Maxwell Viscoelastic Model Research Articles

Reconciling viscoelastic models of postseismic and interseismic deformation: Effects of viscous shear zones and finite length ruptures

AbstractWe have developed a suite of earthquake cycle models for strike‐slip faults to investigate how finite ruptures and lithosphere‐scale viscous shear zones affect interseismic deformation. In particular, we assess whether localized and stationary interseismic deformation and large‐scale, rapidly decaying postseismic transients may be explained with models incorporating either or both of these features. Models incorporating viscous shear zones give more stationary interseismic deformation than layered half‐space, Maxwell viscoelastic models producing similar early postseismic deformation in the near field. This tendency is accentuated when the effective viscosity per unit width of the shear zone increases either with depth or with interseismic time. Models with finite (200 km long) ruptures produce time‐dependent deformation similar to that from models with infinitely long ruptures, though with smaller‐magnitude and somewhat more localized surface velocity fluctuations. Models incorporating a stiff lithosphere, a low‐viscosity mantle asthenosphere, and a crust‐ to lithosphere‐scale viscous shear zone can replicate postseismic and interseismic deformation typical of large, strike‐slip earthquakes. Such models require depth‐dependent, power law, or transient rheologies for the viscous shear zone material in the crust, as long as the effective viscosity per unit shear zone width is approximately 1015 Pa s/m at crustal depths during the postseismic interval. Below the Moho, the shear zone effective viscosity per unit width must be higher throughout the seismic cycle, at least over a short depth interval. For example, if shear zone viscosity per unit width is 5×1016 Pa s/m in the mantle lithosphere, a model with a 50 km thick lithosphere and asthenosphere effective viscosities of 1 to 5×1018 Pa s can reproduce reference velocity profiles for both postseismic and later interseismic deformation.

Determining effects of moisture in mastic materials using nanoindentation

Asphalt concrete consists of coarse aggregates coated with asphalt binder, matrix, which is a mixture of binder and fine aggregates, and mastic, which is a mixture of asphalt binder and fines passing number 200 sieve (0.075 mm). In this study, nanoindentation tests were conducted on dry and wet mastic materials to determine the contact creep compliance, which is used to examine the effects of moisture in the mastic materials. Indentation creep data were fitted using viscoelastic mechanical models. Results show that the dry mastic materials exhibits viscoelastic behavior, while the wet mastic materials shows less viscoelastic behavior compared to the dry mastic materials. Moisture reduces retardation time significantly in the wet mastic materials. The dry mastic materials follow the linear Burgers viscoelastic model and the wet mastic materials follow the Maxwell viscoelastic model. Stiffness measured on the surface of the wet mastic materials is higher than that of the dry mastic materials. Due to moisture conditioning, mastic sample surface might have eroded that makes it less viscous or become exposed to mastic aggregate, and therefore exhibits high stiffness. Indentation results reveal that the wet mastic is softer below a certain depth from the surface. This study projects that the indenter needs to penetrate more than 4000 nm to reach softer wet mastic materials. Also indentation creep holding time needs be more than 1200 s to reach that target depth in wet mastic materials.

Monitoring viscosity changes from time‐lapse seismic attenuation: case study from a heavy oil reservoir

ABSTRACTHeating heavy oil reservoirs is a common method for reducing the high viscosity of heavy oil and thus increasing the recovery factor. Monitoring of these viscosity changes in the reservoir is essential for delineating the heated region and controlling production. In this study, we present an approach for estimating viscosity changes in a heavy oil reservoir. The approach consists of three steps: measuring seismic wave attenuation between reflections from above and below the reservoir, constructing time‐lapse Q and Q−1 factor maps, and interpreting these maps using Kelvin–Voigt and Maxwell viscoelastic models. We use a 4D relative spectrum method to measure changes in attenuation. The method is tested with synthetic seismic data that are noise free and data with additive Gaussian noise to show the robustness and the accuracy of the estimates of the Q‐factor. The results of the application of the method to a field data set exhibit alignment of high attenuation zones along the steam‐injection wells, and indicate that temperature dependent viscosity changes in the heavy oil reservoir can be explained by the Kelvin–Voigt model.

A model of viscoelastic ice-shelf flexure

AbstractWe develop a formal thin-plate treatment of the viscoelastic flexure of floating ice shelves as an initial step in treating various problems relevant to ice-shelf response to sudden changes of surface loads and applied bending moments (e.g. draining supraglacial lakes, iceberg calving, surface and basal crevassing). Our analysis is based on the assumption that total deformation is the sum of elastic and viscous (or power-law creep) deformations (i.e. akin to a Maxwell model of viscoelasticity, having a spring and dashpot in series). The treatment follows the assumptions of well-known thin-plate approximation, but is presented in a manner familiar to glaciologists and with Glen’s flow law. We present an analysis of the viscoelastic evolution of an ice shelf subject to a filling and draining supraglacial lake. This demonstration is motivated by the proposition that flexure in response to the filling/drainage of meltwater features on the Larsen B ice shelf, Antarctica, contributed to the fragmentation process that accompanied its collapse in 2002.

Corneal hyper-viscoelastic model: derivations, experiments, and simulations.

The aim of this study is to propose a method to construct corneal biomechanical model which is the foundation for simulation of corneal microsurgery. Corneal material has two significant characteristics: hyperelastic and viscoelastic. Firstly, Mooney-Rivlin hyperelastic model of cornea obtained based on stored-energy function can be simplified as a linear equation with two unknown parameters. Then, modified Maxwell viscoelastic model of the cornea, whose analytical form is consistent with the generalized Prony-series model, is proposed from the perspective of material mechanics. Parameters of the model are determined by the uniaxial tensile tests and the stress-relaxation tests. Corneal material properties are simulated to verify the hyper-viscoelastic model and measure the effectiveness of the model in the finite element simulation. On this basis, an in vivo model of the corneal is built. And the simulation of extrusion in vivo cornea shows that the force is roughly nonlinearly increasing with displacement, and it is consistent with the results obtained by extrusion experiment of in vivo cornea. Conlusions: This paper derives a corneal hyper-viscoelastic model to describe the material properties more accurately, and explains the mathematical method for determination of the model parameters. The model is an effective biomechanical model, which can be directly used for simulation of trephine and suture in keratoplasty. Although the corneal hyper-viscoelastic model is taken as the object of study, the method has certain adaptability in biomechanical research of ophthalmology.

Elevated temperature nanoindentation characterization of poly(para-phenylene vinylene) conjugated polymer films

Temperature dependent mechanical properties of poly(p-phenylene vinylene) (PPV) were investigated using quasi-static (QS) and dynamic nanoindentation (NI) at temperatures over the range of 25 to 100 °C. The reduced modulus decreased from about 4.40 GPa to 3.64 GPa over this temperature range. The plasticity indices at all measurement temperatures were lower than the critical value of 0.875, characterizing material “sink-in”, rather than “pile-up” during measurements. The plasticity index showed a non-monotonic trend, with a minimum value at around 70 °C. Analysis of indentation stress relaxation data, obtained at different temperatures, was also performed using generalized Maxwell viscoelastic models. From these analyses, a relaxation mode, with a characteristic relaxation time of approximately 0.5 s, was evident. The characteristic time remained relatively unchanged over the temperature range of 25 to 100 °C. However, the relaxation modulus associated with this mode showed the expected decrease with increase in temperature.

Finite element simulation of the slumping process of a glass plate using 3D generalized viscoelastic Maxwell model

Glass slumping process is widely developed toward several industrial applications and also for high-technology uses. It is however difficult to properly predict the glass behavior during the forming process using numerical methods. This study aims at better understanding some of the issues related to finite element modeling of the slumping process. A numerical implementation of the generalized Maxwell model (GMM) is performed to reproduce the 3D viscoelastic behavior of a soda–lime–silica glass plate during the process. The material routine is first verified against experimental and analytical quasistatic isothermal creep recovery tests. Then, FE simulations of glass slumping reveal two important points: for the so-called sag-bending process, the simple Maxwell rheological model is sufficient to take into account the deviatoric part of the glass behavior, whereas a Zener model is required for the bulk behavior. However, in the Drop-Ring-Glass-Slumping application, working with a simple Maxwell or the 6 arms GMM does not give the same results. The applied severe contact condition or the fact that axisymmetric simulation is retained could be the origin of this discrepancy.

Mechanical modelling of confined cell migration across constricted-curved micro-channels.

Confined migration is a crucial phenomenon during embryogenesis, immune response and cancer. Here, a two-dimensional finite element model of a HeLa cell migrating across constricted-curved micro-channels is proposed. The cell is modelled as a continuum with embedded cytoplasm and nucleus, which are described by standard Maxwell viscoelastic models. The decomposition of the deformation gradient is employed to define the cyclic active strains of protrusion and contraction, which are synchronized with the adhesion forces between the cell and the substrate. The micro-channels are represented by two rigid walls and exert an additional viscous force on the cell boundaries. Five configurations have been tested: 1) top constriction, 2) top-bottom constriction, 3) shifted top-bottom constriction, 4) embedded obstacle and 5) bending micro-channel. Additionally, for the first four micro-channels both sub-cellular and sub-nuclear constrictions have been obtained, while for the fifth micro-channel three types of bending have been investigated ('curved', 'sharp' and 'sharper'). For each configuration, several parameters such as the cell behaviour, the covered distance, the migration velocity, the ratio between the cell and the nucleus area as well as the cell-substrate and cell-channel surfaces forces have been evaluated. The results show once more the fundamental role played by mechanics of both the cell and the environment.

Nonrigid Registration of Monomodal MRI Using Linear Viscoelastic Model

This paper describes a method for nonrigid registration of monomodal MRI based on physical laws. The proposed method assumes that the properties of image deformations are like those of viscoelastic matter, which exhibits the properties of both an elastic solid and a viscous fluid. Therefore, the deformation fields of the deformed image are constrained by both sets of properties. After global registration, the local shape variations are assumed to have the properties of the Maxwell model of linear viscoelasticity, and the deformation fields are constrained by the corresponding partial differential equations. To speed up the registration, an adaptive force is introduced according to the maximum displacement of each iteration. Both synthetic datasets and real datasets are used to evaluate the proposed method. We compare the results of the linear viscoelastic model with those of the fluid model on the basis of both the standard and adaptive forces. The results demonstrate that the adaptive force increases in both models and that the linear viscoelastic model improves the registration accuracy.

A New Model for Plastic-Viscoelastic Magnetohydrodynamic (MHD) Flow with Radiation Thermal Transfer

Abstract This paper presents a research for plastic-viscoelastic magnetohydrodynamic (MHD) flow with radiation thermal transfer. A new model is firstly proposed by modifying the classical Maxwell viscoelastic fluid model with the power-law fluid as the damping fluid to substitute the Newtonian fluid as a dashpot. The serial connection gives the nonlinear partial differential equation models. Taylor expansion is used to reduce the nonlinear equation models into linear ones and the analytical solutions are obtained for velocity and temperature fields in terms of G function and Pochhammer polynomial. Moreover, some graphs are presented for associated parameters and the corresponding flow and heat transfer characteristics are analyzed. The ideas and methods of this paper may be used to study other fluid model, such as the generalized Oldroyd-B Fluid or the generalized Burgers' fluid.

A noniterative design procedure for supplemental brace–damper systems in single‐degree‐of‐freedom systems

SUMMARYIn this paper, a method for designing supplemental brace–damper systems in single‐degree‐of‐freedom (SDOF) structures is presented. We include the effects of the supporting brace stiffness in the dynamic response by using a viscoelastic Maxwell model. On the basis of the study of an SDOF under ground excitation, we propose a noniterative design procedure for simultaneously specifying both the damper and the brace while assuring a desired structural performance. It is shown that to increase the damper size beyond the value delivered by the proposed criteria will not provide any improvement but actually worsen the structural response. The design method presented here shows excellent agreement with the FEMA 273 design approach but offers solutions closer to optimality. Copyright © 2013 John Wiley & Sons, Ltd.

Transient shear banding in viscoelastic Maxwell fluids

The fluidization of complex fluids is studied in the context of a Maxwell viscoelastic structural fluid model and compared to the purely viscous case. Solving iteratively the structural models together with the Navier–Stokes equation for the circular Couette flow gives spatially and temporally resolved velocity fields closely resembling those found experimentally for viscoelastic carbopol gels. Namely, transient shear banding is found during the initial fluidization phase. Although both structural models show transient shear bands, the viscoelastic one captures the experimental observations in greater detail, showing, for instance, the elastic backward flows during the transient shear band initialization stage.

Modelling and simulation of the coefficient of restitution of the sliotar in hurling

The coefficient of restitution (COR) is one of the most important standards in all bat-and-ball sports to study the ball suitability and bouncing characteristics. Currently, a demand exists for an improved analytical model for computing the COR, particularly when a new material is introduced. A viscoelastic ball known as a sliotar is taken from the Irish bat-and-ball sport of hurling and used in this paper as a case study. For the theoretical approach, a modified Maxwell's viscoelastic model was used to derive an analytical formula to predict the COR. The developed formula confirms that the damping ratio and the dependency of the COR on the impact speed are the most dominant factors. A 3-dimensional finite element (FE) model was developed to simulate the sliotar impact test to estimate the COR, and to provide a simple tool for further studies on the sliotar. A high-speed camera was used to film the impact event to validate the models' behavior. The close correlation of 9% and 5% between the experimental and the developed analytical and FE models, respectively, indicates that the developed models can successfully identify the COR of the ball in the game of hurling and similar bat-and-ball sports.

Timing of the most recent Neoglacial advance and retreat in the South Shetland Islands, Antarctic Peninsula: insights from raised beaches and Holocene uplift rates

The timing of the most recent Neoglacial advance in the Antarctic Peninsula is important for establishing global climate teleconnections and providing important post-glacial rebound corrections to gravity-based satellite measurements of ice loss. However, obtaining accurate ages from terrestrial geomorphic and sedimentary indicators of the most recent Neoglacial advance in Antarctica has been hampered by the lack of historical records and the difficulty of dating materials in Antarctica. Here we use a new approach to dating flights of raised beaches in the South Shetland Islands of the northern Antarctic Peninsula to bracket the age of a Neoglacial advance that occurred between 1500 and 1700 AD, broadly synchronous with compilations for the timing of the Little Ice Age in the northern hemisphere. Our approach is based on optically stimulated luminescence of the underside of buried cobbles to obtain the age of beaches previously shown to have been deposited immediately inside and outside the moraines of the most recent Neoglacial advance. In addition, these beaches mark the timing of an apparent change in the rate of isostatic rebound thought to be in response to the same glacial advance within the South Shetland Islands. We use a Maxwell viscoelastic model of glacial-isostatic adjustment (GIA) to determine whether the rates of uplift calculated from the raised beaches are realistic given the limited constraints on the ice advance during this most recent Neoglacial advance. Our rebound model suggests that the subsequent melting of an additional 16–22% increase in the volume of ice within the South Shetland Islands would result in a subsequent uplift rate of 12.5 mm/yr that lasted until 1840 AD resulting in a cumulative uplift of 2.5 m. This uplift rate and magnitude are in close agreement with observed rates and magnitudes calculated from the raised beaches since the most recent Neoglacial advance along the South Shetland Islands and falls within the range of uplift rates from similar settings such as Alaska.

Dynamic mechanical analysis of magnetic tapes at ultra‐low frequencies

AbstractA custom, ultra‐low frequency, dynamic mechanical analyzer (ULDMA) has been developed to study the correlated effects of temperature and frequency on the viscoelastic behavior of magnetic tapes. It has been used to acquire data needed for the development of future magnetic tapes that require an archival life of up to 100 years. A range of elevated temperatures is used to simulate real‐world storage environments, which enables the investigation of how the viscoelastic characteristics of tape samples influence the extent to which the tape deforms. The experiments and subsequent analysis examine the influence of the molecular structure on the viscoelasticity of magnetic tapes. Experiments were performed on a variety of magnetic tapes, including poly(ethylene terephthalate) (PET), poly(ethylene naphthalate) (PEN), metalized PET (M‐PET), and metalized Spaltan (M‐SPA). Additional experiments examined PEN and PET substrates by removing the front and back magnetic layers from the tape sample. Because of the viscoelastic behavior of the tapes, a time delay was present between the strain and stress signals, which was determined using a Fourier transform program. The elastic modulus (E), storage modulus (E′), loss modulus (E″), and loss tangent (tan δ) were obtained from the time delay for each of the ULDMA experiments at 25, 50, and 70°C over the frequency range of 0.0100–0.0667 Hz. Plots of these mechanical characteristics demonstrate the ability of frequency and temperature to affect trends associated with mechanical and thermal properties. Finally, some samples displayed an initial relaxation during the ULDMA experiments, which, when modeled using Maxwell's viscoelastic model, provided an insight into the relaxation characteristics of the samples. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013

Mechanical properties of human tympanic membrane in the quasi-static regime from in situ point indentation measurements

The tympanic membrane is a key component of the human auditory apparatus. Good estimates of tympanic membrane mechanical properties are important to obtain realistic models of middle ear mechanics. Current literature values are almost all derived from direct mechanical tests on cut-out strips. For a biomedical specimen like the tympanic membrane, it is not always possible to harvest strips of uniform and manageable geometry and well-defined size suitable for such mechanical tests.In this work, elastic and viscoelastic properties of human tympanic membrane were determined through indentation testing on the tympanic membrane in situ. Indentation experiments were performed on three specimens with a custom-built apparatus that was also used in previously published works. Two types of indentation tests were performed on each specimen: (i) sinusoidal indentation at 0.2 Hz yielding the quasi-static Young's modulus and (ii) step indentation tests yielding viscoelastic properties in the quasi-static regime (0–20 Hz).In the cyclic indentation experiments (type i), the indentation depth and resulting needle force were recorded. The unloaded shape of the tympanic membrane and the membrane thickness were measured and used to create a specimen-specific finite element model of the experiment. The Young's modulus was then found through optimization of the error between model and experimental data; the values that were found for the three different samples are 2.1 MPa, 4.4 MPa and 2.3 MPa. A sensitivity analysis showed that these values are very sensitive to the thickness used in the models.In the step indentation tests (type ii), force relaxation was measured during 120 s and the relaxation curves were fitted with a 5 parameter Maxwell viscoelastic model. The relaxation curves in the time domain were transformed to complex moduli in the frequency domain, yielding viscoelastic properties in the quasi-static regime only.

MHD Peristaltic Motion of a Maxwell Fluid in Axisymmetric Tube

Peristaltic motion of a Maxwell fluid in an axisymmetric tube with a sinusoidal wave train traveling along the wall is studied under the long wavelength and low Reynolds number approximation. This analysis can model the transport of blood in small vessels by using the viscoelastic Maxwell model. The fluid is electrically conducting fluid in the presence of a magnetic field. Analytical solutions for the stream velocity, the axial velocity and the pressure gradient have been obtained. The effects of the Hartman number, the flow rate and the amplitude ratio on the pressure gradient, the pressure rise and the frictional force per wavelength are discussed through numerical computations.

A Phenomenological Thermal-Mechanical Viscoelastic Constitutive Modeling for Polypropylene Wood Composites

This paper presents a phenomenological thermal-mechanical viscoelastic constitutive modeling for polypropylene wood composites. Polypropylene (PP) wood composite specimens are compressed at strain rates from 10<sup >&#x2212;4</sup> to 10<sup >&#x2212;2</sup>&#x2009;s<sup >&#x2212;1</sup> and at temperature of <svg style="vertical-align:-0.20474pt;width:32.787498px;" id="M1" height="13.1875" version="1.1" viewBox="0 0 32.787498 13.1875" width="32.787498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.875)"><path id="x39" d="M244 635q90 0 143 -72t53 -177q0 -133 -65 -229.5t-171 -139.5q-79 -32 -140 -32l-5 30q109 18 185 91t101 186l-68 -36q-29 -16 -60 -16q-79 0 -129 51.5t-50 130.5q0 80 57 146.5t149 66.5zM228 602q-52 0 -78 -45.5t-26 -98.5q0 -69 36.5 -115.5t97.5 -46.5&#xA;q53 0 90 28q4 31 4 66q0 51 -9.5 95.5t-39 80.5t-75.5 36z" /></g><g transform="matrix(.017,-0,0,-.017,8.222,12.875)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> <g transform="matrix(.012,-0,0,-.012,16.375,4.712)"><path id="x2218" d="M326 255q0 -56 -41 -96t-99 -40t-99 40t-41 96t41 96t99 40t99 -40t41 -96zM274 255q0 38 -25.5 65t-62.5 27t-62.5 -27t-25.5 -65t25.5 -65t62.5 -27t62.5 27t25.5 65z" /></g> <g transform="matrix(.017,-0,0,-.017,21.412,12.875)"><path id="x43" d="M614 175l29 -10q-33 -109 -57 -154q-121 -26 -184 -26q-90 0 -160.5 29t-112.5 77t-63.5 105.5t-21.5 119.5q0 157 108 253t277 96q36 0 71.5 -5t69 -13.5t36.5 -8.5q15 -102 20 -150l-29 -8q-20 79 -66.5 114t-128.5 35q-119 0 -187.5 -86t-68.5 -207&#xA;q0 -140 73.5 -227.5t188.5 -87.5q73 0 119.5 37.5t86.5 116.5z" /></g> </svg>, <svg style="vertical-align:-0.20473pt;width:40.950001px;" id="M2" height="13.1875" version="1.1" viewBox="0 0 40.950001 13.1875" width="40.950001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.875)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,8.222,12.875)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.017,-0,0,-.017,16.381,12.875)"><use xlink:href="#x30"/></g> <g transform="matrix(.012,-0,0,-.012,24.538,4.712)"><use xlink:href="#x2218"/></g> <g transform="matrix(.017,-0,0,-.017,29.575,12.875)"><use xlink:href="#x43"/></g> </svg>, and <svg style="vertical-align:-0.20473pt;width:40.950001px;" id="M3" height="13.1875" version="1.1" viewBox="0 0 40.950001 13.1875" width="40.950001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.875)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,8.222,12.875)"><path id="x37" d="M447 623l8 -12l-283 -613l-74 -10l-7 11q174 283 297 551h-216q-48 0 -62.5 -12t-33.5 -63h-29q10 60 18 148h382z" /></g><g transform="matrix(.017,-0,0,-.017,16.381,12.875)"><use xlink:href="#x30"/></g> <g transform="matrix(.012,-0,0,-.012,24.538,4.712)"><use xlink:href="#x2218"/></g> <g transform="matrix(.017,-0,0,-.017,29.575,12.875)"><use xlink:href="#x43"/></g> </svg>, respectively. The mechanical responses are shown to be sensitive both to strain rate and to temperature. Based on the Maxwell viscoelastic model, a nonlinear thermal-mechanical viscoelastic constitutive model is developed for the PP wood composite by decoupling the effect of temperature with that of the strain rate. Corresponding viscoelastic parameters are obtained through curve fitting with experimental data. Then the model is used to simulate thermal compression of the PP wood composite. The predicted theoretical results coincide quite well with experimental data. The proposed constitutive model is then applied to the thermoforming simulation of an automobile interior part with the PP wood composites.

Homogenization of processes in nonlinear visco-elastic composites

The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation e − A(x) : ∫ t 0 σ (x, τ ) dτ ∈ α(σ, x), which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here α(·, x) is a (possibly multivalued) maximal monotone mapping, σ is the stress tensor, e is the linearized strain tensor, and A(x) is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, − div σ = # f . Existence and uniqueness of the weak solution are proved for a boundary-value problem. Convergence to a two-scale problem is then derived for a composite material, in which the functions α and A periodically oscillate in space on a short length-scale. It is proved that the coarse-scale averages of stress and strain solve a single-scale homogenized problem, and that conversely any solution of this problem can be represented in that way. The homogenized constitutive relation is represented by the minimization of a time-integrated functional, and is rather different from the above constitutive law. These results are also retrieved via De Giorgi’s notion of %-convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis. Mathematics Subject Classification (2010): 35B27 (primary); 49J40, 73E50, 74Q (secondary).

Non-elastic models of interaction of an impactor and an Uflyand–Mindlin Plate

This paper is devoted to the mathematical modeling of the low velocity impact of a solid body upon a thin elastic isotropic plate whose dynamic behavior is described by wave equations. As a method of analysis the ray method, the Laplace transform method for the viscoelastic impact and the method of spliced asymptotic expansions received for small times in the contact area and outside it are used. The local deformations in the contact area are taken into account in terms of two elastoplastic models (Aleksandrov–Kadomtsev and Kilchevskii models) and the viscoelastic Maxwell model. The stiffness of the Maxwell model’s element is expressed in terms of the integral operator with the exponential kernel of relaxation and thus the relationship for the contact force takes the integral form with a function of relaxation. The results of the present investigation can be used for the projecting and analysis of the planar constructions and their elements to impact loading in civil engineering and automotive industry.