Linear Viscoelastic Constitutive Model Research Articles

Dynamic mechanical response and failure behavior of solid propellant under shock wave impact

Solid rocket motor propellants are typically subjected to shock wave loads during ignition. To analyze the dynamic mechanical response and failure behavior of hydroxyl-terminated polybutadiene (HTPB) propellant under varying shock intensities, this study innovatively employed a shock tube apparatus in conjunction with schlieren imaging and 3D-DIC techniques. Shock loading experiments were conducted on propellant samples of three different thicknesses under pressures ranging from 0.3 to 0.7 MPa, capturing the dynamic deformation processes. Results revealed that deformation initiated at the clamped region, extending towards the center and forming a parabolic shape. The maximum deformation of the samples shows an approximately linear relationship with the launch pressure and decreases with increasing sample thickness. The 5 mm sample failed under a 0.6 MPa shock pressure, with electron microscopy images identifying matrix damage, particle debonding, and particle fracture as the principal failure mechanisms. Finite element simulations based on the generalized Maxwell linear viscoelastic constitutive model were conducted, demonstrating good agreement with experimental data. The critical stress for fracture was determined to be 4.12 MPa, and the ultimate shock wave pressures for 3, 6, 9, and 12 mm thicknesses were approximately 0.3 MPa, 0.65 MPa, 1.1 MPa, and 1.4 MPa, respectively. These findings provide valuable insights for assessing the structural integrity of solid rocket motors under ignition shock conditions.

The role of unit cell topology in modulating the compaction response of additively manufactured cellular materials using simulations and validation experiments

Additive manufacturing has enabled a transformational ability to create cellular structures (or foams) with tailored topology. Compared to their monolithic polymer counterparts, cellular structures are potentially suitable for systems requiring materials with high specific energy-absorbing capability to provide enhanced damping. In this work, we demonstrate the utility of controlling unit-cell topology with the intent of obtaining a desired stress–strain response and energy density. Using mesoscale simulations that resolve the unit-cell sub-structures, we validate the role of unit-cell topology in selectively activating a buckling mode and thereby modulating the characteristic stress–strain response. Simulations incorporate a linear viscoelastic constitutive model and a hyperelastic model for simulating large deformation of the polymer under both tension and compression. Simulated results for nine different cellular structures are compared with experimental data to gain insights into three different modes of buckling and the corresponding stress–strain response.

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.

Viscoelastic vapor bubble collapse near solid walls and corresponding shock wave formation

This study investigates the influence of viscoelasticity on the collapse of aspherical vapor bubbles near a solid boundary through numerical simulations. A fully compressible three-dimensional finite volume method is employed, incorporating a single-fluid homogeneous mixture cavitation model and the simplified linear Phan-Thien Tanner viscoelastic constitutive model. The collapse dynamics, liquid jetting, shock wave formation, and associated pressure impact are analyzed, and the viscous and viscoelastic stress fields are presented. A comparison of viscoelastic to Newtonian dynamics reveals significant differences in collapse behavior and shock wave formation due to viscoelasticity. Viscoelasticity can induce jet piercing, which is not observed in the Newtonian collapse, and increases vapor re-evaporation after the first collapse. The effect of changing the initial standoff distance is examined for both viscoelastic and Newtonian fluids, where a second jet formation is present only for the viscoelastic collapse, and the second collapse's intensity is increased due to increased vapor production during rebound. Additionally, the variation of elasticity in the viscoelastic case demonstrates a correlation between the amount of vapor produced during rebound and the relaxation time for the investigated cases.

Investigation of the Effect of Axial Gap on the Mechanical Response of a Cartridge-Loaded CMDB Propellant Grain under Vibration Loads

In this study, the effect of the axial gap on the mechanical response of a cartridge-loaded propellant grain under vibration loads is investigated. The wide strain rate range of uniaxial compression tests ( 1.7 × 10 − 3 ~ 4 × 10 3 s − 1 ) on the composite modified double base (CMDB) propellant was carried out by using a universal testing machine, a hydraulic testing machine, and a split Hopkinson pressure bar system, respectively. A linear viscoelastic constitutive model of the CMDB propellant was developed by using the experimental measurements. The results show the studied CMDB propellant has a strong strain rate dependence, exhibiting an initial linear elasticity followed by a strain hardening region. The dynamic process of collision between the propellant grain and the motor case in the axial direction induced by vibration loads was simulated with the developed constitutive model by using the finite element method. The effects of the gap size between the propellant grain and the case and the vibration frequency on the mechanical response of the grain were studied. This shows that with a constant vibration frequency, the stress of the grain increases first and then decreases with increasing gap size. Moreover, the stress increases with increasing vibration loads.

Computational Technique for Crack Propagation Simulation in Viscoelastic Solid Propellant

To further investigate the fracture response in propellant grain, numerical methodology is proposed to cope with crack propagation simulation especially for the mixed mode condition. The numerical discrete scheme of the propellant linear viscoelastic constitutive model is proposed, which provides a key means for the simulation of crack propagation. In order to simulate the cohesive traction distribution on the new crack surface, the extrinsic Park-Paulino-Roesler (PPR) cohesive zone model (CZM) is introduced. To let the crack propagate along any direction determined, element splitting technique and its corresponding topological operations are proposed step by step. Then, computational simulation implementation process is explained in greater detail. Typical fracture problem, single edge-notched tension test (SENT) is solved to demonstrate the efficiency and accuracy of the proposed method. In addition, double edge-notched tension test (DENT) as well as plate tension test with a slant crack is conducted to show the special fracture characters in viscoelastic solid propellant, like time dependence. Computational results reveal that the method proposed can be utilized in further fracture investigation in solid propellant combined with the experimental findings.

Mechanical response of HTPB propellant grain under ultra-high overload

In this study, the mechanical response of HTPB propellant grain under ultra-high loads is investigated. The low, intermediate and high strain rate of uniaxial compression tastings (1.7×10−4~3×103s−1) of the hydroxyl terminated polybutadiene (HTPB) propellant at room temperature were carried out by using the universal testing machine and the split Hopkinson pressure bar (SHPB) system, respectively. The uniaxial compression tests of the cellular rubber, the EPDM rubber and felted wool were carried out. A linear viscoelastic constitutive model of the HTPB propellant was developed by using the test data. The mechanical property of the cellular rubber, the EPDM rubber and the felted wool were studied by the test data. The dynamic mechanical response of the propellant grain under high overloads was simulated with the developed constitutive model by using the finite element method. The deformation of the grain under high overload was studied. The result shows when the overload value is maximum, the stress value of the grain is maximum. The location where the stress value of the grain is maximum is the center of the inner hole of the cushion. The maximum stress value is 20.17MPa.

Investigations of the mechanical response of dummy HTPB propellant grain under ultrahigh acceleration overload conditions using onboard flight-test measurements

In this paper, to study the mechanical responses of a solid propellant subjected to ultrahigh acceleration overload during the gun-launch process, specifically designed projectile flight tests with an onboard measurement system were performed. Two projectiles containing dummy HTPB propellant grains were successfully recovered after the flight tests with an ultrahigh acceleration overload value of 8100 g. The onboard-measured time-resolved axial displacement, contact stress and overload values were successfully obtained and analysed. Uniaxial compression tests of the dummy HTPB propellant used in the gun-launched tests were carried out at low and intermediate strain rates to characterize the propellant's dynamic properties. A linear viscoelastic constitutive model was employed and applied in finite-element simulations of the projectile-launching process. During the launch process, the dummy propellant grain exhibited large deformation due to the high acceleration overload, possibly leading to friction between the motor case and propellant grain. The calculated contact stress showed good agreement with the experimental results, though discrepancies in the overall displacement of the dummy propellant grain were observed. The dynamic mechanical response process of the dummy propellant grain was analysed in detail. The results can be used to estimate the structural integrity of the analysed dummy propellant grain during the gun-launch process.

Creep of an FRP-strengthened metallic beam under cyclic temperature and cyclic load

Metallic beams can be strengthened using fibre reinforced polymer (FRP) plates that are adhesively bonded to their soffit. This relies critically upon the bonded adhesive joint, and because the adhesives are viscoelastic it is known that creep at warm services temperatures can potentially result in large slips that can reduce the effectiveness of the strengthening. This paper examines the impact of adhesive creep upon the performance of an FRP-strengthened metallic beam when subjected to daily temperature cycles and daily load cycles. A finite element (FE) study was conducted that incorporates a linear viscoelastic constitutive model for the adhesive, temperature and load cycles of up to 50 years were considered, and the behaviour of the strengthened beam was examined. Differential thermal expansion between the FRP plate and metallic beam is shown to be significant because it results in both (a) additional instantaneous shear stress across the adhesive joint and (b) the shear stress due to sustained differential thermal expansion results in additional creep of the adhesive that can further reduce the effectiveness of the strengthening. An equivalent cumulative time approach is demonstrated to predict the creep response of the strengthened metallic beam without the need for a complex cyclic analysis.

Investigation of acoustic characterization of ultrasonic air coupling testing of NEPE propellant under strain control

To explore the acoustic characterization of composite solid propellant materials, the wave equation of the viscoelastic body was derived based on the linear viscoelastic constitutive model, and the theoretical relationship between ultrasonic sound velocity and attenuation coefficient, which are closely related to mechanical properties, such as elastic modulus, viscosity coefficient, and complex modulus, was obtained. Ultrasonic air-coupled testing experiments of the NEPE (Nitrate Ester Plasticized Polyether) propellant under different tensile strain states were carried out to obtain the variation law of ultrasonic transmission wave peak amplitude, sound velocity, and attenuation coefficient with strain variables in combination with the variation in propellant porosity under tensile strain. Furthermore, the acoustic parameters with strain and porosity were fitted by the regression equation, and the effect of mesoscopic damage on macroscopic mechanical properties was analyzed. The results show that with the increase in NEPE propellant tensile strain, the porosity increases exponentially. With the increase in porosity, the sound velocity decreases by first polynomials, the wave amplitude decreases by second polynomials, and the attenuation coefficient increases by third polynomials.

A comparison of the influence of nonlinear and linear creep on the behaviour of FRP-bonded metallic beams at warm temperatures

This study investigates whether it is necessary to consider nonlinear (rather than linear) adhesive viscoelasticity when considering the behaviour of an FRP-bonded metallic beam at warm service temperatures (25 °C, 30 °C and 40 °C). The paper presents finite element analyses that compare the effects of nonlinear and linear creep in the bonded joint for two different FRP-strengthened metallic beams (steel and cast-iron). This modelling work incorporates a viscoelastic adhesive constitutive model determined using a dynamic mechanical analyser for a commonly used ambient cure structural epoxy adhesive. The nonlinear viscoelastic behaviour of the strengthening adhesive was characterised using time-temperature superposition combined with the parallel rheological framework model. The study found that a model incorporating nonlinear viscoelastic creep only leads to a slightly larger joint slip (maximum 1.0% after 1 year) and a slightly lower CFRP axial stress (maximum 2.4% after 1 year), compared to an equivalent model using linear creep. This has a limited impact on the structural performance for the cases studied. In most cases, the simpler linear viscoelastic constitutive model is sufficient to analyse the behaviour of the FRP-bonded metallic beam in a warm environment.

A viscoelasticity model for polymers: Time, temperature, and hydrostatic pressure dependent Young's modulus and Poisson's ratio across transition temperatures and pressures

Capturing and fundamentally understanding the variation of a polymer's mechanical properties with time, temperature, and hydrostatic pressure is essential for accurate constitutive modeling. In this paper, a new linear viscoelastic constitutive model is developed that regards Young's modulus and Poisson's ratio as functions of time, temperature, and hydrostatic pressure. Modified logistic functions are employed, which provide appealing mathematical and physical simplicity. Aided by time/temperature/hydrostatic-pressure superposition, these functions can be transformed from the time domain to the temperature or pressure domain, or the converse. Our model is successfully benchmarked against numerous sets of experimental data. Our model is also correlated to polymer processing through continuous cooling transformation (CCT) diagrams. Since CCT diagrams help determine a polymer's microstructure, which in turn determines its macroscopic mechanical properties, our model can be used as a guide for tailoring the manufacturing process (e.g., controlling quenching temperature and speed) to obtain a targeted set of mechanical properties. Our new model can also be embedded within existing nonlinear viscoelastic constitutive frameworks to capture the mechanical behavior of polymers across a wide range of strain rates, temperatures, and pressures.

Dynamic behaviour of coupled soil-structure systems by means of FEM analysis for the seismic risk mitigation of INGV building in Catania (Italy)

Seismic design of new structures, as well as retrofitting and/or improving of existing ones should be definitely considered a multidisciplinary subject, which depends on many factors, such as: local site effects and the dynamic interaction between the foundation soil and the structure. The accurate investigation on the structure and the surrounding soil is the first fundamental step for a realistic evaluation of the structure seismic performance. The present paper deals with the Dynamic Soil Structure Interaction (DSSI) analysis concerning the INGV (National Institute of Geophysics and Volcanology) building in Catania, by means of a FEM 2D modeling. The building is a prestigious masonry structure situated in an area characterized by a high seismic hazard. Several accelerograms scaled at the same PHA, with reference to the estimated seismicity of Catania, have been adopted. Soil properties were carefully investigated by means of static and dynamic in-situ and laboratory tests. Many investigations were also performed on the structure. Equivalent linear visco-elastic constitutive models have been adopted both for the soil and the structure. For considering soil nonlinearity, degraded shear modula ( G ) and increased soil damping ratios ( D ) have been evaluated for all the involved soil layers, according to two different approach es. Firstly, soil nonlinearity has been modeled basing on the EC8 [2003] suggestions; secondly, it has been modeled choosing the values of G and D according to the effective strain levels obtained for each soil layer and for each different input, by means of an iterative sub-routine . The dynamic response of the system has been analyzed in the time and frequency domains. Results are presented in terms of: acceleration amplification factors, Fourier and response spectra, amplification functions and shear forces per floor. The main goals of the paper are: i) to investigate the acceleration profiles along the soil and the structure considering and not considering the DSSI; ii) to investigate the soil filtering effect in terms of predominant frequency considering and not considering the DSSI; iii) to compare the obtained results with the ones given by a simpler 1D free-field soil analysis; iv) to compare the soil amplification factors and the response spectra obtained by 1D and 2D models with that by the Italian technical code [NTC, 2008]; v) to highlight the influence of DSSI in the seismic response of the structure; vi) to evaluate the influence of different modeling of soil nonlinearity on the dynamic response of the soil and structure.

Prediction of brain deformations and risk of traumatic brain injury due to closed-head impact: quantitative analysis of the effects of boundary conditions and brain tissue constitutive model.

In this study, we investigate the effects of modelling choices for the brain-skull interface (layers of tissues between the brain and skull that determine boundary conditions for the brain) and the constitutive model of brain parenchyma on the brain responses under violent impact as predicted using computational biomechanics model. We used the head/brain model from Total HUman Model for Safety (THUMS)-extensively validated finite element model of the human body that has been applied in numerous injury biomechanics studies. The computations were conducted using a well-established nonlinear explicit dynamics finite element code LS-DYNA. We employed four approaches for modelling the brain-skull interface and four constitutive models for the brain tissue in the numerical simulations of the experiments on post-mortem human subjects exposed to violent impacts reported in the literature. The brain-skull interface models included direct representation of the brain meninges and cerebrospinal fluid, outer brain surface rigidly attached to the skull, frictionless sliding contact between the brain and skull, and a layer of spring-type cohesive elements between the brain and skull. We considered Ogden hyperviscoelastic, Mooney-Rivlin hyperviscoelastic, neo-Hookean hyperviscoelastic and linear viscoelastic constitutive models of the brain tissue. Our study indicates that the predicted deformations within the brain and related brain injury criteria are strongly affected by both the approach of modelling the brain-skull interface and the constitutive model of the brain parenchyma tissues. The results suggest that accurate prediction of deformations within the brain and risk of brain injury due to violent impact using computational biomechanics models may require representation of the meninges and subarachnoidal space with cerebrospinal fluid in the model and application of hyperviscoelastic (preferably Ogden-type) constitutive model for the brain tissue.

Viscoelastic and Damage Model of Polyethylene Pipe Material for Slow Crack Growth Analysis

Polyethylene (PE) pipe is widely used for oil and gas transportation. Slow crack growth (SCG) is the main failure mechanism of PE pipes. Current SCG resistance testing methods for PE pipes have significant drawbacks, including high cost, time-consuming, and uncertain reliability. Alternative method is in need to reduce the testing time and cost. In this paper, a numerical model is proposed by taking the viscoelastic and damage effect of PE material into account. The material behavior is described on the basis of linear viscoelastic integral constitutive model, along with the damage effect in effective configuration concept. A three-dimensional (3D) incremental form of a viscoelastic and damage model is derived and implemented by abaqus UMAT. It is found that the curve of tensile displacement versus time, as well as the curve of crack opening displacement (COD) versus time from numerical results fit well with those from the standard Pennsylvania Notch Test (PENT; ASTM 1473). Based on the proposed model, SCG failure process is analyzed, and the effects of damage parameters on SCG process are furtherly studied and discussed.

Characterizing viscoelastic mechanical properties of highly compliant polymers and biological tissues using impact indentation

Precise and accurate measurement of viscoelastic mechanical properties becomes increasingly challenging as sample stiffness decreases to elastic moduli <1 kPa, largely due to difficulties detecting initial contact with the compliant sample surface. This limitation is particularly relevant to characterization of biological soft tissues and compliant gels. Here, we employ impact indentation which, in contrast to shear rheology and conventional indentation, does not require contact detection a priori, and present a novel method to extract viscoelastic moduli and relaxation time constants directly from the impact response. We first validate our approach by using both impact indentation and shear rheology to characterize polydimethylsiloxane (PDMS) elastomers of stiffness ranging from 100 s of Pa to nearly 10 kPa. Assuming a linear viscoelastic constitutive model for the material, we find that the moduli and relaxation times obtained from fitting the impact response agree well with those obtained from fitting the rheological response. Next, we demonstrate our validated method on hydrated, biological soft tissues obtained from porcine brain, murine liver, and murine heart, and report the equilibrium shear moduli, instantaneous shear moduli, and relaxation time constants for each tissue. Together, our findings provide a new and straightforward approach capable of probing local mechanical properties of highly compliant viscoelastic materials with millimeter scale spatial resolution, mitigating complications involving contact detection or sample geometric constraints. Statement of significanceCharacterization and optimization of mechanical properties can be essential for the proper function of biomaterials in diverse applications. However, precise and accurate measurement of viscoelastic mechanical properties becomes increasingly difficult with increased compliance (particularly for elastic moduli <1 kPa), largely due to challenges detecting initial contact with the compliant sample surface and measuring response at short timescale or high frequency. By contrast, impact indentation has highly accurate contact detection and can be used to measure short timescale (glassy) response. Here, we demonstrate an experimental and analytical method that confers significant advantages over existing approaches to extract spatially resolved viscoelastic moduli and characteristic time constants of biological tissues (e.g., brain and heart) and engineered biomaterials.

Comparison of rheological properties of chosen elastomers used in production of bridge bearings

The paper presents the results of rheological testing performed with the use of DMA (Dynamic Mechanical Analyser) rheometer on specimens made of natural rubber (NR) and synthetic rubber (CR) elastomeric materials typically used for production of bridge bearings. The experiments provided the values of complex stiffness modulus and phase shift angles over a very wide temperature and frequency range, enabled determination of glass transition temperatures and plotting the master curves of the respective materials and, moreover, indentation tests were carried out with a spherical indenter pressed down with linearly increasing force. The test results can be used for both qualitative and quantitative evaluation of the materials and for determining certain mechanical parameters of linear viscoelastic constitutive models formulated for the infinitesimal strain theory. The indentation test was conducted to obtain values which can be used for indirect validation of constitutive hyperelastic and visco-hyperelastic models built up for continuum large deformation theory. Moreover, the paper partly evaluates the effect of long-term (natural) ageing of materials on their rheological properties.

Viscoelastic braided stent: Finite element modelling and validation of crimping behaviour

Braided stents are fabricated at their relaxed dimensions and need to be crimped in order to fit into the catheter for deployment. Once the covering sheath is removed, the braided stent self-expands at the diameter of the diseased lumen, the radial force restoring the luminal patency. The mechanical properties of the viscoelastic monofilament used in the fabrication of the braided stents are analysed. The monofilament is subjected to both uniaxial tensile and extensional creep testing, with the purpose of predicting the material's time dependent characteristics. An accurate finite element model is required to enable correct estimation of the mechanical behaviour of braided stents subjected to crimping. In this report, Abaqus (v6.14) was used to develop a linear viscoelastic constitutive material model, where the creep compliance function and stress relaxation function were derived from experimental data. The material model was validated by performing simulation of radial compression for a braided stent and a comparison between the experimentally evaluated radial forces of the braided stent and the finite element prediction is presented. The regression analysis performed for the viscoelastic material model led to an R-squared value of 94.4%, showing a very good agreement between the predicted values and the experimental test data.

Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation

Compliant impacts can be modeled using linear viscoelastic constitutive models. While such impact models for realistic viscoelastic materials using integer order derivatives of force and displacement usually require a large number of parameters, compliant impact models obtained using fractional calculus, however, can be advantageous since such models use fewer parameters and successfully capture the hereditary property. In this paper, we introduce the fractional Chebyshev collocation (FCC) method as an approximation tool for numerical simulation of several linear fractional viscoelastic compliant impact models in which the overall coefficient of restitution for the impact is studied as a function of the fractional model parameters for the first time. Other relevant impact characteristics such as hysteresis curves, impact force gradient, penetration and separation depths are also studied.

Linear orthotropic viscoelasticity model for fiber reinforced thermoplastic material based on Prony series

Material properties description and understanding are essential aspects when computational solid mechanics is applied to product development. In order to promote injected fiber reinforced thermoplastic materials for structural applications, it is very relevant to develop material characterization procedures, considering mechanical properties variation in terms of fiber orientation and loading time. Therefore, a methodology considering sample manufacturing, mechanical tests and data treatment is described in this study. The mathematical representation of the material properties was solved by a linear viscoelastic constitutive model described by Prony series, which was properly adapted to orthotropic materials. Due to the large number of proposed constitutive model coefficients, a parameter identification method was employed to define mathematical functions. This procedure promoted good correlation among experimental tests, and analytical and numerical creep models. Such results encourage the use of numerical simulations for the development of structural components with the proposed linear viscoelastic orthotropic constitutive model. A case study was presented to illustrate an industrial application of proposed methodology.