Section 3: Nonlinear Analysis

Transcription

1 Autodesk Simulation Workshop Section 3: Nonlinear Analysis This section presents the theory and methods used to perform nonlinear analyses using Autodesk Simulation Multiphysics. Nonlinear phenomena can be due to nonlinear material behavior, large displacements, stress dependency, and intermittent contact. Each of these nonlinear phenomena will be treated separately or jointly in the six modules in this section. The first module provides an overview of nonlinear analysis features found in Autodesk Simulation Multiphysics and other nonlinear finite element programs. Modules 2 through 4 cover three of the more common non-linear materials models: elastic-plastic, hyperelastic, and viscoelastic. Some of the problems used to demonstrate how to setup and Modules Contained in Section 3 1. Overview of Nonlinear Analysis 2. Elastic-Plastic Material Analysis 3. Hyperelastic Material Analysis 4. Viscoelastic Material Analysis 5. Riks Analysis 6. Impact perform analyses containing these materials also contain other nonlinear phenomena such as large displacements. The Riks method covered in Module 5 is a type of analysis used to determine the load-deflection response of systems that become unstable during the solution process. Finally, Module 6 addresses contact and related modeling issues. The intent is to present the material in such a way that a senior level undergraduate engineering student or a practicing engineer could use the modules as a learning resource. Standard matrix notation is used instead of the more concise indicial notation taught in graduate school. The content is presented at a depth that an interested student can work through the mathematics and obtain insight into the methods and workings of a commercial program. Each subsequent module contains just enough theory to teach a student or practicing engineer how to set up and perform each analysis type.

2 Important Note on Archived Datasets The datasets associated with each module in Section 3 have been Archived to facilitate downloading. An Archived dataset is a compressed file created by Autodesk Simulation Multiphysics to reduce the overall size of the file. The Archived files do not contain solution results, and it will be necessary to execute the analysis in order to obtain the results. In most cases there is a Begin and an End dataset file. The Begin file contains the problem data prior to setting the analysis parameters, and will not run properly. It provides a starting point for the module. The videos show how to change the analysis parameters in the Begin datasets to those in the End datasets. The End dataset can be executed to obtain the analysis results. Note that Module 1 only has one archive file, and does not require a Begin and End file. An Archived dataset can be retrieved by selecting the Autodesk Simulation Icon in the upper left corner of the screen, selecting Archive in the drop down menu, and then selecting Retrieve. Autodesk Inventor Files The Inventor part or assembly files are also included as part of the datasets. In some cases the Inventor files are used as in the videos or PowerPoint presentations. For example, the intent of the video may be to show how to go from Autodesk Inventor into Autodesk Simulation Multiphysics. In other cases the Inventor files are not part of the videos or PowerPoint presentations, but having them available will enable instructors or students to develop additional problems or exercises.

3 Table of Contents Click below to jump to the current Module: 1. Module 1: Overview of Nonlinear Analysis Methods Module 2: Elastic-Plastic Material Analysis Module 3: Hyperelastic Material Analysis Module 4: Viscoelastic Material Analysis Module 5: Riks Analysis Module 6: Impact... 15

4 1. Module 1: Overview of Nonlinear Analysis Methods Introduction This module presents an overview of the nonlinear analysis capabilities found in Autodesk Simulation. The PowerPoint slides provide an introduction to the types of things that can cause a problem to be nonlinear. These four things are: 1) material behavior, 2) large displacements and/or rotations, 3) stress stiffening or softening, and 4) intermittent boundary conditions. The correlation between these four phenomena and the other modules in this section is also explained (Table 1). The video shows where the nonlinear analysis types are found, and where nonlinear material models are selected. Subsequent modules provide additional detail on nonlinear analysis types and material models. Execution 1) The types of nonlinear analyses available in Autodesk Simulation are found by selecting the analysis type tab in the browser. The nonlinear tab in the popup menu shows four types of analyses. a. MES with nonlinear material models b. Static analysis with nonlinear material models c. Natural frequency with nonlinear material models, and d. Riks analysis. 2) An MES analysis with nonlinear material models is the most general analysis type. It can be used to model real world events containing any combination of the four nonlinear phenomena. An MES analysis is dynamic in nature and must be used when time dependent material models are used. 3) A Static analysis with nonlinear models is limited to non-dynamic events and cannot be used with time dependent materials. 4) A Natural frequency analysis with nonlinear material models allows the natural frequencies of a deformed or stressed system to be computed. This would be used for example to determine the natural frequencies of suspension lines whose stiffness is very dependent on the tension in the cables. 5) A Riks analysis is used to determine the buckling load. It differs from a linear load factor analysis in that it can handle components with nonlinear materials. This is important for metal compression members that yield prior to buckling. 6) The various modules contained in this section provide examples of all of these analysis types except Natural frequency with nonlinear material models (Table 1). However, a similar analysis with linear materials is included as part of Section II. 7) The various nonlinear material models available in Autodesk Simulation can be seen by selecting an element definition tab in the browser. Different elements have different material models available, so the list may differ depending on the element type that was selected. A 2D element type is used in the video. 8) The nonlinear material models are organized within two main categories: 1) Elasticity, and 2) Plasticity. 9) Elasticity is associated with materials that return to their original configuration when unloaded. They exhibit no residual stress and retain their original shape. The Elasticity material models are organized into four subgroups: a) elastic, 2) hyperelastic, 3) foam, and 4) viscoelastic.

5 a. Elastic materials contain isotropic and orthotropic linear elastic materials. b. Hyperelastic material models are typically used to model elastomers (rubbers) and biological material (ligaments and tendons). c. Foam material models have similarities with hyperelastic material models in the way they are formulated. The major difference is that hyperelastic materials are incompressible or nearly so, while foam undergoes large volumetric deformation. d. Viscoelastic material models allow the several of the other elastic materials to have time dependent responses such as creep or relaxation. 10) Plasticity is associated with materials that do not return to their original shape when deformed. Their final shape is not the same as the original shape and residual stresses are often present. The most common examples of plasticity are metals that have been loaded beyond their yield stress and soils that compact. 11) All of the things that can make a problem nonlinear can be present in a single analysis. 12) Nonlinear problems require iterative solution methods. In most cases they also require that the load be applied in increments with equilibrium iterations at the end of each increment. Table 1: Nonlinear Phenomena included in Subsequent Modules Nonlinear Analysis Type Static stress with nonlinear material models MES with nonlinear material models MES with nonlinear material models Riks Analysis Module Module 2 Elastic- Plastic Material Analysis Module 3 - Hyperelastic Material Analysis Module 4 Viscoelastic Material Analysis Module 5 Riks Analysis Material Behavior X Large Displacements and/or Rotations Stress Stiffening or Softening Intermittent Boundary Conditions X X X X X X X X MES with nonlinear material models Module 6 Impact X X X

6 2. Module 2: Elastic-Plastic Material Analysis Section 3: Nonlinear Analysis Introduction This module presents an overview of the issues encountered in performing a nonlinear analysis that includes an elastic-plastic material model. The PowerPoint slides provide an introduction to concepts associated with monotonic and cyclic stress-strain curves, the Bauschinger effect, the von Mises yield criterion, and isotropic and kinematic hardening. The PowerPoint slides also show how these concepts relate to setting up an analysis. The video shows how to perform an analysis in which yielding occurs at a stress concentration point. The redistribution of stresses around the yielded material can easily be seen in the contour plots. Localized yielding around stress concentrations and the associated redistribution of stresses is a very common problem found in industry. Execution 1) The analysis type must be set to Static Stress with Nonlinear Material Models. If one of the nonlinear material analysis types is not selected the user will not be able to select any of the nonlinear material models. The analysis type can be selected during the startup process or by selecting it under Analysis Parameters in the FE Editor browser. 2) Elastic-plastic material models can be used with several element types. In the analysis being performed, the 3D brick element is used. The analysis could also be done with a 2D plane stress element since the plate is very thin and there is little stress variation through the thickness. 3) The type of nonlinear material model that will be used is selected when defining the element definition data. Either of the kinematic or isotropic hardening models could be used. The top two shown in the selection table are bilinear models that use a constant strain hardening modulus. The bottom two use stress-strain curves that can entered into a table manually or imported from an Excel spreadsheet. Since a monotonic type load will be applied either of the isotropic or kinematic hardening models can be used. Both will give the same results. The only time a distinction has to be made between isotropic and kinematic hardening is when cyclic loads are being used. Isotropic hardening will not compute a Bauschinger effect, while the kinematic hardening model will. 4) There are two methods that can be used to capture steep stress gradients: 1) use many lower order elements and/or 2) increase the order of the interpolation functions used in the elements. A combination of these two approaches is used in the example problem. Refinement points are used to increase the number of brick elements in the area of the high stress gradients. In addition, mid-side nodes are included when defining the element data. Mid-side nodes add quadratic displacement terms that provide linear strain results. The mesh density used in the analysis is based on preliminary runs made to investigate convergence of the stress results. 5) The explicit material used in the analysis is cold rolled 1020 steel. This material has a yield strength of 50.8 ksi. The work hardening modulus is automatically returned from the material database by Simulation because the bilinear kinematic hardening model was selected. If the curve kinematic hardening model had been selected, a

7 table of stress-strain values describing the true stress-strain curve for the material would have been returned. It is important that true stress-strain curves be used when performing plasticity calculations. The theory is based on true stress and strain measures. 6) The default analysis parameters can be used for this analysis. The load curve defaults to a 0 to 100% linear increase in load. Loads should be applied incrementally when performing this type of analysis. For simplicity purposes, the load was applied in 20 increments (5% of total each increment). Simulation will automatically decrease the load increment if it is not converging quickly enough. In this problem, the load increment is automatically decreased as the load approaches 100%. This is when the material has reached the point in which yielding is occurring across the total cross section and the displacements are increasing rapidly. 7) An alternative method of performing this analysis that may require less computer time would be to break the problem into two load cases. The first load case would apply the load necessary to reach the onset of yielding in one load increment. The next load case would increase the load through the elastic-plastic portion of the loading. The first load case would be used as a restart file for the second load case. 8) When elastic analyses of parts or systems having stress concentrations are performed it is not uncommon for large stresses that significantly exceed the yield stress to be computed. These elastic stresses are not realistic and an elastic-plastic analysis must be performed if more accurate stress values are required. As this example problem demonstrates, the stresses in the vicinity of the stress concentration will spread out over a larger area as yielding progresses. This redistribution of stress around stress concentrations will generally not affect the overall strength of a large complex system. However, the stress concentrations do provide locations where fatigue cracks can initiate and can decrease the life of the system.

8 3. Module 3: Hyperelastic Material Analysis Section 3: Nonlinear Analysis Introduction This module presents an overview of issues associated with performing an analysis using hyperelastic material models found in Simulation. Hyperelastic materials are routinely used in applications involving larger deformations, and the strain energy density functions used to define the material models are based on finite deformation quantities. Therefore, an introduction to finite deformation quantities used with hyperelastic materials is provided. The PowerPoint slides show how to set up an analysis of an o-ring. There are two videos associated with this module. The first video shows how to set up an axisymmetric model, define the hyperelastic material properties for a 2-parameter Mooney-Rivlin material, and then how to setup the analysis parameters. The portion of the video associated with setting up an axisymmetric model is not peculiar to a hyperelastic material and the process can be used with any of the other material types. Execution 1) The analysis type must be set to Static with nonlinear material models. This can be done either when Simulation is started or it can be changed by Editing the Analysis Type in the FEA editor. 2) The modeling plane used for an axisymmetric model is the y-z plane. The orientation of the part in the coordinate system is very important in this case. The part must be set up so that the z-direction is the axis of symmetry (longitudinal direction) and the y- direction is the radial direction. The x-direction is in the circumferential direction. It is easiest if the CAD model has a cut surface lying on the y-z plane. There are commands in Simulation that can be used to rotate and/or project a surface to make it correspond to the y-z surface. The project command is used in this analysis. 3) It is customary to define the element type as one of the first steps in creating an FE model. When setting up an axisymmetric model is easier to first generate the mesh and then define the element types. Unless the x-coordinate for all vertices lying on the y-z plane is identically equal to zero, Simulation will not let 2D elements be defined. 4) The wedge model used in this analysis has surfaces that are not needed in the axisymmetric model. All surfaces are initially set to have a mesh generated on them. The surfaces for each part can be seen by expanding the surfaces in the FEA Editor. Each of the surfaces has an option that will allow it to be excluded from the meshing process. Each surface not lying on the y-z plane must have this option deselected. 5) The default meshing process is set to mesh a solid part Plane/Shell option must be selected on the 3D Mesh Settings screen to create a 2D or axisymmetric model. 6) The Automatic Mesh Geometry Function is turned off under the Model button on the 3D Mesh Settings screen. This will allow a uniform mesh to be created using an absolute mesh size. 7) The Absolute Mesh size specified for the model is set to 0.01 on the Surface screen of the 3D Meshing settings. This mesh size was found based on trial-and-error to give adequate results for the shaft and housing. A finer mesh is used for the o-ring.

9 8) The o-ring mesh size is selected by editing the CAD Mesh parameters for the o-ring in the FEA Editor browser. An absolute mesh size of inch was found to give acceptable results. Although the runtimes for non-linear analyses can be significant, it is important to verify that final results are not sensitive to the size of the mesh used in the analyses. 9) Once the mesh has been generated, all of the vertices can be forced to lie exactly on the y-z plane by: 1) selecting all vertices, and 2) right clicking and selecting the Project command. The x-value should be set equal to zero. Even if an x-value is 1.0e-15, Simulation won t allow 2D elements to be defined. The Project command is the easiest way to ensure that this condition is met. 10) Now, that the mesh is defined the element types can be set. All parts use the 2D element type. 2D elements can be used to model plane strain, plane stress, and axisymmetric problems. 11) The Element Definition data allows the type of 2D element and the specific class of material to be specified. The 2D element geometry parameter for each of the parts should be set to axisymmetric. 12) The material class for the shaft and housing are set to Linear Isotropic. 13) The material class for the o-ring is set to Hyperelastic, Mooney-Rivlin. The other hyperelastic material types can be seen in the list used to select the Mooney-Rivlin. The Mooney-Rivlin material is selected for this problem because the stretches are expected to be moderate (i.e. less than 200%). 14) The material constants for each part are defined by editing the material properties for each part. A two-parameter Mooney-Rivlin model is used in this analysis. Higher order Mooney-Rivlin material models are available in Simulation. The material library values for the two parameters are used in the analysis. These parameters are associated with the deviatoric or distortional deformation of the material. 15) The Bulk Modulus is also required and is associated with the volumetric deformation. A truly incompressible material will have an infinite bulk modulus. A method for estimating the bulk modulus for a nearly incompressible material is contained in the PowerPoint slides. A value of 400,000 psi is used in the analysis. This corresponds to a Poisson s ratio of ) The vertices at the compression end of the shaft and housing are set to zero in the z- direction. There is no need to define a displacement constraint in the y-direction because of the strain constraints inherent to an axisymmetric model. 17) There is contact between the o-ring and the shaft, and the o-ring and the housing. The contact is defined by: 1) selecting each pair of parts involved in the contact, 2) right clicking and selecting contact. Surface contact is defined for each pair of parts. 18) By selecting the contact pairs in the browser, and right clicking on the Settings, the parameters for each contact pair can be edited. In this analysis, the frictional contact option is used. The coefficients of friction (static and dynamic) are set to The tangential stiffness ratio parameter is set to The addition of friction often adds stability to an analysis. The absence of friction is not realistic unless significant lubrication is used. The coefficients of friction were chosen to simulate rubber to metal contact with light lubrication. 19) The element edges where the pressure load is to be applied to the o-ring are classified as lines. They must be converted to surfaces so that the pressure load can be applied to one side of the o-ring. All of the lines associated with the element pressure loads should be selected. By right clicking and selecting Edit Attributes,

10 the lines can changed to a single surface. The selection option must now be switched from Lines to Surfaces so that the new surface can be selected. 20) Select the new surface, right click and select Add, then select Pressure/Traction. The magnitude of the pressure to be applied is 500 psi. The analysis is now ready to be submitted. 21) Once complete the results can be reviewed in the results section. Of particular importance are the two zones of deformation seen in the results. The top zone has very little deformation and is experiencing principally volumetric type deformation due to hyrdrostatic loading. This causes the von Mises stress levels in this area to be low. The von Mises strain and shear stress are also low because of the hydrostatic loading. The lower zone has significant deformation since it is not constrained by the pressure loading. The distortion causes the von Mises stresses and strains, and the shear stresses to be much higher than in the lower zone. 22) The Penetration Probe showing the amount of penetration at the contact surface between the o-ring and gland may appear on the screen during the solution. This can be turned off by going into the Results Inquire tab, and clicking on the move button under Probes, and then choosing Contact Diagnostic Probes

11 4. Module 4: Viscoelastic Material Analysis Section 3: Nonlinear Analysis Introduction This module presents an overview of issues associated with performing an analysis using viscoelastic material models found in Simulation. Viscoelastic materials demonstrate timedependent stress-strain relationships that lead to creep, relaxation, and hysteresis phenomena. The difference between linear and nonlinear viscoelastic material behavior is discussed in the PowerPoint presentation. In Simulation, linear viscoelastic data can be coupled with linear or nonlinear instantaneous material models. In the PowerPoint presentation and the video an example of a cantilevered sandwich beam is presented that uses linear viscoelastic relaxation data in conjunction with a nonlinear hyperelastic material model. Execution 1) The analysis type must be set to MES with Nonlinear Material Models when Simulation is started. This is one of the nonlinear analysis types. The analysis type can be selected during the start-up process or selected under Analysis Parameters in the FE Editor browser. 2) Viscoelastic material models can be used with several element types. In the analysis used in the example, a 2D plane strain element is used. The width of the beam is relatively large compared to its thickness and a plane strain formulation is appropriate. 3) Module 3 shows how to create a 2D axisymmetric model from a 3D CAD file. The method used to create a 2D plane strain model is very similar. The basic steps are to include only those surfaces that lie on the yz-plane in the meshing operation. Once the yz-plane surfaces are meshed using the plate/shell option in the 3D Meshing Options tab, the vertices are projected onto the yz-plane by setting the x-coordinate to zero using the project command. The element types can then be set to 2D. 4) The top and bottom plates are made of aluminum and a linear isotropic material is selected in the element definition dialog box. Mid-side nodes should also be selected on the element definition dialog box. Mid-side nodes increase the order of the interpolation functions used in the element formulation and will ensure that the stress variation through the thickness of the beam is captured correctly. 5) The adhesive is a commercial adhesive known as ISR This adhesive was selected because detailed relaxation data is contained in the reference given in the PowerPoint presentation. Hyperelastic and viscoelastic data for specific polymers can be difficult to find, and independent testing programs are often required to support analysis activities. The relaxation data presented in the reference was obtained using a Dynamic Material Analyzer technique in conjunction with a shift function. 6) The adhesive is modeled as a Mooney-Rivlin hyperelastic viscoelastic material. The Mooney-Rivlin constants define the instantaneous shear modulus of the material. The reference for ISR gives the instantaneous tension modulus and the PowerPoint presentation shows how to convert this to the instantaneous shear modulus. The relationship between the shear modulus and the Mooney-Rivlin constants is G=2(C 10 +C 01 ). This equation and the assumption that the ratio of the

12 Mooney-Rivlin constants (C 10 /C 01 ) is equal to four are used to determine C 10 and C 01 in the example. This assumption would not be required if a Mooney plot for the adhesive material were available. 7) The bulk modulus is estimated by assuming that Poisson s ratio is An incompressible material has a Poisson s ratio of 0.5 and an infinite bulk modulus. A Poisson s ratio of makes this material nearly incompressible. 8) The Prony series data can be entered using two forms. One form uses the instantaneous shear modulus, G 0, as the multiplier, while the other uses the relaxed modulus, G, as the multiplier. In the example problem, the first form is used and this form must be selected prior to entering the Prony series relaxation data. 9) Simulation allows the relaxation data to either be applied to both the volumetric and deviatoric properties. One option allows the same relaxation data to be applied to both. Another option allows different relaxation data to be applied to the volumetric and deviatoric components. In the example, it is assumed that volumetric response is nearly incompressible and remains so throughout the analysis. Therefore, there is no relaxation of the bulk modulus. The volumetric Prony series data is set to zero. The deviatoric or shear Prony series is entered into the table. Note that the Relaxation Modulus column contains relaxation moduli that have been normalized by the instantaneous shear modulus. They are non-dimensional and the sum must not exceed a value of one. When their sum is equal to one, the material will relax to a value of zero at infinite time. 10) The only analysis parameters that must be edited prior to submitting the simulation is to set all of the multipliers in the load curve table to ones. This will result in a step input. The capture rate is set to 500 which results in a time step of seconds. If too large of a time step is selected, Simulation will automatically decrease the time step size if the number of iterations becomes excessive. 11) Simulation has the ability to embed a graph in the results screen that is updated as the analysis progresses. The video shows how this is done. This is a very useful feature that allows any variable that is computed (e.g. stress of displacement) to be viewed as a function of time during the analysis. The video shows only one graph being embedded. However, multiple graphs could have been added to show the history of more than one parameter. 12) The damping observed in the results is a result of the viscoelastic material. The relaxation data completely defines the viscoelastic properties of the material and phenomena such as creep, dynamic damping (hysteresis), or relaxation can be simulated. In the example problem, the dynamic damping properties of the adhesive were demonstrated. The storage modulus and loss factor as a function of frequency can be determined using the relaxation data. The selection of the best damping material (adhesive) for a particular application is one that has a high loss factor at the particular frequency or frequencies of interest. 13) There will also be damping of high frequency modes due to the numerical integration of the equations of motion. Since the response shown in example is due to lower order modes, the damping from numerical integration should be small for this example. 14) The results are also consistent with the step response of a linear spring-mass system. The peak amplitude ratio for such a system is two. The cantilevered beam results are close to this.

13 5. Module 5: Riks Analysis Introduction This module presents an overview of the Riks Analysis option in Autodesk Simulation and shows how to apply the method to determine the buckling load of an elastic-plastic arch. The PowerPoint slides provide an overview of how a Riks Analysis is different than standard static analysis methods. The concept of a load factor is presented. The load factor changes in magnitude throughout the analysis as equilibrium at each load increment is pursued. The load factor can increase, decrease and even go negative as the analysis progresses. The maximum value of the load factor determines the buckling capacity of the system being analyzed. There are two videos associated with this module. The first video shows how to set up a Riks Analysis, while the second shows how to process and interpret the results. Execution 1) The analysis type must be set to Riks Analysis when Simulation is started. This is one of the non-linear analysis types. The analysis type can be selected during the startup process or by selecting it under Analysis Parameters in the FE Editor browser. 2) Elastic-plastic material models can be used with several element types. In the analysis being performed, the 3D brick element is used. One of the issues with brick elements is that they lock up when they get too thin. Precautions will be taken to avoid this problem during the mesh generation phase. 3) An elastic-plastic material with kinematic hardening will be used to define the nonlinear material type. Since strain cycling will not be encountered during the analysis, either the isotropic or kinematic hardening model can be used. The kinematic model is selected in the video, but either hardening model should give similar results. 4) Mid-side nodes are used to increase the flexibility of the brick elements through the thickness direction of the arch. A bending type stress distribution is expected through the thickness of the arch. The derivatives of the displacement interpolation functions used in an eight node brick will not capture this linear strain variation. The derivatives of the displacement interpolation functions for a brick element having midside nodes will capture this linear strain variation. 5) Post-buckling analysis response typically involves large displacements and a Riks Analysis uses the large displacement option. 6) The automatic mesh size function is turned off so that an absolute mesh size can be entered. Specifying an absolute mesh size provides better control over the size and aspect ratios of the brick elements. The automatic mesh size function tends to create elements with large aspect ratios and widely varying size. Specifying an absolute mesh size provides a more uniform mesh. A large aspect ratio (i.e big in area direction and small in the thickness direction) can lead to lock-up of brick elements. 7) An absolute mesh size of inches will give two elements through the thickness. The mid-side nodes may be sufficient to permit one element through the thickness, but two will give more assurance that the stress gradients are properly computed. 8) The intent of the boundary condition specifications at the ends of the arch is to not restrain rotation about the z-axis. This results in a hinge. Only the nodes along the

14 center row of nodes on the end surfaces are restrained. Fixing nodes on the upper and lower edges will restrain rotation of the surface. 9) The x-direction displacement restraints at the center of the arch will ensure a symmetric solution and will also make the analysis converge quicker. Without these constraints the computed buckled shape may be skewed more to one side than the other due to small numerical imperfections in the model. These displacement constraints can be omitted to see what effect they have. The magnitude of the initial load factor will need to be decreased to obtain convergence. 10) The Embed Graph feature is important in this type of analysis. Because Simulation lacks the option to create a plot of the load factor versus increment number or display the load factor on the screen for each load increment, the only place the load factor can be found is in the log file. The log file data can be pasted into Excel to create a plot of the kind seen in the second video and in the PowerPoint slides. The maximum load factor seen in the plot defines the buckling load. 11) The number of time steps (load increments) controls the duration of the analysis. This number needs to be large enough to capture the desired results. Two numbers were used while setting up this example problem, 50 and 100. The lower number, 50, was used to obtain some initial results. After it was determined that 50 captured all of the response of interest, the higher number, 100, was used to determine if the solution had sufficient resolution and accuracy. There was little difference between the two sets of results. The wiggle in the post-buckling response region was a little smaller when 100 load increments were used. The load factor at which buckling occurs did not change. 12) The initial load factor parameter is used to compute a set of displacements that establishes the magnitude of the hypersphere constraint. This magnitude is treated as a constant throughout the analysis. The linear displacement versus load increment plot shown in the second video is a direct result of keeping the constraint magnitude constant. Using a lower initial load factor would result in smaller displacements for each load increment in the analysis. 13) A displacement controlled experiment could be carried out to experimentally determine a load factor versus load increment curve similar to that computed by the Riks Analysis method. This might be necessary to maintain a safe experimental setup. Most real world load phenomena are not displacement controlled and the load increases remains constant throughout the buckling process. The post-buckling response is dynamic in nature, and a full MES Nonlinear Dynamic Analysis is required to capture this dynamic response. 14) The most common example used to demonstrate the Riks method is a shallow elastic arch. This problem experiences snap through, a phenomenon in which the arch snaps from a concave down geometry to a concave up geometry. The example problem covered in this module does not snap through because of the elastic-plastic material model used.

15 6. Module 6: Impact Introduction This module presents an overview of concepts associated with performing a Mechanical Event Simulation with Nonlinear Materials that includes impact. Impact is contact between two parts that happens at high speed. Therefore, the PowerPoint slides also address concepts associated with contact in general. The example problem used in the PowerPoint slides and in the video uses the same sandwich beam used in Module 4: Viscoelastic Materials. A steel cylinder is added that drops under the action of gravity and impacts the beam. The nonlinear response is examined in the video. Execution 1) The analysis type must be set to MES with Nonlinear Material Models when Simulation is started. This is one of the nonlinear analysis types. The analysis type can be selected during the start-up process or selected under Analysis Parameters in the FE Editor browser. 2) The method used to generate a 2D mesh from a 3D CAD file is shown in Video 3A. In this problem a plane strain 2D element is used. Plane strain was selected because the beam is relatively wide compared to its thickness. 3) The mesh uses an absolute element size of 1/64 of an inch. It also uses mid-side nodes to improve the flexibility of the 2D elements in bending. The problem is sensitive to the amount of bending experienced by the beam. If the problem is run without mid-side nodes, it is stiffer and the cylinder will strike the beam at a different location on the second impact and will not roll off the end. Having students run this option and discuss the difference between the results would make a good homework exercise. 4) The method used to specify the viscoelastic properties of the adhesive are discussed in the Module 4 PowerPoint slides and are illustrated in the Video 4. 5) Two different types of contact are encountered in the model used for this analysis. The first type is between the common surfaces of the top and bottom plates and the adhesive. The adhesive doesn t allow relative motion along the shared surface and the default contact type of Bonded is used. Since it is the default type, no action is required to define the contact. Simulation will automatically apply this type of contact when it determines that there two surfaces occupying the same space. 6) The second type of contact is between the perimeter surface of the cylinder and the free edges of the top plate in the cantilevered beam. Surface contact allows parts to separate or come into contact and to have sliding motion between the contact surfaces. This is the most general type of contact found in Simulation. 7) The contact parameters are specified by editing the contact settings. These parameters specify that the analysis involves impact, that no friction is to be included, and that surface to surface contact is desired. 8) The analysis parameters are set similarly to those in Module 4 with the exception that the Event Duration is set to 0.6 seconds. Exploratory analyses showed that this is long enough to have the cylinder impact the beam twice and then roll of the end. 9) A Capture Rate of 500 steps/second is specified. This results in a time step of seconds. This allows seven or eight points per oscillation of the cantilever to be

16 computed. During contact between the beam and cylinder, Simulation will automatically decrease the time step to obtain quicker convergence. 10) The analysis runs approximately one hour. This should provide an appreciation for how long it takes to perform a complicated impact problem (e.g. between an automobile and a rigid obstacle). 11) The time variation of stress or displacement at a particular location can be seen by embedding a plot on the screen. The method used to do this is shown in detail in Video 4, and is illustrated quickly in the video for this module. 12) The best way to examine the results visually is to create an animation. The video contains a good example how much information can be seen using this approach. The vibration of the cantilever beam due to the impact forces imparted by the cylinder can be easily seen. 13) Although the cylinder is dropped along the y-axis, it picks up momentum in the z- direction due to the angle of the surface normal vector caused by the slope of the beam surface. The contact force and contact stiffness are applied normal to the master surface. As the beam deflects the surface normal takes on a small z-direction component.