Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

A concise, self-contained introduction to solid polymers, the mechanics of their behavior and molecular and structural interpretations. This updated edition provides extended coverage of recent developments in rubber elasticity, relaxation transitions, non-linear viscoelastic behavior, anisotropic mechanical behavior, yield behavior of polymers, breaking phenomena, and other fields.

Mechanical properties of solid polymers

A fully three-dimensional finite-strain viscoelastic model is developed, characterized by: (i) general anisotropic response, (ii) uncoupled bulk and deviatoric response over any range of deformations, (iii) general relaxation functions, and (iv) recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin). Continuum damage mechanics is employed to develop a simple isotropic damage mechanism, which incorporates softening behavior under deformation, and leads to progressive degradation of the storage modulus in a cyclic test with increasing amplitude (Mullins' effect). A numerical integration procedure is proposed which trivially satisfies objectivity and bypasses the use of midpoint configurations. The resulting algorithm can be exactly linearized in closed form, and leads to symmetric tangent moduli. Quasi-incompressible response is accounted for within the context of a three-field variational formulation of the Hu-Washizu type.

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On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects

Viscoelastic instabilities are of practical importance, and are the subject of growing interest. Reviewed here are the fresh developments as well as earlier work in this area, organized into the following categories: instabilities in Taylor-Couette flows, instabilities in cone-and-plate and plate-and-plate flows, instabilities in parallel shear flows, extrudate distortions and fracture, instabilities in shear flows with interfaces, instabilities in extensional flows, instabilities in multidimensional flows, and thermohydrodynamic instabilities. Emphasized in the review are comparisons between theory and experiment and suggested directions for future work.

Instabilities in viscoelastic flows

Polymer melts with long-chain side branches and more than one junction point, such as commercial low density polyethylene (LDPE), have extensional rheology characterized by extreme strain hardening, while the shear rheology is very shear thinning, much like that of unbranched polymers. Working with the tube model for entangled polymer melts, we propose a molecular constitutive equation for an idealized polymer architecture, which, like LDPE, has multiple branch points per molecule. The idealized molecule, called a “pom-pom,” has a single backbone with multiple branches emerging from each end. Because these branches are entangled with the surrounding molecules, the backbone can readily be stretched in an extensional flow, producing strain hardening. In start-up of shear, however, the backbone stretches only temporarily, and eventually collapses as the molecule is aligned, producing strain softening. Here we develop a differential/integral constitutive equation for this architecture, and show that it predicts rheology in both shear and extension that is qualitatively like that of LDPE, much more so than is possible with, for example, the K-BKZ integral constitutive equation.

Molecular constitutive equations for a class of branched polymers: The pom-pom polymer

Two simple types of constitutive equations appropriate to materials exhibiting viscoelasticity are presented, one of a basic solid nature and one of a basic fluid nature. The predictions of the equations for a stress relaxation experiment are worked out and compared to the data from some experiments on various elastomers. The fluid theory is shown to be most appropriate in a certain sense.

A Study of Stress Relaxation with Finite Strain

A simple non.equilibrium thermodynamics is developed and a particular example is studied. The theory is formulated to describe a viscoelastic fluid, capable of finite deformation, which need not be locally in or near a state of thermodynamic equilibrium. This fluid may support shear stresses only when away from local thermodynamic equilibrium _ A notion of time-temperature superposition is contained in the formulation of the constitutive equations. Conservation of energy is obeyed and the second law of thermudynamics is satisfied as a consequence of simple requirements on the constitutive relations. In an adiabatic isochoric motion the temperature increases when work is done on the material and decreases when the material does work. For given volume and temperature, e ntropy decrea ses whe n the mate rial is deformed from equ il ibrium . It is s hown in what general way viscos it y depends upon te mperature . For infinites imal s train, the special form of the stress-s train relations are derived in order to de termine how t.e mpe rature and t ime-t e mperature supe rpos ition e nte r in thi s case.

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Thermodynamics of perfect elastic fluids

The theory of perfect elastic fluids deals with the non‐equilibrium thermodynamics of finite viscoelastic deformation. The thermodynamics of finite elastic strain and of perfect elastic fluids are discussed. It is shown that the stress‐strain relations for perfect elastic fluids may be written in a way which appears formally identical to those in elasticity theory, but the form of this dependence depends on past history.

B. Bernstein

Papers

A Study of Stress Relaxation with Finite Strain

Thermodynamics of perfect elastic fluids

Elastic Stress‐Strain Relations in Perfect Elastic Fluids

A Study of Stress Relaxation with Finite Strain