explicit dynamics chapter 9 material models

7/15/2019 Explicit Dynamics Chapter 9 Material Models

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1-1ANSYS, Inc. Proprietary

2009 ANSYS, Inc. All rights reserved.February 27, 2009

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Chapter 9

Material Models

ANSYS Explicit Dynamics


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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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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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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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Training Manual

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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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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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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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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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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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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Training Manual

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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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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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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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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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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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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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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Johnson-Holmquist Strength Segmented

Example: Penetrator dwell

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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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Training Manual

Impact onto plain concrete

Impact onto reinforced concrete

Brittle / Granular

RHT Concrete Strength

Examples

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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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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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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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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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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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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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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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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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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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Training ManualFailure Crack Softening Failure

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:


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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.


Must be used in conjunction with a Plasticity model

explicit dynamics chapter 9 material models

Documents

material models1

formulation of material

single material model

stress tensor

selection of models

stress deformation

stress deviators

stress state