Explicit Dynamics Chapter 1 Intro to Exp Dyn

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In t rodu c t ion to Exp l ic i t Dynamics

1-2 ANSYS, Inc. Proprietary 2009 ANSYS, Inc. All rights reserved.

February 27, 2009Inventory #002665

Training ManualWelcome! Welcome to the ANSYS Expl ic i t Dynamics introductory training

course!

This training course is intended for all new or occasional ANSYSExpl ic i t Dynamics users, regardless of the CAD software used.

Course Objectives:

Introduction to Explicit Dynamics Analyses. General understanding of the Workbench and ExplicitDynamics (Mechanical) user interface, as related to geometryimport and meshing.

Detailed understanding of how to set up, solve and post-process Explicit Dynamic analyses.

Utilizing parameters for optimization studies.

Training Courses are also available covering the detailed use of otherWorkbench modules (e.g. DesignModeler, Meshing, Advancedm eshing , etc . ) .


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Training ManualCourse Materials

The Training Manual you have is an exact copy of theslides.

Workshop descriptions and instructions are included inthe Workshop Supplement.

Copies of the workshop files are available on the ANSYSCustomer Portal ( www.ansys.com ).

Advanced training courses are available on specifictopics. Schedule available on the ANSYS web pagehttp://www.ansys.com/ under Solutions> Services andSupport> Training Services .
http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/


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Training ManualA. About ANSYS, Inc.ANSYS, Inc.

Developer of ANSYS family of products Global Headquarters in Canonsburg, PA - USA (south of Pittsburgh) Development and sales offices in U.S. and around the world Publicly traded on NASDAQ stock exchange under ANSS For additional company information as well as descriptions and

schedules for other training courses visit www.ansys.com


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Training ManualCourse Overview Chapter 1: Introduction to Explicit Dynamics Chapter 2: Introduction to Workbench Chapter 3: Engineering Data Chapter 4: Explicit Dynamics Basics Chapter 5: Results Processing Chapter 6: Explicit Meshing

Chapter 7: Body Interactions Chapter 8: Analysis Settings Chapter 9: Material Models Chapter 10: Optimization Studies


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Training ManualWhy Use Explicit Dynamics? Implicit and Explicit refer to two types of time integration

methods used to perform dynamic simulations

Explicit time integration is more accurate and efficient for simulationsinvolving

Shock wave propagation Large deformations and strains

Non-linear material behavior Complex contact Fragmentation Non-linear buckling

Typical applications Drop tests Impact and Penetration


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Training Manual

Solution Impact Velocity(m/s)

Strain Rate (/s) Effect

Implicit 12000 > 10 8 Vaporization of collidingsolids

Impact Response of Materials

Why Use Explicit Dynamics?


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Training Manual

VELOCITY LOW HIGH

Deformation Global Local

Response Time ms - s s - ms

Strain 50%

Strain Rate < 10 s-1

> 10000 s-1

Pressure < Yield Stress 10-100 x Yield Stress

Typical Values for Solid Impacts

Why Use Explicit Dynamics?


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Training ManualWhy Use Explicit Dynamics?

Electronics Applications


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Aerospace Applications


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Applications in Nuclear Power safety


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Applications in Homeland Security


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Sporting Goods Application


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Training ManualExplicit Solution Strategy Solution starts with a mesh having assigned material

properties, loads, constraints and initial conditions.

Integration in time, produces motion at the mesh nodes Motion of the nodes produces deformation of the elements

Element deformation results in a change in volume and densityof the material in each element

Deformation rate is used to derive strain rates (using variouselement formulations)

Constitutive laws derive resultant stresses from strain rates Stresses are transformed back into nodal forces (using various

element formulations)

External nodal forces are computed from boundary conditions,loads and contact

Total nodal forces are divided by nodal mass to produce nodalaccelerations

Accelerations are integrated Explicitly in time to produce newnodal velocities

Nodal velocities are integrated Explicitly in time to produce newnodal positions

The solution process (Cycle) is repeated until the calculationend time is reached


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Training ManualBasic Formulation Implicit Dynamics The basic equation of motion solved by an implicit transient dynamic analysis is

where m is the mass matrix, c is the damping matrix, k is the stiffness matrixand F(t) is the load vector

At any given time, t , this equation can be thought of as a set of "static" equilibrium equations that also take

into account inertia forces and damping forces. The Newmark or HHT method is used to solve theseequations at discrete time points. The time increment between successive time points is called theintegration time step

For linear problems: Implicit time integration is unconditionally stable for certain integration parameters. The time step will vary only to satisfy accuracy requirements.

For nonlinear problems: The solution is obtained using a series of linear approximations (Newton-Raphson method), so each

time step may have many equilibrium iterations. The solution requires inversion of the nonlinear dynamic equivalent stiffness matrix. Small, iterative time steps may be required to achieve convergence. Convergence tools are provided, but convergence is not guaranteed for highly nonlinear problems.

)(t F kx xc xm


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Training ManualBasic Formulation Explicit Dynamics The basic equations solved by an Explicit Dynamic analysis express the conservation of mass, momentum

and energy in Lagrange coordinates. These, together with a material model and a set of initial andboundary conditions, define the complete solution of the problem.

For Lagrange formulations, the mesh moves and distorts with the material it models, so conservation ofmass is automatically satisfied. The density at any time can be determined from the current volume of thezone and its initial mass:

The partial differential equations which express the conservation of momentum relate the acceleration tothe stress tensor ij:

Conservation of energy is expressed via:

For each time step, these equations are solved explicitly for each element in the model, based on inputvalues at the end of the previous time step

Only mass and momentum conservation is enforced. However, in well posed explicit simulations, mass,momentum and energy should be conserved. Energy conservation is constantly monitored for feedback onthe quality of the solution (as opposed to convergent tolerances in implicit transient dynamics)

V m

V V 00

z y xb z

z y xb y

z y xb x

zz zy zx z

yz yy yx y

xz xy xx x

zx zx yz yz xy xy zz zz yy yy xx xxe

2221


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Training ManualBasic Formulation Explicit Dynamics The Explicit Dynamics solver uses a central difference time integration scheme (Leapfrog

method). After forces have been computed at the nodes (resulting from internal stress,contact, or boundary conditions), the nodal accelerations are derived by dividing force bymass:

where x i are the components of nodal acceleration (i=1,2,3), F i are the forces acting on the nodes , b i arethe components of body acceleration and m is the mass of the node

With the accelerations at time n - determined, the velocities at time n + are found from

Finally the positions are updated to time n+1 by integrating the velocities

Advantages of using this method for time integration for nonlinear problems are: The equations become uncoupled and can be solved directly (explicitly). There is no requirement for

iteration during time integration No convergence checks are needed since the equations are uncoupled No inversion of the stiffness matrix is required. All nonlinearities (including contact) are included in the

internal force vector

ii

i bm F

x

nni

ni

ni t x x x

2121

21211 nni

ni

ni t x x x


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Training ManualStability Time Step To ensure stability and accuracy of the solution, the size of the time step used in Explicit time

integration is limited by the CFL (Courant-Friedrichs-Levy[1]) condition. This condition implies that thetime step be limited such that a disturbance (stress wave) cannot travel further than the smallest

characteristic element dimension in the mesh, in a single time step. Thus the time step criteria forsolution stability is

where t is the time increment, f is the stability time step factor (= 0.9 by default), h is the characteristicdimension of an element and c is the local material sound speed in an element

The element characteristic dimension, h , is calculated as follows:

[1] R. Courant, K. Friedrichs and H. Lewy, "On the partial difference equations of mathematical physics",

IBM Journal , March 1967, pp. 215-234

m inch

f t

Hexahedral /Pentahedral The volume of the element divided by the square of the longest diagonal andscaled by 2/3

Tetrahedral The minimum distance of any element node to its opposing element faceQuad Shell The square root of the shell area

Tri Shell The minimum distance of any element node to its opposing element edgeBeam The length of the element


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Training ManualStability Time Step The time steps used for explicit time

integration will generally be much smaller

than those used for implicit time integration e.g. for a mesh with a characteristic

dimension of 1 mm and a material soundspeed of 5000 m/s. The resulting stabilitytime step would be 0.18 -seconds. To solvethis simulation to a termination time of 0.1seconds will require 555,556 time steps

The minimum value of h/c for all elementsin a model is used to calculate the timestep. This implies that the number of timesteps required to solve the simulation isdictated by the smallest element in themodel.

Take care when generating meshes forExplicit Dynamics simulations to ensure thatone or two very small elements do notcontrol the time step

h

m inch f t


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Training ManualStability Time Step and Mass Scaling The maximum time step that can be used in explicit time integration is inversely proportional

to the sound speed of the material and therefore directionally proportional to the square rootof the mass of material in an element

where C i j is the material stiffness ( i=1,2,3 ), is the material density, m is the material massand V is the element volume

Artificially increasing the mass of an element can increase the maximum allowable stabilitytime step, and reduce the number of time increments required to complete a solution

Mass scaling is applied only to those elements which have a stability time step less than aspecified value. If a model contains relatively few small elements, this can be a usefulmechanism for reducing the number of time steps required to complete an Explicit simulation

Mass scaling changes the inertial properties of the portions of the mesh to which scaling isapplied. Be careful to ensuring that the model remains representative for the physicalproblem being solved

iiii VC m

C ct

11


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Training ManualElastic Waves

Different types of elastic waves can propagate in solids depending on how the motion of points in the solidmaterial is related to the direction of propagation of the waves [Meyers].

The primary elastic wave is the longitudinal wave. Under uniaxial stress conditions (i.e. an elastic wavetravelling down a long slender rod), the longitudinal wave speed is given by:

For the three-dimensional case, additional components of stress lead to a more general expression for thelongitudinal elastic wave speed

The secondary elastic wave is the distortional or shear wave and its speed can be calculated as

Other forms of elastic waves include surface (Rayleigh) waves, Interfacial waves and bending (or flexural)waves in bars/plates [Meyers]

Meyers M A, (1994) Dynamic behaviour of Materials, John Wiley & Sons, ISBN 0-471-58262-X

E c 0

G K c P

34

GcS


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Training ManualPlastic Waves

Plastic (inelastic) deformation takes place in a ductile metal when the stress in the material exceeds theelastic limit. Under dynamic loading conditions the resulting wave propagation can be decomposed intoelastic and plastic regions [Meyer]. Under uniaxial strain conditions, the elastic portion of the wave travelsat the primary longitudinal wave speed whilst the plastic wave front travels at a local velocity

For an elastic perfectly plastic material, it can be shown [Zukas] that the plastic wave travels at a slowervelocity than the primary elastic wave, so an elastic precursor of low amplitude often precedes the strongerplastic wave

Meyers M A, (1994) Dynamic behaviour of Materials, John Wiley & Sons, ISBN 0-471-58262-XZukas J A, (1990) High velocity impact dynamics, John Whiley, ISBN 0-471-51444-6

d

d c plastic

K c

plastic


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Training ManualShock Waves

Typical stress strain curves for a ductile metal

Uniaxial Stress Uniaxial Strain

Under uniaxial stress conditions, the tangent modulus of the stress strain curve decreases with strain. Theplastic wave speed therefore decreases as the applied jump in stress associated with the stress waveincreases shock waves are unlikely to form under these conditions

Under uniaxial strain conditions the plastic modulus (AB) increases with the magnitude of the applied jumpin stress. If the stress jump associated with the wave is greater than the gradient (OZ), the plastic wave willtravel at a higher speed than the elastic wave. Since the plastic deformation must be preceded by theelastic deformation, the elastic and plastic waves coalesce and propagate as a single plastic shock wave

x

x

z

o

A

B

C


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Training ManualShock Waves A shock wave is a discontinuity in material state (density ( ), energy ( e ), stress ( ), particle velocity

(u ) ) which propagates through a medium at a velocity equal to the shock velocity ( U s )

Relationships between the material state across a shock discontinuity can be derived using theprincipals of conservation of mass, momentum and energy The resulting Hugoniot equations aregiven by:

1 e1

1 u 1

0 e0

0 u 0

U s


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Training ManualShock and Rarefaction Waves

Rarefaction

Shock

Elastic precursor

Shock (compression) andrarefaction (expansion) waves

generated by a pressurediscontinuity

d l


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Training ManualSpatial Discretization

Geometries (bodies) are meshed into a (large) number of smaller elements All elements use in Explicit Dynamics have Lagrange formulations i.e. elements follow the deformation of the bodies

Advanced Explicit Dynamics (AUTODYN) allows other formulations to beused

Euler (Multi-material, Blast) Particle free (SPH)

I d i E l i i D i


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Training Manual

Element formulations for Explicit Dynamics

Solid elements Hexahedral

Exact volume integration Approximate Gauss volume integration

Pentahedral Automatically converted to a degenerate hex

Tetrahedral SCP (Standard Constant Pressure) ANP (Average Nodal Pressure)

Shell elements

Quadrilateral Triangular

Beam (Line) element

Element Formulations

1

2

3

4

1

2

3

I d i E l i i D i


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Hexahedral Solid Elements

Two Formulations:

8 node, exact volume integration, constantstrain element

Single quadrature point with hourglass

stabilization

8 node, approximate Gauss volumeintegration element

LS-DYNA formulation (Hallquist)

Some accuracy is lost for fastercomputation

Single quadrature point with hourglassstabilization


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Training ManualElement Formulations Tetrahedral Solid Elements

Two formulations:

SCP (Standard Constant Pressure)

Textbook 4 noded iso-parametric tetelement

Designed as filler element for hex-dominant meshes

Exhibits volume locking if over constrainedor during plastic flow

ANP (Average Nodal Pressure)

Enhanced 4 noded iso-parametric tetelement (Burton, 1996)

Overcomes volume locking problems

Can be used as a majority mesh element

SCP Tet

ANP Tet

I t d t i t E l i i tD i


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Training ManualElement Formulations Tetrahedral Solid Elements

Pull-out test simulated using bothhexahedral elements (top) and ANP

tetrahedral elements (bottom).Similar plastic strains and material

fracture are predicted for both elementformulations used.

In t rodu c t ionto Expl ic i tDynamics


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Training Manual

Shell Elements

Quadrilateral shell element Belytschko-Tsay, with Chang-Wong correction Co-rotational formulation, bi-linear, 4 noded Single quadrature point with hourglass stabilization Isotropic and layered orthotropic formulations Number of through thickness integration points can be specified

Triangular shell element C0 Triangular Plate Element (Belytscho, Stolarski and Carpenter

1984)

Should be used in quad-dominant meshes

Thickness is a parameter (not modelled geometrically) Actual thickness can be rendered Time step is controlled by the element length, not by thickness

1

2

3

E

1

2

3

4

E




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Training ManualElement Formulations Hourglass Control (Damping) for Hexahedral Solid and Quad Shell Elements

For the hexahedral and quad element formulations, the expressions for strain rates and forces involve onlydifferences in velocities and / or coordinates of diagonally opposite corners of the element If an element distorts such that these differences remain unchanged there is no strain increase in the element and

therefore no resistance to this distortion On the left, the two diagonals remain the same length even though the element distorts. If such distortions occur

in a region of several elements, a pattern such as that shown on the right occurs and the reason for the namehourglass instability is easily understood

In order to avoid such hourglass instabilities, a set of corrective forces are added to the solution Two formulations are available for hexahedral solid elements

AD standard (default) Most efficient option in terms of memory and speed

Flanagan-Belytschko Invariant under rotation Improved results for large rigid body rotations

21

3 4

21

3 4

2D 3D



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Training ManualElement Formulations

Beam (Line) Elements

2 noded Belytschko-Schwer resultantbeam formulation

Extended to allow large axial strains

Resultant plasticity implemented for

range of cross section types Cross-section is a parameter (not

modelled geometrically)

Actual cross section can be rendered

Time step is controlled by the elementlength, not by dimensions of cross-section



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Training Manual

X

Y

Z

11

22

33

Node #1

Node #222

Local 11-Direction

Always defined from node #1 to node#2

Local 22-Direction

Defined by user for Rectangular, I-Beam andGeneral SectionsUser defines initial unit vector 22 at cyclezero. This should lie in plane 11-22

Local 33-Direction

Orthogonal to Local directions 11 and 22

Rin

Rout

aa

A

A ab

A

B

22

a

A

B

22

tw

tf

22

33


Beam cross-sections



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Training ManualElement Usage

What is required for meshing Explicit Applications?

Uniform element size (in finest zoned regions). Smallest element size controls the time step used to advance the solution in time. Explicit analyses compute dynamic stress waves that need to be accurately modeled as they

propagate through the entire mesh.

Element size controlled by the user throughout the mesh.

Not automatically dependent on geometry. Implicit analyses usually have static region of stress concentration where mesh is refined

(strongly dependent on geometry).

In explicit analyses, the location of regions of high stress constantly change as stress wavespropagate through the mesh.

Mesh refinement is usually used to improve efficiency.

Mesh transitions should be smooth for maximum accuracy.

Hex-dominant meshing preferred. More efficient. Sometimes more accurate for slower transients.

Chapter 6 will cover Explicit Meshing in more detail



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Training ManualMaterial ModelingClass of Material Material Effects

Metals ElasticityPlasticity

Isotropic Strain HardeningKinematic Strain HardeningIsotropic Strain Rate HardeningIsotropic Thermal SofteningDuctile FractureBrittle Fracture (Fracture Energy based)Dynamic Failure (Spall)

Concrete / Rock ElasticityPorous Compaction

PlasticityStrain HardeningStrain Rate Hardening in CompressionStrain Rate Hardening in TensionPressure Dependent PlasticityLode Angle Dependent PlasticityShear Damage / FractureTensile Damage / Fracture

Soil / Sand Elasticity

Porous CompactionPlasticityPressure Dependent PlasticityShear Damage / FractureTensile Damage / Fracture

Rubbers / Polymers ElasticityViscoelasticityHyperelasticity

Orthotropic Orthotropic Elasticity

In general, materials have a complexresponse to dynamic loading,particularly when the loading is rapid,intense and distructive

The Material models available forExplicit Dynamics simulations facilitatethe modeling of a wide range ofmaterials and material behaviors, asshown in the table

Chapter 3 will explain how materialdata can be created or retrieved fromlibraries using Engineering Data

The actual material models availablefor Explicit Dynamics analyses arepresented at length in Chapter 6



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Training ManualBasic Formulation

Models available forExplicit Dynamics

Chapter 9 willcover thesematerial models inmore detail

Documents

Explicit Dynamics Chapter 1 Intro to Exp Dyn