Chapter 47: Dynamic Impact of a Rigid Sphere on a Woven Fabric

47 Dynamic Impact of a Rigid Sphere on a Woven Fabric

Summary 972

Introduction 973

Modeling Details 973

Solution Procedure 979

Results 980

Modeling Tips 982

Input File(s) 983

Video 983

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SummaryTitle Chapter 47: Dynamic Impact of a Rigid Sphere on a Woven Fabric

Features Beam-to-beam contact, beam-to-rigid contact, dynamic contact, bilinear Coulomb friction model, isotropic elastic material, nonlinear property extensions to beam elements

Geometry

Material properties ,

Analysis characteristics Nonlinear transient analysis with adaptive time stepping and geometric nonlinearity due to large displacements and large rotations

Boundary conditions Fabric is clamped on all four sides; sliding, frictional contact between the beam elements of the fabric and between the fabric and the sphere.

Applied loads The rigid sphere hits the fabric at the center with an initial velocity of .

Element type 2-node thin elastic beam element with transverse shear effects

FE results 1. Deformed shape and contact status2. History plot of z-displacements of the rigid sphere3. Frictional contact forces

R = 1 cm

E 10GPa= 1500kg m3=

100m s

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IntroductionThis example demonstrates the beam-to-beam contact capabilities of MD Nastran SOL 400. In contrast to the standard grid-to-segment based contact, beam-to-beam contact is a true segment-to-segment contact, in which the beam elements are able to touch each other at arbitrary locations mid-way between the grid points of the elements and can slide along each other, with or without friction. The model consists of a woven fabric which is impacted by a rigid sphere. The fabric is a plane weave and consists of sinusoidally shaped yarns. It is clamped along the four sides

and the yarns are initially in contact at their intersections (see Figure 47-1). The total area of the fabric is . The

sphere, with a radius of and a mass density of , hits the fabric with an initial velocity of at the center.

Figure 47-1 Geometry of the Fabric with the Beam Elements Displayed with the True Cross Section

Modeling DetailsA numerical solution has been obtained with MD Nastran’s SOL 400. The details of the finite element model, contact simulation, material, load, boundary conditions, and solution procedure are discussed below.

The case control section of the input contains the following options for a nonlinear analysis:

BCONTACT = 0SUBCASE 1STEP 1 ANALYSIS=NLTRAN TSTEPNL = 1 BCONTACT = 1 SPC = 1 IC = 2

The analysis is a nonlinear transient analysis and contains a single subcase with one step. The step has time stepping procedure and convergence control settings defined via TSTEPNL, contact table and parameters via BCONTACT, fixed displacements (or single point constraints) via SPC, initial velocity via IC, and the displacements results for the .f06 (output) file.

Large displacement effects are included in the nonlinear analysis using the option:

2 12

6 6 cm2

1cm 981.25 kg m3 100m s

R = 1 cm

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PARAM LGDISP 2

The large strain option is activated for the nonlinear property extensions PBEMN1 to the beam elements, via the option:

NLMOPTS LRGSTRN 1

This option selects (among other things) the updated Lagrange formulation of these elements, which is needed for a proper treatment of the large rotations of the beams.

Element ModelingThe yarns are modelled by 1440 2-node CBEAM elements with an elliptical cross section. The orientation vector that is used to construct the local element y- and z-directions of the beams points in the basic Y-direction for the yarns in the basic X-direction and it points in the basic X-direction for the yarns in the basic Y-direction. The element y-directions of the beams are thus parallel to the basic XY-plane. The major axis of the elliptical cross section coincides with the element y-direction and is also parallel to the basic XY-plane. The minor axis coincides with the element z-direction (see Figure 47-1 and Figure 47-2).

Figure 47-2 Elliptical Cross-Section of the Yarns

The semi-major and semi-minor axes of the cross section are and , so that the area and the moments of inertia of the cross section read:

, (47-1)

, (47-2)

. (47-3)

The cross-section properties for the yarns are defined via the PBEAM option as follows:

PBEAM* 1 1 1.963495408E-06 7.669903939E-13* 1.227184630E-13 0.000000E+00 4.448544285E-13* 0.000000E+00 0.000000E+00*

in which the torsional stiffness of the beam elements is taken as .

v

z-elem

y-elem

a

b

a 1.25mm= b 0.5mm=

A ab 1.9635 106–m

2= =

I14---a

3b 7.6699 10

13–m

4= =

I24---ab

31.2272 10

13–m

4= =

J I1 I2+ 2=

975CHAPTER 47

Dynamic Impact of a Rigid Sphere on a Woven Fabric

The nonlinear extensions to the beam elements can be activated using the PBEMN1 property extension to the regular PBEAM or PBEAML options in the manner shown below:

PBEMN1 1 LS

This PBEMN1 option selects a thin elastic beam element with transverse shear effects, which is similar to the standard CBEAM element with only a PBEAM property, except that the former allows nonlinear material behavior, such as plasticity effects, to be used for the beam elements. In this example, no nonlinear material effects are considered, but the beam elements with and without the property extension will be compared in the elastic regime.

Modeling ContactThe standard contact algorithm of MD Nastran is based on a grid-to-segment approach. The grid points on the surface of the touching (or slave) contact body can touch the segments on the surface of the touched (or master) contact body. Here, the segments of a contact body are, for example, the faces of solid elements, the top and bottom surfaces of shell elements, and the surface of a rigid contact body. The grid-to-segment algorithm works well for contact between solid, shell and rigid contact bodies. It even works fine if the slave body consists of beam elements and the master is a solid, shell or rigid contact body. In that case, the grid points of the beams can touch the segments on the surface of the solid, shell or rigid body.

If both slave and master body consist of beam elements, then the grid-to-segment approach is not very convenient. Beams generally touch each other somewhere in the middle of the element and not necessarily at the grid points. The beam-to-beam contact algorithm of MD Nastran SOL 400 addresses this case. It is a true segment-to-segment contact algorithm, in which the beam elements of the slave contact body can touch the beam elements of the master contact body at arbitrary points mid-way between the grids of the elements. Moreover, beam elements which are in contact can slide along each other with or without friction. The beam-to-beam contact algorithm is activated by the BEAMB option to BCPARA. It supplements the standard grid-to-segment algorithm, that is, the grid points of a beam contact body can touch the surface of solid, shell or rigid bodies through the grid-to-segment algorithm and, if beam-to-beam contact is activated, then the beam elements can also touch beam elements of another (or the same) contact body.

The cross section of the beam elements is taken into account when two beam elements are coming in contact, but the actual shape of cross-section, defined by PBEAM or PBEAML, for example, is ignored. Instead, a circular cross-section is assumed for contact. The radius of the contact cross-section is called the “beam contact radius” and must be defined via the BCBMRAD option. The beam contact radius is defined on a per element basis and may vary from element to element. However, if a beam element is initially in contact with another beam element and during the analysis slides off that element to a third beam element with a different contact radius, the sudden jump in the contact radius may lead to convergence problems. Therefore, the contact surface of the beam elements of a contact body is smoothed by averaging the beam contact radii of the elements at the common grid points. The resulting contact surface for a sequence of beam elements is a piecewise conically shaped surface (see Figure 47-3). Note that the beam contact radius is not used when the grid points of the beam element touch a solid, shell or rigid contact body.

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Figure 47-3 Conically Shaped (Smoothed) Contact Surface Obtained by Averaging the Beam Contact Radii at the Grids

The present example contains three contact bodies. The first two bodies consist of the beam elements representing the yarns in the basic X-direction and the beam elements representing the yarns in the basic Y-direction, respectively (see Figure 47-1). The third contact body is the rigid sphere. The beam-to-beam contact algorithm is used to model contact between the yarns. The standard grid-to-segment based contact algorithm handles contact between the grid points of the yarns and the rigid sphere. Friction is included in the analysis, in the form of the force based, bilinear Coulomb friction model (type 6).

The BCPARA bulk data option defines the number of bodies in contact and contact parameters like the friction type FTYPE and the beam-to-beam contact flag BEAMB.

BCPARA 0 NBODIES 3 BEAMB 1 FTYPE 6

The deformable contact bodies are defined by the bulk data entries BCBODY and BSURF. The BCBODY option defines the contact body with its ID, dimension, type of body etc. and BSURF identifies the elements forming the deformable body.

$ yarns parallel to basic X-directionBCBODY 1 3D DEFORM 1BSURF 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 …$ yarns parallel to basic Y-directionBCBODY 2 3D DEFORM 2BSURF 2 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 …

Note that the dimension of the two deformable contact bodies is set to 3D even though the bodies consist of 1D beam elements. This is because the contact body lives in 3D-space, that is, all grid points have 3 displacement degrees of freedom.

Beam Contact Radius = 0.15 mm

Beam Contact Radius = 0.10 mm

True (smoothed) Contact Surface

977CHAPTER 47

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The rigid sphere is defined as a load-controlled rigid contact body using a BCBODY bulk data option. The BCBODY includes the NURBS surface definition of the sphere. The CONTROL field is set to the ID (1) of the control grid point associated with the body. In contrast to a position or velocity controlled rigid body, the load-controlled body does not have a prescribed motion. Instead, the displacement degrees of freedom of the control grid point are the displacements of the rigid body and can be controlled by single point constraints or loads on the control grid point in the usual way. In this example, the rigid body will be free to move in the basic Z-direction, while the motion in the other two directions will be suppressed via single point constraints (see below).

The sphere is initially located in the positive Z-half space of the basic coordinate system, at some distance from the fabric. During the initial contact search, the body will be moved towards the fabric, such that it just touches the fabric at start of the first time step. This initial contact body approach is activated by the BCONTACT = 0 case control option. During the approach, the rigid body is moved in the direction of the velocity defined by the APPROV section of the BCBODY.

$ rigid sphereBCBODY* 3 3D RIGID 0* 0 0.00000000E+00 0 1* 0 0.00000000E+00 0.00000000E+00 0.00000000E+00* 1.00000000E+00* RIGID 1 1sphere** APPROV* 0.00000000E+00 0.00000000E+00 -1.00000000E-02* NURBS -5 9 3* 3 24 48 0$ control points* 0.00000000E+00 -1.00000000E-02 1.20000000E-02 …

The rigid body represents a solid sphere with a mass density of , a radius of and a total mass of just over four (4) grams. The mass of the sphere can conveniently be assigned to the load-controlled rigid body through a concentrated mass element (CONM2) at the control grid point of the rigid contact body:

CONM2* 2000 1 4.1102503884E-3

To identify how the contact bodies can touch each other, the BCTABLE option is used. BCTABLE with ID 0 is used to define the touching conditions at the start of the analysis, during the initial contact search and the contact body approach. The BCTABLE with ID 1 is the main BCTABLE used to define the touching conditions for later time steps in the analysis, and it is flagged using BCONTACT = 1 in the case control section. The two BCTABLEs are identical and specify that the yarns parallel to the basic X-direction (contact body 1) can touch the yarns parallel to the basic Y-direction (contact body 2) and that the grid points of both beam contact bodies can touch the rigid sphere (contact body 3). The BCTABLEs also define the friction coefficient (0.1) for all possible contact combinations.

$ contact table for initial rigid body approachBCTABLE 0 2 SLAVE 1 0.10 0 0 0 0 MASTERS 2 3 SLAVE 2 0.10 0 0 0 0 MASTERS 3

981.25 kg m3 1cm

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$$ main contact tableBCTABLE 1 2 SLAVE 1 0.10 0 0 0 0 MASTERS 2 3 SLAVE 2 0.10 0 0 0 0 MASTERS 3

The definition of the beam contact radii for the beam elements completes the contact set-up. The radii are defined via the BCBMRAD option. This is a mandatory option if beam-to-beam contact is used. Since the beams generally will touch each other in the direction of the minor axis of the elliptical cross-section of the beam elements (see Figure 47-1), the beam contact radius is set equal to the semi-minor axis for all beam elements in the model.

$ beam contact radiusBCBMRAD 5e-4 ALL

Material ModelingThe isotropic, Hookean elastic material properties of the deformable body are defined using the MAT1 option as follows:

MAT1* 1 1.000000E+10 0.000000E+00* 1.500000E+03 0.000000E+00

Young’s modulus is taken to be and the mass density is set to .

Loading and Boundary ConditionsThe fabric is clamped at all four sides:

SPC1 1 123456 2 3 4 5 124 184 185 186 187 188 189 485 486 487 488 489 490 491 492 493 494 790 791 792 793 794 795 855 856 857 858 859 860 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1461 1462 1463 1464 1465

The rigid sphere is allowed to move only in the basic Z-direction and is given an initial velocity in that direction towards the fabric. As explained in the preceding section, the motion of the sphere is controlled by the displacements of the control grid point of the body, so the displacements of the control grid in the basic X- and Y-direction are suppressed,

SPC1 1 12 1

and the grid is given an initial velocity of in the negative basic Z-direction via the TIC option.

TIC 2 1 3 -100.

The latter is selected via the IC case control option in the step.

a

10GPa 1500 kg m3

100m s

979CHAPTER 47

Dynamic Impact of a Rigid Sphere on a Woven Fabric

Solution ProcedureThe time-stepping procedure to be used is defined through the following TSTEPNL entry:

TSTEPNL 1 400 5e-7 PFNT UV .100 0

In a dynamic contact analysis in MD Nastran SOL 400, the Generalized-Alpha operator with zero spectral radius is automatically chosen by the program. The Generalized-Alpha operator uses two parameters NDAMP and NDAMPM in its formulation. By varying the values of these parameters, the spectral radius can be varied from 0.0 to 1.0. For contact problems, NDAMP is automatically taken as 0.0 and NDAMPM as 1.0, yielding a spectral radius of zero. This is well-suited to damp out high frequencies that are normally excited during the impact process. Other features that are automatically used by the dynamic contact algorithm to avoid high frequency content include the following: There is no projection of the contacting segment onto the contacted segment. A contacting segment that falls within the distance tolerance is simply constrained in its current position. Also, if there is penetration detected during the Newton-Raphson iterations, the maximum penetration is used as a scale-back factor to reduce the time step and restart the increment with the reduced time step.

The TSTEPNL entry controls the time stepping for the solution. Important parameters of the TSTEPNL entry are as follows:

• ID (2nd field of entry 1) - The ID is used as a cross-reference in the case control section to identify the TSTEPNL entry to be used for a particular step.

• NDT, DT, NO (3rd - 5th fields of entry 1) - These parameters control the total simulation time, the initial analysis time step, the output frequency and the maximum possible time step. The product of NDT and DT

defines the total simulation time - in the current problem, the total simulation time comes out to be 2x10-4 s. NO is left as blank in the current problem - the default value of NO is 1 - this implies that for this problem, output is desired at every single step. In addition, the maximum time step cannot exceed NO times DT - which means that for this problem, the maximum time step cannot exceed 5e-7s. In general, for impact problems, given that the energy conversion (from kinetic energy to strain energy and vice-versa) occurs during very small time intervals, it is important to keep tight control over the time-steps.

• METHOD, KSTEP (6th and 7th fields of entry 1) - In the present problem, METHOD is taken as PFNT. FNT or PFNT is a recommended default for contact problems. PFNT denotes Pure Full Newton Technique wherein the operator matrix is reformed at every iteration. KSTEP is left as blank in the present problem, which for the PBEAM + PBEMN1 elements case will default to -1 and for the PBEAM case will default to 1. KSTEP = 1 indicates that the stiffness at the start of the next increment is taken to be the same as the stiffness at the last iteration of the previous increment while KSTEP = -1 indicates that the stiffness is again updated at the start of the next increment.

• CONV (9th field of entry 1) and EPSU (2nd field of entry 2) - In the present problem, this is taken as UV. U indicates displacement control and V indicates the vector component method. The ratio of the maximum iterative change in the displacement over the maximum incremental change in the displacement is calculated. Convergence is established when this ratio is < EPSU (0.1 in the present problem). Note that, by default, for V style checking, separate checks are made over translational degrees of freedom and over rotational degrees of freedom. If the rotational check is deemed to be unnecessary, use can be made of the MDLPRM,MRCONV,N, in which N is set to 2 or 3 to by-pass the rotation check.

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• MAXBIS (2nd field of entry 3) - controls the maximum number of bisections allowed for each time step. In the present problem, this number is set to 0. It should be noted that the penetration check and possible time-step cutback is independent of the value of MAXBIS or DTBIS (the smallest bisection time possible).

• ADJUST (3rd field of entry 3) - controls the time step skip factor for automatic time step adjustment. The blank field allows ADJUST to default to 5 in the present problem. A non-zero ADJUST value allows the following additional checks at the end of an increment:

• After the first 2 increments wherein the user-given time-step is used, the analysis is restarted with either the same time step or possibly a smaller time-step. If the prescribed time step violates frequency-based time step estimates, then the first 2 increments are repeated with the program-evaluated time step. This restart allows good accuracy at the start of the analysis if a high initial time step has been prescribed.

• At a frequency of every ADJUST increments, the dominant frequency of the system is estimated and is used to evaluate the optimal time step. The number of steps (MSTEP) to resolve this dominant period can be defined by the user (4th field of entry 3). MSTEP defaults to 10 (for mildly non-linear) and 20 (for highly non-linear). The time step for subsequent increments is reduced by a factor of ½ or ¼ if the optimal time step is smaller than the current time step. Similarly, the time step for subsequent increments is increased by a factor of 2 or 4 if the optimal time step is larger than the current time step.

• The final optimal time step for the next increment is based on two algorithms - the frequency algorithm (which allows both increase and decrease in time steps and is only checked when ADJUST > 0) and the output algorithm (which is independent of the ADJUST value). After the frequency algorithm comes up with predicted time step, the time step may again be adjusted such that it satisfies the frequency requirement and becomes an even sub-multiple (1, 1/2, ¼, etc.) of the required output time. Note that if the time step is reduced arbitrarily due to a penetration cutback, then the time steps for the next few increments may be changed unevenly before they become regularized.

ResultsFigure 47-4 shows the final deformed shape of the fabric in two views. The contact status is displayed as well. The latter is 1 at the grid points of beam elements in contact and 0 otherwise and indicates that the yarns are in contact at the crossings. The displacement in the basic Z-direction of the rigid sphere is plotted as a function of time in Figure 47-5 for different friction coefficients and for standard beam elements with only a PBEAM property as well as for beam elements with a PBEMN1 nonlinear extension. The first conclusion that can be drawn from this figure is the fact that, in the elastic regime, the standard beam element and the beam element with the nonlinear extension give basically the same results. The difference, of course, is that the beam element with PBEMN1 extension can also be used with material non-linearities, such as plasticity effects. The second thing that stands out is the effect of the friction. Due to friction, the yarns more-or-less stick to each other, so there is less sliding and the fabric behaves stiffer than without friction. This can also be seen from Figure 47-6, in which the final deformed shapes are drawn for the frictionless case and the case with friction.

981CHAPTER 47

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Figure 47-4 Contact Status (red is touching) and Final Deformed Shape of the Fabric

Figure 47-5 Displacement of the Rigid Sphere in the Basic Z-direction

Standard Beam and Beamwith nonlinear extension for Friction Coefficient of 0.2

Standard Beam and Beamwith nonlinear extension for Friction Coefficient of 0.1

Standard Beam and Beamwith nonlinear extension for no Friction

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Figure 47-6 Deformed Shape Without Friction (a) and With Friction Coefficient of 0.2 (b)

Modeling TipsThe beam-to-beam contact algorithm is a symmetric algorithm, in the sense that the same contact condition is found, whether beam element A is touching beam element B, or element B is touching element A. This means that the choice of the slave and master contact bodies on the BCTABLE entry is less important for beam-to-beam contact than it is for the standard grid-to-segment based contact. For the latter, the proper choice of slave and master may be critical in certain cases, particularly if the mesh densities of the bodies differ significantly.

In this chapter, no nonlinear material effects such as plasticity, are considered. The standard CBEAM element with only a PBEAM or PBEAML property supports only elastic material behavior, but if the nonlinear extension PBEMN1 is used in combination with the PBEAML property, nonlinear material effects can be taken into account. The PBEAML can then refer to, for example, a MAT1 material with an associated MATEP entry, to include plasticity effects. Note that the shape of the cross-section must be known to the program to be able to do the cross-section integration, required for nonlinear material behavior. Therefore, a beam element with a PBEAM property cannot support nonlinear material effects, not even with a PBEMN1 extension.

In the present problem, the output frequency NO is defined as 1. This causes output at every step and also prevents the time step from increasing beyond the initial value (5e-7 seconds). In many contact / impact problems, it is beneficial to have a time step value that does not exceed the user-prescribed initial time step value - however one may not desire a NO value of 1 always since that may cause very large output file sizes. For such cases, a larger value of NO (NO = 5, 10, etc.) can be prescribed and ADJUST can be set to 0. The ADJUST = 0 setting forces the program to by-pass the frequency check thereby preventing any time step increase and the output algorithm ensures that the time step is regularized as quickly as possible and that output is produced whenever the time reaches NO times DT.

(a) (b)

983CHAPTER 47

Dynamic Impact of a Rigid Sphere on a Woven Fabric

Input File(s)

VideoClick on the image or caption below to view a streaming video of this problem; it lasts approximately 30 minutes and explains how the steps are performed.

Figure 47-7 Video of the Above Steps

File Description

nug_47a.dat MD Nastran input with standard beam element but without friction

nug_47ax.dat MD Nastran input with beam element with nonlinear extension PBEMN1 but without friction

nug_47b.dat MD Nastran input with standard beam element and friction

nug_47bx.dat MD Nastran input with beam element with nonlinear extension PBEMN1 and friction

R = 1 cm