Presentation on theme: "Modeling of Neo-Hookean Materials using FEM"— Presentation transcript:
1 Modeling of Neo-Hookean Materials using FEM By: Robert Carson
2 Overview Introduction Background Information Nonlinear Finite Element ImplementationResultsConclusion
3 IntroductionNeo-Hookean materials fall under a classification of materials known as hyperelastic materialsElastomer often fall under this categoryHyperelastic materials have evolving material propertiesNonlinear material propertiesOften used in large displacement applications so also can suffer from nonlinear geometriesElastomer mold 
5 Neo-Hookean Material Properties Neo-Hookean Free energy relationship:Note: Neo-Hookean materials only depends on the shear modulus and the bulk modulus constants as material propertiesThe Cauchy stress tensor can be simply found by using a push forward operation to bring it back to the material frameMaterial tangent stiffness matrix can be found in a similar manner as the Cauchy stress tensor
6 Derivation of Weak Form The weak form in the material frame is the same as we have derived in class for the 3D elastic case.
9 Referential Gradient Matrix The referential frame the gradient matrix is a full matrix. However, the shape functions do not change as displacement changes.While, the material frame the gradient matrix remains a sparse matrix. However, the shape functions change as the displacement.
10 Total Lagrangian FormTotal Lagrangian form takes all the kinematic and static variables are referred back to the initial configuration at t=0.By linearizing the nonlinear equations and taking appropriate substeps one can approximate the nonlinear solutionAnother formulation used called the Updated Lagrangian form refers all the kinematic and static variables to the last updated configuration at t=t-1.
11 Newton-Rhapson Method The residual vector shows us how far off the linearized version of the nonlinear model is off from the correct solution.We use the Newton-Rhapson method to approach a solution that is “acceptable.”We define [A] as the Jacobian matrix and will use it to find an appropriate change in the displacements.
12 Jacobian MatrixA common method to compute the Jacobian matrix is by taking the time derivative of the internal forces.The Jacobian matrix for each element is computed and then combine it into a global matrix to find the change in displacements.
14 ANSYS Compression Results Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPaDisplacement in Y direction: -0.2m
15 ANSYS Tension Comparison Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPaDisplacement in Y direction: 0.2m
16 ANSYS Shear Comparison Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPaDisplacement in X direction: 0.5m
17 Material Response Comparison Simple Shear Response of Neo-Hookean and Linear MaterialAxial Loading Response of Neo-Hookean and Linear MaterialMaterial Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
18 Error ComparisonError of Simple Shear Response of Neo-Hookean and Linear MaterialError of Axial Loading Response of Neo-Hookean and Linear Material
19 ConclusionHyperelastic materials are important to model using nonlinear methodsEven at small strains error can be noticeableNonlinear materials can exhibit non symmetric stress responses when loaded in the opposite direction.Their response can be hard to predict without modeling especially under complex loading conditions