SECTION 6 NONLINEARITY

TABLE OF CONTENTS Section Page 6.0 Nonlinearity What Causes Nonlinearities In Structural Mechanics (P  Ku)……………………………………………….. 6-3 Example: Linear Spring…………………………………………………………………………………………… 6-5 Example: Nonlinear Spring……………………………………………………………………………………….. 6-6 Using Finite Element Analysis To Solve Nonlinear Problems: Collapse Example………………………… 6-7 Using Finite Element Analysis To Solve Nonlinear Problems: Gap Contact………………………………… 6-8 Multiple Sources Of Nonlinearity…………………………………………………………………………………. 6-9 Contact Nonlinearity……………………………………………………………………………………………….. 6-10 Contact Element Library: Gap Elements………………………………………………………………………… 6-14 Gap Element Definition……………………………………………………………………………………………. 6-15 Remarks On Gap Elements……………………………………………………………………………………….. 6-18 Geometric Nonlinearity…………………………………………………………………………………………….. 6-19 Material And Geometric Nonlinearity…………………………………………………………………………….. 6-20 Example Of Geometric Nonlinear Analysis:Blade On A Jet Engine………………………………………… 6-21 Example Of Geometric Nonlinear Analysis: Lateral Collapse Of Flat Beam……………………………….. 6-22 Example Of Material, Contact and Geometric Nonlinear Analysis: Punching A Thin Shell….…………... 6-23 Major Difficulties in Nonlinear Analysis…………………………………………………………..l..…………... 6-24

WHAT CAUSES NONLINEARITIES IN STRUCTURAL MECHANICS Material nonlinearities: Material Properties change Plasticity Temperature-dependent properties Hyperelasticity (Rubber) Large Displacements and Strains Contact Structural Instabilities and Collapse Example: Reinforced Rubber seal Problem combines all sources of nonlinearity, which is typical of nowadays real life engineering problems. How do you define temperature-dependent properties in MSC.Patran? Q

WHAT CAUSES NONLINEARITIES IN STRUCTURAL MECHANICS (CONT.) Boundary nonlinearities: Boundary conditions change Contact problems (gaps open and close) Geometric nonlinearities: Configuration changes Large deflections (K=K(u)) Large rotations (bending) Structural Instabilities(buckling) Preloads Static equilibrium: K u = P K = K(P,u) K(P,u) u = P

EXAMPLE: LINEAR SPRING The solution to a linear spring problem is straightforward In a linear case, the displacement, u, is directly proportional to the loading, P. We can write: P = k u We want to solve for u when P is given. The deflection is related to the load by the stiffness. As a result, the final value of u can be easily solved as: u = P / k However, even if the all materials involved are linear and stresses are small one may have large displacements and need to execute a nonlinear analysis. Example: Truss-Spring system Demo e4x6.dat

EXAMPLE: NONLINEAR SPRING u P = k(u) u The stiffness k is known in terms of u, therefore, P can be calculated in terms of u, but an explicit solution for u in terms of P is not known. Therefore, an iterative technique must be employed. The same method used for the regular stiffness coefficients is applied to a nonlinear spring stiffness. The default method in MSC.Marc is the Newton-Raphson method. One necessitates to control that displacement increments do not result in errors in the estimate of P beyond tolerance.

USING FINITE ELEMENT ANALYSIS TO SOLVE NONLINEAR PROBLEMS: COLLAPSE EXAMPLE There are a variety of issues related to Finite Element analysis of structures with nonlinearities Issue #1: What is the source of nonlinearity Issue #2: Why nonlinear modeling is important. Example: Cylindrical Shell Geometric Nonlinearity due to buckling The shell collapses after the critical buckling point load has been exceeded. Without nonlinear analysis the critical load will be overestimates –dangerous.

USING FINITE ELEMENT ANALYSIS TO SOLVE NONLINEAR PROBLEMS: FRICTION CLUTCH Importance of Nonlinearity Classical analysis techniques simply are limited as to their application and problems that fit into linear assumptions Many “Real World” structures exhibit nonlinear behavior Example: Friction Clutch Contact with Friction Nonlinearity This problem actually requires a Coupled Thermal-Structural as the friction generates intense heat producing differential thermal expansion affecting the contact conditions, which simultaneously affects the heat generation.

MULTIPLE SOURCES OF NONLINEARITY Contact is an essential issue in many design problems. Many FE codes offer only limited contact models, such as “gap elements” (Node-to-Node contact in a given direction). MSC.Marc resolves general Body-to-Body contact. Undeformed Outline Impact Contact Pair Hyperelastic (rubber) band Sliding Contact Plastic Seal ANVIL Trigger Hammer Stamp Complex example with multiple sources of nonlinearity: Contact Large displacements Material Hyperelastic plastic

CONTACT NONLINEARITY There are three primary types of contact models: Requiring special elements (use the Properties form in MSC.Patran) 1- Point-to-point contact (gaps) Requiring identifying Contact Bodies (use the Loads/BSc form in MSC.Patran) 2- Rigid body-to-deformable body contact 3- Deformable body-to-deformable body contact Gap: The nodes lateral motion is assumed to be small. Contact is resolved as selected node against another selected node Body-to-body contact: Nodes may move arbitrary distances over surfaces of other contact bodies. The location and extend of contact is found and resolved. LINE VARIABLE SCALE FACTOR 1 X-ANGVEL 1 +1.00E+00

CONTACT NONLINEARITY (CONT.) Undeformed Shell Pod MSC.Marc and MSC.Patran provides contact capabilities for all three types of contact Gaps, in a fixed direction. Elastic body to elastic body contact: surface to surface contact in 2D and 3D. Used for detailed resolution of contact in bearings etc.: can provide very accurate prediction of contact area and pressure. Rigid body to elastic body contact. Many important design problems are otherwise linear with this type of simple surface-to-surface contact. Example: Eye Surgical Implant. Mises Stresses on the elastomeric solid body after three steel pieces pushed on three separate pods. The solid body is completely free of BCs other than the contacts. Other codes require temporary restrictions while MSC.Marc can resolve the problem even when the rigid surfaces are initially separated.

CONTACT NONLINEARITY (CONT.) Rigid surface contact. Contact between a deforming body and rigid body. Very widely used: gaskets, seals, metal forming, et cetera. Rigid surface may be planar or curved in space. Rigid body definition available as rigid line or surface geometry, or using discrete rigid surface elements Ring Example: Multipass Ring Rolling Manufacturing Process Rigid Surface

CONTACT NONLINEARITY (CONT.) Contact may have physical properties that may need to be modeled Contact body interaction can be controlled Contact may be modeled with or without friction. Several (simple) friction models are provided. All contact features work with dynamics (impact) as well as statics. Example: Crash between two Tubes Contact and Nonlinear Transient

CONTACT ELEMENT LIBRARY: GAP ELEMENTS Monitor node-to-node contact Basic types are fixed-direction, true distance, and frictional with bending. Basic variables are the clearance, the direction, and the force transmitted. GAPs account for actual physical surface location (thickness) of structures connected. Default clearance is 0. In MSC.Patran, GAP elements properties are specified under element Properties. There are three type of Gap elements: Fixed Direction True Distance Friction with Bending

GAP ELEMENT DEFINITION GAP’s are created using BAR/2 elements connecting two nodes (one node on each part coming into contact). Fixed Directional GAP elements monitor contact between a node pair in a fixed direction given by a vector. True Distance gaps can be thought as “spherical” gaps (more later). The Gap could initially be “OPEN” or “CLOSED”. If “OPEN”, the “Closure Distance” may be defined to be smaller than the distance between nodes. This is useful to account for the cross section sizes of 1-D beams.

GAP ELEMENT DEFINITION (Cont.) True Distance gaps can be thought as “spherical” gaps. It can also be initially set as “OPEN” or “CLOSED” The “Limiting Distance” is measured spherically with respect to the first node. The “Limit Type” may be set as “MINIMUM”, in which case the gap acts as a Peg. It may be set to “MAXIMUM” in which case the gap act as a String preventing the second node from moving away from the first node beyond a certain distance but allowing the nodes to come together.

GAP ELEMENT DEFINITION (Cont.) Friction with Bending Gaps correspond to Marc element Type 97. It is a special 4-node gap and friction link with double contact and friction conditions. It is designed specifically for use with element types 95 (Axisymmetric Quadrilateral with Bending) and 96 (Axisymmetric, Eight-node Distorted Quadrilateral with Bending).

REMARKS ON GAP ELEMENTS Initial interference can be specified by using a congruent (although not equivalent) mesh with a negative Initial Clearance value If the nodes are not coincident, MSC.Marc will calculate the closure direction as being from node1 to node2 based on their initial positions if you put <0, 0, 0> in as the closure direction For other gap elements, depending on the initial coordinates and the specified clearance, they may start in an open or closed state. Gap Elements have unit area, so the contact “stress” calculated is actually the contact force—Don’t use gap elements for large surface-to-surface contact. Use them when contact is point-to-point For Fixed Directional GAP elements, contact depends only on initial clearance and the relative displacements of the two nodes in the direction of contact. Initial coordinates are not used at all to define initial clearance

GEOMETRIC NONLINEARITY Linear Solution Source of nonlinearity: large displacements (geometric nonlinearity) Geometric nonlinearity occurs whenever the magnitude of the displacements affects the response of the structure. Example: Cantilever Beam loaded with a Tip Load If the deflection is large, the actual, physical beam develops axial strains, which absorbs strain energy: This results in a lesser vertical deflection of the beam tip. Nonlinear Solution

MATERIAL AND GEOMETRIC NONLINEARITY Geometric nonlinearities may include: Preload or stress stiffened structures Buckling, post buckling, “snap thru” or collapse Kinematic or Large Rotations Large Strains MSC.MARC provides a solution methodology which allows modeling of complex geometric nonlinear problems. Example: Rubber Boot Self-Contact and Material Nonlinearity

EXAMPLE OF GEOMETRIC NONLINEAR ANALYSIS: BLADE ON A JET ENGINE Stress stiffening (like a violin string). MSC.MARC lets the user find the natural frequencies of a structure under stress. Example: Jet Engine modes and frequencies of a blade on a jet engine are much higher when the engine is rotating at operating speed than when it is stationary.

EXAMPLE OF GEOMETRIC NONLINEAR ANALYSIS: LATERAL COLLAPSE OF FLAT BEAM Bucking and collapse. The opposite of stress stiffening: shell-like structures under compressive stress can fail suddenly (buckle) and collapse. Very important in many structural design applications (shells). Kinematics (large rotations). Example: Lateral Buckling

EXAMPLE OF MATERIAL, CONTACT ANDGEOMETRIC NONLINEAR ANALYSIS: PUNCHING A THIN SHEET Large strains. Design of rubber components. Design of metal forming processes. Rubber problems are generally straightforward and very useful. Examples: engine mounts, gaskets and seals, solid propellant. Example: Stretching of a thin sheet with a hemispherical punch. (Demo e8x52.dat)

MAJOR DIFFICULTIES IN NONLINEAR ANALYSIS: How to define contact between structures How to create an initial mesh that can undergo large deformations How to define the time steps for transient analysis Defining nonlinear material laws Meshing and Element Technology Large CPU times