s6-1pat328, section 3, march 2001mar120, section 6, december 2001 section 6 nonlinearity

S6-1PAT328, Section 3, March 2001MAR120, Section 6, December 2001

SECTION 6

NONLINEARITY


TABLE OF CONTENTS

Section Page

6.0 NonlinearityWhat Causes Nonlinearities In Structural Mechanics (P Ku)………………………………………………..

6-3 Example: Linear Spring……………………………………………………………………………………………

6-5Example: Nonlinear Spring………………………………………………………………………………………..

6-6Using Finite Element Analysis To Solve Nonlinear Problems: Collapse Example………………………… 6-7 Using Finite Element Analysis To Solve Nonlinear Problems: Gap Contact…………………………………6-8Multiple Sources Of Nonlinearity…………………………………………………………………………………. 6-9Contact Nonlinearity……………………………………………………………………………………………….. 6-10 Contact Element Library: Gap Elements…………………………………………………………………………6-14Gap Element Definition……………………………………………………………………………………………. 6-15 Remarks On Gap Elements………………………………………………………………………………………..

6-18 Geometric Nonlinearity……………………………………………………………………………………………..

6-19Material And Geometric Nonlinearity……………………………………………………………………………..6-20Example Of Geometric Nonlinear Analysis:Blade On A Jet Engine………………………………………… 6-21Example Of Geometric Nonlinear Analysis: Lateral Collapse Of Flat Beam……………………………….. 6-22Example Of Material, Contact and Geometric Nonlinear Analysis: Punching A Thin Shell….…………... 6-23Major Difficulties in Nonlinear Analysis…………………………………………………………..l..…………... 6-24


How do you define temperature-dependent properties in MSC.Patran?Q

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.

S6-4PAT328, Section 3, March 2001MAR120, Section 6, December 2001 S6-4

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.

Spring

P

100 in.

1 in.

Example: Truss-Spring system Demo e4x6.dat


EXAMPLE: NONLINEAR SPRING

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.

P

u

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 ClutchContact 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.


Complex example with multiple sources of nonlinearity:

ContactLarge displacementsMaterial Hyperelasticplastic

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


LINE VARIABLE SCALE FACTOR

1 X-ANGVEL 1 +1.00E+00

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.


Undeformed Shell

Pod

Undeformed Shell

CONTACT NONLINEARITY (CONT.)

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 restrictionswhile MSC.Marc can resolve the problem even when the rigid surfaces are initially separated.


Example: Multipass Ring Rolling Manufacturing Process


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

Rigid Surface

Ring



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 TubesContact 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

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

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.


GEOMETRIC NONLINEARITY

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.

Linear Solution

Nonlinear Solution


Example: Rubber BootSelf-Contact and Material Nonlinearity

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.


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:BLADE ON A JET ENGINE


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 GEOMETRIC NONLINEAR ANALYSIS: LATERAL COLLAPSE OF FLAT BEAM


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)

EXAMPLE OF MATERIAL, CONTACT ANDGEOMETRIC NONLINEAR ANALYSIS: PUNCHING A THIN SHEET


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

s6-1pat328, section 3, march 2001mar120, section 6, december 2001 section 6 nonlinearity

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