Maximum Recoverable Work Research Articles

Free energy and states of fractional-order hereditariness

Complex materials, often encountered in recent engineering and material sciences applications, show no complete separations between solid and fluid phases. This aspect is reflected in the continuous relaxation time spectra recorded in cyclic load tests. As a consequence the material free energy cannot be defined in a unique manner yielding a significative lack of knowledge of the maximum recoverable work that can extracted from the material. The non-uniqueness of the free energy function is removed in the paper for power-laws relaxation/creep function by using a recently proposed mechanical analogue to fractional-order hereditariness.

Thermo-mechanical Behavior of (Meth)Acrylate Shape-Memory Polymer Networks

ABSTRACTOur overall approach is based on developing a photocrosslinkable polymer network with a favorable shape-memory response, using polymer chemistry and crosslinking density to control thermo-mechanical properties. Three polymer networks were created and thermo-mechanically tested, each from tert-Butyl acrylate linear builder co-polymerized with a poly(ethylene glycol) dimethacrylate cross-linker. By systematically altering the molecular weight and the weight fraction of the cross-linker, it was possible to create three polymers that exhibited the same glass transition temperature, but varied by almost an order of magnitude in rubbery modulus. Therefore, the mechanical stiffness could be tailored to suit a given application. Recovery behavior of the polymers was characterized over a range of deformation temperatures. It has been implicitly assumed a linear relationship between Free-Strain (i.e. no actuation force) and Fixed-Stress (i.e. maximum actuation force), however, this has never been confirmed experimentally. The energy per unit volume performed by the shape-memory polymer was quantified, and observed to be a function of strain recovered. The maximum recoverable work was shown to increase with cross-linking density, although the overall efficiency is similar for all materials tested.

Maximum recoverable work in the ionosphere

A general closed expression is derived in the frequency domain for the minimum free energy, related to a state of a linear electromagnetic conductor with memory effects in the constitutive equation of the current density, by evaluating the maximum recoverable work obtainable from the given state. The constitutive equation with memory is given by a linear relation between the current density and the electric field, expressed by a linear functional of the history of this field and a term proportional to its actual value. Another equivalent formulation is also given and used to derive explicit formulae for the particular case of a discrete spectrum model.

Minimum free energy in the frequency domain for a heat conductor with memory

An explicit expression is given in the frequency domain for the minimum free energy related to a particular state of a linear rigid heat conductor with memory effects in the constitutive equations, using the fact that this quantity coincides with the maximum recoverable work obtainable from that state. The constitutive equations for the internal energy and the heat flux are expressed as linear functionals of the histories of temperature and its gradient, respectively, together with the present value of the latter quantity. Another equivalent expression for the minimum free energy is also deduced and used to derive explicit formulae for a discrete spectrum model.

Maximum Recoverable Work for a Rigid Heat Conductor with Memory

In this paper we study the problem of giving an explicit formula for the minimum free energy for a rigid heat conductor, with memory both in the expression of the heat flux and in the one of the internal energy. Two equivalent expressions are derived in terms of Fourier-transformed quantities; one of them is used to obtain the results related to the particular case of a discrete spectrum model.

Complex bodies with memory: Linearized setting

AbstractThe mechanics of complex bodies with memory effects is discussed in linearized setting. The attention is focused on the characterization of free energies in terms of minimum work and maximum recoverable work in the bulk and along a discontinuity surface endowed with its own surface energy, a surface within the body. To this aim, use is made of techniques proposed by Del Piero. Consequences of the Clausius–Duhem inequality are investigated for complex bodies with instantaneous linear elastic response. Copyright © 2008 John Wiley & Sons, Ltd.

Minimum free energy for an electromagnetic conductor with memory

AbstractA closed expression is determined in the frequency domain for the minimum free energy associated with a state of a linear electromagnetic conductor with memory effects, using the fact that this quantity is equal to the maximum recoverable work obtainable from the given state of the material. Another equivalent expression is also derived and applied to evaluate explicit formulae for a discrete spectrum model. In particular, for such a model we present the results corresponding to only one inverse time decay for each of the three kernels of the constitutive equations. These results clearly show the effects of various parameters on the expression for the minimum free energy. Copyright © 2008 John Wiley & Sons, Ltd.

A rigorous construction of maximum recoverable work

A simple and rigorous derivation of the maximum recoverable work is presented. In contrast to previous derivations it is based on simple and rigorous projectional methods. The principle holds if the stress has a non-trivial Newtonian component.

Maximum recoverable work and pseudofree energies for a rigid heat conductor

In this paper, we consider a linear theory for a rigid heat conductor with memory effects for a heat flux in order to deduce explicit formulas for the minimum free energy, which is related to the maximum recoverable work that can be obtained from a given state of a body. Two equivalent forms of this work are given in the frequency domain. Finally, two different expressions for a thermodynamic potential, called pseudofree energy, are introduced for this material.

Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model

The study of the maximum recoverable work is given in terms of Fourier-transformed quantities for incompressible viscoelastic fluids. This maximum is related to the minimum free energy of the system, whose expression is found as a function of the minimal state. Finally, these results are developed for a discrete spectrum model.

Maximum recoverable work in linear thermoelectromagnetism

We give a general closed expression for the minimum free energy in terms of Fourier-transformed quantities for a thermoelectromagnetic conductor with memory effects for the electric current density and the heat flux in the case where the integrated histories of the electric field and of the temperature gradient are chosen to characterize the states of the material. An equivalent formulation is obtained and applied to the discrete spectrum model material response.

Minimum free energy in linear thermoelectromagnetism

A closed expression is given for the minimum free energy of a linear thermoelectromagnetic conductor, whose constitutive equations relative to the electric current density and to the heat flux have memory effects. This expression, derived in the frequency domain, is related to the maximum recoverable work, which can be obtained from a given state of the material. Another equivalent expression of the minimum free energy is deduced; it allows us to give explicit formulae for the case of a discrete spectrum model.

The minimum free energy for an electromagnetic conductor

We consider the linear theory of the electromagnetism characterized by a constitutive equation for the current density with memory effects. We find in the frequency domain a first expression of the minimum free energy, which is the maximum recoverable work we can obtain from a given state of the material. By using another equivalent formulation of the minimum free energy, we give the explicit formulae for the particular case of a discrete spectrum model material response.

Thermal work and minimum free energy in a heat conductor with memory

A rigid linear heat conductor with memory effects is considered. Thus, the behaviour of the material is characterized by a constitutive equation which relates the heat flux to the history of the temperature gradient. A given thermal history is considered and its prolongation via an assigned process is defined. Then, the notion of equivalence is introduced to single out and associate together all those different thermal histories which correspond to the same heat flux. Notably, whenever the heat flux is the same the related thermal work is also the same. An explicit expression of the minimum free energy, related to the maximum recoverable work, is obtained in the frequency domain.

Nonisothermal free energies for linear theories with memory

Explicit expressions for minimum, maximum, and intermediate free energies have recently been given for isothermal linear theories. In this paper, these results are generalized to a nonisothermal, linear model. Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within a linear theory. In particular, an integral equation for the continuation associated with the maximum recoverable work from a given state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, generalizing a result previously derived in the isothermal case. A new variational method developed for the isothermal case is extended to nonisothermal conditions. In the time domain, this approach yields integral equations for both the minimum and maximum free energies associated with a given state. In the frequency domain, explicit forms of a family of free energies, associated with a given state of a discrete spectrum material, are derived. This includes both maximum and minimum free energies. These latter developments, given previously for the isothermal scalar case, are generalized (with some restrictions) in this work to the tensor case. The physical relevance of various results are discussed.

Maximum recoverable work, minimum free energy and state space in linear viscoelasticity

The various formulations of the maximum recoverable work used in the literature are proved to be equivalent. Then an explicit formula of the minimum free energy is derived, starting from the formulation of the maximum recoverable work given by Day. The resulting expression is equivalent to that found by Golden and other authors. However, the particular formulation allows us to prove that the domain of definition of minimum free energy is the whole state space. Finally, the maximum recoverable work is shown to be put as the basis of the thermodymamics of viscoelastic materials under isothermal conditions. In this context the usual relation between the Clausius-Duhem inequality and the dissipation of the material is restored.

Maximum and minimum free energies for a linear viscoelastic material

Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within the theory of linear viscoelasticity. In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.

An Explicit Formula for the Minimum Free Energy in Linear Viscoelasticity

A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positive-definite matrices. The resultant formula suggests how to characterize the state in the sense of Noll in the frequency domain. The property that the maximum recoverable work represents the minimum free energy according to both Graffi's and Coleman-Owen's definitions is used to obtain an explicit formula for the minimum free energy. Detailed expressions are presented for particular types of relaxation function.