Donald E. Carlson

On the derivative of the square root of a tensor and Guo's rate theorems

Formulas for the derivative of the square root of a positive definite, symmetric, second-order tensor are derived and used to obtain expressions for the material time derivatives of the right and left stretch tensors.

A Continuum-Mechanical Theory for Nematic Elastomers

We develop a continuum theory for the mechanical behavior of rubber-like solids that are formed by the cross-linking of polymeric fluids that include nematic molecules as elements of their main-chains and/or as pendant side-groups. The basic kinematic ingredients of this theory are identical to those arising in continuum-level theories for nematic fluids: in addition to the deformation, which describes the trajectories of material particles, an orientation, which delineates the evolution of the nematic microstructure, is introduced. The kinetic structure of our theory relies on the precept that a complete reckoning of the power expended during the evolution of a continuum requires the introduction of forces that act conjugate to each operative kinematic variable and that to each such force system there should correspond a distinct momentum balance. In addition to conventional deformational forces, which expend power over the time-rate of the deformation and enter the deformational (or linear) momentum balance, we, therefore, introduce a system of orientational forces, which expend power over the time-rate of the orientation and enter an additional orientational momentum balance. We restrict our attention to a purely mechanical setting, so that the thermodynamic structure of our theory rests on an energy imbalance that serves in lieu of the first and second laws of thermodynamics. We consider only nematic elastomers that are incompressible and microstructurally inextensible, and a novel aspect of our approach concerns our treatment of these material constraints. We refrain both from an a priori decomposition of fields into active and reactive components and an introduction of Lagrange multipliers; rather, we start with a mathematical decomposition of the dependent fields such as the deformational stress based on the geometry of the constraint manifold. This naturally gives rise to active and reactive components, where only the former enter into the energy imbalance because the latter automatically expend zero power in processes consistent with the constraints. The reactive components are scaled by multipliers which we take to be constitutively indeterminate. We assume constitutive equations for the active components, and the requirement that these equations be consistent with the energy imbalance in all processes leads to the active components being determined by an energy density response function of the deformation gradient, the orientation, and the orientation gradient. We formulate the requirements of observer independence and material symmetry for such a function and provide, as a specialization, an expression that encompasses the energy densities used in the Mooney-Rivlin description of rubber and the Oseen-Zöcher-Frank description of nematic fluids.

On hyperelasticity with internal constraints

By requiring the constitutive equation for the specific internal energy to be such that energy is balanced for all motions compatible with the internal constraint, we are able to infer the exitence and the direction of the reactive stress as well as the usual stress relation for the active stress. In contrast with previous work along this line, our analysis avoids the Lagrange multiplier formalism, and we need not assume that the internal energy response function is extendable off the constraint manifold.

Second-order incompressible elastic torsion

Abstract The problem of second-order torsion of an incompressible, isotropic, homogeneous, hyperelastic cylinder is reduced to a two-dimensional classical linear elasticity traction problem without body force. The case of a rectangular cross section is worked out as an example of the procedure.

Force-Free States, Relative Strain, and Soft Elasticity in Nematic Elastomers

The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity. The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments, has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality, and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the neo-classical model are also discussed.

Controllable states of rigid heat conductors

ZusammenfassungEs werden stationäre Temperaturfelder bestimmt, welche in allen isotropen homogenen starren Körpern ohne äussere Wärmezufuhr möglich sind. Wenn der Wärmestrom nur von der Temperatur und vom Temperaturgradienten abhängt, muss das Temperaturfeld gleichförmig sein. Hängt der Wärmestrom nur vom Temperaturgradienten ab, dann muss das Temperaturfeld schraubenförmig sein. Für den letzten Fall werden Beispiele gegeben.

Inverse deformation results for elastic materials

ZusammenfassungEs wird gezeigt, dass die Umkehrsätze für die Deformation allgemeiner elastischer Körper unmittelbar aus einem Variationsprinzip gewonnen werden können, welches sich auf Gleichgewichtszustände bezieht.

Stress functions for plane problems with couple stresses

ZusammenfassungDie Lösung der zweidimensionalen Gleichgewichtsbedingungen für ein Kontinuum, das Spannungsmomente aufnehmen kann, wird mit Hilfe von willkürlichen Spannungsfunktionen gegeben. Die Spannungsfunktionen werden mit Hilfe der resultierenden Einzelkraft und des resultierenden Momentes gedeutet, welche durch einen Bogen im Körper übertragen werden.

Dependence of linear elasticity solutions on the elastic constants

Necessary and sufficient conditions on the assigned data for the displacement and/or the stress to be independent of the shear modulus in the standard boundary-value problems of three-dimensional classical elastostatics are determined.

Successive approximations in nonlinear thermoelasticity

Abstract Starting with the basic equations of finite thermoelasticity for an incompressible, isotropic, homogeneous body, we generalize the usual method of successive approximations by expanding the relevant fields in terms of two parameters. These parameters separately characterize the mechanical and thermal loading on the body. The successive equations are derived, and due to the incompressibility and isotropy assumptions, it is seen that at each order they are not only linear but also completely uncoupled. The displacement is determined by the equations of linear isothermal elasticity, while the temperature field is determined by the equation of linear heat conduction. As an example, the solution, through second-order terms, is found for the problem of an annular cylinder with axial mechanical loading and radial thermal loading.