The effect of temperature on the viscoelastic behavior of linear low-density polyethylene

Summary Observations are reported on linear low-density polyethylene in isothermal torsional oscillation and relaxation tests at various temperatures ranging from room temperature to 110 C. Constitutive equations are derived for the viscoelastic response of a semicrystalline polymer at small strains. The polymer is treated as an equivalent network of strands bridged by junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is thought of as an ensemble of meso-regions with various potential energies for rearrangement of strands. Two types of meso-domains are introduced: active, where strands separate from temporary junctions as they are excited by thermal fluctuations, and passive, where detachment of strands is prevented by the surrounding macromolecules. The time-dependent behavior of the ensemble reflects separation of active strands from their junctions and merging of dangling strands with the network. Stress–strain relations are developed by using the laws of thermodynamics. The governing equations involve six material constants that are found by fitting the experimental data. The study focuses on the effects of (i) temperature, (ii) the deformation mode (torsion versus bending), and (iii) the loading program (oscillations versus relaxation) on the adjustable parameters.This work was partially supported by the West Virginia Research Challenge Grant Program     

The viscoelastic and viscoplastic behavior of low-density polyethylene

Three series of tensile tests with constant cross-head speeds (ranging from 5 to 200 mm/min), tensile relaxation tests (at strains from 0.03 to 0.09) and tensile creep tests (at stresses from 2.0 to 6.0 MPa) are performed on low-density polyethylene at room temperature. Constitutive equations are derived for the time-dependent response of semicrystalline polymers at isothermal deformation with small strains. A polymer is treated as an equivalent heterogeneous network of chains bridged by temporary junctions (entanglements, physical cross-links and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. The viscoelastic behavior of a polymer is modelled as thermally-induced rearrangement of strands (separation of active strands from temporary junctions and merging of dangling strands with the network). The viscoplastic response reflects mutual displacement of meso-domains driven by macro-strains. Stress–strain relations for uniaxial deformation are developed by using the laws of thermodynamics. The governing equations involve five material constants that are found by fitting the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. It is shown that observations in conventional creep tests reflect not only the viscoelastic, but also the viscoplastic behavior of an ensemble of meso-regions.     

Modelling the viscoplastic response of polyethylene in uniaxial loading–unloading tests

Two series of uniaxial cyclic tests are performed on low-density polyethylene at room temperature. In the first series of experiments, injection-molded specimens are stretched to several maximal strains ε_max in the region of sub-yield deformations with a constant cross-head speed, mm/min, and retracted down to the zero stress with the same strain rate. In the other series, loading–unloading tests are carried out with the maximal strain ε_max=0.10 and cross-head speeds ranging from 5 to 200 mm/min. A constitutive model is derived for the viscoplastic behavior of a semicrystalline polymer at small strains. A polymer is modelled as an equivalent network of chains bridged by permanent junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is treated as an ensemble of meso-regions connected by links (crystalline lamellae). Deformation of a specimen induces sliding of junctions with respect to their reference positions both at active loading and unloading (this process reflects sliding of junctions in amorphous regions and fine slip of crystalline lamellae). At retraction, sliding of junctions is accompanied by mutual displacements of meso-domains (that reflects coarse slip and fragmentation of lamellar blocks). The constitutive equations are determined by 5 adjustable parameters that are found by matching the experimental stress–strain curves.     

The effect of annealing on the viscoplastic response of semicrystalline polymers at finite strains

A constitutive model is developed for the viscoplastic behavior of a semicrystalline polymer at finite strains. A solid polymer is treated as an equivalent heterogeneous network of chains bridged by permanent junctions (physical cross-links, entanglements and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. In the sub-yield region of deformations, junctions between chains in meso-domains slide with respect to their reference positions (which reflects sliding of nodes in the amorphous phase and fine slip of lamellar blocks). Above the yield point, this sliding process is accompanied by displacements of meso-domains in the ensemble with respect to each other (which reflects coarse slip and disintegration of lamellar blocks). To account for the orientation of lamellar blocks in the direction of maximal stresses and formation of micro-fibrils in the post-yield region of deformations (which is observed as strain-hardening of specimens) elastic moduli are assumed to depend on the principal invariants of the right Cauchy–Green tensor for the viscoplastic flow. Stress–strain relations for a semicrystalline polymer are derived by using the laws of thermodynamics. The constitutive equations are determined by six adjustable parameters that are found by matching observations in uniaxial tensile tests on injection-molded isotactic polypropylene at elongations up to 80%. Prior to testing, the specimens were annealed at various temperatures ranging from 110 to 163 °C. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. The effect of annealing temperature on the material parameters is studied in detail.     

Wedging an Orthotropic Body by a Rectangular Wedge of Finite Length

Galin's method is used to derive equations that relate the basic parameters of the problem on wedging an orthotropic space by a rigid rectangular wedge. To eliminate the stress singularity at the wedging crack tips, a Leonov–Panasyuk–Dugdale prefracture zone is assumed to exist at the crack front. The equations are derived using the COD criterion     

A model of traps for yield in amorphous glassy polymers

Summary  Constitutive equations are derived for the viscoelastic and viscoplastic behavior of amorphous glassy polymers at isothermal loading with small strains. The model is based on the trapping concept: a disordered medium is treated as an ensemble of plastic flow units (with the characteristic size of micrometers), which, in turn, consist of a number of cooperative rearranging regions (with the characteristic length of nanometers). The viscoelastic response is described by rearrangement of relaxing regions, whereas the viscoplastic behavior is modeled as irreversible deformation of plastic units. Adjustable parameters are found by fitting observations for aromatic polyesters, nylon-66, polycarbonate block copolymers and an epoxy glass. Fair agreement is demonstrated between experimental data and results of numerical simulation. Received 17 November 1999; accepted for publication 23 March 2000     

Motion of a Mechanical System with Variable Mass–Inertia Characteristics

A closed-form system of dynamic equations describing the free motion of a material system with variable mass–inertia characteristics is derived. The system consists of a carrying body and carried bodies (freight) and undergoes translational–rotational motion in space. The differential equations of motion derived include time-dependent parameters and allow for the inertia and varying mass of the system, etc. It is pointed out that special cases can be derived from the general equations to study various modes of motion and stability phenomena     

The nonlinear viscoelastic response of carbon black-filled natural rubbers

Three series of tensile relaxation tests are performed on natural rubber filled with various amounts of carbon black. The elongation ratio varies in the range from λ=2.0 to 3.5. Constitutive equations are derived for the nonlinear viscoelastic behavior of filled elastomers. Applying a homogenization method, we model a particle-reinforced rubber as a transient network of macromolecules bridged by junctions (physical and chemical cross-links, entanglements and filler clusters). The network is assumed to be strongly heterogeneous at the meso-level: it consists of passive regions, where rearrangement of chains is prevented by surrounding macromolecules and filler particles, and active domains, where active chains separate from temporary nodes and dangling chains merge with the network as they are thermally agitated. The rate of rearrangement obeys the Eyring equation, where different active meso-domains are characterized by different activation energies. Stress–strain relations for a particle-reinforced elastomer are derived by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data. It is demonstrated that the filler content strongly affects the rearrangement process: the attempt rate for separation of strands from temporary nodes increases with elongation ratio at low fractions of carbon black (below the percolation threshold) and decreases with λ at high concentrations of filler.     

A Constitutive Model in Finite Viscoelasticity of Particle-reinforced Rubbers

Two series of tensile relaxation tests are performed on natural rubber filled with high abrasion furnace black. To fit observations, constitutive equations are derived for the nonlinear viscoelastic behavior of a particle-reinforced elastomer. A filled rubber is modeled as a composite medium, where inclusions with low concentrations of junctions are randomly distributed in the host matrix. The inclusions are treated as equivalent networks of macromolecules, where strands can separate from temporary junctions as they are thermally agitated. The bulk medium is thought of as a permanent network of chains. Unlike conventional concepts of transient networks, the concentration of strands in inclusions is assumed to be affected by mechanical factors: under active loading, inter-chain interactions weaken and some strands that were prevented from detachment from their junctions in a stress-free compound become free to separate from the junctions in a deformed medium. Unloading strengthens interactions between macromolecules, which results in an increase in the number of permanent strands. By using the laws of thermodynamics, stress–strain relations for a particle-reinforced rubber are developed. Adjustable parameters in the constitutive equations are found by fitting the experimental data. It is demonstrated that mechanical pre-loading and annealing of specimens at an elevated temperature noticeably affect concentrations of inclusions with various activation energies for rearrangement of strands.     

Short-Term Microdamageability of Laminated Materials under Thermal Actions

The theory of microdamageability of laminated materials is stated with account taken of the thermal effect. Microdamages in the components are simulated by pores empty or filled with particles of damaged material that resist compression. The fracture criterion is assumed to have the Nadai–Schleicher form, which takes into account the difference between the tensile and compressive ultimate loads, with the ultimate strength being a random function of coordinates with a power or Weibull distribution. The stress–strain state and the effective properties of the material are determined from the thermoelastic equations for laminated materials with porous components. The deformation and microdamage equations are closed by the equations of porosity balance corrected for the thermal effect. For various types of loading, nonlinear relations are derived for the coupled processes of deformation of a two-component laminated material and microdamage due to the thermal macrostrain of a component. The effect of physical and geometrical parameters on these processes is studied.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

Formula for the Flow Resistance Factor in a Pipe with a Sudden Expansion at Small Reynolds Numbers

A formula for the flow resistance factors in a pipe with a sudden expansion of the cross section at Reynolds numbers of 0.2 to 10 is obtained by numerical solution of the complete Navier–Stokes equations for incompressible fluids. The flow resistance factors obtained using the derived formula are compared to those found by numerical solution of the Navier–Stokes equations.     

Short-Term Damage Micromechanics of Laminated Fibrous Composites Under Thermal Actions

A microdamage theory is constructed for laminated fibrous materials with transversely isotropic fibers and a porous isotropic matrix under thermal actions. Microdamages in the matrix are simulated by pores, empty or filled with particles of the damaged material that resist compression. The fracture criterion for a microvolume of the matrix is assumed to have the Nadai–Schleicher form, which takes into account the difference between the tensile and compressive ultimate loads, with the ultimate strength being a random function of coordinates with a power or Weibull distribution. The stress–strain state and the effective properties of the material are determined from the thermoelastic equations for laminated fibrous materials with a porous matrix. The deformation and microdamage equations are closed by the porosity balance equations corrected for the thermal effect. For various types of loading, nonlinear relations are derived for the coupled processes of deformation of a laminated fibrous material and microdamage of the matrix due to the thermal macrostrain. The effect of physical and geometrical parameters on these processes is studied.     

Short-Term Microdamageability of Fibrous Materials with Transversely Isotropic Fibers Under Thermal Actions

The theory of microdamageability of fibrous materials with transversely isotropic fibers is stated with account taken of the thermal effect. Microdamages in the isotropic matrix are simulated by pores empty or filled with particles of damaged material that resist compression. The fracture criterion for a microvolume of the matrix is assumed to have the Nadai–Schleicher form, which takes into account the difference between the tensile and compressive ultimate loads, with the ultimate strength being a random function of coordinates with a power or Weibull distribution. The stress–strain state and the effective properties of the material are determined from the thermoelastic equations for fibrous materials with a porous matrix. The deformation and microdamage equations are closed by the equations of porosity balance corrected for the thermal effect. For various types of loading, nonlinear relations are derived for the coupled processes of deformation of a fibrous material and microdamage of the matrix due to the thermal macrostrain. The effect of physical and geometrical parameters on these processes is studied.     

Averaged equations and wave jumps describing the propagation of stokes waves with slowly varying parameters

The equations describing the stationary envelope of periodic waves on the surface of a liquid of constant or variable depth are investigated. Methods previously used for investigating the propagation of solitons [1–5] are extended to the case of periodic waves. The equations considered are derived from the cubic Schrödinger equation assuming slow variation of the wave parameters. In using these equations it is sometimes necessary to introduce wave jumps. By analogy with the soliton case a wave jump theory in accordance with which the jumps are interpreted as three-wave resonant interactions is considered. The problems of Mach reflection from a vertical wall and the decay of an arbitrary wave jump are solved. In order to provide a basis for the theory solutions describing the interaction of two waves over a horizontal bottom are investigated. The averaging method [6] is used to derive systems of equations describing the propagation of one or two interacting wave's on the surface of a liquid of constant or variable depth. These systems have steady-state solutions and can be written in divergence form.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for useful discussions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1989.     

Timoshenko-Type Theory in the Stability Analysis of Corrugated Cylindrical Shells

A technique is proposed for stability analysis of longitudinally corrugated shells under axial compression. The technique employs the equations of the Timoshenko-type nonlinear theory of shells. The geometrical parameters of shells are specified on discrete set of points and are approximated by segments of Fourier series. Infinite systems of homogeneous algebraic equations are derived from a variational equation written in displacements to determine the critical loads and buckling modes. Specific types of corrugated isotropic metal and fiberglass shells are considered. The calculated results are compared with those obtained within the framework of the classical theory of shells. It is shown that the Timoshenko-type theory extends significantly the possibility of exact allowance for the geometrical parameters and material properties of corrugated shells compared with Kirchhoff–Love theory.     

Diffusiophoresis of an aerosol particle in a binary gas mixture

The diffusion force and rate are calculated for the diffusiophoresis of a spherical particle in a binary gas mixture by solving the gas–kinetic equations. Two schemes of diffusiophoresis are considered: constant–pressure diffusion and diffusion of one mixture component through the other fixed component. The problem is solved by the integral–momentum method at arbitrary Knudsen numbers. Diffuse scattering of the gas molecules on the particle surface is assumed. The Lorentzian and Rayleigh models of a binary gas mixture are considered. The dependences of the force and rate of diffusiophoresis on the Knudsen number and the other determining parameters are analyzed. The results obtained are compared with well–known experimental data.     

Microcrack interaction brittle rock subjected to uniaxial tensile loads

A micromechanics-based model is proposed to describe unstable damage evolution in microcrack-weakened brittle rock material. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress–strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to be consistent with the experimental results.     

Spatial motion of a chemically active medium near a caustic

A study is made of the three-dimensional problem of determining the parameters of motion of a gaseous chemically active medium near a caustic, the envelope curve of the rays of the wave fronts in the geometrical acoustics approximation. Two limiting processes whereby perturbations propagate [1] can be distinguished, depending on the ratio of the reaction time of the chemical reaction to a macroscopic time: a quasifrozen process and a quasiequilibrium process. The problem is considered in a linear formulation in [2-6] in the absence of viscosity, thermal conductivity, and chemical reactions. Nonlinear equations are derived in [7–10] for an arbitrary nondissipative medium near a caustic. In the present paper Ryzhov's method [1] is used to derive the nonlinear equations of motion of the medium for both types of process. The pressure distributions near and on the caustic itself are found for an incident step wave. The effect of the chemical reaction on how the flow parameters are distributed in the vicinity of the caustic is ascertained. Equations are derived for an inhomogeneous initially moving fluid near a caustic. A nonlinear equation containing a highest derivative of third order is obtained in the vicinity of the caustic for the case of special media in which the limiting velocities of sound in the mixture at rest are close in value. It is shown that the solution of the corresponding linear equation is expressed in the form of a quadrature from the solution for a chemically inert medium and contains oscillations near the wave fronts.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 81–91, March–April, 1977.