semilunar heart valves. Following the experimentally-guided framework introduced in our previous work (Anssari-Benam et al., 2022 "Modelling the Rate-Dependency of the Mechanical Behaviour of the Aortic Heart Valve: An Experimentally Guided Theoretical Framework," J. Mech. Behav. Biomed. Mater., 134, p. 105341), we derive our proposed Wv function from the experimental data pertaining to the biaxial deformation of the aortic and pulmonary valve specimens across a 10,000-fold range of deformation rate, exhibiting two distinct rate-dependent features: (i) the stiffening effect in σ-λ curves with increase in rate; and (ii) the asymptotic effect of rate on stress levels at higher rates. The devised Wv function is then used in conjunction with a hyperelastic strain energy function We to model the rate-dependent behavior of the valves, incorporating the rate of deformation as an explicit variable. It is shown that the devised function favorably captures the observed rate-dependent features, and the model provides excellent fits to the experimentally obtained σ-λ curves. The proposed function is thereby recommended for application to the rate-dependent mechanical behavior of heart valves, as well as other soft tissues that exhibit a similar rate-dependent behavior.

Experimental and numerical investigation on the ballistic performance of aluminosilicate glass with different nosed projectiles

An experimental-numerical investigation on the ballistic perforation of aluminosilicate glass tiles is presented in this study. A gas gun has been used to launch steel projectiles with assorted nose shapes including flat, spherical, and conical into aluminosilicate glass tiles. It is found that the residual velocity of projectiles and the fragmentation behavior of glass are strongly affected by the shape of the projectile nose. A coupled FEM-SPH numerical model is built to reproduce the ballistic impact behavior of aluminosilicate glass. The Johnson-Holmquist Ⅱ (JH-2) constitutive model is utilized to describe the rate-dependent mechanical behavior of the glass tiles. The glass tile model adopts the transformation of the finite elements into smoothed-particle hydrodynamics (SPH) elements when the failure criterion is met. This numerical method was validated by comparing with quasi-static compression, Brazilian tension, three-point-bending and dynamic SHPB tests data. The ballistic results from the numerical simulations are compared with experimental data showing good performance in predicting the projectile residual speed, glass tile inelastic deformation and fragmentation process. Finally, the effect of projectile deflection angle and tile thickness on the ballistic behavior of glass is investigated via the proposed model numerically.

A three-dimensional fractional visco-hyperelastic model for soft materials

Soft materials have attracted widespread attention and brought a wave of novel potential applications. Accurate mechanical characterization is crucial for improving the ration design of the desired soft elements and understanding the influence of deformation properties on themselves. However, the intrinsic complexity of mixed elasticity and viscosity in terms of their material property, posing new challenges for constitutive modeling. In this work, a fractional visco-hyperelastic constitutive model is developed by introducing the fractional viscoelasticity into finite strain: two branches are incorporated where a hyperelastic spring is used to describe the equilibrium response, and a non-linear fractional dashpot is introduced to characterize the time-dependent material response by employing the 3D fractional viscoelasticity. It provides a method from the perspective of fractional mechanics to avoid the increased number of governing equations and the associated complex calibration process caused by introducing the internal variables. The results show that the proposed model can successfully describe the rate dependent mechanical behavior of soft materials as well as predict the material behavior in different deformation modes with reasonable accuracy.

An electrical-mechanical cohesive zone model combining viscoelasticity and fatigue damage for soft adhesive layer in wearable sensor

The wearable biomedical devices are promising technologies in the field of human healthcare as the sensors are attached to human skin to monitor in real time physiological signal. The signal drift caused by the damage of adhesive layer between the sensor electrode and human skin is an important factor that affects the detection accuracy of the sensor. Considering the viscoelasticity and conductivity of the adhesive layer, a rate-dependent electrical–mechanical cohesive zone model (CZM) was developed to simulate the electrical signal degradation of wearable sensor electrode during cyclic loading condition. In the proposed CZM, the electrical response of the adhesive layer is related to its mechanical behavior and damage state, and the rate-dependent mechanical behavior of the adhesive layer is described by the Maxwell model. In addition, the fatigue damage evolution is also rate-dependent as the threshold of damage initiation and fracture are defined as the function of loading velocity. The proposed CZM was implemented in ABAQUS by the UEL subroutine and applied to simulate the damage and electrical signal degradation of a wearable electrocardiograph (ECG) sensor electrode in working condition. The results demonstrated that the proposed CZM is effective to characterize the rate-dependent fatigue damage behavior and electrical signal degradation behavior of the ECG electrode during cyclic loadings. It is a promising tool for predicting the electrical signal drift of wearable healthcare sensors and other similar components caused by the damage of adhesive layer, which is important for the accurate calibration of the sensors in long-term service.

A Variable-Order Fractional Constitutive Model to Characterize the Rate-Dependent Mechanical Behavior of Soft Materials

Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point.

Rate-dependent mechanical behavior of jointed rock with an impersistent joint under different infill conditions

Transition in the rate-dependent mechanical response of rock was investigated due to the presence of impersistent joint with different infill conditions. Four types of samples, i.e. intact, jointed with no grouting, jointed and grouted with cement, and jointed and grouted with epoxy, were fabricated using model material. A series of dynamic split Hopkinson pressure bar (SHPB) tests was conducted on prepared samples with strain rates varying between 53–130 s −1 along with static uniaxial compression tests (10 −4 s −1 ). Progression of fracture/failure along samples was monitored using high-speed imaging and digital image correlation (DIC). Strength was observed to be significantly lower for jointed samples as compared to intact samples. However, the increasing trend of strength with strain rates remained similar for all types of samples. Epoxy was observed to be a better grout due to greater improvement in the strength of epoxy grouted jointed samples than cement grouted samples under both static and dynamic conditions. Significant changes were observed in fracture behavior (initiation, pattern and mechanism) with strain rate for intact and jointed unfilled/grouted samples. Fracturing was dominated by shear and tensile cracks at high strain rates compared to tensile cracks at low strain rates in all samples. Unlike static loading conditions, the location of cracks initiation shifts away from joint tips with increasing strain rate and depending upon existing infill conditions (unfilled/grouted). • The effect of an impersistent joint on the rate-dependent strength and fracturing behaviour of jointed rocks is investigated and compared with that of intact rocks. • The effect of joint grouting on the rate-dependent behaviour of jointed rocks is investigated and compared with that of jointed rocks with clean joints. • The efficiency of different grout materials in the improvement of the dynamic response of jointed rocks is investigated. • Suggestive guidelines are provided for the selection of grout materials for jointed rocks under static and dynamic conditions.

Modelling the rate-dependency of the mechanical behaviour of the aortic heart valve: An experimentally guided theoretical framework

A theoretical framework, based on extant experimental findings, is presented to devise a novel viscous dissipation function Wv in order to model the rate-dependent mechanical behaviour of the aortic heart valve. The experimental data encompasses Cauchy stress-stretch (σ−λ) curves obtained across a 10,000-fold range of stretch rates (λ˙), from quasi-static (λ˙= 0.001 s−1) to upper-range of physiological (λ˙= 12.4 s−1) deformation rates. The analysis of the data elicits two important trends: (i) the mechanical behaviour of the aortic valve across the tested rates is rate-dependent, with specimens becoming stiffer by increasing rate; and (ii) there appears to be a plateau in the rate-effects on the σ−λ curves; i.e. the rate-effects approach an asymptote with increase in the stretch rate λ˙. Guided by these empirical observations, we devise our new Wv function and demonstrate that the well-known form of the dissipation function commonly used in the literature is a special case of our proposed Wv. The ensuing model is then compared against the experimental σ−λ curves and is shown to provide favourable predictions. An important advantage of the employed modelling framework is that it allows the incorporation of the rate of deformation, which is a direct experimental control parameter, as an explicit modelling variable. The application of the proposed model is thereby recommended for heart valves and other soft tissues that exhibit similar rate-dependent features.