Dynamic loading examples

� Impact loading on body structure during a car crash;

� Punching force on sheet metal in a stamping process;

� Impact force on electrical chips when an iPhone is dropped on the ground;

� Impact force on the head when a person falls on the ground.

Dynamic analysis algorithm:

Explicit vs. Implicit

1. Explicit• More efficient (no stiffness matrix inversion);• Less accurate in stress calculation (no equilibrium iteration);

• Applies to fewer element types;• Not applicable to cyclic loadings and snap through analysis.

2. Implicit• More costly (need to do stiffness matrix inversion at each increment);

• Much more accurate in stress calculation (enforce equilibrium iteration at each increment);

• Applies to many more element types;• Not practical for dynamic analysis of very large model.

ABAQUS Explicit method

• Updates the stiffness matrix at the end of each increment;

• Does not enforce internal force equilibrium (no equilibrium iteration);

• Results are stable as long as the increments are small enough;

• Uses a consistent, large-deformation theory —models can undergo large rotations and large deformation;

• Allows for the definition of very general contact conditions.

Numerical implementation of

ABAQUS Explicit dynamic analysis

Explicit central-difference time integration rule

degree of freedom

i increment number in an explicit dynamics step

mass matrix

applied load vector

internal force vector

Stability condition

Time increment used in an explicit analysis must satisfy a stability condition. Otherwise, the solution becomes unstable – the displacements will oscillate with increasing amplitudes.

Without damping;

With damping.

ωmax: highest frequency of the system;ξmax: critical damping.

Stable time increment size

Stability limit is the smallest time of a dilatational wave across any of the elements in the mesh:

: the smallest element dimension in the mesh;

: dilatational wave speed.

: Lamé's constants of elasticity;

ρ : material density.

Element size effect

• In a two-dimensional analysis refining the mesh by a factor of two in each direction will increase the run time in the explicit procedure by a factor of eight—four times as many elements and half the original time increment size.

• In a three-dimensional analysis refining the mesh by a factor of two in each direction will increase the run time by a factor of sixteen.

Mass scaling

• Artificially increase material density by a factor of f2. Reduce the total time period of the event from T to T/f.

• Has exactly the same effect on inertia forces as speeding up the time of simulation.

• Can be used for rate-dependent materials.

• Must be used with care to ensure that the inertia forces do not dominate and change the solution.

Explicit dynamic procedure implementation

*Dynamic, Explicit, (time increment option)(data line containing four data)

Data line for global or ELEMENT BY ELEMENT estimation:1. Enter a blank field.2. Time period of the step.3. Enter a blank field.4.Maximum time increment allowed. If this value is not specified, no upper limit is imposed.

Data line for fixed time increment using DIRECT USER CONTROL:1. Time increment to be used throughout the step.2. Time period of the step.

General contact

• Is a different contact algorithm other than contact pair algorithm.

• Can be defined either as model data or history data, while contact pair can only be defined as model data.

• Allows very simple definitions of contact with very few restrictions on the types of surfaces involved;

• Can be used simultaneously with the contact pair algorithm (i.e., some interactions can be modeled with the general contact algorithm, while others are modeled with the contact pair algorithm);

• Can be used only with three-dimensional surfaces;• Can be used only in mechanical finite-sliding contact

analyses.

General contact keywords

*CONTACT*CONTACT INCLUSIONSsurf_1, *CONTACT EXCLUSIONSsurf_a, surf_b

*CONTACT, OP=MOD*CONTACT INCLUSIONSsurf_2, surf_3 *CONTACT EXCLUSIONSsurf_a, surf_c

*CONTACT, OP=NEW*CONTACT INCLUSIONSsurf_1,surf_2, surf_3 *CONTACT EXCLUSIONSsurf_a, surf_b surf_a, surf_c

Include surf_1 as self-contact. Exclude contact between surf_a and surf_b.

Further include contact between surf_2 and surf_3. Further exclude surf_c as self-contact.

This step is equivalent in contact definition to the above two steps.

Energy output

• It is particularly important in checking the accuracy of the solution in an explicit dynamic analysis.

• The “artificial” energies, such as the artificial strain energy (ALLAE), the damping dissipation (ALLVD), and the mass scaling work (ALLMW) should be negligible compared to “real” energies such as the strain energy (ALLSE) and the kinetic energy (ALLKE).

• In a quasi-static analysis the value of the kinetic energy (ALLKE) should not exceed a small fraction of the value of the strain energy (ALLIE).

An example of dynamic explicit analysis

Steel support (1mm thick, elastic)

Steel punch (1mm thick, elastic)

Square-box beam (aluminum alloy, 2.5mm thick)

311 MPa

500 MPa

0.12

V (mm/s)

t (s)

0.5

5000

Result