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Chapter 1 Introduction to Explicit Dynamics
ANSYS Explicit Dynamics
Introduction to Explicit Dynamics
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Training Manual Welcome!
• Welcome to the ANSYS Explicit Dynamics introductory training
course!
• This training course is intended for all new or occasional ANSYS
Explicit Dynamics users, regardless of the CAD software used.
• Course Objectives:
• Introduction to Explicit Dynamics Analyses.
• General understanding of the Workbench and Explicit
Dynamics (Mechanical) user interface, as related to geometry
import 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 other
Workbench modules (e.g. DesignModeler, Meshing, Advanced
meshing, etc.).
Introduction to Explicit Dynamics
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Training Manual Course Materials
• The Training Manual you have is an exact copy of the
slides.
• Workshop descriptions and instructions are included in
the Workshop Supplement.
• Copies of the workshop files are available on the ANSYS
Customer Portal (www.ansys.com).
• Advanced training courses are available on specific
topics. Schedule available on the ANSYS web page
http://www.ansys.com/ under “Solutions> Services and
Support> Training Services”.
Introduction to Explicit Dynamics
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Training Manual A. 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
Introduction to Explicit Dynamics
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Training Manual Course 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 Manual Why 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 simulations
involving
– Shock wave propagation
– Large deformations and strains
– Non-linear material behavior
– Complex contact
– Fragmentation
– Non-linear buckling
• Typical applications
– Drop tests
– Impact and Penetration
Introduction to Explicit Dynamics
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Training Manual
Solution Impact Velocity
(m/s)
Strain Rate (/s) Effect
Implicit <10-5 Static / Creep
< 50 10-5 - 10-1 Elastic
50 -1000 10-1 - 101 Elastic-Plastic (material
strength significant)
1000 - 3000 105 - 106
Primarily Plastic (pressure
equals or exceeds material
strength)
3000 - 12000 106 - 108
Hydrodynamic (pressure
many times material
strength)
Explicit > 12000 > 108 Vaporization of colliding
solids
Impact Response of Materials
Why Use Explicit Dynamics?
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Training Manual
VELOCITY LOW HIGH
Deformation Global Local
Response Time ms - s µs - ms
Strain <10% >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?
Introduction to Explicit Dynamics
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Training Manual Why Use Explicit Dynamics?
• Electronics Applications
Introduction to Explicit Dynamics
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Training Manual Why Use Explicit Dynamics?
• Aerospace Applications
Introduction to Explicit Dynamics
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Training Manual Why Use Explicit Dynamics?
• Applications in Nuclear Power safety
Introduction to Explicit Dynamics
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Training Manual Why Use Explicit Dynamics?
• Applications in Homeland Security
Introduction to Explicit Dynamics
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Training Manual Why Use Explicit Dynamics?
• Sporting Goods Application
Introduction to Explicit Dynamics
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Training Manual Explicit 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 density of the material in each element
• Deformation rate is used to derive strain rates (using various element 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 nodal accelerations
• Accelerations are integrated Explicitly in time to produce new nodal velocities
• Nodal velocities are integrated Explicitly in time to produce new nodal positions
• The solution process (Cycle) is repeated until the calculation end time is reached
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Training Manual Basic 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 matrix
and 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 these equations at discrete time points. The time increment between successive time points is called the integration 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.
)(tFkxxcxm
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Training Manual Basic 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 and boundary conditions, define the complete solution of the problem.
• For Lagrange formulations, the mesh moves and distorts with the material it models, so conservation of mass is automatically satisfied. The density at any time can be determined from the current volume of the zone and its initial mass:
• The partial differential equations which express the conservation of momentum relate the acceleration to the 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 input
values 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 on the quality of the solution (as opposed to convergent tolerances in implicit transient dynamics)
V
m
V
V00
zyxbz
zyxby
zyxbx
zzzyzxz
yzyyyx
y
xzxyxxx
zxzxyzyzxyxyzzzzyyyyxxxxe
222
1
Introduction to Explicit Dynamics
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Training Manual Basic 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 by mass:
where xi are the components of nodal acceleration (i=1,2,3), Fi are the forces acting on the nodes, bi are
the components of body acceleration and m is the mass of the node
• With the accelerations at time n - ½ determined, the velocities at time n + ½ 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
Fx
nn
i
n
i
n
i txxx 2121
21211 nn
i
n
i
n
i txxx
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Training Manual Stability 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 the time 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 for solution stability is
where Δt is the time increment, f is the stability time step factor (= 0.9 by default), h is the characteristic dimension 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
min
c
hft
Hexahedral /Pentahedral The volume of the element divided by the square of the longest diagonal and
scaled by √2/3
Tetrahedral The minimum distance of any element node to it’s opposing element face
Quad Shell The square root of the shell area
Tri Shell The minimum distance of any element node to it’s opposing element edge
Beam The length of the element
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Training Manual Stability 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 sound
speed of 5000 m/s. The resulting stability
time step would be 0.18 µ-seconds. To solve
this simulation to a termination time of 0.1
seconds will require 555,556 time steps
• The minimum value of h/c for all elements
in a model is used to calculate the time
step. This implies that the number of time
steps required to solve the simulation is
dictated by the smallest element in the
model.
– Take care when generating meshes for
Explicit Dynamics simulations to ensure that
one or two very small elements do not
control the time step
h
min
c
hft
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Training Manual Stability 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 root of the mass of material in an element
where Cij is the material stiffness (i=1,2,3), ρ is the material density, m is the material mass and V is the element volume
• Artificially increasing the mass of an element can increase the maximum allowable stability time 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 a specified value. If a model contains relatively few small elements, this can be a useful mechanism 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 is applied. Be careful to ensuring that the model remains representative for the physical problem being solved
iiiiVC
m
Cct
11
Introduction to Explicit Dynamics
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Training Manual Wave Propagation
• Explicit Dynamics computes wave propagation in solids and liquids Average Velocity
Velocity at Gauge 1
Constant pressure applied to left surface for 1 ms
Rarefaction
Shock
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Training Manual Elastic Waves
• Different types of elastic waves can propagate in solids depending on how the motion of points in the solid material 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 wave travelling 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 the longitudinal elastic wave speed
• The secondary elastic wave is the distortional or shear wave and it’s 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
Ec 0
GKcP
34
GcS
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Training Manual Plastic Waves
• Plastic (inelastic) deformation takes place in a ductile metal when the stress in the material exceeds the
elastic limit. Under dynamic loading conditions the resulting wave propagation can be decomposed into
elastic and plastic regions [Meyer]. Under uniaxial strain conditions, the elastic portion of the wave travels
at 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 slower
velocity than the primary elastic wave, so an elastic precursor of low amplitude often precedes the stronger
plastic wave
Meyers M A, (1994) “Dynamic behaviour of Materials”, John Wiley & Sons, ISBN 0-471-58262-X
Zukas J A, (1990) “High velocity impact dynamics”, John Whiley, ISBN 0-471-51444-6
dd
c plastic
Kc plastic
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Training Manual Shock 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. The plastic wave speed therefore decreases as the applied jump in stress associated with the stress wave increases – shock waves are unlikely to form under these conditions
• Under uniaxial strain conditions the plastic modulus (AB) increases with the magnitude of the applied jump in stress. If the stress jump associated with the wave is greater than the gradient (OZ), the plastic wave will travel at a higher speed than the elastic wave. Since the plastic deformation must be preceded by the elastic 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 Manual Shock Waves
• A shock wave is a discontinuity in material state (density (ρ), energy (e), stress (σ), particle velocity
(u) ) which propagates through a medium at a velocity equal to the shock velocity (Us)
• Relationships between the material state across a shock discontinuity can be derived using the
principals of conservation of mass, momentum and energy The resulting Hugoniot equations are
given by:
•
1
e1
1
u1
0
e0
0
u0
Us
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Training Manual Shock and Rarefaction Waves
Rarefaction
Shock
Elastic precursor
Shock (compression) and
rarefaction (expansion) waves
generated by a pressure
discontinuity
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Training Manual Spatial 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 be
used
– Euler (Multi-material, Blast)
– Particle free (SPH)
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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
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• Hexahedral Solid Elements
– Two Formulations:
• 8 node, exact volume integration, constant strain element
– Single quadrature point with hourglass stabilization
• 8 node, approximate Gauss volume integration element
– LS-DYNA formulation (Hallquist)
– Some accuracy is lost for faster computation
– Single quadrature point with hourglass stabilization
Element Formulations
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Training Manual Element Formulations
• Tetrahedral Solid Elements
– Two formulations:
• SCP (Standard Constant Pressure)
– “Textbook” 4 noded iso-parametric tet element
– Designed as “filler” element for hex-dominant meshes
– Exhibits volume locking if over constrained or during plastic flow
• ANP (Average Nodal Pressure)
– Enhanced 4 noded iso-parametric tet element (Burton, 1996)
– Overcomes volume locking problems
– Can be used as a majority mesh element
SCP Tet
ANP Tet
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Training Manual Element Formulations
• Tetrahedral Solid Elements
Pull-out test simulated using both
hexahedral elements (top) and ANP
tetrahedral elements (bottom).
Similar plastic strains and material
fracture are predicted for both element
formulations used.
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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
Element Formulations
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Training Manual Element 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 only
differences 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 name
“hourglass 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 Manual Element Formulations
• Beam (Line) Elements
– 2 noded Belytschko-Schwer resultant
beam 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 element
length, not by dimensions of cross-
section
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Training Manual
X
Y
Z
11
22
33
Node #1
Node #2 22’
Local 11-Direction Always defined from node #1 to node#2
Local 22-Direction Defined by user for Rectangular, I-Beam and General Sections
User defines initial unit vector 22’ at cycle zero. This should lie in plane 11-22
Local 33-Direction Orthogonal to Local directions 11 and 22
Rin
Rout
a a
A
A a b
A
B
22
a
A
B
22
tw
tf
22
33
Element Formulations
• Beam cross-sections
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Training Manual Element 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 waves propagate 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 Manual Material Modeling
Class of Material Material Effects
Metals Elasticity
Plasticity
Isotropic Strain Hardening
Kinematic Strain Hardening
Isotropic Strain Rate Hardening
Isotropic Thermal Softening
Ductile Fracture
Brittle Fracture (Fracture Energy based)
Dynamic Failure (Spall)
Concrete / Rock Elasticity
Porous Compaction
Plasticity
Strain Hardening
Strain Rate Hardening in Compression
Strain Rate Hardening in Tension
Pressure Dependent Plasticity
Lode Angle Dependent Plasticity
Shear Damage / Fracture
Tensile Damage / Fracture
Soil / Sand Elasticity
Porous Compaction
Plasticity
Pressure Dependent Plasticity
Shear Damage / Fracture
Tensile Damage / Fracture
Rubbers / Polymers Elasticity
Viscoelasticity
Hyperelasticity
Orthotropic Orthotropic Elasticity
• In general, materials have a complex response to dynamic loading, particularly when the loading is rapid, intense and distructive
• The Material models available for Explicit Dynamics simulations facilitate the modeling of a wide range of materials and material behaviors, as shown in the table
• Chapter 3 will explain how material data can be created or retrieved from libraries using Engineering Data
• The actual material models available for Explicit Dynamics analyses are presented at length in Chapter 6
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Training Manual Basic Formulation
• Models available for Explicit Dynamics
– Chapter 9 will cover these material models in more detail