A geometric approach to Hu-Washizu variational principle in nonlinear elasticity
aa r X i v : . [ m a t h - ph ] S e p A geometric approach to Hu-Washizu variational principle innonlinear elasticity
Bensingh Dhas ∗ and Debasish Roy † Center of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore 560012,India Computational Mechanics Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore560012, India
Abstract
We discuss the Hu-Washizu (HW) variational principle from a geometric standpoint. Themainstay of the present approach is to treat quantities defined on the co-tangent bundles ofreference and deformed configurations as primal. Such a treatment invites compatibility equa-tions so that the base space (configurations of the solid body) could be realised as a subset ofan Euclidean space. Cartan’s method of moving frames and the associated structure equationsestablish this compatibility. Moreover, they permit us to write the metric and connection using1-forms. With the mathematical machinery provided by differentiable manifolds, we rewritethe deformation gradient and Cauchy-Green deformation tensor in terms of frame and co-framefields. The geometric understanding of stress as a co-vector valued 2-form fits squarely withinour overall program. We also show that for a hyperelastic solid, an equation similar to theDoyle-Erciksen formula may be written for the co-vector part of the stress 2-form. Using thiskinetic and kinematic understanding, we rewrite the HW functional in terms of frames anddifferential forms. Finally, we show that the compatibility of deformation, constitutive rulesand equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functionalwhen varied with respect to traction 1-forms, deformation 1-forms and the deformation. Thisnew perspective that involves the notion of kinematic closure precisely explicates the necessarygeometrical restrictions on the variational principle, without which the deformed body may notbe realized as a subset of the Euclidean space. It also provides a pointer to how these restrictionscould be adjusted within a non-Euclidean setting.
Keywords: non-linear elasticity, differential forms, Cartans’ moving frame, kinematic closure, Hu-Washizu variational principle
Differential forms provide for a natural descriptor of many phenomena of interest in science andengineering [8]. The rules for combining and manipulating differential forms were developed byHermann Grassmann; however it was only in the work of ´Elie Cartan that differential forms found ∗ [email protected] † [email protected] et al. [13], where stress was interpreted as a co-vector valued 2-form. Ourkinematic reformulation is also dual to the understanding of stress as a 2-form proposed by Kanso et al. . Using this kinematic-cum-kinetic formalism, we rewrite the HW energy functional and showthat it recovers the equilibrium equations as well as compatibility and constitutive rules.The goals, just stated, have consequences for computation as well. Mixed finite element methodsalready discussed in the literature may now be realised as tools that place geometry and deforma-tion on an equally important footing. The algebraic and geometric structures brought about bydifferential forms are instrumental in this. Numerical techniques developed in the form of vectorfinite elements like the Raviart-Thomas, N´ed´elec [19, 16, 17] and other carefully handcrafted finiteelement techniques are now being unified under the common umbrella of finite element exteriorcalculus, where differential forms play a significant role [5, 3, 2, 11]. Within nonlinear elasticity,techniques based on mixed methods are already the preferred choice [21, 22, 1] for large deformationproblems. Motivated by the algebra of differential forms, techniques to approximate differentialforms outside the conventional framework of finite elements [12, 25] are also being explored. Thesetechniques can be better developed and interpreted using the geometric approach we propose forthe HW functional, when applied to nonlinear elasticity problems.The rest of the article is organized as follows. A brief introduction to differential forms is givenin Section 2. Kinematics of an elastic body is presented in Section 3, where important kinematicquantities like deformation gradient and right Cauchy-Green deformation tensor are reformulated interms of frame and co-frame fields. This reformulation is facilitated by Cartan’s method of movingframes. This section also contains a discussion on affine connections using connection 1-forms.In Section 4, we introduces stress as a co-vector valued differential 2-form; this interpretation isoriginally due to Kanso et al. . Then a result like the Doyle-Ericksen formula is presented whichyields a constitutive description for traction 1-forms in the presence of a stored energy function. InSection 5, we rewrite the HW variational principle in terms of differential forms using the kinematicsand kinetics of deformation developed in Sections 3 and 4. Then we show that variations of theHW functional with respect to different input arguments lead to the compatibility of deformation,constitutive rule and equations of equilibrium. We also remark on the interpretation of stress asa Lagrange multiplier enforcing compatibility of deformation. Finally, in Section 6, we remarkon the usefulness of geometrically reformulated HW variational principle in constructing efficientnumerical schemes for non-linear elasticity and its extension to other theories in nonlinear solidmechanics. We now provide a brief introduction to differential forms; the material is standard and can befound in many introductory texts to differential geometry and manifold theory, e.g. [24, 9]. Let M be a smooth manifold; at any point of M one may define the tangent space, which is a vectorspace. We denote it by T x M . The dual space to this vector space is denote by T ∗ x M . If α ∈ T ∗ x M and v ∈ T x M , then the action α ( v ) is a real number. The collection of T x M for each x ∈ M is called the tangent bundle T M := S x ∈M T x M ; similarly we may define the cotangent bundleas T ∗ M = S x ∈M T ∗ x M . If one picks at each point in M an element of T ∗ x M , we say that a1-form is defined on M . In other words 1-forms are sections from T ∗ M , just as a vector field isa section from T M . Force is an important example whose differential geometric representationis a 1-form; a force 1-form (or co-vector) acts on a velocity vector to produce power. Electric3eld is another example which can be represented as a 1-form. Conventionally, in mechanics, thedistinction between a 1-form and a vector is ignored allowing one to think of both force and velocityas vectors. This lack of distinction between vectors and 1-forms sacrifices important algebraic andanalytical properties pertaining to the exterior algebra, exterior derivatives and integrability ofdifferential forms. In particle mechanics, the integrability of 1-forms (exact forms) translates to theexistence of a potential for a force. An alternative and geometric approach to understand n -formis to think of them as objects that can be integrated over an orientable n sub-manifold to producea real number. In other words, an n -form is a map from an n − dimensional sub-manifold to realnumbers. The degree of a differential form is an important property; it is the dimension of thesub-manifold on which the form has to be integrated to produce a real number; it can vary fromzero to the dimension of M . The set of all differential forms of degree n over M is denoted byΛ n ( M ); we may sometimes suppress the argument whenever it is clear where the differential formsare defined. The collection of all Λ n over n is denoted by Λ( M ) = S i =1 ,...,n Λ i ( M ). Here n is thedimension of the manifold M . Λ( M ) is often called the exterior algebra over M .An important algebraic operation on differential forms is the wedge or skew product. Usingthe wedge product, one can combine two differential forms to produce a differential form of higherdegree. This algebraic operation is denoted by ∧ ; it is antisymmetric and bilinear. It takes twodifferential forms of degrees m and n to produce a differential form of degree m + n . For anytwo differential forms α and β of degree m and n , the wedge product between them satisfies thefollowing anti-symmetric relationship. α ∧ β = ( − mn β ∧ α (1)Let M and N be two smooth manifolds and ϕ be a diffeomorphism between them; ϕ : M → N and α ∈ Λ n ( N ). The pull back of α to the co-tangent space T ∗ M is denoted by ϕ ∗ ( α ) and is givenby the following relationship, ϕ ∗ ( α )( v , ..., v n ) = α (d ϕv , ..., d ϕv n ) , v i ∈ T M (2)Here, d ϕ is the differential of ϕ ; it maps T M to T N . (2) must be understood as a point-wiserelationship at each tangent space of M and ϕ ( M ) respectively. Under the pull-back map, thewedge product is distributive. If α, β ∈ T ∗ N , then ϕ ∗ ( α ∧ β ) = ϕ ∗ ( α ) ∧ ϕ ∗ β (3)where ϕ ∗ ( . ) is defined as in (2).On a smooth manifold, one can define a notion of differentiation for differential forms called theexterior differentiation. It turns out that this notion of differentiation is coordinate independent.If α is a differential form, then its exterior derivative is denoted by d α . The exterior derivativeoperator is a linear map and increases the degree of a differential form by 1. An important propertyof the exterior derivative is that d ◦ d = 0; the second exterior derivative of a differential form isidentically zero. Under a diffeomorphism, the exterior derivative commutes with its pull back. Thisrelationship may be written as, ϕ ∗ d α = d( ϕ ∗ α ) (4)Exterior derivative is also distributive when applied to the wedge product of two differential forms.For differential forms α and β of degree n and m , the exterior derivative of wedge product betweenthem is given by, d( α ∧ β ) = d α ∧ β + ( − n α ∧ d β (5)4sing exterior differentiation, one can define closed and exact forms. We say that a differentialform α ∈ Λ n is exact if there exists a differential form β ∈ Λ n − such that α = d β . The differentialform α is closed if d α = 0. If we assume M to be a simply connected subset of R n , Poincar´e lemmaestablishes that closed forms are also exact. The failure of a closed form to be exact is measuredby the co-homology group. If C ⊂ M is a hyper-surface of dimension m and α is a differential formof degree m −
1, Stokes theorem relates d α to the trace of α on the boundary of C denoted by ∂ C .Stokes theorem can be written as, Z C d α = Z ∂ C α (6)We now assume that the manifold M is equipped with a metric g and d V denotes the volume formgenerated by the metric. In such a case, one can define a linear isomorphism between differentialforms of degree m and n − m ( n denotes the dimension of M ), for each n . This isomorphism isdefined by the following relationship, α ∧ β = h ⋆ α, β i g d V ; α ∈ Λ n , β ∈ Λ n − m (7)In the above equation, h ., . i g is the inner-product induced by the metric on Λ n − m . This isomorphismis called the Hodge star map; ⋆ : Λ m → Λ n − m . It turns out that this map is useful in definingstresses and computing the variation of the HW energy functional. The reference configuration of the body is identified with a smooth manifold with boundary. Thismanifold is denoted by B and its boundary by ∂ B . Similarly, the deformed configuration and itsboundary are denoted by S and ∂ S respectively. These configurations are endowed with a C ∞ chartfrom which they inherit their smoothness. Following the usual notation in continuum mechanics,we label the material points of the reference and deformed configurations by their position vectors.The position vectors of a material point in the reference and deformed configuration are denotedby X and x with coordinates X i and x i . The tangent and co-tangent spaces at each point X ∈ B isdenoted by T X B and T ∗ X B respectively. The deformation map relating the reference and deformedconfigurations is denoted by ϕ : B → S . The tangent and co-tangent spaces of the deformedconfiguration at a point x = ϕ ( X ) is denote by T ϕ ( X ) S and T ∗ ϕ ( X ) S respectively. At each tangent space of B , we choose a collection of orthogonal vectors; we denote these vectorsby E i . Note that the orthogonality here is with respect to the Euclidean inner product. In otherwords, we have assumed that each tangent space of the reference or deformed configuration isendowed with a metric tensor, which is Euclidean. We call this collection of vector fields a framefield to the reference configuration and it is denoted by F B = { E , ..., E n } . It is clear that the framefields at point X span T X B . Similarly, the orthonormal frame field associated with the deformedconfiguration is denoted by F S = { e , ..., e n } , where e i are sections from T S . The natural (algebraic)duality between tangent and cotangent spaces induces co-frames on the cotangent bundles of B and S . The co-frames of the reference and deformed configurations are denoted by F ∗B = { E i , ..., E n } and F ∗S = { e , ..., e n } respectively, where E i and e i are sections from the cotangent bundles of the5eference and deformed configurations. The natural duality between frame and co-frame fields ofthe reference and deformed configurations may be written as, E i ( E j ) = δ ij ; e i ( e j ) = δ ij E i ∈ T ∗ B , e i ∈ T ∗ S (8)The differential of the position vector of a material point in the reference configuration is denotedby d X and is given by, d X = E i ⊗ E i (9)Similarly, the differential of a position vector in the deformed configuration in terms of the frameand co-frame fields is given by, d x = e i ⊗ e i (10)From the definition of d X in (9), it can be seen that tangent vectors from the reference configurationare mapped to itself under d X . To see this, choose V ∈ T X B with V = c i E i . Substituting thelatter and using the definition of d X , we arrive at d X ( V ) = c j E i E i ( E j ). Using the duality betweenthe frame and co-frame fields, we conclude d X ( V ) = V . Similarly, d x maps a tangent vector fromthe deformed configuration on to itself. The differential of a position vector in the reference anddeformed configurations are thus identity maps on the respective tangent spaces. The importanceof the differential of position will be clarified when we discuss deformation gradient.Similar to the differential to position, one can also define the differential of a frame field. Thedifferential of the frame fields in the reference configuration is given by,d E i = γ ji ⊗ E j (11)where, γ ij is called the connection matrix; it contains 1-forms as its entries. Because of the orthog-onality of the frame fields, the connection matrix is skew symmetric, i.e. γ ij = − γ ji . Similarly, thedifferential of the frame fields associated with the deformed configuration is given by,d e i = ¯ ω ji ⊗ e j (12)¯ ω ij is the connection matrix associated with the deformed frame fields and it is also skew symmetric,¯ ω ij = − ¯ ω ji .For a given choice of connection 1-forms and position 1-forms, there are certain compatibilityconditions (Poincar´e relations) guaranteeing the existence of position vectors. These equations arecalled Cartan’s structure equations. The first compatibility condition establishes the torsion freenature of a configuration. For the reference and deformed configurations, this condition may bewritten as, d X = 0; d x = 0 (13)Plugging (9) and (10) into the above equation leads to,d E i = γ ij ∧ E j ; d e i = ¯ ω ij ∧ e j (14)The second compatibility condition establishes that the reference and deformed configurations arecurvature-free. This leads to the following conditions on the reference and deformed frame fields,d E i = 0; d e i = 0 (15)6sing the differential of the frame fields in the above equations leads to,d γ ij = γ ik ∧ γ kj ; d¯ ω ij = ¯ ω ik ∧ ¯ ω kj (16)For a simply connected body, the structure equations as above for the reference and deformedconfigurations provide the necessary kinematic closure to ensure that the configurations can beembedded with an Euclidean space. Indeed, without this closure effected by the structure equations,a model cannot in general produce a deformed configuration which is Euclidean embeddable. In the previous sub-section, we introduced the notion of differentials to position vectors and framefields. We now present the geometric meaning of these infinitesimal quantities. Consider thedifferential of the position vector in the reference configuration given in (9). For a given coordinatesystem, the position vector X is a smooth function of its coordinates ( X , X , X ). Let Γ be aparametrized curve, Γ : [ a, b ] → B . For convenience, we assume Γ to be along the first coordinatedirection, which is obtained by freezing the other two coordinate functions to some constant. Wealso assume that the frame fields E i are constructed by a GramSchmidt procedure on the tangentvectors of the co-ordinate lines at each material point X . Fig. 1 shows how the position vector andthe frame change as one moves along the curve Γ. From our assumptions on Γ, it may be seen thatthe tangent vector to Γ is given by the cE , where c is a real valued function. Now, the differentialof position, which is a vector valued 1-form, can be integrated along the curve Γ to produce avector. This is nothing but the vector between the material points X ( a ) and X ( b ), which may beformally written as, X ( b ) − X ( a ) = Z ba d X ( cE )d s = Z ba cE i E i ( E )d s, = Z ba cE d s, (17)In the above equation, E and c can vary along the curve Γ. The above interpretation of d X isvery similar to that of 1-forms as real numbers defined on curves.We now consider the differential of the frame field, integrating which along Γ leads to the relativerotation of the frame F between the points X ( a ) and X ( b ). Fig. 1(b) demonstrates the rotationexperienced by the frame as one traverses along the curve Γ. E i = Z ba d E i ( E )d s ;= Z ba E j γ ji ( E )d s ; (18)From the discussion, it is clear that the displacement (rotation) vector between two points (frames)in a configuration depends on the position 1-form and the curve or path used for integration.However, the displacement vector between two points on a configuration should be independent ofthe path chosen to integrate the 1-forms. This condition is exactly what the structure equationenforces. 7igure 1: The coordinate lines and the frame field generated from these coordinate lines are shownin (a). The frame fields at X ( a ) and X ( b ) are shown in (b); we have moved the frames to the samepoint so that it is convenient to interpret the change. We have used the notation E i ( a ) and E i ( b )to indicate the frame at the material points X ( a ) and X ( b ). As discussed earlier, the deformation map sends the position vector of a material point in thereference configuration to its corresponding position vector in the deformed configuration. Thedifferential of the deformation map or the deformation gradient, denoted by d ϕ , maps the tangentspace of the reference configuration to the corresponding tangent space in the deformed configura-tion. For an assumed frame field (for both reference and deformed configurations), the differentialof the deformation map can be obtained by pulling back the co-vector part of the differential of thedeformed position vector. d ϕ = e i ⊗ ϕ ∗ ( e i )= e i ⊗ θ i ; θ i ∈ T ∗ B , e i ∈ F S (19)In writing (19), we have introduced the following definition: θ i := ϕ ∗ ( e i ). In our construction,the 1-forms θ i contain local information about the deformation map ϕ . A primitive variable inour theory, we call this the deformation 1-form. From (19), we see that the vector leg of thedeformation gradient is from the deformed configuration, while the co-vector leg is from the referenceconfiguration. If V ( X ) ∈ T X B , the action of the deformation gradient on V is given by,d ϕ ( V ) = e i θ i ( V ) (20)Since θ i ( V ) are real numbers, the above equation is a linear combination of tangent vectors fromthe deformed configuration. 8 .4 Pull-back of structure equations Having introduced the deformation gradient and the deformation 1-form in the last subsection, weare now ready to rewrite the referential version of structure equations in the deformed configuration.These equations are obtained by pulling back the structure equations for the deformed configurationunder the deformation map: ϕ ∗ (d e i ) = ϕ ∗ (¯ ω ij ∧ e j )d θ i = ω ij ∧ θ j (21)where, ω ij := ϕ ∗ (¯ ω ij ) are called pulled back connection 1-forms. The key fact used in obtaining theabove equation is that exterior derivative and wedge product commute with pull back. Using asimilar argument, the pull back of the second compatibility condition is given by,d ω ij = ω ik ∧ ω kj (22)For concreteness, we present the components of the structure equations in three spatial dimensions.Each structure equation constitutes a system of three equations. The matrix form of the firststructure equation (referentially pulled back) may be written as, d θ d θ d θ = ω − ω − ω − ω ω ω ∧ θ θ θ (23)Note that the combining rule for the elements of the matrix and the vector on the right hand sideof the above equation is via the wedge product. The component form of the second compatibilityequations is given by, d ω = ω ∧ ω d ω = ω ∧ ω (24)d ω = ω ∧ ω The notion of length is central to continuum mechanics; important kinematic quantities like strainand rate of deformation are derived from it. Indeed, it may not be possible to assess the state ofdeformation without the metric structure defined on both reference and deformed configurations.The metric structure of a configuration is defined by a symmetric and positive definite tensor, whichencodes the notion of length (in that configuration). We denote the metric tensor of the referenceand deformed configurations by G and g respectively; G : T X B × T X B → R and g : T x S × T x S → R .These tensor valued functions pertain to the idea of infinitesimal lengths at the tangent spaces of B and S . As such, the notion of length is an additional structure placed on B and S . In this work,we assume these metrics to be Euclidean. In terms of the co-frame field, the metric tensor of thereference configuration is given as, G = ( E j ⊗ E j ) : ( E i ⊗ E i )= E i ⊗ E i (25)9he double contraction in the above equation is calculated using the inner product induced by theEuclidean metric. Similarly, the metric tensor in the deformed configurations may be written as, g = ( e i ⊗ e j ) : ( e j ⊗ e i )= e i ⊗ e i (26)In terms of the frame fields, the inverses of the metric tensors for the reference and deformedconfigurations may be written as, G − = E i ⊗ E i ; g − = e i ⊗ e i (27)The right Cauchy-Green deformation tensor is obtained via the pull back of the metric tensor ofthe deformed configuration to the reference configuration. In terms of the the co-frame fields, thisrelationship can be written as, C = ϕ ∗ ( g )= ϕ ∗ ( e i ⊗ e i )= θ i ⊗ θ i (28)An alternate way to compute the C is to use the usual definition in continuum mechanics, C =d ϕ t d ϕ . Here, the ( . ) t is understood to be the adjoint map induced by the metric structure. Usingthe orthonormality of the frame field we arrive at, C = ( θ i ⊗ e i )( e j ⊗ θ j )= θ i ⊗ θ i (29)The calculations leading to (28) and (29) are exactly the same; only the sequence in which pull backand inner product are applied differs. The Green-Lagrangian strain tensor may now be written as, E = 12 ( C − G )= 12 [( θ i ⊗ θ i ) − ( E i ⊗ E i )] (30)The the first invariant of the right Cauchy-Green tensor is given by, I = h θ i , θ i i G (31)Here h ., . i G denotes the inner product induced by the metric tensor G . The area forms induced bythe co-frame of the reference configuration are given by,d A = E ∧ E ; d A = E ∧ E ; d A = E ∧ E ; (32)Similarly, the area forms induced by the co-frame of the deformed configurations are given by,d a = e ∧ e ; d a = e ∧ e ; d a = e ∧ e ; (33)We also define the pulled back area forms from those of the deformed configuration to the referenceconfiguration. These area forms are obtained as,d A = θ ∧ θ ; d A = θ ∧ θ ; d A = θ ∧ θ ; (34)10n writing the above equations, we have used d A i := ϕ ∗ (d a i ) and the definition of the deformation1-form. The second invariant of C is now given by, I = h d A i , d A i i G (35)In terms of co-frame fields, the volume forms of reference and deformed configurations may bewritten as, d V = E ∧ E ∧ E ; d v = e ∧ e ∧ e (36)The pull back of d v to the reference configuration is,d V = θ ∧ θ ∧ θ (37)The third invariant of C is then given by, I = ( ⋆ d V ) (38) An affine connection on a smooth manifold is a device used to differentiate sections of vector andtensor bundles in a co-ordinate independent manner. This differentiation is often referred to ascovariant. The choice of an affine connection determines the covariant differentiation. For a smoothmanifold with a metric, the metric induces a unique covariant derivative and we denote it by ∇ .For vector fields v, w and a real valued function f , the covariant derivative satisfies the followingproperties. ∇ fe i w = f ∇ e i w ; f ∈ Λ ∇ e i ( w + v ) = ∇ e i w + ∇ e i v (39) ∇ e i ( f w ) = e i [ f ] w + f ∇ e i we i [ f ] in the above equations denotes the action of a vector field on a real valued function f . We nowshow how the connection 1-forms encode the affine connection. Let w = P i =1 w i e i is an arbitraryvector field defined on S . Then the covariant derivative of w in the direction of e i is given by, ∇ e i w = d w j ( e i ) e j + w j ¯ ω kj ( e i ) e k (40)Having defined the covariant derivative of a vector field in the direction of frame fields, it is nowpossible to extend the above definition of covariant differentiation to arbitrary vector fields usingthe defining properties given in (39). In the previous section, we have reformulated the kinematics of continua using differential forms.The right Cauchy-Green deformation tensor and the deformation gradient were the key objectsin the reformulation. In this section, we present a geometric approach to stress, originally dueto Kanso et al. [13]. Even though this approach is intuitive and geometric, it was never put touse. In classical dynamics [15], force is understood as a co-vector so that its pairing with velocityproduces power. This understanding of force as dual to velocity does not require the metric tensor to11igure 2: Piola Transform: notice that the length and direction of the force acting an the deformedand reference area is preserved under Piola transform.compute power and it must be contrasted with the usual understanding of of both velocity and forceas vectors. Extending this concept of force as a co-vector to the continuum mechanical definition ofstress is our present goal. As a consequence, we also show that the description of power/work usedin continuum mechanics may be undertaken without the notion of a metric. However, it shouldalso be noted that a formulation of continuum mechanics without the metric tensor may not befeasible as the notion of strain crucially depends on the metric.We denote the Cauchy stress tensor by σ . The traction ’vector’ acting on an infinitesimal areawith unit normal n is denoted by t , given by the well known formula t = σn . The traction t is aforce which depends on the material point at which it is evaluated and the area which sustains it.The relation between the normal and the traction is postulated to be linear. In the language ofdifferential forms, an infinitesimal area is regarded as a 2-form, while from classical dynamics wealso know that force is a co-vector or a 1-form. Putting these two ideas together, we are led to ageometric definition of Cauchy stress given by, σ = t i ⊗ d a i ; (41)In the above equation, d a i is the area 2-form, which sustains the traction vector t i . The tensorproduct in the above equation is due to the linearity between the traction and area forms. Fromthis equation, it is easy to see that the area forms change orientation if the order the co-vectors arereversed. Geometrically, if there are n linearly independent area forms at a point on the manifold,the stress tensor assigns to each area form a 1-form called traction. With this understanding,Cauchy stress may now be identified with a section from the tensor bundle Λ ⊗ Λ which has thedeformed configuration as its base space.In contemporary continuum mechanics, Nanson’s formula describes how unit normals transformunder the deformation map [18]. Geometrically, Nanson’s formula is nothing but the pull back ofa 2-form under the deformation map. The pull back of area forms generated by the co-frame isalready described in (34). The first Piola stress may now be obtained by pulling back the area legof the Cauchy stress under the deformation map. This partial pull back of the Cauchy stress istermed the Piola transform. This relationship may be formally written as, P = t i ⊗ ϕ ∗ (d a i )= t i ⊗ d A i (42)12ote that the traction 1-form in both the definitions of Cauchy stress and first Piola stress are thesame, i.e. the pull back does nothing to the traction 1-form. In the above discussion, contraryto convention, Cauchy and first Piola stresses are identified as third order tensors, not the usualsecond order. This ambiguity can be removed if one applies Hodge star on the area leg of these twostresses. In three dimensions, the Hodge star establishes an isomorphism between forms of degree2 and 1. Applying Hodge star to Cauchy and first Piola stresses leads to, σ = t i ⊗ ( ⋆ d a ) i (43) P = t i ⊗ ( ⋆ d A ) i (44)Kanso et al. made a distinction between the stress tensors given in (41), (42) and (43), (44).However we do not see a need for it, since both the usual and geometric definitions of stress containexactly the same information; only the ranks of these tensors are different. The Doyle-Ericksen formula is an important result in continuum mechanics [6], which states thatthe Cauchy stress understood as a 2-tensor and the metric tensor of the deformed configurationare work conjugate pairs. For a stored energy density function W , Doyle-Ericksen formula gives usthe following relationship, σ = 2 ∂W∂g (45)In writing (45), we have assumed that W depends on the differential of deformation through theright Cauchy-Green deformation tensor, i.e. W is frame-indifferent. From the discussion presentedin the previous section, it can be seen that the area leg of the Cauchy stress tensor is determinedby the choice of coordinate system for the tangent bundle of the deformed configuration. On theother hand, the area leg of the first Piola stress is determined by both the co-ordinate system forthe tangent bundle of the deformed configuration and the deformation map. Clearly, the area leg ofa stress tensor does not require a constitutive rule, the only component requiring so is the traction1-form.We now claim that for a stored energy function W , the traction 1 − form has the followingconstitutive rule, t i = 1 J ∂W∂e i (46)The last equation is in the same spirit as the Doyle-Ericksen formula. To establish the result statedin (46), we first compute the directional derivative of W along e i , ∂W∂e i = ∂W∂ (d ϕ ) ∂ (d ϕ ) ∂e i (47a)= ( t j ⊗ ⋆ d A k )( e i ⊗ e j ⊗ θ k ) (47b)= h ⋆ d A J , θ J i t i (47c)= t i⋆ d V (47d)= J t i (47e)We used chain rule to arrive at the right hand side of (47a). In (47b), the expression for Piolastress (as a two tensor) in terms of W and the directional derivative of d ϕ along e i are used13o get the right hand side and performing the required contractions lead to (47c). The claim isfinally established by using the definitions of pull back and Hodge star for volume forms. In threedimensions, constitutive relations have to be supplied to the three traction 1-forms each conjugateto vector elements in the frame to close the equations of motion. If the Cauchy stress generatedby a stored energy function is known, then the expression for traction 1-form can be computedusing the simple relationship t i = σn i , where the vector fields n i are chosen to be elements fromthe frame of the deformed configuration. We first present the conventional HW variational principle for a finitely deforming elastic body.The HW variational principle takes the deformation gradient, first Piola stress and deformationmap as input arguments. The referential version of HW functional for a non-linear elastic solidreads, I HW = Z B [ W ( C ) − P : ( F − d ϕ )]d V − Z ∂ B h t, ϕ i d A (48)In the above equation t = P N , is the traction defined on the surface ∂ B . The integration in theabove equation is withrespect to the volume and the area forms of the reference configuration.In (48), deformation gradient is assumed to be independent, this tensor field is denoted by F ,while the deformation gradient computed from deformation is denoted by d ϕ . Another importantpoint to notice is that the the second term in the above equation is bilinear in Piola stress anddeformation gradient. The variation of the HW functional with respect to deformation, deformationgradient and first Piola stress leads to the equilibrium equation, constitutive rule and compatibilityof deformation gradient. HW variational principle have been previously exploited to formulatenumerical solution procedure to solve non-linear problems in elasticity see[1, 22, 21]. From now on we will use the definitions of Cauchy and Piola stresses as given in (41) and (42)respectively. We now rewrite the HW variational principle such that it takes deformation 1-forms,traction 1-forms and deformation map as inputs. We also assume that compatible frame fields forthe reference and deformed configurations are given. The HW functional may be written as, I ( θ i , t i , ϕ ) = Z B W ( θ i )d V − ( t i ⊗ d A i ) ˙ ∧ ( e i ⊗ ( θ i − d ϕ i )) − Z ∂ B h t ♯ , ϕ i d A (49)Note that we did not write the volume form in the second term on the RHS of (49), since theoutcome of ˙ ∧ is a 3-form which can be integrated over the reference configuration to produce workdone by the traction 1-forms on the frame fields. In (49), ˙ ∧ denotes a bilinear map. For α ∈ T ∗ S , v ∈ T S and a, b ∈ Λ( B ) the action of this map is given by ( α ⊗ a ) ˙ ∧ ( v ⊗ b ) = α ( v ) a ∧ b . Note thatthe definition of ˙ ∧ given here is a little different from the one in Kanso et al. ; specifically, we donot use the metric tensor. From the definition of ˙ ∧ , it can be seen that the work done by stresson deformation is metric independent. This property of our current variational formulation bringsthe continuum mechanical definition of stress a step closer to the definition of force as defined inclassical mechanics. Another important point here is that the second term on the RHS in (49) is14quivalent to the second term in (48); however now the relationship between the different argumentsis multi-linear. Remark 1:
In writing (49), we have presumed that the geometry of the body is Euclideanand it is frozen during the deformation process. We believe that this assumption can be relaxedby permitting non-integrability in the connections and deformation 1-forms (i.e. by incorporatingsource terms in the structure equations).
Remark 2:
For a frame to represent Euclidean geometry, it is not required that the connection1-forms be identically zero. It only requires zero source term in Cartan’s structure equations. Anyset of (unit) tangent vectors to a curvilinear co-ordinate system will have non-zero connection1-forms, even as it acts as a frame in the Euclidean space.We now proceed to obtain the Euler-Lagrange equations or the condition for critical points of thefunctional I . We use the Gateaux derivative for this purpose. Let ǫ denote a small parameter andˆ( . ) an increment in quantity with which we are differentiating the functional I . We first calculatethe variation of I with respect to traction 1-forms; t i t i + ǫ ˆ t i , where ˆ t i are assumed to be fromthe tangent space of T ∗ S . I ( ǫ ) = Z B W ( θ i ) − (( t i + ǫ ˆ t i ) ⊗ d A i ) ˙ ∧ ( e j ⊗ ( θ j − d ϕ j )) − Z ∂ B h t ♯ , ϕ i d A (50)Using the definition of ˙ ∧ and Gateaux derivative, we get a vector valued 3-form for each i . Thesethree 3-forms have to be equated to zero to get the condition for critical points in the direction oftraction 1-forms. These conditions can be formally written as, (d A ∧ ( θ − d ϕ )) − (d A ∧ d ϕ ) − (d A ∧ d ϕ ) − (d A ∧ d ϕ ) (d A ∧ ( θ − d ϕ )) − (d A ∧ d ϕ ) − (d A ∧ d ϕ ) − (d A ∧ d ϕ ) (d A ∧ ( θ − d ϕ )) ⊗ e e e = (51)Since e i are elements form the frame, we have g ( e i , e i ) = 1. The above equation can be true onlywhen the coefficient matrix on the LHS is zero, which leads to,(d A ∧ ( θ − d ϕ )) = 0; (d A ∧ d ϕ ) = 0; (d A ∧ d ϕ ) = 0 (52a)(d A ∧ d ϕ ) = 0; (d A ∧ ( θ − d ϕ )) = 0; (d A ∧ d ϕ ) = 0 (52b)(d A ∧ d ϕ ) = 0; (d A ∧ d ϕ ) = 0; (d A ∧ ( θ − d ϕ )) = 0 (52c)The above equations can be recast in a matrix form as, θ − d ϕ d ϕ d ϕ θ θ − d ϕ d ϕ θ d θ d θ − ϕ ∧ θ ∧ θ θ ∧ θ θ ∧ θ = (53)For these equations to hold, the following conditions must be met, θ − d ϕ = 0; θ − d ϕ = 0; θ − d ϕ = 0; (54)The above condition simply states that there exist three zero forms whose exterior derivatives arethe deformation 1-forms; or in other words, deformation 1-forms are exact and ϕ i are the potentialsfor the corresponding deformation 1-forms. 15e now compute the variation of I with respect to deformation 1-forms. Incremental changesin the deformation 1-forms may be written as, θ i θ i + ǫ ˆ θ i , where ǫ ˆ θ i is assumed to be an elementfrom the tangent space of T ∗ B . I ( ǫ ) = Z B W ( θ i + ǫ ˆ θ i ) − ( t i ⊗ d A ( ǫ ) i ) ˙ ∧ ( e j ⊗ θ j ( ǫ )) − Z ∂ B h t ♯ , ϕ i d A (55)Using the definition of Gateaux derivative, for each θ i we have, ∂W∂θ = [ t ( e ) ⋆ ( θ ∧ θ ) − t ( e ) ⋆ (d ϕ ∧ θ ) + t ( e ) ⋆ (( θ − d ϕ ) ∧ θ ) − t ( e ) ⋆ (d ϕ ∧ θ ) − t ( e ) ⋆ ( θ ∧ ϕ ) + t ( e ) ⋆ ( θ ∧ d ϕ ) + t ( e ) ⋆ ( θ ∧ ( θ − d ϕ ))] ♯ (56a) ∂W∂θ = [ t ( e ) ⋆ ( θ ∧ ( θ − d ϕ )) − t ( e ) ⋆ ( θ ∧ d ϕ ) − t ( e ) ⋆ ( θ ∧ d ϕ ) − t ( e ) ⋆ (( θ ∧ d θ ) − t ( e ) ⋆ (d ϕ ∧ θ ) − t ( e ) ⋆ ( θ ∧ θ ) + t ( e ) ⋆ (( θ − d ϕ ) ∧ θ )] ♯ (56b) ∂W∂θ = [ t ( e ) ⋆ (( θ − d ϕ ) ∧ θ ) − t ( e ) ⋆ ( ϕ ∧ d θ ) − t ( e ) ⋆ ( ϕ ∧ d θ ) − t ( e ) ⋆ (d θ ∧ d ϕ ) − t ( e ) ⋆ ( θ − ( θ − ϕ )) + t ( e ) ⋆ ( θ ∧ d ϕ ) + t ( e ) ⋆ ( θ ∧ θ )] ♯ (56c)If we now take into account the compatibility equations previously established in (54), the lastequations reduce to, ∂W∂θ = [ t ( e ) ⋆ ( θ ∧ θ ) + t ( e ) ⋆ ( θ ∧ θ ) + t ( e ) ⋆ ( θ ∧ θ )] ♯ (57a) ∂W∂θ = [ t ( e ) ⋆ ( θ ∧ θ ) + t ( e ) ⋆ (( θ ∧ θ ) + t ( e ) ⋆ ( θ ∧ θ )] ♯ (57b) ∂W∂θ = [ t ( e ) ⋆ ( θ ∧ d θ ) + t ( e ) ⋆ ( θ ∧ d θ ) + t ( e ) ⋆ ( θ ∧ θ )] ♯ (57c)From these equations, it is seen that a 2-form accompanies the components of traction 1-forms;this is indeed true since we use a Piola transform to write the constitutive rule in the referenceconfiguration. From a comparison of (56), where Cartan’s structure equations for kinematic clo-sure have not been enforced, and (57), we note that the expressions for traction in the formerhave additional terms. These additional terms may be related to incompatibilities created by theemergence of defects (such as dislocations) as the deformation evolves. In other words, without anexplicit imposition of the kinematic closure conditions on the flow, the deformed body may neverbe realized as a subset of the Euclidean space.Finally, we compute the variation of I with respect to deformation; ϕ i ϕ i + ǫ ˆ ϕ i , where ˆ ϕ belongs to T B . Using the definition of the superimposed incremental deformation in I and uponcomputing the Gateaux derivative, we have the following equation, Z B ( t i ⊗ d A i ) ˙ ∧ ( e j ⊗ d ˆ ϕ i ) = 0 (58)To complete the variation, we need to shift the differential from ˆ ϕ . We first calculate the following,d( ϕ k t j ( e k )d A j ) = d ϕ k ∧ t j ( e k )d A j + ϕ k d( t j ( e k )) ∧ d A j + ϕ k t j ( e k )d A j (59)This equation invites a few comments. The first is that we are calculating the exterior derivative ofa 2-form, with ϕ k t j ( e k ) being a scalar. Using the product rule of differentiation, we have expanded16he right hand side of (59). The second term in (59) should be evaluated using the connection1-forms since it involves the exterior derivative of a vector. This terms is relevant when one workswith a non-trivial connection for the manifold, examples being a body with dislocations and aKirchhoff shell. If we invoke compatibility of deformation, we have d A = 0, which leaves (59) inthe following form, d( ϕ k t j ( e k )d A j ) = d ϕ k ∧ t j ( e k )d A j + ϕ k d( t j ( e k )) ∧ d A j (60)An expression similar to (60) was utilized by Kanso et al. [13] to define mechanical equilibrium. Theexpression for exterior derivative defined in (60) involves the connection 1-forms of the manifold,which is similar to the covariant exterior derivatives used in gauge theories in physics. Using (59)in (58) leads to, Z B d( ˆ ϕ k t j ( e k )d A j ) − ˆ ϕ k d( t j ( e k )) ∧ d A j = 0 (61)The first term in the above equation can be converted to a boundary term via Stokes’ theoremleading to, Z ∂ B ˆ ϕ k t j ( e k )d A j − Z B ˆ ϕ k d( t j ( e k )) ∧ d A j = 0 (62)Using the arbitrariness of ˆ ϕ k , we conclude that,d( t j ( e k )) ∧ d A j = 0 (63)This is the condition for the critical point of the energy functional in the direction of deformation,which is nothing but the balance of forces. Note that the connection 1-forms of the deformedconfiguration appear through the exterior derivative of the deformed configuration’s frame field. For a hyper-elastic solid, stress is derived from a stored energy function which may be writtenas a function of the deformation gradient. This assumption permits us to write the equations ofequilibrium as the Euler-Lagrange equation of the stored energy functional. In a certain sense,the stress generated in a hyper-elastic solid should satisfy certain integrability condition (i.e. theexistence of the stored energy function). Moreover, if we assume the stored energy function tobe translation and rotation invariant, it implies equilibrium of forces and moments. Thus for thehyper-elastic solid, balances of force and moment are consequences of translation and rotationinvariance; stress is only a secondary variable introduced for writing the equations of equilibriumin a convenient way.When formulated as a mixed problem, the stress tensor has a completely different role. OurHW functional has deformation, deformation 1-forms and stress 2-forms as inputs. For a storedenergy function, viewed as a function of deformation 1-forms, translation and rotation invariancecannot be discussed directly, since nothing about the geometry of the co-tangent bundle from whichthe deformation 1-forms were pulled back is known. In other words, there is nothing in the storedenergy function that requires the base space of the deformed configurations co-tangent bundle tobe Euclidean. The second term in (49) is introduced to imposes this constraint. Observe that, in(49), the second term is multilinear in the input arguments, vis-´a-vis, stress 2-forms, differentialof deformation and deformation 1-forms. The stress 2 − form can now be thought as a Lagrange17ultiplier introduced to impose the equality between differential of deformation and deformation 1-forms. Alternatively, the equality between the differential of deformation and deformation 1-formsimplies that the deformed configuration is Eulclidean. We have formulated the HW variational principle using differential forms. The main tenets ofthis reformulation are Cartan’s method of moving frames and the interpretation of stress as a co-vector valued 2-form. The Euler-Lagrange equations clearly explicate how additional stresses couldbe generated when the kinematic closure or compatibility conditions are not explicitly imposed.These stresses are the result of incompatibilities that develop and co-evolve with the deformationand may render the deformed solid body non-Euclidean. In this sense, the present variationalapproach may be used not only with models that aim at restricting the deformed body as a subsetof the Euclidean space, but also with those where the evolution of incompatibilities, e.g. defects,is of importance.Our novel approach to the HW variational principle also has consequences in the numericalsolution of the equations of nonlinear elasticity. The discretization schemes based on finite elementexterior calculus may now be seamlessly used to approximate the differential forms appearing inthe HW functional. Such an approximation has the advantage of respecting the algebraic andgeometric structures defined by these differential forms even after discretization. Work is currentlyunder way to study such approximations.The kinematic framework developed to describe deformation has a specific advantage in mod-elling the motion of shells. In the case of shells, the differential of the position vector is non-trivialsince the tangent spaces at different material points are not the same. When tied with Cartan’smoving frames, the kinematics of the shell surface can be completely reformulated in terms of dif-ferential forms. The stress resultants may then be understood as bundle valued differential forms.These kinetic and kinematic ideas may then be stitched together using the new HW principle, toarrive at a shell theory that is transparent in its geometric assumptions.
Acknowledgements
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