Finite element formulations for large strain anisotropic material with inextensible fibers
 P. Wriggers^{1}Email author,
 J. Schröder^{2} and
 F. Auricchio^{3}
https://doi.org/10.1186/s4032301600793
© The Author(s) 2016
Received: 6 April 2016
Accepted: 13 July 2016
Published: 11 August 2016
Abstract
Anisotropic material with inextensible fibers introduce constraints in the mathematical formulations. This is always the case when fibers with high stiffness in a certain direction are present and a relatively weak matrix material is supporting these fibers. In numerical solution methods like the finite element method the presence of constraints—in this case associated to a possible fiber inextensibility compared to a matrix—lead to so called lockingphenomena. This can be overcome by special interpolation schemes as has been discussed extensively for volume constraints like incompressibility as well as contact constraints. For anisotropic material behaviour the most severe case is related to inextensible fibers. In this paper a mixed method is developed that can handle anisotropic materials with inextensible fibers that can be relaxed to extensible fiber behaviour. For this purpose a classical ansatz, known from the modeling of volume constraint is adopted leading stable elements that can be used in the finite strain regime.
Keywords
Background
Many different approaches were developed over the last decade to formulate finite elements for anisotropic material with inextensible fibers. The problem is the high stiffness ratio between fiber and matrix material with the limit case of inextensible fibers where this ratio tends to infinity. This is physically related to the exact fulfilment of the kinematic constraint associated with the inextensibility of fibers in certain directions.
Generally the method of Lagrange multipliers provides a possibility to fulfil such constraints for small and finite deformations. In this paper the Lagrange multiplier approach is employed to model anisotropic material behaviour at finite strains. Furthermore a relaxed version, i.e., the perturbed Lagrangian formulation, is used to model extensible fibers as well. Boundary value problems that incorporate extreme constraints cannot be solved using the finite element method with standard displacement interpolations. This leads to well known locking phenomena.
The main source of locking problems is that the mathematical formulation has to deal with constraints or is set up such that constraints are fulfilled approximately, like in penalty or other related methods. These problems are wellanalyzed for geometrically linear problems in the case of volume constraints, see e.g. [4, 13, 28, 30]. They were investigated in the mathematical community quite early, see [3, 7], and are now well understood leading to the Babuska–Brezzi (BB) condition. It can be employed to investigate the stability behaviour of mixed finite elements in the linear range. Within nonlinear problems the BB condition can only be used at certain stages of the analysis, see e.g. [9].
Different strategies were pursued in computational mechanics over the last years in order to circumvent locking effects. It became evident that element ansatz functions that interpolate the deformation or displacement field within an element with first order shape functions (bi or trilinear interpolation) do not converge properly when applied to problems with constraints like incompressibility or distinct anisotropic material behaviour. Thus different variational formulations were explored in order to construct finite elements that can be used for problems with constraints. Approaches include reduced integration and stabilization, see e.g. [31] for the linear case. Many variants can be found in the literature. It was shown that the reduced integration has to be used together with stabilization and can be extended to nonlinear problems, see e.g. [6, 17] leading to elements that are in general locking free for incompressibie deformations. Additionally these elements are very efficient due to reduced integration. However stabilized elements rely on artificial stabilization parameters and thus the numerical solution can depend on theses parameters in certain cases.
Formulations, based on the mixed variational principle of HuWashizu, were developed, e.g. see Simo and coworkers who introduced the enhanced strain elements first for the geometrically linear, e.g. see [24] and then for large deformations, [22, 23]. However, these elements depict nonphysical instabilities at certain deformation states.
Other mixed finite element formulations, that are stable, perform well in the framework of small deformations and isotropy, e.g. see [5, 8]. Extensions to problems undergoing finite deformations are discussed in [1, 2] for the case of incompressibility. For finite strain anisotropic material behavior it is even more complex to find good finite element formulations. Many classical approaches that were designed for fiberreinforced materials depict nonphysical behavior, see e.g. [12, 27]. Discussions related to the correct formulations of the mathematical model for anisotropic behaviour can be found in e.g. [11, 18]. These authors state that all fiberrelated terms have to be provided in the energy by the complete deformation tensor and not by its isochoric part.
Reduced integration schemes using a special stabilization have been successfully applied to the simulation of composite reinforced material, see Hamila and Boisse [10]. Also special interpolations eliminated locking behaviour for composite materials, see ten Thjie and Akkerman [26]. Still many researchers use HuWashizubased displacement, dilatation and pressure formulations, early introduced for incompressible materials by [25], for nearly incompressible materials with highly stiff fibers (like in arterial walls), see [29] and the references therein. However for strongly anisotropic material with inextensible fibers these approaches have limited performance, especially at finite strains.
A new formulation was presented in [21] who introduced a novel finite element formulation that is developed especially for anisotropic materials, based on isotropic tensor functions as discussed in [19, 20]. There the constraints, associated with the anisotropy, are controlled by an additional deformation measure. A secondorder tensorial Lagrangemultiplier was introduced via a discontinous ansatz. This approach offers the opportunity to reduce the interpolation order of the anisotropic part and thus is able to relax the constraints due to anisotropy. This formulation leads to a stable methods for the solution of problems with anisotropic materials undergoing large strains.
In this paper a different approach is followed. Here the constraint of inextensibility in fiber directions is formulated as a constraint and also as a limiting case. For this purpose a constraint equation is introduced within a Lagrange multiplier scheme. This allows to select ansatz functions as well for the displacement field in fiber direction as for the fiber forces. Additionally a perturbed Lagrangian formulation is introduced to relax the constraint condition and to be able to introduce real fiber stiffnesses. Since it can happen that fibers buckle locally when subjected to a compressive force a special form of the constraint is introduced that acts only for tension states. Furthermore this formulation can be used to enforce strain states in fiber direction that can be associated with e.g. muscle contractions in biomechanics applications or specific piezoelectric effects in fibers.
The performance of the developed element formulations is compared to existing formulations using benchmark problems. All numerical results were obtained with the AceGen/AceFEM system developed in [14–16].
Anisotropic material with inextensible fibers behaviour at large strain
In this section a summary of the continuum mechanics background is provided for the formulation of problems exhibiting anisotropic response in finite elasticity. The formulation is reduced to the necessary equations that are needed to formulate the problem in AceGen. This omits many derivations since automatic differentiation is used. All formulations are presented with respect to the initial configuration. The formulation accounts for transversely isotropic material behaviour by using a mixed approach. It is assumed that the material is not extendable in the given fiber direction \({\mathbf {a}}\).
Continuum mechanics
Kinematical anisotropic constraint
Lagrange multiplier formulation

One constraint. The Lagrange multiplier term related to the constraint of a material that is not extendable in the direction \({\mathbf {a}}\) yields with (7)where \(\sigma _{c} \) is the Lagrangian multiplier that physically represents the fiber stress related to the constraint.$$\begin{aligned} W^{tiL}({\mathbf {C}},\sigma _c ) = \sigma _c \,( \text{ tr } [\,{\mathbf {C}} {\mathbf {M}}\,] 1) \end{aligned}$$(9)

Several constraints. For more than one constraint direction one can introduce \(n_c\) additional directional unit vectors \({\mathbf {a}}_i\) and associated structural tensors \({\mathbf {M}}_{i}\) and reformulate (9)$$\begin{aligned} W^{tiL}({\mathbf {C}},\sigma _{c\,i} ) = \sum _{i=1}^{n_c} \sigma _{c\,i} \,( \text{ tr } [\,{\mathbf {C}} {\mathbf {M}}_{i}\,] 1) \end{aligned}$$(10)

Constraints for tension only. In case that the response of the fiber system only occurs in tension states (9) can be rewritten by using the Macauley bracket: \(\langle x \rangle = \frac{1}{2} (x + \Vert x\Vert )\). This choice yieldswhere \(\alpha \) is a positive integer that can be selected in the range \((1,\ldots , 4)\).$$\begin{aligned} W^{tiL}({\mathbf {C}},\sigma _c ) = \sigma _c \,\langle \text{ tr } [\,{\mathbf {C}} {\mathbf {M}}\,] 1\rangle ^\alpha \end{aligned}$$(11)

Constraints for a given stretch. If a stretch \(\bar{\lambda }_c\) is prescribed in a certain direction \({\mathbf {a}}\), then one can formulate, using (8), the constraint$$\begin{aligned} W^{tiL}({\mathbf {C}},\sigma _c ) = \sigma _c \,(\, \text{ tr } [\,{\mathbf {C}} {\mathbf {M}}\,] \bar{\lambda }_c^2\,). \end{aligned}$$(12)
Perturbed Lagrangian formulation
The perturbed Lagrangian formulation can also be used to introduce a fiber stiffness that is related to the physical behaviour of the fiber. In that case \(C_c\) has a physical meaning.
Penalty formulation
Mixed element formulation
For the mixed interpolation tetrahedral and hexahedral elements are selected and compared. For both element formulations a quadratic interpolation for the displacement field \({\mathbf {u}}\) and a linear interpolation for the mixed variable \(\sigma _c\) is selected. This choice is motivated by the classical mixed formulation for the incompressibility constraint. For anisotropic material with inextensive fiberss the variable \(\sigma _c\) is the stress component related to the constraint, e.g. the stress in direction of \({\mathbf {a}}\).
Note that in the mixed form for the incompressibility with the constraint \((J1)\), that is related to the determinant of \({\mathbf {F}}\), a cubic function of the components of the deformation gradient describes this constraint. In the case of the constraint (9) for anisotropic materials this function is only a quadratic form of the components of the deformation gradient. Thus it is not obvious that the same choice for the interpolation of \(\sigma _c\) will be sufficient.^{2}

for a tetrahedron with 10 nodes (\(n_u=10\))with \(\kappa = 1\xi \eta \zeta \) and$$\begin{aligned} N_1= & {} (2\xi 1)\xi ,\,\, N_2 =(2\eta 1)\eta ,\,\,N_3 =(2\zeta 1)\zeta ,\,\,N_4 =(2\kappa 1)\kappa , \nonumber \\ N_5= & {} 4\xi \eta ,\,\, N_6= 4\eta \zeta ,\,\, N_7= 4\zeta \xi ,\,\, N_8= 4\xi \kappa ,\,\,N_9= 4\eta \kappa ,\,\, N_{10}= 4\zeta \kappa , \end{aligned}$$(20)

a hexahedron with 27 nodes (\(n_u=27\))with \(I= 1,\ldots ,27\). \(N_I(s)\) is given for the vertex nodes by$$\begin{aligned} N_I(\xi ,\eta ,\zeta ) = N_I(\xi )\,N_I(\eta )\,N_I(\zeta ) \end{aligned}$$(21)for s being either \(\xi ,\eta \) or \(\zeta \). Here \(s_I\) is related to a specific coordinate of a vertex node of the hexahedron in the space of the reference coordinates \((\xi ,\eta ,\zeta )\) with \(\xi _I = \{1,+1\} \), \(\eta _I = \{1,+1\} \) and \(\zeta _I = \{1,+1\} \), see Fig. 1. For the mid nodes the shape function \(N_I(s)\) are given by$$\begin{aligned} N_I(s) = \frac{1}{2} (1s_I) [s(s1)] +\frac{1}{2} (1+s_I) [s(s+1)] \end{aligned}$$with \(\xi _I = 0 \), \(\eta _I = 0 \) and \(\zeta _I = 0\).$$\begin{aligned} N_I(s) = (1s^2) \end{aligned}$$
Examples

Tetrahedral elements for the constraint formulation (9), (10), (11) and (12) with quadratic ansatz functions (21) for the deformations and linear ansatz, see (23), for the Lagrangian multiplier \(\sigma _c\). These elements are labeled T2A1 in the following.

Tetrahedral elements for the perturbed Lagrangian formulation (15) with quadratic ansatz functions (20) for the deformations and linear ansatz, see (22), for the Lagrangian multiplier \(\sigma _c\). These elements are labeled T2A1P in the following.

Hexahedral elements for the constraint formulation (9), (10), (11) and (12) with quadratic ansatz functions (20) for the deformations and linear ansatz, see (22), for the Lagrangian multiplier \(\sigma _c\). These elements are labeled H2A1 in the following.

Hexahedral elements for the perturbed Lagragngian formulation (15) with quadratic ansatz functions (21) for the deformations and linear ansatz, see (23), for the Lagrangian multiplier \(\sigma _c\). These elements are labeled H2A1P in the following.

Tetrahedral elements based on the quadratic ansatz functions (20) for the deformations. These elements are labeled T2, and the associated penalty ones T2P.

Hexahedral elements based on the quadratic ansatz functions (21) for the deformations. These elements are labeled H2, and the associated penalty ones H2P.
Cook’s membrane problem
Mesh density
N  Mesh division 

2  \(2 \times 2 \times 1\) 
4  \(4 \times 4 \times 2\) 
8  \(8 \times 8 \times 4\) 
16  \(16 \times 16 \times 8\) 
In a first computation a mesh with \(N=16\) was used to obtain the load displacement curve for Cook’s membrane problem. The element used for this simulation was the H2A1P formulation. The load was applied in 10 even load increments \(\lambda \) with \(\Delta \lambda = 0.25\). The parameter for the perturbed formulation was selected as \(C_c=10^6\).
For the computation of the load displacement curve the vertical displacement of the mid node \((X,Y,Z) = (48,52,5)\) of the plane at the right end of the cantilever beam is chosen which is related to the response in the direction of the load \(p_0\), see Fig. 3. The load displacement curve is depicted in Fig. 6. Furthermore the outofplane displacement in zdirection is plotted that shows the outofplane deformation of the cantilever beam due to the anisotropic material.
The deformed mesh on the right in Fig. 6 was computed with a mesh of \(16 \times 16 \times 8\) elements which lead to a total number of 59058 degrees of freedoms. The deformation at the final configuration clearly depicts the twist in the deformed shape due to the anisotopic constraint at large deformations. The solution was computed with several load steps. In total eight load steps were applied for all discretizations reported in Fig. 7. The convergence behaviour was robust, six iterations per load step were needed for all discretizations to obtain convergence. In this solution procedure Newton type convergence was observed. When using the automatic load stepping scheme of AceFEM the total load can be applied in five load steps which reduces the total number of iterations to 33 and thus leads to reductions in computing time by a factor of around 1.5.
A convergence study is performed for the fully constraint case, using the Lagrangian multiplier formulation (9). The element formulations H2A1 and T2A1 are compared. Figure 7 depicts the convergence of the vertical displacement at point (48,60,0).
It can be observed that the hexahedral element performs slightly better for coarse meshes. Here one has to acknowledge that the coarsest mesh (\(N=2\)) of the triangularization for the tetrahedral elements is not symmetric and thus will have a certain bias. Nevertheless the displacement for the coarsest mesh is close to the final result, being approximately only 5% off.
In order to show the dependency of the solution on the penalty or fiber stiffness parameter \(C_c\) a series of computations were performed. The perturbed formulation (14) was used and a mesh division of \(N=8\) selected.
Here it can be observed that the anisotropic constraint in direction of a is not enforced for a penalty parameter \(C_c \le 10\). Then there is an intermediate stage where the stiffness of the fiber changes the deformation state. This is related to parameters between \(10 \le C_c \le 10^5\). Finally from \(C_c > 10^5\) on there is no further change, thus the parameter is sufficient to enforce the constraint. Additionally we note, that for \(C_c > 10^7\) the result is the same as for the pure Lagrangian multiplier formulation (9).
A convergence study is now performed for the perturbed Lagrangian formulation, see (14). The results are compared with the penalty formulation (16) for a parameter of \(C_c=10^6\). The results can be found in Fig. 9.
It can be seen that the penalty formulation does not converge to the same solution as the perturbed Lagragnian formulation. Here a penalty parameter was used that is sufficient to fulfill the constraint, see Fig. 8. Thus it is clear from Fig. 9 that the penalty formulation locks. Furthermore it is interesting to observe that for a penalty parameter of \(C_c > 10^7\) the penalty method for the H2 as well as for the T2 element diverged while the perturbed Lagrangian formulations H2A1P and T2A1P are still robust.
Shear deformation of a beam
The constitutive data are provided for the Lame constants: \(\mu = 500\) and \(\lambda =1000\). The direction of anisotropy is given by \({\mathbf {a}} = \{1,0,0\}\) which enforces the constraint in x direction. The beam is clamped at the left end using the boundary conditions: \(u_x=0\) for all nodes at \(x=0\), \(u_y=0\) for all nodes at \(x=0\) and \(y=0\) and \(u_z=0\) for all nodes at \(x=0\) and \(z=0\). The beam is loaded by a constant traction of \(p_y=5\) at the right end.
For larger loads local buckling occurs. This is due to the high compressive stresses at the bottom of the beam. The load deflection curve in Fig. 11 depicts the nonlinear behaviour and the final deformation of the beam for a mesh with T2A1P elements. The deformed configuration of the beam (no scaling) shows clearly near the clamping local buckles that in the end led to the large deflection of the beam. This is related to a bending torsion state which is triggerd by the local buckling.
It is clear that in reality an internal local buckling of the fibers will occur and thus the formulation (11) has to be applied. This leads then to a bending of the beam without local buckling, since fiber buckling due to compressive stresses is not present anymore. However, since the fibers in tension cannot extend, the deflection related to (11) is smaller than the deflection of a beam under bending without any constraints.
Rolling up of a beam
The selected finite element mesh is depicted on the left side of Fig. 13. The final state of the deformation is shown on the right side of Fig. 13. It is obtained for the load factor \(\beta = 10\).
It is clear that large strain states can be imposed by the formulation (12).
It can be conlcuded that the active enforcement of a given stretch using formulation (12) can be applied to generate arbitray deformation states depending on the selection of the direction vector \({\mathbf {a}}\) and the magnitude of the prescribed stretch \(\bar{\lambda }_c\).
Bias extension test
The plot in Fig. 18 shows the mesh convergence for the T2A1 element formulation using \(N=4,8,16,32\) and 64 elements per side. As can be seen the result is insensitive with respect to the mesh size. The deviation for \(N=4\) is related to the fact that the mesh cannot model the different shear zones, see Fig. 17.
It is worth noting that the final displacement can be reached with the T2A1 element in one single load step for all mesh sizes, while the T2 element needs about 25 load steps to reach the final configuration. Thus the new T2A1 element is a lot more robust than the T2 element for such applications.
Conclusions
Finite elements for large strain anisotropic behaviour were developed in this paper. Special emphasis was put on a formulation that was able to enforce inextensible fiber extensions for anisotropic materials exactly using a constraint formulation. This led to a Lagrange multiplier method with different ansatz spaces for the deformations and the Lagrangian multipliers (fiber stresses). The mixed approach shows a robust convergence behaviour and does not lock. A comparison with standard quadratic elements depicts the locking behaviour of these elements when the constraint was added via a penalty term. Furthermore the mixed approach led to a more robust behaviour in the iterative procedure needed to solve the associated nonlinear problems.
It is well known that illconditioning can occur when a large penalty parameter \(C_c\) is selected. Thus in practise the penalty formulation is only able to approximately enforce the constraint condition (8).
In the linear case both conditions, while being different, yield a linear dependence on the components of the displacement gradient. Thus there the choice of using the same ansatz function for the pressure (incompressibility) and the fiber stress (anisotropy) is justified.
Declarations
Author's contributions
The theoretical derivations are joint work of PW, JS and FA. PW developed the AceGen Code. The Cook’s membrane problem was developed by JS. The shear deformation problem of the beam stems from FA and the third example is due to PW. The final example was suggested by the reviewers. All authors read the approved the final manuscript.
Acknowledgements
The first and second author acknowledge the support of the ”Deutsche Forschungsgemeinschaft” under contract of the SPP 1748, No. WR19/501 and SCHR570/231.
This contribution is dedicated to our friend and exceptional scientist Pierre Ladeveze on behalf of his 70th birthday.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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