A Geometrically Exact Continuum Framework for Light-Matter Interaction in Photo-Active Polymers I. Variational Setting
AA Geometrically Exact Continuum Framework forLight-Matter Interaction in Photo-Active PolymersI. Variational Setting
M Mehnert , W Oates , P Steinmann , Institute of Applied Mechanics, Friedrich-Alexander University Erlangen-Nuremberg,91054 Erlangen, Germany Florida Center for Advanced Aero Propulsion (FCAAP),Department of Mechanical Engineering,Florida A & M and Florida State University,Tallahassee, Florida 32310, USA Glasgow Computational Engineering Centre, University of Glasgow,G12 8QQ Glasgow, United Kingdom
Abstract
Molecular photo-switches as, e.g., azobenzene molecules allow, when embedded into a polymericmatrix, for photo-active polymer compounds responding mechanically when exposed to light of cer-tain wavelength. Photo-mechanics, i.e. light-matter interaction in photo-active polymers holds greatpromise for, e.g., remote and contact-free activation of photo-driven actuators. In a series of earliercontributions, Oates et al. developed a successful continuum formulation for the coupled electric, elec-tronic and mechanical problem capturing azobenzene polymer compounds, thereby mainly focussingon geometrically linearized kinematics [1, 2, 3]. Building on that formulation, we here explore thevariational setting of a geometrically exact continuum framework based on Dirichlet’s and Hamilton’sprinciple as well as, noteworthy, Hamilton’s equations. Thereby, when treating the dissipative case,we resort to incremental versions of the various variational problems via suited incorporation of adissipation potential. In particular, the Hamiltonian setting of geometrically exact photo-mechanicsis up to now largely under-explored even for the energetic case, arguably since the correspondingLagrangian is degenerate in Dirac’s sense. Moreover, in general, the Hamiltonian setting of dissipa-tive dynamical systems is a matter of ongoing debate per se. In this contribution, by advocating anovel incremental version of the Hamiltonian setting exemplified for the dissipative case of photo-mechanics, we aim to also unify the variational approach to dissipative dynamical systems. Takentogether, the variational setting of a geometrically exact continuum framework of photo-mechanicspaves the way for forthcoming theoretical and numerical analyses.
Photosensitive materials possess the ability to convert photonic energy into a mechanical materialresponse, which eliminates the necessity of electric wiring or circuits of conventional smart materialssuch as shape memory alloys, electroceramics or electro-active polymers [4]. The photo-mechanicalcoupling can originate from various, fundamentally different physical effects, depending on the spe-cific material under investigation. In electrically polar solids, the bulk deformation results from theconverse piezoelectric effect in combination with the photovoltaic effect. While this combination ofproperties was discovered in single crystals of SbSI [5, 6], more recently ferroelectric compounds suchas BiFeO [7, 8] and PbTiO in form of thin films [9] have come into focus. The group of polarand non-polar semiconductors show a similar behavior, as the material deformation also originates1 a r X i v : . [ c s . C E ] N ov rom the converse piezoelectric effect. However, in the case of polar semiconductors the necessaryelectric field is generated by light-induced changes in the free surface charges [10, 11] whereas in thecase of non-polar semiconductors such as Germanium, Silicon or Carbon nanotubes, an excess ofelectron hole pairs induces the electric stimulus [12, 13, 14, 15]. Another promising representativeof photo-sensitive materials can be found in organic polymers, in which conformational changes ofmolecular switches, e.g. from rod (low energy state) to kinked (high energy state) shape, are triggeredby light and, when embedded into a polymeric matrix, result in (potentially large) photo-induceddeformation of the resulting (effective) compound material [16, 17]. As an example, in azobenzenephoto-switches, these conformational changes are a consequence of trans-cis (and likewise the reversecis-trans) photo-isomerization depending on the wave-length of the incident light, typically in the ∼ −
500 nm (UV to visible) range [18, 19, 20]. Clearly, photo-sensitive polymers promise fasci-nating applications, e.g., for remote and contact-free activation of optical actuators [21, 22].Recently, in a series of contributions Oates et al. proposed a comprehensive phenomenologicalcontinuum formulation of photo-mechanics that is specifically tailored to capture the light-inducedmechanical response of azobenzene polymers [1, 2, 3]. Therein modeling at continuum length scales,while retaining the specifics of the underlying light-matter interactions in a homogenized sense, relieson the introduction of electronic order parameter fields in addition to the electric and mechanicalfields common in the field of electro-mechanics [23]. Thus, along with the electronic degrees of free-dom, an electronic (micro-force-type) balance equation arises in addition to the common electric andmechanical balances. Taken together, the formulation sub-divides into an electric, an electronic anda mechanical sub-problem. The formulation in [1, 2, 3] focuses mainly on geometrically linearizedkinematics when treating azobenzene-polymer compounds that qualify as mechanically stiff. Inter-estingly, ongoing research in organic chemistry focuses on synthesis and characterization of a varietyof alternative polymer compounds involving various types of molecular photo-switches, thereby alsopromising options for mechanically soft photo-active polymers [24, 25, 26]. Consequently, a geomet-rically exact continuum modeling framework is a necessary tool for the analytical and computationaldesign and optimisation of future photo-mechanical devices.Motivated by this state of affairs, we here pursue a rigorous geometrically nonlinear account on thephenomenological continuum modeling of photo-active polymers, thereby focusing on the variationalsetting. When considering light-matter interaction, the frequency (wave length) of electro-magneticwaves associated with the incident light is orders of magnitude higher (smaller) than that displayedby the mechanical response of matter at the continuum length scale. Thus, for the sake of simplic-ity, we here consider all electric quantities as time-averaged, indeed as quasi-static, and neglect anymagnetic effects. Consequently, only the electric potential and electric Gauss law remain from theelectro-magnetic degrees of freedom and the set of Maxwell equations describing electro-magneto-dynamics.The electronic and mechanical solution fields may, however, display inertia effects, thus requiringtheir incorporation at the continuum length scale. Consequently, after exercising the quasi-staticcase within the realm of Dirichlet’s principle as a preliminary, we treat the dynamic case of thecoupled problem via Hamilton’s principle and, noteworthy, via Hamilton’s equations. The latter issomewhat sophisticated due to the Lagrangian being degenerate in the sense of Dirac’s theory sinceno velocity of the electric potential is involved. Thus, Legendre transformation of the Lagrangian intothe Hamiltonian involves Lagrange multipliers to enforce corresponding constraints on the associatedmomenta [27, 28]. 2mportantly, light-matter interaction is associated with energetic losses, e.g. due to optical scatter-ing and/or photochemical reactions, thus asking for consideration of dissipation. We will thus inparticular demonstrate how to cope with the dissipative case by resorting to incremental variationalsettings. Noteworthy, especially Hamilton’s equations based on an incremental total energy are anovelty also beyond photo-mechanics.We structure this manuscript as follows: Section 2 first introduces the electric, electronic and mechan-ical solution fields and their space-time gradients in the context of a geometrically exact continuumdescription. Based thereon, Section 3 discusses the corresponding contributions to the various in-ternal and external energy densities. These serve as potentials for the explicit constitutive relationsin Section 4. To set the stage for the variational setting of photo-mechanics, Section 5 explores theenergetic and dissipative case of Dirichlet’s principle as a preliminary to a corresponding account onHamilton’s principle in Section 6. Lastly, Section 7 addresses the two-fold challenge associated withHamilton’s equations, i.e. the determination of the Hamiltonian from a degenerate Lagrangian anda proper account for the dissipative case. Eventually, Section 8 concludes the manuscript.
The subsequently formulated continuum framework for the description of light-matter interactioninevitably requires numerous electric, electronic and mechanical quantities. In an attempt to highlightrespective quantities that share similar characteristics in each of these fields, we try to term theseidentically in different fonts. Scalar- and vector-valued electric quantities are written in meager italicand blackboard fonts a, a respectively, bold sans-serif fonts a is used for (vector- and tensor-valued)electronic quantities, and bold italic font a is selected for vector- and tensor-valued mechanicalquantities. In order to facilitate the handling of this work, Table 1 summarizes the terms andexpressions of the respective fields. Modelling light-matter interaction in photo-active polymers based on molecular photo-switches con-sists of coupled electric, electronic and mechanical sub-problems, each expressed in terms of a cor-responding solution field. Subsequently, we shall first briefly introduce the electric, electronic andmechanical solution fields together with their pertinent space-time gradients.
The scalar-valued electric solution field y , parameterized in terms of the material space coordinate X and time t , represents the electric potential y = y ( X , t ) with E := −∇ X y ( X , t ) and v := D t y ( X , t ) . (1)Its (negative) material space gradient renders the nominal (Piola-type) electric field E , its materialtime gradient, introduced here merely for completeness, denotes the material rate v of the electricpotential. 3pace-time quantitiesmaterial identity tensor I spatial identity tensor i material position vector X spatial position vector x time t Electric quantitieselectric solution field y = y ( X , t )electric field E := −∇ X y ( X , t )material rate of the electricsolution field v := D t y ( X , t )Electronic quantitieselectronic solution field y t , y c electronic order parameterfield y := { y t , y c } space gradient of theelectronic order parameterfield F := ∇ X y ( X , t )material rate of the electronicorder parameter field v := D t y ( X , t )Mechanical quantitiesmechanical solution field y deformation gradient F := ∇ X y ( X , t )Jacobian determinant J = det( F )cofactor K = K ( F ) := cof F = J F − T inverse deformation gradient f := F − inverse Jacobian determinant j = J − inverse cofactor k = K − velocity v := D t y ( X , t )Table 1: Summary of the necessary expressions for the description of a photo-mechanical modellingframework The vector-valued electronic solution fields y t and y c , here collectively assembled in the double-vector-valued electronic order parameter field y := { y t , y c } , represent the effective density of, e.g.,vector-valued trans (low energy, rod shape) and cis (high energy, kinked shape) states of polymer-embedded photo-active azobenzene molecular switches y = y ( X , t ) with F := ∇ X y ( X , t ) and v := D t y ( X , t ) . (2)The material space-time gradients F and v of the electronic order parameter field y capture its spatialand temporal changes.In terms of generalized continua, the electronic order parameter(s) contained in y are attached tothe material macro position vector X . They may be thought of as effective micro position vectors(electronic coordinates) obtained by homogenizing micro position vectors connecting to photo-activecharged particles of, e.g., azobenzene molecules within an RVE, see Figure 1.4 NN N N
Azobenzene cis-state Azobenzene trans-statetrans-cisisomerizationcis-transisomerization
Figure 1: Light induced transformation of an azobenzene molecule.
The vector-valued mechanical solution field y represents the nonlinear deformation map of geomet-rically exact continuum kinematics. It maps material position vectors X (material coordinates) ofphysical points in the material (undeformed/reference) configuration into their spatial counterpart x in the spatial (deformed/current) configuration, i.e. x = y ( X , t ) with F := ∇ X y ( X , t ) and v := D t y ( X , t ) . (3)The corresponding material space-time gradients F and v render the deformation gradient (or ratherthe tangent map) and the velocity.Regarding the deformation gradient F as the tangent map d x = F · d X it proves convenient tointroduce its cofactor K as the area map d a = K · d A and its determinant J as the volume mapd v = J d V via K = K ( F ) := cof F = J F − T and J = J ( F ) := det F . (4)Moreover, it is useful to occasionally abbreviate the inverses of F , K , and J as f := F − and k =: K − and j := J − . (5)In the sequel, the following derivatives of f , K and J are needed ∂ f ∂ F = − f (cid:2) f t and ∂ K ∂ F = f t ⊗ K − K (cid:0) f with ∂J∂ F = K , (6)whereby the non-standard dyadic products (cid:2) and (cid:0) expand in Cartesian coordinate representationas [ A (cid:2) B ] ijkl := A ik B jl and [ A (cid:0) B ] ijkl := A il B jk . Composition with the inverse deformation map X = y − ( x , t ) results in the re-parameterized electricsolution field (cid:101) y , i.e. the re-parameterized electric potential, and its corresponding (negative) spatial(space) gradient (cid:101) e , i.e. the true (Cauchy-type) electric field (cid:101) y ( x , t ) := y ( X , t ) ◦ y − and (cid:101) e := −∇ x (cid:101) y ( x , t ) . (7)Likewise, composition with the inverse deformation map renders the re-parameterized electronicsolution fields (cid:101) y t and (cid:101) y c , i.e. the re-parameterized electronic order parameter field (cid:101) y = { (cid:101) y t , (cid:101) y c } , andits corresponding spatial (space) gradient (cid:101) f (cid:101) y ( x , t ) := y ( X , t ) ◦ y − = y ( y − ( x , t ) , t ) and (cid:101) f := ∇ x (cid:101) y ( x , t ) . (8)5inally, composition of (cid:101) e and (cid:101) f with the deformation map x = y ( X , t ) results in the re-parameterizedspatial (space) gradients e and f , i.e. e = e ( X , t ) := (cid:101) e ( x , t ) ◦ y = E · f and f = f ( X , t ) := (cid:101) f ( x , t ) ◦ y = F · f . (9)Note the push-forward relation between E and e as well as between F and f in terms of the inversedeformation gradient f := F − .For the ease of notation we shall from here on use the sloppy notation e and f also for (cid:101) e and (cid:101) f ,thereby ignoring the parameterizations in either spatial or material coordinates x or X , respectively,if there is no danger of confusion. Any of the variational settings as discussed below build on properly defined expressions for variouskind of energy densities. We shall thus first discuss these separately for the electric, the electronicand the mechanical sub-problem. Thereby, we distinguish energy densities per unit volume in eitherthe material or the spatial configuration by corresponding sub-scripts, i.e. the material ( • ) m versusthe spatial ( • ) s density. These densities are related by the Jacobian J of the deformation gradientas ( • ) m = J ( • ) s . The electric field penetrates free space and matter likewise, whereby electro-static energy is stored.We shall here denote the electro-statically stored energy as electric internal potential energy withmaterial density e m = J e s . Expressed in terms of the nominal electric field E and the deformationgradient F (that in free space is a suited artificial extension of its counterpart in matter, see [29]),the electric internal potential energy density reads as e m = e m ( E , F ) := − J ε e ( E , F ) · e ( E , F ) = − ε e ( E , F ) · K · E = − ε E · f · K · E . (10)Here, ε denotes the electric permittivity of free space, a natural constant. Note i) the commonquadratic expression of 2 e s = − ε | e | when expressed in terms of the true electric field e , and ii) thenegative sign of e s (thus a Legendre transformation e s − e · ∂ e e s results in a corresponding (dual)energy density with positive sign when expressed in terms of the conjugate variable − ∂ e e e ).For convenience of later analyses, we pre-compute the derivatives of the electric internal potentialenergy density e m with respect to the nominal electric field E and the deformation gradient F as − ∂e m ∂ E = ε e · K and ∂e m ∂ F = [ e s i + ε e ⊗ e ] · K , (11)where i is introduced as the spatial identity tensor. Observe the term ε e representing the freespace electric flux density (electric displacement) as well as the so-called energy-momentum formatof e s i + ε e ⊗ e representing the free space Maxwell stress (both of Cauchy-type).6urthermore, we introduce the electric external potential energy densities v elecm and (cid:98) v elecm in the bulkof matter and at the boundary between matter and free space, respectively, as v elecm = v elecm ( y ) := q fm y and (cid:98) v elecm = (cid:98) v elecm ( y ) := (cid:98) q fm y. (12)Here, q fm and (cid:98) q fm are the externally prescribed electric free charge densities per unit volume and unitarea, respectively, in the material configuration. The electronic solution fields y t and y c are associated with effective charge densities interacting withthe electric field, thereby storing energy. For the sake of terminological consistency, we shall denotethe corresponding stored energy as electronic internal potential energy with density c m = J c s , itreads as c m = c m ( y , E , F ) := − J ω y · e ( E , F ) = − ω y · K · E . (13)Here, ω = { ω t0 , ω c0 } denote the effective charge densities bound to the electronic order parameter(s),whereby we shall assume ω as given and constant in order to avoid the necessity to include internalvariables within a variational setting (we shall do so in a separate contribution). Different modeloptions of time-varying effective charge densities are pursued in [1, 30].For convenience of later analyses, we pre-compute the derivatives of the electronic internal potentialenergy density c m with respect to the nominal electric field E , the electronic order parameter(s) y and the deformation gradient F as − ∂c m ∂ E = ω y · K and − ∂c m ∂ y = J ω e and ∂c m ∂ F = [ c s i + ω e ⊗ y ] · K . (14)Observe the term ω y as a contribution to the polarization in matter, the term ω e as an electronic(Lorentz-type) dipole force density in matter as well as the energy-momentum format of c s i + ω e ⊗ y as a contribution of polarization to the Maxwell stress in matter.Furthermore, we introduce the electronic external potential energy densities v tronm and (cid:98) v tronm in thebulk of matter and at the boundary between matter and free space, respectively, as v tronm = v tronm ( y ) := − b m · y and (cid:98) v tronm = (cid:98) v tronm ( y ) := − t m · y . (15)Here, b m and t m , introduced for the sake of completeness, are externally prescribed electronic forcedensities per unit volume and unit area, respectively, in the material configuration.The electronic kinetic energy density k tronm = J k trons captures inertia of the electronic modes in termsof the material velocity v of the electronic order parameter(s) and the electronic inertia density (cid:37) m = J (cid:37) s , a phenomenological parameter, as k tronm = k tronm ( v ) := 12 (cid:37) m v · v . (16)Finally, we introduce the electronic dissipation potential density p m = J p s in order to account forenergy losses, e.g. due to optical scattering and/or photochemical reactions p m = p m ( v ) := 12 J γ v · v . (17)7ere, γ denotes a phenomenological damping parameter related to the material velocity v of theelectronic order parameter(s). Its inverse relates to a time constant characterizing the relaxation ofthe material from a higher to a lower energetic excitation state. We shall denote the energy that is mechanically stored in matter as mechanical internal potentialenergy with material density w m = J w s . Expressed in terms of the electronic order parameter(s) y ,their material gradient F and the deformation gradient F , i.e. in terms of the micro and the macrodeformation, the mechanical internal potential energy density reads generically as w m = w m ( y , F , F ) := w ◦ m ( y , F ) + w • m ( F ) . (18)Here, we distinguish the contribution w ◦ m due to the electronic order parameter(s) and the defor-mation gradient and the contribution w • m due to the material gradient of the electronic order pa-rameter(s). The former captures mechanically stored energy of e.g. amorphous azobenzene-polymerblends (as azobenzene-polyimide polymer networks), whereas the latter describes e.g. acrylate-basedazobenzene-polymer blends displaying liquid crystal domain formation (as azobenzene-LCNs).Furthermore, we introduce the mechanical external potential energy densities v mechm and (cid:98) v mechm in thebulk of matter and at the boundary between matter and free space, respectively, as v mechm = v mechm ( y ) := − b m · y and (cid:98) v mechm = (cid:98) v mechm ( y ) := − t m · y . (19)Here, b m and t m are externally prescribed mechanical force densities per unit volume and unit areain the material configuration and (cid:98) v mechm is given in energy per area.Finally, the mechanical kinetic energy density k mechm = J k mechs captures inertia of the mechanicalmodes in terms of the material velocity v of the deformation map and the mechanical inertia density ρ m = J ρ s as k mechm = k mechm ( v ) := 12 ρ m v · v . (20)This concludes the energetic characterization of the electric, electronic and mechanical sub-problems. The derivatives of the various electric, electronic and mechanical energy densities with respect to theirarguments (state variables) define constitutive expressions for their energetically conjugate quantities(state functions). Collectively, the state variables and state functions constitute the state quantitiesthat describe the state of a system. We shall here introduce these constitutive relations as definitionsfor the convenience of later considerations, thereby distinguishing between nominal (Piola-type) andtrue (Cauchy-type) quantities.
The nominal (Piola-type) electric flux density (or rather electric displacement) in free space, thenominal polarization and the nominal electric flux density in matter follow as D ε := − ∂e m ∂ E = ε e · K and P := − ∂c m ∂ E = ω y · K and D := D ε + P . (21)8oreover, these nominal quantities relate to the true (Cauchy-type) electric flux density in free space,the true polarization and the true electric flux density in matter via a Piola transformation, i.e. aright-sided push-forward with the inverse cofactor k , to render d ε := − ∂e s ∂ e = ε e and p := − ∂c s ∂ e = ω y and d := d ε + p = D · k . (22)Finally, the nominal electric external source density, i.e. the free bulk charge, and the nominal electricexternal flux density, i.e. the free surface charge, derive form the electric external potential energydensities as q fm := ∂v elecm ∂y and (cid:98) q fm := ∂ (cid:98) v elecm ∂y . (23)The free bulk charge and the free surface charge are here considered as given, i.e. as externallyprescribed data. The electronic nominal (Piola-type) stress in matter follows as the derivative P := ∂w m ∂ F = ∂w • m ∂ F . (24)Moreover, it relates to the electronic true (Cauchy-type) stress via a Piola transformation to render s := ∂w m ∂ F · k = ∂w • m ∂ F · k . (25)Likewise, we define nominal energetic, dissipative, and total electronic internal source densities as s (cid:93) m := ∂w m ∂ y = ∂w ◦ m ∂ y and s (cid:91) m := ∂ p m ∂ v = J γ v and s m := s (cid:93) m + s (cid:91) m . (26)Then, their true counterparts, i.e. the true energetic, dissipative, and total electronic internal sourcedensities compute as s (cid:93) s := ∂w s ∂ y = ∂w ◦ s ∂ y and s (cid:91) s := ∂ p s ∂ v = γ v and s s := s (cid:93) s + s (cid:91) s = j s m . (27)Next, we introduce the nominal exterior electronic external source density and the nominal electronicexternal flux density b (cid:78) m := − ∂v tronm ∂ y and t m := − ∂ (cid:98) v tronm ∂ y . (28)The nominal exterior electronic external source density b (cid:78) m is complemented by its interior coun-terpart b (cid:77) m that is due to the coupling of the electronic order parameter(s) with the electric field.Collectively, these render the total electronic external source density b m , i.e. b (cid:77) m := − ∂c m ∂ y and b m := b (cid:78) m + b (cid:77) m . (29)The nominal exterior electronic external source density and the nominal electronic external fluxdensity are here considered as given, i.e. as externally prescribed data.Finally, the nominal electronic momentum p m := ∂k tronm ∂ v = (cid:37) m v (30)is the conjugate quantity to the material velocity v of the electronic order parameter(s).9 .3 Mechanical Problem The electric and electronic (Maxwell stress) contributions to the nominal (or rather Piola) stressderive from the electric and electronic internal potential energy densities P elec := ∂e m ∂ F = [ e s i + e ⊗ d ε ] · K and P tron := ∂c m ∂ F = [ c s i + e ⊗ p ] · K . (31)Likewise, the mechanical contribution to the nominal (or rather Piola) stress derives from the me-chanical internal potential energy density P mech := ∂w m ∂ F . (32)Collectively, these result in the total nominal (or rather Piola) stress as P = P elec + P tron + P mech = (cid:2) [ e s + c s ] i + e ⊗ d ] · K + ∂w m ∂ F . (33)Piola transformation then renders the electric and electronic contributions to the true (or ratherCauchy) stress s elec := ∂e m ∂ F · k = [ e s i + e ⊗ d ε ] and s tron := ∂c m ∂ F · k = [ c s i + e ⊗ p ] . (34)together with the mechanical contribution to the true (or rather Cauchy) stress s mech := ∂w m ∂ F · k . (35)Again, collectively, these result in the total true (or rather Cauchy) stress s = s elec + s tron + s mech = (cid:2) [ e s + c s ] i + e ⊗ d ] + ∂w m ∂ F · k = P · k . (36)Next, the nominal mechanical external source density, i.e. the volume-distributed body force, andthe nominal mechanical external flux density, i.e. the area-distributed surface traction, derive as b m := − ∂v mechm ∂ y and t m := − ∂ (cid:98) v mechm ∂ y . (37)Finally, the nominal mechanical momentum is the conjugate quantity to the material velocity v ofthe deformation map p m := ∂k mechm ∂ v = ρ m v . (38)This concludes the constitutive characterization of the electric, electronic and mechanical sub-problems. For quasi-static situations without any inertia effects, Dirichlet’s principle of stationary potential en-ergy renders the pertinent equilibrium equations in the bulk and at the boundary, here for the electric,the electronic and the mechanical sub-problems. Traditionally, Dirichlet’s principle is restricted toconservative, i.e. energetic cases void of dissipation. However, when expressed as an incrementalvariational problem in terms of the incremental work, also non-conservative, i.e. dissipative cases canbe considered when properly incorporating a dissipation potential. In the sequel, we will demonstratethe variational setting of energetic and dissipative cases when modelling light-matter interaction inphoto-active polymers. 10 .1 Energetic Case
For the energetic case we first expand on the total potential energy densities before examining thepertinent variational setting.
For the sake of convenience, we introduce the total potential energy density u m = J u s in the bulk ofmatter as the summation of the corresponding internal electric, electronic and mechanical potentialenergies as well as the total external potential energy density u m = u m ( y, y , y , E , F , F ) := e m ( E , F ) + c m ( y , E , F ) + w m ( y , F , F ) + v m ( y, y , y ) . (39)Thereby, the total external potential energy density v m = J v s in the bulk of matter is the summationof the corresponding external electric, electronic and mechanical potential energy densities v m = v m ( y, y , y ) := v elecm ( y ) + v tronm ( y ) + v mechm ( y ) . (40)Moreover, we abbreviate the total external potential energy density (cid:98) v m = (cid:98) J (cid:98) v s at the boundarybetween matter and free space as the summation of the corresponding external electric, electronicand mechanical potential energy densities (cid:98) v m = (cid:98) v m ( y, y , y ) := (cid:98) v elecm ( y ) + (cid:98) v tronm ( y ) + (cid:98) v mechm ( y ) . (41)The total potential energy density u m in the bulk of matter as well as the total external potentialenergy density (cid:98) v m at the boundary between matter and free space together with the electric internalpotential energy density e m in the bulk of free space contribute to the potential energy functional asdiscussed in the sequel. To begin with, we define the potential energy functional for Dirichlet’s principle as U = U ( y, y , y ) := (cid:90) B m u m ( y, y , y , E , F , F ) d V + (cid:90) ∂ B m (cid:98) v m ( y, y , y ) d A (42)+ (cid:90) S m e m ( E , F ) d V. Then, Dirichlet’s principle requires stationarity of the potential energy functional upon admissible,i.e. (space) boundary conditions satisfying material variation D δ (i.e. variation at fixed materialposition X ) of the solution fields y, y , y as U ( y, y , y ) → stationary point . (43)Concretely, the stationarity condition for the potential energy functional expands asD δ U = (cid:90) B m D δ u m ( y, y , y , E , F , F ) d V + (cid:90) ∂ B m D δ (cid:98) v m ( y, y , y ) d A (44)+ (cid:90) S m D δ e m ( E , F ) d V . = 0 ∀ D δ y, D δ y , D δ y . δ y, D δ y , D δ y andusing the constitutive relations as introduced in the above, results eventually in the Euler-Lagrangeor rather equilibrium equations Div D − q fm = 0 in B m and [[ D ]] · N = (cid:98) q fm at ∂ B m (45a)Div D = 0 in S m (45b)Div P + b m = s (cid:93) m in B m and P · N = t m at ∂ B m (45c)Div P + b m = in B m and − [[ P ]] · N = t m at ∂ B m (45d)Div P = in S m (45e)Observe that the Neumann-type boundary condition for the mechanical sub-problem appears as jumpcondition for the total Piola stress P at the boundary between matter and free space, thus involvingthe Maxwell stress P elec as present in the free space and exerted on the continuum body as thecorresponding Maxwell traction [23, 31]. Consequently, for polymers with low relative permittivity ε r in the order of some 10 − , the free space sub-problem is indeed non-negligible.Furthermore, note that the equilibrium equations follow in terms of flux and source densities per unitarea and volume, respectively, in the material configuration. For completeness, Piola transformationthen renders the entirely equivalent expressions in terms of flux and source densities per unit areaand volume, respectively, in the spatial configurationdiv d − q fs = 0 in B s and [[ d ]] · n = (cid:98) q fs at ∂ B s (46a)div d = 0 in S s (46b)div s + b s = s (cid:93) s in B s and s · n = t s at ∂ B s (46c)div s + b s = in B s and − [[ s ]] · n = t s at ∂ B s (46d)div s = in S s (46e) Based on the dependency of the potential energy functional on the solution fields and their material spacegradients, variational calculus results in the following Euler-Lagrange equations: • Euler-Lagrange equations in the bulk of matterDiv ∂u m ∂ ∇ X y = ∂u m ∂y , Div ∂u m ∂ ∇ X y = ∂u m ∂ y , Div ∂u m ∂ ∇ X y = ∂u m ∂ y . • Euler-Lagrange equations in the bulk of free spaceDiv ∂e m ∂ ∇ X y = 0 , Div ∂e m ∂ ∇ X y = . • Euler-Lagrange equations at the boundary between matter and free space (cid:20) ∂u m ∂ ∇ X y − ∂e m ∂ ∇ X y (cid:21) · N = − ∂ (cid:98) v m ∂y , ∂u m ∂ ∇ X y · N = − ∂ (cid:98) v m ∂ y , (cid:20) ∂u m ∂ ∇ X y − ∂e m ∂ ∇ X y (cid:21) · N = − ∂ (cid:98) v m ∂ y Identifying the individual terms with the constitutive relations as in the above renders the result. s in the equilibrium equation related to the deformation mapin the bulk of matter renders div s = div( c s i + s mech ) + ∇ x e · p + q fs e (47)thereby clearly identifying the classical Lorentz-type volume forces ∇ x e · p + q fs e [32, 33] due to thebound and free charge densities together with an additional pressure-like term c s i in the Cauchystress that is due to the additional solution field or rather electronic order parameter(s) y .The equilibrium equations, expressed in terms of flux and source densities per unit area and volume,respectively, in either the material or the material configuration complete the variational setting ofthe energetic case for quasi-static situations.Obviously, which of the equivalent alternative versions is used for solving coupled boundary valueproblems of light-matter interaction in photo-active polymers is largely a matter of taste. For the dissipative case we consider incremental work densities as basis ingredients for the pertinentvariational setting that allows inclusion of a dissipation potential.
For an extension towards the dissipative case, we first introduce and abbreviate the increments ofthe electric, electronic and mechanical solution fields as w := d y, w := d y , w := d y . (48)Then, the incremental work density u m = J u s in the bulk of matter follows as the increment d u m ofthe total potential energy density evaluated at fixed configuration space y, y , y as u m ( w, w , w , ∇ X w, ∇ X w , ∇ X w ) := (cid:2) d u m ( y, y , y , E , F , F ) (cid:3) fixed configuration space . (49)As a result, and incorporating the constitutive relations as introduced in the above, the explicitrepresentation of the incremental work density in the bulk of matter follows as u m ( w, w , w , ∇ X w, ∇ X w , ∇ X w ) = (50)[ D · ∇ X w + q fm w ] + [ P : ∇ X w − [ b m − s (cid:93) m ] · w ] + [ P : ∇ X w − b m · w ] . A step-by-step derivation using ∇ x e · d − d · ∇ x e = curl e × d = o and thus likewise ∇ x e · p − p · ∇ x e = o isdiv s = div s mech − d · ∇ x e − e · ∇ x p + ∇ x e · d + q fs e , = div s mech + curl e × d − e · ∇ x p + q fs e , = div s mech + ∇ x e · p − [ p · ∇ x e + e · ∇ x p ] + q fs e , = div s mech − ∇ x ( p · e ) + ∇ x e · p + q fs e , = div( c s i + s mech ) + ∇ x e · p + q fs e . (cid:98) v m = (cid:98) J (cid:98) v s at the boundary between matterand free space computes at fixed configuration space y, y , y as (cid:98) v m ( w, w , w ) := (cid:2) d (cid:98) v m ( y, y , y ) (cid:3) fixed configuration space = (cid:98) q fm w − t m · w − t m · w . (51)Finally, the electric incremental work density e m = J e s in the bulk of free space expands at fixedconfiguration space y, y as e m ( ∇ X w, ∇ X w ) := (cid:2) d e m ( E , F ) (cid:3) fixed configuration space = D · ∇ X w + P : ∇ X w . (52)The total incremental work density u m in the bulk of matter as well as the external incremental workdensity (cid:98) v m at the boundary between matter and free space together with the electric incrementalwork density e m in the bulk of free space contribute to the incremental work functional as discussedin the sequel. The incremental work functional allows inclusion of a dissipation potential p m and reads U = U ( w, w , w ) := (cid:90) B m [ u m ( w, w , w , ∇ X w, ∇ X w , ∇ X w ) + p m ( w / d t ) d t ] d V + (cid:90) ∂ B m (cid:98) v m ( w, w , w ) d A (53)+ (cid:90) S m e m ( ∇ X w, ∇ X w ) d V. Note that in order to obtain an incremental quantity, the dissipation potential p m (which is a power-like quantity of dimension incremental work density per time) is multiplied by d t , whereby basedon the increment w := d y its argument expresses as v := d y / d t ≡ w / d t . Recall that here, i.e. inquasi-static situations, time is merely a parameter that orders the sequence of external loading.Then, the incremental Dirichlet principle requires stationarity of the incremental work functionalupon admissible, i.e. (space) boundary conditions satisfying material variation D δ of the incrementalsolution fields w, w , w as U ( w, w , w ) → stationary point . (54)Concretely, the stationarity condition for the incremental work functional expands asD δ U = (cid:90) B m [ D δ u m ( w, w , w , ∇ X w, ∇ X w , ∇ X w ) + D δ p m ( w / d t ) d t ] d V + (cid:90) ∂ B m D δ (cid:98) v m ( w, w , w ) d A (55)+ (cid:90) S m D δ e m ( ∇ X w, ∇ X w ) d V . = 0 ∀ D δ w, D δ w , D δ w . Requiring stationarity of the incremental work functional for arbitrary admissible D δ w, D δ w , D δ w and using the explicit expressions for the incremental work densities as introduced in the above14esults eventually in the Euler-Lagrange or rather equilibrium equations Div D − q fm = 0 in B m and [[ D ]] · N = (cid:98) q fm at ∂ B m (56a)Div D = 0 in S m (56b)Div P + b m = s m in B m and P · N = t m at ∂ B m (56c)Div P + b m = in B m and − [[ P ]] · N = t m at ∂ B m (56d)Div P = in S m (56e)Expanding the equilibrium equation related to the electronic order parameter(s) (micro deformation)in the bulk of matter and neglecting external electronic sources, eventually allows term-by-termcomparison with the formulation outlined in [1]Div ∂w • m ∂ F + J ω e = ∂w ◦ m ∂ y + J γ D t y = ⇒ div (cid:18) ∂w • m ∂ F · k (cid:19) + ω e = ∂w ◦ s ∂ y + γ D t y . (57)Thereby, the first terms left and right capture electronic forces associated with structured andamorphous regions, respectively, whereas the second terms on left and right describe the electronic(Lorentz-type) dipole force density in matter and the energy losses, e.g. due to optical scatteringand/or photochemical reactions, respectively.For completeness, Piola transformation of the equilibrium equations then renders the equivalentexpressions in terms of flux and source densities per unit area and volume, respectively, in thespatial configuration div d − q fs = 0 in B s and [[ d ]] · n = (cid:98) q fs at ∂ B s (58a)div d = 0 in S s (58b)div s + b s = s s in B s and s · n = t s at ∂ B s (58c)div s + b s = in B s and − [[ s ]] · n = t s at ∂ B s (58d)div s = in S s (58e) Based on the dependency of the incremental work functional on the incremental solution fields and their materialspace gradients, variational calculus results in the following Euler-Lagrange equations: • Euler-Lagrange equations in the bulk of matterDiv D = q fm , Div P = s m − b m , Div P = − b m . • Euler-Lagrange equations in the bulk of free spaceDiv D = 0 , Div P = . • Euler-Lagrange equations at the boundary between matter and free space[[ D ]] · N = (cid:98) q fm , P · N = t m , [[ P ]] · N = − t m . s s := s (cid:93) s + s (cid:91) s , consisting of energetic and dissipativecontributions, that appears in the equilibrium equation related to the electronic order parameter(s).This concludes derivation of the equilibrium equations for the electric, electronic and mechanicalsub-problems, embracing energetic as well as dissipative cases, from Dirichlet’s principle. For dynamic situations, Hamilton’s principle of least action states that the dynamics of a systembetween two given points in time, captured by the evolution of the state space coordinates, rendersthe action integral, i.e. a functional over the state space, a stationary value upon material variationsof the state space coordinates. In the sequel, we will demonstrate the variational setting and theensuing balance equations for energetic and dissipative cases when modelling light-matter interactionin photo-active polymers.
For the energetic case we first expand on the total Lagrangian energy density before examining thepertinent variational setting.
We introduce the total Lagrangian energy density l m = J l s in the bulk of matter as the differencebetween the total kinetic energy density k m and the total potential energy density u m , thus l m = l m ( y, y , y , E , F , F , v , v ) := k m ( v , v ) − u m ( y, y , y , E , F , F ) . (59)Thereby, the total kinetic energy density k m = J k s in the bulk of matter consists of electronic andmechanical contributions k m = k m ( v , v ) := k tronm ( v ) + k mechm ( v ) . (60)The total Lagrangian energy density l m in the bulk of matter as well as the (negative) total externalpotential energy density (cid:98) v m at the boundary between matter and free space together with the (neg-ative) electric internal potential energy density e m in the bulk of free space contribute to the actionfunctional as discussed in the sequel. To begin with, we define the action functional for Hamilton’s principle as A = A ( y, y , y ) := (cid:90) T (cid:20) (cid:90) B m l m ( y, y , y , E , F , F , v , v ) d V − (cid:90) ∂ B m (cid:98) v m ( y, y , y ) d A (61) − (cid:90) S m e m ( E , F ) d V (cid:21) d t. δ of the solution fields y, y , y as A ( y, y , y ) → stationary point . (62)Concretely, the stationarity condition for the action functional expands asD δ A = (cid:90) T (cid:20) (cid:90) B m D δ l m ( y, y , y , E , F , F , v , v ) d V − (cid:90) ∂ B m D δ (cid:98) v m ( y, y , y ) d A (63) − (cid:90) S m D δ e m ( E , F ) d V (cid:21) d t . = 0 ∀ D δ y, D δ y , D δ y . Requiring stationarity of the action functional for arbitrary admissible D δ y, D δ y , D δ y and using theconstitutive relations as introduced in the above, results eventually in the Euler-Lagrange or rather balance equations Div D − q fm = 0 in B m and [[ D ]] · N = (cid:98) q fm at ∂ B m (64a)Div D = 0 in S m (64b)Div P + b m = s (cid:93) m + D t p m in B m and P · N = t m at ∂ B m (64c)Div P + b m = D t p m in B m and − [[ P ]] · N = t m at ∂ B m (64d)Div P = in S m (64e)For completeness, Piola transformation then renders the equivalent expressions in terms of momen-tum, flux and source densities per unit area and volume, respectively, in the spatial configuration Based on the dependency of the action functional on the solution fields and their material space-time gradients,variational calculus results in the following Euler-Lagrange equations: • Euler-Lagrange equations in the bulk of matter0 = ∂l m ∂y − Div ∂l m ∂ ∇ X y , D t ∂l m ∂ v = ∂l m ∂ y − Div ∂l m ∂ ∇ X y , D t ∂l m ∂ v = ∂l m ∂ y − Div ∂l m ∂ ∇ X y . • Euler-Lagrange equations in the bulk of free space0 = − Div ∂e m ∂ ∇ X y , = − Div ∂e m ∂ ∇ X y . • Euler-Lagrange equations at the boundary between matter and free space − (cid:20) ∂e m ∂ ∇ X y + ∂l m ∂ ∇ X y (cid:21) · N = − ∂ (cid:98) v m ∂y , − ∂l m ∂ ∇ X y · N = − ∂ (cid:98) v m ∂ y , − (cid:20) ∂e m ∂ ∇ X y + ∂l m ∂ ∇ X y (cid:21) · N = − ∂ (cid:98) v m ∂ y d − q fs = 0 in B s and [[ d ]] · n = (cid:98) q fs at ∂ B s (65a)div d = 0 in S s (65b)div s + b s = s (cid:93) s + j D t p m in B s and s · n = t s at ∂ B s (65c)div s + b s = j D t p m in B s and − [[ s ]] · n = t s at ∂ B s (65d)div s = in S s (65e)Note the additional inertia contributions for the electronic and the mechanical sub-problem as com-pared to the quasi-static situation. For the dissipative case we consider the incremental action density as basis ingredient for the pertinentvariational setting that allows inclusion of a dissipation potential.
For an extension towards the dissipative case, we first introduce and abbreviate the velocities (ma-terial time derivatives) of the incremental electronic and mechanical solution fields asD t w := D t d y = d D t y ≡ d v , D t w := D t d y = d D t y ≡ d v . (66)The equivalence between the velocities of the increments and the increments of the velocities (of theelectronic and mechanical solution fields) relies on the commutativity D t d( • ) ≡ d D t ( • ) of materialtime derivatives and increments.Then, the incremental action density l m = J l s in the bulk of matter follows as the increment d l m ofthe total Lagrangian energy density evaluated at fixed state space y, y , y , v , v as l m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , D t w , D t w ) := (cid:2) d l m ( y, y , y , E , F , F , v , v ) (cid:3) fixed state space . (67)As a result, and incorporating the constitutive relations as introduced in the above, the explicitrepresentation of the incremental action density in the bulk of matter follows as l m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , D t w , D t w ) = (68) − [ D · ∇ X w + q fm w ] − [ P : ∇ X w − [ b m − s (cid:93) m ] · w − p m · D t w ] − [ P : ∇ X w − b m · w − p m · D t w ] . The incremental action density l m in the bulk of matter as well as the (negative) external incrementalwork density (cid:98) v m at the boundary between matter and free space together with the (negative) electricincremental work density e m in the bulk of free space contribute to the incremental action functionalas discussed in the sequel. 18 .2.2 Variational Setting The incremental action functional allows inclusion of a dissipation potential p m and reads A = A ( w, w , w ) := (cid:90) T (cid:20) (cid:90) B m [ l m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , D t w , D t w ) − p m ( w / d t ) d t ] d V − (cid:90) ∂ B m (cid:98) v m ( w, w , w ) d A (69) − (cid:90) S m e m ( ∇ X w, ∇ X w ) d V (cid:21) d t. Then, the incremental Hamilton principle requires stationarity of the incremental action functionalupon admissible, i.e. incremental space-time boundary conditions satisfying material variation D δ ofthe incremental solution fields w, w , w as A ( w, w , w ) → stationary point . (70)Concretely, the stationarity condition for the incremental action functional expands asD δ A = (cid:90) T (cid:20) (cid:90) B m [ D δ l m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , D t w , D t w ) − D δ p m ( w / d t ) d t ] d V − (cid:90) ∂ B m D δ (cid:98) v m ( w, w , w ) d A (71) − (cid:90) S m D δ e m ( ∇ X w, ∇ X w ) d V (cid:21) d t . = 0 ∀ D δ w, D δ w , D δ w . Requiring stationarity of the incremental action functional for arbitrary admissible D δ w, D δ w , D δ w and using the explicit expressions for the incremental action densities as introduced in the aboveresults eventually in the Euler-Lagrange or rather balance equations Based on the dependency of the incremental action functional on the incremental solution fields and their materialspace-time gradients, variational calculus results in the following Euler-Lagrange equations: • Euler-Lagrange equations in the bulk of matterDiv D = q fm , Div P + b m = s m + D t p m , Div P + b m = D t p m . • Euler-Lagrange equations in the bulk of free spaceDiv D = 0 , Div P = . • Euler-Lagrange equations at the boundary between matter and free space[[ D ]] · N = (cid:98) q fm , P · N = t m , [[ P ]] · N = − t m . D − q fm = 0 in B m and [[ D ]] · N = (cid:98) q fm at ∂ B m (72a)Div D = 0 in S m (72b)Div P + b m = s m + D t p m in B m and P · N = t m at ∂ B m (72c)Div P + b m = D t p m in B m and − [[ P ]] · N = t m at ∂ B m (72d)Div P = in S m (72e)Expanding the balance equation related to the electronic order parameter(s) (micro deformation)in the bulk of matter and neglecting external electronic sources, eventually allows term-by-termcomparison with the formulation outlined in [1]Div ∂w • m ∂ F + J ω e = ∂w ◦ m ∂ y + J γ D t y + (cid:37) m D tt y = ⇒ div (cid:18) ∂w • m ∂ F · k (cid:19) + ω e = ∂w ◦ s ∂ y + γ D t y + (cid:37) s D tt y . (73)The first and second terms left and right are as in the quasi-static situation, for the dynamic situ-ation the third term on the right captures in addition the inertia of the electronic order parameter(s).For completeness, Piola transformation of the balance equations then renders the equivalent expres-sions in terms of momentum, flux and source densities per unit area and volume, respectively, in thespatial configurationdiv d − q fs = 0 in B s and [[ d ]] · n = (cid:98) q fs at ∂ B s (74a)div d = 0 in S s (74b)div s + b s = s s + j D t p m in B s and s · n = t s at ∂ B s (74c)div s + b s = j D t p m in B s and − [[ s ]] · n = t s at ∂ B s (74d)div s = in S s (74e)Note the total electronic internal source density s s := s (cid:93) s + s (cid:91) s appearing in the balance equation relatedto the electronic order parameter(s).This concludes derivation of the balance equations for the electric, electronic and mechanical sub-problems, embracing energetic as well as dissipative cases, from Hamilton’s principle. Hamilton’s equations are alternative to Hamilton’s principle of least action when describing thedynamics of a system, however in terms of the evolution of the phases space coordinates rather thanthe state space coordinates. Thereby, the Hamiltonian follows from a Legendre transformation ofthe Lagrangian in order to exchange the velocities in state space by their corresponding momenta inphase space. The Hamiltonian setting proves beneficial for dynamical systems with symmetries, i.e.when certain momenta are conserved. In the sequel, we will derive Hamilton’s equations and thusthe ensuing balance equations for energetic and, as a novelty per se, dissipative cases when modellinglight-matter interaction in photo-active polymers.20 .1 Energetic Case
For the energetic case we first expand on the total Hamiltonian energy density before examining thepertinent variational setting.
Formally, the total Hamiltonian energy density h λ m = J h λ s in the bulk of matter (for the notation seebelow) follows from Legendre transformation of the total Lagrangian energy density l m exchangingthe velocities v, v , v for the corresponding momenta p m , p m , p m , i.e. by seeking for the supremumsup v, v , v { p m v + p m · v + p m · v − l m ( y, y , y , E , F , F , v , v ) } . (75)The total Lagrangian energy density l m does however not depend on the velocity v of the electricpotential y , thus according to the Dirac theory [27] l m qualifies as degenerate . Consequently, themomentum p m conjugate to v satisfies the constraint p m . = 0 (76)that requires enforcement via an additional Lagrange multiplier λ . It should be noted that, whilethe electric potential may vary over time, there is no resulting impuls as no corrolated mass exists.Thus, the explicit representation of the total Hamiltonian energy density h λ m in the bulk of matter(whereby the notation h λ m shall indicate inclusion of the Lagrange multiplier λ ) reads h λ m ( y, y , y , E , F , F , p m , p m , p m , λ ) = t m ( p m , p m ) + u m ( y, y , y , E , F , F ) + λ p m . (77)Here, the total dual kinetic energy density t m = J t s in the bulk of matter that is parameterized inthe momenta p m and p m rather than in the velocities v and v , respectively, follows likewise fromLegendre transformation and reads explicitly t m = t m ( p m , p m ) := sup v , v { p m · v + p m · v − k m ( v , v ) } = 12 1 (cid:37) m p m · p m + 12 1 ρ m p m · p m . (78)Finally, the total Hamiltonian energy density e λ m = J e λ s in the bulk of free space follows from Legendretransformation of the electric energy density e m exchanging the velocities v, v (see below) for thecorresponding momenta p m , p m , i.e. by seeking for the supremumsup v, v { p m v + p m · v + e m ( E , F ) } . (79)The electric energy density e m does, however, not depend on the velocities v and v of the electric po-tential y and the deformation map y , respectively. Thus, in line with the Dirac theory for degenerateLagrangians, the momenta p m and p m conjugate to v and v , respectively, satisfy the constraints p m . = 0 and p m . = (80)that require enforcement via additional Lagrange multipliers λ and λ , respectively.21hus, the explicit representation of the total Hamiltonian energy density e λ m in the bulk of free space(whereby the notation e λ m shall indicate inclusion of the Lagrange multipliers λ and λ ) reads e λ m ( E , F , p m , p m , λ, λ ) = e m ( E , F ) + λ p m + λ · p m . (81)The total Hamiltonian energy densities h λ m and e λ m in the bulk of matter and free space, respectively,as well as the total external potential energy density (cid:98) v m at the boundary between matter and freespace contribute to the Hamiltonian energy functional as discussed in the sequel. To begin with, we define the Hamiltonian energy functional eventually rendering Hamilton’s equa-tions as H = H ( y, y , y , p m , p m , p m , λ, λ ) := (cid:90) B m h λ m ( y, y , y , E , F , F , p m , p m , p m , λ ) d V + (cid:90) ∂ B m (cid:98) v m ( y, y , y ) d A (82)+ (cid:90) S m e λ m ( E , F , p m , p m , λ, λ ) d V. Then, with admissible material variations D δ of the phase space y, y , y , p m , p m , p m , Hamilton’s equa-tions result from requiringD { tδ } (cid:20)(cid:90) D m [ y p m + y · p m ] d V + (cid:90) B m y · p m d V (cid:21) . = D δ H ∀ D δ y, D δ y , D δ y , D δ p m , D δ p m , D δ p m , (83)whereby D m := B m ∪ S m denotes the entire solution domain and D { tδ } ( • , ◦ ) defines the Poisson-bracket-type combination D t ( • ) D δ ( ◦ ) − D δ ( • ) D t ( ◦ ) of material time derivatives and variations . Interestingly, in terms of the symplectic matrix, D { tδ } ( • , ◦ ) := D t ( • ) D δ ( ◦ ) − D δ ( • ) D t ( ◦ ) expresses asD { tδ } ( • , ◦ ) := [ D t ( • ) , D t ( ◦ )] (cid:20) − (cid:21) (cid:20) D δ ( • )D δ ( ◦ ) (cid:21) = − [ D δ ( • ) , D δ ( ◦ )] (cid:20) − (cid:21) (cid:20) D t ( • )D t ( ◦ ) (cid:21) . Furthermore, the variation D δ H of a generic Hamiltonian function H = H ( • , ◦ ) reads asD δ H = (cid:20) ∂H∂ ( • ) , ∂H∂ ( ◦ ) (cid:21) (cid:34) D δ ( • )D δ ( ◦ ) (cid:35) = [ D δ ( • ) , D δ ( ◦ )] ∂H∂ ( • ) ∂H∂ ( ◦ ) . Thus, finally, due to the skew-symmetry of the symplectic matrix and the arbitrariness of the admissible variations,Hamilton’s equations eventually result as (cid:20) D t ( • )D t ( ◦ ) (cid:21) = (cid:20) − (cid:21) ∂H∂ ( • ) ∂H∂ ( ◦ ) with (cid:20) − (cid:21) (cid:20) − (cid:21) = − (cid:20) (cid:21) . In case of a Hamiltonian functional, variational derivatives substitute the partial derivatives of H . δ H = (cid:90) B m D δ h λ m ( y, y , y , E , F , F , p m , p m , p m , λ ) d V + (cid:90) ∂ B m D δ (cid:98) v m ( y, y , y ) d A (84)+ (cid:90) S m D δ e λ m ( E , F , p m , p m , λ, λ ) d V. Evaluating the above (Hamiltonian) requirement for arbitrary admissible D δ y, D δ y , D δ y , D δ p m , D δ p m , D δ p m and using the constitutive relations as introduced in the above, Hamilton’s equationsresult in the following balance equations λ = D t y in B m (85a)Div D − q fm = 0 in B m and [[ D ]] · N = (cid:98) q fm at ∂ B m (85b) λ = D t y in S m (85c)Div D = 0 in S m (85d) p m /(cid:37) m = D t y in B m (85e)Div P + b m = s (cid:93) m + D t p m in B m and P · N = t at ∂ B m (85f) p m /ρ m = D t y in B m (85g)Div P + b m = D t p m in B m and − [[ P ]] · N = t at ∂ B m (85h) λ = D t y in S m (85i)Div P = in S m (85j) Based on the dependency of the Hamiltonian energy functional on the phase space coordinates and their materialspace gradients, Hamilton’s equations, when using the abbreviations p := p m , p := p m and p := p m , read as: • Hamilton equations in the bulk of matter (cid:20) D t y D t p (cid:21) = (cid:20) O I − I O (cid:21) δh λ m δy∂h λ m ∂p , (cid:20) D t y D t p (cid:21) = (cid:20) O I − I O (cid:21) δh λ m δ y ∂h λ m ∂ p , (cid:20) D t y D t p (cid:21) = (cid:20) O I − I O (cid:21) δh λ m δ y ∂h λ m ∂ p . • Hamilton equations in the bulk of free space (cid:20) D t y D t p (cid:21) = (cid:20) O I − I O (cid:21) δe λ m δy∂e λ m ∂p , (cid:20) D t y D t p (cid:21) = (cid:20) O I − I O (cid:21) δe λ m δ y ∂e λ m ∂ p . • Hamilton equations at the boundary between matter and free space − (cid:20) ∂e λ m ∂ ∇ X y − ∂h λ m ∂ ∇ X y (cid:21) · N = − ∂ (cid:98) v m ∂y , ∂h λ m ∂ ∇ X y · N = − ∂ (cid:98) v m ∂ y , − (cid:20) ∂e λ m ∂ ∇ X y − ∂h λ m ∂ ∇ X y (cid:21) · N = − ∂ (cid:98) v m ∂ y jλ = j D t y in B s (86a)div d − q fs = 0 in B s and [[ d ]] · n = (cid:98) q fs at ∂ B s (86b) jλ = j D t y in S s (86c)div d = 0 in S s (86d) j p s /(cid:37) s = j D t y in B s (86e)div s + b s = s (cid:93) s + j D t p m in B s and s · n = t s at ∂ B s (86f) j p s /ρ s = j D t y in B s (86g)div s + b s = j D t p m in B s and − [[ s ]] · n = t s at ∂ B s (86h) j λ = j D t y in S s (86i)div s = in S s (86j)Note the extra equations relating either the momenta or the Lagrange multipliers (that enforce theconstraints p m = 0 in D m and p m = in S m ) to their conjugate velocities. For the dissipative case we consider the incremental total energy density as basis ingredient for thepertinent variational setting that allows inclusion of a dissipation potential.
For an extension towards the dissipative case, we first introduce and abbreviate the increments ofthe electric, electronic and mechanical momenta as q := d p m , q := d p m , q := d p m . (87)Next, the incremental ’inertial work density’ , instrumental for Legendre transforming the Lagrangianinto the Hamiltonian, follows as the increment of twice the total kinetic energy densityd[ p m v + p m · v + p m · v ] = p m D t w + p m · D t w + p m · D t w + v q + v · q + v · q . (88)Then, the incremental total energy density h λ m = J h λ s in the bulk of matter follows formally fromLegendre transformation of the incremental action density l m exchanging the incremental velocitiesD t w, D t w , D t w for the corresponding incremental momenta q, q , q , thus resulting in h λ m = h m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , q, q , q , λ ) := (89)d[ p m v + p m · v + p m · v ] − l m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , D t w , D t w ) . Alternatively with the constitutive relation p m /(cid:37) m =: v and p m /ρ m =: v , the incremental total energy densityfollows from h λ m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , q, q , q , λ ) := (cid:2) d h λ m ( y, y , y , E , F , F , p m , p m , p m , λ ) (cid:3) fixed phase space . Here, the constraint p m . = 0 eliminates the incremental Lagrange multiplier d λ . p m . = 0 and the explicit result for theLagrange multiplier v = λ as well as the explicit representation of the incremental action density l m in the above, the explicit representation of the incremental total energy density in the bulk of matterfollows as h λ m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , q, q , q , λ ) = (90)[ D · ∇ X w + q fm w + λ q ] + [ P : ∇ X w − [ b m − s (cid:93) m ] · w + v · q ] + [ P : ∇ X w − b m · w + v · q ] . In an entirely analogous fashion, the incremental total energy density e λ m = J e λ s in the bulk of freespace computes as e λ m ( ∇ X w, ∇ X w , q, q , λ, λ ) := (91) (cid:2) d e λ m ( E , F , p m , p m ) , λ, λ (cid:3) fixed phase space = [ D · ∇ X w + λ q ] + [ P : ∇ X w + λ · q ] . The incremental total energy densities h λ m and e λ m in the bulk of matter and free space, respectively,as well as the external incremental work density (cid:98) v m at the boundary between matter and free spacecontribute to the incremental total energy functional as discussed in the sequel. The incremental total energy functional allows inclusion of the dissipation potential p m and reads H = H ( w, w , w , q, q , q , λ, λ ) := (cid:90) B m [ h λ m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , q, q , q , λ ) + p m ( w / d t ) d t ] d V + (cid:90) ∂ B m (cid:98) v m ( w, w , w ) d A (92)+ (cid:90) S m e λ m ( ∇ X w, ∇ X w , q, q , λ, λ ) d V. Then, with admissible material variations D δ of the incremental phase space w, w , w , q, q , q , Hamil-ton’s equations result from requiring incrementally D { tδ } (cid:20)(cid:90) D m d[ y p m + y · p m ] d V + (cid:90) B m d[ y · p m ] d V (cid:21) . = D δ H ∀ D δ w, D δ w , D δ w , D δ q, D δ q , D δ q . (93)Concretely, the material variation of the incremental total energy functional expands asD δ H = (cid:90) B m [ D δ h λ m ( w, w , w , ∇ X w, ∇ X w , ∇ X w , q, q , q , λ ) + D δ p m ( w / d t ) d t ] d V + (cid:90) ∂ B m D δ (cid:98) v m ( w, w , w ) d A (94)+ (cid:90) S m D δ e λ m ( ∇ X w, ∇ X w , q, q , λ, λ ) d V. The increments of the integrands on the left-hand side expand asd[ y p m + y · p m ] = w p m + w · p m + y q + y · q and d[ y · p m ] = w · p m + y · q . Thus, application of D { tδ } for material time derivatives of the incremental phase space w, w , w , q, q , q and the phasespace y, y , y , p m , p m , p m as well as for material variations D δ w, D δ w , D δ w , D δ q, D δ q , D δ q of only the incrementalphase space (i.e. with vanishing material variations D δ y, D δ y , D δ y , D δ p m , D δ p m , D δ p m ), renders explicitlyD { tδ } d[ y p m + y · p m ] = D t y D δ q + D t y · D δ q − D δ w D t p m − D δ w · D t p m and D { tδ } d[ y · p m ] = D t y · D δ q − D δ w · D t p m . δ w, D δ w , D δ w ,D δ q, D δ q , D δ q and using the expressions for the incremental total energy densities as introduced inthe above, Hamilton’s equations result eventually in the following balance equations λ = D t y in B m (95a)Div D − q fm = 0 in B m and [[ D ]] · N = (cid:98) q fm at ∂ B m (95b) λ = D t y in S m (95c)Div D = 0 in S m (95d) v = D t y in B m (95e)Div P + b m = s m + D t p m in B m and P · N = t at ∂ B m (95f) v = D t y in B m (95g)Div P + b m = D t p m in B m and − [[ P ]] · N = t at ∂ B m (95h) λ = D t y in S m (95i)Div P = in S m (95j)Note again the total electronic internal source density s s := s (cid:93) s + s (cid:91) s appearing in the balance equationrelated to the electronic order parameter(s).For completeness, Piola transformation then renders the equivalent expressions in terms of momen-tum, flux and source densities per unit area and volume, respectively, in the spatial configuration Based on the dependency of the incremental total energy functional on the incremental phase space coordinatesand their material space gradients, Hamilton’s equations read as: • Hamilton equations in the bulk of matter (cid:34) D t y D t p (cid:35) = (cid:20) O I − I O (cid:21) δ h λ m δw∂ h λ m ∂q , (cid:34) D t y D t p (cid:35) = (cid:20) O I − I O (cid:21) δ h λ m δ w ∂ h λ m ∂ q , (cid:34) D t y D t p (cid:35) = (cid:20) O I − I O (cid:21) δ h λ m δ w ∂ h λ m ∂ q . • Hamilton equations in the bulk of free space (cid:34) D t y D t p (cid:35) = (cid:20) O I − I O (cid:21) δ e λ m δw∂ e λ m ∂q , (cid:34) D t y D t p (cid:35) = (cid:20) O I − I O (cid:21) δ e λ m δ w ∂ e λ m ∂ q . • Hamilton equations at the boundary between matter and free space − (cid:20) ∂ e λ m ∂ ∇ X w − ∂ h λ m ∂ ∇ X w (cid:21) · N = − ∂ (cid:98) v m ∂w , ∂ h λ m ∂ ∇ X w · N = − ∂ (cid:98) v m ∂ w , − (cid:20) ∂ e λ m ∂ ∇ X w − ∂ h λ m ∂ ∇ X w (cid:21) · N = − ∂ (cid:98) v m ∂ w λ = j D t y in B s (96a)div d − q fs = 0 in B s and [[ d ]] · n = (cid:98) q fs at ∂ B s (96b) jλ = j D t y in S s (96c)div d = 0 in S s (96d) j p s /(cid:37) s = j D t y in B s (96e)div s + b s = s s + j D t p m in B s and s · n = t s at ∂ B s (96f) j p s /ρ s = j D t y in B s (96g)div s + b s = j D t p m in B s and − [[ s ]] · n = t s at ∂ B s (96h) j λ = j D t y in S s (96i)div s = in S s (96j)This concludes derivation of the balance equations for the electric, electronic and mechanical sub-problems, embracing energetic as well as dissipative cases, from Hamilton’s equations. Continuum modeling of light-matter interaction in photo-active polymers is instrumental when de-signing and optimising devices with the technologically attractive capacity for remote and contact-freeactuation by light. Current research in organic chemistry focusses on synthesis and characterizationof a variety of polymer compounds involving different molecular photo-switches. Thus, there is greatpromise for future development of photo-mechanically coupled material compositions. These alsoinclude options for mechanically soft photo-active polymers, thus asking for a continuum frame-work providing geometrically exact description of the deformation and related kinematic quantities.Continuum formulations are key pre-requisite for computational simulations of devices based on,e.g., variational (Galerkin-type) approaches such as the finite element method. In this contributionwe provide the necessary preliminaries, such as continuum formulations of the solution fields, energydensities and constitutive relations, entering the electric, electronic and mechanical sub-problems, fora comprehensive account on the variational setting of photo-mechanics. Based thereon, we demon-strated how to variationally cast the pertinent equilibrium and balance equations for energetic aswell as, noteworthy, dissipative cases. In combination with Hamilton’s equations, especially the lattercase is a novelty per se. In conclusion, we established a geometrically exact variational continuumframework of light-matter interaction in photo-active polymers that allows for analytical and com-putational investigations of photo-mechanical devices. Forthcoming contributions will focus on thecorresponding continuum thermodynamics and the computational setting, among further extensions.
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