## Abstract

Using the classical theory of invariants for the specific class of graphene's symmetry, we constitutively characterize electro-magneto-mechanical interactions of graphene at continuum level. Graphene's energy depends on five arguments: the Finger strain tensor, the curvature tensor, the shift vector, the effective electric field intensity and the effective magnetic induction. The Finger strain tensor describes in- surface phenomena, the curvature tensor is responsible for the out-of-surface motions, while the shift vector is used due to the fact that graphene is a multilattice. The electric and the magnetic fields are described by the effective electric field intensity and the effective magnetic induction, respectively. An energy with the above arguments that also respects graphene's symmetries is found to have 42 invariants. Using these invariants, we evaluate all relevant measures by finding derivatives of the energy with respect to the five arguments of the energy. We also lay down the field equations that should be satisfied. These are the Maxwell equations, the momentum equation, the moment of momentum equation and the equation ruling the shift vector. Our framework is general enough to capture fully coupled processes in the finite deformation regime.

## 1. Introduction

From the theoretical point of view, the electronic properties of graphene are due ultimately to its crystal structure, which is a honeycomb lattice of carbon atoms that can be regarded as a hexagonal 2-net [1–8]. The *s*^{2}*p*^{2} configuration of the atomic carbon hybridizes in graphene into a configuration in which the 2s, 2p_{x} and 2p_{y} orbital of each carbon are *sp*^{2}-hybridized to form in-plane *σ* bonds with its three nearest neighbours. The remaining p_{z} orbital is oriented along the *z*-direction, perpendicular to graphene's plane, and forms *π*-bonds which merge with neighbouring 2p_{z} orbitals to form delocalized states across the graphene plane. The ease of move of electrons in these *π* states is responsible for graphene's electrical conductivity [9].

The extraordinary mechanical and electronic properties together with optical transparency make graphene suitable for a broad range of applications, especially in optoelectronics. However, many of graphene's foreseen applications require precise control of its electronic structure so as to meet the demands of the customer. These modifications range from controlling the fine tuning of local pools with different charge carriers concentrations (the so-called charge puddles), to altering the band structure altogether and opening a band-gap. Regarding the latter, several approaches have been used to date such as one-dimensional confinement into nanoribbons [10], chemical functionalization [11] and mechanical strain [13,12]. More possibilities for this task are present in bi- or trilayer graphene, where a symmetry breaking caused by different charge carrier concentration in the layers, e.g. by applying a vertical electric field [14], can lead to the opening of a band-gap.

Although a band-gap opening in monolayer graphene by the sole application of a uniaxial strain is hardly feasible (e.g. [15–17]), the situation could be more promising when a combination of uniaxial and shear strains is applied. However, the theory using a tight binding model with suitable parametrized hopping integrals still predicts large strains (between 12 and 17% to open a band-gap [18]). In bilayer or trilayer graphene, the situation is more promising, at least at the theoretical level. Even though uniaxial stress is not capable of opening up a band gap at reasonably low strain level in a bilayer graphene [19–21], inhomogeneous or out-of-plane strains are considered as viable tools [21–23]. It is worth adding here that applied compressive strain perpendicular to the bilayer graphene plane, i.e. which alters the interlayer distance, strongly enhances band overlap [24]. Also, strain can be used to generate energy beam collimation, one-dimensional channels, surface states and confinement [25]. Lattice deformations modify the band structure in the sense that they change the distance between ions leading to shifts in the on-site energies of the *p*_{z}-orbital [26].

All the above indicate that strains are known to offer many different ways to control and modify graphene's electronic structure. It is the purpose of the present contribution to lay down a continuum framework for describing the effects mechanical deformations have on the electromagnetic properties of graphene and vice versa. In our aim, we are guided by the work of Novoselov *et al.* [27], where graphene's resistivity with applied voltage is measured and the outcome is seen schematically in figure 1. The result of this analysis indicates that graphene is a semiconductor that has high resistivity for zero voltage, while for applied voltage greater than 30 V (or less than −30 *V*) graphene's resistivity goes to zero. Thus, graphene's resistivity starts from a large value when there is no applied voltage and decreases rapidly with application of voltage. For small applied voltages (in the vicinity of 0 V), graphene's resistivity is high (of approx. 6 k*Ω*); thus, at this case, it seems reasonable to assume that graphene behaves as an insulator [28].

The interaction of continua with electromagnetic fields for an insulator has attracted the interest of researchers for a long time. Some of the earlier and fundamental works we can mention are, for example, [29–36] (see the earlier versions of that book as well). A list of some more recent works on electro- and magneto-elasticity can be found in the book by Dorfmann & Ogden [37]. In this communication, we assume graphene as a nonlinear thermo-electro-magneto-mechanical body,^{1} and we use as a theoretical basis the work by Santapuri *et al.* [38].

The above models of electro-magneto-elasticity are at the continuum level. At the discrete level, graphene is a multilattice [1–39], i.e. it is a hexagonal 2-lattice and there is a standard procedure for obtaining a continuum model starting from lattice considerations. The two basic assumptions for this transition are:

The fact that graphene is a multilattice implies that there is a continuum energy that has the shift vector as an argument as well. This, coupled with the structure tensor that describes graphene symmetries [1–5], renders the complete and irreducible representation for the energy function describing graphene at the continuum level. Also since graphene is a real 2-d material the curvature tensor should be included in the list of independent arguments of the energy to take into account out-of-surface motions. This framework tackles solely the mechanical effects [1–8]. To include the electric effects, we first need an expression of the Cauchy–Born rule for the electronic structure of the crystal. An extension of the Cauchy–Born rule for the electronic structure of smoothly deformed crystals that are insulators is treated elegantly by Lu [45,46]. In some sense, their main result is a sharp continuation theorem for the electronic structure of the equilibrium state of a crystal: knowing the electronic structure of the equilibrium state they enquire how it changes when the crystal is deformed smoothly. This is an extension of the Cauchy–Born rule for the electronic structure of crystal lattices under sharp stability conditions. This way a nonlinear elasticity model can rigorously derived.

In our analysis, here we couple our previous approaches [1,3,4] with that of Santapuri *et al.* [38] for the electro-magneto-mechanical modelling. In this way, we obtain an expression for the energy that describe electro-magneto-mechanical interactions in graphene. As in [1,3], the structure tensor describes the anisotropy of graphene, while the presence of the shift vector **P**^{sv} is due to the fact that graphene is a multilattice. Also the curvature tensor should be included in the list of energy's arguments. Following one of the possible options given by Santapuri *et al.* [38], the electric and the magnetic effects are introduced by assuming the energy to depend on the effective electric field intensity **e***, and the effective magnetic induction **b***, respectively. On the other hand, the mechanical field is described by the Finger surface strain tensor, **B**^{F}, the curvature tensor **B**^{c} and the shift vector, **P**^{sv}.

Using this energy, we evaluate all relevant constitutive quantities: the stress tensor, the couple stress tensor, the stress-like quantity related with the shift vector, the effective electric polarization, **p*** and the effective magnetic field intensity, **h***. These are work conjugate with the independent variables (**B**^{F},**B**^{c},**P**^{sv},**e***,**b***). The constitutive expressions together with the Maxwell equations, the momentum equation, the moment of momentum equation and the equation ruling the shift vector, render a closed set of equations capable of describing fully coupled electro-magneto-mechanical modelling for graphene at the continuum level. In the generic framework of [38], the temperature *θ* is also used together with its dual counterpart: the entropy, *η*. We here assume that we are at the isothermal and isentropic cases, so we disregard quantities related with these fields. It is necessary to indicate that the resistivity versus voltage diagram of figure 1 could be strongly dependent on temperature; but in this work, we freeze temperature and entropy in order to model at the continuum level figure 1. Also, in [38] the effective free current density **j*** is expressed as a function of the state variables; we here adopt a simple assumption for the free current density. Essentially, our assumption stipulates that the free current density is derived from a scalar potential as the sum of its derivatives with respect to the electric and the magnetic fields.

In this analysis, an issue which is important pertains the piezoelectric effects. Following [47] graphene exhibits inversion symmetry and thus it is not piezoelectric intrinsically. To induce piezoelectricity, these authors accomplish adsorption of atoms on the surface of graphene on only one side and this way the inversion symmetry is broken. For our analysis, we assume that graphene is piezoelectric in order to have a more general framework, and also to account for the fact that even in high voltages graphene's resistivity is not absolutely zero. For magnetization effects, we take a similar point of view. Recent works [48] reveal that placing graphene on an insulating magnetic substrate can make the material ferromagnetic without disturbing its exceptional conductivity. Thus, while pristine graphene sheets are intrinsically non-magnetic, graphene can be made ferromagnetic by doping the material with magnetic impurities. However, the dopants can be detrimental to graphene's highly sought-after electronic properties. We include magnetization effects to take into account cases where graphene is made magnetizable after some appropriate process and for having a most generic modelling framework. We nevertheless adopt the framework of [38], where the effective electric field intensity and the effective magnetic induction are used as arguments on the energy potential (see the fourth line in table 7 of [38]). The reason for using these quantities is that they can be better calculated theoretically using the electric scalar potential and the magnetic vector potential. Then, the effective electric polarization can be evaluated from the constitutive law, while the effective magnetization from the equation **m***=(1/*μ*_{0})**b***−**h***, **h*** being determined from the constitutive law and *μ*_{0} is the permittivity of the free space.

All the above considerations pertain to insulators and we assume that graphene behaves as such for zero voltages since its resistivity is high. When voltages less than −30 *V* or greater than 30 V are applied, figure 1 shows that graphene's resistivity goes to zero. For modelling conductors at the continuum level, there are only few works available; a notable example is the work of Maugin (see ch. 5 in [31]), where conductors are treated within the framework of elasticity. The free charges of a conducting material move very easily inside it due to the very low resistivity. Thus, the coupling between the electro-mechanical fields is only on the one way: the strain field can affect the electric field, but not vice-versa. Certainly, this holds true for perfect conductors which are mathematical idealizations of materials with very low resistivity.

To model graphene as a conductor, we can also use the framework of [38]. In contrast to insulators, the conductor's effective free charge density, *σ**, and the effective free current density, **j*** are not zero, thus allowing movement of free electrons within the material. Thus, for the modelling of graphene as a conductor, one assumes that the free charges and the free current density are different than zero, measuring the movement of free electrons. For the insulator case, these two quantities are both equal to zero, which affects the equations of Maxwell. The constitutive laws are affected as well since the effective electric field intensity and the effective magnetic induction are affected by the movement of free electrons. Thus, these quantities for the conductor case contain the effect of free electrons, an effect which is not present for an insulator. From the mathematical point of view, the constitutive laws remain of the same form but the inherent meaning of the terms **e***,**b*** is different. Terms **e***,**b*** for insulators measure the electric and the magnetic effects of the valence electrons while the effect of free electrons should also be included when **e***,**b*** are related with measurements for conductors.

In the physics literature for semiconductors, it is assumed that there are current carriers of two types [49]: electrons in the conduction band and positive holes in the filled valence band. In general, electrons and holes in condensed matter physics are described by separated equations which are not in any way connected. By contrast, electron and holes states in graphene are interconnected [27]. At the continuum level, we may tackle electrons and holes separately by using two different arguments on the continuum energy: one for the effective electric and magnetic fields of the electrons

Two remarks are in order here. Firstly, we note that for conductors there is no available extension in the literature on the Cauchy–Born rule similar to that of [45,46] for insulators. It seems even possible that the Cauchy–Born rule fails due to the ease of movement of free electrons. Certainly, in lack of an hypothesis like the Cauchy–Born rule, it is difficult to decide whether and how any continuum theory such as nonlinear electro-magneto-elasticity can describe the behaviour of a material. To be able to frame our theory, we tacitly make an assumption analogous to the Cauchy–Born rule, even though there is no evidence so far that such an assumption is valid. Secondly, we note that graphene's change from a conductor to insulator is an interesting case of a phase transition that may be described by the loss of positive-definiteness of some of the elasticities deriving from our energy. At the purely mechanical level, criteria for the initiation of configurational weak phase transitions [8] are given in [7].

The paper is structured as follows. Section 2 tackles the field equations: the momentum, moment of momentum equations and the equation ruling the shift vector come from our previous relevant works on the topic [1,3,4]. The equations governing the electromagnetic fields are the Maxwell equations and we choose the appropriate expression from the work of [38]. Section 3 introduces the independent arguments of graphene's energy together with the structure tensor describing graphene's anisotropy: the zig-zag and the armchair direction. Using the classical theory of invariants, we then evaluate the complete and irreducible representation for this energy under graphene's symmetry. We confine ourselves in the isothermal and isentropic case, and also adopt a specific assumption for the free current density. Using this energy, we evaluate the stress tensor, the couple stress tensor, the effective polarization and the effective magnetic field intensity. We also present a pathway of how some of the material constants of our framework can be measured using either molecular mechanics or ab-initio calculations. In §4, we clarify how our generic framework applies to graphene as an insulator or as a conductor according to the applied voltage. The article ends up with some concluding remarks in §5.

As far as notation is concerned, we note that we use tensor notation almost throughout the paper. Contraction in one index is denoted by ⋅, contraction in two indices is denoted by :, while contraction in more than two indices by •. The symbol ⊗ stands for the tensor product. All quantities are two dimensional, so all operators are surface operators: the gradient operator, grad, the divergence operator, div and the curl operator, curl.

All in all, while strains are known to offer many different ways to control and modify graphene's electronic and magnetic structure (see the citations on the second and the third paragraph of this section), these are works which assume partial couplings between electro-magneto-mechanical fields. So, while novel and interesting these approaches are, they adopt a relatively narrow point of view targeting at specific problems, of valuable interest, nevertheless. The main intention of the present work is to fill the gap in literature regarding the constitutive modelling of graphene when electric and magnetic fields are taken into account at the continuum level. The present framework adopts a generic point of view which is general enough to capture fully coupled processes in the finite deformation regime. Compared to our previous works on graphene [1,3,4], the present framework adopts a multiphysics point of view, where not only the mechanical field is present, but also the electric and the magnetic fields. Thus, while in our previous works only the mechanical effects were taken into account, here the electric and the magnetic fields are coupled to the mechanical field.

## 2. Basic equations

### (a) Kinematics

At the continuum level (topologically speaking), graphene is modelled as a two-dimensional smooth surface embedded in a three-dimensional Euclidean space. Position vectors on the reference configuration *Θ*^{α},*α*=1,2 as
*t* being time. Covariant surface base vectors are then defined as

The surface deformation gradient **F** reads
**F** are

### (b) Field equations

For the description of the field equations, we combine the approach of [38] with our previous works on the topic [1,3,7] by treating not only the mechanical field, but also its coupling with the electric and the magnetic fields. Essentially, the field equations are the Maxwell equations for the electromagnetic fields and the momentum, moment of momentum and the equation ruling the shift vector for the mechanical fields.

The Maxwell equations are [38]

where **v** the velocity. The effective magnetic induction is denoted by **b***, **e*** is the effective electric field intensity, **d*** the effective electric displacement, *σ** the effective free charge density, **h*** the effective magnetic field intensity and **j*** the effective free current density. The relationship between effective and standard quantities depends on the observer and we refer the reader to [38] for the specific mathematical relationship between them.

The conservation of mass reads
**f**^{m} are the mechanical body forces while the electromagnetic body forces and couples are given by the relations [38]
**p*** and **m***, respectively. The couple stress tensor is denoted by **M**. The Cauchy stress tensor is denoted by **T** and related with the first Piola–Kirchhoff stress tensor **P**^{P-K} by the formula

For an internal energy *ϵ* depending on arguments (**F**,*θ*,**p***/*ρ*,**m***/*ρ*), Santapuri *et al.* [38] give the following expression of the first law of thermodynamics:
*ρ*_{0} is the mass density in *r*^{t} is the thermal supply rate, while **q** is the heat flux vector. The Clausius–Duhem inequality can then take the form [38]
*θ* and entropy by *η*.

Since graphene is a multilattice to these equations, one should add the equation ruling the shift vector [1–5]:
**L** is constitutively characterized in terms of the shift vector,

All in all, the field equations are the equations of Maxwell, the momentum, the moment of momentum, and the equation ruling the shift vector. The Clausius–Duhem inequality is used to derive the constitutive laws. Our framework is restricted to the isothermal and isentropic cases, in this way thermal effects and their products are neglected.^{2} For the case of graphene, we render the form of the constitutive laws in the next section. The constitutive equations together with the field equations form a closed set of equations which, when solved, gives the fields (**B**^{F},**B**^{c},**P**^{sv},**e***,**b***) for every time and for all the surface of the thin film. These equations are strongly coupled in the sense that Maxwell equations contain terms from all the mechanical fields (**B**^{F},**B**^{c},**P**^{sv}) and vice-versa the mechanical balance laws contain terms from the electromagnetic fields (**e***,**b***).

## 3. Constitutive equations

The framework of [38] provides a comprehensive catalogue of free energy's dependent and independent quantities in order to model thermo-electro-magneto-mechanical interactions. We choose from their analysis the description that uses (**F**,**e***,**b***,*θ*) as independent arguments (the fourth line of table 7 of [38]) and we form for our purposes the potential

The full formulation of [38] considers for the energy the following form:
*ϵ*=*ϵ*(**F**,*η*,**p***,**h***) is the internal energy, while the energy *E*^{Fθeb}=*E*(**F**,*θ*,**e***,**b***) is a Helmholtz-like energy which is produced from *ϵ* using the Legendre transformation with respect to the variables *η*,**p***,**h***.

Instead of working with the deformation gradient one may work with the Finger deformation tensor **B**^{F}. Then, by neglecting thermal effects the energy, *E*^{Fθeb}, depends on the triplet (**B**^{F},**e***,**b***). To include out-of-surface motions, we should introduce the curvature tensor **B**^{c}, as an additional argument of the energy in line with our previous works on the topic [1–5]. In addition, since graphene is a multilattice, a 2-lattice in particular, the shift vector, **P**^{sv}, should be an argument of the energy as well in line with well-established theories of multilattices [40–43].

So, to constitutively describe graphene at the continuum level, we assume that energy *E*^{Fθeb} of [38] is for us denoted by *W*^{anis} and has the following arguments:
**x**,*t*). From the physical point of view, the term **B**^{F} describes in-surface mechanical changes, **B**^{c} describes out-of-surface mechanical changes, **P**^{sv} describes changes in the shift vector, **e*** describes the effect of the electric field and **b*** describes the effect of the magnetic field.

We also disregard thermal effects and we may also assume that **j***=**j***(**B**^{F},**B**^{c},**P**^{sv},**e***,**b***). We may go one step further by assuming that there exists a scalar function, say *Π* = *Π*(**B**^{F},**B**^{c},**P**^{sv},**e***,**b***), such that the current free density has the form
*Π* is a given function of the invariants.

The energy of equation (3.2) is the anisotropic expression for graphene's energy. We may take an isotropic scalar-valued energy function at the expense of using the tensor of anisotropy as an extra argument in line with the principle of isotropy of space [52,53]. For the specific case of graphene, the tensor of anisotropy is the structure tensor for the group **U**=**a**_{1}⊗**a**_{1}−**a**_{2}⊗**a**_{2}, **L**=**a**_{1}⊗**a**_{2}−**a**_{2}⊗**a**_{1} and {**a**_{1},**a**_{2}} is an orthonormal basis vector. It can also be written as
*θ*=2*π*/6, we have

Since all our energy arguments are in the Eulerian expression we should push-forward the tensor ^{3} of equation (3.1) [52,53]
**A** and a vector **z** the following definitions:

The invariants *I*_{1},*I*_{2} describe terms related with **B**^{F} solely, while *I*_{4},*I*_{5} terms related with **B**^{c} solely. The invariant *I*_{3} describes the effect of anisotropy on **B**^{F}, while *I*_{6} the effect of anisotropy on **B**^{c} coupled with **B**^{F}. The invariant *I*_{7} is a coupling term between **B**^{F} and **B**^{c}. The invariants *I*_{8},*I*_{9} describe coupling terms of **B**^{F} and **B**^{c} together with the effect of anisotropy. The invariants *I*_{10},*I*_{11} and *I*_{12} describe the effect of anisotropy on **B**^{F} coupled with **P**^{sv},**e*** and **b***, respectively. The invariants *I*_{13},*I*_{14} and *I*_{15} describe the effect of anisotropy on **B**^{c} coupled with **P**^{sv},**e*** and **b***, respectively. The invariants *I*_{16},*I*_{17} and *I*_{18} describe coupling between **B**^{F} and **P**^{sv},**e*** and **b***, respectively. The invariants *I*_{19},*I*_{20} and *I*_{21} describe coupling between **B**^{c} and **P**^{sv},**e***,**b***, respectively. The invariants *I*_{22},*I*_{23} and *I*_{24} describe coupling terms between **B**^{F} and the pairs (**P**^{sv},**e***),(**P**^{sv},**b***) and (**e***,**b***), respectively. The invariants *I*_{25},*I*_{26} and *I*_{27} describe coupling terms between **B**^{c} and the pairs (**P**^{sv},**e***),(**P**^{sv},**b***) and (**e***,**b***), respectively. Terms *I*_{28},*I*_{29} and *I*_{30} are due solely to **P**^{sv},**e*** and **b***, respectively. The terms *I*_{31},*I*_{32} and *I*_{33} are coupling terms for the pairs (**P**^{sv},**e***),(**P**^{sv},**b***) and (**e***,**b***), respectively. The invariants *I*_{34},*I*_{35} and *I*_{36} describe the effect of anisotropy on **P**^{sv},**e*** and **b***, respectively. Finally, *I*_{i}, *i*=37,38,…,42 describe the effect of anisotropy on the coupling of pairs (**P**^{sv},**e***),(**P**^{sv},**b***) and (**e***,**b***).

The list of invariants of equation (3.10) provides an exhaustive description of all possible couplings between all involved fields, namely **B**^{F},**B**^{c},**P**^{sv},**e*** and **b*** together with the effect of graphene's anisotropy. This is why the present framework provide a basis for a fully coupled approach. The energy of equation (3.9) can be written as

To proceed, the Cauchy stress tensor reads
**M** is the couple stress tensor and we also need the term **L**=∂*W*/∂**P**^{sv} related with the shift vector.

For the necessary derivatives after some straightforward calculations, we find

This term should be used in equation (3.16) for the evaluation of the first Piola–Kirchhoff stress tensor.

The term that appears in the definition of the couple stress tensor reads

The term that appears in the definition of the effective polarization reads

The term that appears in the definition of the effective magnetic–electric field intensity reads

The term related with the shift vector, **L**, has the expression

When equations (3.16)–(3.20) are substituted into equations (3.13)–(3.15), we obtain the constitutive equations that describe the electro-magneto-mechanical interactions for graphene at the continuum level. The constitutive equations are the Cauchy stress tensor **T**, the couple stress tensor **M**, the effective polarization **p***, the effective magnetic field intensity **b*** and the term ∂*W*/∂**P**^{sv} related with the shift vector.

For the free current density **j*** of equation (3.3), we may assume for *Π* the simplified dependence of the form
**j*** in the form

To bring our framework closer to more applied approaches and in order to present a path for measuring some of the material constants of the present framework, we further assume that *W* is a linear function of the invariants, i.e.
*W*/∂*I*_{i} in the previous equations, should be substituted by constants *c*_{i},*i*=1,…,42. To present some possible ways for measuring these material constants, we combine our previous similar work [6] with the work of Scott [56] as well as some basic knowledge from *π* and *σ* orbitals [9]. On the one hand, our previous paper [6], while confined to the geometrically and materially linear regime, it uses molecular mechanics calculations to evaluate material constants at the continuum level by keeping a clear distinction between the shift vector and the strain tensor. What one should keep from this analysis is that the radial distribution diagram (abbreviated as RDF henceforth) can differentiate between two distances in the unit cell of graphene: the shift vector as the 1.42 Å peak and the skeletal lattice vector as the 2.42 Å peak. So, for our analysis here, motivated from [6] all invariants containing the shift vector **P**^{sv} should be mechanically measured through changes of the 1.42 Å peak in the RDF diagram. Invariants containing term **B**^{F} should be related to changes of the 2.42 Å peak in the RDF diagram. The curvature effect can be introduced starting from a non-flat graphene with known curvature. So, the work [6] can motivate the measurable quantities in the RDF diagram for the purely mechanical fields.

The experiments that should be performed in our finite deformation regime are motivated from the approach of Scott [56]. This author gives some plausible definitions for the incremental bulk modulus, the incremental Young modulus and the incremental Poisson ratio, that we adopt here as well for some of the material constants related with our list of invariants. Essentially, these incremental measures of [56] are derivatives of the applied stress with respect to strain, so they measure stress–strain changes in an incremental form. For our analysis, for the role of strain one should be careful to distinguish the 1.42 Å peak of the RDF that corresponds to invariants containing the shift vector from the 2.42 Å peak that corresponds to invariants that contain term **B**^{F}.

For the electric field, we note that this is measured for graphene by the *π* and *σ* orbitals [9]. The *π* orbitals are out-of-plane functions rendering the most probable place occupied by graphene's electrons. The *σ* orbitals are in-plane functions measuring the same quantity but in-plane only. For our analysis, here we assume that when out-of-surface mechanical fields should be quantified for their effect on the electric field, then only the *π* orbital changes should be measured. When there are only in-surface changes the *π* orbitals remain unchanged, but *σ* orbitals should change. So, when **B**^{F} is coupled with **e***, the invariant involved should describe how the *σ* orbital changes with loading, while when **B**^{c} is coupled with **e***, the corresponding invariant should measure the effect loading has on the *π* orbitals. We note that such electronic measurements of the *π* and *σ* orbitals should be done using quantum methods (e.g. the ab initio method [57] or tight-binding methods [58]).

So, for our case, we first note that the first three invariants of equation (3.10) are of pure mechanical nature, since only the mechanical field **B**^{F} is present and the anisotropy. So, for material constant *c*_{1} we propose to measure in an RDF diagram how the 2.42 Å peak changes with in-plane shear strain in the armchair direction. This would then be a generalization of the incremental shear modulus of [56] for graphene where the two peaks in the RDF are different [6]. For constant *c*_{2} we are motivated from the incremental bulk modulus defined in [56] and measure the effect a pressure experiment has on the 2.42 Å peak of the RDF. Invariant *I*_{3} differs from *I*_{1} due to the anisotropy term, so we propose constant *c*_{3} should be defined as constant *c*_{1} but the loading now should be along the zig-zag direction. The remaining purely mechanical constants that can be seen under the above perspective are *c*_{16},*c*_{28},*c*_{34}. For *c*_{16} we are motivated from the incremental Young modulus of [56] and propose this quantity to measure the effect axial tension along the armchair direction has on the 2.42 Å peak of the RDF. For *c*_{28} we propose this quantity to measure the effect axial tension along the armchair direction has, but now in the 1.42 Å peak of the RDF diagram. For *c*_{34} we propose an analogous definition with *c*_{28} but the loading should now be along the zig-zag direction. We note, that in line with the approach of [56] the above-defined material constants are derivatives of the applied stress with respect to one of the two peaks of graphene in the RDF, i.e. the material constants discussed so far are quantities of the form

Invariants that combine the elastic field with the electric field are *I*_{11},*I*_{14},*I*_{17},*I*_{20},*I*_{22},*I*_{25} to name a few. For *c*_{11} we propose it to be a measure of how the *σ* orbitals change with the change in the 2.42 Å peak when the loading is simple shear along the zig-zag direction. This is essentially motivated from the definition of the incremental shear modulus of [56] coupled with the electric field. For *c*_{14} we propose a similar definition as that of *c*_{11} but now the *π* orbitals should be measured. For material constant *c*_{17} we propose that it should measure changes the *σ* orbitals suffer with changes in the 2.42 Å peak when the loading is an axial tension along the armchair direction. This is essentially motivated from the incremental Young modulus of [56] for the case when the electric field is present as well. The material constant *c*_{20} then can have a similar definition as that of *c*_{17} but now the *π* orbitals are measured. For the material constant *c*_{31} one may measure the changes in the *σ* orbital with changes in the 1.42 Å peak when graphene's sheet is under axial tension along the armchair direction. Material constant *c*_{39} can have the same definition as *c*_{31} but the loading should be along the zig-zag direction. We note that all the quantities defined for the coupling of the electric field with the mechanical fields are essentially chain rules of the form

To summarize, the field equations are equations (2.9), (2.10), (2.12), (2.13) and (2.19). These together with the constitutive laws of equations (3.13)–(3.15) (with the help of equations (3.16)–(3.20)) provide a closed set of equations which coupled with boundary conditions render the fields **B**^{F},**B**^{c},**P**^{sv},**e***,**b*** for all times and for all of graphene's surface.

## 4. Special cases for the constitutive equations: graphene as an electric conductor and as an electric insulator

All the above are generic expressions valid for both cases where graphene is an insulator or a conductor. But there are also fundamental differences between these two characters that graphene may have. For conductors, free electrons move easily within graphene, a situation which is no longer true for insulators. Thus, for insulators, one should set in the field equations of §2 the free charge density *σ** and the free current density **j*** equal to zero. The constitutive laws of §3 remain as they are but **e*** and **b*** measure the electric and magnetic effects stemming from the movement of valence electrons only; free electrons do not move in an insulator, so they do not contribute when measuring **e*** and **b***. On the other hand, for a conductor *σ**≠0,**j***≠0 as an outcome of the movement of free electrons. So, these terms should be kept intact at the field equations of §2. The constitutive laws of §3 remain as they are, but now **e*** and **b*** measure the electric and magnetic effects generated not only from valence electrons, but also from free electrons.

In the physics literature for semiconductors, it is assumed that there are current carriers of two types [49]: electrons in the conduction band and positive holes in the filled valence band. In general, electrons and holes in condensed matter physics are described by separated equations which are not in any way connected. By contrast, electron and holes states in graphene are interconnected [27]. At the continuum level, we may tackle electrons and holes separately by using two different arguments on the continuum energy: one for the effective electric and magnetic fields of the electrons

## 5. Conclusion

Motivated by the fact that pristine graphene behaves as a semiconductor, we build a mathematical model to describe the effects of electro-magneto-mechanical fields on graphene. By proper definition of graphene's energy arguments and implementation of graphene's symmetries, we find that 42 invariants explain the actions of the external fields to material. We give an explanation of the coupling between the external fields and the way that this is expressed through each invariant. We close by giving the appropriate field equations, i.e. the Maxwell equations, the momentum equation, the moment of momentum equation and the equation ruling the shift vector.

The novelty and importance of our approach is multiple: firstly, we provide a solid framework for the nonlinear electro-magneto-elasticity of semiconductors, by differentiating and studying both regimes, i.e. the conducting behaviour for high voltage and the insulating behaviour of low voltage. Secondly, our model can work as a basis for studying various types of doped graphene, which exhibits a number of important properties that do not appear in pristine graphene. More specifically, properly doped graphene can exhibit piezoelectric [47] and ferromagnetic [48] behaviour. Both pristine graphene and doped graphene have exceptional and extraordinary thermal, electric and magnetic properties so a mathematical model is fundamental to the design and manufacturing of future devices. Finally, our analysis implies that a certain number of parameters can describe fully the coupled problem and our work can motivate carefully selected experiments and simulations in order to measure these parameters.

As for future directions, we consider that numerical investigation of the coupling between the electric and the mechanical field using, e.g. density functional calculations, constitutes a highly challenging theoretical and numerical issue that would highlight the coupling between these fields and measure some of the material parameters introduced here. Solving some simple but fundamental boundary value problems constitutes a worth working task also for graphene. For such an endeavour, one should start working with a simplified model (in line with our previous purely mechanical approach [3]) and expect to have more than one solution as in [3].

## Authors' contributions

All authors contributed equally to this work.

## Competing interests

Does not apply to this work.

## Funding

No funding was received for this work.

## Acknowledgements

Some of the material presented here was written while D.S. was a visitor at the Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, Okinawa, Japan. D.S. acknowledges the kind hospitality of Prof. Eliot Fried in the OIST during his visit. Prof. V. K. Kalpakides (Ioannina, Greece) is thanked for many valuable discussions on the topic.

## Footnotes

↵1 In this communication, we assume that the dissipation of energy only happen due to heat transfer and electric dissipation.

↵2 Although we expect the resistivity versus voltage diagram to be strongly dependent on temperature.

↵3 We have used the classical invariants presented, for example, in [52,53]; however, a new set of invariants for coupled problems have been recently investigated by Shariff and co-workers [54]. Such invariants are defined with the principal directions of the deformation, and it has been shown recently that for problems where we have deformations and magnetic or electric fields, such new invariants may have clearer physical meanings than the classical invariants by Spencer [55], however, since such representation theories are very new, in the present work we do not include them in our analysis.

- Received October 28, 2015.
- Accepted March 10, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.