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Many materials such as biological tissues, polymers, and metals in plasticity can undergo large deformations with very little change in volume. Low-order finite elements are also preferred for certain applications, but are well known to behave poorly for such nearly incompressible materials. Of the several methods to relieve this volumetric locking, the method remains popular as no extra variables or nodes need to be added, making the implementation relatively straightforward and efficient. In the large deformation regime, the incompressibility is often treated by using a reduced order or averaged value of the volumetric part of the deformation gradient, and hence this technique is often termed an approach. However, there is little in the literature detailing the relationship between the choice of and the resulting and stiffness matrices. In this article, we develop a framework for relating the choice of to the resulting and stiffness matrices. We examine two volume-averaged choices for , one in the reference and one in the current configuration. Volume-averaged formulation has the advantage that no integration points are added. Therefore, there is a modest savings in memory and no integration point quantities needed to be interpolated between different sets of points. Numerical results show that the two formulations developed give similar results to existing methods.

Many materials exhibit nearly incompressible responses over some part of their deformation. Biological tissues, rubbers, metals undergoing plastic deformation, and soils under undrained conditions are examples. To model these materials in a finite element (FE) context, care has to be taken to use elements designed for nearly incompressible behavior. The fact that low-order finite elements perform poorly for nearly incompressible media is well known, and treatment for this issue has received significant attention. There are several ways to view the problem, but one is that standard elements have too many incompressibility constraints which overwhelm the shear response. While higher-order elements can relieve this, even they do not behave optimally. And for certain classes of problems, such as some dynamic problems or problems involving jumps in the displacement field, low-order elements are generally preferred.

Many solutions have been proposed to relieve this volumetric locking effect in low-order elements, particularly quadrilaterals and hexahedra. Reduced integration [

Because they can be implemented relatively simply in a displacement-only framework, the so-called

For the finite deformation case, these methods have been extended using a so-called

In this paper, we develop an 8-node hexahedral element that averages the Jacobian over the element. While the integration procedure adds some modest complexity to the computation time to the code, it avoids tracking variables at an extra integration point. Extra integration points can add memory costs. They can also add significant complexity in multiphysics problems or history-dependent materials, though we do not investigate those here. We examine integral averages in both the reference and current configurations, and detail the relationship between the choice of

While beyond the scope of this paper, it bears mentioning that the

The remainder of the paper is organized as follows: Section 2 discusses integral averages for

A few notes on notation: We employ the summation convention in this paper, that a repeated index within a single term of an expression has an implied sum over the range of the index. For example

Outer products are represented with the ‘

First- and second-order tensors are denoted in bold, e.g.

We use curly braces for second-order tensors converted to vectors in Voigt notation. Kinematic quantities have doubled off-diagonal terms unless stated otherwise, while other quantities do not. For example

Similarly, fourth-order tensors converted to Voigt-form matrices are denoted with brackets, e.g.

The basis of the

Here,

where

Many forms for

An alternative approach is to use an averaging scheme or stabilization procedure. We adopt the former in the paper. Specifically, we examine two possibilities

and

The first is a volume average of the Jacobian over the reference volume of the element

Note: As discussed in [

The appropriate function for the

where

First, we must derive an expression for the modified velocity gradient

The expression for

Here we define

Note that this has the familiar form of the small strain

For the choice of

Hence

Curiously, averaging J over the reference configuration results in

Averaging J over the current configuration results in a somewhat more complicated formulation. In this case

Hence

In a finite element context, we can factor out the element nodal velocity vector implicit in the equations from the finite element shape functions. Recall that

where

Here,

In other words, it is the tensor equivalent of the standard updated Lagrangian

where

Similarly, for the Jacobian averaged in the current configuration

Therefore

With the correct expression for

In implementation, this expression is generally rearranged into a vector form such that

In this case, the nodal submatrix

The full

where

We assume that the gradient of the weighting function, or virtual velocity vector, takes the same modifications as that used in the physical velocity gradient. Hence the element internal force vector may be written

This derivation may also be carried out in a total Lagrangian setting, however, the volumetric part does not separate out as cleanly and the formula is more complicated.

Note: It is not strictly necessary to use the same averaging for the virtual velocity gradient as for the actual, or, in fact, any averaging. The authors in [

The resulting formulation is somewhat simpler, but is also nonsymmetric. As the authors point out in that article, some materials have a nonsymmetric tangent stiffness already, and for these materials little is lost. However, for materials with a symmetric tangent stiffness, the proposed formulation creates a symmetric stiffness matrix, which has advantages in storage and solution time.

The stiffness matrix may be derived by taking a directional derivative, or another pseudo-time derivative. The derivation is relegated to the appendix, where the latter approach is taken. If

where

The formula is general, and can be applied for any choice of

For the Jacobian averaged in the current configuration

Clearly the second formula is more complex than the first. These formulas are derived in the appendix.

In this section, the performance of the proposed

In the first three numerical examples, a nearly incompressible nonlinear elastic block fixed on bottom and subjected to three different loading conditions are presented. The block is 2 meters high with a square cross section of 1 meter, shown in

For this set of examples, the material model used is a Neo-Hookean model that follows the decomposition of strain energy function into isochoric and volumetric components:

in which U is the penalty function enforcing incompressibility defined as

where K is a generalization of the linear bulk modulus. The isochoric part may be defined as

In which

The material parameters of

This set of examples investigates the performance of three-dimensional mean dilatation 8-node

This example is used to assess performance of near incompressible limit material subjected to a side pressure. Follower forces are used. In this block, displacement is fully constrained in bottom and a side pressure of 2 kPa is uniformly applied to the front left edge. Three meshes respectively consisting of 2, 16 and 128 elements are considered to investigate the convergence. For each mesh, the solution for the two element shapes is obtained to ensure that elements with nonconstant initial element Jacobians perform well (for clarity, the element Jacobian is the volume ratio between the physical volume and the volume in the bi-unit cube in the parent domain, and is different from the Jacobian of the deformation which we have been discussing throughout this paper). For each mesh, regular and irregular hexahedra, and each refinement, four cases are run: standard displacement elements with no special treatment of the volumetric deformation,

From the obtained results we can see that average volume

Standard | ANSYS | ||||||||
---|---|---|---|---|---|---|---|---|---|

Method | in reference | in current | Solid 185 | ||||||

configuration | configuration | ||||||||

Element type | cubes | irreg. | cubes | irreg. | cubes | irreg. | cubes | irreg. | |

3*d_{x} | 2 elements | −0.0001 | −0.0001 | 0.0007 | −0.0024 | 0.0006 | −0.0027 | 0.0003 | −0.0029 |

16 elements | −0.0021 | −0.0026 | 0.0044 | 0.0039 | 0.0045 | 0.0039 | 0.0044 | 0.0039 | |

128 elements | −0.0030 | −0.0035 | 0.0029 | 0.0027 | 0.0029 | 0.0027 | 0.0029 | 0.0028 | |

3*d_{y} | 2 elements | 0.0065 | 0.0061 | 0.0970 | 0.0979 | 0.0975 | 0.0985 | 0.0979 | 0.0988 |

16 elements | 0.0154 | 0.0160 | 0.0675 | 0.0677 | 0.0673 | 0.0675 | 0.0678 | 0.0679 | |

128 elements | 0.0293 | 0.0294 | 0.0669 | 0.0670 | 0.0667 | 0.0668 | 0.0670 | 0.0671 | |

3*d_{z} | 2 elements | −0.0798 | −0.0800 | −0.4674 | −0.4646 | −0.4717 | −0.4690 | −0.4658 | −0.4631 |

16 elements | −0.0938 | −0.0944 | −0.3087 | −0.3086 | −0.3091 | −0.3090 | −0.3085 | −0.3084 | |

128 elements | −0.1279 | −0.1281 | −0.3260 | −0.3259 | −0.3263 | −0.3262 | −0.3259 | −0.3258 |

iteration number | residual norm |
---|---|

1 | 5.10E + 0 |

2 | 1.700E − 2 |

3 | 2.822E − 4 |

4 | 1.657E − 8 |

This example is presented to assess performance of the proposed

As illustrated in this figure and the table, displacement results are larger with

In this example, performance of the proposed

Cook’s membrane is a classical plane strain problem used to test performance of elements in nearly incompressible problems under moderate distortion. Here we apply the problem in three dimensions, with one layer of elements in depth and all nodes constrained out of plane. We follow the nonlinear example in [

The geometry is shown in ^{2}, and ^{2}.

We compare meshes of 16, 64, and 256 elements. The deformed meshes for the standard and

For quantitative comparison, the vertical displacements at the upper right node are recorded. For verification, ANSYS is again run with the Solid 185 element. The results are shown in

Standard | ANSYS | |||
---|---|---|---|---|

Method | in reference | Solid 185 | ||

configuration | ||||

128 elements | d_{x} | 0.0226 | 0.0176 | 0.0181 |

d_{Y} | −0.0829 | −0.1264 | −0.1264 | |

d_{Z} | 0.0152 | 0.0175 | 0.0170 |

Standard | ANSYS | |||
---|---|---|---|---|

Method | in reference | Solid 185 | ||

configuration | ||||

number of elements | 16 | 2.1611 | 6.0320 | 6.0250 |

64 | 2.1949 | 6.6318 | 6.6283 | |

256 | 2.2836 | 6.8279 | 6.8230 |

In the final example, the incompressible behavior of a three-dimensional mouse cornea subjected to pressure loading is studied by means of standard and

The mouse cornea is a nearly incompressible soft biological tissue consisting of five layers of which the stroma is the thickest and stiffest. For simplicity, only the stiffness of the stroma is considered in the constitutive model development. The stroma is composed of oriented and dispersed collagen fibrils embedded in nearly incompressible matrix. The material model used is an anisotropic hyperelastic model adapted from Pandolfi and Holzapfel [

in which U is the penalty function enforcing incompressibility constraints of the cornea defined as

where

for a given unit orientation vector

The unit vector

cumferential orientation around the cornea edge. Again, in the transition region, a linear interpolation of the two directions is assumed.

Spheres with different radii of curvature for outer and inner surfaces of the cornea result in varying thickness throughout. For the mouse cornea model, a thickness of 0.3 mm at the apex and 0.1 mm at the edge was assumed. An initial cornea height of 0.84 mm and the distance from cornea center to the inner and outer edges of respectively 1.33 mm and 1.39 mm were used.

The finite element simulation is performed using the standard 8-node hexahedral elements and the proposed

The results of vertical displacement at the apex versus the number of element layers through corneal thickness are shown in

fewer layers, and the standard elements appear to exhibit some locking.

In summary, we have developed a

We have examined two possible choices for the integral averaging of the Jacobian, over the reference configuration and over the current configuration. While there may be some justifications for the current configuration as more natural, the formulation is more complex. Since the volume change in nearly incompressible materials

λ (kPa) | μ_{0} (kPa) | k_{1} (kPa) | k_{2} (unitless) |
---|---|---|---|

5500 | 60 | 20 | 400 |

is small, one does not expect there to be a great difference in the results. This observation is confirmed by numerical examples.

Since the formulation is general, it can be applied to other choices of

The authors gratefully acknowledge the support of the U.S. National Institutes of Health grant 1R21EY020946- 01.

Craig D.Foster,Talisa MohammadNejad, (2015) Trilinear Hexahedra with Integral-Averaged Volumes for Nearly Incompressible Nonlinear Deformation. Engineering,07,765-788. doi: 10.4236/eng.2015.711067

As mentioned previously, the stiffness matrix may be derived using a directional derivative or a pseudo-time derivative. Here, we take the latter approach. We examine the nodal subvector of the element internal force vector

where, as mentioned previously,

For convenience, for the remainder of the derivation, we will move the nodal subscript of the

The time derivative of the nodal subvector of the element internal force vector, then, can be written

Examining the quantity inside the integral

We examine the second term first. Recall that

rather than

Taking the second and third quantities one at a time

The third quantity becomes

Next we evaluate

Adding up all these quantities, we find that

and hence the nodal submatrices of the element stiffness matrix may be written

The first term is the common “material stiffness” with the modified

Therefore,

The process for the Jacobian averaged in the current configuration is the same, though the calculations are more tedious.

Hence, the derivative we seek must be