A constitutive model for describing decoupled material behavior in thickness and in‐plane directions

An increasing demand for paper and paperboard in engineering applications has emerged throughout the years. This leads to a growing interest in simulation tools to predict the material behavior, which is challenging due to the anisotropic structure of the material. It can be divided into in‐plane and out‐of‐plane behavior, which is mostly independent of each other resulting from the microstructure. Thus, a constitutive material model that captures in‐plane and out‐of‐plane behavior in a consistent manner is desired to simulate complex deformation states. The decoupling will be incorporated by a decomposition of the kinematics and the energy formulation into in‐plane and out‐of‐plane terms by use of structural tensors. This approach is flexible and can easily be extended to inelastic material behavior. In this work, the introduction of the decoupling into a constitutive material model for inelastic behavior is described. Visco‐elasticity is included through inelastic potentials. Three different potential formulations are given where focus is on the third, novel formulation, which defines inelastic strain evolution only in out‐of‐plane directions. The capabilities of this formulation are compared to the remaining potentials by simulations of a pinched cylinder that is loaded in the out‐of‐plane direction.

ip n 1 n 2 n 3 op F I G U R E 1 Illustration of preferential material directions -in-plane area () underlined in grey and out-of-plane direction ().material parameters, for example, the Poisson's ratio, to zero (cf.[3,13]), which is restrictive from a continuum mechanical point of view and limits the applicability to linear problems.In ref. [5], the authors introduce the decoupling through structural tensors aligned with the preferential material directions in the elastic material formulation.Nevertheless, the demand for a constitutive material model that captures the decoupling effect in a flexible manner such that an extension to inelastic material behavior is straightforward is desired.Thus, a decoupled material model for visco-elastic behavior using structural tensors is presented.This model is in alignment with the one introduced in ref. [14], which is a flexible model that can easily be extended to inelastic material behavior.This extension is shown by including visco-elasticity that is incorporated by different inelastic potential formulations that present different inelastic strain evolutions.In this work, an additional potential is introduced that defines inelastic strain evolution only for the out-of-plane direction.This is motivated by the densification effect and internal friction, which are most dominant in the out-of-plane direction (cf.[11]).Therefore, a model that captures inelastic strain evolution in the out-of-plane direction separately is straightforward.

DECOUPLING OF IN-PLANE AND OUT-OF-PLANE MATERIAL BEHAVIOR
The decoupling will be included by changing only the kinematics and the energy formulation.These two modifications are included easily in the model and lead to an independent material response in both directions.The split of the kinematics and the energy formulation is a general approach, which makes the model flexible for an extension to inelastic material behavior or adaption to different materials.

Structural tensors
The anisotropy is incorporated by structural tensors aligned with the three preferential material directions in analogy as in, for example, refs.[15][16][17][18].For paper and paperboard, we define the machine direction (MD), cross machine direction (CD) and thickness direction (ZD).Their orientation with respect to a paper roll is given in Figure 1.
We introduce director vectors of unit length   where the index  = 1, 2, 3 is related to the three material directions MD, CD and ZD, respectively.The out-of-plane vector,  3 , is defined orthogonal to the in-plane ones yielding to the relations  3 =  1 ×  2 and  3 ⋅  2 =  3 ⋅  1 = 0. From this, we define structural tensors by dyadic products of the corresponding director vectors, that is,   =   ⊗   .As we want to decompose the material answer into out-of-plane and in-plane behavior, we further introduce an in-plane structural tensor by  =  −  3 , which can be assumed to include all material directions except the out-of-plane one.Thus, we have the two structural tensors  and  3 to define the material behavior separately.

Modified invariants
The Helmholtz free energy is commonly described in terms of invariants of strain measures, more precisely, in terms of the right Cauchy-Green deformation tensor .This motivates the introduction of modified invariants to decompose the energy formulation into in-plane and out-of-plane terms in a simple manner.Thus, we apply the in-plane structural tensor, , onto the right Cauchy-Green tensor, , and formulate modified invariants by are the eigenvalues of .With this, the out-of-plane part is reduced in the formulation and only in-plane terms are described.The in-plane energy can be defined analogously to classical material formulations with these modified invariants, which makes the physical interpretation easy to follow.

Isoplanar-area-changing split
An additional approach that we adapt is the classical isochoric-volumetric split.We introduce an isoplanar-area-changing split where we further decompose the deformations into area-preserving and area-changing terms.
In the classical approach, the third invariant,  ∶=  3 = det(), is used for the decomposition into volume-preserving and volume-changing deformations as it defines the squared volume ratio.Analogously, the second modified invariant  ′ 2 is used in our approach as it is the squared area ratio, which can be shown by Nansons formula.It is further denoted by   ∶=  ′ 2 for comparability to the squared volume ratio  used in the isochoric-volumetric split.The decomposition into isoplanar and area-changing deformations is defined by ( where  iso  refers to the isoplanar term and   to the area-changing one.The area-preserving characteristic can be shown by inserting  iso  into the second modified invariant   instead of the total right Cauchy-Green tensor, which yields Thus, we have decomposed the kinematics into in-plane and out-of-plane deformations by use of structural tensors and further split into isoplanar and area-changing terms.

INELASTIC MATERIAL FORMULATION
A multiplicative decomposition of the deformation gradient of  =     into elastic terms,   , and inelastic ones,   , is introduced.This results in an additional configuration, which is named the intermediate configuration.Inelastic material models are commonly derived in this configuration but pull-back operations are required for the implementation of these models as the intermediate configuration is not uniquely defined.This non-uniqueness motivates the introduction of a co-rotated intermediate configuration which results from the multiplicative decomposition of   =     into its rotational part   , which is an orthogonal tensor, and the stretch tensor   , which is symmetric and positive definite.The co-rotated intermediate configuration is defined by a pull-back from the intermediate configuration by   .With this, we can define co-rotated elastic right Cauchy-Green tensor by C ∶=  −1      =  −1   −1  .The overbar ( •) indicates a quantity defined in the co-rotated intermediate configuration.The eigenvalues and symmetry properties between both intermediate configurations remain equal but the co-rotated one is not affected by the rotational non-uniqueness.Thus, all quantities in the co-rotated intermediate configuration are uniquely defined and can be implemented directly using automated differentiation tools such as Ace Gen [19,20].For a more detailed discussion on this topic, the reader is referred to [21].
The two in-plane director vectors are assumed to evolve with the deformation gradient, and thus, are defined in the intermediate configuration by ñ1 = (   1 )∕(|   1 |) and ñ2 = (   2 )∕(|   2 |) (cf.[22]).These definitions and a pullback operation to the co-rotated intermediate configuration lead to the description of the out-of-plane structural tensor in this configuration as We define a Helmholtz free energy in terms of the elastic right Cauchy-Green tensor and the out-of-plane structural tensor in the co-rotated intermediate configuration, that is,  =  * ( C , M3 ), and insert its time derivative into the isothermal Clausius-Planck inequality, − ψ +  ∶ 1∕2 Ċ ≥ 0. Applying the procedure of Colemann and Noll (cf.[23]) leads to the definition of the second Piola-Kirchhoff stress tensor, , and the reduced dissipation inequality,  red , that is, where L ∶= U  −1  holds and Γ is a stress-like quantity which is symmetric (cf.[17,22,24]).Hence, by defining D ∶= sym( L ), the reduced dissipation inequality can be reformulated to  red = Γ ∶ D ≥ 0, which must be fulfilled for thermodynamical consistency.Therefore, we introduce a potential ḡ as a positive, convex, zero-valued and scalar-valued isotropic function of the stress-like quantity Γ and define the evolution equation of the inelastic strain by where  is a relaxation time.

Choice of potentials
We introduce several potentials ḡ to capture different effects and introduce them in the current configuration by  for physical interpretation.Thus, they are described in terms of the Kirchhoff stress tensor  and the out-of-plane structural tensor in the current configuration M3 .In ref. [14], it is shown that tr The first two potentials are in alignment with ref. [14] where the first one describes inelastic strain evolution in in-plane directions when in-plane loadings are applied.It reads The second potential is defined analogously but with an additional separate out-of-plane term such that inelastic strains might also evolve in the out-of-plane direction but remain decoupled from the in-plane behavior.Thus, we define In this work, we investigate the effect of a third potential, which is only defined for out-of-plane inelastic strain evolution.The densification and internal friction effect that are most dominant in the out-of-plane direction (cf.[25][26][27][28]) motivate the introduction of an inelastic potential defined only for an out-of-plane evolution.Thus, it is defined by F I G U R E 2 Geometry and boundary conditions of the pinched cylinder.

Specific choice of the Helmholtz free energy
The Helmholtz free energy is additively split into an elastic and inelastic term,   and   , which both are further decomposed into isoplanar in-plane, area-preserving in-plane and out-of-plane parts denoted by  iso  ,   and  op , respectively.This yields The elastic energy is defined by whereas the inelastic one reads op =  3 4 The relations C , and Ω =  − M3 have been used here.
The same material parameters for the elastic and inelastic formulation are used here for simplicity but can be adjusted in future works.

NUMERICAL EXAMPLES
In the following, the influence of the third potential is compared to the first and second one.For this, the numerical example of a pinched cylinder was simulated as already given in ref. [14] for the first two potentials.The geometry and boundary conditions are given in Figure 2.Only one eighth of the cylinder was simulated due to symmetry effects.The out-of-plane direction was assumed pointing outwards -normal to the cylinder surface -and the specimen was fixed in  1 and  2 direction at  3 = 300 mm.A displacement  = 100 mm was applied over 100 s at the node at  2 = 300 mm and  The aim of these simulations was to investigate the inelastic strain evolution with the three different potentials.Figure 3 (upper line) shows the in-plane component of the inelastic right Cauchy-Green deformation tensor,    , computed with potentials one to three from left to right.It can be seen that the first two potentials lead to an inelastic strain evolution due to the loading.No inelastic strains in the in-plane directions can be observed for the third potential, which has been newly introduced here and is defined for out-of-plane inelastic strain evolution only.
The lower line in Figure 3 illustrates the out-of-plane inelastic right Cauchy-Green deformation    .The results on the left, which show calculations with the first potential, indicate no evolution as the potential only defines in-plane inelastic strain evolution.The out-of-plane part of the second and third potential (Figure 3 lower line, middle and right) are defined in a similar manner (cf.Equations ( 9) and ( 10)), leading to similar localized inelastic strain evolutions.Thus, the introduction of the third potential shows the possibility of defining out-of-plane inelastic strain evolution separately from the in-plane one.

CONCLUSION
A model for decoupled material behavior in thickness (out-of-plane) and in-plane directions has been presented that is motivated by the macroscopic material behavior of paper and paperboard.The decoupling has been introduced by a decomposition of the kinematics and the Helmholtz free energy into in-plane and out-of-plane parts.The kinemat-ics, precisely the right Cauchy-Green deformation tensor, is further decomposed into area-preserving and area-changing deformations leading to the isoplanar-area-changing split, which is in alignment with the classical isochoric-volumetric split.Including the decoupling only by decomposing the right Cauchy-Green deformation tensor and the Helmholtz free energy leads to the flexibility of this model for adjustments to different material responses, and thus, to extending it to inelastic material behavior, which is a major advantage.To prove this flexibility, a visco-elastic material formulation has been introduced in terms of a co-rotated intermediate configuration.The latter one has been used as it is uniquely defined in contrast to the intermediate configuration, and therefore, quantities defined within this configuration can be implemented directly using automated differentiation tools.The model presented is in alignment with the one depicted in ref. [14].In this work, we present an additional inelastic potential that defines inelastic strain evolution only in the out-of-plane direction when out-of-plane loading is applied.The influence of the potential is shown by simulations of a pinched cylinder that have been presented in ref. [14] for the first two potentials and are compared to the newly introduced potential.It is shown that by the specific choice of the potential different material behaviors can be obtained.The introduction of the third potential given here indicates inelastic material behavior for the out-of-plane direction only, which is an advantage for adjusting the material model more precisely to the material behavior of paper and paperboard in the following since the out-of-plane response shows inelastic behavior and is independent of the in-plane one.

A C K N O W L E D G M E N T S
() = tr( Γ) and  ∶ M3 = (1∕) Γ ∶ sym( −1  M3 ) holds, where  = tr( −1  M3 ).By use of these relations, the potentials introduced in the current configuration,   , are then reformulated in terms of the co-rotated intermediate configuration, ḡ .

F I G U R E 3
In-plane (upper line) and out-of-plane (lower line) component of the inleastic right Cauchy-Green deformation tensor computed with Potential 1 (left), Potential 2 (middle) and Potential 3 (right).