Consistent hypo-elastic behavior using the four ...

Noname manuscript No. (will be inserted by the editor)

B. Panicaud · E. Rouhaud · G. Altmeyer · M. Wang · R. Kerner · A. Roos · O. Ameline

Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry Received: date / Accepted: date

B. Panicaud Universit´e de Technologie de Troyes (UTT) Institut Charles Delaunay (ICD) - CNRS UMR 6279 Laboratoire des Syst`emes M´ecaniques et d’Ing´enierie Simultan´ee (LASMIS) 12, rue Marie Curie - CS 42060 10004 Troyes cedex - France Tel.: +33-(0)325718061 Fax.: +33-(0)325715675 E-mail: [email protected] E. Rouhaud Universit´e de Technologie de Troyes (UTT) Institut Charles Delaunay (ICD) - CNRS UMR 6279 Laboratoire des Syst`emes M´ecaniques et d’Ing´enierie Simultan´ee (LASMIS) 12, rue Marie Curie - CS 42060 10004 Troyes cedex - France Universit´e Pierre et Marie Curie (UPMC) Laboratoire de Physique Thorique de la Matire Condens´ee (LPTMC) Tour 23-13, Boite Courrier 121, 4 Place Jussieu, 75005 Paris - France E-mail: [email protected] G. Altmeyer Universit´e de Technologie de Troyes (UTT) Institut Charles Delaunay (ICD) - CNRS UMR 6279 Laboratoire des Syst`emes M´ecaniques et d’Ing´enierie Simultan´ee (LASMIS) 12, rue Marie Curie - CS 42060 10004 Troyes cedex - France E-mail: [email protected] M. Wang Universit´e de Technologie de Troyes (UTT) Institut Charles Delaunay (ICD) - CNRS UMR 6279 Laboratoire des Syst`emes M´ecaniques et d’Ing´enierie Simultan´ee (LASMIS) 12, rue Marie Curie - CS 42060 10004 Troyes cedex - France E-mail: [email protected] R. Kerner Universit´e Pierre et Marie Curie (UPMC) Laboratoire de Physique Th´eorique de la Matire Condens´ee (LPTMC) Tour 23-13, Boite Courrier 121, 4 Place Jussieu, 75005 Paris - France E-mail: [email protected]

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Abstract The covariance principle of the theory of Relativity within a fourdimensional framework ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space-time and the use of 4D tensors, operations and operators. This 4D formalism enables also to clearly distinguish between the covariance principle (i.e. frame-indifference) and the material objectivity principle (i.e. indifference to any rigid body motion superposition). We propose and apply here a method to build a constitutive relation for elastic materials using such a 4D formalism. The present article is specifically devoted in the application of this methodology to construct hypo-elastic materials with the use of the 4D Lie derivative. It enables thus to obtain consistent non-dissipative models equivalent to (hyper)elastic ones. Keywords Constitutive models ; Covariance principle ; Material objectivity principle ; Continuum mechanics ; Hypo-elastic behavior ; Hookean-like materials 1 Introduction In the present article, we are interested in discussing frame-indifference issues in continuum mechanics, in relation with material objectivity (i.e. indifference to any rigid body motion superposition). Moreover, it is focused primarily on the construction of constitutive models. The aim is to place the mathematical framework within the four-dimensional formalism of the General Relativity theory. Such an aim has non-negligible interest, for it involves in principle non-trivial issues. Frame-indifference of material constitutive models is essential in three-dimensional continuum mechanics in the case of finite transformations, in solids or fluids [1–4]. In general, the superposition of a rigid body motion of the matter, i.e. active transformations, and/or the motion of the chosen frame of reference, i.e. passive transformations restricted to Euclidean transformations, may impose constraints on the constitutive models [2, 5–7]. Because the two notions, i.e. material objectivity and frame-indifference, are similar in threedimensional continuum mechanics, it has lead to strong debates [8, 9] to decide whether this principle has to be used or not, how and to which kind of materials. A definition of the frame-indifference principle is contained for example in the classical monograph of Truesdell and Noll [7], while recent A. Roos Universit´e de Technologie de Troyes (UTT) Institut Charles Delaunay (ICD) - CNRS UMR 6279 Laboratoire des Syst`emes M´ecaniques et d’Ing´enierie Simultan´ee (LASMIS) 12, rue Marie Curie - CS 42060 10004 Troyes cedex - France E-mail: [email protected] O. Ameline Universit´e de Technologie de Troyes (UTT) Institut Charles Delaunay (ICD) - CNRS UMR 6279 Laboratoire des Syst`emes M´ecaniques et d’Ing´enierie Simultan´ee (LASMIS) 12, rue Marie Curie - CS 42060 10004 Troyes cedex - France

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discussions of the encountered problems can be found in [2–4]. Differential geometry has proved to be the most useful mathematical formalism for the description of transformations of material continua. It has been applied, in particular, to the treatment of finite transformations of solids [5, 6,10]. More recently, several authors have made a systematic use of this mathematical tool within a three-dimensional (3D) space for which Stumpf and Hoppe have proposed a review [11]. Complements can be also found in [12, 7]. Roug´ee has shown the interest of such an approach for the ”intrinsic Lagrangean” description of a continuum [13,14]. This approach has also been systematically applied to the general description of elastic solids [12,15–19]. In a recent paper, Xiao et al. have investigated the problem of inelasticity [20]. Besides, differential geometry has found a major application in physics with the theory of General Relativity for the description of the four-dimensional (4D) space-time, with applications to gravity [21]. In this context, the covariance principle guarantees the frame-indifference of any physical relations in any kind of systems of reference, not only the rigid body-like Euclidean frames. This principle has been developed and reviewed by Landau and Lifshitz [22]. It seems thus interesting to write constitutive models in continuum mechanics using differential geometry (with or without gravity), but within this 4D formalism to take advantage of the inherent frame-indifference or covariance principle. Unlike the 3D approach, indifference to any superposition of rigid body motions, and frame-indifference to any arbitrary changes of 4D coordinate systems have now to be distinguished within that 4D formalism as it has been proved in [23]. Moreover, the use of this 4D formalism enables to define specific derivatives that can be used either for balance equations or constitutive models, as detailed in [24]. For incremental formulations, the use of Lie derivative is justified because it is the only objective transport that corresponds to a derivative of a tensor with respect to time. It ensures simultaneously indifference to any rigid body motion superposition and frame-indifference. In order to further prove its interest, the problems of hypo-elastic modeling is detailed in the present article. The advantages of a 4D tensor formalism is also demonstrated and brings a consistent solution to the above mentioned difficulties. Then a method will be proposed to construct 4D hypo-elastic models. Application of this method will be done for specific elastic materials, such as non-linear Hookean-like models. The use of such constitutive models will be numerically tested for different solicitations and then compared to results obtained from a reference model. 2 Comparison of 3D and 4D formulations for finite deformations 2.1 Difficulties faced with a 3D formulation for finite deformations A first problem occurs when considering the entities that serve to construct 3D constitutive models for finite deformations. Indeed, several constraints need to be considered, like for instance the indifference with respect to the superposition of rigid body motion. However, in 3D this property coincides

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with the frame-indifference principle, because any rigid body can be seen as an observer in the 3D sense. Consequently, if this property is not imposed, then the principle of frame-indifference cannot be validated. This major difficulty with the notion of material objectivity has provoked many debates [25, 8, 26, 9, 27]. Other difficulties also appear while building 3D constitutive models for finite deformations. For example, when dealing with material objectivity, choosing a correct transport is not a trivial matter. This has been frequently discussed elsewhere [28–30]. Indeed, when computing a time transport without taking into account the local material motion, some wrong predictions can be made. Furthermore, in 3D, many objective transports can be defined among countless possibilities [31,32, 29]. This has two consequences. First, among the possible transports, some, even if they are objective, lead to unrealistic behaviors (see for example the Jaumann rate under shear loads oscillating at large deformations [33]). Second, for elastic models, the use of transports which are not true time derivatives (i.e. not defined as true time differentials) cannot be mathematically integrated into an (hyper)elastic model. In other words, such kind of hypo-elastic models does not present a reversible behavior. This problem will be detailed further. Another question is raised when dealing with finite deformations in 3D: how to choose between an Eulerian and a Lagrangean description? This problem is complicated by the possible use of convected coordinates for which a correspondence can be established between the components of Eulerian and Lagrangean tensors [28]. This means that a deeper link exists between the two descriptions. 2.2 Advantages of a 4D formulation for finite deformations Extend the number of dimensions by including time as an extra dimension offers an interesting framework to consistently deal with the kinematics of finite deformations. The 4D formalism clearly distinguishes the covariance principle and the indifference with respect to the superposition of a rigid body motion [24,23]. The latter is finally a property of the material that can be taken into account, or not, whereas the covariance principle is intrinsically verified through the use of 4D tensors and 4D operators. The problems concerning time derivatives are now solved by using the natural 4D time derivatives: the 4D covariant derivative in the direction of the velocity and the 4D Lie derivative in the velocity field. The former is not invariant with respect to the superposition of a rigid body motion, whereas the latter is. Both are frame-indifferent. Other properties of these derivatives for applications to non-relativistic transformations of continuum mechanics can be found in [24]. These derivatives corresponds to true derivatives with respect to time that can be properly integrated to obtain reversible elastic behaviors. This is actually the aim of the present article and will especially be detailed for the Lie derivative in the next sections. The 4D description also encompasses both the Eulerian and Lagrangean descriptions of material transformations (a proof can be found in [23]). The 4D tensorial constitutive model is intrinsic and corresponds to an expression

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that is frame-indifferent. An inertial frame corresponds to a specific choice of 4D coordinates for which the constitutive model has a given expression; this specific case is often described as the Eulerian description. The Lagrangean case corresponds to another choice of 4D coordinates: a specific choice of observer linked to the material, i.e. to the convected 4D frame where the constitutive model can be described.

3 Four-dimensional description of space-time Differential geometry proposes a general mathematical context for the description of tensors and the associated algebras. The present section introduces definitions that are necessary for the rest of this work; a more detailed presentation may be found in [34,35]. Classical notions of 4D physics are also reviewed in order to introduce specific vocabulary and notations. Further details concerning these subjects are proposed for example in [36, 37] where the general concepts are given, while the theory of General Relativity applied to physical fields is presented by Landau and Lifshitz [22] and Weinberg [21].

3.1 4D coordinates and their transformations As opposed to classical mechanics point of view, space-time is described by a four-dimensional differentiable manifold. The coordinates of a point within this manifold are parameterized by a set of four real numbers ξ µ . This point is called an event and corresponds to a given position and instant of time. The coordinates are denoted with a common index, so that: ξ µ = (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = (ξ i , ξ 4 )

(1)

In this work, the index notation is used and Einstein’s summation convention is adopted. Greek indices µ, ν... run from 1 to 4 and label a four-dimensional entity. Latin indices i, j run from 1 to 3 and label the spatial part of this entity. The fourth coordinate ξ 4 = ct denotes the time t multiplied by a constant reference speed c to make its dimension homogeneous with the remaining three space coordinates. In the theories of Relativity, c is the universal constant speed of light in vacuum. Other sets of coordinates could be indifferently chosen to parameterize the points of the manifold. Consider two possible sets of coordinates noted ξ µ and ξeµ . The coordinate transformation from ξ µ to ξeµ (ξ ν ) implies that: dξeµ = The matrix

∂ ξeµ ∂ξ ν

∂ ξeµ ν dξ ∂ξ ν

(2)

is the Jacobian matrix of the coordinate transformation and µ e the determinant of this Jacobian matrix is noted ∂∂ξξ ν .

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3.2 4D invariant interval and metric tensor An invariant interval ds is defined as a generalized length element, such that: ds2 = −(dζ 1 )2 − (dζ 2 )2 − (dζ 3 )2 + (dζ 4 )2 = ηµν dζ µ dζ ν

(3)

where the coordinates ζ µ represent the 4D coordinates of an event and ζ i corresponds to the classical 3D cartesian coordinates, as further discussed in section 4.1. This particular coordinate system is also said to be standard, Galilean or inertial and ηµν is the Minkowski metric such that:  −1 0 0 0  0 −1 0 0  = 0 0 −1 0  0 0 0 +1 

ηµν

(4)

The group of Lorentz transformations corresponds to the set of coordinate systems for which the components of the metric tensor are equal to the Minkowskian metric ηµν . Our application concerns non-relativistic phenomena, thus the speed of any material point is always very small with respect to the speed of light. To properly construct a 4D description of a continuum, a constant speed is used in the definition of 4D tensors. This reference speed has to be much larger than any of the speed of the material points, so that we use c. To interpret the final results, the terms that are very small within the framework of our hypotheses can be neglected at the end of the derivation. As a 4-scalar, the interval ds is thus left unchanged under any change of 4D coordinate systems. One has thus, for a coordinate system ξ µ and using Eqs. 2 and 3:   λ κ ∂ζ ∂ζ 2 ηλκ dξ µ dξ ν ds = ∂ξ µ ∂ξ ν = gµν dξ µ dξ ν (5) which defines gµν as the covariant components of the metric tensor for a general coordinate system ξ µ . Classically, the covariant and contravariant components of the metric tensor are such that: gµλ g λν = gµ ν = δµν

(6)

where δµν is Kronecker’s symbol. A metric symmetric connection is chosen, which is often called Christoffelian connection. In the present work, gravitational fields are not taken into account and the components of the metric tensor depend only on the choice of the coordinates. A covariant derivative is defined and the coefficients of the affine connection can be identified with the Christoffel’s symbols. For a second-rank tensor, it corresponds to: ∇λ αµν =

∂αµν µ κν ν κ µν + Γκλ α + Γκλ αµκ − W Γκλ α ∂ξ λ

(7)

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α where Γβγ are the Christoffel’s symbols defined as (see for example details in [21]): α Γβγ =

∂gδβ ∂gβγ 1 αδ ∂gδγ − ) g ( β + γ 2 ∂ξ ∂ξ ∂ξ δ

(8)

W is the weight of the tensor density [35]. Further, for every point of spacetime, there exists a set of coordinates with a Minkowskian metric in which all the Christoffel’s symbols vanish. As a consequence, the Riemann 4D curvature tensor of this particular 4D space is equal to zero, such that the considered space-time is Euclidean.

4 Description of the deformation of a continuum within Newtonian hypotheses and a 4D formalism The goal of this work is to describe the finite deformations of a material body as it is traditionally studied in continuum mechanics. An observer evaluates physical entities characterizing the state of the material for each instant of time and point of space. These observed transformations could be schematically described as ranging from slow infinitesimal elastic perturbations undergone by a solid material, to, at the other extreme, a gas explosion in a rocket. In any case, the speed of the material is always small with respect to the speed of light c anywhere in the matter. Thus, the physical hypotheses that are retained correspond, at the limit, to the classical hypotheses of Newtonian physics. It is further possible to describe the physics of phenomena for which the velocity is small with respect to the speed of light using the 4D formalism of the theory of Relativity. The description of a deformable continuum within a four-dimensional and relativistic context has been proposed by many authors; see for example [38–50]. Their work has been essentially developed to describe relativistic phenomena for astrophysics applications or were adapted to the theory of elementary particles and fields. Nevertheless, it is useful for development of the present proposition, because we can rely on these formulations for the construction of the non-relativistic 4D tensors.

4.1 Observers and frames of reference A set of four base vectors is defined in the tangent space at each point of the manifold under consideration. The four orthonormal vectors eµ represent the base vectors associated to the 4D inertial coordinate system ζ µ defined in section 3.2. Because space-time is Euclidean, it is possible to define the base vectors associated to any coordinate system as: ∂ζ µ eµ ∂ξ ν ∂ξ ν g ν (ξ κ ) = µ eµ ∂ζ g ν (ξ κ ) =

(9)

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where the vectors g ν and g ν represent respectively the covariant and contravariant local base vectors associated to the general coordinate system ξ κ . Within this 4D context, because the fourth coordinate ζ 4 represents time, an observer may be defined as a set of four base vectors associated to the chosen coordinate system for each event of the manifold. The observers are thus completely defined once the coordinate system is chosen. In other words, the choice of the 4D coordinate system is equivalent to the choice of the observers and 4D coordinate transformations describe changes of observers. A frame of reference is further defined as a class of equivalence of observers. A review and extensive definitions on this subject have been proposed by Arminjon and Reifler [51]. An inertial (or Galilean or standard) frame of reference can then be defined as the set of observers with Minkowskian metric. This corresponds to the 3D classical definition, as understood in Newtonian physics. Such inertial observers exist for the Euclidean space-time within the range of the considered hypotheses. The four base vectors eµ are associated to an inertial 4D coordinate system ζ µ and correspond to an inertial observer. The four base vectors g ν represent any observer associated to a curvilinear 4D coordinate system ξ κ . The coordinates ξ κ may be function of time. Thus, the observers represented by these base vectors g ν (ξ κ ) may be accelerating or rotating.

4.2 Tensors and indifference to coordinate transformations Tensor density of weight W can then be defined over the points of the manifold. For the sake of generality, they are noted α to represent any given tensor density. They are function of the considered coordinates and a tensor density field is defined and noted α(ξ µ ). If α is a second-rank tensor density, it expanded as: α = αµν g µ ⊗ g ν = αµν g µ ⊗ g ν = αµν g µ ⊗ g ν = αµν g µ ⊗ g ν eµ ⊗ g eν = α eµ ⊗ g eν = α eµ ⊗ g eν = α eµ ⊗ g eν =α eµν g eµν g eµν g eµν g

(10)

where αµν , α eµν are the contravariant components of this tensor density, αµν , eµν its mixed compoeµν or α α eµν its covariant components, and αµν , αµν , α nents. Classically, upper indices denote the contravariant components of the tensor, while lower indices denote its covariant components. The components of a tensor density α always transform through a change of coordinates from ξ µ to ξeµ as [34]: α W ∂ξ α e = ∂ ξeβ α W ∂ξ α eµν = ∂ ξeβ α W ∂ξ α eµν = ∂ ξeβ

α

(11)

∂ ξeµ ∂ ξeν λκ α ∂ξ λ ∂ξ κ

(12)

∂ξ λ ∂ξ κ αλκ ∂ ξeµ ∂ ξeν

(13)

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respectively for a scalar density and the contravariant or covariant components of a second-rank tensor density. It is possible to write similar equations for the components of tensor densities of any rank and for any components types. The use of tensor densities of weight W is not common in classical continuum mechanics, although it is mentioned in [5]. It allows to appropriately correct the components of the tensors for a possible change in scale or dimension when using curvilinear coordinates. The definition of weighed quantities is further useful for the study of material deformations because, in particular, any tensor carrying the dimension of stress is in fact a tensor density of weight equal to one (W = 1).

4.3 Material transformations Consider a finite deformation of a material body. A material particle is defined as an elementary 3D volume of this material. This definition, classically found in fluid mechanics, is equivalent with the definition of a ”material point” in physics or with the definition of a ”representative elementary volume” in solid mechanics. The reference configuration is defined as the volume of space that is occupied by the particles of matter of the studied continuum at the chosen time of reference. The spatial coordinates of the particle in the initial configuration are noted X i in a 4D coordinate system ξ µ ; this defines the material or Lagrangian coordinates. After deformation takes place, a deformed configuration is defined at the current time t. The spatial coordinates of the particles are then noted xi in the same 4D coordinate system ξ µ ; they correspond to the spatial or Eulerian coordinates. Then, the deformations of the continuum are described with mappings: xµ = xµ (X k , t) =



xi = xi (X k , t) x4 = ct

(14)

such that xi (t) = xi (X k , t) is bijective with existing and continuous derivatives to verify the axiom of continuity. The description of the deformation corresponds to a diffeomorphism. For simplification purposes, and without much loss of generality, the time of reference is chosen to be the initial time t = 0. It is also usual in classical continuum mechanics to choose an inertial frame of reference and an orthonormal coordinate systems for ξ µ . Then the coordinates of the particles are noted z µ in the inertial system ζ µ , as previously introduced. Similarly to the 3D case, we have thus defined 4D material configurations corresponding in the space-time formulation to space-like hyper-surfaces (3D volumes) of the space-time manifold. The motion of the material body is described by the specification of the events undergone by the particles within this 4D manifold. The motion of the particles corresponds then to a set of world-lines (trajectories) that spans a connected open domain of the manifold.

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4.4 Newtonian motion The interval dτ along the motion of a particle corresponding to: dτ 2 = −(dz 1 )2 − (dz 2 )2 − (dz 3 )2 + (cdt)2 = ηµν dz µ dz ν = gµν dxµ dxν

(15)

has the same value in any 4D coordinate system as defined by Eqs. 3 and 5. This can further be expressed as:   v2 2 2 (16) dτ = (cdt) 1 − 2 c i

where v = kv i k with v i = dz dt the 3D material speed with respect to the inertial frame. This implies that the definition of time could vary. Within a relativistic formulation, time is thus said to be absolute when expressed in the inertial system and proper when expressed in a coordinate system moving with the particle (for which v i = 0). The proper time t0 is such that τ = ct0 . It is not here the purpose to evaluate the proper time of a particle. Within the range of our hypotheses, vc  1 for any particle and for any event of any motion. In the rest of this work, this defines what is referred to a Newtonian motion. Then, with Eq. 16, dτ /c = dt0 ≈ dt. In other words, the proper time, in these circumstances, is infinitely close to the absolute time. Consequently, in this work, t always refers to the absolute time. A reference instant can then be chosen to define the set of material particles that constitutes the material body. The reference instant is further supposed to be the initial time of the chosen chronology, and this, for all observers. The definition of the invariant interval leads to Eq. 16, which is very important for the interpretation of 4D rates. The interval dτ ≈ cdt can be interpreted both as a total increment of the coordinates with Eq. 15, and, as representing an increment of the absolute time t (multiplied by c) with Eq. 16.

4.5 Choice of 4D coordinates: Newtonian observers Among the possible coordinate systems that could be chosen, we are going to restrict the choice to the ones that are in practice useful for the description of a continuum within the hypotheses of classical continuum mechanics. Although theoretically possible, there is currently no practical interest for engineering materials in describing phenomena with an observer whose speed is close to the speed of light. Thus, the practical coordinate systems will define 4D Newtonian observers. Following the considerations of section 4.1, a 4D Newtonian observer is defined as a set of base vectors g ν (ξ κ ) for which the measure of time is absolute such that: ξ 4 = ζ 4 = ct

(17)

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Further, these observers may be accelerated or rotated with respect to the inertial observers, but their speed has to remain small with respect to the speed of light such that: ds ≈ cdt (18) These observers are thus only defined locally in space-time to ensure that the components of the associated metric are not singular. The description of a motion for which the speed of every material particle is small with respect to the speed of light with observers whose speed is small with respect to the speed of light constitutes the Newtonian hypotheses, adopted in this work. 4.6 4D deformation gradient General definitions for 4D deformation and strain tensors have been proposed by Lamoureux-Brousse [43] for General Relativity applications. They are applied here to the 4D non-relativistic limit. To take a full advantage of the 4D formalism, this author has proposed to compare two different motions xµ (X i , x4 ) and x0µ (X 0i , x04 ) of the same material body. These two motions are described within two different coordinate systems respectively ξ µ and ξ 0µ , associated to the two respective covariant components of the metric tensor 0 . The body is defined at a given absolute instant, the time of gµν and gµν reference, with a unique reference configuration that is described by X µ or X 0µ , within each respective coordinate system. To compare two motions of a given material body, a 4D generalization of the deformation gradients is defined such that: ∂xµ ∂x0ν ∂x0µ F 0µν (x0α , xβ ) = ∂xν F µν (xα , x0β ) =

(19) (20)

We further define the quantity J (4D Jacobian of the deformation) as: r g J = F αβ (21) g0 0 with g = |gµν | and g 0 = |gµν |. J is a scalar with a weight equal to zero. It should be noted that J is linked to the 4D determinant of the 4D deformation gradient, following the classical 3D continuum mechanics notation. It can be also shown that [5,22]:

J=

dΩ dΩ 0

(22)

where dΩ and dΩ 0 represents the respective four-volume forms. If ρ and ρ0 are the associated respective densities, then: c dt dV c dt dV 0 ρ 1 ⇒ ≈ 0 J ρ J≈

(23)

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where dV and dV 0 are the respective spatial volume variations, such that J is also a measure of the dilatation and of the variation of the local density with respect to the reference motion, under the hypotheses of this proposition i.e. non-relativistic motions.

4.7 Definitions of 4D strain and deformation tensors It is now possible to define the 4D strain tensor eµν , following LamoureuxBrousse [43], as: eµν =

1 (gµν − bµν ) 2

(24)

where bµν is the 4D generalization of the inverse of the left Cauchy-Green or Cauchy’s deformation tensor, constructed from the inverse of the material transformation gradient matrix F 0µν , as: 0 bµν = F 0αµ F 0βν gαβ

(25)

The tensors above have been chosen because they offer a correct description of finite deformations similar to the ones used for the 3D Eulerian description, for modelling fluids and metal forming processes [53,54]. We also define the 4D equivalent of the ”material” counterparts of the above strain tensors as: Eµν =

1 0 (Cµν − gµν ) 2

(26)

where Cµν is the 4D generalization of the right Cauchy-Green or Green’s deformation tensor, defined from the material transformation gradient matrix F µν , as: Cµν = F αµ F βν gαβ

(27)

Note that we have preserved, with the capital letters, the classical mechanics notations for the definitions of the 4D deformation/stretch and strain tensors. In the present work, the 4D approach is similar to the 3D classical case transplacements from reference to current configurations. The choice is formal but the underlying physics has to be handled with care. In particular, in the General Relativity theory which includes the present treatment, there is no notion of placement at a given time; rather there are world-lines, instead of places and times. Eq. 24 is interpreted as a comparison between the reference configuration (static isometric 4D motion given by Eq. 25) and the world-lines of the corresponding particles motions in the current configuration.

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4.8 Four-velocity The definition of the 4D velocity u is: uµ =

dxµ dτ

(28)

One has, using Eq. 16:  dxi dxµ =  q dt dτ c 1−

 v2 c2

,q

1 1−

v2 c2



(29)

Note that when the velocity of the particle of matter is small with respect to the speed of light, its four-velocity becomes:   1 dxµ 1 dxi ≈ = ,1 (30) c dt c dt because the four-velocity is a dimensionless quantity. 5 Rates within a 4D formalism 5.1 Derivative with respect to time following a motion Consider a 4D tensor field α(ξ µ ) that is itself indifferent to the superposition of rigid body motions. The coordinates of the particle are xµ in the coordinate system ξ µ . Within a Newtonian context, for a small increment of time (t0 −t), the coordinates of the particle go from xµ to x0µ with the velocity uµ . The value of the tensor density as seen by the particle goes then from α(xµ ) to α(x0µ ). Following the trajectory of the particle, the coordinates of the particle could also be written x0µ = xµ + uµ (t0 − t), at first order, such that: α(x0µ ) = α (xµ + uµ (t0 − t))

(31)

The variation of the tensor following the motion of the particle is then: α (xµ + uµ (t0 − t)) − α(xµ ) α (x0µ ) − α(xµ ) = lim (32) t0 →t t →t t0 − t t0 − t This is the only possible definition for the variation of a tensor with respect to time along a trajectory and corresponds to a 4D generalization of a material derivative. For a second-rank tensor, this can be written as (following Eq. 10): lim 0

αµν (x0µ ) g µ (x0µ ) ⊗ g ν (x0µ ) − αµν (xµ ) g µ (xµ ) ⊗ g ν (xµ ) (33) t →t t0 − t concerning its contravariant components. Depending on the choice that is made for the variation of the base vectors, two types of time derivatives may be defined. First, if the base vectors are considered constant, a covariant rate is defined. Second, if the base vectors are transformed by the motion to represent the variation as seen by the moving particle, the Lie derivative in the velocity field is defined. Both rates are detailed in the following sections. Illustration of the space-time description, the flow and the tangent spaces at each time can be seen in figure 1. lim 0

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Fig. 1 Illustration of the space-time description, the flow and the tangent spaces at each time

5.2 Covariant rate An operator n∇α is constructed from the definition of the covariant derivative ∇α [24] to evaluate the variations of the tensor density α along a direction given by a unitary vector n of the 4D space-time. Consider now a given particle moving in space-time with a four-velocity u. The evaluation of the variations of α along the direction u leads to the construction of the covariant rate operator u∇α. This covariant rate is expressed in the chosen curvilinear coordinates and is itself a tensor of the 4D space-time [22]. It corresponds to an increment of the tensor density as seen by a point of the manifold and following the direction of the motion. Within a Newtonian D context, it becomes the operator noted Dt in this work, such that: α(x0µ ) − α(xµ ) Dα = lim = uλ ∇λ α 0 t →t Dt t0 − t

(34)

The contravariant components of the covariant rate of a second-rank tensor density take the form: Dαµν dxλ dαµν dxλ µ κν ν κ µν = ∇λ αµν = + ( Γκλ α + Γκλ αµκ − W Γκλ α ) (35) Dt dt dt dt Similar operators may be defined for tensor densities of other ranks. The covariant rate of a tensor density (Eq. 35) is further decomposed to interpret the meaning of this rate such that:  µν   i  ∂α dx ∂αµν dxλ µ κν Dαµν ν κ µν = + + (Γκλ α + Γκλ αµκ − W Γκλ α ) i Dt ∂t dt ∂x dt (36) The different terms of Eq. 36 could then be interpreted as: – Term between double brackets: the partial time derivative of the tensor density.

15

– Term between brackets: the time derivative of the tensor density due to the fact that the particle could have a 3D-velocity with respect to the observer. – The terms with Christoffel’s symbols correspond to the increment of the tensor that is due to the fact that the chosen coordinates could possibly depend on the event (for curvilinear frames). This decomposition shows that the operator is constructed with the total d time variation of the tensor within the coordinate system (the operator dt in Eq. 35), corrected with terms that give the variation of the tensor due to the possible variations of the coordinates with respect to time. It could be interpreted as the 4D generalization of the 3D material time derivative. This term is frame-indifferent and covariant (according to his name), in the sense that it can be shifted by the metric tensor to pass from contravariant to covariant components. The last terms of the decomposition of Eq. 36 proves that this rate is indeed frame-indifferent, but depends on the superposition of a rigid body motion. 5.3 Lie derivative The possibility to define a 4D rate operator whose result is both frameindifferent and indifferent with respect to superposition of rigid body motions is now detailed. We wish to establish the value of the local instantaneous total time variation of α as seen by the moving particle. The 4D operator corresponding to the variation of the tensor as seen by the particle within its motion is then:
µ 0 µ α xµ + dx dt (t − t) − α(x ) (37) Lu (α) = lim t0 →t t0 − t This is, by definition, the Lie derivative of α along the velocity field u noted Lu (α) as demonstrated, for example, in [35]. This operator could be interpreted as a Lagrangean entity (it is defined for a particle of matter), computed within an Eulerian formalism (it is defined at a given event of space-time). It can be further shown that the Lie derivative of a second-rank tensor density of contravariant components αµν take the following form, within the Newtonian hypotheses [35]: ∂uµ ∂uν ∂uλ ∂αµν − αλν λ − αµλ λ + W αµν λ λ ∂ξ ∂ξ ∂ξ ∂ξ µ ν λ µν ∂u ∂u ∂u dα − αλν λ − αµλ λ + W αµν λ = dt ∂ξ ∂ξ ∂ξ

Lu (α)µν = uλ

(38) (39)

Similar formulations exist for the other types of components and other tensor ranks [35]. The Lie derivative is a general operator that could be computed along a vector field that is not necessarily the velocity field. For short in this work, we will nevertheless refer to “the Lie derivative” when it should be read “the 4D Lie derivative computed within the velocity field of the continuum and within the hypotheses and definitions of sections 3 and 4”.

16

The Lie derivative represents the variation of a tensor as seen by the particle of matter within its motion. It is an operator that acts in fact on two tensors: the velocity and the tensor itself. It acts on the whole tensor and not only on its components. The Lie derivative can be proved to be frame-indifferent and indifferent to the superposition of rigid body motion [24]. Eq. 39 could also be defined in 3D, but the partial time derivative ought to be added differently (non-autonomous Lie derivative). It means that a 4D formalism is natural to express a full derivative including all the time effects, for all the coordinate systems including the curvilinear ones. The Lie derivative corresponds to a convective transport, when the covariant rate corresponds to a material derivative. 3D classical convective transports can be derived as a limit case of 4D convective transports for non-relativistic applications. Details on these aspects can be found in [24]. 5.4 Useful properties of the Lie derivative 5.4.1 Isometry transformation One interesting property of the Lie derivative is linked to its indifference to any rigid body motion superposition. For the Lie derivative, it is possible to prove that [12]: L(a1 u1 +a2 u2 ) (α)µν = a1 Lu1 (α)µν + a2 Lu2 (α)µν

(40)

As an example, for a linear relation between two second-rank tensors that describes the mechanical behavior (or a part of it) of the material: σ µν = M µναβ gαβ

(41)

where σ is a stress-like tensor and g is the metric tensor, useful for the transformations of solids. Applying the chain’s rule for the Lie derivative to Eq. 41 [12], it leads to: Lu (σ)µν = Lu (M )µναβ gαβ + M µναβ Lu (g)αβ

(42)

It is worth noting that the Lie derivative of M µναβ is not zero. It is possible to decompose the velocity field in an isometry part uξ and a non-isometry part u∗ . The isometry fields correspond to rigid body motions that keep the body undeformed. Indeed, stress has not to be produced by such a motion, for the considered materials. Those fields are the Killing vectors of the metric, for which the Lie derivative is zero as a definition [35]. Applying the relation of Eq. 40 for the decomposition of the velocity field in Eq. 42, it leads to: Lu (σ)µν = Lu (M )µναβ gαβ + M µναβ La1 uξ +u∗ (g)αβ = Lu (M )µναβ gαβ + a1 M Luξ (g)αβ + M µναβ Lu∗ (g)αβ = Lu (M )µναβ gαβ + M µναβ Lu∗ (g)αβ

(43)

For pure isometry for which u∗ = 0, the considered behavior model is simply equal to Luξ (σ)µν = Luξ (M )µναβ gαβ . Indeed, the Lie derivative is indifferent

17

to the superposition to rigid body motions. Other derivatives add terms that depend on rigid body motion. Moreover, the covariant rate with Christoffel’s connection given by Eq. 34 is such that, for all u (isometric or not), the covariant derivative is zero. Consequently, such a rate is not able to describe constitutive model of deformed materials with a rate formalism. Only the 4D Lie derivative enables to build consistent constitutive model. 5.4.2 Covariant and contravariant components of the Lie derivative The expressions of the Lie derivative include a change in the sign of the corrective terms depending on the variance of the tensor components [24]. This leads to the fact that, the contravariant and covariant components of the Lie derivative are not equal, even when they are expressed within an Cartesian set of base vectors. The Lie derivative is not covariant in the sense that it cannot be shifted by the metric tensor to pass directly from contravariant to covariant components. To illustrate this point, we calculate the Lie derivative of the following relation: Lu (αµν ) = Lu (g µκ g νλ ακλ ) = Lu (g µκ )g νλ ακλ + g µκ Lu (g νλ )ακλ + g µκ g νλ Lu (ακλ )

(44)

The contravariant and covariant components of the Lie derivative may be shifted only when the motion is an isometry (Lu∗ (gµν ) = 0). This is not of particular interest for the considered applications of deformed materials. In conclusion, it is necessary to be cautious when passing from the contravariant components to the covariant components in the constitutive models to be constructed below. 5.4.3 Integration The last property concerns integration of the Lie derivative over a closed path. Some authors argue that the integration of some objective derivatives over a closed path of a quantity α will not lead to zero [52]. However, because the Lie derivative corresponds to an exact or total derivative, its integration over a circular path is necessarily zero. We can give some elements of proof. For example, for very small displacements such that ξ 0µ = ξ µ + uµ (ξ)∆t and W = 0, it is easy to prove, from Eq. 38, that:   µ ν µν λν ∂u µλ ∂u µν λ ∂α −α −α g µ ⊗ g ν (45) ∆t Lu (α )g µ ⊗ g ν = ∆t u ∂xλ ∂xλ ∂xλ     ∂ακβ ∂uµ κβ λ µ ≈ α (ξ) + u ∆t δκ − ∆t κ ∂xλ ∂x   ν ∂u δβν − ∆t β g µ ⊗ g ν − αµν (ξ)g µ ⊗ g ν (46) ∂x ≈ αµν (ξ 0 )g 0µ ⊗ g 0ν − αµν (ξ)g µ ⊗ g ν (47) = ∆α (48)

18

where ⊗ is the tensorial product and δνµ means for the Kronecker’s symbol. Consequently, it leads to: I I Lu (α)dt = Lu (αµν )g µ ⊗ g ν dt (49) I = dα = 0 (50) The integral is strictly zero, which proves the affirmation. This demonstration can be easily extended for tensor densities for which W 6= 0. It can be emphasized that Eqs. 49 and 50 are correct even for finite deformations. Generally, the use of convective integration (such as for Lie derivative) needs to take care. A general relation can be obtained for each kind of components and rank of any tensors. It thus enables to calculate the integration of Lie derivative, whatever the path, as follows (for contravariant components and W = 0): Z t  µ ν ∂ξ µ ∂ξ ν ˜ ∂ξ ∂ξ ds0 Lu (ααβ )(ξ) (51) αµν (ξ) = ααβ (ξ0 ) α β + ∂ξ0 ∂ξ ∂ ξ˜α ∂ ξ˜β t0 0

Eq. 50 can be also found as a particular case of Eq. 51 for a closed path. 5.5 Lie derivative and 4D strain rate tensor It is also necessary to introduce a 4D strain rate tensor Dµν . As presented in [21], the 4D strain rate is defined as: Dµν =

1 (∇ν uµ + ∇µ uν ) 2

(52)

where we have used the covariant derivative acting on the 4-velocity uµ . It is also possible to calculate the Lie derivative of the 4D strain tensor defined by Eq. 24. We can then prove that: 1 Lu (gµν ) 2 = Dµν

Lu (eµν ) =

(53) (54)

To obtain Eq. 53, we have used the relation Lu (bµν ) = 0 as it corresponds to an isometry of the reference configuration. To obtain Eq. 54, the following relations have been considered:  skew 1 ∂gµν κ ∂gµκ κ λ κ Γκν u gλµ = u + u (55) 2 ∂xκ ∂xν 1 ∂gµν κ λ κ ⇒ (Γκν u gλµ )sym = u (56) 2 ∂xκ Because of Eq. 44, it is easy to prove that Lu (bµν ) 6= 0 and Lu (eµν ) 6= Dµν . It is now possible to ask whether the Lie derivative of the contravariant components of a specific tensor can be equalized to Dµν ? Because Lu ((b−1 )µν ) =

19

0 and Lu (g µν ) = −2Dµν , it can be demonstrated that the strain tensor 1 −1 µν ) − g µν ) is the corresponding tensor, which is different to the ten2 ((b sors previously defined (Eqs. 24 and 26). It is worth noting that the Lie derivative depends on the variance of the tensor. Such a remark is also true for any tensor, including stress tensor. The use of the 4D Lie derivative is the key point to simultaneously recover the 3D classical convective transport (Eq. 53), as well as the correct description of the time variation from a mathematical point of view. According to the demonstration done in section 5.4.3, it is important to note that, contrarily of what can be sometimes found in bibliography [52], H Dds = 0 if D is understood as a Lie derivative.

6 Construction of a 4D (hyper)elastic constitutive model An elastic model, compatible with the proposed 4D formalism, can be constructed in several ways, by : – – – –

3D analogy; 4D thermodynamics; representation theory in 4D, between strain and stress tensors; definition of a 4D stiffness for linear behaviors.

These approaches could be used either for the isotropic or for the anisotropic cases. First, a 3D analogy can be used to extend the 4D relations. This methodology is interesting, but it can suffer from a lack of generality. This will be demonstrated for hypo-elastic materials in section 7. Second, the 4D thermodynamics aims to mimic the 3D thermodynamics approach. Some works have been already carried out [55,49, 56] trying to extend this domain, essentially for relativistic applications. However, it is important to note that their work is still subject to intense debate, especially coupling different phenomena or concerning irreversible processes. For simple elastic behavior and isothermal conditions, this approach can be used as shown in [23]. Third, the representation theory can be extended towards 4D. The coupling between the 4D elastic strain and 4D stress tensors is based on the definition of 4D invariants of tensorial relations. This way is very efficient for isotropic behavior [28]. For anisotropic problems, this approach enables us to generalize some specific behaviors (transverse isotropy...), but is more difficult to extend for general anisotropy. Fourth, when considering the more general linear coupling between a second-order stress tensor and second-order elastic strain tensor, it is possible to give a specific relation for the 4D stiffness tensor. The latter approach is directly used in the present article. As a consequence, a specific model is proposed for isotropic behavior, with isothermal conditions at a macroscopic scale. We consider a model for which a linear behavior is expected in inertial frames. It satisfies the following relations, whatever the chosen frame:
σ µν = 2a1 gαβ eαβ g µν + 2a2 eµν (57)
αβ ⇔ σµν = 2a1 g eαβ gµν + 2a2 eµν (58)

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This model will be called reference model to compare further with hypoelastic ones. The present method is based on the linear combination between stress and strain tensors defined on different frames of reference. A more systematic procedure [7] ought to be used, but the present method is sufficient to find the class of investigated elastic relations. The tensor variance (co or contravariant) is of primary importance to find the elastic models in the present method. Indeed, the coupling between different variances is the way to obtain the different possible relations. From a mathematical point of view, it is always possible to find an equation to relate covariant or contravariant stress tensor to a specific covariant or contravariant strain tensor (with action of the shifter gµν if necessary). 7 Construction of hypo-elastic models 7.1 Definition of hypo-elastic materials in 3D It is now possible to define a method for construction of hypo-elastic models. It is useful to define clearly what means for hypo-elasticity. Hypo-elastic models could be defined in 3D in two ways which are not strictly equivalent. First, according to [5], a 3D hypo-elastic model is associated to the ordinary differential equations system with the following form: X D3D σij = F3D (σij , Dij )

(59)

X is an where Dij is the components of the 3D strain rate tensor and D3D objective 3D operator rate (which is not necessarily a true derivative in the mathematical sense [24]). The dependence on Dij has to be linear as demonstrated in [5]. This relation can be integrated on a closed path but the results is not necessarily zero, meaning a dissipative effect along the path which is a strong problem for a so-called elastic behavior. Second, it is sometimes possible to consider hypo-elasticity as any relation between stress rate and strain rate tensors following: X D3D σij = F3D (Dij )

(60)

For linear relation, it corresponds to a tangent modelling of the considered behavior. Eq. 60 can be seen as a particular case of the more general definition given by Eq. 59, provided that the linear dependence of Dij in Eq. 60 is added. Eq. 60 can be also integrated on a circular path but the results is not necessarily zero, meaning again a dissipative effect along the path which is a problem for a pure elastic behavior. For clarity, we will further consider the first definition given by Eq. 59 as the 3D hypo-elasticity within the present article. 7.2 Construction of 4D hypo-elastic models It is now possible to define methodologies to obtain 4D hypo-elastic models. It can be constructed in several ways, by :

21

– – – – – –

3D analogy; 4D thermodynamics; representation theory in 4D, between strain rate and stress rate tensors; rheologic approaches; integral models (memories...); derivative of the 4D hyper-elastic models.

These approaches can be used either for isotropic or for anisotropic behavior. The three first ones are the same described in section 6. Rheologic approaches constitute an easy way to combine elementary behaviors in order to obtain more complex ones. Integral models is a more general approach that enables to take into account different mechanisms with different characteristic and relaxation times leading to memories effects. Those two last approaches will not be investigated in the present paper. Last, the derivative of (hyper)elastic models can be easily done provided a choice of the derivative. The last and first ones will be considered in the next sections. That first approach leads directly to the generalization of relation 60 for a restricted class of 4D hypo-elastic models using a frame-indifferent 4D X derivative operator D4D : X µν D4D σ = F4D (g µν , Dµν )

(61)

Whereas the generalization of Eq. 59 is such as: X µν D4D σ = F4D (g µν , σ µν , Dµν )

(62)

This is the definition of hypo-elasticity that will be considered in 4D. It can also be seen as a generalization of Eq. 61. The frame-indifferent 4D derivative X operator D4D can be a priori either the 4D covariant rate or the Lie derivative, defined respectively by Eqs. 35 or 38. According to the previous arguments, for applications to deformed materials, only the 4D Lie derivative will be considered. Indeed, by focusing in the present article on material constitutive models, we are thus interested in frame-indifferent tensors which are also indifferent to the superposition of rigid body motion. Therefore, only the Lie derivative of the 4D stress tensor has to be considered. Moreover, as previously shown in section 5.4.1, only the 4D Lie derivative Lu (σ µν ) is able to reproduce a consistent behavior of deformed materials.

7.3 Rate models for Hookean-like behaviors In order to illustrate the different approaches and possible models, we consider the reference model given by Eq. 57 that will help us to obtain different rate models. In the next sections, integrability will be used to classify the models. It corresponds to the property allowing to obtain an equivalent (hyper)elastic model by integration of the hypo-elastic one.

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7.3.1 Non-integrable hypo-elastic models First, it is possible to obtain a specific hypo-elastic relation just by changing the strain and the stress, by respectively the strain rate and a stress rate. All calculations are performed in a general frame. Therefore, we obtain:
X µν D4D σ = 2a1 g αβ Dαβ g µν + 2a2 g µα g νβ Dαβ (63) The relation can be easily projected in inertial or convected frame. Similar relations can be established for other kind of components. This model is in general non integrable, but corresponds to the definition of a 4D hypo-elastic model as given by Eq. 62. This kind of model will be called rate 1. 7.3.2 Integrable non-hypo-elastic models As previously mentioned, another approach can be obtained by derivative of the reference model. It means that each side of the equations is obtained by applying the same derivative. Consequently, and providing a correct definition of the initial conditions, these incremental models are directly equivalent to the corresponding (hyper)elastic ones. All calculations are performed in a general frame. For example:

X µν X D4D σ = D4D 2a1 g αβ eαβ g µν + 2a2 g µα g νβ eαβ (64) In other words, the model is integrable because of the use of a true derivative (covariant rate or Lie derivative) on each side of the equation. This kind of model will be called rate 2. However, Eq. 64 is not an hypo-elastic model since it cannot be expressed within the general form of Eq. 62. Let us consider a general (hyper)elastic relation such that : σ µν = f µν (g αβ , eλκ )

(65)

Eq. 57 is a particular case of Eq. 65. Derivative of this equation with the covariant rate leads to (W = 1 for the stress tensor): X µν X D4D σ = D4D (f µν (g αβ , eλκ ))) (66) µν λ dσ dξ µ κν ν κ µν ⇔ + (Γκλ σ + Γκλ σ µκ − Γκλ σ )= dt dt dξ λ µ κν df µν ν µκ κ µν + (Γκλ f + Γκλ f − Γκλ f ) (67) = dt dt µν µν αβ dσ df (g , eλκ ) ⇔ = (68) dt dt ∂f µν (g αβ , eλκ ) ∂σ µν = (69) ⇔ ∂t ∂t Last equation can be simplified for the same reason, as we know the (hyper)elastic corresponding model. Same demonstration could be performed with choice of the Lie derivative, which is of more useful interest. It shows that such a model can be expressed and thus computed through different ways, and especially simple ones. However, such an expression (Eq. 69) even simple do not preserve general frame-indifference. For Newtonian physics, it is not of particular disturbance.

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7.3.3 Integrable hypo-elastic model Consequently, it wonders if Eq. 64 can be formally expressed as a hypoelastic model according to the definition of Eq. 62. Due to explicit derivative of Eq. 64, some terms depending on eµν appears. It is thus necessary to invert the (hyper)elastic relation from which the derivative is performed. In the present case, it leads to, for a general frame: eµν = −


1 a1 gαβ σ αβ gµν + gµα gνβ σ αβ a2 (2N a1 + 2a2 ) 2a2

(70)

where N is a parameter depending on the dimension that will be discussed further. It is now necessary to choose a derivative. As previously said, we consider only the Lie derivative, which is independent to any rigid body superposition. It leads to:


Lu (σ µν ) = Lu 2a1 g αβ eαβ g µν + Lu 2a2 g µα g νβ eαβ (71) Calculations of the different terms have been performed, then Eq. 70 is used to replace eµν . The following model is obtained: Lu (σ µν ) = ϕ1 g µν + ϕ2 g µα g νβ Dαβ + ϕ3 σ µν − 4g µα Dαβ σ βν

(72)

2

2a1 αβ 2(a1 ) (g αβ Dαβ )(g αβ σαβ ) + 2a1 (g αβ Dαβ ) − (σ Dαβ ) a2 (N a1 + a2 ) a2 2a1 and ϕ2 = (g αβ σαβ ) + 2a2 N a1 + a2 and ϕ3 = g αβ Dαβ

with ϕ1 =

This expression follows the definition of hypo-elastic materials as given by Eq. 62. Moreover, it is an equivalent incremental formulation of the given elastic model. This model will be called rate 3. It can be emphasized that rate 2 and rate 3 models are different forms of the same model, when using the 4D Lie derivative for rate 2 model. 8 Numerical applications 8.1 Description of material properties and loadings To test the models, numerical simulations have been carried out. Only one kind of material has been tested: an isotropic material. This later has a Young modulus of 200000M P a and a Poisson ratio of 0.3, which corresponds to 2a1 = 115384M P a and 2a2 = 153846M P a. Several models have been compared for different loading paths. First, a small deformation model in 3D is considered, i.e. the classical Hooke rate model:   dεij dεij dσ ij = 2a1 δij δ ij + 2a2 (73) dt dt dt

24

Where ε is the small deformation tensor [33]. Kirchhoff-Saint-Venant rate model is also computed, which holds for finite deformation in 3D:   dΣ ij dE ij dE ij = 2a1 δij δ ij + 2a2 (74) dt dt dt ⇒ σ ij = (det(F ))−1 Fia Fib Σ ab

(75)

Where E ij is the 3D Green-Lagrange tensor [33]. Objective Eulerian rate models are also used for comparison. One has been tried with Jaumann 3D objective transport [31] and another with GreenNaghdi or polar 3D objective transport [32]. The two considered models are:   deij deij dJ σ ij = 2a1 δij δ ij + 2a2 (76) dt dt dt   dGN σ ij deij deij = 2a1 δij δ ij + 2a2 dt dt dt

(77)

Where eij is the 3D Euler-Almansi tensor [33]. Expressions for the different transport rates can be found in [24]. In 3D, no difference remains between co and contravariant components because of the use of Cartesian coordinates. However, to compare with the 4D models, one has to take care of the sign of the metric over the spatial components. Then, the 4D models are considered. Calculations have been made in the inertial frame. The reference model is directly given by Eq. 57:
σ µν = 2a1 ηαβ eαβ η µν + 2a2 eµν (78) For inertial coordinates, rates 1, 2 and 3 models take respectively the forms:
Lu (σ µν ) = 2a1 η αβ Dαβ η µν + 2a2 η µα η νβ Dαβ (79)

Lu (σ µν ) = Lu 2a1 η αβ eαβ η µν + 2a2 η µα η νβ eαβ

(80)

Lu (σ µν ) = ϕ1 η µν + ϕ2 η µα η νβ Dαβ + ϕ3 σ µν − 4η µα Dαβ σ βν

(81)

For each of them, only the 2D-spatial components of the Cauchy stress is plotted for comparison, according to the considered loading paths. Therefore, we have tested three loadings: – simple shear ; – uniaxial extension ; – a complex combination : it corresponds to a shear loading then unloading (during 2 seconds), followed by a simple extension loading then unloading (during 2 more seconds).

25

This loading path has been applied through the transformation tensor Fij in 3D or F µν in 4D. It corresponds to a gate signal of 4s duration. The c This commercial code enables calculations have been performed by Zset . to calculate finite deformations with complex behavior models in 3D, so the present models have been projected in 3D before computing. Despite the possibility to compute calculations on a structure, the present results under concerned correspond only to a material point.

8.2 Numerical results for simple shear The loading is presented in figure 2. The results are shown in figures 3, 4 and 5, which presents the evolution of the stress components with time.

12 Fig. 2 Loading versus time, for simple Fig. 3 Cauchy stress σ versus time, for simple shear shear

Fig. 4 Cauchy stress σ 11 versus time, Fig. 5 Cauchy stress σ 22 versus time, for simple shear for simple shear

For simple shear loading, it is confirmed that the 3D models are not able

26

to reproduce the behavior of the reference model, including Jaumann and polar rate models. This can be observed for all the components. Moreover, when looking at 11 and 22 components, it appears that the 4D Lie rate 1 model is not correct, because it is not an integrable model. According to the definition, the 4D Lie rate 1 and 2 models present a correct behavior and reproduce the reference model, and therefore finite elasticity. 8.3 Numerical results for uniaxial extension The loading is presented in figure 6. The results are shown in figures 7, 8 and 9, which presents the evolution of the stress components with time. For

Fig. 6 Loading versus time, for uniaxial Fig. 7 Cauchy stress σ 11 versus time, extension for uniaxial extension

Fig. 8 Cauchy stress σ 12 versus time, Fig. 9 Cauchy stress σ 22 versus time, for uniaxial extension for uniaxial extension

uniaxial extension loading, it is again confirmed that the 3D models are not able to reproduce the behavior of the reference model, including Jaumann and polar rate models. This can be observed for all the components (except

27

12 which is zero for all the models). Moreover, when looking at 11 and 22 components, it appears that the 4D Lie rate 1 model is not correct, because it is not an integrable model. According to the definition, the 4D Lie rate 1 and 2 models present a correct behavior and reproduce the reference model, and therefore finite elasticity. 8.4 Numerical results for complex load The loading is presented in figure 10. The results are shown in figures 11, 12 and 13, which presents the evolution of the stress components with time. Because the common 3D models are definitively wrong, only the 4D models are compared.

Fig. 10 Loading versus time, for com- Fig. 11 Cauchy stress σ 11 versus time, plex load for complex load

Fig. 12 Cauchy stress σ 12 versus time, Fig. 13 Cauchy stress σ 22 versus time, for complex load for complex load

The observation of the different components shows that the 4D Lie rate 1 model if definitively not able to reproduce the reference model. Non-

28

integrable models produce wrong behaviors. Rate 2 model is correct. Two tests have been performed for the rate 3 model, for N = 3 and N = 4, which corresponds to the parameter in Eq. 72. 4D hydrostatic pressure and 3D hydrostatic pressure are not the same for deformed materials, even at the non-relativistic limit. It means that the choice of the parameter has to be done carefully when comparing the components of the stress tensor for different models (3D or 4D). Here to reproduce the reference model, only the choice N = 3 is correct. Indeed, the definition of the material parameters a1 and a2 has been performed in comparison to the usual 3D parameters as explained at the beginning of section 8.1. The correct identification of material parameters in relation with 4D models will be address in a future communication. Eventually, the treatment of complex loading is possible for different rate models hypo-elastic or not, using the 4D formalism and the Lie derivative. 9 Conclusion The application of the covariance principle of General Relativity ensures the frame-indifference of any physical law within arbitrary frame of reference. A law should thus be written using 4D tensors and using operations and operators involving exclusively 4D tensors. This principle has been applied to the construction of material constitutive models. Within this formalism, the constitutive model may or may not be indifferent with respect to rigid body superposition in the non-relativistic limit, may or may not be isotropic, but is, in any case, frame-indifferent. Next, we have proposed a 4D formulation for some 4D constitutive models. In this paper, the problem of hypo-elastic behavior has been investigated in the framework of the 4D differential geometry. It is thus possible to express such hypo-elastic behaviors by different ways. But only, one leads to an integrable and hypo-elastic model. Methodology can be carried out by derivative of a (hyper)elastic relation ; then, inversion of the (hyper)elastic expression to obtain a relation between the stress rate, the stress and the strain rate tensors but without explicitly the strain tensor. An important point mentioned is the choice of Lie derivative as transport operator to construct the hypo-elastic model. This choice ensures that: – the transport operator corresponds to a true time derivative. – the transport operator includes naturally the effect of time variation. – the stress transport is invariant with respect to superposition of rigid body motions. – the stress transport is frame-indifferent. When considering finite transformations of elastic materials (such as elastomers), it is also important to properly choose the correct entity as well as the correct relation. From infinitesimal transformations, it is possible to extend an elastic behavior to finite transformations by using at least three different kind of modelling. First, it is possible to directly replace the quantity by adapted ones copying the existing relation. Second, it is possible to consider a thermodynamical approach which is based on a non-dissipative

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condition that leads to the partial derivative of thermodynamic potentials with respect to physical variables. Third, it is possible to consider variations of quantities. A reference model given by the first kind of modelling and three models given by the third kind of modelling have been compared. Numerical investigations have been performed to confirm equivalence of the reference model with rate 2 and rate 3 models, when using the 4D Lie derivative for rate 2 model. The two later are also equivalent because they simply correspond to two different forms of the same model, but only one can be strictly said as 4D hypo-elastic. Rate 1 model has a different behavior with inelastic response for the considered loading paths (simple shear and uniaxial expansion), meaning that it cannot be used as a correct elastic models since it is non-integrable. Numerical results has proved the interest of the integrable model with 4D Lie derivative, whatever is its form (i.e. respecting the definition of hypo-elasticity or not). In a future work, it would be interesting to test the combination with viscosity and its capability to reproduce experimental data. Anyway, from a theoretical point of view, the present 4D approach has proved its capability to solve finite transformations problems of continuum mechanics, especially for reversible elasticity behavior. Acknowledgements Authors are indebted to C. Gay, F. Sidoroff and C. Stolz for their help through fruitful propositions, discussions and exchanges. We thank the R´egion Champagne-Ardenne and the Scientific Council of the Universit´e de Technologie de Troyes for their trust and financial support.

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