Viscoelastic and Electromagnetic Materials with Nonlinear Memory.

A method is presented for generating free energies relating to nonlinear constitutive equations with memory from known free energies associated with hereditary linear theories. Some applications to viscoelastic solids and hereditary electrical conductors are presented. These new free energies are then used to obtain estimates for nonlinear integro-differential evolution problems describing the behavior of nonlinear plasmas with memory.

1. Introduction

The theory of materials with memory was designed to provide a wide range of material models, including models of viscoelastic materials, dielectrics and heat conductors with memory. The great majority of them can be characterized by memory functionals: reversible changes are described by the instantaneous response while dissipativity is expressed by the dependence on histories (see [1,2,3,4] and references therein). Usually, the Boltzmann superposition principle was employed to derive a linear constitutive equation for the response of the materials [5]. However, when the behavior of a given material is nonlinear, the Boltzmann superposition principle is not applicable; so a constitutive equation has to be sought by other means.

While more than one approach to nonlinear viscoelasticity was being explored in the literature, it is worth mentioning here the pioneering work of Schapery [6]. By means of a time scale shift factor, he derived single-integral nonlinear viscoelastic constitutive equations from the thermodynamic theory of irreversible processes; these equations are very similar to the Boltzmann superposition integral form and, similar to linear viscoelasticity, still consider the strains to be infinitesimal. A different approach has been employed by Findley et al. [7]. To describe nonlinear memory, they considered a polynomial expansion of a multiple-integral expression so that the dependence of strain on the stress history, and vice versa, is nonlinear even for small deformations. The mathematical complexity of this formulation is too complicated for use in many situations. Another approach involves objective rate equations by means of a thermodynamically consistent scheme which naturally allows the construction of nonlinear viscoelastic models (see [8]). More recently, in [9,10], nonlinear mechanical viscous effects are described by assuming a semi-continuum theoretical model with a geometric nonlinearity.

Although many nonlinear models with memory have been developed, this topic is open to many important improvements. A method is presented here for generating nonlinear constitutive equations from known linear theories characterized by memory functionals. The novelty of our procedure is based on the properties of the free energy functionals.

Memory response functionals lead to the difficulty of determining coherent free energy functionals even if the material response is linear [11,12]. A fundamental property of materials with memory is that there is in general no unique free energy density (we henceforth omit “density”) associated with a given state but rather a convex set of functionals which obey the requirements of a free energy ([13,14] and earlier references therein).

In recent years, explicit formulae have been given for the minimum free energy associated with linear constitutive equations [15,16,17]. The case of fluids is discussed with in [18]. In addition, for relaxation functions given by sums of decaying exponentials (discrete spectrum model) and generalizations of these, explicit expressions have been presented for the maximum and intermediate free energies [19,20,21]. Based on this work, an expression for a more centrally located free energy has been presented [21]. Both the isothermal and non-isothermal cases have been considered [22,23].

If the relaxation function is an integral over decaying exponentials (continuous spectrum model and generalizations), then an explicit expression can be given for the minimum free energy [24]. In this case, the maximum free energy is the work function [21,23,25].

Free energies relating to heat conductors with memory are considered in [26]. There are similarities between such materials, as described in [27], and electrical conductors with memory, which are of interest in the present work.

The concept of equivalent classes of states or minimal states, based on the work of Noll [28], is explored in the context of linear models [29,30,31]. One recent result is that for materials with relaxation functions given by sums of exponentials and generalizations of these, the minimal states are usually non-singleton, while for integrals over exponentials, they are singleton [21,23,25]. A more general point of view on this topic is adopted on page 365 of [1].

In the early sections, free energies with quadratic memory terms—which yield constitutive equations with linear memory terms—are discussed.

A method is presented for generating free energies and nonlinear constitutive equations from known free energies associated with linear theories ([1], p. 112). Examples are discussed for nonlinear viscoelastic theories. Such new free energies are then used to prove a boundedness result for integro-differential equations describing the behavior of nonlinear electromagnetic systems, specifically electrical conductors with memory.

In Section 2, the central result concerning free energies for nonlinear systems is proved on a general vector space—which may be taken as relating to mechanical, thermal or electromagnetic systems or a combination of these. Minimal states are defined and discussed. In Section 3, free energies with quadratic memory terms and higher-order contributions are considered. The issue of whether the quadratic memory terms are positive definite or positive semi-definite is relevant for applications in later sections. This question is explored in Section 4 in the context of minimal states. In Section 5, examples relating to viscoelastic materials are discussed, while in Section 6 and Section 7, electrical conductors with memory are considered, and we obtain estimates for nonlinear integro-differential evolution problems describing the behavior of nonlinear plasmas with memory. While the discussion of these sections is assumed to apply to a small element of the body centered at a point , we shall omit explicit mention of . Material deformation, while implicitly included in the general developments of Section 2, are not of primary interest in the present work.

On the matter of notation, vectors are denoted by lowercase and uppercase boldface characters and scalars by ordinary script. The real line is denoted by

2. Free Energies for Nonlinear Systems

We consider, in this section, a method for generating free energies relating to systems with nonlinear, memory-dependent constitutive equations. Independent and dependent field variables will be defined on a general vector space . Let and be the history and present value of the independent field variable, where the standard notation is understood. The dependent field variable is given by a constitutive relation

We will assume that belongs to a suitable function space with norm . As noted earlier, the free energy associated with a given state of the material is not in general a uniquely defined quantity. Free energies associated with a given state form a bounded convex set [14]. Let us denote by a member of . We have where is a nonlinear continuously differentiable function of the present value and a nonlinear functional of the history with a Fréchet differential that is continuous in its arguments. The static or equilibrium history is given by

In particular, denotes the zero history, where is the zero vector in . Note that the requirement is imposed on all free energies. This eliminates the arbitrary constant associated with all physical energies. However, the intrinsic arbitrariness associated with free energies of materials with memory remains (see [32,33,34]).

We define the equilibrium free energy by it is assumed that this is a positive definite function of .

The quantity is the work function, which represents the work completed on the element up to time t. It is assumed to be given by where the superimposed dot represents time differentiation and the centered dot represents the scalar product in . The convergence of the integral in relation (6) imposes restrictions on the behavior of in the distant past. The work completed over the time interval is given by

Based on consequences of the second law of thermodynamics derived in [32,35], in a continuum mechanics context (and easily generalizable to other dissipitive systems, including electromagnetism [36]), we assign the following (defining) properties to a free energy [14,32,35]:

where

For any history

Finally, assuming that

These are referred to as the Graffi conditions for a free energy. The relationship between them and the Coleman and Owen [37] definition of a free energy are explored, for a linear theory, by Del Piero and Deseri [30,31]; see also [15].

Note that equality occurs by definition in Property 2 for the static history. Together with the condition (4), Property 2 implies that all free energies are non-negative-valued functions.

We say that a non-negative-valued function

We can write (10) as an equality in the form where is the rate of dissipation.

The work function has Properties 1–3 of a free energy, though for zero rate of dissipation. However, it does not have a fourth property, as discussed on page 435 of [1], which applies to all other known free energies. This relates to the fact that for a long established periodic history, we must have , where T is the period of the history. This may not crucially affect the usage of as a free energy in all circumstances, but it should be considered with caution.

Let us now state and prove the central result of this section. Let , , be a set of n free energies relating to a state in a given material or perhaps in different materials at time t. To allow for the latter possibility, we assign to each , different constitutive equations and work functions , where

Accordingly, and

If all the free energies belong to the same material, the dependent field variables are all equal, and the index i refers to different free energies of the same material.

The quantity

is a free energy for the state

provided that

and

We have where is defined by (14). Moreover, from (11), we obtain the rate of dissipation

These relations essentially state Property 3. In addition, by virtue of (12) and (14), which is Property 1. It follows from (13), by taking the stationary limit of the history , that which is Property 2. Finally, by virtue of (16), satisfies the normalization condition (4) since every , has this property. □

This result allows us to build free energies and constitutive dependent fields relating to nonlinear systems from those associated with basic constitutive equations with linear memory terms (for which many explicit forms exist [15,16,17,18,19,20,21,22,38]) though, in fact, the may be any choice of free energies. Specifically, the result can be used as follows: assume we have a nonlinear dependent field variable of the form (14), where f obeys (15) and is determined by (12). Then, (13) immediately yields a free energy with a rate of dissipation given by (17) and dependent variable generated through (18). This is the way it is used in Section 5.1 and Section 6.1.

Taking f to be an analytic function of its arguments at the origin, we can write

A constant term is excluded by (4) and (16). If we omit higher powers, assumption (15) takes the form

In particular, taking to be a linear combination of the , it follows that

If we are dealing with free energies relating to the same linear material with dependent field , then , , and (14) becomes

In the linear case, and then . When higher powers are neglected in (20), this gives which, together with (21), amounts to convexity.

Relations (

2.1. Relative Histories

We can write (2) in the form where is the relative history defined by

Property 1, given by (8), becomes where the second term on the left is defined by the requirement that, for any ,

The quantity is the Fréchet differential of at in the direction which denotes a static history given by (3), for arbitrary . We put where , defined by (5), is given here by and represents the history-dependent part of the free energy.

2.2. Summed Histories

It is sometimes the case, as in one of the physical configurations dealt with in later sections (and in [26]), that the work function has the form with no time derivative on the independent field. Formally, we transform this into (6) as follows: define the summed past history by

Then, we can write (29) in the form and treat the quantity as the independent field variable. Let

Note that the relative summed history has the opposite sign to defined by (24), which is a choice that is more convenient in this context.

Dependence on cannot occur in the free energy or any other physical quantity. This is because the summed past history depends on the choice of the origin of the time variable. Thus,

In addition, drops out (see (28)) and

Property 1 as given by (25) simplifies to while (10) in Property 3 is replaced by

Property 2 is given by (9) where the quantity is zero.

2.3. Minimal States

We now introduce the concept of a minimal state. This is an equivalence class of histories defined as follows [20,28,29,30,31]. The state of our system at a fixed time t is specified by the history and present value . Let two states , have the property that then , are said to be in the same equivalence class or minimal state. The latter terminology was introduced in [20]. Thus, if they have the same output from time t onwards, they are equivalent histories. The derivatives in (37) arise from the definition of a process in terms of the independent field variable ([14] for example). Requirement (37) means that , differ by at most a constant for . Note that, for two equivalent states, we have where is the quantity defined by (7) for and is that quantity for .

A fundamental distinction in the present work is whether the material under discussion has minimal states that are singletons, i.e.,

Observe that property (36) requires that for , in the same minimal state.

3. Linear and Nonlinear Memory Models

We now consider free energies with quadratic memory terms, which produce linear memory constitutive equations. Let where . There is no loss of generality in taking

Assuming that is integrable on , we have

Applying Fubini’s theorem and (40), it follows that . In addition, with similar limits holding at large u for fixed s. An alternative form of (39) is

The form (39) emerges by expanding the general functional in (27) to include quadratic terms and neglecting any dependence on in the kernel ([23], for example and [1], p. 149). The linear term is omitted because it is inconsistent with the requirement that be positive definite. The quantity will be a valid free energy provided certain conditions are imposed on the kernel , which in particular must be a non-negative operator so that the second term on the right of (43) is non-negative.

Noting (41), we define where the prime indicates differentiation with respect to the argument. The constitutive relation has the form where

Causality requires that vanishes on [39]. An alternative form of (46) is

The standard choice for is given by so that from (47), we obtain . Thermodynamic arguments can be used to show that using an adaption of a technique described in [14], for example. Here, the second property is a consequence of (45). In earlier work on free energies, involving tensor constitutive relations [15,18,19,20,22,23,40], it is also assumed that a condition which cannot be deduced from (40) or from thermodynamics.

Note that (39) can be put in the form

It follows that

By differentiating (43) with respect to t and using (11), we obtain [17,23]

Thus, because of (10), must be a non-negative operator.

It is assumed that so that the Fourier transform of exists. We have ([1], page 161) where and are the Fourier cosine and sine transforms. The latter vanishes at . It is a consequence of the second law that ([14,36], for example) for dissipative materials.

Nonlinear Models

Let , , be given as in (39), and satisfy Properties 1–3. The simplest nonlinear model is obtained from a quantity of the form which is a free energy by Proposition 1. In particular, taking into account that where ⊗ denotes the dyadic product, we can write where , , and is a fourth-order tensor belonging to . With a little abuse of notation, the double dot here denotes the scalar product in .

From (14), we have where denotes the dependent field related to , , which is given by

The special case where and is the basis of developments in Section 5.1 and other sections.

Moreover, according to (17), the nonlinear rate of dissipation has the form where

being related to as indicated by (41).

A more general expression, say , can be obtained by the functional Taylor expansion of and in (27) and neglecting the third-order terms because must be non-negative. We let where and are second and fourth-order tensors on , respectively. Any dependence of on is neglected.

Further constraints must be placed on and to ensure that has the required Properties 1–3 of a free energy. Here, we will limit ourselves to observing that (57) is recovered from (59) by letting

When summed past histories are involved, reduces to , as stated in (34), and therefore, we can simply choose where denotes the reduced summed history. In particular, (57) is recovered provided that

4. Minimal States and Quadratic Free Energies

Let us consider the concept of a minimal state in the context of linear memory constitutive equations. Applying the definition (36) to (46), we find [23] that and are equivalent, or in the same minimal state, if and only if provided the equilibrium quantity has a unique inverse, and where is the linear functional [16,20,25,30,31]

In the case where vanishes (see (33)), which is the case of primary interest here, there is no requirement that (61) holds, although must always be true. We introduce the relation as an extra condition in the definition of a minimal state. It follows from (64) that (61) holds. We shall sometimes refer to the equivalence or otherwise of histories, omitting the mention of present values, when the former are central to the argument.

The linearity of the functional means that the requirement of the equivalence of and is the same as that be equivalent to the zero history. Thus, if the minimal state including the zero history is singleton (non-singleton), then all minimal states are singleton (non-singleton).

In the arguments that follow, we introduce certain results obtained for the minimum, maximum and other free energies in [15,17,20] and related work, without developing the detailed apparatus.

Let in (27) have the form where is a linear functional of the history with the property that if and only if and are equivalent histories. The quantity is thus a functional of the minimal state. We have

If states are equivalent to the zero state, usually in the context of the difference of two equivalent states, the present value is zero, and a distinction between actual and relative histories is unnecessary.

The form (66) applies to the minimum, maximum and a family of intermediate free energies given in [

Let so that where is the Hermitian conjugate of . It follows from (66) and (67) that the free energy is itself a functional of the minimal state so that if , are equivalent states, then

Relation (38) follows automatically, but it is not necessary to assume (71) for this relation to be true. Let us introduce the scalar product notation

The free energy is given by where S is defined by (51).

We now prove certain results for free energies, using this bracket notation ([1], p. 173).

If the free energy is a functional of the minimal state and if

The first equality in (74) follows from the definition of equivalence, on noting that S, and more obviously , are equal for the states , at time . We also have since is equivalent to the zero state. Thus, the last equality in (74) can be deduced using the bilinearity of the scalar product. □

It follows from Proposition 2 that

For a free energy with a history-dependent part of the form (66), the statement that

If, for the non-zero history , the quantity vanishes for ; in other words, if is equivalent to the zero history and the minimal states are non-singleton, then, from (75), vanishes at and is non-negative.

If vanishes for the non-zero history , then, from (66), and by (68), we have that vanishes, and the minimal states are non-singleton since is non-zero. □

In [21,23] (and also [1], p. 168), materials are characterized by the singularity types in the complex frequency plane of the Fourier transform of the derivative of the relaxation function. If this quantity has only isolated singularities (corresponding to a relaxation function consisting of sums of decaying exponentials, possibly multiplying polynomials and trigonometric functions) then minimal states are non-singleton. If the singularities characterizing a material include branch cuts, then the minimal states are singletons [24] (see also [36], p. 499). This is the case of main interest in the present work.

We adopt a different viewpoint on free energies and constitutive equations in this work. The standard thermodynamical point of view is to specify a free energy and deduce a constitutive relation from this. Alternatively, an applications-oriented approach, which is now adopted, involves deciding on a constitutive equation and searching for a free energy that yields this relation. This latter step may not be easy.

4.1. Quadratic Free Energies for Singleton Materials

We make the assumption in the following sections that the materials are such that their minimal states are singletons. This implies that the free energies, at least in the categories specified in Proposition 3, are positive definite functionals of the history.

4.1.1. The Graffi Free Energy

Let (46)–(47) be the constitutive relations of the dependent field on a general vector space. A corresponding free energy is the Graffi functional, which is given by

It satisfies Properties 1–3 of a free energy only if so that these conditions are assumed to hold. The rate of dissipation is

Equation (77) can be written in the form (39) as indicated on page 238 of [1].

Let us assume that

It will be true, for instance, if consists of sums (or integrals) of decaying exponentials multiplying non-negative coefficients (or a non-negative function) with dominant term proportional to . It follows that

The Graffi free energy is not in general a functional of the minimal state [30]. It is, however, a positive definite functional of the history, by virtue of the first inequality in (78) and a positive definite function of the present value by virtue of the assumption after (5).

4.1.2. The Work Function

Recalling the first equality in (46) we put

Using (47), the total work performed on the material, given by (6), can be expressed in the form where . Then, we conclude that

This is a special example of (39) with . However, has singular delta function behavior [40] and is therefore not bounded. We emphasize that the work function obeys the properties of a free energy with zero dissipation rate, , [33,41,42].

We denote by the work function, given by (82), since it is the maximum free energy for singleton materials, but it is not in general a functional of the minimal state [30]. It is a positive definite function of and a positive definite functional of the history, which is clear from its representation in the frequency domain [15]. In particular, for singleton materials, it can be shown that

5. Viscoelastic Systems with Memory

In the sequel, the vector space is , the subspace of symmetric second-order tensors on . In addition, memory kernels take values in , which represents the space of fourth-order tensors. Let and denote the strain and the stress, respectively. Using the notation (1), denotes the strain history and denotes the constant strain history of value ,

A material is viscoelastic if the stress tensor not only depends on the current value of the strain but also on its history:

The dependence of and on the space variable is understood but not written.

The linear constitutive equation for a viscoelastic body is given by where the memory kernel is a summable and continuous fourth-order tensor-valued function. It is of interest to compare this with the more general relation (46). Let

We can rewrite (83) as where . Since corresponds to in (24) if , then corresponds to in (46) with . Accordingly, corresponds to . The consequence of the second law stated by (56) takes the form in the present context. Moreover, from thermodynamic arguments [14], it follows that

According to Properties 1–3 and (4), a functional is said to be a free energy for the (possibly nonlinear) stress functional if it fulfills: for all . The term is related to the Fréchét differential through the representation formula (see (26))

The free energy (39) becomes where . Moreover, (11) takes the form where is the rate of dissipation, which is given in general by (53) and here by

Since Bearing in mind that a double integration by parts, with respect to and , yields

We have the correspondences between and and between and noted after (84). In addition, the work function (82) becomes

Since , it follows from (88) that .

Graffi’s free energy takes the form

In addition,

For the functional specified by (89) to be a free energy, it is required that (see (78))

5.1. Nonlinear Constitutive Equations

Special cases of Proposition 1 are now considered in the context of viscoelasticity.

Let be the given linear constitutive functional (84) and any related free energy functional with kernel . Let be any given smooth function such that

Then, nonlinear stress–strain constitutive equations can be obtained by considering the memory relation and the corresponding nonlinear free energy is

Indeed, we have and

For example, we can choose , and . When , then (92) and (93) yield

In fact, we should replace 1 in the relation for by a constant with the dimensions of free energy so as to maintain explicitly correct dimensionality in each expression. What we are doing here is choosing units such that this constant has a value of 1.

More generally, let us consider memory kernels , satisfying (85) and . Let denote the related i-th linear model and let be any associated free energy satisfying (86). Thus, we have

In particular, for any given pair of kernels and , we can construct a nonlinear stress–strain functional of the form which admits the following free energy functional

In addition, given a nonlinear function g obeying the relations specified in (91), we can generalize (95) and (96) as follows: for any fixed pair of integers . By virtue of (88), we have

5.2. A Nonlinear Viscoelastic Model Based on Graffi’s Free Energy

Letting and , as given by (89) and (84), respectively, we obtain from (94) a nonlinear constitutive equation of the following form and a related free energy given by

Moreover, the corresponding rate of dissipation is

For isotropic viscoelastic materials, the kernel and the relaxation modulus take the special form where is the unit second-order tensor and the unit fourth-order tensor. Here , so that and , . Moreover, and , which are the conditions (90). Using the decomposition, where tr stands for the trace and the subscript ∘ denotes the deviatoric part of the tensor, we rewrite the nonlinear stress–strain relation in the form where and denote the bulk elastic kernel and the bulk relaxation modulus, respectively.

The related free energy takes the form

5.3. A One-Dimensional Example

Consider one-dimensional models associated with strain and applied traction in the direction so that

The symbol T for the component of is consistent with the engineering stress considered in the literature as the ratio of the axial force over the reference area. Moreover, for simplicity, let where and . Letting , after differentiating equation (84) with respect to time, we obtain which represents the well-known equation for a standard linear solid (or Zener model). The corresponding Graffi’s free energy satisfies a similar differential equation,

The parameter represents the reciprocal of the characteristic relaxation time of the material. Equations (98) and (99) are not invariant under the time transformation and hence, they describe a rate-dependent material behavior. In particular, they predict different linear elastic behavior as (very fast processes) and (very slow processes); since we have (up to additive constants)

Accordingly, from (94), it follows

The asymptotic traction responses for T and is plotted in Figure 1 for both limit cases.

Figure 1

Asymptotic responses in the -plane as (dashed) and (solid), both for linear (on the right) and nonlinear (on the left) constitutive relations with and .

6. Electric Conductors with Memory

In this section, the vector space is . In addition, kernels take values in , which is the space of second-order tensors. Let denote the subspace of which contains all symmetric tensors and contains the convex set of positive-definite symmetric tensors (a tensor is positive definite if for all non-zero ).

In accordance with the notation specified by (1), let denote the electric field at time t and denote its past history. The constant history equal to is given by

An electric conductor is hereditary if the current vector depends on the electric field history:

As previously, the dependence of and on the space variable is understood and not written.

We start from a basic, linear constitutive equation for the current where the memory kernel is a summable, continuous and positive-definite tensor-valued function. Let . At any constant history , we have

This relation resembles Ohm’s law and is referred to as the relaxation conductivity tensor.

Note that the common form of Ohm’s law, , is actually recovered if in (100) we (formally) choose the kernel equal to times the Dirac mass at zero, .

If we introduce the magnetic vector potential and assume the vanishing of the electric scalar potential (as usual in electric conductors), we obtain so that the relative past history of , given by (24), equals the relative summed past history of the electric field in the notation (32), since

Thus, and after an integration by parts, we can rewrite (100) in the form where . The history , given by (101), corresponds to in (24) if . In addition, we see that corresponds to in (46) with . Alternatively, taking and , then corresponds to ; given by (32) and from (31), it follows that

Condition (56), which is a consequence of the second law, takes the form [43] in the present context. It is satisfied if is positive-definite.

A functional of the relative past history of the magnetic vector potential, , is said to be a free energy for the (possibly nonlinear) current functional , if it fulfills Properties 1–3 and (4), as adapted as in Section 2.1 and Section 2.2. We write these as follows: where , . Note that we omit any dependence of on as in (35). The term is related to the Fréchét differential through the representation formula (see (26)) for any choice of . In the context of quadratic free energies, the operation is simply differentiation with respect to the explicit occurrence of in as given in (101).

Firstly, the general form of the free energy (39) reduces to where . Moreover, we can write (105)4 in the form: where is the rate of dissipation, which is given in general by (53) and here by

Letting subject to (104), the work function (82) becomes

There are many choices of free energy that can be used in this context. For example, there is the explicit form for the minimum free energy relating to continuous spectrum materials, which is derived in [24]. Minimal states are singletons for such materials. However, we will opt for algebraic simplicity by choosing the Graffi free energy (77). This takes the form

For this functional to be a free energy, it is required that

6.1. Nonlinear Electric Conductors

We now consider special cases of Proposition 1 in the context of electrical conductors with memory.

Let be a given linear constitutive functional (103) and any related Graffi free energy functional with kernel . Various forms of nonlinear constitutive equation can be obtained by taking the nonlinear current to be where f satisfies (91). Due to the condition , the linear constitutive equation for gives the first-order approximation to (107). The expression for the corresponding nonlinear free energy is

Indeed, we have and

More generally, let denote the linear models whose kernels are compatible with thermodynamics in the sense that they obey (104), and let be any free energy associated with the i-th model satisfying (105). Thus, we have

For any given pair of kernels and , we can construct a nonlinear current functional and a free energy functional of the form

6.2. Integral Representations of the Current

The goal of this subsection is to establish a connection between the nonlinear constitutive functionals proposed in Section 6.1 and constitutive functionals in the form of single, double and triple integrals of the kind proposed by Graffi in [44,45] (see also [7]).

Following the suggestions of Graffi’s paper, we assume a nonlinear constitutive equation of the following general form where , and are a second, third and fourth-order tensor-valued function, respectively. A somewhat similar expansion was used in (59). For arbitrary vectors , we have

The quantities and can be taken to be invariant under any permutation of the arguments provided the same permutation is applied to the subscripts and .

For simplicity, let us assume that the constitutive relation for is isotropic. It follows that vanishes, and , must be isotropic tensors. Thus, where are scalar functions of the elapsed time s. This implies and so that

The symmetry properties of gives relations between , , and : which are apparent in any case from (108) and among other similar relations. We finally obtain where

Because of the first equality in (109) we have . Hence, assuming the factorization it follows that . Letting in (106) we obtain and (110) becomes where

Finally, letting , we obtain the corresponding free energy

It is worth noting that this choice of is not unique, but it is the simplest one. Indeed, in general for any pair , such that , we have

For the static history , , we have where and are defined by

In particular, by applying the Graffi free energy functional, , we have

If we restrict our attention to the one-dimensional case, the nonlinear current response to the application of the static -valued history is plotted in Figure 2.

Figure 2

Nonlinear current responses at different constant values of the static history with and .

7. Some Applications to Nonlinear Evolution Problems

The vector space is now taken to be . In addition, kernels take values in , which is the space of second-order tensors. In this section, we consider some nonlinear evolution problems arising from the coupling of hereditary models for the electric current with Maxwell equations. This kind of problem describes electromagnetic phenomena in the ionosphere and, more generally, in a nonlinear model of plasma (see [36,46,47], for instance). The corresponding linear model has been scrutinized in [48,49] and extended to explain electromagnetic behavior in a conducting (or imperfect) dielectric such as water (see [36,43,50], for example).

7.1. An Energy Inequality for a Nonlinear Plasma

Let be a bounded region occupied by a plasma. According to the Minkowski approach, Maxwell’s equations take the same form in the whole space , namely

Hereafter, denotes the gradient.

Let denote the magnetic field. The magnetic induction and displacement vector are given by where and are positive constants which stand for the electric permittivity and magnetic permeability of the material, respectively. As usual, where is the vacuum permittivity and is the electric susceptibility of the material. Thus, we can write by introducing the polarization . Similarly, where is the vacuum permeability and is the relative permeability of the material so that, after introducing the magnetization , we can write .

We assume that the constitutive equation of the electric current of the plasma is given by where is a (linear or nonlinear) functional of the electric field history. Let be a free energy functional satisfying (105).

We first establish a local energy inequality for the plasma evolution system. Multiplying the first equation of the system (111) by and the second by , we obtain by virtue of (112) the local energy balance

From the last inequality in (105) it follows that

According to [51], we assume the natural boundary conditions at a free plasma surface,

Initial conditions, including the past history of the electric field , must also be assigned;

Now, integrating (113) over , it follows

being the norm in . This is due to the well-known property of the Poynting vector, along with the first boundary condition in (114). Accordingly, the norms of , and the function are bounded for all t by their initial values,

7.2. Boundedness of the Electric Current

By exploiting this energy estimate, we establish here some results about the boundedness of the electric current functional. It is noteworthy that linear and nonlinear models yield different consequences on the norm of the electric current.

Let be a linear isotropic functional, which is given by and is a related quadratic free energy, so that its integral over is equivalent to a weighted -norm. For instance, when , and the Graffi functional is the chosen free energy, then where denotes the weighted Hilbert space . Moreover, letting , we have and then, the norm of the electric current is bounded for all as well as ,

On the other hand, let be a nonlinear functional, for instance

We are forced to restrict our analysis to to prevent the electric current from assuming singular or non-zero constant values when the null constant history of the electric field is considered. Accordingly,

Moreover, applying Hölder’s inequality and then, we have the following estimate of the norm of the electric current,

This allows us to establish the following result.

Let (112) and (117) be constitutive relations for and . In addition, let the initial data be such that and are bounded. Furthermore, we assume that Maxwell’s equations (111) with boundary condition (114) admit solutions. Then, for all ,

Item (i) follows from the energy inequality (116). In order to establish item (ii), we first assume . By virtue of (119), there is a positive constant such that applying first the Hölder inequality and then the generalized Young inequality to the last term of (118), we obtain where denotes the Lebesgue measure of and stands for a positive constant dependent on and . These estimates are similar to those obtained in the linear case. More generally, when , we put and applying the Minkowski inequality , , we obtain hence, there exists some positive constant such that

Moreover, if , then and the generalized Young inequality gives in particular, when , we have