Mathematical Modelingof Additive Manufacturing Technologies

Alexander V. Manzhirov,Member, IAENG,and Sergei A. Lychev

Abstract—Mathematical modeling of additive manufacturingtechnologies is aimed at improving the performance of device,machine, and mechanism parts. These technologies includestereolithography, electrolytic deposition, thermal and laser-based 3D printing, 3D-IC fabrication technologies, etc. They arebooming nowadays, because they can provide rapid low-costhigh-accuracy production of 3D items of arbitrarily complexshape (in theory, from any material). However, deformationand strength problems for products manufactured with thesetechnologies yet remain to be solved. The fundamentally newmathematical models considered in the paper describe theevolution of the end product stress-strain state in additive manu-facturing and are of general interest for modern technologies inengineering, medicine, electronics industry, aerospace industry,and other fields.

Index Terms—Additive manufacturing technology, mathe-matical modeling, mechanics of growing solids, stress-strainstate, deformation, strength.

I. I NTRODUCTION

M ATHEMATICAL modeling of a variety of naturalphenomena and technological processes requires tak-

ing into account thematerial evolutionand remodelingof asolid, which can be associated with creation and annihilationof material points or with internal constraint redistributionin the bulk of the solid. If such changes are accompaniedwith deformation of the entire solid, then what we deal withis a growing solid, whose properties are highly unusual.Models of winding and welding, vapor deposition, photo-polymerization, and ion implantation processes can serveas examples [1], [2]. The solid material composition insuch processes is changed either by adding macroscopicvolumes, whose locally thermostatic states can be describedby statistical parameters such as temperature, distortion, andtension, or by implanting individual atoms or molecules(referred to asextra substancein [3]), which from themacroscopic viewpoint leads to distributed defect evolutionin the boundary layer. Winding and welding are examples ofthe former, and ion implantation is an example of the latter.Sometimes both mechanisms should be taken into account, asis the case with the vapor deposition process, which involvesthe adherence of atomic clusters consisting of large numbersof coupled atoms as well as the adherence of individualatoms of ions bombarding the growth surface.

Manuscript received March 14, 2014; revised March 31, 2014. Supportedin part by the Russian Science Foundation under grant 14-19-01280, bythe Russian Foundation for Basic Research under grants 13-01-92693and 14-01-00741, and by the Branch of Power Engineering, MechanicalEngineering, Mechanics, and Control Processes of the Russian Academy ofSciences under Program No. 12.

A. V. Manzhirov is with the Institute for Problems in Mechanics andBauman Moscow State Technical University, Moscow, Russia (phone:+79262332511; fax: +74997399531; e-mail: manzh@inbox.ru).

S. A. Lychev is with the Institute for Problems in Mechanics andBauman Moscow State Technical University, Moscow, Russia (e-mail:lychevsa@mail.ru).

We point out that growth is often closely associatedwith defects formation processes. In particular, vapor phasedeposition causes continuous defect formation in growingstructures, which can readily be shown by estimating thecrystal growth rate. Indeed, the atoms condensed from vaporto a crystal surface with regular atomic structure are onlyweakly coupled with the surface and evaporate back withhigh probability. But if there is an unfinished atomic planeon the growth surface, then the atoms that hit the plane edgebecome strongly coupled. This forces the unfinished atomicplane to be completed rapidly, and the crystal growth stopsuntil there is formed a sufficiently large nucleus of a newatomic plane. One can estimate the probability of such anucleus to appear as well as the resulting crystal growthrate, which proves to be many orders of magnitude smallerthan observed in experiments. This apparent paradox canbe explained by assuming that there are a large number ofdefects continuously formed on the crystal surface, whichplay the role of nuclei for independently growing islands ofatomic planes [4]. With this growth mechanism, the forceinteractions arising between these islands result in a residualstress field.

Kroner showed in his pioneering paper [3] that the residualstresses insimple materials[5] can be represented in termsof the incompatibility of the local distortion field definedin the reference description by methods of non-Euclideangeometry. Thus, the geometric language of the theory ofsmooth manifolds can be used to describe not only solidswith distributed defects but also growing solids.

Stress-strain state analysis for growing solids has beencarried out in numerous papers [6]–[11], where a number oftrends in generalizing classical continuum mechanics havebeen used. One of these is developed in the framework ofthe theory ofinhomogeneity(structural heterogeneity) arisingfrom a special connection of parts of a body rather thanfrom distinctions in the physical properties of the materials ofthese parts [12], [13]. This kind of structural inhomogeneityalso arises in bodies made of a single material, which arehomogeneous in the classical sense. To distinguish betweenthese two kinds of inhomogeneity, we use the termmaterialuniformity [5] for the latter one.

The growth of a solid is usually viewed as a processwhere additional material is joined to the solid, which isdeformed in the process. It is assumed that the additionalmaterial may have the form of material surfaces, threads, ordrops and be deposited to the main body in some stressedstate. Moreover, the growing body, together with additionalmaterial, can be viewed as a single body represented bymultiple solid components, and the growth process can betreated as the generation of constraints providing that thenumber of connected components of the solid decreases.This leads to a change in the topology of the body. In

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particular, boundary points become interior points. Here theprocess of gluing a bundle of paper sheets can be usedas an example. Prior to gluing, we have a set with many(possibly, very many) connected components, but after gluingwe have a connected set. If each sheet were subjected tosome deformation from the standard (uniform) state in thegluing process, no smooth deformation of the final body aftergluing would be able to bring all the sheets to the standardstate simultaneously. The response of a local part of the bodyto external loadings for the case in which it is defined by theelasticity tensor would vary from point to point of the body inany configuration defining an immersion in Euclidean space.Thus, the body is inhomogeneous even if it is made froma single material (i.e., is materially uniform). This exampleillustrates the existence of a special type of inhomogeneity,which is studied in [12], [14], [15].

Thus, inhomogeneity results from the growth process andis related to varying physical and mechanical propertiesof the material. One can say that it arises in special as-sembling scenarios. To describe the response of the solidto external inputs, one can either treat it as a nonuniformsolid, which results in a complicated description of theconstitutive equations, or somehow reconstruct the naturalglobal configuration of the solid and use it as a referenceconfiguration. For simple materials, the latter can be done ifone allows embedding the reference shapes in a space witha more flexible definition of geometric properties (e.g., in anaffine connection space) and defining a global natural shapewith additional geometric parameters such as the torsion,curvature, and nonmetricity of the connection.

We point out that the inhomogeneity in solids can bedescribed without using the ideas of non-Euclidean geometry.Clearly, if a configuration is an embedding of a body in aspace that is necessarily Euclidean, then an inhomogeneoussolid does not have a global natural configuration; i.e., anyconfiguration is not free of residual stresses. At the sametime, we have to use a stressed configuration as a reference,which complicates the statement of constitutive relations.In particular, they have one more tensor argument knownas implant [9], [10], [14] which characterizes the initiallocal deformation. However, the geometric meaning of theimplant becomes clear if we treat it as the initial (“assembly”)local deformation of the element in the natural state, whichdirectly leads to the notion of local transformation of thenatural frame used in the geometry of a space with absoluteparallelism and thus introduces the concept of non-Euclideangeometry. Therefore, we prefer to use the geometric languagefrom the very beginning.

II. B ODY AS A SMOOTH MANIFOLD

In what follows, we use the concept of a body as anabstract smooth manifold, that is, an open subset of sometopological space equipped with a specialmaterial connec-tion. This concept allows one to describe the inhomogene-ity phenomenon in materially uniform bodies in a ratherelegant geometric way. The foundations of the theory ofinhomogeneity have been laid down in the milestone work byNoll [12] and developed by Wang, Epstein, and Maugin [13]–[15]. Since inhomogeneity results from an accretion process,we can hope that this geometric approach will be effectivefor the problems considered.

We treat a bodyB as a smooth manifold without boundary.This means thatB is a set equipped with a topologysatisfying the separation axiom and can be covered by finitelymany overlapping open setsUk ⊂ B homeomorphic toopen subsets inRn. The homeomorphisms are establishedby coordinate mappings (charts)χk : Uk → R

n such thatfor every intersectionUk∩Up the corresponding compositionχk χ−1

p : Rn → R

n is continuous and has sufficientlymany derivatives. Note thatn can be1, 2, or 3 dependingon whether the body is a fiber, a membrane, or a solid,respectively. We refer ton as thedimensionof the body.The material points are elements of the setB and can beidentified by their coordinates provided by the chartsχk.The collection of chartsχk

lk=1 defines an atlas (of order

l) of the manifold. If a manifold can be covered by an atlasof the first order, then this manifold istrivial . The needfor nontrivial atlases is clear for one- and two- dimensionalsolids. (A sphere is a simple example.) At first glance, itseems that the three-dimensional case is different, and onlytrivial atlases are needed. Indeed, a three-dimensional solidembedded in Euclidean space can be modeled by a trivialmanifold covered by a single chart whose values are just theCartesian coordinatesof the points that constitute the body.But this impression is wrong! In fact, the structure of theatlas should be consistent with thematerial connection(seebelow), and this consistency may require nontrivial atlases;Wang [13] showed this by examples (one of which is thefamous “Mobius crystal”) Of course, all such bodies havenontrivial inhomogeneity structure. We point out that thesebodies can be created by appropriate growing processes that“sew” these three-dimensional bodies from two-dimensionalsurfaces. Thus, the notion of atlas plays a significant role inthe theory of growing bodies.

Connectionin general is a rule that determines the trans-formation of a vector as it moves along a path (curve)on B that carries the vector from one fiber to another. Alinear (affine) connection determines the linear transforma-tion under infinitesimal transport, i.e., a mapping∇ :TX(B) → L(TX(B), TX(B)). In a local chart, one has∇∂α

∂β = Γγαβ∂γ , where theΓγ

αβ are the Christoffel symbolsof the connection. A linear connection∇ is said to becompatiblewith a metric g on the manifold if the innerproduct of two arbitrary vectors remains the same after theparallel transport of these vectors along an arbitrary curve.It can be shown that∇ is compatible withg if and only if∀u∇ug = 0. Consider an n-dimensional manifoldB with ametricg and a connection∇. The triple(B,∇, g) is calleda Riemann–Cartan manifold.

The torsion of a connection is the mapT : TX(B) ×TX(B)→ TX(B) defined by

T(u,v) = ∇uv −∇vu− [u,v].

In the components in a local chart, one hasTαβγ = Γα

βγ−Γαγβ.

A connection is said to besymmetricif it is torsion free, thatis, if ∇XY −∇Y X = [X,Y ].1 TheRiemannian curvatureis the mapR : TX(B) × TX(B) → L(TX(B), TX(B))

1One can show that on every Riemannian manifold(B, g) there existsa unique torsion-free linear connection∇ compatible withG, namely, theLevi-Civita connection.

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Fig. 1. Configurations and Deformations

defined by

R(u,v) = ∇u∇v −∇v∇u −∇[u,v].

A metric-affine manifold is a manifold equipped with both aconnection and a metric,(B,∇, g). If the connection is met-rically compatible, then the manifold is called aRiemann–Cartan manifold. If the connection is torsion free but hasnontrivial curvature, thenB is called aRiemannian manifold.If the curvature of the connection vanishes but the torsionis nontrivial, thenB is called a Weitzenbock manifold. Ifboth the torsion and the curvature vanish, thenB is a flat(Euclidean) manifold.

Let us explain the concept ofmaterial connectionin a fewwords. It implements the idea of a local uniform referenceconfiguration that carries an infinitesimal neighborhood ofa material point to some uniform (typically natural) strainstate. In the simplest but frequently studied cases, one cancarry the whole body to a uniform state by some globalconfiguration. In this case, the connection turns out to beEuclidean, and the theory becomes trivial. In general, theredoes not exist a smooth mapping transforming the infinites-imal neighborhoods of all material points to a uniform statesimultaneously. That is why one equips a materially uniformbody, i.e., a body all of whose material points are of the samekind, with some artificial (or structural) inhomogeneity.From the mechanical viewpoint, such bodies do not haveshapes free from residual stresses. The only way to returnthe neighborhood of each material point to a natural stateand hence relax the residual stresses is to cut the body intoinfinitely many parts and allow them to deform independently(Fig.1). This fictitious process in some sense is reciprocal tothe growing process. One can find a detailed statement ofthe theory in [8], [11].

III. G ROWING BODY

Now let us give a precise definition of growing body.We consider growing bodies that can be represented ascontinuous families of nested bodies. Recall that, accordingto the definitions given above, the manifolds that representbodies have no boundaries. Certainly, physical bodies haveboundaries. The boundary points are not included in theopen setB, but their union is the set∂B = B \B, whichrepresents the boundary of the bodyB. We can assume that∂B is a smooth manifold whose dimension is the dimensionof B minus one. Finally, we believe that the inclusion ofmaterial can be represented as a continuous adjunction of

material surfaces(in the sense of [16]) to the boundary∂B.These considerations can be summarized as follows:

A layerwisegrowing body is a continuous monotone (withrespect to inclusion) one-parameter family of manifolds

C = Bαα∈I , ∀α < β ∈ I Bα ⊂ Bβ , (1)

where I = (a, b) ⊂ R is an open interval, such that thefollowing property holds:

∀α ∈ I ∀X ∈ Bα \Ba ∃γ ∈ I X ∈ ∂Bγ . (2)

We refer roBa as to initial body and toBb as to final body.Since the family (1) represents some evolution process, werefer to α as theevolution parameter. Property (2) statesthat any interior point that belongs to the adjoined part ofthe body has been an element of the boundary manifold atsome stage of growth, and so the topological dimension ofits neighborhood has changed. This is specific for growingbodies.

We single out a particular class of growth, namely,com-plete surface growth, by the following condition:

∀α < β ∈ I ∂Bβ 6∈ Bα,

or, equivalently,∀α < β ∈ I ∂Bβ∩∂Bα = ∅. This propertyensures that for eachBα ∈ C there exists a continuousprojection ofBα \Ba onto an intervalI ′ ⊂ I exists, so thatone can interpret the manifoldBα \Ba as a trivial bundleover I ′ whose fibers are the manifolds∂Bγ , γ ∈ I ′.

The idea of continuity can be formalized by means ofsome measure (such as volume or mass) on the manifoldsBα. If a certain measuremes(. . .) has been introduced, thenthecontinuityof the family (1) is equivalent to the followingproperty:

∀ε ∃δ ∀α < β ∈ I β − α < δ ⇒ mes(Bβ \Bα) < ε.

It is also possible to interpret the layerwise character ofgrowth in terms of measure as follows:

limβ→α

mes(Bβ \Bα)

β − α= k,

0 < k <∞, ∃Ω ⊂ Bβ \Bα dimΩ = dimBα − 1.

The latter condition states that the infinitesimal increment,which is the set difference between two nearby instances ofthe familyC representing the growing body, is asymptoticallyequivalent to a body of dimension less by one. For example,if the Bα are three-dimensional manifolds, thenΩ is two-dimensional, and its mechanical response can be describedby relations suitable for membranes, shells,etc.

Clearly, the definition given above does not cover allpossible ways of surface growth that can be implemented asa continuous process of joining surfaces, fibers, or drops. Anappropriate classification can be obtained on the geometricbasis. In this framework, we treat a growing body as a bundleof smooth manifolds; the topological structure of the bundle,in particular, the dimensions of the base and a typical fiber,depends on the accretion process. See [8], [11] for details.

It is conventional in rational mechanics [5, p. 35] thatbodies are presented in the physical spaceE as shapesBα ⊂E . On the one hand, shapesBα are connected subsets of thephysical space, and on the other hand, every shape is theimage of a configurationκα : Bα → Bα and belongs to

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the class of admissible configurations that equip the bodyBα with the structure of a smooth manifold. We associatetwo shapes with each element of the family (1), the referenceshapeBR

α = κRαBα and the actual shapeBα = καBα. Thus,

the family (1) induces the corresponding families of referenceand actual configurations as well as families of reference andactual shapes. From now on, we use exactly these definitionsof configurations and shapes. Note that the reference shapeis not stress free in general.

Note that the conventional notation for the position of amaterial pointXα in the reference configuration isTE ∋Xα = κ

Rα (X), and xα is used for the position in the

actual configuration; i.e.,TE ∋ xα = κα(X). The com-position ϕα = κα (κ

Rα )

−1 of these configurations, i.e.,ϕα : Xα 7→ xα, is the deformation. The derivative inthe mappingϕ, which exists owing to the differentiabilityof the configurations and their inverses, is known as thedeformation gradient(which is actually not a gradient at all)F = ∂xm/∂Xnem⊗en ≡ ∂xm/∂Xn∂m⊗dxn. We alwaysassume thatJ = detF > 0.

In general, the configuration of a growing body is a smoothmapping of the material manifold equipped with a materialconnection onto the physical manifold, whose connection issubstantially different.

We assume that the bodiesBα are materially uniform,simple and elastic [5], so their response can be defined bythe response functional,

T α = S(Hα). (3)

HereS(. . .) is the response functional, which is nonlinear inthe general case. We assume thatS(. . .) does not explicitlydepend on the evolution parameterα. The tensorT α is somekind of stress tensor field. (To be definite, we use the termCauchy stresses). Assume thatS satisfy the principle offrame indifference and has been calibrated,

S(0) = 0, limdetH→0

|S(H)| =∞.

The tensorHα is a smooth tensor field representing the localdistortion. It can be written in the form of multiplicativedecomposition

Hα = Fα Kα, (4)

whereF α is the conventional deformation gradient, i.e., thelinearization of the mappingγα : BR

α → Bα, which can berepresented by the relative gradient∇

κRα

as follows [5]:

F α = ∇κ

Rαγα. (5)

It is important to note that the tensor fieldFα is compatiblein the following sense: there exists a vector field whosegradient givesF α. Note that this property does not holdin general for the second factor on the right-hand side in (4),namely, for the smooth tensor fieldKα. This field wasdubbed theimplant field in [14]. Indeed,Kα is a field oflinear transformations acting on the undistorted incompatibleinfinitesimal parts and joining them without gaps into aglobal reference configuration.

By virtue of its incompatibility, the implant fieldKα

induces an inhomogeneity that can be represented by anon-Euclidean material connection, which is a certain typeof affine connection admissible on the manifoldBα. Inabstract terms, this connection can be defined as a field of

Fig. 2. Correspondence Between Local Shapes and Space with AbsoluteParallelism

operatorsΓα that map some tangent vectorh onto a linearoperator [12],

Γαh = K−1α ∇Kα

(Kαh)Kα,

It is clear that the material connection is actually a geometricrepresentation of the implant fieldKα. The non-Euclideanfeatures of such a connection are represented by the torsiontensor fieldTα,

Tα(h,p) = (Γαp)h − (Γαh)p− [h,p].

Here[·, ·] denotes the Lie bracket, which represents the com-mutator of tangent vector fields [12], [17]. One can expressthese relations in terms of the natural frame∂ν induced bya certain coordinate map, sayκR

α , on the manifoldsBα inthe form [14]

eβ = (Kα)ν·β∂ν , (Γα)

βγν =

(K−1

α

)ρ·γ,ν

(Kα)β·ρ,

(Tα)βγν = (Γα)

βγν − (Γα)

βνγ ,

where theeβ combine to form a nonholonomic system offrames corresponding to the “implantation” byKα.

The connection turns the manifoldBα into a Cartan space(space with absolute parallelism, teleparallel space [17])(Fig.2). Indeed, using the operatorsΓα, we can define arule of parallel transport onBα. The implant fieldKα

defines arbitrary affine transformations of the natural frame∂ν at all points ofB, and so the frame fieldeβ becomesnonholonomic. The rule of parallel transport can be stated asfollows. A vector is transported parallelly if its projectionsonto the frameseβ remain invariant. Since the fieldKα

defines the inhomogeneity, we see that this situation hasa vivid physical interpretation; i.e., and observer travelingwith the moving frame sees no inhomogeneities, just as ageodesic observerdoes not “feel” any gravitation field ingeneral relativity; e.g., see [14]. Actually, one can representthe local shapes as a continuous family of stressed referenceshapes (Fig.3).

We have already pointed out that the response functionalS does not depend onα explicitly. Formally, this meansthat, for everyα < β ∈ I, the response of the bodyBα ata fixed interior material pointx ∈ Bα is equal to that of itssuccessorBβ . From the physical viewpoint, this means thatthe properties of already accreted material do not change asthe growth process continues. The “implantation” parameterscorresponding to a material point are completely determinedat the accretion time and remain unchanged ever after. Inother words, the inhomogeneity arises owing to the growthprocess on the boundary and does not develop further in the

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Fig. 3. Representation for Local Shapes as a Family of Reference Shapes

bulk. In terms of [15], this means that remodeling does notoccur in the bulk. Thus, we exclude processes like plasticityor shrinkage from consideration. We shall refer to growingbodies under such condition asnonrearranged:

A layerwise growing body isnonrearrangedif

∀α < β ∈ I ∀x ∈ Bα Tβ(x) = Tα(x). (6)

Note that the property of nonrearrangement in the bulk istypical of surface growth models as opposed to so-calledvolume growth models [15].

It is not difficult to take into account the natural geometricdefinition of Cartan connection and its torsionTα, but thiscan require some additional calculations. Thus, in many casesit is preferable to deal directly with the implant fieldKα

inducing certain type of connection. To this end, we canchoose ahereditaryfamily of reference shapes, i.e., a familythat satisfies the following relation:

∀α < β ∈ I Bα = κRβ|αBα,

where κRβ|α is the restriction of the mappingκR

β to thedomainBα ⊂ Bβ . This means that the family of referenceshapes can be treated as a sequence of continuations of the setBRa to the setBR

b . From the mechanical viewpoint, this meansthat the shapeBR

a continuously increases by the addition ofa flux of material surfaces to its boundary, while the stateof already adhered material particles remains unchanged.Obviously. these shapes are not stress free and can be inequilibrium only under special external fields of bulk forcesand surface forces on the boundary (Eshelby forces). In thiscase, the condition (6) can be represented in terms of theimplantKα as follows:

∀α < β ∈ I ∀x ∈ Bα Hα(x) = Hβ(x).

Furthermore, if the familyC (1) admits differentiation withrespect to the parameterα, then condition (6) can be rewrit-ten in terms of the field equation,

K = 0, (7)

where the symbolK denotes the mappingK : I ∋ α 7→Kα and the dot stands for differentiation with respect to theparameterα.

IV. B OUNDARY VALUE PROBLEM IN THE HYPERELASTIC

CASE

Let us introduce the elastic potentialWκR

κ, that is, the

elastic energy per unit volume in the reference stateκR,

which can be interpreted as a function of three argumentsF , K, andX [14],

WκR

κ(K, F ,X) = J−1

KWκC

κ(H ,X) = J−1

KWκC

κ(K·F ,X).

One can express the Piola stress tensorTκR

κcorresponding

to κR by the formula

TκR

κ=

∂WκR

κ

∂F ∗ = J−1K

K∗ ·∂WκC

κ

∂H∗ .

The stress tensorTκC

κcorresponding toκC can be defined

fiberwise as follows:

TκC

κ=

∂WκC

κ

∂H∗ .

The boundary value problem for an accreted solid isdetermined by the equations of equilibrium inV (t) withboundaryΩ(t) whose parametrically depends on time,

∇κR·

[J−1K

K∗ ·∂WκC

κ(H ,X)

∂H∗

∣∣∣H=K·F

]+ b = 0, (8)

and the boundary conditions onΩ(t),

nκR·

[J−1K

K∗ ·∂WκC

κ(H ,X)

∂H∗

∣∣∣H=K·F

] ∣∣∣∣∣Ω(t)

= p.

At first glance, the formal statement of the boundary valueproblem differs from the classical one only in that the bound-ary of the domain depends parametrically on time. However,there is a more profound difference: the elastic potentialdepends on the distortion tensor field, whose determinationrequires additional conditions. The particular form of theseconditions depends on the geometric structure of the adheringelements, that is, essentially, on the structure of the bundleof material manifolds. If the growth of a body is due toa continuous influx of prestressed material surfaces to thisbody, then this condition can be written in the form

PκR·

[J−1K

K∗ ·∂WκC

κ(H,X)

∂H∗

∣∣∣H=K·F

]·PκR

∣∣∣∣∣Ω(t)

= T .

HereP = (E − n⊗ n) is the projection onto the tangentplane toΩ(t). This equation expresses the fact that the fibersalign with the specified tension determined by the surfacetensorT , i.e., two-dimensional tensor of second rank definedon the tangent space to the adhering material surface.

If the growth results from continuous adherence of pre-stressed surfaces, then the equation for the distortion tensorK can be obtained from the relations of the theory ofmaterial surfaces (theory of solids with material bound-ary [16]). The effect of material surface adhering leadsto an infinitesimal change in the stress–strain state of theaccreted solid, but since the elementary adhesion act occurson an infinitesimal time interval, the stress rate proves to befinite. This rate can be found from the equations of contactinteraction between the 3D body and the adhering materialsurface. The equilibrium equation for the physical boundary(which is a bounding surface of the body in its actual statefrom the geometric viewpoint and a thin film in a membranestress state from the mechanical viewpoint) can be writtenas∇s ·T + bs = nκR

·TκR

κ

∣∣∣Ω(t)

, where∇s is the surface

nabla operator andbs is the surface density of external forcesacting onΩ(t). To complete the statement of the boundary

Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K.

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value problem, we should pose a condition on the curvilinearboundary∂Ω(t) of the surfaceΩ(t). This condition can be

of the form n ·T∣∣∣∂Ω(t)

= f . Here n is the external unit

normal to the curve∂Ω(t) in the tangent plane andf is alinear density of forces distributed on∂Ω(t).

V. NOTE ON THE REPRESENTATIONS OF STRESSES

In classical linear algebra, associated with a vector spaceis the dual space, i.e., the space of linear functionals (cov-ectors). The distinction between vectors and covectors iscommonly ignored in continuum mechanics. However, asuperstructure on a vector space in the form of a linearfunctional space seems more natural from the physical view-point. If all the variables of vector nature are defined inthe same vector space, which is usually equipped with aninner product, then there is a quite justified wish to treat thevelocity vector and the force vector as elements of the samevector space, whose inner product is a scalar, i.e., power,exerted by the force on the velocity. On the other hand,taking into account their equivalence as elements of the samevector space one can formally consider their sum which isdeprived of any physical meaning. The situation is fixed bythe argument that the force should be a covector and thevelocity should be a vector. The covector representation ofthe force treat it as a linear functional whose action on thevelocity vector gives a scalar, i.e., power. Of course, the sumof a vector and a co-vector is not defined.

Following [18], we use the statement of continuum me-chanics in terms of vector- and covector-valued forms (exte-rior forms), which are taken as replacements for the standardstresses. This statement seems to be more natural from thegeometric viewpoint. It permits one to present the conceptof stresses on a manifold with non-Euclidean connection.

We take an approach to stresses that treats them ascovector-valued two-forms and regard them as fundamentalvariables,

T : Bα → TB ⊗ Ω2, T = ∗2σ = σ·ji· e

i ⊗ (∗ej),

P : BRα → TB ⊗ Ω2, P = ∗2P = p·ji·e

i ⊗ (∗ej).

Here∗ denotes the Hodge star operator.Physically T and P can be interpreted as follows. The

stress upon pairing with a velocity field provides an area-form that is ready to be integrated over a surface to give therate of work done by the stress on that surface [18]. Theabove-mentioned considerations remain valid on a manifoldwith non-Euclidean connection [19]. All one needs to definestress covector-valued forms in such cases is a volume 3-form.

Equations (8) can also be transformed to local form bythe Cartan calculus machinery. Consider the Cartan exteriorderivative

d : T ∗E ⊗ Ωk−1 → T ∗E ⊗ Ωk; T 7→ dT

∀u udT = d(uT )−∇u∧T ,

where ∧ is, by definition, a pairing on the first elementsof dyadic decomposition and a wedge product on the rest.Note that fork = 0, d is reduced to the regular covariantderivative, while fork = 3, d is identically zero.

Since the pullback of forms commutes with the exteriorderivative, it is possible to used in the definition of aderivativeD on elements of the spaceT ∗E ⊗ Ωk−1 suchthat the following diagram commutes [18]

T ∗E ⊗ Ωk−1(BR)ϕ∗2

←−−−− T ∗E ⊗ Ωk−1(B)yD

yd

T ∗E ⊗ Ωk(BR) ←−−−−ϕ∗2

T ∗E ⊗ Ωk(B)

In terms of exterior calculus, the balance equations appearin the following form:

dT +G⊗ ε = 0, DP + g ⊗ ε = 0.

Here ε is the volume form corresponding to a certainparametrization of physical space. We again point out thatthe above-mention considerations remain valid on a manifoldwith a non-Euclidean connection. All one needs to definestress covector-valued forms in such cases is a volume 3-form.

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