RD-E: 2500 Spring-back

An explicit stamping simulation is followed by a spring-back analysis using implicit or explicit solvers for stress relaxation.

The spring-back simulation of sheet metal bent into a hat-shape is studied. The problem is one of the famous tests from the Numisheet'93. As spring-back is generally a quasi-static unloading, the use of the Radioss implicit solver is justified. The Radioss explicit solver is also used to compare the method's efficiency. However, for the stamping phase only the explicit solver is used, as the forming process is highly dynamic.

The example illustrates how to link up the explicit computations. It highlights the efficiency of the implicit solver for the spring-back simulation.

Options and Keywords Used

Explicit stamping simulation, implicit / explicit spring-back simulation, and stress relaxation
Implicit strategy and time step control by arc-length method
Anisotropic elasto-plastic material law (/MAT/LAW43 (HILL_TAB)) and Hill model
Orthotropic shell formulation, QEPH, progressive plastification, and iterative plasticity
Interface (/INTER/TYPE7), Penalty method, and friction
Concentrated load (/CLOAD)
Dynamic relaxation (/DYREL)
Implicit parameters (Implicit Solution)
Implicit spring-back (/IMPL/SPRBACK)
Imposed velocity (/IMPVEL)
Rigid body (/RBODY)

Simulation deals with:

Stamping simulation by explicit: from the beginning up to 960 ms.
Spring-back simulation:
- using explicit (dynamic approach): from 960 ms to 6000 ms:
  - From 960 ms to 2000 ms: Stamping tools are slowly withdrawn because the quasi-static analysis requires dynamic effects to be minimized during spring-back. Thus, the interfaces are not deleted. Options are defined in the *_0002.rad Engine file.
  - From 2000 ms to 6000 ms: A dynamic relaxation (/DYREL) is activated in the *_0003.rad Engine file in order to converge towards quasi-static equilibrium.
- using implicit (static approach): from 960 ms to 1000 ms:
  - The input implicit options are added in the *_0002.rad Engine file. Stress relaxation is activated using the /IMPL/SPRBACK keyword. All interfaces are deleted and specific boundary conditions are added on the stamping tools. Tools are not withdrawn.

In the simulation, the tools are modeled using rigid bodies (/RBODY) as shown in Figure 2.

An automatic main node is chosen. The center of gravity is computed using the main and secondary node coordinates and the main node is moved to the center of gravity where is placed mass and inertia (ICoG is set to 1). No mass or inertia are added to the rigid bodies.

A quarter of the structure is modeled in order to limit the model size and to eliminate rigid body modes for implicit computation. Symmetry planes are defined along the y axis = 0.

The nodes on the longitudinal plane are fixed in the Y translation and X, Z rotations.

For the other symmetry plane, the nodes are fixed in the X translation and Y, Z rotations.

Stamping tools are restricted to moving only along the Z-axis. The boundary conditions are applied on the main nodes of the rigid bodies, including the parts (Figure 3).

For the numerical simulation of the implicit spring-back, additional conditions must be added in the *_0002.rad Engine file in order to remove the rigid body modes that are not permitted in the implicit approach. The stamping tools are fully fixed (X, Y, Z translations and X, Y, Z rotations). The translation of the ID 427 node is fixed along the Z-axis allowing the sheet metal to move towards the final shape without rigid body mode.

Imposed velocities are applied on the stamping tools via the main nodes of the rigid bodies. The velocity of the punch is controlled by a specific input curve, as shown in Figure 5 and Figure 6. During implicit spring-back, all velocities are set to zero. Explicit spring-back computation up to 6000 ms necessitates imposed velocities on tools in order to withdraw them as of 1000 ms.

" Punch part …………… Curve 1, scale factor set to -1.

" Die part …………… . Curve 2, scale factor set to 1.

" Blank holder part ……. Curve 2, scale factor set to -1.

The stamping is performed by explicit simulation up to 960 ms using Curve 1. The implicit simulation is carried out only for the spring-back stage from 960 ms to 1000 ms. Curve 2, therefore, is only defined for explicit spring-back simulation.

Considering the symmetries, a constant concentrated load of 612.5 N is vertically applied on the blank holder via the main node of the rigid body. The load is set to zero from 960 ms before studying the spring-back.

Implicit spring-back analysis is launched using /IMPL/SPRBACK.

A solver method is required to resolve Ax=b in each iteration of the nonlinear cycle. It is defined using /IMPL/SOLVER.

The nonlinear implicit parameters used are:

`/IMPL/NONLIN/1 #L_A Itol Toli Tolj Tolk 2 0 0.025`	Implicit type Static nonlinear Nonlinear solver `N`=1 Modified Newton Tolerance `Toli`=0.025 Update of stiffness matrix `L_A`=2 Maximum iterations
`/IMPL/DTINI 0.08`	Initial time step 0.08 ms
`/IMPL/DT/STOP # DT_MIN DT_MAX 1e-5 0.0`	Minimum time step `DT_MIN`=10^-5 ms Maximum time step no
`/IMPL/DT/2 #lt_w L_arc L_dtn dt_sca dt_sca_max 6 0.0 20 0.67 0.0`	Time step control method Norm displacement (arc-length) Desired convergence iteration number `It_w`=6 Maximum convergence iteration number `L_dtn`=20 Decreasing time step factor `dt_sca`=0.67 Maximum increasing time step scale factor `dt_sca_max`=0.0 Arc-length `L_arc`=0 Automatic computation
`/IMPL/SPRBACK`	Spring-back option Activated

Refer to Radioss Starter Input for more details about implicit options.

A solver method is required to resolve Ax=b in each iteration of the nonlinear cycle. It is defined using /IMPL/SOLVER.

/IMPL/SOLVER/2
# Iprec It_max Itol Tol
5 0 0 0.

Linear solver method: N=2; Direct solver MUMPS
Precondition methods: I_cpre=5; Factored approximate inverse
Maximum iterations number: It_max=0; System dimension (NDOF)
Stop criteria: Itol=0; Relative residual on matrix
Tolerance for stop criteria: Tol=0; Machine precision

Refer to Radioss Starter Input for more details about implicit options.

Explicit spring-back analysis uses the dynamic relaxation in the *_0003.rad Engine file from 2000 ms.

The explicit time integration scheme starts with nodal acceleration computation. It is efficient for the simulation of dynamic loading. However, a quasi-static simulation via a dynamic resolution method is needed to minimize the dynamic effects for converging towards static equilibrium, the final shape achieved after spring-back.

The dynamic effect is damped by introducing a diagonal damping matrix proportional to mass matrix in the dynamic equation.(1)

M \ddot{u} + C \dot{u} + K u = F

(2)

C \dot{u} = \frac{2 β}{T} M

Where,

$β$: Relaxation value which has a recommended default value
$T$: Period to be damped (less than or equal to the highest period of the system)

The inputs of the relaxation dynamics are:

Relaxation factor: 1
Period to be damped: 1000 ms

This option is activated using the /DYREL keyword (inputs: $β$ and $T$ ).

Input Files

The input files used in this example include:

Explicit spring-back: <install_directory>/hwsolvers/demos/radioss/example/25_Spring-back/Explicit_spring-back/DBEND_44*
Implicit spring-back: <install_directory>/hwsolvers/demos/radioss/example/25_Spring-back/Implicit_spring-back/DBEND_44*

Model Description

This example deals with the numerical simulation of a stamping process, including the spring-back.

This refers to one of the sheet metal stamping tests "2D Draw Bending" indicated in Numisheet’93. The final shape of the formed sheet metal, after releasing all constraints on the blank sheet is studied. During the spring-back simulation, an explicit-to-implicit sequential solution method is used, where a dynamic forming process using the explicit solver is used first, followed by an implicit modeling of the spring-back deformations by statically removing the stamping stress.

Two analyses are performed in order to compare the performances and the accuracy of results:

Explicit stamping and implicit spring-back simulations
Explicit stamping and explicit spring-back simulations (using dynamic relaxation)

The spring-back simulation of the forming sheet metal uses an elasto-plastic nonlinear approach. The implicit input options and the incremental strategy used are described in the modeling section.

A numerical simulation of stamping is performed up to 960 ms. Spring-back computation is carried out from 960 ms to 1000 ms for implicit (static approach) and to 6000 ms for explicit (quasi-static approach).

A standard stamping operation is studied. The stamping tools include a punch, a die and a blank holder.

Units: mm, ms, g, N, MPa.

A load F of 1225 N is vertically applied on the blank holder in order to flatten the sheet metal against the die. The load is removed before spring-back simulation.

The sheet metal stamping operation is managed using a variable imposed velocity applied on the punch with a maximum set to 0.1 ms^-1. The tools are withdrawn after the stamping phase in order to enable the spring-back to be observed.

The main geometrical dimensions of the parts are:

Radius of die's corners: 5 mm
Radius of punch's corners: 5 mm
Width of punch: 50.4 mm
Sheet metal dimensions: 35 mm x 175 mm

The thickness of the sheet metal is defined at 0.74 mm. The Coulomb friction coefficient between the sheet metal and the die is defined at 0.129.

The stamping tools' material undergoes a linear law using:

Material Properties
Initial density: 8x10^-3 $[\frac{g}{m m^{3}}]$
Note: The density is artificially increased from 8x10^-3 $[\frac{g}{m m^{3}}]$ to minimize the run time.
Young's modulus: 206000 $[MPa]$
Poisson ratio: 0.3

The material of the sheet metal under the roller has distinct characteristics of anisotropy. Its anisotropic elasto-plastic behavior can be reproduced by a Hill model (/MAT/LAW43). This law can be considered as a generalization of the von Mises yield criteria for anisotropic yield behavior.

The yield stress is defined according to a user function and the yield stress is compared to equivalent stress:(3)

σ_{e q} = \sqrt{A_{1} σ_{1}^{2} + A_{2} σ_{2}^{2} - A_{3} σ_{1} σ_{2} + A_{12} σ_{12}^{2}}

The $A_{i}$ coefficients are determined using Lankford's anisotropy parameters range. Angles for Lankford parameters are defined with respect to orthotropic direction 1.

$R = \frac{r_{00} + 2 r_{45} + r_{90}}{4}$	$H = \frac{R}{1 + R}$
$A_{1} = H (1 + \frac{1}{r_{00}})$	$A_{2} = H (1 + \frac{1}{r_{90}})$
$A_{3} = 2 H$	$A_{12} = 2 H (r_{45} + 0.5) (\frac{1}{r_{00}} + \frac{1}{r_{90}})$

A hardening coefficient is used to describe the hardening model as full isotropic (value set to 0) or based on the Prager-Ziegler kinematic model (value set to 1). Hardening can be interpolated between the two models, if the coefficient value is between 0 and 1.

Material Properties
Initial density: 8x10^-3 $[\frac{g}{m m^{3}}]$
Young's modulus: 206000 $[MPa]$
Poisson ratio: 0.3
Lankford 0 degrees: r₀₀= 1.73
Lankford 45 degrees: r₄₅ = 1.34
Lankford 90 degrees: r₉₀= 2.24

The yield curve used is shown in Figure 9. Failure is not taken into account.

Model Method

Taking symmetry into account, only a quarter of the structure is modeled. The symmetry plane is along axis y = 17.5 mm and x = 0 mm.

The punch is shown in purple, the blank holder in green and the die in red. The sheet metal (blue) is modeled using 4-node shell elements.

The sheet metal is discretized by a non-regular mesh and a fine mesh is used for parts to be plastically deformed. The smallest size of the shell element is 1.5 mm.

In order to achieve accurate simulation results, the QEPH shell element formulation is used in explicit and implicit analyses. A Lagrangian formulation is adopted.

In accordance with the elasto-plastic Hill model for the material law, the sheet metal is described by the shell elements using the orthotropic property (TYPE9). The shell characteristics are:

Five integration points (progressive plastification)
Interactive plasticity with three Newton iterations (I_plas = 1)
Thickness changes are taken into account in stress computation (I_thick = 1)
Initial thickness is uniform, equal to 0.74 mm
Orthotropy angle: 0 degree
Reference vector: (1 0 0)

The input components of the reference vector $\vec{V}$ are used to define direction 1 of the local coordinate system of orthotropy. The orthotropy angle, in degrees defines the angle between direction 1 of the orthotropy and the projection of the vector $\vec{V}$ on the shell.

Three TYPE7 interfaces using the Penalty method are employed to model contacts between the stamping tools and the sheet metal. The parameters defining the contact are:

Material Properties
Coulomb friction: 0.129
Gapmin: 0.37
Critical damping coefficient on interface stiffness: 1
Critical damping coefficient on interface friction: 1 (default)

In the implicit approach, the contact using the Penalty method with fictional springs is stored in a separate stiffness matrix to the main one. Therefore, supplementary memory is needed and information of the second contact stiffness will be printed when contact is active.

Critical damping coefficients (inputs) description:

The normal force computation is indicated by: (4) $F_{n} = K_{s} P + C \frac{d P}{d t}$

Figure 13.

Where,

$K_{S} = K_{0} (\frac{G a p}{G a p - P})$

$C = V I S_{S} \sqrt{2 K_{S} M}$

$K_{0}$

Initial interface spring stiffness

$V I S_{S}$

Critical damping coefficient on interface stiffness (default value: 0.05)
The tangential force computation is indicated by:(5) $F_{t} = \min (μ F_{n}, C_{t} V_{t})$
Where,

$C_{t} = V I S_{F} \sqrt{2 K_{S} M}$

$V I S_{F}$

Critical damping coefficient on interface friction (default value: 1)
For spring-back computation by implicit, the removing of the stamping tools is taken into account by deleting all interfaces using the input option in the second *_0002.rad Engine file as:
```
/DEL/INTER
1 2 3
```
Interfaces ID 1, 2 and 3 are deleted.

Results

In the metal stamping operation, the highly nonlinear deformation processes tend to generate a large amount of elastic strain energy in the metal material in addition to some of the plastic deformed areas. The internal energy, which is stored in the sheet metal during stamping, is subsequently released once the stamping pressure has been removed. This energy released is the driving force of the spring-back in the sheet metal forming process. Therefore, the spring-back deformation for sheet metal forming is mainly due to the amount of elastic energy stored in the part while it is being plastically deformed.

The material density has been multiplied by 10,000 to obtain a reasonable computation time using explicit simulations. An additional time period is also required for slowly withdrawing the tools, prior to the explicit spring-back simulation in order to achieve a good result. Thus, explicit stamping takes longer than stamping followed by implicit spring-back computation.

Figure 14 shows the deformed configurations using implicit simulation. The symmetrical part is added.

Stamping is performed from the beginning up to 960 ms. The final shape after the spring-back process is achieved after 1000 ms using the implicit solver and after 6000 ms using the explicit solver.

The animations in Figure 16 include the results of the spring-back during simulation. There is an increasing number of stresses in the sheet metal from the start up to 960 ms, after which, the stresses begin to decrease as a result of the spring-back (stress relaxation).

Figure 17 shows the internal energy stored in the sheet metal during the stamping.

The dynamic relaxation used in the explicit spring-back computation enables to improve convergence towards quasi-static solution. The variation of the kinetics energy on the sheet metal in the explicit spring-back simulation is depicted in Figure 18 (from 960 ms up to 6000 ms):

Comparison with experimental data on geometry after spring-back is shown in Table 1.

Table 1. Simulation Results Compared to Experiment
	$θ$ ₁	$θ$ ₂
Experiment (means values)	105.7	77.7
Implicit	100.8	78.6
Explicit	106.1	78

The performance results are presented in Table 2.

Table 2. Implicit/Explicit Computation Time
	Stamping CPU (cycles)	Spring-back Cycles (iter. Num.)	Spring-back CPU (CPU per cycle)	Total CPU
Explicit	1160 (92326)	229379 (-)	2698 (0.01)	3858
Implicit	-	120 (354)	1589 (13.2)	2749

The implicit simulation for spring-back is performed from 960 ms to 1000 ms. Explicit spring-back simulation is performed until the kinetics energy on the sheet metal reaches a minimum value (quasi-static equilibrium). The final computation time is set to 6000 ms.

Explicit and implicit analysis' both obtain good results in this test, with implicit computation being 40% faster than the explicit computation. The implicit approach is; however, 1320 times more expensive per step than the explicit solver. The use of the implicit approach allows you to economize on the overall computation time.