1. Element Formulations and Accuracy:
Shear Locking, Assumed Strain, Herrmann, Fully Integrated and Reduced Integration Elements

2. Fundamental Characteristics Of Bending (One Element)
Linear variation of axial strain, exx, through the thickness (y direction).
No strain in the thickness direction, eyy, (if we take Poisson’s ratio as zero).
No membrane shear strain.

3. Shear Locking
Fully integrated 1st order 4 node quad elements, such as Type 3, in bending:
The element detects shear strains that are physically non-existent but are present solely because of the numerical formulation used.
To control this problem there are special element types called Assumed Strain (AS) and Reduced Integration (RI) elements
RI, AS, and quadratic elements will be discussed later
The axial strain can be viewed as the change in length of the horizontal lines through the Integration points. The thickness strain is the change in length of the vertical lines, and the shear strain is the change in the angle between the horiz. and vert. lines
Shear Locking

4. Shear Locking The element cannot bend without shear.
The negative consequence—significant effort (strain energy) goes into shearing the element rather than bending it.
Leads to overly stiff behavior.
Reduced integration elements will correct shear locking on a problem like this:
However a new problem develops called, Hourglassing (discussed later) which can be controlled by using Assumed Strain elements
Thus shear-locking is controlled with RI elements but RI elements introduced hourglassing, which is in turn controlled by AS elements. h l P
Standard Orthogonal elements

5. Shear Locking
Don’t use first order fully integrated elements in regions dominated by bending. Instead, try AS elements.
Linear Shape-function elements
Quad4 (plane strain, plane stress) (element 3,7)
Hex8 (element 11)
Assumed Strain elements will produce inaccurate results on skewed meshes.
The further away from 90 the edge angle, the worse the results
Both parallelogram or trapezoidal -shape are bad (figures) h l P q
Skewed parallelogram elements l P q
Skewed trapezoidal elements h

6. Assumed Strain Elements

7. Assumed Strain Elements
Conventional isoparametric four-node plane stress and plane strain, and eight-node brick elements behave poorly in bending
In reality an applied pure bending moment will create only bending stress and no shear
Because of the linear assumption of these conventional elements, they cannot “bend” – any deformation is due to shear
This additional “non-physical” shear stress tends to store more energy than bending stress and is called “Parasitic Shear”
It generates a stiffer solution in general
For these elements, the shape functions have been modified such that shear strain variation can be better represented
This modification is optional and is termed “assumed strain”

8. Assumed Strain Elements
Disadvantages:
The substantially improved accuracy of the solution is at the expense of a slightly increased computational costs during the stiffness assembly
They are sensitive to distortion and cannot capture bending behavior well in distorted meshes
Advantages:
These are the most cost effective continuum elements for bending dominated problems
Can model bending with only one element through the thickness
They have no hourglass modes
They can be used confidently with plasticity and contact
The most benefit of the assumed strain procedure is obtained for coarse meshes. The effect decreases with mesh refinement
If a fine mesh is used, then the assumed strain option could be left off (and providing a slightly faster analysis)

9. Assumed Strain Elements: Example
Section A-A b a
Cantilever Beam example using conventional and assumed strain fields
Result table shows the normalised tip deflection (FE:Theory)
Note the effectiveness of the 3x1 and 6x1 mesh when assumed strain is invoked

10. Assumed Strain Elements
In both Mentat and Patran, assumed strain may be invoked from:
The main Jobs forms as shown here
As a geometric property specifically assigned to elements

11. Herrmann Elements

12. Volumetric Locking
The element in the corner has only one node that is not constrained. The incompressibility constraint restricts the path of motion of that node to those paths that do not violate that constraint. Unless the loading causes the node to move exactly along that path the element “locks”.

13. Volumetric Locking
Imagine a square “hole” filled with an incompressible material. Whether the pressure is 1 psi or 1,000,000 psi there will be no deformation of the material and we have a non-unique (singular) solution. The Herrmann variable allows a unique solution of the problem by tracking the “internal” pressure in the element.
Pressure

14. Herrmann Elements
Incompressible Materials cause severe numerical constraints and Standard Elements perform poorly due to Volumetric Locking.
The volume of an element of incompressible material must remain fixed, causing severe constraints on the kinematically admissible displacement fields.
For example, in a fine mesh of standard Hex 8: Each element has, on average, 3 DOF (1 node/elem), but 8 constraints. For, the volume at each of the 8 integration point must remain fixed. Hence, the mesh is over-constrained - it “locks”.

15. Herrmann Elements
Herrmann Elements have been formulated to handle fully or nearly incompressibility
Available for small or large strain analyses
Include plane stress/strain, axisymmetric, and three-dimensional elements
Benefits:
Good (obtainable!) results for incompressible materials
Enable Poisson’s ratio of 0.5
Quadratic rate-of-convergence during solution
Used also for compressible elastic materials, since their hybrid formulation usually gives more accurate stress prediction

16. Herrmann Elements
Support both the Total and Updated Lagrange framework
Can be used in conjunction with other material models (e.g. elastic-plastic)
Can be used in rigid-plastic flow analysis problems and coupled soil-pore pressure analyses
Pressure is treated as an independently interpolated solution variable
Actually…the standard displacement formulation is modified based on the “Herrmann” formulation
Generally:
Lower-order elements have an additional node which contains the pressure (Lagrange multipliers)
Higher-order elements have the pressure at each corner node
Elements 155 (plane strain triangle), 156 (axisymmetric triangle), and 157 (3-D tetrahedron) are exceptions - they have an additional node located at the centre of the elements and have Lagrange multipliers at each corner node
Elements (tri3/tet4) can be used for large strain plasticity

17. Reduced Integration Elements

18. Reduced Integration And Hourglassing
First Order Reduced integrated Elements have only one integration point. These are cheaper, and often better in performance too.
These elements have the following bending behavior:
The elements should detect and prevent strain at the corners, but do not.
The deformation is a spurious zero energy mode, shaped like an hourglass, hence the name (Also called “keystoning” because of the trapezoidal shape.)
Hourglassing

19. Reduced Integration Elements: Example
Cantilever Beam
Normalised Tip Deflection (FE:Theory)
Note the effectiveness of the 3x1 and 6x1 mesh when reduced integration is used – even compared to the assumed strain elements…
Mesh
Quad4
(RI)
(AS)
(Standard)
3X1
0.973
0.932
0.025
6X1
0.994
0.952
0.093
12X2
0.983
0.291
24X4
0.997
0.621
48X8
0.999
0.998
0.868
96X16
1.000
1.004
0.963
Section A-A b a

20. Constant Dilatation Elements

21. Constant Dilatation Elements
This is the remedy to the typical element locking experienced during fully plastic (incompressible) material behaviour
It is a modified variational principle that imposes a constant dilatational strain constraint on the element
It is known as a “modified volume strain integration”, “mean dilatation” or “B-bar” approach
This procedure has been implemented for a number of lower-order elements in Marc (type 7, 10, 11, 19 and 20)
The combination of these elements and the constant dilatational option is recommended for inelastic (e.g. plasticity and creep) analysis where incompressible or nearly incompressible behaviour occurs and should always be used in metal forming analysis
The use of finite strain plasticity using an additive decomposition of the strain rates obviates the need of the constant dilatation parameter
For materials exhibiting large strain plasticity with volumetric changes (for example, soils, powder, snow, wood) the use of constant dilatation (or plasticity) will enforce the incompressibility condition and, in such materials, yield incorrect and non-physical behavior

22. Guidelines for Selecting Elements

23. Guidelines for Selecting Elements
Choose quad’s over tri’s
Choose bricks over wedges
Avoid low order tetrahedral elements wherever possible
The element exhibits slow convergence with mesh refinement
This element provides accurate results only with very fine meshing
This element is recommended only for filling in regions of low stress gradient in meshes of Hex8 elements, when the geometry precludes the use of Hex8 elements throughout the model
For tetrahedral element meshes the second-order element should be used
Full integration, first order elements in conjunction with the assumed strain procedure work well for most applications
Use low order quad’s and hex’s with assumed strain with reduced integration if the mesh is refined, of high quality and will not deform very badly
Use 2nd order reduced integration quads and solids if the mesh is coarse
Don’t use 2nd order elements if there are gaps in the simulation
Contact fully supports lower and higher order elements

24. Guidelines for Selecting Elements
During plastic deformation, metals exhibit incompressible behavior
The incompressible behavior can lead to certain types of elements being over-constrained
This leads to an overly stiff behavior (volumetric locking)
Turn ON the “Constant Dilatation” option to correct for this
During hyperelastic deformation, rubbers exhibit incompressible behavior
Use the corresponding Hermann element to correct for this
Second order elements are susceptible to volumetric locking when modeling incompressible materials
In general, avoid using them to model hyperelasticity and plasticity
For large strain analyses, lower-order elements are recommended
Second order elements should be used with caution above 20% strain

25. Element Quality

26. Element Quality
Good
Good
Not good
Not good

27. Element Quality: Distorted (don’t use AS) (Use high order + RI instead)P q
Skewed parallelogram elements
Skewed trapezoidal elements
Standard Orthogonal elements h
Vertical Displacement of Tip for a 6x1 assumed strain quad mesh

28. Element Quality: Distorted MeshP q
Skewed parallelogram elements
Skewed trapezoidal elements
Standard Orthogonal elements h
Vertical Displacement of Tip for a 6x1 assumed strain + reduced integration quad mesh