Mooney-rivlin was named after the surnames of two mechanics, namely M. Mooney and R. S. rivlin. 1940, Mooney published a paper entitled Theory of Large Elastic Deformation in the famous Journal of Applied Physics. Eight years later, 1948, rivlin published an article entitled "Large Elastic Deformation of Heterogeneous Materials" in the Journal of Philosophy of the Royal Society of London. So there is today's Mooney-rivlin model, which once dominated the whole study of rubber mechanics. At the same time, it also lays a foundation for other models with strain tensor invariants as the core. Another kind of hyperelastic model is the model with principal stretching as the core, such as Ogden model, which we will introduce in future articles.
Mooney was born in Kansas City, Missouri, USA from 65438 to 0893. At the age of 24, he received a bachelor's degree from the University of Missouri and a doctorate from the University of Chicago at the age of 30. He is a member of the National Research Council of the United States and a physicist of the Western Electric Company and the American Rubber Company. Like Rivlin we introduced before, Dr. Mooney devoted his life's work and research to the mechanics of polymer materials. Of course, Mooney is over 20 years older than Rivlin. Dr. Rivlin introduced it in the last article, so I won't repeat it here.
Like other hyperelastic models, we use elastic strain energy to characterize mechanical properties. According to the order of Mooney-Rivlin, there are four common types: 2-parameter, 3-parameter, 5-parameter and 9-parameter strain energy.
The elastic strain energy of Mooney-rivlin 2 parameter is:
The elastic strain energy of Mooney-rivlin 3 parameter is:
The elastic strain energy of Mooney-rivlin 5 parameter is:
The elastic strain energy of Mooney-rivlin 9 parameter is:
As can be seen from the above four elastic strain energy formulas:
1. Higher-order strain energy model can simulate more complex stress-strain curves, but it also means more calculation, experiment and parameter fitting. At the same time, it increases the burden of nonlinear solver, which may lead to more difficult convergence.
2. Mooney-rivlin model is a special form of polynomial model. When N= 1, the polynomial model is reduced to two-parameter Mooney-Rivlin, and when N=2, the polynomial model is reduced to five-parameter Mooney-Rivlin; When N=3, the polynomial model is simplified to 9-parameter Mooney-Rivlin.
3.2 In the parametric model, when the parameter C0 1 is zero, it is simplified to Neo-Hookean model (the coefficient of C 10 is twice). Non-zero C0 1 term makes the deformation prediction under uniaxial tension more accurate, but the model cannot accurately simulate the multi-axial stress data. Or the data obtained from a deformation test cannot be used to predict other types of deformation.
4.2 The shear modulus of parametric model is constant coefficient \mu=2(C 10+C0 1), which is not suitable for simulating carbon black filled vulcanized rubber. C 10 and C0 1 are positive definite constants. For most rubbers, when c10/c01≈ 0.1~ 0.2, a reasonable approximation can be obtained in the strain range of 150%.
5. Three or more Mooney-Rivlin models can describe the unsteady shear modulus. However, it needs to be calculated carefully after introducing higher-order terms, because unstable strain energy values may be produced and non-physical results beyond the test range may be obtained.
Which one should be used for the actual simulation of Mooney-Rivlin model of these four functions? It is often determined according to the stress-strain curve of material experiments. For example, the stress-strain curve with single curvature (without inflection point) can use two or three parameters. Double curvature (including an inflection point) can use 5 parameters. Three curvatures (including two inflection points) can be selected as the 9-parameter model.
At the same time, in order to produce effective and correct superelastic material properties, Mooney-Rivlin parameters must meet specific positive definite requirements. If these constraints cannot be met, the solution may not converge. For Mooney-Rivlin with different parameters, the positive parameter constraint requirements are met as shown in the figure.
Generally speaking, Mooney-rivlin model has been widely recognized and applied. Especially in the small strain range (0~ 100% tension and 30% compression), it can better characterize the mechanical behavior of rubber materials. Models with different parameters also provide users with more choices of different working conditions. However, Mooney-rivlin also has some limitations:
1. is not applicable to the condition that the deformation exceeds 150%.
2. Because there are many parameters, it is relatively difficult to obtain high-order Mooney-Rivlin parameters from manuals or documents, and it is necessary to fit the experimental data with curves.
3. It is not suitable for analyzing compressible hyperelastic materials, such as foam materials.
4. The prediction error of experimental data outside the input data range is large.
In this example, we will use the five-parameter Mooney-Rivlin to analyze the compression state of rubber materials.
Defining Mooney-Rivlin Hyperelastic Materials
Here we simulate rubber material, and the input parameters are: C 10 =-0.55 MPa, C0 1 = 0.7 MPa, C20 = 1.7 MPa, c1= 2.5mpa, and CO2 =-0.
Build a model
Establish a cylinder with a diameter of 10 mm and a height of 10mm, and divide the grid. Fixed bottom constraint. And a downward displacement of 5mm was applied to the top surface.
Find a solution
Because of strong nonlinearity, we set 30 substeps. Then click the Solve button.
inspection result
The equivalent stress distribution is shown in the figure. It can be found that the stress increases nonlinearly under the condition of equal step displacement.
Here is an operation video for your reference.
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