Chapter 2: 3-D Punch (Rounded Edges) Contact

2 3-D Punch (Rounded Edges) Contact

Summary 70

Introduction 71

Requested Solutions 71

FEM Solutions 71

Results 74

General Analysis Tips 77

Input File(s) 78

Video 78

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SummaryTitle Chapter 2: 3-D Punch (Rounded Edges) Contact

Contact features • Axisymmetric/3-D contact• Analytical deformable body contact• Friction along deformable-deformable contact plane• Comparison of linear and parabolic elements

Geometry Axisymmetric and 3-D continuum elements (units: mm)

• Punch Diameter = 100• Punch Height = 100• Foundation Diameter = 200• Foundation Height = 200• Fillet radius at edge of punch contact = 10

Material properties

Analysis type • Linear elastic material• Geometric nonlinearity• Nonlinear boundary conditions

Boundary conditions • Symmetry displacement constraints in 3-D model (quarter symmetry)• Noncontacting surface of the foundation is fixed

Applied loads A uniform pressure (distributed load) is applied to the punch in the axial direction,

Element type Axisymmetric• 4-node linear elements• 8-node parabolic elements

Contact properties Coefficient of friction and FE results 1. Plot of contact pressure versus radius

2. Plot of contact normal force and friction force versus radius3. Plot of radial displacement and relative tangential slip versus radius

Epunch 210kN mm2= Efoundation 70kN mm2= punch foundation 0.3= =

ux uy uz 0= = =

P 100N mm2=

0.0= 0.1=

FrictionNAFEMS No Friction

0 20 40 60 80 100

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Friction

No Friction

Radial Displacement (mm)Radius (mm)

3-D continuum• 8-node linear elements

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3-D Punch (Rounded Edges) Contact

IntroductionAn axisymmetric steel punch is compressed on an aluminium cylinder. It is assumed that the material behavior is linear

elastic. The punch is loaded by a uniform pressure with magnitude in the axial direction. The effect of friction is studied along the contact zone. Axisymmetric 2-D solutions are used to serve as a target solution for a 3-D analysis. For the 3-D solutions, one quarter of the assembly is modeled, using symmetry conditions. (Ref: NAFEMS, 2006, Advanced Finite Element Contact Benchmarks, Benchmark 2, 3-D Punch (Rounded Edges) Contact)

Requested SolutionsBoth 2-D (axisymmetric) and 3-D solutions are requested. Two solutions, one frictionless and the other using a friction coefficient of 0.1 between the punch and foundation, are requested. The displacement, force, and stress fields in the contact zone (contacting surface of the punch and contacted surface of the foundation) are of interest and are obtained with both linear and parabolic elements in the axisymmetric case and with linear elements in the 3-D case. The SOL 400 elements specified through suitable extensions to the PLPLANE or PSOLID entries are demonstrated. In the 3-D case, solutions obtained with these elements are also compared to those obtained using existing HEX elements.

The solutions presented include:

• Radial displacement of top contact surface of punch as function of coordinate.• Contact force, friction force, and contact pressure distributions as a function of coordinate.

FEM SolutionsNumerical solutions have been obtained with MD Nastran’s solution sequence 400 for multiple 2-D axisymmetric and 3-D cases. The axisymmetric cases include linear and parabolic elements, with and without friction. The 3-D case includes linear elements with and without friction.

The contact, material, geometry, convergence, and other parameters are explained below - primarily with respect to the axisymmetric linear element case and are representative for both 2-D and 3-D cases.

Contact ParametersThe element mesh using axisymmetric linear elements is shown in Figure 2-1 and is further described as follows: Two contact bodies, one identified as the punch and the other identified as the foundation, are used. Pressure is applied at the top of the punch in the axial direction. The bottom of the punch, in turn, compresses the foundation. Typical element length along the punch and foundation is 4 mm and 3.5 mm, respectively. Contact body ID 4 is used to identify the punch and body ID 5 is used to identify the foundation.

BCBODY 4 2D DEFORM 4 0 .1 -1BSURF 4 1 2 3 4 5 6 7........

BCBODY 5 2D DEFORM 5 0 .1BSURF 5 229 230 231 232 233 234 235..........

P 100N mm2=

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BCBODY with ID 4 is identified as a two-dimensional deformable body with BSURF ID 4 and friction coefficient of 0.1. Furthermore, -1 on the 8th field indicates that BCBODY 4 is described as an analytical body, wherein the discrete facets associated with the element edges are internally enhanced by using cubic splines. Since the punch has rounded edges in the contact zone, using an enhanced spline representation of the punch yields better accuracy. The minus sign indicates that the nodal locations defining the spline discontinuities are automatically determined. Note that since the foundation is a rectangular shape with sharp angles, using the spline option with this body is not necessary since it would only increase the computational cost without an associated improvement in accuracy.

Figure 2-1 Element Mesh used for Axisymmetric Case in MD Nastran (Benchmark 2)

The BCTABLE bulk data entries shown below identify the touching conditions between the bodies:

BCTABLE 0 1 SLAVE 4 0. 0. .1 0. 0 0. 0 0 0 MASTERS 5 BCTABLE 1 1 SLAVE 4 0. 0. .1 0. 0 0. 0 0 0 MASTERS 5

BCTABLE with ID 0 is used to define the touching conditions at the start of the analysis. It should be noted that this is a required option that is required in SOL 400 for contact analysis. It is flagged in the case control section through the optional BCONTACT = 0 option. Note that BCTABLE 0 and other contact cards with ID 0 (e.g., BCPARA 0) would be applied at the start of the analysis even without the BCONTACT = 0 option. For later increments in the analysis,

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3-D Punch (Rounded Edges) Contact

BCONTACT = 1 in the case control section indicates that BCTABLE with ID 1 is to be used to define the touching conditions between the punch and the foundation.

The BCPARA bulk data entry shown below for the frictional linear axisymmetric case defines the general contact parameters to be used in the analysis:

BCPARA 0 NBODIES 2 MAXENT 84 MAXNOD 84 FTYPE 6 BIAS 9.0E-01 ISPLIT 3 RVCNST 1.0E-04

Note that ID 0 on the BCPARA option indicates that the parameters specified herein are applied right at the start of the analysis and are maintained through the analysis unless some of these parameters are redefined through the BCTABLE option. Important entries under BCPARA option include FTYPE - the friction type, RVCNST - the slip-threshold value and the BIAS - the distance tolerance bias. As per general recommendation, BIAS is set to 0.9 (note that the default value of BIAS is 0.9). For the frictional case, FTYPE is set to 6 (bilinear Coulomb model) and RVCNST is set to 1e-4 (this is a non-default value that is used in this particular problem - the need for a non-default value is discussed in more detail later). Note that when other parameters on the BCPARA option like ERROR (distance tolerance), FNTOL (separation force) are not specified, left as blank or specified as 0, program calculated defaults are used. It should also be noted that while the BCPARA parameters generally apply to all the bodies throughout the analysis, some of the parameters like ERROR, BIAS, FNTOL can be redefined via the BCTABLE option for specific body combinations and for specific times through the analysis.

Material/Geometry ParametersThe two material properties used herein for the punch and foundation are isotropic and elastic with Young’s modulus and Poisson’s ratio defined as

$ Material Record : steelMAT1 1 210000. .3$ Material Record : aluminumMAT1 2 70000. .3

For the 2-D case, axisymmetric elements are chosen via the CQUADX option pointing to a PLPLANE entry which inturn, points to an auxiliary PSHLN2 entry as shown below.

PLPLANE 1 1PSHLN2 1 1 1 + + C4 AXSOLID L ++ C8 AXSOLID Q

where the C4 entries indicate that linear 4-noded full integration axisymmetric solid elements are to be used and the C8 entries indicate that parabolic 8-noded full integration axisymmetric solid elements are to be used. Note that the PSHLN2 entry enables SOL 400 to access a robust 2-D element library featuring linear and parabolic plane stress, plane strain or axisymmetric elements. Multiple element topologies (4-noded, 6-noded, 8-noded) can be defined as plane stress, plane strain, or axisymmetric through the PSHLN2 options. These elements which can be used for isotropic/orthotropic/ anisotropic elastic/elasto-plastic applications augment previous SOL 400 hyperelastic element technology that could be used in conjunction with the PLPLANE and MATHP options.

For the 3-D case, hex elements are chosen via the CHEXA option pointing to a PSOLID entry. For elastic or small strain applications, the user has two choices: Use existing 3-D solid elements with just the PSOLID option or use 3-D solid element technology accessed by the PSOLID entry pointing to an auxiliary PSLDN1 entry. For large strain elasto-plastic applications, the user should always use the 3-D solid elements; i.e., the primary usage of the 3-D solid

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elements is for large strain elasto-plasticity for which the PSLDN1 + NLMOPTS,LRGSTRN,1 bulk data entry is recommended. However, as in the current example, these elements can also be used for elastic applications when used in conjunction with PSLDN1 and with NLMOPTS,ASSM,ASSUMED entry.

Convergence ParametersThe nonlinear procedure used is defined through the NLPARM entry:

NLPARM 1 10 PFNT 0 25 UP YES

where 10 indicates the total number of increments; PFNT represents Full Newton-Raphson Technique, wherein the stiffness is reformed at every iteration; KSTEP = 0 in conjunction with PFNT indicates that the program automatically determines whether the stiffness needs to be reformed after the previous load increment is completed and the next load increment is commenced. The maximum number of allowed recycles is 25 for every increment and if this were to be exceeded, the load step would be cut-back and the increment repeated. UP indicates that convergence will be checked using both displacements (U) and residual criteria (P). YES indicates that intermediate output will be produced after every increment (note that this has been turned to NO for the 3-D case due to voluminous output). The second line of NLPARM is omitted here, which implies that default convergence tolerances of 0.01 will be used for U and P. It should be noted that the PFNT iterative method used conducts checking over incremental displacements and is generally more stringent than for the FNT iterative method which convergence is checked over weighted total displacements.

Case Control ParametersSome of the case control entries to conduct these analyses are highlighted as follows: SUBCASE 1 indicates the case being considered. There are no STEP entries in this analysis since a single loading sequence is being considered. For multiple loading sequences that follow one another, STEP entries can be used within a single SUBCASE to identify each sequence. BCONTACT = 1 is used to indicate the contact parameters for SUBCASE 1. NLPARM = 1 is used to flag the nonlinear procedure for SUBCASE 1. In addition to regular output requests like DISPLACEMENTS, STRESSES, the option that is required for contact related output in the F06 file is BOUTPUT. It should be noted that with the BOUTPUT option, one can obtain normal contact forces, frictional forces, contact normal stress magnitudes, and contact status for the contact nodes.

ResultsThe radial displacements obtained for the frictionless and frictional cases for the linear axisymmetric element case are compared in Figure 2-2. The results match very well with the corresponding NAFEMS results (Benchmark 2 of NAFEMS 2006).

It is noteworthy to study the effect of the slip threshold value, RVCNST, on the friction results. The radial displacements for two different values of RVCNST are compared in Figure 2-3. It is seen that RVCNST has a significant influence on the radial displacements. It should be noted that the default value of RVCNST is calculated as 0.0025 times the average edge length of all elements that can participate in contact. For the linear axisymmetric problem, the default RVCNST is of the order of 0.015. Relative radial displacements which are smaller than this value imply a transition

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zone and the frictional force linearly increases from 0 to the peak value within this zone. In order to capture the frictional force and the relative sliding more accurately, a smaller value of RVCNST (= 1e-4) is required in this problem. In general, for friction problems, a good check to be made from the f06 file or by postprocessing is whether the friction force is of the order of , where is the friction coefficient and is the nodal contact normal force.

Figure 2-2 Radial Displacement as Function of the Radial Coordinate (friction coefficient =0.0 and 0.1) Obtained with Linear Axisymmetric Elements

Figure 2-3 Effect of slip threshold value, RVCNST, on Radial Displacement

The contact normal force and friction force along the punch for the linear axisymmetric element is plotted in Figure 2-4. It is instructive to check that equilibrium is well-maintained (the sum of the contact forces transmitted via the punch should be equal to the total force being applied to the punch). It can be shown that the sum of all contact forces at the punch-foundation interface is within .03% of the total force applied on the punch

. Also, the friction forces are about 0.1 times the contact normal forces.

Fn Fn

FrictionNAFEMS No Friction

0 20 40 60 80 100

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Friction

No Friction

Radial Displacement (mm)Radius (mm)

0 20 40 60 80 100

-0.020

-0.015

-0.010

-0.005

0.000

0.005

μ = 0.1 RVCNST=default

μ = 0.1 RVCNST=1e-4

No Friction

Radial Displacement (mm)Distance (mm)

=PRpunch2 100502 7.85e5N= =

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The contact pressure is plotted for the contacting nodes for both the linear and parabolic axisymmetric elements of the punch in Figure 2-5. The parabolic solution shows a rather oscillating type of behavior. Also, as may be expected, the parabolic solution shows a more localized stress peak. These trends are consistent with the NAFEMS benchmark 2 results. The oscillatory behavior can be improved by refining the mesh in the contact zone (and the surrounding part assuring connection with the remaining part of the structures).

Figure 2-4 Contact Normal Force and Friction Force at Punch as a Function of Radial Coordinate Along Punch-Foundation Contact Interface

Figure 2-5 Variation of Contact Normal Stress Along Radial Coordinate of Punch for Linear and Parabolic Axisymmetric Elements

The displacement contours in the punch for the 3-D frictional case are shown in Figure 2-6. The left-hand side shows the solution for the 3-D solid elements identified through the PSOLID + PSLDN1 options. The right-hand side shows

0 10 20 30 40 50 600

50000

100000

150000

200000

250000

300000

350000

Contact Friction Force

Contact Normal Force

Force (N)

Distance (mm)

0 10 20 30 40 50 600

100

200

300

400

500

600

700

800

Linear Elements

Parabolic Elements

Contact Normal Stress (N/mm )

Distance (mm)

2

77CHAPTER 2

3-D Punch (Rounded Edges) Contact

the solution for the existing 3-D solid elements identified through the PSOLID options only. As seen, the solutions are very close to each other.

Figure 2-6 Comparison of Punch Displacement Contours in Two different Solid Elements Available in SOL 400

General Analysis Tips• While the contact checking algorithm in SOL 400 provides a number of options for the searching order via the ISEARCH parameter on the BCTABLE option, the user should be aware of a few recommendations regarding the touching (slave) body and the touched (master) body: The touching body should be convex, generally be less stiff, and be more finely meshed than the touched body. This allows for better conditioning of the matrices and provides for better nodal contact. Note that these recommendations may not all be satisfied at the same time; for example, in this benchmark, the punch which has been identified as the first body is convex and smaller than the foundation but has a slightly coarser mesh and is somewhat stiffer than the foundation.

• The accuracy of the friction solution should be judged by checking that the frictional forces at the nodes are generally equal to . If this is violated, the slip-threshold value, RVCNST, may need to be adjusted. Note

also that to ensure a quality solution with friction, in general, the incremental displacements need to converge well. This can be ensured by using PFNT on the NLPARM option and checking on U.

• The PSHLN2 entry in conjunction with PLPLANE entries allows the users to flag 2-D elements for plane stress, plane strain, or axisymmetric applications with isotropic/orthotropic/ anistropic elastic/elasto-plastic materials. Similarly, PSLDN1 entries in conjunction with PSOLID entries allows the users to flag nonlinear 3-D solid continuum elements. The 2-D elements offer a range of abilities for small strain and large strain elastic/elasto-plastic analysis. The fundamental application of the 3-D elements is for large strain elasto-plastic applications, wherein use should be made of the NLMOPTS,LRGSTRN,1 option to flag appropriate element behavior. It should be noted that the 3-D elements can also be used in the elastic regime (as in this current example - see nug_02em.dat). In such situations, it is highly recommended that one not use NLMOPTS,LRGSTRN,1 but use NLMOPTS,ASSM,ASSUMED to ensure better behavior in elastic bending. Existing 3-D element technology for SOL 400 can be used for elastic applications too (see nug_02en.dat for example). In this case, one simply uses PSOLID without the PSLDN1 addition.

Fn

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• For the axisymmetric case, the pressure load is applied through PLOADX1. It should be noted that the pressure

value to be specified on the PLOADX1 option is not the force per unit area but the pressure over a circular ring of angle Accordingly, on the LOAD bulk data entry, the pressure load is scaled by a value of

Input File(s)

VideoClick on the image or caption below to view a streaming video of this problem; it lasts approximately 18 minutes and explains how the steps are performed.

Figure 2-7 Video of the Above Steps

File Description

nug_02am.dat Axisymmetric Linear Elements Without Friction

nug_02bm.dat Axisymmetric Linear Elements With Friction

nug_02cm.dat Axisymmetric Parabolic Elements Without Friction

nug_02dm.dat Axisymmetric Parabolic Elements With Friction

nug_02em.dat 3-D Linear Elements Without Friction - PSLDN1 used along with PSOLID to flag nonlinear HEX elements

nug_02en.dat 3-D Linear Elements Without Friction - existing HEX element technology flagged through PSOLID

nug_02fm.dat 3-D Linear Elements With Friction - PSLDN1 used along with PSOLID to flag nonlinear HEX elements

nug_02fn.dat 3-D Linear Elements With Friction - existing HEX element technology flagged through PSOLID

100N mm2

2

2

FrictionNAFEMS No Friction

0 20 40 60 80 100

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Friction

No Friction

Radial Displacement (mm)Radius (mm)