Week-6 Calculate the Stretch Ratio by comparing the ELFORM (-2,-1,1,2) with Ogden_Material Model.
INTRODUCTION:
Ogden material model is a hyperelastic material model used to define the non-linear behaviour of materials like rubber and polymers. This material model is generally used for materials showing incompressible, isotropic and non-linear stress-strain behaviour.
In this study, a solid cube of rubber material modeled by Ogden material model is simulated for tensile behaviour. Different element formulations are used to compare the stress vs stretch behaviour.
THEORY:
Engineering problems can be categorized based on the time scale of physics. Generally, slow processes are time-independent where loads are applied sowly and inertia effects are neglected. Whereas fast processes are time-dependent where inertial effects can not be neglected. The non-linearity increases as the problems become dynamic. Two methods are used to solve these kinds of implicit and explicit method.
IMPLICIT METHOD:
This method is generally used for solving slow processes where inertia effects can be neglected. Here, the problem is solved for the static equiibrium where the sum of the forces is equal to zero. This method solves the problem by inverting the stiffness matrix. We get a solution in one step for simple linear problems but equillibrium.
Iterations are performed within a time-step if the problem is non-linear. The solution is unconditionaly stable but at the time steps should be small enough to achieve the convergence for non-linear problems. This method can be used for static and quasi static problems.
EXPLICIT METHOD:
This method is generally used for solving fast processes where inertial effects are important. The problem is solved for the dynamic equillibrium where the sum of forces are equal to mass time acceleration. This method only increments without any iterations. It is conditionally stable, the time step should be smaller than the critical time step size (Courant-Friedrich-Levy time step). This method is generally used to solve dynamic problems because it handles non-linearities (material, contact and geometrical) with ease.
The tensie behaviour of cube is time-independent and a slow process. The inertial effects are also no important here. So, the implicit method is the best choice for the given problem. The material used for cube is defined by Ogden material model. This has a non-linear stress-strain behaviour and so this cannot be solved in one step but equillibrium iterations would be used to solve this problem.
ELEMENT FORMULATION:
Cube is modelled with a solid element with different element formulations.These are discussed below,
Constant stress solid element (ELFORM 1) is the under-integrated solid element. It needs hourglass control for large deformation problems due to under-integrated formulation. It is efficient, accurate and has low computational cost compared to other solid elements.
Fully integrated S/R solid element (ELFORM 2) is a selective reduced integrated element. The volumetric part of the stiffness matrix has reduced integration formulation which can alleviate volumetric locking issues. But the deviatoric part has fuly integrated formulation so, these are susceptible to shearlocking.
To resolve this shear locking problem, ELFORM -1 and ELFORM -2 are developed which are simiar to EFORM 2. These solid elements have selective reduced integration formulation which is used to prevent volumetric locking. These solid elements work wellfor poor aspect ratio as transverse shear energy is not affected by a poor aspect ratio for this formulation. ELFORM -1 has efficient formulation and EFORM -2 has accurate formulation. EFORM -2 has higher computational cost than ELFORM -1
PROCEDURE:
The input keyword file with Ogden material card is given with the problem statement, this keyword file is opened in LS-PrePost.
At first, a solid cube is created in LS- PrePost by using Shape Mesher option in mesh utility. The cube with edge length 10mm is divided into 10 blocks in all three directions. Finally,we get a cube with 1000 solid elements.
Now to model a tensie behaviour the cube is held from one side and given displacement from the other side. For this single point constraint and prescribed motion boundary conditions are applied through create entity utility. Now, for a tensie test in reality the material (cube) will elongate and because of this the cube edges and faces will shrink which are normal to the applied displacement, so these nodes are free to move in those directions and not constrained. Also, the centre node of that face will remain in the same position until the end and will not translate anywhere. Due to the tensie force generated, cube tends to rotate along the axis so even these degree of freedoms are constrained as shown in the fol owing figures.
Objective:
To simulate the tensile behaviour of a solid block with different element formulations and compare the stretch ratios from each case.
Step 1: Solid block creation and section definition: The given keyword file, which contains the Ogden Materialdefinition is opened in LS Dyna. The default values of the hyperelastic materialdefinitioncan be retained.
A FE modelof a solid box is created using Shape mesher option under Element and Mesh panel as shown below.
Section definition: The section properties for the solid block are assigned as shown below.The different element formulations, that need to be tested are ELFORM 1,2, -1 and -2.
Part Definition
The material and the section are then assigned to the part ID of the solid block as shown below.
The definitions with different ELFORMS are saved separately in a keyword file and simulated individually to obtain the results.
Step 2: Boundary conditions SPC - X direction
The nodes of the box on the YZ plane are selected and their motion is constrained in the X-direction, since they should be fixed.
SPC - Z Direction
The motion of the middle nodes in the Y direction is constrained in the Z - direction, to avoid vertical displacement.
SPC - Y Direction
The motion of the middle nodes in the Z direction is constrained in the Y - direction, to avoid lateral displacement.
Prescribed Motion:
The nodes on the opposite side of the YZ plane are selected and a displacement is given to them according to the minimum load to be applied on the movingsurface to achieve a stretch ratio of 5, therefore a minimum value of 50 mm is taken to define theload curve.
The above shownload curve is assigned to the nodes using the appropriate LCID as shown below.
Step 3: Control cards: A control_implicit_auto card, with IAUTO set to 1, is used for automatic timestep adjustment.
A control_implicit_general card is used with IMFLAG = 1 to specify that an implicit solving scheme needs to be used.The initialtimestep value DTO is set to 0.01.
A control_implicit_solution card is used to specify whether a linear or a non linear solution is desired. This card is used with default parameters.
The simulation time is set as 1 ms using the control_termination card.
Step 4:Database cards: A database_binary_ D3 plot is defined with an output frequency DT of 0.01 ms.
Under DATABASE_ASCll, GLSTAT and ELOUT are selected with the same value of DT.
A DATABASE_ EXTENT_ BINARY card is used with STRFLG = 1, in order to compute the elastic strain in the model.
The keyword file is checked for errors using the Model Checker and saved before startingthe simulation.
Results: The simulation is completed with the fol owing message showing a NORMAL TERMINATION
X-Stress: The uniaxial stress plots of the 4 cases can be plotted as shown below and the maximum stress in the X-direction can be visualized.
ELFORM 1
ELFORM 2
ELFORM -1
ELFORM -2
The peak stresses are as follows
ELFORM 1 - 5.024 MPA
ELFORM 2 - 5.031 MPA
ELFORM -1 - 5.024 MPA
ELFORM -2 - 5.024 MPA
X-Strain
The uniax ial strain plots of the 4 cases can be plotted as shown below and the maximum strain in the X-direction can be visualized.
ELFORM 1
ELFORM 2
ELFORM -1
ELFORM -2
The peak strains are as follows
ELFORM 1 - 1.788
ELFORM 2 - 1.789
ELFORM -1- 1.788
ELFORM -2 - 1.789
Calculation of engineering strain, engineering stress and stretch ratio
The output of the D3 plot in LS Dyna gives true stress and true strain. The engineering stress and strain can be calculated using the relations below.
True stress = Engineering Stress (1 + Engineering Strain)
True strain = ln (1 + Engineering Strain)
Stretch ratio = 1+ Engineering Strain
The values of the true stress and strain are imported into an excel sheet and the computation of the engineering stress and strain for the 4 cases are done as shown below. The plot of Engineering stress against the stretch ratio for each case is also shown for each case.
Comparison:
From the graphs of Engineering stress vs stretch ratio, the max value at a stretch ratio of 5 is approximately 0.8 MPA. The values from the 4 cases are almost the same, due to the simplicity of the model considered.
Conclusion:
Thus the FE modelof a solid block has been simulated for uniaxial loading with various element formulations and the results of engineering stress vs stretch ratio have been compared.
Thank and regards,
Shubham Sharma