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Chapter 9
Material Models
ANSYS Explicit Dynamics
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Material Models
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Training ManualMaterial Behavior Under Dynamic Loading
In general, materials have a complex response to dynamic loading
The following phenomena may need to be modelled
Non-linear pressure response
Strain hardening
Strain rate hardening
Thermal softening Compaction (porous materials)
Orthotropic behavior (e.g. composites)
Crushing damage (e.g. ceramics, glass, geological materials, concrete)
Chemical energy deposition (e.g. explosives)
Tensile failure
Phase changes (solid-liquid-gas)
No single material model incorporates all of these effects
Engineering Data offers a selection of models from which you can choosebased on the material(s) present in your simulation
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Training ManualModeling Provided By Engineering Data
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Training Manual
Material deformation can be split into two independent parts
Volumetric Response - changes in volume (pressure)
Equation of state(EOS)
Deviatoric Response - changes in shape
Strength model
Also, it is often necessary to specify a Failure model as materialscan only sustain limited amount of stress / deformation before theybreak / crack / cavitate (fluids).
Material Deformation
Change in
VolumeChange in
Shape
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Training ManualPrincipal Stresses
A stress state in 3D can be described by a tensor with six stresscomponents
Components depend on the orientation of the coordinate system used. The stress tensor itself is a physical quantity
Independent of the coordinate system used
When the coordinate system is chosen to coincide with theeigenvectors of the stress tensor, the stress tensor is represented bya diagonal matrix
where 1, 2 , and 3, are the principal stresses (eigenvalues).
The principal stresses may be combined to form the first, second andthird stress invariants, respectively.
Because of its simplicity, working and thinking in the principalcoordinate system is often used in the formulation of material models.
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Training Manual
Elastic Response
For linear elasticity, stresses are given by Hookes law :
where l and G are the Lame constants (G is also known as the Shear Modulus) The principal stresses can be decomposed into a hydrostatic and
a deviatoric component :
whereP
is the pressure ands
i are the stress deviators
Then :
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Training Manual
Hookes LawGeneralized Non-Linear
Response
Equation of State
Strength Model
Non-linear Response
Many applications involve stresses considerably beyond the elastic
limit and so require more complex material models
Failure Model i(max,min) = f
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Training ManualModels Available for Explicit Dynamics
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Training ManualElastic Constants
ShearModulus G
YoungsModulus E
PoissonsRatio n
BulkModulus K
Shear Modulus
Youngs Modulus
Shear Modulus
Poissons RatioShear Modulus
Bulk Modulus
Youngs Modulus
Poissons Ratio
Youngs ModulusBulk Modulus
Poissons Ratio
Bulk Modulus
E - 2G
2G
GE
3 (3G - E)
2G (1 + n)2G (1 + n)
3 (1 - 2n)9KG
3K + G
3K - 2G
2 (3K + G)
E
2 (1+ n)
E
3 (1 - 2n)
3EK9K - E
3K - E6K
3K (1 - 2n)
2 (1 + n)3K (1 - 2n)
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Training ManualPhysical and Thermal Properties
Density
All material must have a validdensity defined for Explicit
Dynamics simulations.
The density property defines the
initial Mass / unit volume of a
material at time zero
This property is automatically
included in all models
Specific Heat
This is required to calculate the
temperature used in material modelsthat include thermal softening
This property is automatically
included in thermal softening models
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Training ManualLinear Elastic
Isotropic Elasticity
Used to define linear elastic materialbehavior
suitable for most materials subjected to
low compressions.
Properties defined
Youngs Modulus (E)
Poissons Ratio ()
From the defined properties, Bulk modulus
and Shear modulus are derived for use in
the material solutions.
Temperature dependence of the linear
elastic properties is not available for explicit
dynamics
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Training ManualLinear Elastic
Orthotropic Elasticity
Used to define linear orthotropic elasticmaterial behavior
suitable for most orthotropic materials
subjected to low compressions.
Properties defined
Youngs Modulii (Ex, Ey, Ez)
Poissons Ratios (xy, yz, xz)
Shear Modulii (Gxy, Gyz, Gxz)
Temperature dependence of the propertiesis not available for explicit dynamics
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Training ManualLinear Elastic
Viscoelastic
Represents strain rate dependent elastic behavior
Long term behavior is described by a Long Term
Shear Modulus, G.
Specified via an Isotropic Elasticity model or
Equation OF State
Viscoelastic behavior is introduced via an
Instantaneous Shear Modulus, G0and a
Viscoelastic Decay Constant .
The deviatoric viscoelastic stress at time n+1 is
calculated from the viscoelastic stress at time n
and the shear strain increments at time n:
Deviatoric viscoelastic stress is added to the
elastic stress to give the total stress
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Training Manual
Stress
Time
Strain
Time
s = Constant= Constant
Stress Relaxation Creep
Viscoelastic
Linear Elastic
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Training ManualHyperelastic
Several forms of strain energy potential () areprovided for the simulation of nearly
incompressible hyperelastic materials.
Forms are generally applicable over differentranges of strain.
Need to verify the applicability of the modelchosen prior to use.
Currently hyperelastic materials may only beused for solid elements
0.00
1.00
2.00
3.00
4.00
5.00
6.00
0 1 2 3 4 5 6 7 8
Eng. Strain
Eng.
Stress(MPa)
Mooney-Rivlin
Arruda-BoyceOgden
Treloar Experiments
Tensile tests on vulcanised rubber
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Examples of Hyperelasticity
Hyperelastic
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Training ManualPlasticity
If a material is loaded elastically and subsequently unloaded, all the distortion energy isrecovered and the material reverts to its initial configuration.
If the distortion is too great a material will reach its elastic limit and begin to distort plastically.
In Explicit Dynamics, plastic deformation is computed by reference to the Von Mises yield
criterion (also known as PrandtlReuss yield criterion). This states that the local yieldcondition is
where Y is the yield stress in simple tension. It can be also written as
or(since )
Thus the onset of yielding (plastic flow), is purely a function of the deviatoric stresses(distortion) and does not depend upon the value of the local hydrostatic pressure unless theyield stress itself is a function of pressure (as is the case for some of the strength models).
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Training ManualPlasticity
If an incremental change in the stressesviolates the Von Mises criterion theneach of the principal stress deviators
must be adjusted such that the criterionis satisfied.
If a new stress state n + 1 is calculatedfrom a state n and found to fall outsidethe yield surface, it is brought back to theyield surface along a line normal to theyield surface by multiplying each of the
stress deviators by the factor
By adjusting the stresses perpendicular
to the yield circle only the plasticcomponents of the stresses are affected.
Effects such as work hardening, strainrate hardening, thermal softening, e.t.c.can be considered by making Yadynamic function of these
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Training ManualPlasticity
Bilinear Isotropic / Kinematic Hardening
Used to define the yield stress (Y
) as a linear functionof plastic strain, p
Properties defined
Yield Strength (Y0)
Tangent Modulus (A)
Isotropic Hardening
Total stress range is twice the maximum yield stress, Y
Kinematic Hardening
Total stress range is twice the starting yield stress, Y0
Models Bauschinger effect
Often required to accurately predict response of thin
structure (shells)
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Training ManualPlasticity
Isotropic vs Kinematic Hardening
1
2
1
2
Initial Yield surface
Current Yield surface
Isentropic Hardening (3 = 0) Kinematic Hardening (3 = 0)
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Training ManualPlasticity
Multilinear Isotropic / Kinematic Hardening
Used to define the yield stress (Y) as apiecewise linear function of plastic strain, p
Properties defined
Up to ten stress-strain pairs
Isotropic Hardening Total stress range is twice the maximum yield
stress, Y
Kinematic Hardening
Can only be used with solid elements
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Training ManualPlasticity
Johnson Cook Strength
Used to model materials, typically metals, subjected tolarge strains, high strain rates and high temperatures.
Defines the yield stress, Y, as a function of strain, strain rate
and temperature
p = effective plastic strain
p*= normalized effective plastic strain rate (1.0 sec-1)
TH = homologous temperature = (T - Troom) / (Tmelt - Troom)
The plastic flow algorithm used with this model has an
option to reduce high frequency oscillations that are
sometimes observed in the yield surface under high
strain rates. A first order rate correction is applied by
default.
A specific heat capacity must also be defined to enable
the calculation of temperature for thermal softening
effects
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Normal impact of tungsten
sphere on thick steel plate
at 10 kms-1
Lagrange Parts used with
erosion
Johnson-Cook strength
model used to model
effects of strain hardening,
strain-rate hardening and
thermal softening
including melting
Effects of Strain Hardening (Johnson-Cook Model)
Hypervelocity Impact
Plasticity
M i l M d l
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Training ManualPlasticity
Cowper Symonds Strength
Used to define the yield strength of isotropicstrain hardening, strain rate dependant materials.
Hardening term is same as that used in the JohnsonCook Model
Strain rate dependent term has different form
No thermal softening term
The plastic flow algorithm used with this model
has an option to reduce high frequencyoscillations that are sometimes observed in theyield surface under high strain rates. A first orderrate correction is applied by default.
Strain rate properties should be input assumingthat the units of strain rate are 1/second.
M t i l M d l
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Training ManualPlasticity
Steinberg Guinan Strength
Computes the shear modulus and yield strength as functionsof effective plastic strain, pressure and internal energy(temperature)
Fits experimental data on shock-induced free surfacevelocities
Yield Stress and Shear modulus increase with increasingpressure and decreases with increasing temperature
Yield stress reaches a maximum value which is subsequentlystrain rate independent.
subject to Y0 [1 + ]
n
Ymax
= effective plastic strain
t = temperature (degrees K)
= compression = v0 / v
Primed parameters (with subscriptsPand ) are derivatives
with respect to pressure and temperature
Constants for 14 metals in the library.
M t i l M d l
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Training ManualPlasticity
Zerilli Armstrong Strength
Used to model materials subjected to large strains, high strain ratesand high temperatures.
Based on dislocation dynamics.
Applicable to a wide range ofbcc (body centered cubic) and fcc (face
centered cubic) metals.
For fcc metals (e.g. Copper, Nickel, Platinum),
set C1 = 0
For bcc metals (e.g. Iron, Chromium, Tungsten,
Vanadium), set C2 = 0
A specific heat capacity must also be defined
to enable the calculation of temperature for
thermal softening effects
bcc
fcc
M t i l M d l
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Training ManualBrittle / Granular
Drucker-Prager Strength
Yield stress is a function of Pressure
Used for dry soils, rocks, concrete and
ceramics where cohesion and compaction
cause increasing resistance to shear up to a
limiting value of the yield stress.
Three forms
Linear
Original Drucker-Prager model
Stassi
Constructed from yield strengthsin uniaxial compresion and tension
Piecewise
Yield stress is a piecewise linear
function of pressure
M t i l M d l
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Training ManualBrittle / Granular
Johnson-Holmquist Strength
Use to model brittle materials (glass,ceramics) subjected to large pressures,
shear strain and high strain rates
Combined plasticity and damage model
Yielding is based on micro-crack growth
instead of dislocation movement (metallicplasticity)
Fully cracked material still retains some
strength in compression due to frictional
effects in crushed grains
Yield reduced from intact value to
fractured value via a Damage function
Damage accumulates due to effective
plastic strain
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Training ManualBrittle / Granular
Johnson-Holmquist Strength Continuous (JH2)
Strength is modeled as smoothly varying functions ofintact strength, fractured strength, strain rate and
damage via dimensionless analytic functions
Damage is accumulated as ratio of incremental
plastic strain over a pressure-dependant fracture
strain
Two methods for application of damage
Gradual (default)
Damage is incrementally applied as it accumulates
Instantaneous Damage accumulates over time, but is only applied to failure
when it reaches 1.0
Can be used with a Linear or Polynomial Equation of
State
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Training ManualBrittle / Granular
Johnson-Holmquist Strength Segmented (JH1)
Strength is modeled using piecewise linear segments
Damage is always applied instantaneously Damage accumulates over time, but is only applied to failure
when it reaches 1.0
Can be used with a Linear or Polynomial Equation of
State
The gradual softening in the more recent continuous
model (JH2) has not been supported by experimental
data, so this earlier variant is still commonly used
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Training ManualBrittle / Granular
Johnson-Holmquist Strength Segmented
Example: Penetrator dwell
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Training ManualBrittle / Granular
RHT Concrete Strength
Advanced plasticity model for brittle materials developed by Riedel,Hiermaier and Thoma at the Ernst Mach Institute (EMI)
Models dynamic loading of concrete and other brittle materials suchas rock and ceramic.
Combined plasticity and shear damage model in which the deviatoricstress in the material is limited by a generalised failure surface of theform:
Represents the following response of geological materials
Pressure hardening
Strain hardening
Strain rate hardening in tension and compression
Third invariant dependence for compressive and tensile meridians Strain softening (shear induced damage)
Coupling of damage due to porous collapse
Input can be scaled with compressive strength, fc
Data for 35MPa and 140MPa in the distributed material library
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Impact onto plain concrete
Impact onto reinforced concrete
Brittle / Granular
RHT Concrete Strength
Examples
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Training ManualBrittle / Granular
MO Granular
Extension of the Drucker-Prager model
Takes into account effects associated with granular materials such aspowders, soil, and sand.
In addition to pressure hardening, the model also represents densityhardening and variations in the shear modulus with density.
Yield stress has two components, one dependent on the densityand one dependent of the pressure
Where Y , p , and denote the total yield stress, the pressure yieldstress and the density yield stress respectively.
The un-load / re-load slope is defined by the shear moduluswhich is defined as a function of the density of the material atzero pressure
The yield stress is defined by a yield stress pressure and ayield stress density curve with up to 10 points in each curve.
The shear modulus is defined by a shear modulus densitycurve with up to 10 points.
All three curves must be defined.
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Training ManualEquation of State
Equation of State Properties
Bulk Modulus
A bulk modulus can be used to define a linear,
energy independent equation of state
Combined with a Shear modulus property, this material
definition is equivalent to using an Isotropic Linear
Elastic, model
Shear Modulu s
A shear modulus must be used when a solid or
porous equation of state are selected.
To represent fluids, specify a small value.
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Training ManualEquation Of State
Mei-Gruneisen form of Equation of State
Covers entire (p,v=1/,e) space using a 1st
order Taylor expansionfrom a reference curve
Reference Curves
The shock Hugoniot
A standard adiabat
The 0 K isotherm The isobar p = 0
The curve e = 0
The saturation curve
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Training ManualEquation of State
Polynomial EOS
A Mie-Gruneisen form of equation of state that expresses
pressure as a polynomial function of compression
(density)
> 0 (compression):
< 0 (tension):
Commonly found in early papers
Shock EOS is more commonly used today
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Training ManualEquation of State
Shock EOS
A Mie-Gruneisen form of EOS that uses the shock
Hugoniot as a reference curve
The Rankine-Hugoniot equations for the shock jump
conditions defining a relation between any pair of the
variables (density), p (pressure), e (energy), up(particle
velocity) and Us(shock velocity).
Us - up space is used to define the Hugoniot
In many dynamic experiments, measuring up and Us, it has
been found that for most solids and many liquids over a wide
range of pressure there is an empirical linear relationship
between these two variables:
Us = C1 + S1up
Gruneisen Coefficient, G, is often approximated usingG = 2s1 - 1
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Training ManualEquation of State
Shock EOS Linear
Lets you define a linear or a quadratic relationship
Us = C1 + S1Up
Us = C1 + S1Up + S2Up2
Shock EOS Bilinear Lets you define a bilinear relationship
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Training ManualPorosity
Some materials exhibit irreversible compaction
due to pore collapse
Examples
Foam
Powders
Concrete
Soils
Porous materials are extremely effective in
attenuating shocks and mitigating impact
pressures.
Compact to solid density at relatively low stress
levels
Volume change is large
Significant amount of energy is irreversibly
absorbed
Four models are available in Explicit Dynamics
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Training ManualPorosity
Crushable Foam
Relatively simple strength model designed to represent
the crush characteristics of foam materials under impact
loading conditions (non-cyclic loading).
Must be used with Isotropic Elasticity
automatically included
Compaction curve is defined as a piecewise linear
principal stress vs volumetric strain curve.
Youngs Modulus, E, is used for unloading / re-loading
Maximum Tensile Stress provides a tension cutoff
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Training ManualPorosity
Compaction EOS Linear
Plastic compaction path is defined as a piecewiselinear function ofPressure vs Density
The elastic unloading / reloading path is defined
via a piecewise linear function ofSound Speed vs
Density
The Bulk Modulus of the material is calculated from
Model can be combined with a variety of strength
and failure models
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Training ManualPorosity
Compaction EOS Non-Linear
Plastic compaction path is defined as a piecewiselinear function ofPressure vs Density
Elastic unloading / reloading path is defined via a
piecewise linear function ofBulk Modulus vs
Density
For non-linear unloading, if the current pressure is
less than the current compaction pressure, the
pressure is obtained from the bulk modulus using:
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Training ManualPorosity
P-alpha EOS
Crushable Foam and Compaction EOS give good results for low
stress levels and for materials with low initial porosities, but theymay not do well for highly porous materials over a wide stressrange
Herrmanns P- alpha EOS is a phenomenological model whichgives the correct behavior at high stresses but at the same timeprovides a reasonably detailed description of the compactionprocess at low stress levels.
Principal assumption is that specific internal energy is the same for
a porous material as for the same material at solid density atidentical conditions of pressure and temperature.
Solid EOS
Porous EOS
whereV is the specific volume of the porous material and Vs
is thespecific volume of the solid material
= g (p,e) (fitted to experimental data)
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Training ManualFailure
Material failure has two components
Failure initiation
When specified criteria are met within a material, a
post failure response is activated
Post failure response
After failure initiation, subsequent strengthcharacteristics will change depending on the type of
failure model
Instantaneous Failure
Deviatoric stresses are immediately set to zero and remain so
Only compressive pressures are supported
Gradual Failure (Damage)
Stresses are limited by a damage evolution law
Gradual reduction in capability to carry deviatoric and / or
tensile stresses
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Training ManualFailure
Plastic Strain Failure
Models ductile failure
Failure occurs if the Effective Plastic Strain in the
material exceeds the Maximum Equivalent Plastic
Strain
Material fails instantaneously
This failure model must be used in conjunction with
a plasticity or brittle strength model
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Principal Stress / Strain Failure
Models brittle failure or ductile failure (Strain only)
Failure is based on one of two criteria
Maximum Tensile Stress / Principal Strain
Maximum Shear Stress / Shear Strain
from the maximum difference in the principal stresses / strains
Failure is initiated when either criteria is met
Material fails instantaneously
If used in conjunction with a plasticity model, deactivate
Maximum Shear Stress / Strain criteria
specify a value of +1.0e20
then shear response is handled by the plasticity model.
Crack Softening Failure can be combined with these model
for fracture energy based softening
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Stochastic Failure
Real materials have inherent microscopic flaws, which
cause failures and cracking to initiate. Stochastic Failure
reproduces this numerically by randomizing the Failure
stress or strain of a material
Can be used with most other failure models
Mott distribution is used to define the variance in failure
stress or strain. Stochastic Variance must be specified
Distribution Type
Fixed
The same random distribution is used for each Solve
Random
A new distribution is calculated for each Solve
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Stochastic Failure
Example: Fragmenting Ring
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Tensile Pressure Failure
Used to represent dynamic spall (or cavitation) Tensile pressure is limited by
If the pressure (P)becomes less than the MaximumTensile Pressure (Pmin
), failure occurs
Material instantaneously fails.
If Material also uses damage evolution, theMaximum Tensile Pressure is scaled down as the
damage, D, increases from 0.0 to 1.0
Can only be applied to solid bodies.
Can be combined with Crack Softening Failure toinvoke fracture energy based softening
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Fracture energy based damage model which provides a gradual reduction in theability of an element to carry tensile stress.
Primarily used to investigate failure of brittle materials
Applied to other materials to reduce mesh dependency effects.
Failure initiation based on any of the standard tensile failure models
On failure initiation, a linear softening slope is used to reduce the maximum possibleprincipal tensile stress in the material as a function of crack strain
Softening slope is defined as a function of the local cell size and the Fracture Energy Gf Fracture energy is related to the fracture toughness by Kf
2 = EGf
After failure initiation, a maximum principal tensile stress failure surface is defined tolimit the maximum principal tensile stress in the cell and a Flow Rule is used toreturn to this surface and accumulate the crack strain
Flow Rule: No-Bulking (Default)
Associative in -plane only
Good results for impacts onto brittle materials such as glass, ceramics and concrete
Radial Return
Non-associative in - and meridional planes
Bulking Associative Associative in - and meridional planes
Can only be used with Solid elements
Can be used in combination with any solid equation of state, plasticity model orbrittle strength model.
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Example : Impact on Ceramic Target
1449m/s impact of a 6.35mm diameter steel
ball on a ceramic target
Johnson-Holmquist Strength model used inconjunction with Crack Softening
Experiment (Hazell)
Simulation
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Johnson Cook Failure
Used to model ductile failure of materialsexperiencing large pressures, strain rates and
temperatures.
Consists of three independent terms that define
the dynamic fracture strain (f) as a function ofpressure, strain rate and temperature:
Can only be applied to solid bodies.
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Grady Spall Failure
Used to model dynamic spallation of metals under shock
loading. Critical spall stress for a ductile material is calculated
using:
r is the densityc is the bulk sound speed
Y is the yield stressec is a Critical Strain Value If maximum principal tensile stress exceeds the critical
spall stress (S), instantaneous failure of the element isinitiated.
Typical value for the Critical Strain is 0.15 for Aluminum.
Can only be applied to solid bodies.
Must be used in conjunction with a Plasticity model