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Week - 10 Hyperelastic Material Models

Aim: To calculate the Mooney Rivlin and Ogden material constants and compare both using stress-strain data from a Dogbone specimen tensile test with 100 percent strain. The given material data is the engineering stress-strain in MPa/(mm/mm). The comparison should be shown from…

sriram srikanth
updated on 25 Oct 2021

Aim:

To calculate the Mooney Rivlin and Ogden material constants and compare both using stress-strain data from a Dogbone specimen tensile test with 100 percent strain.
The given material data is the engineering stress-strain in MPa/(mm/mm).
The comparison should be shown from the d3hsp file and using simulation.

Theory:

Hyper-elastic Material:

Hyperelastic material models are regularly used to represent the large deformation behaviour of materials with FEA. They are commonly used to model mechanical behaviours of unfilled/filled elastomers. In addition to elastomers, hyperelastic material models are also used to approximate the material behaviour of biological tissues, polymeric foams, etc. Linearly elastic materials are described through two material constants (like Young’s modulus and Poisson ratio). In contrast, hyperelastic materials are described through a strain-energy density function. The strain-energy density can be used to derive a nonlinear constitutive model (i.e., stresses as a function of large strain deformation measures like deformation) which are derived by researches and some acceptable models we are using popularly such as

Neo-Hookean (first order reduced polynomial)
Mooney-Rivlin (first order complete polynomial)
Signorini (second order reduced polynomial)

Hyperelastic materials are the special class of materials which respond elastically even when they are subjected to large deformations. They show both a nonlinear material behavior as well as large shape changes. They are characterized by:

Can undergo large elastic deformations in order of 100 to 700% which are fully recoverable, i.e., the initial shape is recovered when load is removed.
Are nearly incompressible, which means that they change can their shape but overall volume remains almost constant.
Show a highly nonlinear stress-strain relation.
The material softens and then becomes stiffer again when applied to tension whereas, under compression, they have a quite stiff response.

Many polymers show hyperelastic behavior, such as elastomers, rubbers, and other similar soft flexible materials.

Hyperelastic materials are mostly used in applications where high flexibility, in the long run, is required, under the presence of high loads. Some typical examples of their use are as elastomeric pads in bridges, rail pads, car door seal, car tires, and fluid seals.

In finite element analysis, the hyperelasticity theory is used to represent the non-linear response of hyperelastic materials at large deformations. Hyperelasticity is popular due to its ease of use in finite element models. Usually, stress-strain curve data from experimental tests is used to fit the constants of theoretical models, thus approximating the material response.

The choices of hyperelasticity models which are available in SimScale platform are:

Neo-Hookean (first order reduced polynomial)
Mooney-Rivlin (first order complete polynomial)
Signorini (second order reduced polynomial)

Calculating the Stress-Strain Relation From Strain Energy Density Function

The stress-strain relation for hyperelastic materials is normally calculated with a strain energy density function. A brief theoretical description follows.

Consider a solid body subjected to a great deformation. A point inside the body with position $μ 1$ with respect to the original configuration is displaced to the position $α 1$ in the final configuration through the displacement vector u:

x = X + u

The tensor of the gradient of the deformation F relating the initial configuration to the final configuration is given by:

F = \partial x \partial X = I + \nabla X u

The local change of volume J is given by its determinant J=det(F). This tensor turns out to not be the best way to describe large deformations, so the right Cauchy-Green tensor is used instead:

C = F T F

This tensor is symmetrical, with its invariants given by:

I C = t r (C)

I I C = 1 2 (t r (C) 2 - t r (C 2))

I I I C = d e t (C)

The third invariant IIIC relates to the change in volume, and can also be written as:

I I I C = d e t (F) 2 = J 2

In the case of an incompressible material, J=1.

Of high interest for our purpose is to express the right Cauchy-Green tensor and its invariants in terms of the principal stretches λi:

λ i = 1 + ε i

C i i = λ 2 i

C i j = 0, i \neq j

I C = λ 21 + λ 22 + λ 23

I I C = λ 21 λ 22 + λ 22 λ 23 + λ 23 λ 21

I I I C = λ 21 λ 22 λ 23

where εi are the nominal strains in the principal directions (measured in the original configuration). Notice that the stretches λi are the components of the gradient of deformation tensor F.

It is assumed that a strain energy density function U exists such that the hyperelastic (second) Piola-Kirchhoff stress tensor S in the material can be related to the right Cauchy-Green deformation tensor:

S = 2 \partial U \partial C

The true stress tensor on the material is related to the second Piola-Kirchhoff stress tensor. It can be expressed (after some algebra, not shown here) in terms of the right Cauchy-Green tensor and the strain energy density function:

σ i j = - p δ i j + 2 \partial U \partial I 1 C i j - 2 \partial U \partial I 2 C - 1 i j

Here, p is the hydrostatic external pressure, which causes pure volumetric change, and deltaij is the kronecker delta. Notice that due to the incompressibility assumption, the J term is dropped.

For the material models available in SimScale, the strain energy density function is given by:

U = C 10 (I 1 - 3) + C 01 (I 2 - 3) + C 20 (I 1 - 3) 2 + 1 2 K (J - 1) 2

I 1 = I c J - 2 3

I 2 = I I c J - 4 3

J = I I I 1 2 c

You will find that in the user interface, the compressibility is controlled by the D1 constant, given by:

D 1 = 2 K

Particular Strain Energy Density Forms of the Available Hyperelasticity Models Before giving the appropriate material parameters to define specific hyperelastic materials, one should know the strain energy density forms of the hyperelasticity models. Following are the strain energy density forms of all the available hyperelasticity models on the SimScale platform (as mentioned above). NEO-HOOKEAN With C 01 = C 20 = 0 in the above formulation, one obtains the most basic form known as neo-Hookean, given by: U = C 10 (I 1 - 3) + 1 D 1 (J - 1) 2 MOONEY-RIVLIN With C 20 = 0 in the above formulation, one obtains the enhancement to neo-Hookean form known as Mooney-Rivlin, given by: U = C 10 (I 1 - 3) + C 01 (I 2 - 3) + 1 D 1 (J - 1) 2 SIGNORINI Strain energy density function for Signorini is represented as: U = C 10 (I 1 - 3) + C 01 (I 2 - 3) + C 20 (I 1 - 3) 2 + 1 D 1 (J - 1) 2

Procedure:

Experimental Data:

The given data of Engg stress and Engg strain of hyperelastic material is shown in the image given below are plotted in excel.

Part Definition:

To verify and validate the given material data with material models of Mooney-Rivlin and Ogden a dogbone specimen is taken for the analysis.

To find the Mooney-Rivlin and Ogden constants the value of N is set to 1-3. The value of density and Poisson’s ratio is taken as the generic value for the rubber material.

For N>0, data from a uniaxial test are used.

SGL Specimen gauge length
SW Specimen width
ST Specimen thickness

The value of SGL, SW, and ST is set to unity (1.0), then the curve LCID1 is defined using *DEFINE_CURVE engineering stress versus engineering strain.

The section property of the dogbone specimen is assigned as a shell element with 2 mm thickness and ELFORM=2.

The Material and Section are assigned to the dogbone specimen part.

Boundary Condition:

The nodes at the fixed end are constrained in the X and Z direction only and the Y direction is not constrained because of lateral expansion during the tensile test.

The nodes at the middle is constrained in the Y direction since the neutral axis passes through the middle of the specimen in the X-direction.

The nodes of the pulling end are assigned with a boundary prescribed motion in X direction using displacement load curve LCID.

Control Cards:

*CONTROL_IMPLICIT_GENERAL

Activate implicit analysis and define associated control parameters. This keyword is required for all implicit analyses

IMFLAG: Implicit/Explicit analysis type flag: 1- implicit analysis
DT0: Initial time step size for implicit analysis: 0.01
IMFORM: Element formulation flag for seamless spring back: 2 retain original element formulation (default)

*CONTROL_IMPLICIT_SOLUTION

The linear equation solver performs the CPU-intensive stiffness matrix inversion

NSOLVR: Linear equation solver method: 12 (Nonlinear with BFGS updates + optional arclength)

*CONTROL_IMPLICIT_AUTO card has been defined to adjusts the time step size

IAUTO: set to 1 which automatically adjusts the timestep size.
ITEOPT: optimizes the operation count step:11
ITEWIN: is the iteration window: 5

*CONTROL_TERMINATION:

This card is used to mention the termination time of the simulation.

ENDTIM: 1 (Termination time)

Analysis setup for Material model validation:

1) The d3hsp file for each material model is opened in notepad++ and the material constants of Mooney-Rivlin and Ogden as well as final fit data of stretch and Engg. stress is obtained for each case.

From the output files of material verification, the d3hsp file is opened in notepad++. The Mooney-Rivlin constants obtained are

C₁₀ = c1 = 0.4419E-01

C₀₁ = c2 = 0.3686E-01

For material model validation, the value of N is changed to 0 in the material card. These values are inputted to the material card MAT_77_H. The data obtained from the d3hsp file is an extension (stretch) and True stress which is converted to Engg. strain and Engg, stress to verify with experimental data.

From the d3hsp output file of the Ogden material model the Ogden constants obtained are

$mu1$ =2.2769006165419E-01

$alpha1$ =1.3427238801312E+00

These values are inputted to the material card MAT_77_O. The data obtained from the d3hsp file is Engg Stress and stretch ratio which is converted to Engg strain to verify with experimental data.

5) The keywords used for material model validation is the same as that of material model verification except for the changes to the material card(N=0) and addition of database extent binary to compute elastic strain. The keyword file is saved and made to run in the LS-DYNA program manager to get the requested output file.

X-Stress Plot:

The maximum X-stress developed for Mooney-Rivlin material model(N=0) is 0.4346 MPa.

The maximum X-stress developed for Mooney-Rivlin material model(N=1,2,3) is 0.4342,0.4295,0.4297 MPa.

maximum X-stress developed for Ogden material model(N=0) is 0.4312 MPa.

maximum X-stress developed for Ogden material model(N=1,2,3) is 0.4301,0.4302,0.4303 MPa.

X-Strain Plot:

The maximum Lower Ipt X-strain developed for Mooney-Rivlin material model(N=0) is 7.347e-01 and maximum Lower Ipt X-strain developed for Ogden material model(N=0) is 7.357e-01.

Material Model Verification:

From the data obtained from the d3hsp file for different polynomial values i.e, (N=1,2,3) curves are plotted with Engg. stress vs Engg. strain in excel is as shown in the below images. It is observed from the graph, that the curves for different polynomial are superimposing and are better adjusted with the experimental data. The Ogden material model gives a close fit compared to the Mooney-Rivlin material model.

Comparison of Mooney-Rivlin and Ogden material model: From this data, we can easily found that Young's Modulus value for hyperelastic and Ogden material. The Ogden material model gives a close fit compared to the Mooney-Rivlin material model.

CONCLUSION: -

The Mooney-Rivlin and Ogden material constants are obtained from the d3hsp file by inputting the value of polynomial N=1 in the material card MAT_77_H and MAT_77_O
The stretch stress curve data obtained from d3hsp for both hyperelastic constitutive models were verified with experimental material data and found to be a close fit.
The simulation results of the tensile test for both material models with experimental data showed a slight deviation due to the high non-linear behavior of the given hyperelastic material and the generic values of density and Poisson’s ratio.
From the simulation results, both hyperelastic constitutive material models were found to be valid to describe the behavior of the given material data.
The Ogden material model has more polynomials ranging from 1 to 8 and is easy to use and gives better adjustment to experimental data at the cost of computational time.
Learned to verify and validate the hyperelastic material model for the given material data.