The MAT9 entry is used to define general three-dimensional anisotropic constitutive
stress-strain relationship using the anisotropic material matrix, [G]. You can use MAT9 to define an anisotropic material property for the CHEXA, CPENTA, and CTETRA solid elements. You can also use MAT9 to define a three-dimensional orthotropic material.
The stress-strain constitutive relationship is:
Equation 7-17.
where:
{σ} stress
{ε} strain
{A} thermal expansion coefficients
E Young’s modulus
ν Poisson’s ratio
G shear modulus
(T − Tref) change in element temperature used to calculate initial element thermal expansion
The MAT9 entry is also used to define three-dimensional orthotropic constitutive stress-strain relation using the anisotropic material matrix [G]. The stress-strain relationship is given by:
Equation 7-18.
If you want to use the MAT9 entry to define an orthotropic material, the terms of the material matrix can be generated by using Eq. 7-19:
Equation 7-19.
υij = Poisson’s ratios
Ex, Ey, Ez = Young’s modulus in the x-, y- and z-directions Gxy, Gyz, Gzx = shear moduli
Δ =
also
G44 = Gxy G55 = Gyz G66 = Gzx and
G14 = G15= G16= 0.0 G24 = G25= G26= 0.0 G34 = G35= G36= 0.0 G45 = G46= G56= 0.0
MAT9 Format
The format of the MAT9 Bulk Data entry is as follows:
1 2 3 4 5 6 7 8 9 10
MAT9 MID G11 G12 G13 G14 G15 G16 G22
G23 G24 G25 G26 G33 G34 G35 G36
G44 G45 G46 G55 G56 G66 RHO A1
A2 A3 A4 A5 A6 TREF GE
Field Contents
MID Material identification number.
Gij Elements of the 6 × 6 symmetric material property matrix in the material coordinate system.
RHO Mass density.
Ai Thermal expansion coefficient.
TREF Reference temperature.
When you use the MAT9 entry, you should define a material coordinate system on the PSOLID entry (using field 4). For solid elements, NX Nastran outputs stresses in the material coordinate system, which by default, is the basic coordinate system. In general, for solid elements, it isn’t easy to determine the orientation of the element coordinate system.
Defining Temperature- and Stress-Dependent Properties for MAT2
With the MAT9 entry, you can use related bulk data entries to define temperature- or stress-dependent properties.
• You can use the MATT9 entry to designate certain material properties on the MAT9 entry as temperature-dependent by referencing TABLEMi entries.
• In SOLs 106 and 129, you can use the MATS1 entry to designate certain material properties on the MAT9 entry as stress-dependent.
See also
• “MAT9” in the NX Nastran Quick Reference Guide
• “PSOLID” in the NX Nastran Quick Reference Guide
• “MATT9” in the NX Nastran Quick Reference Guide
• “MATS1” in the NX Nastran Quick Reference Guide MAT9 Example
The following example illustrates the use of the anisotropic material property using the MAT9 material entry. This example consists of three parts as follows:
1. Develop a material property matrix using a pseudo-anisotropic model consisting of a single CHEXA element with some bars along the edges and across the diagonals.
2. Verify that the material matrix developed in part 1 is correct.
3. Use the material matrix in a distorted element.
This three-part approach is analogous to constructing an anisotropic material test coupon and then testing that coupon in the test lab to obtain a strain matrix. From the strain matrix, you develop the material matrix. Then, you verify the material matrix by modeling the test coupon.
Finally, you use the newly developed material matrix in the actual structure.
Part 1 – Develop the Material Matrix
To simulate a three-dimensional anisotropic material, this example uses a single CHEXA element with CBAR elements of various cross-sectional areas and bending stiffnesses along the edges and across the diagonals as shown inFigure 7-3(a). The cross sections of the bars aren’t important, but if you’re interested in the actual dimensions, refer to the input file “anis1.dat” in the Test Problem Library located on the delivery media. The dimensions of the cube are 1 · 1 · 1 inches. To constrain the cube, one of the corner grid points is fixed in all six component directions.
Six self-equilibrating load cases are used to represent each of the six stress components. Each direction was applied as a separate subcase. A self-equilibrating normal load consists of applying equal and opposite forces to opposite faces of the cube (i.e., forces F1, F2, and F3).
For the shear load, a self-equilibrating load consists of applying four forces around the cube in order to place the cube in a state of pure shear. Note that for all self-equilibrating load cases, the net resultant should be zero; hence, the SPC force should also be zero. Because of the dimensions chosen for CHEXA, each load case represents a unit stress, and the resulting strains are the strains due to unit stress.
Figure 7-3. Pseudo-Anisotropic Model
The strains at the center of the CHEXA were extracted from the output file and summarized in Table 7-4.
Table 7-4. Strains Due to Unit Loads on Pseudo Anisotropic Model Applied Unit Load
FX FY FZ FXY FYZ FZX
εX 0.155 -0.034 -0.056 -0.016 0.002 -0.011
εY -0.034 0.072 -0.018 -0.005 0.001 -0.003
εZ -0.056 -0.018 0.107 -0.008 0.003 -0.005
γXY -0.016 -0.005 -0.008 0.197 -0.011 0.009
γYZ 0.002 0.001 0.003 -0.011 0.219 -0.021
Resulting Strains (10–6)
γZX -0.011 -0.003 -0.005 0.009 -0.021 0.180
Table 7-4represents a strain matrix that, when inverted, is the material matrix [G] introduced inEq. 7-19. Note that since the strain matrix is symmetric, the material matrix [G] is also symmetric.
The result is the material matrix [G] for our anisotropic material as shown inEq. 7-20.
Equation 7-20.
Part 2 – Verify the Material Matrix
Now that the material matrix has been generated, the next step is to verify that the matrix is entered on the MAT9 entry correctly. To do this, we return to the 1 · 1 · 1 inch cube. However, this time the analysis will not include the CBAR entries as shown inFigure 7-3(b). The same six load cases are applied as in Part 1.
If the material matrix is correct, the strains produced should be numerically the same as those shown inTable 7-4.
The input file for the verification model is shown inListing 7-2.
Listing 7-2. Anisotropic Verification Input File (Continued)
PLOAD4 1 13 1. 2 12 +
MAT9 2 10.67+6 6.91+6 6.89+6 1.25+6 -0.09+6 0.86+6 19.03+6 6.90+6 1.27+6 -0.12+6 0.83+6 14.28+6 1.23+6 -0.15+6 0.82+6 5.27+6 0.21+6 -0.11+6 4.63+6 0.52+6 5.72+6
SPC1 1 123 6
SPC1 1 12 1
SPC1 1 2 2
ENDDATA
Listing 7-2. Anisotropic Verification Input File
You should notice a few items in the verification model input file. The six loads are applied separately, each with their own subcase. The constraints are applied to three separate grid points and are non-redundant. The non-redundant constraint set allows the element to expand in all directions without imposing any constraint forces. However, rigid body modes are constrained, which is a requirement for static analysis. The last item to note is the use of the basic coordinate system for the material definition.
The resulting center strains for the verification model are shown inFigure 7-4. As can be seen, there is good agreement compared to the strain matrix shown inTable 7-4.
Figure 7-4. Verification Model Strain Output Part 3 - Use the Anisotropic Material in the Actual Structure
The last step in the example is to use the material properties to model an actual structure as shown inFigure 7-5. The structure is very coarse since it is used for demonstration purposes only. The model consists of four elements with their material coordinate system oriented 45 degrees with respect to the basic coordinate system.
Figure 7-5. Anisotropic Solids Model
The input file is shown inListing 7-3. Note the local coordinate system used to define the material orientation. The displacement results are shown inFigure 7-6.
$ FILENAME - ANIS3.DAT
TITLE = USING THE MATERIAL MATRIX LOAD = 1
CORD2R 1 0 0.0 0.0 0.0 .70710680.0 .7071068+
.70710680.0 -.707107
PSOLID 1 2 1 GRID SMECH
$
MAT9 2 10.67+6 6.91+6 6.89+6 1.25+6 -0.09+6 0.86+6 19.03+6 6.90+6 1.27+6 -0.12+6 0.83+6 14.28+6 1.23+6 -0.15+6 0.82+6 5.27+6 0.21+6 -0.11+6 4.63+6 0.52+6 5.72+6
$
SPC 1 51 123456 0.0
SPC 1 52 123456 0.0
SPC 1 56 123456 0.0
SPC 1 55 123456 0.0
$ ENDDATA
Listing 7-3. Structure with Anisotropic Material Input File
Figure 7-6. Output for Structure with Anisotropic Material
7.5 Material Properties for Heat Transfer
You define material properties for heat transfer analysis are supplied on MAT4, MAT5, MATT4, MATT5, RADM and RADMT entries.
• The MAT4, MAT5, and RADM entries provide temperature independent (constant valued) material properties.
• The MATT4, MATT5, and RADMT entries supply information about temperature-dependent properties through reference to material tables (TABLEMi (i = 1, 2, 3, 4)).
With heat transfer material properties:
• Although the surface characteristics for free convection (heat transfer coefficient) and
radiation (emissivity and absorptivity) are not normally considered to be material properties, they are included in the bulk data entries described here.
• Temperature dependence of the quantity of interest is ultimately defined on a TABLEMi entry. For example, the table is connected to the MAT4 through the MATT4 entry. Most material properties are directed toward a structural element and are referenced by the property entry for the element of interest.