Robert F. Landel

Stored energy function and compressibility of compressible rubberlike materials under large strain

By using new invariants in the theory of finite elasticity an expression is obtained for the stored energy function of slightly compressible materials in which the effects of the distortional change (change of shape) and of the volume change are clearly separated. The volume‐related terms are expressed as a function of the third invariant Γ3, the classical compressibility K0, and an induced anisotropy of the effective compressibility which is due to the large deformations. After evaluating the terms, using data on pressure, volume, uniaxial stress‐strain, and ΔV/V0 versus strain data on natural rubber from the literature, it is shown that the volume change contribution to the total stress observed in a simple tensile experiment can be clearly separated from the distortional contribution, even at finite strains.

Synthesis and Properties of a New Class of Potential Biomedical Polymers

Abstract Both low molecular weight compounds and some polyelectrolytes containing quaternary nitrogen atoms in their structure have been known for some time to possess important biological activity. Aliphatic ammonium compounds are used as bacteriocides and muscle relaxants, and quinolinium compounds have recently been found to be excellent antileukemic agents. In addition, a considerable amount of evidence has been accumulated to show that organic ammonium salts combine with heparin, and the resulting complex may be used as a coating to render polymers blood-compatible. However, the introduction of ammonium groups into the backbone of a polymer exhibiting good mechanical properties offers certain advantages.

Rheological behavior of progressively shear‐thickening solutions

FM‐9 (the commercial name of an ICI product) in organic solvents or poly(methacrylic acid) (PMMA) in aqueous solution exhibit strong time‐dependent shear‐thickening behavior. The induction time needed for the onset of the thickening depends markedly on shear rate, solvent used, concentration, and molecular weight. Slightly altering the nature of the solvent profoundly affects the behavior. The effects are similar to those obtained by changing the concentration or molecular weight. The nature of thickening behavior is not well understood. The effect will be illustrated (using FM‐9 solutions) as a function of shear rate, concentration, and temperature. Defining a critical shear rate γc as that at which the induction time for thickening becomes so short as to be immeasurable within the resolution of the viscometer, γc for this polymer is found to be inversely proportional to the zero shear‐rate viscosity of the solution for the temperature and concentrations studied.

Stored Energy Function of Rubberlike Materials Derived from Simple Tensile Data

An explicit formulation to obtain the stored energy function W from simple tension experiments alone, based on the Valanis‐Landel separable symmetric stored energy function, W(λ1, λ2, λ3) = w(λ1) + w(λ2) + w(λ3), is developed. For a simple extension stress‐strain law of the form λσ=Ee, the analytical formula of W in the limited range 1≤λ≤2.5 is found to be W=E ∑ i=l3 [λi−1−lnλi−16 (lnλi)2+118 (lnλi)3−1216 (lnλi)4], where the limitation originates with the stress‐strain law. The expression is used to verify the validity of the Valanis‐Landel postulation through prediction of the stress‐strain behavior in multiaxial deformations.

Viscoelastic Properties of Rubberlike Composite Propellants and Filled Elastomers

Abstract The mechanical properties of rubberlike composite propellants and similar filled elastomers are determined largely by the volume fraction of filler, the visco-elastic properties of the binder, and the interactions between the binder and filler particles. The ratio of the quasi-equilibrium modulus of the composite to that for the unfilled elastomer increases with the volume fraction of the filler, apparently according to an equation of the form proposed by Eilers and Van Dyck. However, the same ratio for the dynamic storage modulus decreases as the frequency is increased or the temperature is decreased. The time-dependent tensile properties can be characterized by stress-strain curves measured at different strain rates and temperatures. Both the small deformation and ultimate properties can be represented by master curves, which are functions only of the experimental time scale, along with a temperature function which is a near-universal function of the glass temperature. Propellants under constan...

Mechanical properties of a polyurethane elastomer in the rubber-to-glass transition zone

Abstract The complex shear compliance of a polyurethane elastomer has been measured in the transition region from rubbery to glasslike properties with a Fitzgerald transducer. Values for the real and imaginary components of the complex shear compliance J ′ and J ″ are reported over a frequency range of 45 to 6000 cycles/sec, and a temperature range of −16 to 39°C. Composite curves of J ′ and J ″ reduced to a standard temperature were obtained by the method of reduced variables. The temperature shift factors followed the WLF equation. The data do not extend to high enough reduced frequencies to approach the glassy compliance, but at low frequencies the dynamic compliance is essentially equal to the equilibrium rubbery value. The loss tangent J″ J′ passes through an extremely broad maximum with increasing frequency. The shape of the retardation distribution function resembles that of polyisobutylene; however, the value of the ratio of the maximum in the retardation distribution function to the equilibrium compliance is not the same for the two polymers. The value for the polyurethane agrees instead with that previously found for cellulose tributyrate and polyvinyl chloride gels (which are crosslinked by crystallization). For several decades at the long time end of the transition zone the relaxation distribution function has the slope of −1 2 prescribed by the Rouse theory, as does a 20.4% cellulose tributyrate gel. At shorter times the experimental curve rises more steeply than predicted by theory, just as do the corresponding curves for vinyl polymers. However, the total deviation from the linear theoretical curve is small and the overall shape of the relaxation distribution function for the polyurethane more nearly resembles that found for disubstituted than for monosubstituted vinyls.

Equations of State and Constitutive Equations

Since equations of state relate only pressure, volume, and temperature, they exclude stress fields other than pure hydrostatic pressure/tension. As such—they are not of themselves useful in describing the general stress‐deformation‐temperature response. Constitutive equations are required. The latter are, in turn, subsets of still more general expressions, the free energy functions. Rheological constitutive equations incorporate time, but introduce the need to carefully distinguish the independent variable as the stress or the deformation. Moreover, for glassy solids they must incorporate at least two distinct, yet interacting, memory functions, while providing for the physical aging process. The paper comments on thermodynamics and free energy functions and on some conceptual difficulties, including the definition of reference states for strain, especially volumetric strain. For glasses, the volumetric reference state is a particular problem because the unloaded state is metastable, and physical aging ca...

Cell Model and Elastic Moduli of Disordered Solids: Low Temperature Limit

The cell theory has been previously employed to compute the equation of state of a disordered condensed system. It is now generalized to include anisotropic stresses. We adopt the condition of affine deformation, transforming an originally spherical into an ellipsoidal cell. With a Lennard‐Jones n−m potential between non‐bonded centers, the formal expression for the deformational free energy is derived. It is to be evaluated in the limit of the linear elastic range. Since the bulk modulus in this limit is already known, it is convenient to consider a uniaxial deformation. To begin with, we restrict ourselves to the low temperature limit in the absence of entropy contributions. Youngs modulus and Poissons ratio then follow.

A Simple w′(λ) Function for The Valanis-Landel Form of Stored Energy Function

Abstract In the Valanis-Landel formulation of the stored energy function W, stresses depend on the function w′(λ)(=dw/dλ). This function exhibits strong curvature, making it difficult to represent analytically with good accuracy. It is found for both SBR and NR that the function λw′(λ) is not only far less curved, it is essentially linear in λ for the range of about 0.4 < λ < 2.0. The long range of simple proportionality to strain invites examination of molecular theories of rubberlike elasticity. Above the linear range the response can be approximated by kλn. These simplifications should make it easier to convert w′(λ) to the W1 and W2 functions employed in finite element analysis.

Problems Encountered with Conventional Fiber-Reinforced Composites

The problems encountered in dealing with composites can be usefully classified as preparational, computational and operational. That is, we first ask how it is made and the effects of process variables, including the resultant chemistry, on the initial properties.* Then, knowing (or presuming) the component properties and attributes, one may try to calculate the properties of any given composite structure made from these and at the same time estimate its ability to withstand some generalized stress field, i.e., some combination of mechanical, thermal or environmental loads for some stated period of time. Finally, we can ask what problems are encountered under use conditions.