Week - 10 Hyperelastic Material Models

                                                                  Hyperelastic Material Modelling from Raw data 

Objective: To prepare a hyperelastic material model card in LS-Dyna from the given engineering stress-strain data (see Figure 1). The material cards created are then correlated against the input data through a uniaxial tensile test on a dogbone specimen. 

                                                                     Figure 1: Given constitutive material data 

Theory:  Hyperelastic materials are a  special class of materials that exhibit non-linear elastic behaviour. The elastic region of the stress-strain curve of such materials is also very large i.e. it deforms elastically up to extremely large strain values of 100-500% or more. Such deformations are recoverable on the removal of the applied load. Most hyperelastic materials are incompressible in nature, implying that their volume remains constant even when they undergo shape change. They exhibit high stiffness in compression and in tension, they become softer initially and then stiffer on being loaded in tension again.   

                                                            Figure 2: Sample Stress-Strain Plot of a Hyperelastic Material [1]

The constitutive relations of Hyperelastic materials are generally computed from strain energy density functions i.e. strain energy per unit volume (which is the area under the stress-strain curve). The general form of strain energy density function is given as [1]:

                                                               .............................(i)

where W = Strain Energy Density, Cijk = describes the shear behaviour of the material, I1, I2 & I3 = Principal invariants of the deviatoric strain. If the material is considered incompressible or if Poisson's ratio is used along with the material model, then Equation (i) reduces to [1]:

                                                                           ......................................(ii)

The most commonly used material models for hyperelastic materials and which are available in LS-Dyna material library are:

• Neo-Hookean: This model is formulated using a first-order polynomial with only one constant (Equation(iii)). The relation between shear stress and deformation is linear and does not accurately capture the behaviour of hyperelastic materials at large deformations. [1]

                                                                                    ...............................................(iii)

• Mooney-Rivlin: This involves the second invariant of strain as well as an additional constant in its first-order polynomial function, thereby providing a better ability to fit to material data. [1]

                                                                           .................................(iv)

• Yeoh: This model uses a third-order polynomial and therefore provides an even better ability to adjust the strain energy density function to stress-stretch data. It uses three constants but only the first invariant of the deviatoric strain. [1]

                                                             .....................(v)

• Ogden: This material describes strain energy density in terms of the principal stretches. The constants here are mu and alpha instead of C. The 3 in Equation(vi) ensures that the strain energy is zero in case the principal stretch `lambda` is 1. [1]

                                                            .....................(vi)

For this project, the hyperelastic models under consideration here are Mooney-Rivlin and Ogden.

Uniaxial Tensile Test Setup: The dogbone specimen to be used for the simulation was 145mm long and 20mm wide (at the necking region). The specimen was meshed with 632 quad elements and 720 nodes with a Belytschko-Tsay element shell formulation. A thickness of 2mm with 3 integration points through its thickness was assigned to the mesh. Moreover, to assist in convergence, the shear factor was dropped from 1.0 to 0.83.  

                                                                Figure 3: Dogbone specimen for uniaxial tensile test

The center nodes at the extreme ends of the X-axis were constrained in translation along the Y-direction only. All the nodes of one end (marked in black in Figure 4) were constrained along all degrees of freedom except Y-translational. All the nodes of the other end of the specimen (marked in black in Figure 5) were given a displacement of 5mm at a rate of 0.5mm/ms along the positive X-direction.

              Figure 4: Nodes with only Y-translational free                                     Figure 5: Nodes with only Y-translational free

         Figure 6: Nodes with displacement along positive X                               Figure 7: Displacement magnitude along positive X

The simulation was run for 10ms in explicit mode with Binary and ASCII  output files being requested at every 0.1ms. 

Constitutive Material Modelling: MAT_Hyperelastic_Rubber (077_H) is used to model Mooney Rivlin. For the initial run, the number of constants N is set to 1. The engineering stress-strain data is defined and input as the constitutive material curve and the simulation is run for a few cycles before terminating. The d3hsp file was then referred to, to find the Mooney-Rivlin Constants as computed by the solver. The fitted stress-extension curve data was also noted and further processed to obtain the engineering stress-strain curve. The input plot and the LS-Dyna fitted curve were then superimposed to check for satisfactory correlation. A similar procedure was followed for Ogden but with the material card MAT_Ogden_Rubber (077_O) and N=1. 

                             Figure 7: Input Stress-Strain Curve                       Figure 8: Correlation of Mooney-Rivlin fitted curve with input curve

                                                                    Figure 9: Correlation of Ogden fitted curve with input curve

The resulting material properties for Mooney-Rivlin and Ogden have been tabulated in Table 1 and Table 2 below:

                                                      Table 1: Material Properties for Mooney Rivlin Card

                                                     Table 2: Material Properties for Ogden_Rubber Card

After establishing the card parameters, the tensile test simulation was run again with the derived constants instead of the material data curve. 

Results: The von-mises fringe plot shows uniform stress in the narrow-necked region of the specimen indicating an accurate representation of a tensile test.

                                                                   Figure 10: Uniaxial Tensile Test Fringe Plot animation

The X-strain for the mean integration point as well as the Equivalent Von-Mises stress for element S298 was recorded. From these true stress-strain values, the engineering stress-strain was derived using the formulae: 

                                                               true strain = ln (1 + engg. strain).....................................(i) 

                                                               true stress = engg. stress * (1+ engg strain).....................(ii)

The resulting data and plots (with respect to the input data and fitted curve) for the two material cards are shown in Figures 11 & 12.

        Figure 11: Stress-Strain plot from Mooney-Rivlin material card            Figure 12: Stress-Strain plot from Ogden material card

The max engineering stress at 100% strain was 8.28E-01 MPa for both the material cards. In fact, as can be observed from Figure 13 below, the constitutive plots from the two materials are identical. 

                                                 Figure 13: Comparison of input and post-processed stress-strain curves

Conclusion: From Figure 13, it can be observed that both Mooney-Rivlin and Ogden (1st order) produce identical results and do not accurately capture the input curve although the maximum error in stress for a given strain is 4.1% only. Hence, both models can be considered a good fit for the given data.   

Reference: 

1. Non-Linear Elastic Deformations, Ogden R.W., Dover Publications Inc., New York - 1997.