On Purely Mechanical Simple Kinematic Internal Constraints

The classic purely mechanical approach to materials with simple kinematic internal constraints is supplemented. A right Cauchy–Green tensor which locally represents the kinematically admissible restricted domain of a finite hyperelastic stress response function is constructed explicitly. It satisfies all imposed constraints identically. It is obtained by a procedure which annihilates the banned modes of deformation in the actual placement. A unique direct sum based stress decomposition is obtained. Further, a procedure is provided, to seamlessly relax constraints, or reversibly, that allows constraints to smoothly develop under loading starting from an unconstrained description. The involved relaxation of internal constraints is briefly illustrated herein. References to published full feathered applications are given where the method is used and verified in finite element form.


Introduction
The standard reference to the classic purely mechanic theory of simple kinematic constraints is Truesdell and Noll [1,Sect. 30]. It covers the historical aspects and provides an axiomatic presentation of theory with examples to incompressibility and simple inextensibility. It defines the mechanical reactions to imposed purely kinematic constraints as the energy conjugate stress, that is workless in all deformations satisfying the constraint. Further, the classic theory defines the constitutively determinate work performing stress, called the extra stress, as the difference between the total stress and the workless, constitutively indeterminate, stress. Thus, it introduces an additive decomposition of the total stress. Relying on these definitions, the so-called Principle of material determinism for simple materials subject to B A. Zdunek (C 1 ) modification of an actual Cauchy-Green tensor to become an element of the constraint manifold, and for, (C 2 ) relaxing constraints 1 by turning them into variable substitutions, making the material 'nearly' constrained, and finally by, (C 3 ) indicating the impact of these two contributions on the solutions of boundary value problems in computational solid mechanics.
Using the modified Cauchy-Green tensor as an argument of the hyperelastic stored energy function confines its domain to the actual constrained situation. 2 Using the Clausius-Planck form of the second law of thermodynamics (for isothermal processes) and employing the famous Coleman Noll procedure [2] induces automatically the proper direct sum decomposition of the stress into a workless reactive part and a work performing active part. 3 Turning constraint equations into variable substitutions (C 2 ) allows a seamless description from unconstrained to fully constrained and reverse under loading for example. Finally, contribution (C 3 ) bridges material mechanics and computational mechanics.
Incompressibility, inextensibility and preservation of the angle between two material elements is considered here. Combinations of them are considered as example applications. Put in other words, (C 1 ) allows to state an internally constrained finite hyperelastic (boundary value) problem in an inner product vector space that preserves volume, length or angle, simply or in combination. The construction of the constrained right Cauchy-Green stretch tensor is performed by a rank-one update procedure annihilating one target banned mode at the time in the Eulerian convected metric tensor representing the actual transplacement, in the sense of Green an Zerna [15]. In other words, the annihilation is done point-wise by composition in the local material fibre triad. The result is pulled back to the reference placement where the constrained right Cauchy-Green stretch tensor is the Lagrangian form of the Eulerian convected metric tensor. It should not be confused with the augmenting internal variable representing physical metric tensor introduced by Valanis and Valanis and Panoskaltsis [25,26]. The right Cauchy-Green tensor and the convected Lagrangian metric tensor are equivalent designations used interchangeably.
Following the fibre deformation in terms of convected local fibre triads is a key ingredient providing maximum possible direct insight into a phenomenologic type of description, [16]. It takes advantage of given preferred directions. The verification of the finite element realisation of the stiffening two-fibre family reinforcement (Example 2 herein), can be found in the recently published work [31]. To my best knowledge, this work is the first of its kind.
This contribution is concluded in that spirit by briefly presenting the option (C 2 ) to relax internal constraints keeping 4 the direct sum decomposition of the stress used in the fully constrained case. Tested and verified finite element implementations which can be interpreted in terms of the so-called generalised metric approach are available. The 3-field, displacement, dilatation and pressure formulation by Simo, Taylor and Pister [14, (1985)] for near incompressibility is the first to be mentioned. The following formulations [27][28][29][30] for near incompressibility and/or strongly transversely isotropic finite hyperelasticity are further realizations.
The outline of the remainder of this contribution is: The notation used is given in a short paragraph in the Appendix A. A condensed exposition of the kinematics used is given in 1 Called extrapolating off the manifold by Carlson et al. [7]. 2 For example, the unimodular Cauchy-Green tensor for incompressibility. 3 For example, decomposition in a spherical and a traceless part for incompressibility. 4 Transforming the Lagrange multiplier to a constitutively determinate stress response function.
Sect. 2. The kinematic aspects of simple internal constraints are gathered in Sect. 3. The procedure to construct the constrained right Cauchy-Green tensor is presented in Sect. 3.2, (C 1 ). The constitutive aspects of simple internal constraints are illustrated for finite hyperelasticity in Sect. 4. The relaxation of internal constraints, in other words, the procedure to connect with the related unconstrained material is briefly presented in Sect. 5. The claimed generalisation of the classic theory is discussed in Sect. 6. The note is summarised and concluded in Sect. 7. Small examples are used to illustrate concepts and show applications throughout the note. Finite element applications are found in [27][28][29][30][31].

Kinematics
Adopting the continuum mechanics view, let κ t : B → E be a smooth time-dependent embedding of a material body B into the differentiable Euclidean point manifold E . The translation space at instant t ≥ 0 associated with E is denoted U t . The placement of, or region occupied by, the material body at time t is the image P t := κ t (B). The position of a material particle X ∈ B is given by x t := κ t (X) ∈ P t . The placement P 0 := κ 0 (B) at the instant t = 0 is used as a reference. The reference position is denoted X := κ 0 (X). The embedding is invertible, for X ∈ P 0 , X = κ −1 0 (X) ∈ B. The abbreviated notation x := κ t (X) is used for the actual position and the actual placement is denoted P = κ t (B). The composed point mapping, ϕ(X, t) := κ t • κ −1 0 : P 0 → P, defines a common referential description of the deformation of a material body used in continuum mechanics. It is assumed to be bijective and continuously differentiable with respect to the time parameter t as many times as needed. The tangent map is the linear transformation F := Grad ϕ : T X P 0 → T x P.
Parametrisation It is henceforth assumed that a locally bijective and smooth parametrization ξ A (X) around X ∈ P 0 is given, or that a material curve can be reconstructed from a given distinct unit tangent fieldM(X) in P 0 , called a preferred direction in the phenomenologic theory of anisotropic solids, see Spencer [16], Boehler [17] and Zhang and Rychlewski [18]. A convective local parameterization ξ a (x) around x ∈ P is chosen, cf. Green and Zerna [15]. Then the natural tangent base vectors {G A } and {g a } on T X P 0 and T x P are defined in a standard fashion by, At least one of these base vectors will correspond to a preferred direction,M in the presence of an anisotropic constraint. Transverse isotropy caused by a simple inextensibility involves a single preferred direction. The local convected triad is completed by constructing two perpendicular base vectors in the isotropic plane. Local orthotropy, caused by two simple inextensibilities, involves two distinct, in general oblique, preferred directions. The third base vector is constructed using the vector cross product. For isotropy the preferred local triad is constructed to be orthogonal. An initially isotropic solid may in principle develop inextensibility under load. Arterial soft tissue is known to develop Fung-like exponential stiffening.
The co-vector bases on the dual spaces T * X P 0 and T * x P are denoted {G ∼ A } and {g ∼ a } respectively. They are determined with the aid of the duality pairing as follows, where δ A B and δ a b are Kronecker delta symbols. The metric tensors on the referential and actual tangent spaces are defined by, They are linear maps that are assumed to be symmetric and positive definite 5 by definition, i.e., G ∈ Sym + (T * X P 0 ⊗ T * X P 0 ) and g ∈ Sym + (T * ϕ(X) P ⊗ T * ϕ(X) P), respectively. Using the convected parametrisation the deformation gradient becomes, F = δ a · A g a ⊗ G ∼ A . Summing up, the deformation gradient F, its adjoint F * and their inverses F −1 and F − * take the forms: The basis vectors G A and g a henceforth represent tangents to material curves at X ∈ P 0 and at ϕ(X, t) ∈ P. In case of anisotropy one or more of these represent convected preferred directions.
In view of (2.4a), the tangent mapping F : G A → g a is written; where the so-called convected metric tensor g = g(ϕ) has the explicit form: where λ 2 a := g a 2 are stretches and α ab := λ −1 a λ −1 b g a · g b are the direction cosines, i.e., the shears, between the material line elements g a and g b for a, b = 1, 2, 3 with b > a, in T ϕ(X) P, respectively. Further, is noted that the referential value of convected metric tensor (2.6) 2 has to be provided. It is constructed exploiting any given preferred direction as outlined in the paragraph Parametrisation above.
Take for example strong transverse isotropy, caused by an inextensibility constraint, one of the stretches in (2.6) measures the length ratio in the associated preferred direction. Likewise, a direction cosine will measure the actual angle between two provided preferred directions. The parametrisation by local convected triads provides the maximum possible insight into the local mechanics relying on a phenomenologic theory. See also the full feathered examples in [31].
The last identity in (2.5) follows from the definition of duality and gives rise to the introduction of the symmetric and positive definite right Cauchy-Green tensor C ∈ Sym + (T * X P 0 ⊗ T * X P 0 ); Finally, it is noted that the right Cauchy-Green tensor is the pull-back of the convected metric tensor g(ϕ) to the reference configuration. Moreover, the possibility of using it as a Lagrangian metric at X ∈ T X P 0 is noted. The postulate of material impenetrability takes the following form in convected frames, where J is the volume ratio. Further, the stretch λ F of a material line element in the material where the right handside is an example of using C as the Lagrangian metric, cf. (2.5). Following Spencer [16], Boehler [17] and Zhang and Rychlewski [18], a stress-like Lagrangian structural tensor, is determined by a given preferred directionM . Here, it will be employed as a basis vector in the local triad, i.e.,M ∈ {G A }, by construction. A structural tensor is idempotent, A 2 F = A F , which follows from the normalisation, G F = 1.
Further, the standard set of admissible transplacements ϕ(X, t) is defined by, where ∂P 0D denotes the part of the boundary where the transplacement is prescribed. Moreover, the tangent set to C is introduced as the standard set of admissible material velocities,φ(X, t); (2.11) It is also known as the standard set of admissible variations due to the connections δϕ ∼ = ϕ =φ t , see Simo and Hughes [24, Sect. 7.2.1]. Thus, a material velocity (increment or variation) must respect homogeneous Dirichlet boundary conditions in order to be identified with a test function or variation in the standard Bubnov-Galerkin sense. This requirement becomes clear linearising a weak formulation. 6 The rate form (or first variation) of the condition of impenetrabilityJ (ϕ) > 0, which here is expressed in terms of the strain rateĊ = DC(ϕ) [φ] at ϕ ∈ C . Finally, we may note that C is a submanifold of the Euclidean point manifold E and that T ϕ C is a submanifold of the associated translation space U .

Simple Kinematic Internal Constraints
In this section the kinematic aspects of subsidiary kinematic conditions are considered. Consequences on the work conjugate stresses are derived from the constraint rates.

Constraint Manifold Theory
This subsection is essentially an adaption of the constraint manifold theory following Podio-Guidugli and associates [4][5][6]. Some observations are added. The constraint manifold is embedded into the set of admissible transplacements (2.10) and its tangent is embedded into the set of admissible variations (2.11). It provides a bridge to computational mechanics. The embedding couples the restrictions imposed by a kinematic constraint on the strain C(ϕ) and its rate or first variation, to the transplacement ϕ and its rateφ or variation δϕ, respectively.
Many materials vigorously resist certain transplacements to such an extent that it is a perfect approximation to exclude them from the standard set of admissible transplacements (2.10) C (ϕ). Classic examples concern, pure dilatation J 1/3 (C) and uniaxial stretch λ F (C; A F ) in a material direction given by A F . An exclusion of a class of transplacements from C may provide considerable simplification in solving boundary-value problems analytically, while the same restriction may call for a special formulation solving them numerically. Incompressibility is the leading example with vast literature in both branches. The kind of additional subsidiary kinematic conditions considered here, appended to the set of admissible transplacements C , are called internal constraints, since they act in the domain occupied by the body, and in addition require a modification of the material description.
Here we consider the restriction of the purely mechanical theory of a non-aging, i.e., time invariant, so-called simple material in the sense of Noll [19,20], to the subclass of hyperelasticity. Basically, for hyperelasticity, there are three material stretches and three direction cosines in the actual placement that may be constrained point-wise, see the convected metric tensor g(ϕ) (2.6). That is, we need to consider the actual value of its Eulerian form g(ϕ), or equivalently, its Lagrangian form in the reference placement, C(ϕ), at ϕ(X, τ ) for τ = t . In order to determine the work-less and the work performing stresses the strain rateĊ is needed. Simple internal constraints in conjunction with thermoelasticity and in general thermodynamics for simple materials are out of the scope for this contribution. Extensions are dealt with by, Gurtin and Podio-Guidugli [5], Negahban [10] and by Bertram and Glüge [8], respectively.
The dimension of symmetric tensors is six. Consequently, there can be at most six independent internal constraints simultaneously. A fully constrained material is rigid. The convected metric tensor for a rigid material coincides with the referential metric tensor G for t ≥ 0. An isotropic internal constraint can be expressed in terms of the principal invariants of the Lagrangian metric tensor C alone, while an anisotropic internal constraint, in the sense of Spencer [16], involves the joint invariants of C and a set of structural tensors defining the anisotropy, of the type (2.9). Definition 3.1 (Simple internal constraint as a constraint manifold) Following Podio-Guidugli and associates [4][5][6], a simple, time invariant, internal constraint, may be considered in terms of a differentiable manifold, where ϕ ∈ C is an admissible transplacement, and where the constrained quantity, φ, takes on the given constant value φ 0 := φ(G(X); S ) ∈ R it has in the referential placement, see (2.7). Further, S is a non-empty set, containing the given referential metric tensor G −1 and possibly a given referential structural tensor field, e.g., in the form (2.9). See [16][17][18] for the use of structural tensors for anisotropic solids. Finally, the function φ is supposed to be continuously differentiable sufficiently many times with respect to C.
The following basic two families of constrained quantities φ i (C(ϕ)) ∈ F i , i = 1 or i = 2 are used: whereλ a := J − 1 /3 λ a , a = 1, 2, 3 are the so-called volume-preserving stretches. These two families separate the classes of compressible and incompressible materials, see [29]. The second family is subject to the normalisation condition det[J − 1 /3 F] ≡ 1. A collection of 1 < n ≤ 6 compatible simple internal constraints is denoted the constraint manifold, see also Steigmann [11]. In passing it is noted that C is frame invariant and that the presence of the referential metric tensor G −1 implies that the identity mapping I ∈ M . Finally, a simple minded view suggests to consider a kinematic constraint φ(C; S ) = φ 0 as a level set in R 6 . The recent contribution by Vianello [6] discourages from such a departure, however.

Just as a function
. From a simple minded geometric point of view, a constraint may be seen as an iso-surface in R 6 . There are however cases where this analogy breaks down, see [6].
The requirements of linear independence of multiple constraints from Carlson et al. [7] and Negahban [10] are recalled. Reinforcement with inextensible fibre families, requires that their preferred directions are distinct.

Definition 3.3 (Linear independent constraints) A number of constraints
. . , n ≤ 6 define a linear independent set of tensors. That is, if their linear combination is the zero tensor, if and only if all scalar multipliers are zero; In analogy with the gradient ∂ X F (X) to a surface F (X) = 0 at X in R 3 , the tensor gradient ∂ C φ(C) to the constraint φ(C) = φ 0 at C in R 6 , may be conceived as a normal, respectively.
the set of normals to the constraint manifold M evaluated for a ϕ ∈ C a at C(ϕ), (3.4), is recalled, cf. [7,Eqs. (2.4)]: (3.6) Note that for a simple inextensibility in the preferred directionM(X), the normal to the inextensibility constraint is A(X) see (2.9).
Further, the time rate of a constraint is essential for the development. It is computed by the chain rule of differential calculus as, Here DC(ϕ)[φ] is the first variation of the right Cauchy-Green stretch tensor at ϕ in the direction of the material velocityφ. It provides a constraint onφ(X, t) or equivalently on the strain rate at ϕ,Ċ(ϕ) = DC(ϕ) [φ]. The collection of the constraints on the strain rate is added to the standard set of admissible material velocities (2.11).
8) where the strain rateĊ(ϕ) = DC(ϕ) [φ] is computed at ϕ ∈ C a . Then, the augmented set of admissible material velocities which may be called the augmented set of admissible variations is formed as, The definition of a stress tensor which is workless in all admissible transplacements ϕ(X, t) ∈ C a with admissible material velocityφ(X, t) ∈ T ϕ C a at X ∈ P 0 corresponding locally to a stretch rate DC(ϕ)[φ] ∈ T C M is also needed. The phenomenologic description of kinematics in the presence of simple internal constraints is completed. The workless stress, in other words the stress reaction, is neither an element of the kinematics nor of the constitutive part of the stress response. It is energy conjugate to the kinematically inadmissible modes of deformation, akin to reactions in rigid-body mechanics. Definition 3.6 (Workless stress tensor) For any admissible material velocityφ(X, t) ∈ T ϕ C a corresponding to an admissible strain rateĊ ∈ T C M (Definition 3.5), and a tensor S R ∈ N C M (Definition 3.4), the vanishing of the scalar product, defines an energy conjugate workless stress tensor which, for a set of (Lagrange) multipliers i ∈ R, i = 1, 2, . . . , n ≤ 6, has the representation, For n = 6 any material is rigid. There is no stretch or shear C = G. The stress response is in the form (3.10). For n < 6, the difference between the total stress 7 and the workless stress, S − S R =: S E may be non-trivial, for non-trivial C ∈ M . According to the principle of determinism for simple materials subject to internal constraints [1,Sect. 30,p. 70] the so-called extra stress S E can be computed from the history of the right Cauchy-Green stretch tensor C t (s) ∈ M for objective simple materials in the sense of Noll [20], see also Day [19]. For hyperelasticity we only need that C(t) ∈ M at the current instant t and thaṫ C(t) ∈ T C M . The proposed additive decomposition of the total stress [1,3,7] follows from the projection theorem in linear algebra Bowen and Wang [21,Th. 17.3] (provided in Appendix D) applied to the set of symmetric tensors S ∈ Sym(T X P 0 ⊗ T X P 0 ), as the direct sum Sym = Sym R ⊕ Sym E . The classic theory [1] is coherent with the constraint manifold theory [3,7]. The latter goes one step further. It removes a possible scalar valued indeterminacy in the extra stress S E 8 by normalisation. The so-called active stress S A := S − S R is introduced. There, by postulate, S A is constructed orthogonal to S R in the sense of the Lagrangian inner product (S R , S A ) (T X P 0 ,C) = 0, see Appendix B. Imposing this scalar valued subsidiary condition makes the stress decomposition unique.

Construction of a Constrained Cauchy-Green/Lagrangian-Metric Tensor
In this subsection the procedure to construct a constrained Cauchy-Green tensor denoted C ∈ M by modifying a given actual Cauchy-Green tensor C is presented, i.e., item C 1 in the Introduction, the first of the main contributions of this work. The resulting constructed tensorĈ =Ĉ(C) induces a unique direct sum decomposition of the total stress into workless and work performing parts of S R and S A corresponding to the actual constraint manifold. Using the constrained Cauchy-tensor bypasses the need to assume a normalisation condition, e.g. (S R , S A ) C = 0 as in [3,7]. In view of the definition (2.7), the constrained Cauchy-Green tensorĈ may be interpreted as a constrained convected Lagrangian metric tensor. The constrained convected Eulerian metric tensor denotedĝ is obtained simultaneously. Definition 3.7 (Constrained Cauchy-Green tensor) For a given arbitrary Cauchy-Green tensor C, a constrained Cauchy-Green tensorĈ =Ĉ(C) is constructed such that, Remark: In view of the Definition (3.1) the constructed Cauchy-Green tensor satisfies each constraint as an identity; i.e., Further,Ĉ has the following two properties,Ĉ (3.12) 1 implies that no modification is performed if given C ∈ M , and where (3.12) 2 implies that the constrained convected metricĈ ∈ M does not depend on any of the constrained quantities.
Let f i (φ i ) : T ϕ(X) P → T ϕ(X) P be a bijective linear transformation on the actual placement depending on the single constrained quantity φ i ∈ F 1 or φ i ∈ F 2 , of the type defined by Definition 3.1. The composition of n ≤ 6 such mappings is introduced: Proposition 3.1 (Construction of a constrained Lagrangian convected metric) Given the constraint manifold in the sense of Vianello [6] M , the mapping f (3.13), the actual tangent map F and the background convected metric tensor g(ϕ); the following composition defines the admissible tangent map: where the associated constrained Eulerian and Lagrangian convected metric tensors are identified;Ĉ Proof In Appendix C it is shown by calculation thatĈ ∈ M for the three basic cases incompressibility, inextensibility and preservation of angle.
The constrained Lagrangian convected metric/Cauchy-Green tensorĈ ∈ M provides the proper referential objective strain measure for the internally constrained situation. Example 2 (Two oblique plain extensibility constraints) See Appendix C.2. For two oblique plain extensibility constraints, φ i (C; G −1 , A i ) = φ 0 i , φ i = λ i and φ 0 i = 1 for i = 1, 2, the constrained Lagrangian metric tensor becomes;

16)
Example 3 (Angle preservation/unshearability) See Appendix C.3. The constraint fixing the referential direction cosine φ 0 = α 0 ab between two material line elements G a and G b in T X P 0 is; Note that G a and G b are employed as referential basis vectors and that the second parametric argument is a structural tensor in the sense of Spencer [16]. The annihilation mapping for this case f(γ ab ; g a , g ∼ b ) : T ϕ(X) P → T ϕ(X) P and its inverse is formed in terms of the spatial natural convected basis vectors: 9 f γ ab ; g a , g The expression for γ ab is; For a = 1 and b = 2 the resulting constrained Lagrangian metric tensor is obtained as: (3.20) Further, using the constrained convected metric approach the constraint rates are identically satisfied, by construction. Proof Time differentiating the defining identity ofĈ (3.11), Since,Ĉ ∈ M , we have that ∂Ĉφ i ∈ N C M and in view of (3.22) the strain rate d tĈ is identically perpendicular to the constraint normal ∂Ĉφ i . Thus, d tĈ ∈ T C M .
(3.25) Proposition 3.3 (Strain rate projection)P * is idempotent, Proof Multiplying both sides of (3.25) byP * and simplifying proves the proposition for strain ratesĊ ∈ T C M : It follows that it is a projection on the space of strain rates: Further, it follows from the identityP * * =P thatP is also idempotent.

Proposition 3.4 (Stress projection)P is a projection on the stress space:
Proof The result follows from inspection of the scalar product, Further, let subspaces Sym R and Sym A denote the kernel and the range ofP respectively. The projection theorem in linear algebra, provides the direct sum decomposition, (3.28) Assuming the workless stress S R in the form (3.10) the constrained convected metric approach yields;P for a set of non-trivial scalars i , i = 1, 2, . . . , n.
One of the main results of this note can be stated. Namely, the constrained convected metric formulation induces a direct sum based decomposition of the stress. The result is stated by the following proposition; Proposition 3.5 (Stress decomposition) For any total stress tensor S ∈ Sym(T X P 0 ⊗ T X P 0 ) and projectionP := [∂ CĈ ] * by the projection theorem [21,Th. 17.3] in linear algebra we have, (3.30) Proof S A =PS ∈ Sym A and S R = (I −P)S ∈ Sym R respectively. Here, the workless stress S R is spanned by the normals to the constraint manifold using (3.10).
Note, that we have not yet involved any constitutive assumption for the complementary (working) stress S A .
Next, the projection is constructed explicitly. To that end, both sides of the definition P * := ∂ CĈ are multiplied from the right by the tensor ∂ φ i C representing the sensitivity of the Lagrangian metric tensor with respect to the i-th constrained mechanical quantity. Simplifying using (3.12) 2 we obtain, which means that ∂ φ i C is in the left null-space ofP. An essential auxiliary result is derived.
Proposition 3.6 (Induced duality) The tensors ∂ φ i C ∈ T C M and ∂ C φ i ∈ N C M are dual: Proof The result follows from evaluating the partial derivative, ∂ φ i φ j . Its rationale is evident recalling that the constraints in M are required to be linearly independent.
The explicit form of the constrained convected metric induced stress projectionP can now be stated.

Proposition 3.7 (Explicit projection representation) The explicit form of the stress projec-tionP is;P
Proof It is deduced using the definition of the null-space (3.24) and that of the left nullspace implied by (3.31). Using (3.32) it is readily checked that it is idempotent and has the right kernel and range.
Example 2 (Two oblique plain extensibility constraints) For two oblique plain extensibility constraints the explicit expression for the stress projectionP becomes, where H = 2, 1 for F =1, 2. Using idempotency and the left null-space property (3.31) the working stress S A =PS becomes complementary to the workless stress (Definition 3.6), see Proposition 3.5.

Example 3 (Angle preservation/unshearability) For the angle preservation constraint (3.17)
with a = 1 and b = 2 the stress projectionP assumes the form, With the explicit form of the stress projectionP in hand theory of the constrained convected metric method can be finalized by interpreting the mechanical meaning of the introduced Lagrange multipliers { i } n i=1 . Further the projection can be characterized as orthogonal or oblique. Remarkably, the interpretation of the multipliers can be established without assuming any particular constitutive form of the material. Proof The relation is obtained using the derived decomposition of the stress (3.30), the representation (3.10) for the workless stress and the duality (3.32). This proposition concludes the unique stress decomposition using the constrained convected metric approach.
The result deserves a pair of illustrations.
Example 1 (Incompressibility) Inserting the expression ∂ J C from Example 1 into (3.37) the Lagrange multiplier is determined as the mean hydrostatic stress,

38)
Example 2 (Two oblique plain extensibility constraints) Using the expression (3.35) 2 the Lagrange multipliers are determined by; For oblique constraints shear stresses are involved.
Example 3 (Angle preservation/unshearability) Using the expression (3.36) 3 the Lagrange multiplier for angle preservation in the 12-plane (a = 1 and b = 2) becomes the associated shear stress; The last task that needs to be investigated is whether the stress projectionP is self-adjoint or not. That is, ifP is an orthogonal or oblique projection. The complementary projection tô P, Q := I −P will be a sum of component projectionsQ i . The total projection will be selfadjoint if all the components are self-adjoint. To simplify matters a bit a single component of the complementary projection is studied, For arbitrary tensors S, T ∈ Sym(T X P 0 ⊗ T X P 0 ) self-adjointness follows if, the left and right hand-sides below are equal, i.e., LHS = RHS where: LHS: where ( * , * ) C is the proper Lagrangian inner product (see Notation (N.2) and Appendix B), and where the definition T := CTC and the identity T , S = S , T is used.

43)
where β i > 0. The multiplier β i can be determined taking the scalar product on both sides with ∂ C φ i and using (3.32); Proof The result is deduced by inspection of (3.42).
Example 1 (Incompressibility) For incompressibility, n = 1: β = 3 /4J 2 , The spherical projection in incompressibility is shown to be self-adjoint. Therefore, the stress decomposition in incompressibility is governed by an orthogonal projection.
Example 2 (Two oblique plain extensibility constraints) For two oblique plain extensibility constraints, n = 2, F =1, 2: (3.46) where H = 2, 1 for F = 1, 2. The constrained spatial material/convected metric tensorĝ(ϕ) will be oblique, in general. The stress decomposition induced by the constrained convected metric approach will therefore be governed by an oblique projection in case of two oblique simple extensibility constraints.
Thus, it can be concluded that the constrained convected metric approach induced decomposition of the stress may be orthogonal or oblique. The outcome is case dependent. 10

Working Stress Determination; Hyperelasticity, Clausius-Planck Procedure
In this section the consequences of the of simple internal constraints on the constitutive aspects are considered. The determination of the constitutive expression for the working stress induced by the constrained convected metric approach is outlined in a purely mechanical situation, using the Clausius-Planck procedure. For brevity and transparency the procedure is illustrated assuming a hyperelastic material response.
is assumed. Parametric dependence on structural tensors is suppressed for transparency. The strain energy is stated in the following augmented Lagrange multiplier form; Inserting this ansatz into the Lagrangian form of the Clausius-Planck law, yields the following expression for the total stress induced by the constrained convected metric approach; whereP is the stress projection, see Propositions 3.4 and 3.7.
Proof The result is obtained applying the celebrated Coleman and Noll procedure [2]. It states that the arguments ofĊ and˙ i must vanish independently. The argument ofĊ yields Remark 2 SinceP 2 =P is idempotent one may write, by assumption. The implication of this form is two-fold: first, notably the argument within brackets is now the unconstrained form of a hyperelastic solid, second, the form underlines the fact that the constrained convected metric approach separates the kinematic aspect from the constitutive one. In other words, an unconstrained material may for example be subjected to a testing programme that keeps the stretch fixed in a given direction. It will respond essentially as if it was subject to an internal extensibility constraint. Recall that the projection P * := ∂Ĉ/∂C is induced by the constructedĈ ∈ M . Note that, it is not claimed that the constructed constrained Lagrangian convected metric tensorĈ is unique. However, in view of the projection theorem (see Appendix D, and Oden and Demkowicz [32, Prop. 2.7.1, p. 172]) P induces a unique decomposition of the hyperelastic stress response function ∂Ψ (C)/∂C. Given the constraint manifold it determines the corresponding work performing stress.
Given a referential preferred unit directionM 1 ∈ P 0 the associated stretch α in the transplacement ϕ is calculated as α 2 =M 1 · CM 1 . Using isotropic scaling by α −1 yieldŝ F 1 = α −1 1F, such that,F 1M 1 = α −1 αm 1 =m 1 , where m 1 = 1. Thus,F 1 is an admissible deformation gradient for a simple inextensibility in the preferred directionM 1  (1) The isotropic scaling changes the volume ratio by the factor α −3 , while the advocated method scales the actual volume ratio by the factor α −1 . Assume that the actual deformation gradient is a simple extension without lateral contraction in the preferred directionM 1 . The presented construction method in Sect. 3.2 properly yields an identity mappingF 2 = I and the right volume ratio det [F 2 ] = 1, while the isotropic scaling version yields an admissible but physically not acceptable result,F 1 = α −1 I + (1 − α −1 )m 1 ⊗M ∼ 1 and a wrong, non-trivial, modified volume ratio, (2) Further it is evident that the isotropic scaling method can in general only provide a candidate admissible deformation gradient for a case with a single constraint.
In conclusion, the presented method in Sect. 3.2 constructs an admissible deformation gradient which is physically acceptable. It has been demonstrated also for cases with two simultaneous constraints, see Example 2 and Remark 7. Here it is shown, that other constructed candidates may be members of the constraint manifold, but may not necessarily be physically acceptable.

Extension to Nearly-Constrained Materials and Unconstrained Ones-The Generalised Metric Approach
It would be preferred that an unconstrained material gradually transforms into the corresponding internally constrained material as its response in a certain mode of deformation declines when penalized by increasing the governing material parameter. The literature is however filled with separate constitutive postulates for uncoupled nearly-constrained materials. Again, near incompressibility relying on the celebrated Flory split of the deformation gradient F = J 1/3F provides a rich spectrum of such contributions. A prominent one is the pressure, dilatation, unimodular Cauchy-Green tensor, (p, J,F * gF) three field formulation of the virtual work principle by Simo, Taylor and Pister [14]. This class of constitutive assumptions can be bypassed. It has no explicit part in the relaxation of constraints presented here. In this section a target component constraint manifold M φ i is (on-off) relaxed. Or, when an additional physical quantity φ(C) becomes involved in a generalised constraint.

The Generalised Metric Approach
In this subsection the procedure to construct a generalised Cauchy-Green/Lagrangianmetric tensor denotedC by modifying a given constrained Cauchy-Green/Lagrangianmetric tensorĈ is presented, i.e., item C 2 in the Introduction, the second main contributions of this work. A question is whether the constrained convected metricĈ can be relaxed keeping the same stress decomposition, and simultaneously transforming an associated Lagrange multiplier to a material response function. The answer is affirmative and the solution is strikingly simple, close and rational. The generalising steps are outlined very briefly, omitting technical details. The generalised manifold is marked by a tilde signM φ ; Remark 4 It is noteworthy that the simple internal constraint (3.1) is replaced by a variable substitution (5.1). Formally, the parametric dependence on a referential value φ 0 is replaced by an independent variableφ. However, in the left hand-side ϕ ∈ C while the independent field in the right hand-side is in an auxiliary scalar point field. The matter is clear considering a weak formulation, see [31].

Remark 5
In computational mechanics, i.e., in finite precision arithmetic, a relaxed constraint manifold is often employed approximating the associated limiting one, Not all constraints need to be relaxed, r ≤ n. For transparency and simplicity, however, it is assumed henceforth that all constraints are relaxed, r = n: Each variable substitution is satisfied as an identity, point-wise for X ∈ P 0 ; φ i (C; S ) ≡φ i ⇒C ≡ C, (5.5) in the strong sense, whereC should be understood as the value of the tensor functioñ C(C(ϕ),

Definition 5.2 (Generalised Lagrangian metric) A generalised Lagrangian metric tensor,
A mode of deformation which was extracted/annihilated from the spatial metric (Proposition 3.1) can be reinstated using the same mapping (3.13). The change of variable (5.1) can be performed. The following reinstatement mapping is introduced to that end; Proof The generalised convected Lagrangian metricC ∈M , by (5.5) φ i (C) ≡φ i for all variable substitutions i; for all variable substitutions i.
The equivalence (5.5) implies the rate equivalence, From the time rate of (5.1) and (5.9) the following identity it follows that, where the under-braced conclusion is obtained in view of (5.10). Using (5.11) the duality (3.32) can readily be generalised to, Computing the partial derivative ∂φ iφ j (C;φ j ).

Proposition 5.3 (Generalised strain rate and stress projection)
The generalised metric approach induces a projection of the strain rate; (5.14) Idempotency [P * ] 2 =P * follows using the generalised duality (5.13). The stress decomposition is obtained as the dual projection,P =P * * : Proof The identity (5.14) 2 follows from (5.12) and (5.10). The r-relaxed components in S R are no longer reactions but constitutively determinate.
Proof There is nothing really to prove in this procedure. The outcome of Step 3 is stated in the next Proposition 5.5. Step 4 is given by Proposition 3.5 exchangingP byP.
To each auxiliary kinematic variableφ i a work-conjugate material stress response function, denoted˜ i is associated, which previously was the Lagrange multiplier, i , see Definition 3.6. Their determination (Step 3 in Proposition 5.4) assuming hyperelasticity is outlined. Example 1 (Incompressibility) Applying (5.16) to relaxed incompressibility and simplifying one then obtains the classic constitutive expression for the volumetric response function: Example 2 (Two oblique plain extensibility constraints) Applying (5.16) relaxing two oblique plain extensibility constraints and simplifying one then obtains the constitutive expression for the fibre tension response functions: Example 3 (Angle preservation/unshearability) Relaxing an angle preservation constraint in the 12-plane (a = 1 and b = 2) one obtains the constitutive expression for the associated shear stress,˜ Remark 6 In view of the folding property of projections 11 the r relaxed responses can be removed from the decomposition in the strong point-wise sense. Considering a weak formulation it is however an advantage, or even the purpose, to keep the set of auxiliary variables {φ i ,˜ i } r i=1 separate allowing a mixed formulation. The finite element realisation of a reinforcement of a material by two oblique strongly anisotropic fibre families is considered in the recently published work [31].

Discussion
The presented constrained convected metric approach to hyperelastic solids with simple internal constraints is coherent in essence with the classic theory Truesdell and Noll [1] and with the constraint manifold approach Podio-Guidugli and associates [4][5][6] and Carlson et al. [7]. From a historical point of view, the idea developed from two observations. The first was the view that the right Cauchy-Green tensor may be seen to be the referential Lagrangian convected metric tensor, obtained by pull-back of the spatial metric tensor, 12 the second was that its unimodular version F * ḡ F =F * gF 13 satisfies the incompressibility constraint identically, by construction. The employment of given preferred directions in the 11 P + Q = I. 12 Follows on using tensor algebra on manifolds [12,13], where C : T X P 0 → T * X P 0 , is metric like, while in classic continuum mechanics [1] it is introduced as F T F : T X P 0 → T X P 0 with no immediate connection to the spatial metric tensor. 13 See the connection to the celebrated split of the deformation gradient F = J 1/3 1F due to Flory [22] used in the Simo, Taylor and Pister [14] three field formulation of the virtual work principle. sense of Spencer [16] constructing a special-purpose local-triad based convected parametrisation is a key element to handle especially multiple anisotropic constraints with oblique preferred directions, [31].
The principle of determinism for simple materials subject to internal constraints [1,Sect. 30,p.70] introduces an additive decomposition of the stress. Without a normalisation condition it is not unique. In the classic theory the normalisation condition is optional; it is suggested by example [1,Sect. 30,Eq. (30.17)] see also Spencer [16,p. 184]. It serves two purposes. First it is used to remove indeterminacies from the work-performing stress. Having done that, it enables the mechanical interpretation of the Lagrange multiplier(s), e.g. as the mean hydrostatic stress or as a uniaxial tension.
In the constraint manifold theory Carlson et al. [7] it is assumed that the work-performing stress is perpendicular to the workless stress. It must be emphasized that this is an assumption. To my best knowledge there is no generally accepted principle that prescribes it. One of the main results of this contribution is that the relation between the work-performing stress and the workless stress is determined by the constructed constrained Cauchy-Green tensor to be an element of the constraint manifold, starting from an arbitrary actual Cauchy-Green tensor. It is found that these stresses are not always perpendicular. Sometimes they are oblique. Two examples with possible oblique stress projections (decompositions) are provided. Inextensibility in two oblique preferred directions and a preservation of the angle between two oblique preferred directions. The consequences are interesting and reaching beyond the scope of this work.
Further, it is noted that the direct sum decomposition of the Lagrangian stretch rate is induced by the fact it is an element of the tangent set to the constraint manifold. The direct sum decomposition of the total stress follows from duality. Remarkably, it can be completed formally without any reference to a particular material law, in line with the observation by Carlson et al. [7]. The determination of the actual work performing stress response evidently requires a material law. The Clausius-Planck procedure is used herein for that purpose. It is a restriction to the purely mechanical situation of the celebrated Coleman-Noll approach based on the Clausius-Duhem law, see Coleman and Noll [2]. The restriction to finite hyperelasticity is done purely for transparency.
The classic theory of internal constraints requires the Cauchy-Green tensor to be in the constraint manifold. Adding this requirement to the the set of admissible transplacements (Definition 3.2) shows that a trial solution is requested to satisfy each kinematic constraint point-wise. Further, the classic theory of internal constraints requires the time rate of the Cauchy-Green stretch to be in the tangent set associated with the constraint manifold, at the admissible strain. Adding this requirement to the the set of admissible variations (Definition 3.5) shows that a test function δϕ ∼ = ϕ =φ t is requested to satisfy the first variation of each constraint, point-wise. From the computational point of view, this is very hard to realise. It is known that it is difficult to construct test functions in three-space which are exactly solenoidal point-wise. A closer look at equation (3.23) indicates that the use of the modified Cauchy-Green tensor (Definition 3.7) enables the use of standard test functions.
The generalisation of the method for unconstrained materials is almost trivial. It relies on the observation that the Lagrange multiplier method can be changed into a variable substitution. A constraint is transformed into a variable substitution, introducing auxiliary kinematic variables. In essence, the transformation yields a Hu-Washizu approach. Relaxing the constraints turns Lagrange multipliers into material response functions. The number of constraints is reduced accordingly. Point-wise they may be folded on account of the projection theorem in linear algebra [21,Th. 17.3]. The relaxed degrees of freedom may be kept providing a basis together with the fully constrained ones for a mixed formulation. This possibility is exercised in the recently published work [31]. It should be noted that the compositions employed to annihilate and reinstate deformation modes are performed in the spatial configuration, without the introduction of an intermediate configuration, as opposed to other celebrated multiplicative splits of the deformation gradient, e.g. the one used for elasto-plasticity due to Lee [23].
Finally, the presentation of the constrained convected metric method is confined to simple or composed elementary constraints concerning compressibility, extensibility in a given preferred direction, or to the change of the angle between a pair of preferred directions. More general simple internal constraints, like for example the Ericksen constraint, C, G −1 = 3, fall outside the scope of this work.

Summary, Conclusion and Outlook
The purely mechanical consequences of imposing simple kinematic internal constraints are discussed. The constraint manifold restricting the set of admissible transplacements is formed. A Lagrangian setting using the reduced, objective, form of a constraint in terms of the right Cauchy-Green stretch tensor is employed. Starting from an arbitrary Cauchy-Green tensor a modified Cauchy-Green tensor is constructed that satisfies all the subsidiary conditions in the constraint manifold as identities. It is shown that the time rate of a subsidiary condition written in terms of the modified Cauchy-Green tensor induces a projection. It defines the admissible Lagrangian strain rate as an element in the tangent set to the constraint manifold. By duality it defines a corresponding stress decomposition. It defines a work performing stress. A duality condition defining the complementarity between the workless part of the stress and the work performing part is identified. It also governs the determination of the Lagrange multipliers in terms of the total stress. The stress decomposition need not be orthogonal. It may be oblique. The sufficient condition for orthogonality (self-adjointness) is derived. Examples of oblique decompositions are provided. The determination of the work performing constitutive stress response function is exemplified using the Coleman and Noll procedure [2] based on the Clausius-Planck law for hyperelasticity. The theory and procedure for the modification of an arbitrary Cauchy-Green tensor such that it satisfies all constraints point-wise for an actual transplacement extends the classic theory [1] and the constraint manifold theory [7], remaining coherent with the classic foundations.
The theory and procedure for the relaxation of constraints seamlessly extending to unconstrained materials is shown. It gives rise to the so-called generalised metric method. A unified theory for a number of verified finite element applications [14,[27][28][29][30][31] is presented. Possible extensions are indicated, for example, formulation of a preservation of the angle between two families of material line elements alone or in combination with inextensibility or incompressibility. Extensions to non-isothermal conditions. Consideration of other not so elementary constraints, for example the Ericksen constraint etc.
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where α (DE) := λ −1 (D) λ −1 (E) (E , D) C is the Lagrangian expression for the corresponding spatial direction cosine, wherein λ (D) := D C and λ (E) := E C are the stretches in these directions. Using spatial unit direction vectors d := λ −1 (D) FD, |d| = 1 and e := λ −1 (E) FE, e = 1, the spatial bijective simple shear-like mapping for this case is given by, (C.8) The right Cauchy-Green stretch tensor (Lagrangian metric tensor) for a simple preservation of angle is computed as,Ĉ =F * gF = F * ĝ F. Evaluating the inner product between E and D in terms of the Lagrangian inshearable metricĈ, and using the expressions (C.8) yields, Inserting (C.9) in (C.6) shows thatĈ ∈ M insh , which completes the proof.

Remark 7
The simultaneous annihilation of a dilatation and a volume preserving stretch in the referential direction D in the deformation ϕ(X, t) with deformation gradient F = Grad ϕ is performed using a simple uniaxial extension with a lateral contraction, confer (C.3). The annihilation mapping used is; where J = det F is the actual dilatation, andλ = d ∼ ,FD is the actual volume preserving stretch in the referential direction D defined asλ := J − 1 /3 λ and whereF := J 1 /3 F is the actual unimodular deformation gradient.