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# Orthotropic materials

## Element coordinate system and invariant node numbering:

Shell elements stress update calculations in LS-DYNA are done in a local element coordinate system, not the global coordinate system. By default, the local x-axis is aligned with the 1-2 edge.

The default element system therefore implies that that the material is tied to and rotates with the 1-2 edge of the element. For anisotropic material, this can cause elements to deform to some nonphysical low energy modes.

Think of a uniaxial tension test with the strong material direction aligned with the test specimen. If this test was meshed such that the 1-2 edge of any element was along the weak axis of the material (transverse to the load direction), it is likely that the element would deform such that 1-2 edge rotated to the load direction so that the element energy would be lower. This of course would twist and stress other elements which might increase their energy, so the 1-2 edge may not rotate very far, but since any finite rotation is wrong, we would have a bad analysis and an ugly mesh.

The invarient node numbering option (see `*CONTROL_ACCURACY`) uses a different averaging method to determine the element system x-axis.

The method is this: two vectors, eta and mu, are defined in the plane of the shell by connecting the midside points of opposite shell edges. In other words, eta bisects the element in one direction and mu bisects the element in the other direction. Halfway between eta and mu is a vector we'll call `eta + mu`. The element x-axis is 45 degrees to one side of `eta + mu` while the element y-axis is 45 degrees to the other side of `eta + mu`. In a rectangular element, the element x is thus parallel to the n1,n2 edge. The invariant node numbering option defines a local element system that rotates by exactly 90 degrees when an element is renumbered such that node numbers 1,2,3,4 are changed to 2,3,4,1. The x-axis is usually not aligned exactly with the N1-N2 edge for non-rectangular elements. The invariant node numbering option defines a local element system that rotates by exactly 90 degrees when an element is renumbered such that node numbers 1,2,3,4 are changed to 2,3,4,1. i An obvious advantage of this approach is that the mesh and material behavior is independent of the way that the elements are numbered. The element coordinate system rotates consistently during in-plane shear and hourglass deformation so that behavior is consistent for direction dependent anisotropic materials. Perhaps more important, the likelihood that the material will deform into a non-physical low energy modes is reduced. While the invariant node numbering option is not perfect, it seems to consistently give better results for anisotropic material. The invariant node numbering option is available for shell element formulations 1, 2, 5, 7, 9, 10, 11, and 16. It is not available for 3, 4, 6, or 8.

In LS-POST, the term "Local" refers to the shell element local coordinate system. This system is determined from the element connectivity, not from some coordinate system defined in the input deck. The node1-to-node2 vector (N1-N2), is the local x-direction, the local z-direction is normal to the shell (cross product of N1-N2 and N1-N4), and the local y-direction is the cross product of z and x.

Defining the material coordinate system:

For shell elements of anisotropic material, there are 3 options for defining the initial direction of the material axes, `AOPT=0`, 2, and 3 (see `*MAT_OPTIONTROPIC`). During the solution, the element coordinate system rotates and transforms with the element, so the angle between the element system and material system can be assumed to remain constant. In other words, the material direction is constantly updated as the element rotates and deforms. It is therefore sufficient to define the material coordinate system in the undeformed geometry.

For this discussion, the material coordinate system will be called the a-b-c system to be consistent with the user's manual. For shell elements, the c-axis coincides with the element normal, the a-axis is in the plane of the shell, and the b-axis is determined by the cross product, `b = c x a`. Actually, for warped elements, the a-axis is not exactly in the element plane, but is projected along the c-axis such that it is orthogonal to c. This projection is also done for the local element system, so a simple 2D transformation between the two systems is possible.

For reasons discussed in the last paragraph, the three available shell element options for defining the a-b-c system boil down to defining the a-axis.

For `AOPT=0`, the a-axis is assumed to be equal to the local element system x-axis.

For `AOPT=2`, the a-axis is defined as the user defined vector a, projected onto the surface of the element. Note that the user defined vector d is not used at all. Also, note that the user defined vector a, is not equivalent to the a-axis of the material unless a is orthogonal to the element normal.

For AOPT=3, the a-axis is defined by the cross product of a user defined vector v with element normal, ie. `a = v x c`. Given the same user defined vector, `AOPT=3` defines a material coordinate system that is rotated exactly 90 degrees from coordinate system defined by `AOPT=2`.

Note:

When `*ELEMENT_SHELL_BETA` is used with an orthotropic material, the material axes as defined by the AOPT option in the material input are rotated by the element PSI angle to get the reference direction for the element. The material axes for the element integration points are then rotated by the integration point beta angles in *section_shell. In summary, AOPT, PSI (or MANGLE in `*MAT_54/55` if `AOPT=3`), and the integration point beta angles go into defining the material directions at each integration point. The User's Manual is incorrect in saying that PSI is always measured from the N1 to N2 axis.

Using beta angles to rotate the material coordinate system:

A beta angle allows the user to reorient the material coordinate system by rotating it about the normal direction vector by some angle, beta. When using `materials 22, 23, 33, 34, 36, 41-50, 54, 55, 56, 59, 103`,the user may define one beta angle for each layer of the element (at through thickness integration points) using the `ICOMP` flag and the `B1`, `B2` etc parameters of `*SECTION_SHELL`. A single beta angle may be defined for each element using the BETA option of `*ELEMENT_SHELL`. Beta angles are summed if defined in both places.

For `materials 2, 21, 86, 117`, a default beta angle for all elements may be defined using the material's beta parameter if the material coordinate system is defined using `AOPT=3`, but not for `AOPT=0` or `AOPT=2`. The BETA option for `*ELEMENT_SHELL` overides the default value. The `ICOMP` flag of `*SECTION_SHELL` cannot be used with these materials, so they are not as easy to use for layered composites.

Using the BETA option of `*ELEMENT_SHELL`, arbitrary material orientation for any geometry can theoretically be defined. However, there is no obvious way to automatically generate the correct beta angles. These angles most likely need to be calculated by hand, or possibly by a customized program if an equation can be written to describe the material direction throughout the domain or subdomains.

Element stress output in the material coordinate system:

To output stress and strain values of anisotropic materials in the material coordinate system, set the `CMPFLG` to 1 (see `*DATABASE_EXTENT_BINARY`). This flag effects the output of shell, solid, and thick shell stresses to the ascii ELOUT file as well as the binary D3PLOT files.

For orthotropic materials, AOPT only goes into establishing the initial material coordinate system. It has no effect on how the material coordinate system is updated through time. With the exception of material type 2 which uses a total Lagrangian formulation, the material coordinate system is updated based on the rotation of the element coordinate system. This can be a real problem for solid elements which shear/distort badly in that the material coordinate system is largely influenced by the element connectivity (except that INN can be set to 3 or 4 in `*CONTROL_ACCURACY` in v. 970 to invoke invar. node numbering for solids). For shell elements, the dependence of material direction on element connectivity is eliminated by flagging invarient node numbering with `*CONTROL_ACCURACY`.

By default, the local element system is: x is along node1-to-node2; y is orthogonal to x and in the plane of (node1, node2, node4); for shells; z is along the shell normal. Again, to make the local element coordinate system insensitive to the order of the nodes in the element connectivity, invoke invarient node numbering by setting `INN = 2` (shells) or `INN = 3` (solids) in `*CONTROL_ACCURACY`. Invarient node number for solids is available only in version 970 or later.

Don't be confused into thinking the local material coordinate system is the same as the local element coordinate system. The two coordinate systems are not necessarily aligned but they are updated in the same manner.

For mat 126, the material coordinate system update is dependent on the element formulation. Special element formulations 0 and 9 for mat 126 update the element coordinate system using different approaches. Formulation 0 is best suited for elements undergoing large shearing deformation.

Elastic constants for orthotropic shells:

Shells are plane stress elements and so the laws governing 3D orthotropy (as in solid elements) don't necessarily apply to shells. For any orthotropic shell, the input constants `EC`, `PRCA`, and `PRCB` are not used. Furthermore, in the case of `*MAT_FABRIC`, `GBC` and `GCA` are not used. We will add a "(not used)" note in the User's Manual soas not to mislead users in the future.

lpb, jpd revised 5/30/03 (LSPOST local system) revised 6/17/03 elastic constants for ortho shells