Monday 12 November 2012

Mechanical Vibrations By G.K. Grover

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Table Of Contents
Fundamental of Vibrations
  • Introduction
  • Definitions
  • Vect6z method of representing harmonic motion
  • Addi!,nn of two simple harmonic motions of the same frequency
  • Pheromelion of beats
  • Complex method of representing harmonic motion
  • Work done by a harmonic force on a harmonic motion
  • Fourier series and harmonic analysis
  • Analytical method for harmonic analysis
  • Numerical method for harmonic analysis
  • Notes on certain type of functions
  • Practice Problems
Undamped Free Vibrations of Single Degree of Freedom System
  • Introduction
  • Derivations of differential equation
  • Solution of differential equation
  • Torsional vibrations
  • Equivalent stiffness of spring combinations
  • Spring in series
  • Springs in parallel
  • Inclined springs
  • Rayleigh's energy method
  • Practice Problems
Damped Free Vibrations of Single legree of Freedom Systems
  • Introduction
  • Different types of damping
  • Free vibrations with viscous damping
  • Over — damped system
  • Critically — damped system
  • Under — damped system
  • Logarithmic decrement
  • Viscous dampers
  • Fluid dashpot
  • Eddy current damping
  • Dry friction or coulomb damping
  • Frequency of damped oscillations
  • Rate of decay of oscillations
  • Solid or structural damping
  • Slip or Interfacial damping
  • Practice Problems

Thursday 1 November 2012

Deep drawing of a square box



Products: Abaqus/Standard  Abaqus/Explicit  

Objectives

This example problem demonstrates the following Abaqus features and techniques:
  • transferring results from Abaqus/Explicit to Abaqus/Standard using the import analysis technique;
  • comparing results from an analysis sequence that uses Abaqus/Explicit for a forming step and Abaqus/Standard for a springback analysis with results obtained using Abaqus/Standard for both the forming and springback steps; and
  • comparing characteristics of different contact formulations with finite sliding, especially with regard to the treatment of surface thickness.

Application description

This example illustrates the forming of a three-dimensional shape by a deep drawing process. In general, the forming procedure involves a forming step followed by a springback that occurs after the blank is removed from the tool. The goal of analyzing the forming procedure is to determine the final deformed shape after springback.

Geometry

The blank is initially square, 200 mm by 200 mm, and is 0.82 mm thick. The rigid die is a flat surface with a square hole 102.5 mm by 102.5 mm, rounded at the edges with a radius of 10 mm. The rigid square punch measures 100 mm by 100 mm and is rounded at the edges with the same 10 mm radius. The rigid blank holder can be considered a flat plate, since the blank never comes close to its edges. The geometry of these rigid parts is illustrated in Figure 1.5.2�1.

Materials

The blank is made of aluminum-killed steel, which is assumed to satisfy the Ramberg-Osgood relation between true stress and logarithmic strain,
with a reference stress value (K) of 513 MPa and a work-hardening exponent (n) of 0.223. Isotropic elasticity is assumed, with a Young's modulus of 211 GPa and a Poisson's ratio of 0.3. An initial yield stress of 91.3 MPa is obtained from these data. The stress-strain behavior is defined by piecewise linear segments matching the Ramberg-Osgood curve up to a total (logarithmic) strain level of 107%, with Mises yield, isotropic hardening, and no rate dependence.

Boundary conditions and loading

Given the symmetry of the problem, it is sufficient to model only a one-eighth sector of the box. However, for easier visualization we have employed a one-quarter model. Symmetry boundary conditions are applied at the quarter edges of the blank. The punch and the blank holders are allowed to move only in the vertical direction. Allowing vertical motion of the blank holders accommodates changes in the blank thickness during forming.

Interactions

Contact interaction is considered between the blank and the punch with a friction coefficient of 0.25 and between the blank and the die with a friction coefficient of 0.125. The contact interaction between the blank and the blank holders is assumed to be frictionless.

Abaqus modeling approaches and simulation techniques

The most efficient way to analyze this type of problem is to analyze the forming step using Abaqus/Explicit and to import the results in Abaqus/Standard to analyze the springback. For verification purposes the complete analysis is also carried out with Abaqus/Standard. However, this is computationally more expensive and will be prohibitively more expensive for simulation of the forming of realistic, complex components.
This problem is used in Nagtegaal and Taylor (1991) where implicit and explicit finite element techniques for forming problems are compared. The computer time involved in running the simulation using explicit time integration with a given mesh is directly proportional to the time period of the event, since the stable time increment size is a function of the mesh size (length) and the material stiffness. Thus, it is usually desirable to run the simulation at an artificially high speed compared to the physical process. If the speed in the simulation is increased too much, the solution does not correspond to the low-speed physical problem; i.e., inertial effects begin to dominate. In a typical forming process the punch may move at speeds on the order of 1 m/sec, which is extremely slow compared to typical wave speeds in the materials to be formed (the wave speed in steel is approximately 5000 m/sec). In general, inertia forces will not play a dominant role for forming rates that are considerably higher than the nominal 1 m/sec rates found in the physical problem. Therefore, explicit solutions are obtained with punch speeds of 10, 30, and 100 m/sec for comparison with the static solution obtained with Abaqus/Standard. In the results presented here, the drawing process is simulated by moving the reference node for the punch downward through a total distance of 36 mm in 0.0036 seconds. A detailed comparison of analyses of various metal forming problems using explicit dynamic and static procedures is discussed in the paper by Nagtegaal and Taylor (1991).
Although this example does not contain rate-dependent material properties, it is common in sheet metal forming applications for this to be a consideration. If the material is rate-dependent, the velocities cannot be artificially increased without affecting the material response. Instead, the analyst can use the technique of mass scaling to adjust the effective punch velocity without altering the material properties. Rolling of thick plates,” Section 1.3.6, contains an explanation and an example of the mass scaling technique.

Summary of analysis cases

Forming analysis with Abaqus/Explicit.Case 1a
Using the general contact capability.
 Case 1b
Using the kinematic contact pairs.
 Case 1c
Using penalty contact pairs.
 Case 1d
Forming analysis of a fine mesh case using the general contact capability (included for the sole purpose of testing the performance of the Abaqus/Explicit code).
 Case 1e
Forming analysis of a fine mesh case using kinematic contact pairs (included for the sole purpose of testing the performance of the Abaqus/Explicit code).
Springback analysis with Abaqus/Standard.Case 2a
Abaqus/Standard springback analysis using the *IMPORTUPDATE=NO option.
 Case 2b
Abaqus/Standard springback analysis using the *IMPORTUPDATE=YES option.
 Case 2c
Springback analysis of a fine mesh case (included for the sole purpose of testing the performance of the Abaqus/Standard code) using the *IMPORTUPDATE=YES option.
Forming and springback analysis with Abaqus/Standard.Case 3aUsing the surface-to-surface contact formulation.
 Case 3bUsing the node-to-surface contact formulation.

Analysis types

As described earlier, the import capability in Abaqus is utilized to run the forming step as an explicit dynamic analysis followed by a static stress analysis using Abaqus/Standard for calculating the springback. For comparison, results from a complete static stress analysis using Abaqus/Standard for both the forming and the springback steps are presented.

Analysis techniques

The import feature in Abaqus is used for transferring results from Abaqus/Explicit to Abaqus/Standard.

Mesh design

The blank is modeled with 4-node, bilinear finite-strain elements (type S4R); while the punch, die, and the blank holder are meshed using 4-node, three-dimensional rigid surface elements (type R3D4). The mesh design for the various parts is shown in Figure 1.5.2�1 and Figure 1.5.2�2.

Loads

The blank is held between the blank holders by applying a concentrated load of 22.87 kN. Further loading on the blank is applied by contact forces with the punch in the forming step.

Analysis steps

Using Abaqus/Explicit for the forming procedure involves a single forming step where the rigid punch is pushed against the blank while the blank is held by the blank holders by applying a concentrated load. This description applies to Cases 1a–1e. For the import analysis in Abaqus/Standard a single step is used to calculate the springback as in Cases 2a–2c. For the complete analysis in Abaqus/Standard as in Cases 3a and 3b, the following steps are adopted:
  • First step: the blank holders are brought in contact with the blank by applying a small displacement to the reference point of one of the rigid blank holders.
  • Second step: a concentrated load is applied to the reference point of the blank holder to hold the blank in place while maintaining contact.
  • Following steps: the forming is effected by pushing the rigid punch against the blank.
  • Final two steps: the springback is analyzed by deactivating the contact pairs.

Output requests

The output variables STH for shell thickness and PEEQ for equivalent plastic strain are specifically requested along with preselected variables. Further, the history of reaction force and displacement for the punch is also requested.

Case 1a: Explicit forming analysis using general contact

This analysis pertains only to the forming step. For the complete analysis the forming step in this case needs to be followed by a springback analysis (either Case 2a or Case 2b).

Interactions

General contact is used (see the *CONTACT option and the related suboptions in the input file) to define contact interactions in this case. This allows very simple definitions of contact with very few restrictions on the types of surfaces involved (see Defining general contact interactions in Abaqus/Explicit,” Section 34.4.1 of the Abaqus Analysis User's Manual). However, general contact does not account for changes in shell thickness by default. Consequently, a nondefault option is specified to account for thinning of the blank (see*SURFACE PROPERTY ASSIGNMENT in the input file).

Case 1b: Explicit forming analysis using kinematic contact pairs

This analysis again pertains only to the forming step. For the complete analysis the forming step needs to be followed by a springback analysis (either Case 2a or Case 2b).

Interactions

Contact pairs are defined to include blank interaction with the punch, die, and the blank holder separately with appropriate friction behavior as previously specified. The contact pair algorithm, which is specified using the *CONTACT PAIR option, has more restrictions on the types of surfaces involved and often requires more careful definition of contact (seeDefining contact pairs in Abaqus/Explicit,” Section 34.5.1 of the Abaqus Analysis User's Manual). Contact interactions are defined between all element-based surfaces in the model.

Case 1c: Explicit forming analysis using penalty contact pairs

This analysis pertains only to the forming step. The springback calculations have to be done separately (Case 2a or Case 2b).

Interactions

Penalty contact is specified for contact pairs to include blank interaction with the punch, die, and the blank holder separately with appropriate friction behavior.

Case 1d: Explicit forming analysis using general contact with a refined mesh

In this case the mesh for the blank is uniformly refined so that the number of elements in each direction is twice the number in the previous cases. This case is run to purely benchmark the efficiency of performing an explicit analysis.

Interactions

The contact interactions are exactly the same as in Case 1a.

Case 1e: Explicit forming analysis using kinematic contact pairs with a refined mesh

In this case the refined mesh defined in Case 1d is utilized for performing the explicit forming analysis.

Interactions

The contact interactions are exactly the same as in Case 1b.

Case 2a: Static springback analysis with UPDATE=NO during import

For running this case, a prior explicit forming analysis (Case 1a, Case 1b, or Case 1c) should have been completed for importing results into Abaqus/Standard. By settingUPDATE=NO on the *IMPORT option, the displacements are the total values relative to the original reference configuration before the forming analysis. This makes it easy to compare the results with the analysis in which both the forming and springback are analyzed with Abaqus/Standard.

Boundary conditions

Boundary conditions are imposed in the Abaqus/Standard analysis to prevent rigid body motion and for symmetry. The node at the center of the box is fixed in the z-direction.

Interactions

No contact interactions are used in this analysis once the deformed sheet with its material state at the end of Abaqus/Explicit is imported.

Case 2b: Static springback analysis with UPDATE=YES during import

Similar to Case 2a, a prior explicit forming analysis (Case 1a, Case 1b, or Case 1c) should have been completed for importing results into Abaqus/Standard. However, usingUPDATE=YES on the *IMPORT option implies that the displacements are relative to the deformed configuration at the end of the forming analysis. The boundary conditions and interactions are exactly the same as Case 2a.

Case 2c: Static springback analysis using a refined mesh with UPDATE=YES during import

For running this case, Case 1d or Case 1e for explicit forming analysis should have been completed for importing results into Abaqus/Standard. Here again, using UPDATE=YESon the *IMPORT option implies that the displacements are relative to the deformed configuration at the end of the forming analysis. The boundary conditions and interactions are exactly the same as Case 2a.

Case 3a: Static analysis of forming and springback using surface-to-surface contact

In this analysis both the forming and the springback steps are analyzed in Abaqus/Standard.

Interactions

In this case the surface-to-surface contact formulation is invoked by using *CONTACT PAIRTYPE=SURFACE TO SURFACE. Since double-sided surfaces are not available in Abaqus/Standard, two single-sided surfaces are used to model the blank when the forming step is modeled in Abaqus/Standard: one surface to model the top of the blank and one to model the bottom of the blank. The surface-to-surface contact formulation considers the original shell thickness by default throughout the analysis. There is no option to consider the current shell thickness instead of the original shell thickness.

Solution controls

Contact stabilization using *CONTACT CONTROLSSTABILIZE is used to avoid chattering between the blank and the rigid surfaces it is in contact with. In addition, the adaptive automatic stabilization scheme is applied to improve the robustness of the static analysis.

Case 3b: Static analysis of forming and springback using node-to-surface contact

As in Case 3a, both the forming and the springback steps are analyzed in Abaqus/Standard.

Interactions

In this case the node-to-surface contact formulation is used. Since, shell thickness cannot be considered by node-to-surface finite-sliding contact, “softened” contact is used to approximate the thickness (see the *SURFACE BEHAVIOR option in the input file).

Discussion of results and comparison of cases

Figure 1.5.2�3Figure 1.5.2�5, and Figure 1.5.2�4 show contours of shell thickness in the blank at the end of the forming step before springback in Abaqus/Explicit (Case 1a) and Abaqus/Standard analyses (Case 3a and Case 3b), respectively. Figure 1.5.2�6Figure 1.5.2�7, and Figure 1.5.2�8 show contours of equivalent plastic strain in the blank in the final deformed shape for the Abaqus/Explicit and the two Abaqus/Standard analyses, respectively. The predicted results are very similar. The Abaqus/Explicit results match the surface-to-surface contact formulation in Abaqus/Standard more closely than the node-to-surface results in Abaqus/Standard. This observation is true for both the equivalent plastic strain contours and shell thickness contours and is a consequence of the intrinsic differences between the various contact formulations. The node-to-surface formulation in Abaqus/Standard accounts for the shell thickness indirectly by using carefully specified pressure-overclosure relationships (soft contact). The other analyses use contact formulations that account for shell thickness directly. Despite the fact that the surface-to-surface formulation in Abaqus/Standard uses the original shell thickness throughout the analysis, the results correlate well.
Closer inspection of the results reveals that the corners of the box are formed by stretching, whereas the sides are formed by drawing action. This effect leads to the formation of shear bands that run diagonally across the sides of the box, resulting in a nonhomogeneous wall thickness. The material draws unevenly from the originally straight sides of the blank. Applying a more localized restraint near the midedges of the box (for example, by applying drawbeads) and relaxing the restraint near the corners of the box is expected to increase the quality of the formed product.
Figure 1.5.2�9 shows the reaction force on the punch, and Figure 1.5.2�10 shows the thinning of an element at the corner of the box. Here again, the results from the surface-to-surface formulation in Abaqus/Standard match those from Abaqus/Explicit better than the node-to-surface contact formulation in Abaqus/Standard. In spite of the approximate treatment of surface thickness via the pressure-overclosure relationship for the node-to-surface formulation, the shell thicknesses predicted by Abaqus/Explicit and the node-to-surface formulation in Abaqus/Standard differ only by about 4%, reflecting the overall quality of the results.
The springback analysis runs in 6 increments for both of the contact formulations in Abaqus/Standard. Most of the springback occurs in the z-direction, and the springback is not significant. The corner of the outside edge of the formed box drops approximately 0.35 mm, while the vertical side of the box rises by approximately 0.26 mm. Figure 1.5.2�11shows a contour plot of the displacements in the z-direction obtained from the springback analysis using the node-to-surface formulation.
The analysis with UPDATE=NO on the *IMPORT option yields similar results. However, in this case the displacements are interpreted as total values relative to the original configuration.

Files

Case 1a: Explicit forming analysis using general contact
deepdrawbox_exp_form.inp
Input file for the explicit forming step.
Case 1b: Explicit forming analysis using kinematic contact pairs
deepdrawbox_exp_form_cpair.inp
Input file for the explicit forming step.
Case 1c: Explicit forming analysis using penalty contact pairs
deepdrawbox_exp_form_plty_cpair.inp
Input file for the explicit forming step.
Case 1d: Explicit forming analysis using general contact with a refined mesh
deepdrawbox_exp_finemesh.inp
Input file for the explicit forming step.
Case 1e: Explicit forming analysis using kinematic contact pairs with a refined mesh
deepdrawbox_exp_finemesh_cpair.inp
Input file for the explicit forming step.
Case 2a: Static springback analysis with UPDATE=NO during import
deepdrawbox_std_importno.inp
Input file for the static springback step.
Case 2b: Static springback analysis with UPDATE=YES during import
deepdrawbox_std_importyes.inp
Input file for the static springback step.
Case 2c: Static springback analysis using a refined mesh with UPDATE=YES during import
deepdrawbox_std_finesprngback.inp
Input file for the static springback step with a refined mesh for the blank.
Case 3a: Static analysis of forming and springback using surface-to-surface contact
deepdrawbox_std_both_surf.inp
Input file for the complete static analysis.
deepdrawbox_std_both_surf_stabil_adap.inp
Input file for the complete static analysis with adaptive stabilization.
Case 3b: Static analysis of forming and springback using node-to-surface contact
deepdrawbox_std_both.inp
Input file for the complete static analysis.

References


Abaqus Keywords Reference Manual

Other
  • Nagtegaal J. C. and L. M. Taylor, Comparison of Implicit and Explicit Finite Element Methods for Analysis of Sheet Forming Problems, VDI Berichte No. 894, 1991.

Figures

Figure 1.5.2�1 Meshes for the die, punch, and blank holder.
Figure 1.5.2�2 Undeformed mesh for the blank.
Figure 1.5.2�3 Contours of shell thickness with Abaqus/Explicit.
Figure 1.5.2�4 Contours of shell thickness with Abaqus/Standard using surface-to-surface contact formulation.
Figure 1.5.2�5 Contours of shell thickness with Abaqus/Standard using node-to-surface contact formulation.
Figure 1.5.2�6 Contours of equivalent plastic strain with Abaqus/Explicit.
Figure 1.5.2�7 Contours of equivalent plastic strain with Abaqus/Standard using surface-to-surface contact formulation.
Figure 1.5.2�8 Contours of equivalent plastic strain with Abaqus/Standard using node-to-surface contact formulation.
Figure 1.5.2�9 Reaction force on the punch versus punch displacement.
Figure 1.5.2�10 Shell thickness of the thinnest part of the blank versus time.
Figure 1.5.2�11 Contour plot showing the springback in the z-direction.

Springback of two-dimensional draw bending



Products: Abaqus/Standard  Abaqus/Explicit  
This example illustrates the forming and springback analysis of a two-dimensional draw bending process. The forming analysis is performed using Abaqus/Explicit, and the springback analysis is run with Abaqus/Standard using the *IMPORT option.

Problem description

The example described here is one of the benchmark tests reported at the Numisheet '93 Conference. The benchmark contains a series of six problems performed with three different materials and two different blank holder forces. One of the six problems is described here. The simulations for all the problems are described in the paper by Taylor et al. (1993).
The blank initially measures 350 mm by 35 mm and is 0.78 mm thick. The problem is essentially a plane strain problem (the out-of-plane dimension for the blank is 35 mm). A cross-section of the geometry of the die, the punch, the blank holder, and the blank is shown in Figure 1.5.1�1. The total blank holder force is 2.45 kN, and a mass of 5 kg is attached to the blank holder. A coefficient of friction of 0.144 is used for all interacting surfaces.
The blank is made of mild steel. The material is modeled as an elastic-plastic material with isotropic elasticity, using the Hill anisotropic yield criterion for the plasticity. The following material properties are used:
Young's modulus = 206.0 GPa
Poisson's ratio = 0.3
Density = 7800.
Yield stress  = 167.0 MPa
Anisotropic yield criterion: =1.0, =1.0402, =1.24897, =1.07895,
=1.0, =1.0
The problem is symmetric about a plane through the center of the punch, and only half of the problem is modeled. The blank is modeled with a single row of 175 first-order shell elements. Symmetry boundary conditions are applied on the plane of symmetry, and boundary conditions are applied on all the nodes of the blank to simulate the plane strain conditions. The out-of-plane dimension for the blank in the model is 5 mm; thus, the blank holder force is scaled appropriately.
The forming process is simulated in two steps with Abaqus/Explicit. The blank holder force is applied in the first step of the analysis. The force is ramped on with a smooth step definition to minimize inertia effects. In the second step of the analysis the punch is moved down 70 mm by prescribing the velocity of the rigid body reference node for the punch. The velocity is applied with a triangular smooth step amplitude function, starting and ending with zero velocity, and with a peak velocity occurring at the middle of the time period.
A significant amount of springback occurs in this case. Because the blank is very flexible and the fundamental mode of vibration is low, it would take a long simulation to obtain a quasi-static solution of the springback analysis in Abaqus/Explicit.
The springback analysis is performed with Abaqus/Standard using the *IMPORT option. The results from the forming simulation in Abaqus/Explicit are imported into Abaqus/Standard, and a static analysis calculates the springback. During this step an artificial stress state that equilibrates the imported stress state is applied automatically by Abaqus/Standard and gradually removed during the step. The displacement obtained at the end of the step is the springback, and the stresses give the residual stress state.
The UPDATE parameter on the *IMPORT option determines the reference configuration. When the UPDATE parameter is set equal to YES on the *IMPORT option, the deformed sheet with its material state at the end of the Abaqus/Explicit analysis is imported into Abaqus/Standard and the deformed configuration becomes the reference configuration. This procedure is most convenient if, during postprocessing, the displacements due to springback need to be displayed. When the UPDATE parameter is set equal to NO on the*IMPORT option, the material state, displacements, and strains of the deformed sheet at the end of the Abaqus/Explicit analysis are imported into Abaqus/Standard, and the original configuration remains as the reference configuration. This procedure should be used if it is desirable to obtain a continuous displacement solution.
In this two-dimensional draw bending problem significant springback occurs, and large-displacement effects are included in the calculations by including the NLGEOM parameter on the *STEP option.

Results and discussion

The optimum peak velocity for the punch (the value that gives quasi-static results at least cost) is determined by running the explicit analysis with peak velocities of 30 m/s, 15 m/s, and 5 m/s. The energy histories are shown in Figure 1.5.1�2Figure 1.5.1�3, and Figure 1.5.1�4, respectively. From these results it is evident that the amount of kinetic energy in the model is too large at a peak velocity of 30 m/s for the analysis to simulate the quasi-static forming process, while at a peak velocity of 5 m/s the kinetic energy is virtually zero. A peak velocity for the punch of 15 m/s is chosen for the forming analysis, as the kinetic energy for this case is considered low enough not to affect the results significantly. For accurate springback analysis it is important that stresses are not influenced by inertia effects.
The blank at the end of the Abaqus/Explicit forming analysis is shown in Figure 1.5.1�5. The shape after springback is shown in Figure 1.5.1�6. The results compare well with the reported experimental data. In the numerical results the angle between the outside flange and the horizontal axis is 22� in the Abaqus/Explicit analysis and 17.1� in the Abaqus/Standard analysis. The differences in the results are due to differences in the contact calculations. In Abaqus/Explicit the change in shell thickness is accounted for during contact calculations, and the surface-to-surface contact formulation in Abaqus/Standard does the same. However, for the node-to-surface contact formulation in Abaqus/Standard when a shell is pinched between two surfaces, it is necessary to use “softened” contact to account for the shell thickness and, consequently, an approximation is made. A modified Abaqus/Explicit analysis that uses softened contact and zero shell thickness (NO THICK parameter on the *SURFACE option) has been set up to compare the results from Abaqus/Explicit and Abaqus/Standard directly. The predicted results match closely. The average angle measured in the experiments is 17.1�, with a range from 9� to 23� in the experimental results. The results of the springback analysis when the reference configuration is updated are nearly identical to the results when the reference configuration is not updated.

Input files

Forming analysis in Abaqus/Explicit with a punch velocity of 15 m/s using contact pairs.
Forming analysis in Abaqus/Explicit with a punch velocity of 15 m/s using general contact.
Springback analysis in Abaqus/Standard with the *IMPORTUPDATE=YES option.
Springback analysis in Abaqus/Standard with the *IMPORTUPDATE=NO option.
Input data used with Abaqus/Standard for both the forming and the springback analyses.
Input data used with Abaqus/Standard for both the forming and the springback analyses using the surface-to-surface contact formulation.
Modified forming analysis in Abaqus/Explicit with a punch velocity of 15 m/s using softened contact and contact pairs as in springback_std_both.inp.
Forming analysis in Abaqus/Explicit with a punch velocity of 30 m/s using contact pairs.
Forming analysis in Abaqus/Explicit with a punch velocity of 5 m/s using contact pairs.
Forming analysis in Abaqus/Explicit with a punch velocity of 30 m/s using general contact.
Forming analysis in Abaqus/Explicit with a punch velocity of 5 m/s using general contact.

Reference

  • Taylor,  L. M., J. Cao, A. P. Karafillis, and M. C. Boyce, Numerical Simulations of Sheet Metal Forming,” Proceedings of 2nd International Conference, NUMISHEET 93, Isehara, Japan, Ed. A. Makinovchi, et al.
  • Figures

    Figure 1.5.1�1 Cross-section showing the geometry of the die, the punch, the blank holder, and the blank.
    Figure 1.5.1�2 Energy history for forming analysis: 30 m/s peak velocity of punch.
    Figure 1.5.1�3 Energy history for forming analysis: 15 m/s peak velocity of punch.
    Figure 1.5.1�4 Energy history for forming analysis: 5 m/s peak velocity of punch.
    Figure 1.5.1�5 Blank at the end of the forming analysis in Abaqus/Explicit.
    Figure 1.5.1�6 Blank after springback in Abaqus/Standard.

    Postbuckling and growth of delaminations in composite panels



    Products: Abaqus/Standard  Abaqus/Explicit  

    Objectives

    This example illustrates the application of the use of VCCT fracture criterion in both Abaqus/Standard and Abaqus/Explicit to predict the postbuckling response, onset, and growth of delaminations in laminated composite panels.

    Application description

    Delaminations are a primary failure mode for laminated composite materials. The delamination growth is more prominent under compressive loading since it results in buckling of a sublaminate leading to the delamination growth. The particular problem considered here is described in Reeder (2002). The results from the VCCT debond approach in Abaqus are compared to the experimental results.

    Geometry

    A flat 9.0 in (228.6 mm) � 4.5 in (114.3 mm) composite panel with a centrally located 2.5 in (63.5 mm) diameter delamination is studied in this example, as shown in Figure 1.4.9�1.

    Materials

    The panel is made of a AS4/3501-6 graphite/epoxy composite material system for which the typical lamina properties are given in Table 1.4.9�1. The laminate stacking sequence for the panel is [(�45/90/0)2/�60/�15]S. The critical fracture toughness for Modes I, II, and III at the delamination interface are also given in Table 1.4.9�1.

    Boundary conditions and loading

    The panel is subjected to compressive loading along its long axis. The overall dimensions of the model with boundary conditions and loading can be seen in Figure 1.4.9�1.

    Abaqus modeling approaches and simulation techniques

    The delamination is placed at the interface between the 5th and 6th ply (between 45� and �45�). The delamination region is modeled using two superimposed shell elements with contact constraints defined to prevent penetration of elements. The Benzeggagh-Kenane mixed-mode failure criterion (Benzeggagh and Kenane, 1996) is used to determine the growth of delamination based on the strain energy release rate computed using VCCT.

    Analysis types

    Both static and dynamic analyses are performed.

    Mesh design

    The finite element model is created with fully integrated first-order shell elements (S4). The finite element mesh for the model is shown in Figure 1.4.9�2 with the circular delamination at the center.

    Loads

    The loading consists of a prescribed displacement of 0.03 in (0.76 mm) at the top edge of the panel.

    Solution controls

    To achieve a stable delamination growth in Abaqus/Standard, a small amount of damping is specified for the interface where delamination growth occurs (see Automatic stabilization of rigid body motions in contact problems” in “Adjusting contact controls in Abaqus/Standard,” Section 34.3.6 of the Abaqus Analysis User's Manual). There are no solution controls specified in the Abaqus/Explicit analysis.

    Results and discussion

    The final deformed configuration of the composite panel obtained from Abaqus/Standard is shown in Figure 1.4.9�3. The postbuckling in the panel section cut along the long axis is illustrated in Figure 1.4.9�4. A contour plot of the bond status variable BDSTAT obtained from Abaqus/Standard indicating the growth of delamination is shown in Figure 1.4.9�5. The load-strain predictions are compared with the experimental data presented by Reeder (2002) in Figure 1.4.9�6. The predictions from VCCT are in agreement with the experimental results. The onset of delamination growth predicted by the VCCT debond approach in Abaqus is within 10% of the experimental data. The energy dissipated to stabilize the delamination growth is less than 4% of the total strain energy, as shown in Figure 1.4.9�7.
    The deformed configurations obtained from Abaqus/Explicit are shown in Figure 1.4.9�8 and Figure 1.4.9�9. Delamination growth obtained from the Abaqus/Explicit analysis is shown in Figure 1.4.9�10. The force-displacement responses differ slightly from the Abaqus/Standard results, as shown in Figure 1.4.9�11, but show reasonable agreement.

    Input files

    Postbuckling analysis of a composite plate using Abaqus/Standard.
    Postbuckling analysis of a composite plate using Abaqus/Explicit.

    References



    Other
    • Benzeggagh, M., and M. Kenane, Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus, Composite Science and Technology, vol. 56, p. 439, 1996.
    • Reeder, J., S. Kyongchan, P. B. Chunchu, and D. R. Ambur, Postbuckling and Growth of Delaminations in Composite Plates Subjected to Axial Compression, 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, vol. 1746, p. 10, 2002.

    Table

    Table 1.4.9�1 Properties for AS4/3501-6 graphite/epoxy material.
    PropertyValue
    18.500 � 106 lb/in2 (127.554 kN/mm2)
    1.640 � 106 lb/in2 (11.307 kN/mm2)
    0.871 � 106 lb/in2 (6.005 kN/mm2)
    0.871 � 106 lb/in2 (6.005 kN/mm2)
    0.522 � 106 lb/in2 (3.599 kN/mm2)
    0.30
    0.46863 lb/in (0.08207 N/mm)
    3.171825 lb/in (0.55546 N/mm)
    3.171825 lb/in (0.55546 N/mm)


    Figures

    Figure 1.4.9�1 The flat composite panel.
    Figure 1.4.9�2 The meshed NASA panel model.
    Figure 1.4.9�3 The final deformed configuration (Abaqus/Standard).
    Figure 1.4.9�4 The postbuckling in the panel section (Abaqus/Standard).
    Figure 1.4.9�5 The growth of delamination (Abaqus/Standard).
    Figure 1.4.9�6 The load-strain predictions compared with experimental data.
    Figure 1.4.9�7 The energy dissipated to stabilize the delamination growth.
    Figure 1.4.9�8 The final deformed configuration (Abaqus/Explicit).
    Figure 1.4.9�9 The postbuckling in the panel section (Abaqus/Explicit).
    Figure 1.4.9�10 The growth of delamination (Abaqus/Explicit).
    Figure 1.4.9�11 The force-displacement response comparison between Abaqus/Explicit and Abaqus/Standard.