A Weyl geometric model for thermo-mechanics of solids with metrical defects
aa r X i v : . [ c ond - m a t . o t h e r] A p r A Weyl geometric model for thermo-mechanics of solids withmetrical defects
Bensingh Dhas ∗ , Arun R Srinivasa † and Debasish Roy ‡ Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science,Bangalore 560012, India Computational Mechanics Lab, Department of Civil Engineering, Indian Institute ofScience, Bangalore 560012, India Department of Mechanical Engineering, Texas A&M University, College Station, Texas77843-3123
Abstract
This work seeks a rational route to large-deformation, thermo-mechanical modeling of solidcontinua with metrical defects. It assumes the geometries of reference and deformed configura-tions to be of the Weyl type and introduces the Weyl one-form – an additional set of geometri-cally transparent degrees of freedom that determine ratios of lengths in different tangent spaces.The Weyl one-form prevents the metric from being compatible with the connection and enablesexploitation of the incompatibility for characterizing metrical defects in the body. When such abody undergoes temperature changes, additional incompatibilities appear and interact with thedefects. This interaction is modeled using the Weyl transform, which keeps the Weyl connectioninvariant whilst changing the non-metricity of the configuration. An immediate consequence ofthe Weyl connection is that the critical points of the stored energy are shifted. We exploitthis feature to represent the residual stresses. In order to relate stress and strain in our non-Euclidean setting, use is made of the Doyle-Ericksen formula, which is interpreted as a relationbetween the intrinsic geometry of the body and the stresses developed. Thus the Cauchy stressis conjugate to the Weyl transformed metric tensor of the deformed configuration. The evolu-tion equation for the Weyl one-form is consistent with the two laws of thermodynamics. Ourtemperature evolution equation, which couples temperature, deformation and Weyl one-form,follows from the first law of thermodynamics. Using the model, the self-stress generated by apoint defect is calculated and compared with the linear elastic solutions. We also obtain condi-tions on the defect distribution (Weyl one-form) that render a thermo-mechanical deformationstress-free. Using this condition, we compute specific stress-free deformation profiles for a classof prescribed temperature changes.
The inelastic response of a material body is brought about by anomalies in their internal structure –what is loosely referred to as ’geometric frustration’. In materials with structural order (crystallinematerials, to wit), these anomalies are identified with dislocations and disinclinations. Anomalies ∗ [email protected] † [email protected] ‡ [email protected]
1n structured solids have been studied using tools from differential geometry over a long time. Thework by Bilby et al. [2] is a classical example where geometric tools were employed to study dislo-cations in a crystalline solid. They addressed the technologically important problem of describingdislocations using a continuous field and constructed a geometric space with an asymmetric affineconnection whose torsion was related to the density of dislocations; see [14] for a differential ge-ometric description. To generalize such a description of defects, Wang [24] introduced the notionof a material connection, which was characterized using the material uniformity field. The par-allel transport of vectors between different tangent spaces was aided by this material uniformityfield. Invariants of the material connection describing the intrinsic geometry of the material body(torsion, curvature etc.) were related to the density of defects.Inelasticity at the continuum scale, on the other hand, is often modelled with internal vari-ables. Phenomena such as thermo-elasticity, visco-elasticity, visco-plasticity, growth, damage etc.fall within the scope of such a description; a brief introduction to a class of these theories may befound in Gurtin et al. [12]. These theories introduce scalar, vector or tensor internal variables, withlittle or no geometric (kinematic) interpretation. As a consequence, the deformation field is oftencoupled with these internal variables somewhat arbitrarily. Moreover, the idea of an intermediateconfiguration, frequently used with these theories, is hard to appreciate from both physical andmathematical standpoints. The elasto-plastic decomposition, used in finite deformation plasticity,is an example that exploits the notion of an intermediate configuration via a multiplicative de-composition of the deformation gradient; see [20] for a brief historical review. As the geometry ofthe intermediate configuration is opaque, its introduction also corrupts notions of differentiationfor vector and tensor fields. On the whole, the clarity offered by geometrically motivated micro-mechanical theories of Kondo, Bilby and others is often lost in such a description of inelasticity atthe continuum scale.The assumption of a fixed relaxed configuration has been questioned before by Eckart [7] .He used an anelasticity tensor to characterize the anelastically deformed solid; this tensor wasdetermined based on an evolution equation for the metric tensor.More recently Rajagopal andSrinivasa [18] used a notion of evolving natural configurations to describe inelastic response ofa variety of materials. On similar lines, a geometrically founded continuum thermo-mechanicaltheory was proposed by Stojanovitch and co-workers [22]. It was based on the decompositionof the deformation gradient into elastic and thermal parts. The individual parts were assumedto be non-integrable, even as the total deformation gradient remained integrable. A frame fieldhaving non-zero anholonomy was defined using the thermal part of the deformation gradient. TheWeitzenbok connection was constructed by demanding the frame fields to be parallel. It should bementioned that the connection utilized by Stojanovitch and co-workers was the same used by Bilbyand others to describe a dislocated solid. This modeling approach does not affect the metric onthe body manifold. This implies that the distance between material points do not change as thetemperature of the body is varied from the reference temperature; hence no strains are introduced.Because of this property, the Weitzenbok connection does not appear to offer an attractive routefor modelling thermo-elasticity. Ozakin and Yavari [17] have proposed an alternate geometricframework for thermo-elastic deformations. Their work is based on a hypothesized material orrelaxed manifold whose geometry is non-Euclidean. In such a configuration, a thermally strainedbody is stress free. They implement this hypothesis by allowing the metric tensor of the materialmanifold temperature dependent. Deformation now becomes a map between the material andspatial configurations. The difference in the geometry of spatial and material manifolds leads tostresses in the former. Using this setting, they obtain deformation and temperature fields whichdo not produce stresses. The condition for a temperature change to be stress free is worked out asthe vanishing of the curvature tensor of the material manifold. However, the free energy is not a2unction of the curvature of the material manifold. A major problem with the worldview of Ozakinand Yavari is posed by the material manifold – the premise that the material manifold is stress-freecannot be verified since the notion of stress for the material manifold is undefined. Moreover,having a configuration which is virtual (i.e. physically unrealizable) only increases the opacity ofthe theory. The geometric description of thermo-elastic bodies discussed in [17] has been extendedto include time evolution by Sadik and Yavari [21].Considerable effort has gone in the modelling of point defects, biological growth and thermalstrains through an internal variable perspective. Models belonging to this genre try to capture thelocal zones of expansion or contraction using internal variables. Cowin and Nunziato [5] introducedthe void volume fraction as an additional kinematic variable which followed a second order evolutionrule. Based on this, they demonstrated that a porous material could support two kinds of waves:one determined by its elastic properties and the other by properties related to porosity. Similarly,Garikipati et al. [9] developed a continuum formulation for point defects within the framework oflinear elasticity. Point defects were understood as centers of expansion or contraction. A formationvolumetric tensor was introduced to characterize point defects; the dipole tensor conjugate to theformation volumetric tensor described the forces required to keep the point defect in mechanicalequilibrium. The linear nature of the theory permitted the use of Green’s functions to determine thestresses due to the point defects. In a more recent work, Moshe et al. [16] has developed a metricdescription for defects in amorphous solids. Here, the absence of long range order in the materialmakes it impossible to define Burgers’ vector in terms of the torsion tensor. However, using theLevi-Civita parallel transport, the authors arrive at a notion for Burgers’ vector. They also showthat, in a two dimensional setting, a conformal change of the metric tensor is sufficient to representdislocation- and disinclination-like defects in amorphous solids. The question of restructurabilityof geometrically frustrated solids is addressed by Zurlo and Truskinovsky [29]. They develop asurface deposition protocol to additively manufacture these residually stressed bodies. Yavari andGoriely [28] formulate a geometric theory for point defects. They postulate that the stress-freeconfiguration of a body with point defects may be identified with a Weyl manifold. The density ofdefects in the body is defined as the deviation of the volume form of the material manifold from theEuclidean volume form. Using this, they compute the stresses generated by a shrink-fit in the finitedeformation setting. To represent the shrink fit, the volume form is assumed to be discontinuousacross the shrink fit surface; this is however not in conformity with the structure of a differentiablemanifold. They also employ the usual equilibrium equations without explicating on its variationalstructure, even though the divergence of stress depends on the Weyl connection. Moreover, withthe focus being only on analytical solutions, the constitutive framework adopted by Yavari andGoriely remains somewhat simplistic. For instance, no information on distant curvature was madeuse of in the model.Though an important problem, the modelling of the mechanical response of solids with metricaldefects and temperature change is inadequately addressed in the literature. By metrical defectswe mean the anomalies in the material body, which modify the local notion of length. Pointdefects and growth are examples of metric anomalies. A key component in building such a theoryis a geometric space that consistently incorporates the modified notion of length due to defectsand temperature change. This is what we set out to pursue, using a Weyl geometric setting.Weyl manifolds are geometric spaces where the metric is incompatible with the connection. Thisincompatibility has previously been exploited to describe point defects [28, 19]; however there areconsiderable differences in our present worldview. The Weyl transform which keeps the connectioninvariant but alters the metric by a positive factor is now thought of as a modification introduced inthe configuration of the body due to temperature change. Identifying the reference and deformedconfigurations as Weyl manifolds make it possible to represent defects and temperature changes in3 consistent manner. This modeling perspective also has an important advantage: it completelyavoids the mysterious intermediate configuration. The inelastic response of the body arising outof the defect distribution is buried within the connection associated with the configuration. Animportant consequence of having a connection different form the flat Levi-Civita is that the criticalpoints of the stored energy function are non-trivially modified. This modification may be exploitedto represent configurations which are residually stressed. This fact, to the best of our knowledge,remains unexplored in the literature on defect mechanics. The description has a parallel withthe idea of ’pseudo-’forces encountered in particle mechanics; these forces arise only when thecomponents of the connection are non-trivial. This approach is used to predict the stresses createdby a diffused shrink fit. The prediction of the radial stress by our methodology is found to be inline with that of the linear elastic solution; however the hoop stress is found to be quite different.The deviation in the hoop stress is essentially due to the diffused representation of the shrink fit.Further, the model is applied to arrive at a condition for a body to be in a state of zero stressunder a prescribed temperature change. Using this condition we also recover a well know stressfree configurations known from linear elasticity.This article is organized into eight sections and an appendix. Section 2 gives a brief introductionto a Weyl manifold and discusses its key invariants. Kinematics of a body with metrical defectsare discussed in section 3. Kinetics required to describe the dynamics of the defective body areconsidered in section 4; an important aspect being the adaptation of the classical Doyle-Ericksenformula within the Weylian setting. The equilibrium equation of the Weyl one-form is also derivedin this section. In section 5, restrictions imposed by the laws of thermodynamics on the constitutiverelations are discussed. Restricting the theory only to local equilibrium considerations, self stressesgenerated due to a point defect are discussed in section 6. Conditions for a configuration to bestress free state is discussed in section 7. Section 8 concludes the article with a few observationsand comments on further possible applications of the present model. The appendix briefly discussesthe numerical procedure used to solve the diffused shrink fit problem.
The defining hypothesis of a Riemannian manifold is that the positive definite metric is preservedunder parallel transport. H Weyl [26, 25], in an effort to include electromagnetism within theframework of general relativity, made the following hypothesis: the parallel transport of the metricalong a curve is proportional to the metric itself. In other words, the ratios of length betweendifferent tangent spaces also need to be parallel transported. This hypothesis introduces an addi-tional degree of freedom which generalizes the Riemannian geometry by an independent scale ateach tangent space of the manifold. Even though Weyl’s relativity did not succeed in unifying gen-eral relativity and electro-magnetism, the resulting mathematical tool was used to describe otherphysical phenomena (for recent developments in Weyl relativity, see [1]). We now briefly reviewthe construction of the Weyl geometry. Let M be an n dimensional differentiable manifold with aRiemannian metric g and γ be a curve in M given by, γ : (0 , → M . Using coordinates, the curvemay be given by ( x ( t ) , .., x n ( t )) ∈ M , t ∈ (0 , ddt g ( V, W ) = φ ( γ ′ ( t )) g ( V, W ) (1)
V, W ∈ T x M , where T x M denotes the tangent space at x ∈ M and γ ′ ( t ) is the vector field tangentto γ at ( x ( t ) , .., x n ( t )). The one-form φ is introduced to describe the scale degrees of freedomat each tangent space. Note that as φ is set to zero, Eq. 1 reduces to the defining hypothesis4f Riemannian geometry. We now formally define a Weylian manifold to be the triplet ( M , g , φ ),consisting of a differential manifold M , a Riemannian metric g and a one-form φ . On integratingEq. 1 along the curve γ , we arrive at a relation between inner-products at T x ( γ (0)) M and T x ( γ ( t )) M given by, g ( V ( t ) , W ( t )) = g ( V (0) , W (0)) e R t φ ( γ ′ ( t )) dt . (2)We denote the length of a vector V by l = g ( V, V ). To obtain the variation of l along C , we set V = W in Eq. 2, leading to the following relationship, l ( s ) = l (0) e R t φ ( γ ′ ( t )) dt (3)If we choose the curve to be closed, i.e. γ (0) = γ (1), the transport of the inner-product given inEq. 2 becomes, g ( V (1) , W (1)) = g ( V (0) , W (0)) e H φ ( γ ′ ( t )) dt (4)Using Stokes’ theorem for the line integral, the equation above may be written as, Z Ω dφ = I γ φ (5)where, Ω is the area enclosed by the closed curve γ . If the line integral in Eq. 4 vanishes for anyclosed curve γ , then we have dφ = 0. Now using Poincar´es’ lemma, we have an exact Weyl one-form φ : I φ ( γ ′ ( t )) dt = 0 = ⇒ φ = df (6)where f is a scalar valued function and df denotes the differential of f . Thus integrability ofthe Weyl one-form ensures that the parallel transport of the inner-product is path independent.We also define an integrable Weyl manifold as one whose metric has a path independent paralleltransport. Formally, an integrable Weyl manifold is denoted by ( M , g , df ). If φ does not satisfythe integrability condition given in Eq. 6, then the parallel transport is path dependent; we callsuch a manifold non-integrable Weyl. An affine connection is an additional structure defined on a smooth manifold, using which one candifferentiate vector and tensor fields. It also defines a notion of parallel transport on a tangentbundle. On a differentiable manifold, one can define infinitely many connections. However, weconsider the unique torsion free connection which is natural to Weyl’s hypothesis. In terms ofcovariant derivatives, Weyl’s hypothesis may be written as, ∇ g = φ g (7)where, ∇ ( . ) is the covariant derivative relative to the Weyl connection. In component form, theequation may be written as, g ij ; k = φ k g ij (8)We introduce a new tensor field Q := φ ⊗ g , called the non-metricity tensor, which describes theincompatibility of the Weyl connection with the metric. In the equation above, g ij ; k denotes thecovariant derivative of the metric tensor. With respect to an arbitrary connection, g ij ; k may bewritten as, g ij ; k = ∂ k g ij − Γ lki g lj − Γ lkj g il (9)5ere Γ kij denotes the Christoffel symbol of the second kind. The assumption that the connectionis torsion free implies a symmetry condition on the lower two indices of the Christoffel symbol,Γ kij = Γ kji . Using Eq. 9 in Eq. 8, along with the torsion free assumption leads to the followingexpression for the coefficients of the Weyl connection.Γ lmi = 12 g kl ( ∂ m g ik + ∂ i g km − ∂ k g mi ) −
12 ( φ m δ li + φ i δ lm − φ j g mi g jl ) (10)If the Weyl one-form is integrable then we have,Γ lmi = 12 g kl ( ∂ m g ik + ∂ i g km − ∂ k g mi ) −
12 ( ∂ m f δ li + ∂ i f δ lm − ∂ j f g mi g jl ) (11)Note that for any Weyl manifold (integrable or non-integrable), the connection may be written asthe sum of the Levi-Civita connection and a (1,2) tensor. This (1,2) tensor is determined by themetric and the Weyl one-form. Another important property of a Weyl connection is the invarianceunder Weyl transformation. A Weyl transformation is defined by, g ij → e s g ij ; φ → φ + ds (12)where, s is a real valued function. One may immediately verify the invariance of the connection bysubstituting Eq. 12 in Eq. 10. However, if we define ¯ g ij := e s g ij , ¯ φ := φ + ds and substitute themin Eq. 9, we arrive at the following, ¯ g ij ; k = ¯ φ k ¯ g ij (13)The invariance of the Weyl connection under Weyl transformation is a cornerstone in ourthermo-mechanical theory; section 3 discusses this aspect in detail. The notion of an integrable Weyl manifold was established at the beginning of this section. We nowdiscuss its equivalence with a Riemannian manifold. This equivalence is established by showing thatWeyl’s connection boils down to the Levi-Civita connection when the Weyl one-form is exact. Forthe sake of completeness we record the Levi-Civita connection induced on a Riemannian manifoldby the metric g . The Christoffel symbols associated with the Levi-Civita connection are given by,Γ lmi = 12 g kl ( ∂ m g ik + ∂ i g km − ∂ k g mi ) (14)By setting the Riemannian metric as e f g and computing the Christoffel symbols of the associatedLevi-Civita connection, we are led to the connection coefficients of an integrable Weyl connectiongiven in Eq. 11. This result implies that an integrable Weyl manifold is a Riemannian manifoldwith a modified metric. On a Weyl manifold, the presence of a metric permits us to define the arc length for any curve onthe manifold. This is similar to the case with Riemannian manifolds. Let γ be a curve on M ; theexpression for the arc-length of γ is given by, l = Z γ e s s φ (cid:18) dγdt (cid:19) g (cid:18) dγdt , dγdt (cid:19) dt (15)6 ( ., . ) is the Riemannian metric associated with a Weyl manifold, dγdt is the vector field tangent to γ and s is the variable of the Weyl transformation. Note that s weights the metric at each pointon the curve. We choose the following three-form as the volume form, d V = e ns √ gdx ∧ ... ∧ dx n (16)where n is the dimension of the manifold and g the determinant of the Riemannian metric. Notethat the volume-form is not compatible with the Weyl connection. Apart from non-metricity, theWeyl connection may also have curvature. As with a Riemannian manifold, the curvature operatoris defined as the non-commutativity of second covariant derivatives. R ( X, Y )( Z ) = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z (17)In the equation above, X, Y and Z are vector fields on M . The curvature operator is linear in allits input arguments; it is also anti-symmetric in X and Y . If one writes the curvature operator ina coordinate (holonomic) basis, the last term in the above equation vanishes. The components ofthe curvature tensor are given by, R likj = ∂ k Γ lji − ∂ j Γ lki + Γ lka Γ aji − Γ lja Γ aki (18)The Ricci curvature is given by, R ij = R kikj (19)In contrast to the Riemannian case, the Ricci tensor of a Weyl manifold is unsymmetric and thusmay be decomposed into its symmetric and antisymmetric parts. R ij = ˆ R ij + K ij (20)ˆ R ij and K ij respectively denote the symmetric and antisymmetric parts of the Ricci tensor. Theantisymmetric tensor K is sometimes called the distant curvature. It is given by the exteriorderivative of the Weyl one-form. K = dφ (21)For an integrable Weyl manifold, the distant curvature vanishes making the Ricci tensor symmetric;this follows from the identity dd = 0. Using the symmetric part of the Ricci tensor, the followingscalar curvature can be defined, R = g ij R ij (22)Note that R does not depend on the distant curvature. The scalar measure extracted form theanti-symmetric distant curvature is K = g ik g jl K ij K kl (23)Using the machinery of Weyl manifolds discussed so far, we now proceed to develop a thermo-mechanical theory for a solid body with metrical defects. We assume the reference and deformed configurations to be Weylian manifolds. These configura-tions are respectively denoted by the triplets ( B , G , ξ ) and ( S , g , φ ), where B and S are smooththree-manifolds. The metric tensors of the reference and deformed configurations are denoted by g and G and the associated Weyl one-forms by η and φ respectively. We introduce an additionalkinematic variable in the deformed configuration called the Weyl scaling, which is denoted by s .7 is identified as the variable of Weyl transformation discussed in the previous section. The Weyltransformation modifies the deformed metric by scaling it by a positive scalar and changing theone-form by an exact one-form ds . In this work, we assume that Weyl transformation modelsthe thermal strain introduced in the body due to temperature change. Since the Weyl connectionis invariant under Weyl transformation, the Weyl scaling does not modify the connection. How-ever the metric and the non-metricity are modified due to temperature change. This modificationrepresents the change in the strain field and defect distribution in the material body. The Weyltransformed metric and the one-form of the deformed configuration are denoted by ¯ g := e s g and¯ φ = φ + ds respectively. We relate s to the temperature field of the deformed configuration throughthe following relationship, s = α ( T − T ) . (24)The equation above is a constitutive relation chosen for simplicity, although other relationshipsare possible. The variables α , T and T respectively denote the co-efficient of thermal expansion,temperature field and reference temperature of the body. Throughout this work, we assume themetric tensors G and g as Euclidean. The deformation map relating the reference and deformedconfigurations is given by, ϕ : B → S . (25)The derivative of the deformation map or the tangent map is denoted by F ; it maps tangentvectors from the reference configuration to those of the deformed configuration. If we denote theco-ordinates of the reference and deformed configurations by ( X , ..., X ) and ( x , ..., x ), then thedeformation gradient can be expressed as, ∂∂x i = F Ii ∂∂X I (26)where, ∂∂X I and ∂∂x i are the coordinate bases for T X B and T ϕ ( X ) S respectively. These basis vectorsmay be identified with vectors tangent to the coordinate lines at each point of the manifold. When-ever the deformed configuration has a metric tensor, the right Cauchy-Green deformation tensor C : T X B × T X B → R can be defined by pulling back the metric tensor of S to B : C = ϕ ∗ g . (27)In this equation, ϕ ∗ ( . ) denotes the pull-back map. The covariant components of C are thus givenby, C IJ = F iI g ij F jJ (28)As the temperature of the body changes from the reference, the metric and the Weyl one-form ofthe deformed configuration change via the Weyl transformation. In the presence of thermal strains,the Cauchy-Green deformation tensor denoted by ¯ C is given by,¯ C IJ = F iI ¯ g ij F jJ = e s F iI g ij F jJ (29)From now on, we will use the expression in Eq. 29 as the definition of the right Cauchy-Greendeformation tensor. Note that Eq. 29 reduces to the usual definition of the right Cauchy-Greendeformation tensor when s is set to zero. This pertains to the case of no temperature change inthe body. Components of the Green-Lagrangian strain in the presence of temperature change aregiven by, E IJ = 12 ( ¯ C IJ − G IJ ) (30)8s a consequence of Weyl transformation, the principal stretches are scaled by exp (cid:0) s (cid:1) . If λ i arethe principal stretches of C , it follows that,¯ λ i = exp( s ) λ i (31)¯ λ i are the principal stretches of ¯ C . The invariants of ¯ C and C are related in the following way,¯ I = exp ( s ) I ; ¯ I = exp (2 s ) I ; ¯ I = exp (3 s ) I , (32)Here, I i and ¯ I i are the principal invariants of C and ¯ C respectively. In the literature, the Green-Lagrangian strain tensor is sometimes defined as half the difference between the deformed and thereference metrics (see [10], [8]). Although this definition might make sense in a small deformationsetting, it is problematic since the addition of metric tensors is not well defined. To be more precise,the reference metric operates only on tangent vectors of the reference configuration, while thedeformed metric operates on tangent vectors of the deformed configuration. In some cases, by the’deformed metric’, the pulled-back metric is implied. The distinction between the deformed metricand the pulled back metric is quite important in a geometrically motivated continuum theory, sincethe metric tensor may be construed as a separate dynamical field describing the intrinsic geometryof the body.The Jacobian of deformation in the presence of temperature change is denoted by ¯ J and is givenby, ¯ J = exp (cid:18) s (cid:19) r gG det F (33)where, g and G denote the determinants of the reference and deformed metric tensors. The non-metricity tensors of the reference and deformed configurations are denoted by H and Q respectively.These tensors are defined by, H = ξ ⊗ G ; Q = φ ⊗ g (34)In this work, the non-metricity tensor of the deformed configuration completely characterizes theresidual stresses developed in the body due to inelastic effects. The initial configuration mayalso be trivially viewed as a deformed configuration with identity map as deformation. With thisunderstanding, the tensor H has significance well beyond what the metric tensor of the initialconfiguration offers. Specifically, as a conjugate to the kinematic variable ξ , one may talk aboutresidual stresses using just one configuration. This may be contrasted with elastic stresses (stressesdue to deformation) whose description necessitates both the reference and deformed configurations.The pull-back of the deformed configurations Weyl one-form, denoted as φ ′ ∈ T ∗ B , is given by, φ ′ = ϕ ∗ φ (35)Similarly, the pulled-back non-metricity tensor Q ′ is defined by Q ′ := ϕ ∗ ( Q ), which leads to thefollowing expression, Q ′ = φ ′ ⊗ ¯ C (36)This expression is obtained by applying the pull-back map to the Weyl one-form and the metrictensor. Similarly, the pull-back of the distant curvature is denoted by K ′ := ϕ ∗ ( K ). Since thepull-back map commutes with exterior derivative, we can write K ′ as, K ′ = ϕ ∗ dφ = d ( ϕ ∗ φ )= dφ ′ (37)9he spatial rate of deformation tensor d , characterizes the relative velocity of the material in asmall neighbourhood around a point in the deformed configuration and is given by, d = 12 L v g (38)where, L v ( . ) denotes the Lie derivative. When the geometry of the material body is Riemannian,the formula above reduces to d ij = ( v i ; j + v j ; i ). An index after a semicolon denotes the covariantderivative (given by the appropriate connection). The Lie derivative of the metric tensor is, L v c ∂ c ( g ab ) = v c ∂ c g ab + g cb ∂ a v c + g ca ∂ b v c (39)By adding and subtracting γ cda v d and γ cdb v d in the last equation and using the definition of covariantderivative, we arrive at, L v c ∂ c ( g ab ) = v c ∇ c g ab + g cb ∇ a v c + g ca ∇ b v c (40)Now, using the properties of Weyl connection, the equation above may be rewritten as, L v c ∂ c ( g ab ) = φ c v c g ab + g cb ∇ a v c + g ca ∇ b v c (41)= φ c v c g ab + ∇ a v b + ∇ b v a − ( φ a v b + φ b v a ) (42)where, v a = g ab v b and the relation ∇ a v c = ∇ a g cb v c + g cb ∇ a v c have been used. Using Eq. 42, therate of deformation tensor may now be computed. Substituting Eq. 42 in Eq. 38, we arrive at rateof deformation tensor of a body whose configurational geometry is Weylian. d ij = 12 (cid:16) v i ; j + v j ; i − ( φ i v j + v i φ j ) + φ k v k g ij (cid:17) (43)It should be noted that the rate of deformation tensor for a body with Weylian geometry has threeadditional terms compared with the Riemannian case. These additional terms are a consequenceof the incompatibility of the connection with the metric. The connection of the reference configuration is assumed to be time independent, while that of thedeformed configuration evolves with time. The connection coefficients of reference and deformedconfigurations are denoted by Γ
CAB and γ cab respectively. If the reference configuration has a non-trivial connection (different from the flat Levi-Civita), then the body has a distribution of defects,which may require residual stresses for mechanical equilibrium to be maintained. The connectionof the deformed configuration is determined by the evolution of the Weyl one-form, which in turnis coupled to the evolving temperature field. This evolution models the generation, motion andextinction of metrical defects due to externally applied thermo-mechanical stimuli. The balance of linear momentum in the reference configuration is given by, ∇ . P + B = ρ d V dt (44)Here, P is the first Piola stress and ∇ ( . ) is the divergence operator determined by the Weylconnections of the reference and deformed configurations. The Piola stress P is related to the10auchy stress σ through the Piola transform (which is applied to the second index of the Cauchystress). We define the symmetric Piola (second Piola) stress as the pull-back of the first index ofthe first Piola stress. The relation between the Cauchy stress and second Piola stress is thus givenby, S = J ϕ ∗ ( σ ) (45) The equilibrium condition for the Weyl one-form is obtained as a critical point of the free-energywith respect to the one-form φ . In the reference configuration, this condition may be written as, δ φ ˆ ψ = 0; δ φ ˆ ψ := δ φ Z B ψ dV (46)where, ˆ ψ is the free energy of the body and δ φ ( . ) denotes the Gˆateaux or the variational derivative.Assuming that the free energy depends on F , Q ′ , K ′ and T , its variation may be evaluated as, δ φ ψ = ∂ψ ∂ F : δ φ F + ∂ψ ∂ Q ′ ˙: δ φ Q ′ + ∂ψ ∂ K ′ : δ φ K ′ (47) δ φ ( A ) denotes the directional derivative of the tensor A with respect to φ . Since the deformationgradient does not depend on φ , we have δ φ F = . We also have δ φ K = dδφ and hence δ φ K ′ = d ( δφ ) ′ ;here ( δφ ) ′ := ϕ ∗ ( δφ ). Similarly, δ φ Q ′ = ( δφ ) ′ ⊗ ¯ C . We also introduce the following definitions fora simpler exposition, M := ∂ψ ∂ Q ′ ; N := ∂ψ ∂ K ′ (48)Using these results and definitions, the variation of the free energy may be written as, Z B (cid:16) M IJK C JK ( δφ ) ′ I + N IJ ∂ [ I ( δφ ) ′ J ] (cid:17) dV = 0 (49)Applying the relation between ( δφ ) ′ and δφ and using the divergence theorem on the second term,we arrive at, ∇ J N IJ − M JKI ¯ C JK = 0 (50) In continuum mechanics, the Doyle-Ericksen formula [6, 15] is an important result, which relatesstress (Cauchy stress) with the metric of the deformed configuration. Through the metric, thisformula is thus able to relate the intrinsic geometry of the deformed configuration to the stressesdeveloped. The Doyle-Ericksen formula is given by, σ ij = 2 ρ ∂ψ∂g ij (51) ρ and ψ are the mass density and free energy density of the deformed configuration. An interestingaspect of the formula is that it emphasizes length as a fundamental quantity. If the last statementis understood abstractly, then there is no reason to restrict length just to its Euclidean notion. Toobtain a relationship between the Cauchy stress and the scaled metric ¯ g , we replace g in Eq. 51by ¯ g , so the desired relation is given by, σ ij = 2 ρ ∂ψ∂ ¯ g ij (52)11n the reference configuration, the equation above provides a relation between the second Piolastress and the right Cauchy-Green deformation tensor, S IJ = 2 ρ ∂ψ ∂ ¯ C IJ (53)Note that this equation is related to Eq. 52. To prove the equivalence, one starts by noticing that, ϕ ∗ ∂ψ ∂ ¯ C IJ = ∂ψ∂ ¯ g ij (54)Moreover, the second Piola stress is related to the Cauchy stress though the push-forward of its firstindex and an inverse Piola transform applied to the second index. Formally, these two operationsmay be written together as, σ ij = 1 J ϕ ∗ ( S IJ ) (55)Eqs. 54 and 55 together lead us to the required equivalence. In the present context, the notionof stress is understood in the sense of Eq. 51. The positive scalar factor introduced in the metricby the Weyl transformation does not create a new notion of stress, unlike what was conceivedof by Gurtin. Gurtin in [11] postulated the existence of new stresses called micro-stresses anda new balance rule called the micro-force balance. These stresses were shown as energeticallyconjugate to the internal variables introduced to describe the inelastic deformation. Moreover, thedivergence-type balance law for the internal variables was restrictive and a physical meaning forthese micro-stresses remained elusive. Identifying thermal strains as a geometric object permitsus to reinterpret the classical geometric results seamlessly. This is often impossible with internalvariable theories, which require additional postulates for closure. In linear thermo-elasticity, the Duhamel-Neuman hypothesis is commonly adopted to model themechanical response of solids in the presence of temperature change. It postulates the existence of anew tensor-valued internal variable called the thermal strain whose evolution is directly proportionalto the temperature change in the body. The total stain ǫ Tot , defined as the symmetric part of thedisplacement gradient, is recovered by an additive splitting, ǫ Tot = ǫ Mec + ǫ The . (56)The mechanical and thermal parts of the strain are denoted by ǫ Mec and ǫ The respectively. For abody admitting isotropic thermal expansion, ǫ The relates to the change in temperature ( T − T ) inthe following way, ǫ The = α ( T − T ) I (57)Several extensions of this additive splitting to finite deformation thermo-elasticity exist in theliterature. Splitting of the deformation gradient in thermal and elastic parts comes in two forms: F = F e F T and F = F T F e . Implicit in these settings is the notion of an intermediate configurationto describe the incompatibility created by thermal deformation. Yet another extension of theDuhamel-Neuman hypothesis to finite deformation is through an additive decomposition of therate of deformation tensor into mechanical and thermal components d = d e + d T (for details see[15]), where, d = 12 ρ (cid:20)(cid:18) ∂ χ∂ τ : L v τ (cid:19) + ∂ χ∂ τ ∂T ˙ T (cid:21) . (58)12nd d e and d T are given as, d e = ρ (cid:18) ∂ χ∂ τ : L v τ (cid:19) ; d T = ρ ∂ χ∂ τ ∂T ˙ T (59)Here, χ denotes the complementary free energy and τ the Kirchhoff stress. The thermo-elasticsplitting given in Eq. 58 is verifiable using Legendre transform; hence it is not a hypothesis. Itonly requires the existence of a temperature dependent free energy. In any case, the splitting ofthe rate of deformation tensor cannot account for the incompatibility due to thermal strain sincethe geometry of the body remains essentially Euclidean.With our present formulation, we are able to avoid the notion of an intermediate configurationwhilst incorporating the incompatibility due to thermal strains in a consistent manner. The rateof deformation tensor ¯ d = L v ¯ g may be written as,¯ d = e s d + 12 v [ s ]¯ g (60)The equation above follows from the properties of the Lie derivative, v [ . ] denotes the action of avector field on a function, d := L v g and ¯ d := L v ¯ g . The current geometric methodology has theadded advantage that the rate of deformation tensor may be determined kinematically, without theuse of inverse elastic relations. Neither complementary free energy nor Legendre transform is usedto arrive at the equation. However, the decomposition given in Eq. 58 may also be adopted in thepresent setting. We use a local equilibrium thermodynamics framework to determine the restrictions imposed onthe constitutive functions by the laws of thermodynamics. In line with the standard hypothesis,we postulate the existence of state variables called the specific internal energy density and specificentropy, denoted by U and η respectively. U is assumed to depend on the right Cauchy-Greendeformation tensor, non-metricity tensor, distant curvature and specific entropy. U may also dependon co-ordinates of the reference configuration; however we choose to work with an internal energywhich is homogeneous, i.e. the explicit spatial dependence is ignored. The balance of energy, whichis the statement of the first law of thermodynamics, may be written as, ddt Z B ρ (cid:18) U + 12 G ( V , V ) (cid:19) dV = Z B h B , V i dV + Z ∂ B h t , V i dA − Z ∂ B h q , n i dA (61)In the above equation, h ., . i denotes the natural pairing between cotangent and tangent vectors.The heat flux vector and the unit vector normal to the boundary ∂ B are denoted by q and n respectively. Kanso et al. [13] have interpreted the stress tensor as a bundle valued two-form.Such a description has the advantage of having the integrals in the balance of energy consistentwith the integration of differential forms. However, introducing such notions of stress do not affectthe resulting equations; hence we do not dwell on such technical details here. The temperaturegradient of the deformed configuration is given by ¯ g ij ∂ j T . To determine the heat flux vector of thereference configuration, one may adopt one of the following two routes. In the first, one exploits theconstitutive relation between temperature gradient and heat flux in the deformed configuration anduse the Piola transform to bring the heat flux vector to the reference configuration. Alternatively,one may use the Piola transform on the temperature gradient of the deformed configuration and13se the referential version of the constitutive rule connecting heat flux and temperature gradient.The two routes are equivalent if the constitutive rule is transformed carefully.The equilibrium equation for the one-form given in Eq. 50 does not account for time depen-dent relaxation experienced by defects. We model time dependent relaxation by adjoining theequilibrium equation for the Weyl one-form with an additional viscous term. Such a procedure toobtain the evolution rule for states that relax to equilibrium has been employed in arriving at theCahn-Hillard and Ginsberg-Landau equations. Geometric evolutions like the Ricci flow and meancurvature flow equations also have a similar variational structure. The equilibrium equation for theone-form with relaxation may be written as, L V ( φ ′ ) = 1 m δ φ ′ ˆ ψ (62) L V ( . ) denotes the Lie derivative with respect to the velocity field, m is a constant describing thetime dependent relaxation experienced by defects. In addition to the balance of energy for thewhole body, we also postulate that a local form of energy balance holds (the fields used in theintegral statement of Eq. 61 are assumed sufficiently smooth so that localization theorem can beapplied), which may be written as, ρ ˙ U = 12 S IJ ˙¯ C IJ + N IJ ˙ φ ′ [ I,J ] + M IJK ¯ C JK ˙ φ ′ I − ∂ I q I + ρ R − m ˙ φ ′ I ˙ φ ′ I (63)where, R denotes the heat source and ( . ) [ I,J ] the anti-symmetrization with respect to the indices I and J . The last equation is obtained from Eq. 61 through the use of the balance of linear momen-tum, divergence theorem and localization theorem. We impose the second law of thermodynamicsthrough the Clausius-Duhem inequality; its local form is given by, ρ ˙ η ≥ ρ RT − ∂ I (cid:18) q I T (cid:19) (64)Applying the Legendre transform ψ = U − T η with respect to the conjugate variables η and T ,we arrive at, ˙ η = 1 T ( ˙ U − ˙ ψ − ˙ θη ); η = − ∂ψ ∂T . (65)The reference free energy density is assumed to be a function of the right Cauchy-Green deformationtensor, non-metricity tensor, distant curvature tensor and temperature. The rate of Helmholtz freeenergy density is calculated as,˙ ψ = ∂ψ ∂ ¯ C IJ ˙¯ C IJ + ∂ψ ∂Q ′ IJK ¯ C JK ˙ φ ′ I + ∂ψ ∂Q ′ IJK φ ′ I ˙¯ C JK + ∂ψ ∂K ′ IJ ˙ φ ′ [ I,J ] + ∂ψ ∂T (66)In arriving at the equation, use is made of the relation ˙ Q IJK = ˙ φ ′ I ¯ C KL + φ I ˙¯ C KL . Using therelationship between entropy, internal energy and free-energy density in Eq. 64, we have thefollowing form of second law, ρ ( ˙ U − ˙ ψ − ˙ T η ) + ∂ I q I − q I T ∂ I T − ρ R ≥ (cid:18) S IJ − ρ (cid:18) ∂ψ ∂ ¯ C IJ + ∂ψ ∂Q ′ KIJ φ ′ K (cid:19)(cid:19) ˙¯ C IJ + (cid:18) M IJK − ρ ∂ψ ∂Q ′ IJK ¯ C JK (cid:19) ˙ φ ′ I + (cid:18) N IJ − ρ ∂ψ ∂K ′ IJ (cid:19) ˙ φ ′ [ I,J ] − (cid:18) ∂ψ ∂T + η (cid:19) ˙ T − m ˙ φ ′ I ˙ φ ′ I − q I T ∂ I T ≥ S IJ = 2 ρ (cid:18) ∂ψ ∂ ¯ C IJ + ∂ψ ∂Q ′ KIJ φ ′ K (cid:19) (69)Similarly, we also have, M IJK = ∂ψ ∂Q ′ IJK ; N IJ = ∂ψ ∂K ′ IJ (70)The last two relations were established through a variational argument in the previous Section 4.1.Using these constitutive relations in the dissipation inequality leads to, − (cid:18) m ˙ φ ′ I ˙ φ ′ I + q I T ∂ I T (cid:19) ≥ m < q I = − L IJ ∂ J T , where L is the thermal conductivity which ispositive definite. The temperature evolution equation is obtained by substituting the constitutions for second Piolastress and entropy into the local form of energy balance. Using Eq. 65, the rate of entropyproduction ˙ η is given by,˙ η = − (cid:18) ∂ ψ ∂T ˙ T + ∂ ψ ∂ ¯ C ∂T : ˙¯ C + ∂ ψ ∂ ¯ Q ′ ∂T ˙: ˙¯ Q ′ + ∂ ψ ∂ ¯ K ′ ∂T : ˙¯ K ′ (cid:19) (72)substituting Eqs. 72, 65 and 69 into the energy balance leads to the following evolution equationfor temperature. − ρ T (cid:18) ∂ ψ ∂T ˙ T + ∂ ψ ∂ ¯ C ∂T : ˙¯ C + ∂ ψ ∂ ¯ Q ′ ∂T ˙: ˙¯ Q ′ + ∂ ψ ∂ ¯ K ′ ∂T : ˙¯ K ′ (cid:19) = −∇ . q + ρ R (73) An accurate prediction of residual stresses requires complete information on the distribution ofdefects. As assumed, the Weyl one-form of the deformed configuration encodes the metrical defectspresent in the body. This one-form may or may not evolve, depending on the thermo-mechanicalprocesses the body is subjected to. Assume the initial configuration B as unloaded and withoutdeformation, but with defects. The no-deformation assumption implies that F = I , whereupon itfollows that Cauchy, first and second Piola stresses are indistinguishable or, in other words, thePiola transform and pull-back operation are trivial (identities). P = S ; S = σ (74)The condition of no external traction or displacement on B implies that the residual stresses haveto be self equilibrating. σ IJ ; J = 0 (75)In addition, if we assume that residual stresses are obtainable from a stored energy function, thenthe stress tensor may be written as, σ IJ = 2 ρ ∂ψ R ∂G IJ (76)15here, ψ R is the free energy due to defects. The last equation is the Doyle-Ericksen formuladiscussed in the previous section. The metric tensor appearing in the formula is one on B . Theassumption that residual stresses are characterizable using a free energy, is grounded in the physicalreality that a residually stressed body, on being cut, would deform. This implies that the energystored in the body due to the presence of defects may be converted to strain energy through adeformation process. This property is often used in an experimental characterization of residualstresses through destructive testing. It should be noted that the free energy for the residual stressfield need not be the same as the stored energy used to determine a deformation process. Dis-tributions of the Weyl one-form and the stored energy ψ R together determine the residual stressdistribution on B .When B is subjected to deformation, the defects present in the body evolve, which along withthe deformation-induced stresses equilibrates the externally applied loads. Thus we postulate thatthe free energy of the body at a configuration has two components: one due to deformation andtemperature, and the other due to defects. Using a referential description, the component offree energy due to defects is written as a function of the pulled back non-metricity tensor and thepulled back distant curvature; we denote this component by ψ R . The deformation and temperaturedependent component of free energy is denoted by ψ ϕ ; it is assumed to be a function of thedeformation gradient, metric tensor of the deformed configuration and temperature. ψ = ψ ϕ ( F , g , T ) + ψ R ( Q ′ , K ′ ) (77)Assuming the Doyle-Ericksen formula to hold in the deformed configuration, the second Piola stressis now given by, S = ∂ψ ϕ ∂ ¯ C + ∂ψ R ∂ Q ′ ∂ Q ′ ∂ ¯ C (78)The first term on the right hand side is the second Piola stress caused by deformation, while thesecond term is due to the defects. Note that, when the Weyl one-form vanishes, so does the defectcomponent of the free energy leading to the usual expression for the second Piola stress. Similarly,when deformation vanishes, F = I and the right Cauchy-Green deformation tensor reduces to themetric of the deformed configuration leading to Eq. 76. Having dwelt on the thermodynamic framework and the form of the constitutive rules in theprevious subsections, we now make a specific choice for the constitutive relations. We assumethe deformation part of the free energy to be of the compressible Neo-Hookian type. Indeed, anyhyperelastic free energy function could be used in its place. The stored energy for a compressibleneo-Hookian material is given by, ψ ϕ = µ I − − µ log( ¯ J ) + λ J ) (79)Note that the principal invariants used in the stored energy density are Weyl transformed. For thedefect part of the free energy, we assume the following form, ψ R = 12 (cid:0) k ¯ Q ′ IJK G IL G JM G KN ¯ Q ′ LMN + k G KM G LN K ′ KL K ′ MN (cid:1) (80)In the last equation, k and k are material constants characterizing the defect-induced free energy.The defect free energy may be further simplified as, ψ R = 12 (cid:16) k ¯ C IJ ¯ C IJ φ ′ K φ ′ K + k K ′ MN K ′ MN (cid:17) (81)16n the above expression, the definitions ¯ C IJ = ¯ C KL G IK G JL and K ′ IJ = G IK G JL K ′ KL are used.The second Piola stress generated from such a free energy is given by, S IJ = µ G IJ + 12 ( − µ + λ log( ¯ J ))( ¯ C − ) IJ + k ¯ C IJ φ ′ K φ ′ K (82)Using Eq. 48, tensors M and N for the assumed free energy function are given by, M IJK = k ¯ C IJ φ ′ K ; N IJ = k K ′ IJ (83) We present the calculation of self stresses due to a point defect using the geometric theory con-sidered in the preceding sections. Presently, we ignore defect evolution and focus only on themechanical equilibrium. This description exploits the fact that the equilibrium configuration of abody with a nontrivial Weyl connection is different from that with a flat Levi-Civita connection.For a body with a hyperelastic stored energy and flat Levi-Civita connection, the identity deforma-tion is always a critical point under zero traction and zero boundary displacement. In other words,the reference configuration (identity deformation) is a natural state. In the present approach, theidentity deformation is stress-free but not the only critical point of the stored energy function: weconjecture that this critical point is unstable when the body is defective. In the presence of pointdefects, the connection is modified locally and the Weyl one-form encodes this information.In the linear elasticity setting, point defects are analogous to a spherical inclusion forced intoa spherical cavity of slightly different diameter. The difference in the volume of the cavity andinclusion results in stresses in the inclusion and the matrix surrounding it. The infinitesimal versionof the inclusion problem introduces the notion of a dipole tensor to characterize the pre-stress causedby the point defect. This description may be referred to as Eshelby’s method of eigenstrain in theinfinitesimal setting. For a recent review on modelling point defects based on the elasticity theory,see [4]. The accuracy of linear elastic predictions for point defects depends on how closely the dipoletensor represents the point defect and the dependence of the dipole tensor on the energy functional.Different approaches have been suggested for obtaining the dipole tensor from molecular dynamicsimulations of point defects. These methods use the atomistic stress, displacement or the Kansakiforce to deduce the elastic dipole tensor. A main disadvantage of the linear elastic approach topoint defects is that it cannot be extended to include nonlinear interactions.We now consider a spherical body with radius one and a point defect placed at the origin.As opposed to linear elasticity, the point defect is modelled by a one-form which is localized nearthe origin and reaches zero asymptotically away from the origin. This description is similar to ashrink-fit problem with a diffused shrink-fit surface. A spherical coordinate system is used for thereference and deformed configurations of the body; these coordinates are denoted by ( R, Θ , Ξ) and( r, θ, ξ ) respectively. We adopt the stored energy function given in Eq. 79 and set the coefficient k to zero. The Weyl one-form is assumed to be integrable. The metric tensor in the sphericalcoordinate system is given by, G IJ = R sin Ξ 00 0 R ; g ij = r sin ξ
00 0 r , (84)We choose f to be of the form, f = L − kR (85)17ere, L and k are parameters controlling the point defect distribution. We may mention that theequation above is a choice and other choices are possible permitting us to model different kinds ofpoint defects like extra matter and vacancy. Note that Eq. 85 depends only on the radial coordinate.We assume that no displacement or traction conditions are applied to the body. Because of thespherical symmetry, the equilibrium configuration is also assumed to be radially symmetric. Thisleads to the following expressions for the deformation gradient and right Cauchy-Green deformationtensor, F Ii = ∂r∂R ; C IJ = (cid:0) ∂r∂R (cid:1) r
00 0 r . (86)The trace of C , C − and the Jacobian of deformation are given by, G IJ C IJ = (cid:18) ∂r∂R (cid:19) + 2 (cid:16) rR (cid:17) ; G IJ ( C − ) IJ = (cid:18) ∂r∂R (cid:19) − + 2 (cid:16) rR (cid:17) − ; J = ∂r∂R (cid:16) rR (cid:17) (87)The equilibrium configuration for the assumed Weyl connection is obtained by solving the Euler-Lagrange equation for the stored energy; these equations are nothing but the equilibrium equationsgiven in Eq.44. We use a finite element based numerical procedure to minimize the stored energywhich is discussed in Appendix-A. The radial component of the Weyl one-form and the radialdisplacement at equilibrium are shown in Fig. 1. Fig.2 compares the radial and hoop stressescomputed using linear elasticity as well as the present methodology. The linear elastic solutionswere obtained by Teodosiu [23]. The stresses in the linear elastic case was predicted by assumingthe elastic constants of the inclusion and the matrix to be same; the values of λ and µ were assumedto be 1 and 0 . L and k were in Eq. 85 are assumed to be 1.35 and 17.5 respectively. The radial and hoop stress stressdistributions for the matrix and inclusion in the linear elastic case are given by [23], σ rr = ( − µCr ; 0 ≤ r ≤ r − µCr (cid:16) r R (cid:17) ; r ≤ r ≤ R (88) σ θθ = ( − µCr ; 0 ≤ r ≤ r µCr (cid:16) r R (cid:17) ; r ≤ r ≤ R (89)where, r is the radius of the cavity into which the inclusion is forced, R the radius of the sphere, C = − v ′ π ( µ K ′ ) and v ′ the difference in volume between the inclusion and the cavity. From Fig 2,it is seen that the radial stress predicted by the present method compares well with the linear elasticapproach, even though the former produces a milder gradient. This may be owing to a diffusedrepresentation of the interface between the inclusion and the matrix. The diffused representationmanifests in the hoop stress as well; the unphysical discontinuity in the hoop stress via linearelasticity is absent in our predictions. Higher hoop stresses are predicted by our method, which isessentially due to the smoothness enforced by the diffused representation of the shrink-fit interface.The diffused representation has also the important advantage that interfaces need not be trackedexplicitly. 18 Figure 1: Radial displacement at equilibrium for the assumed one-form df (a) Radial stress (b) Hoop stress Figure 2: Comparisons of radial and hoop stresses computed by the present method and linearelasticity. The linear elastic solution was obtained from [23]19
Stress-free configurations
We now discuss a framework to calculate non-trivial configurations which are in a stress-free state inthe presence of metrical defects. This problem is technologically important, since a manufacturingprocess may, in principle, be tuned to optimize the defect distribution such that during operationit is close to a state of zero stress. Arteries are a good example of systems which are known tooptimize their stress levels by accumulating extra-matter in the form of growth. In the contextof zero stress configurations, we ask if, for a given deformation, it is possible to put the body ina state of zero stress by a distribution of metrical defects. Using our thermo-mechanical theory,we obtain a condition for the one-form such that body is in a state of zero stress. Applying thiscondition, we compute non-trivial configurations which are in a state of zero stress for a giventemperature change. For the sake of analytical tractability, we assume the body to be in a stateof thermal equilibrium. The relaxation experienced by the defects to reach the equilibrium state isalso ignored. For the calculations performed in this section, we adopt the constitutive rule discussedin Section 5.3.We now deduce the constraint imposed on the one-form by the condition of zero stress; thiscondition is obtained by setting the second Piola stress to zero. Even though any stress measuremay be utilized to obtain the zero stress condition, expressing it in terms of the second Piola stressis more convenient as the free energy is postulated in the reference configuration. For the assumedfree energy, the zero stress condition is given by, µ G IJ + 12 (cid:0) − µ + λ ln( ¯ J ) (cid:1) ( ¯ C − ) IJ + k ¯ C IJ φ ′ K φ ′ K = 0 . (90)It should be noted that defects influence the stresses only through φ ′ I φ ′ I , which is nothing but themagnitude of the one-form given by G − ( φ ′ , φ ′ ). Contracting the last equation with the referencemetric tensor, we arrive at the following equation relating φ ′ I φ ′ I and the components of the rightCauchy-Green deformation tensor, φ ′ I φ ′ I = k ¯ C IJ ¯ G IJ (cid:20) µ (cid:0) − µ + λ ln( ¯ J ) (cid:1) ( ¯ C − ) IJ ¯ G IJ (cid:21) . (91)If the deformation is assumed to be known, then the equation above may be thought of as aconstraint on the Weyl one-form. The equilibrium distribution of the Weyl one-form may then bearrived at based on the minimization problem for φ given in Eq. 46 with the last equation actingas a constraint.On the other hand, if one assumes the defect density to be given and ask if there exists adeformation which renders the body stress free, Eq. 90 becomes a condition on the deformationgradient. However, the deformation gradient computed from the Eq. 90 need not be integrable,i.e. there may not exist a deformation map whose derivative is the F computed from Eq. 90.Integrability of F is guaranteed, if the anholonomy associated with the tangent vector in the rangeof F vanishes. If we denote the tangent vector in the range of F by { e , e , e } , then e i = F Ii ∂∂X I . (92)We are asking for the condition under which e i = ∂∂x i for some coordinates x i . The anholonomyassociated with the vector fields e i is given by,[ e i , e j ] = A kij e k (93)20 ., . ] denotes the Lie bracket between the vector fields. If all the Lie brackets vanish, then one canfind a coordinate system such that the each vector of the frame field can be described as a tangentto some coordinate line. In terms of the deformation gradient, the components of anholonomy maybe found as, A kij = F Ji ∂F kj ∂X J − F Jj ∂F ki ∂X J ! . (94)The condition for integrability of F may now be written as, A kij = 0 (95)For multiply connected bodies, additional conditions are required depending on the muti-connectednessof the domain. For the integrability of the deformation gradient in such a domain, see [27]. In thefollowing, we explore simple distributions of defects which keep the body in a stress free state. We specialize the zero stress equations obtained in the previous sub-section to incompatible strainscaused by temperature change. Assuming the temperature field to be given, the variable of Weyltransformation s is determined using Eq. 24. It is well established that not all temperaturedistributions can be made stress-free by applying deformation [3]. The necessary condition for abody with temperature change to be stress free is Eq. 90, with the one-form given by φ = d s .Substituting the condition above into Eq. 90 leads to, µ G IJ + 12 ( − µ + λ log( ¯ J ))( ¯ C − ) IJ + k ¯ C IJ G KL s ,L s ,K = 0 . (96)The last equation is an implicit relation for the deformation gradient. In addition, the integrabilityof F discussed in Eq. 95 also needs to be imposed.Now, we compute the deformed shape of the body with a constant temperature field imposed,i.e. s is a constant in Eq. 90. Assuming a Cartesian coordinate system for the reference anddeformed configurations, the metric tensors in these configurations may be written as δ IJ and e s δ ij respectively. From Eq. 96, we arrive at the following relation for the deformation gradient, F T F = 1 µe s (cid:18) µ − λs − λ log(det( F )) (cid:19) I . (97)We assume a solution of the form F = c Λ , Λ ∈ SO(3) and c ∈ R . Substituting these in Eq. 90,we arrive at the following equation for c , c = 1 µe s (cid:18) µ − λs − λ log c (cid:19) (98)The SO(3) part is irrelevant since we are looking at a body with no displacement and tractionconditions applied. If we set s = 0 in the above equation, it reduces to c + λµ log c = 1. Thesolution to this equation is c = 1. This solution recovers the reference configuration of the body.Non-trivial stress-free configurations for different values of s may be computed through a numericalroot finding technique, applied to Eq. 98. The values of c computed for different values s areshown in Fig. 3. The curve shown in Fig. 3 is slightly nonlinear; this is because of the nonlinearconstitutive rule assumed for stress. Having computed the deformation gradient, one needs toestablish that F is compatible. For the present scenario, compatibility is not an issue as the21 c Figure 3: Value of c for different values of s ; the material constants λ and µ are both taken as 1.deformation gradient is constant. The above calculation also establishes the classical result that,for an isotropic body subjected to constant temperature field and free-free boundary conditions,the deformation is volumetric and without stress. The constitutive relation for stress also plays amajor role in determining stress free configurations. It may so happen that certain constitutiverules would not permit any solution for a stress free configuration and constructing such constitutiverules should be an interesting exercise. The role played by constitutive inequalities is also worthexploration. We have reported a geometric theory for thermo-mechanical deformation of bodies with metricaldefects. We have exploited the geometric setting initially proposed by Weyl in the context ofgeneral relativity to represent bodies with metrical defects. These defects are identified with theincompatibility of the connection and the metric. Our proposal on the thermo-mechanics of defect-mediated deformation has completely dispensed with the notion of an intermediate configuration.The defect equilibrium, described by the Weyl one-form, is obtained as a critical point of the freeenergy. The Weyl transformation is used to model the interaction of incompatibilities introducedby temperature change. To include dissipation in the defect evolution, we introduced a viscous termin the defect equilibrium equation, which made the evolution of one-form a gradient flow. Using thelaws of thermodynamics, restrictions have been obtained on the constitutive rules. The importantproblem of relating stress and strains in a non-Euclidean setting is resolved via the Doyle-Ericksenformula.We have applied our theory to a diffused version of the shrink-fit problem which representspoint defects. With the equilibrium configurations computed using a minimization procedure, wehave shown that the shifted critical point of the stored energy in the presence of a Weyl connectionresults in residual stresses. This solution could also recover aspects of the linear elastic shrink-fit problem. Other than mathematical expedience, a diffused representation is also practicallymeaningful, say in the context of additive manufacturing processes that materials in the form ofthin layers. In addition, we have derived conditions for a defective body to be in a state of zerostress. Using this condition, we have been able to recover zero stress configurations resulting fromsimple temperature distribution. Also recovered in the process are the well known linear elastic22esults, albeit in a nonlinear setting.We have focused mostly on the geometric aspects in the model and addressed only a few simpleboundary value problems. We have left unaddressed the characterization of the Weyl one-formto a particular kind of point defect. This characterization can be done using molecular dynamiccalculations or from density function theory calculations. Such characterizations do exist for thedipole tensor used in the linear elastic case. A more interesting problem would be to study theinteraction of dislocations and point defects within a similarly geometric framework. This wouldrequire the connection to have torsion. The numerical solution procedure presently adopted issomewhat simplistic and does not respect all the geometric features of the model. Geometricallymotivated discretization schemes based on finite element exterior calculus might be a very goodapproach to arrive at efficient and accurate numerical solution schemes.
A Numerical solution procedure
This section discusses the numerical procedure used to obtain the solution of the minimizationproblem discusses in Section 6. We formulate the (mechanical) equilibrium of a body with a pointdefect as a minimization problem over a Weyl manifold, with the Weyl connection specified. Asdiscussed earlier, the connection on the manifold modifies the critical points of the stored energyfunction. These configurations are computed using a finite element based discretization procedure.Since we are looking at a radially symmetric problem, only the radial displacement is discretizedusing a three noded quadratic Lagrange finite element. The minima of the discretized stored energyfunction is computed using Newtons’ method which requires the first and the second derivativesof the discrete stored energy. The first derivative of the stored energy is often called the residualforce, we denote it by R . The condition R = represent the equilibrium of forces at the node.The discrete equilibrium is analogous to the linear momentum equation given in Eq. 44. Since thebody has non-trivial connection, we use following equation to compute an incremental change inthe deformation. F aA = H aA + (cid:18) γ abc H bA u c + ∂u a ∂X A (cid:19) (99)In the above equation, u is an incremental displacement field superimposed on an deformation ϕ with deformation gradient H and γ ijk are the connection coefficients of the deformed configuration.The second term in Eq. 99 accounts for non-trivial connection associated with the configuration ϕ , the proof of the above equation can be found in [15]. The components of the connections inspherical coordinate system is given as, γ r = − ∂ r f − ∂ θ f − ∂ ξ f − ∂ θ f − r sin ξ − ∂ ξ f − r ; γ θ = r − ∂ r f r − ∂ r f − ∂ θ f cot ξ − ∂ ξ f ξ − ∂ ξ f ; γ ξ = r − ∂ r f − sin ξ cos ξ − ∂ θ f r − ∂ r f − ∂ θ f − ∂ ξ f (100)The residual force can then be computed using the relation, R = ddǫ | ǫ =0 W ( ǫ ) (101)Where, W ( ǫ ) denotes the perturbed stored energy about the configuration ϕ , which is obtainedusing the relation ϕ + ǫ u , here ǫ is a small parameter. The perturbed deformation gradient is23btained using the relation Eq. 99. The finite element approximation for the incremental radialdisplacement u r is denoted by u hr and it is given by, u hr = i = N X i =1 N i u ir (102)In the above equation u ir and N i denotes the incremental radial displacement and shape functionat the i th node. The incremental radial displacement and the nodal shape function at all nodes aredenoted by u h and Υ respectively. In terms of the nodal displacement and stored energy function,the condition for discrete mechanical equilibrium can be written as, ∂W∂ u h = (103)The above equation is nothing but the necessary condition of an extrema for a finite dimensionalextremization problem. For the assumed stored energy function, the discrete equilibrium equationis given by, µ ∂I ∂ u h + 1 J ( λ log( J ) − µ ) ∂J∂ u h = 0 (104)The variables ∂I ∂ u h and ∂J∂ u h denotes the directional derivatives computed using the perturbationgiven in Eq. 99, whose expressions are given as, ∂I ∂ u = 2 (cid:16) drdR − u r dfdR drdR + du r dR (cid:17) (cid:16) − dfdR drdR Υ + d Υ dR (cid:17) + r R (cid:16) u r (cid:16) r − dfdR (cid:17)(cid:17) (cid:16) r − dfdR (cid:17) Υ (105) ∂J∂ u = (cid:18)(cid:16) u r (cid:16) r − dfdR (cid:17)(cid:17) (cid:16) − dfdR drdR Υ + d Υ dR (cid:17)(cid:19) (cid:0) rR (cid:1) + 2 (cid:16) u r (cid:16) r − dfdR (cid:17)(cid:17)(cid:16) drdR − u r drdR dfdR + du r dr (cid:17) (cid:16) r − dfdR (cid:17) (cid:0) rR (cid:1) Υ (106)In the above equation, u r denotes the displacement computed from the last iteration and r denotesthe last converged radius and d Υ dR denotes the derivative of the nodal basis with respect to the radialco-ordinates. The Hessian (tangent stiffness matrix) required for Newtons’ method for the assumedstored energy function is given by, ∂ W∂ u ∂ u = µ ∂ I ∂ u ∂ u + 1 J ( λ + µ − λ log J ) ∂J∂ u ⊗ ∂J∂ u (107)The second derivatives of I and J are computed as, ∂I ∂ u ∂ u = 2 (cid:18) − drdR Υ + d Υ dR (cid:19) ⊗ (cid:18) − drR dfdr Υ + d Υ dR (cid:19) + 4 r R (cid:18) r − dfdR (cid:19) Υ ⊗ Υ (108) ∂J∂ u ∂ u = 2 (cid:18) u r (cid:18) r − dfdR (cid:19)(cid:19) (cid:18) r − dfdR (cid:19) (cid:18) Υ ⊗ (cid:18) − drdR Υ + d Υ dR (cid:19) + (cid:18) − drdR Υ + d Υ dR (cid:19) ⊗ Υ (cid:19) +2 (cid:18) r − dfdR (cid:19) (cid:18) drdR − u r drdR dfdR + du r dR (cid:19) (cid:18) r R (cid:19) Υ ⊗ Υ (109)24 eferences [1] C. Barcel´o, R. Carballo-Rubio, and L. J. Garay. Weyl relativity: a novel approach to weyl’sideas. Journal of Cosmology and Astroparticle Physics , 2017(06):014, 2017.[2] B. A. Bilby, R. Bullough, and E. Smith. Continuous distributions of dislocations: a newapplication of the methods of non-riemannian geometry.
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