Explicit Dynamics Chapter 9 Material Models

Chapter 9 Material Models

ANSYS Explicit Dynamics

ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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February 27, 2009 Inventory #002665

Material Models

Material Behavior Under Dynamic Loading

Training Manual

• 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 choose based on the material(s) present in your simulation ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Modeling Provided By Engineering Data


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Training Manual


Material Models

Material Deformation

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 materials can only sustain limited amount of stress / deformation before they break / crack / cavitate (fluids). Change in Volume


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Change in Shape


Material Models

Principal Stresses

Training Manual

• A stress state in 3D can be described by a tensor with six stress components – 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 the eigenvectors of the stress tensor, the stress tensor is represented by a diagonal matrix

where σ1, σ2 , and σ3, are the principal stresses (eigenvalues).

• The principal stresses may be combined to form the first, second and third stress invariants, respectively.

• Because of its simplicity, working and thinking in the principal coordinate system is often used in the formulation of material models. ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Elastic Response

Training Manual

• For linear elasticity, stresses are given by Hooke’s 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 :

where P is the pressure and si are the stress deviators

• Then :


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Material Models

Non-linear Response

Training Manual

• Many applications involve stresses considerably beyond the elastic limit and so require more complex material models Hooke’s Law

Generalized Non-Linear Response

Equation of State

Strength Model

Failure Model ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

σi (max,min) = f 1-7


Material Models

Models Available for Explicit Dynamics


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Training Manual


Material Models

Elastic Constants

Training Manual

Shear Modulus G

Young’s Modulus E

Shear Modulus Young’s Modulus Shear Modulus Poisson’s Ratio

2G (1 + n)

Shear Modulus Bulk Modulus

9KG 3K + G

Young’s Modulus Poisson’s Ratio

E 2 (1+ n)

Young’s Modulus Bulk Modulus

3EK 9K - E

Poisson’s Ratio Bulk Modulus

3K (1 - 2n) 2 (1 + n)


Poisson’s Ratio n

Bulk Modulus K

E - 2G 2G

GE 3 (3G - E) 2G (1 + n) 3 (1 - 2n)

3K - 2G 2 (3K + G) E 3 (1 - 2n) 3K - E 6K

3K (1 - 2n)

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Material Models

Physical and Thermal Properties

Training Manual

• Density – All material must have a valid density 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 models that include thermal softening • This property is automatically included in thermal softening models


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Material Models

Linear Elastic

Training Manual

• Isotropic Elasticity – Used to define linear elastic material behavior • suitable for most materials subjected to low compressions. – Properties defined • Young’s Modulus (E) • Poisson’s Ratio (ν) – 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Linear Elastic

Training Manual

• Orthotropic Elasticity – Used to define linear orthotropic elastic material behavior • suitable for most orthotropic materials subjected to low compressions. – Properties defined • Young’s Modulii (Ex, Ey, Ez) • Poisson’s Ratios (νxy, νyz, νxz) • Shear Modulii (Gxy, Gyz, Gxz) – Temperature dependence of the properties is not available for explicit dynamics


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Material Models

Linear Elastic

Training Manual

• 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, G0 and a Viscoelastic Decay Constant β.

– 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Linear Elastic

Training Manual

• Viscoelastic s = Constant

e = Constant

Strain

Stress

Time

Time

Creep

Stress Relaxation ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Hyperelastic

Training Manual

• Several forms of strain energy potential (Ψ) are provided for the simulation of nearly incompressible hyperelastic materials. •

• Forms are generally applicable over different ranges of strain. 6.00

Tensile tests on vulcanised rubber 5.00 Mooney-Rivlin Arruda-Boyce

Eng. Stress (MPa)

4.00

Ogden Treloar Experiments

3.00

2.00

1.00

0.00 0

1

2

3

4

5

6

7

8

Eng. Strain

• Need to verify the applicability of the model chosen prior to use. • Currently hyperelastic materials may only be used for solid elements ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Hyperelastic

Training Manual

Examples of Hyperelasticity


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Material Models

Plasticity

Training Manual

• If a material is loaded elastically and subsequently unloaded, all the distortion energy is recovered 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 Prandtl–Reuss yield criterion) . This states that the local yield condition 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 the yield stress itself is a function of pressure (as is the case for some of the strength models). ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity

Training Manual

• If an incremental change in the stresses violates the Von Mises criterion then each of the principal stress deviators must be adjusted such that the criterion is satisfied. • If a new stress state n + 1 is calculated from a state n and found to fall outside the yield surface, it is brought back to the yield surface along a line normal to the yield surface by multiplying each of the stress deviators by the factor

• By adjusting the stresses perpendicular to the yield circle only the plastic components of the stresses are affected.

• Effects such as work hardening, strain rate hardening, thermal softening, e.t.c. can be considered by making Y a dynamic function of these ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity

Training Manual

• Bilinear Isotropic / Kinematic Hardening – Used to define the yield stress (Y) as a linear function of plastic strain, εp

– 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) ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity

Training Manual

• Isotropic vs Kinematic Hardening σ2

σ2

Current Yield surface

σ1

σ1

Initial Yield surface

Isentropic Hardening (σ3 = 0)


Kinematic Hardening (σ3 = 0)

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Material Models

Plasticity

Training Manual

• Multilinear Isotropic / Kinematic Hardening – Used to define the yield stress (Y) as a piecewise linear function of plastic strain, εp

– 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity

Training Manual

• Johnson Cook Strength – Used to model materials, typically metals, subjected to large strains, high strain rates and high temperatures. • Defines the yield stress, Y, as a function of strain, strain rate and temperature

εp = effective plastic strain εp* = 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity

Training Manual

Effects of Strain Hardening (Johnson-Cook Model) Hypervelocity Impact • 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


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Material Models

Plasticity

Training Manual

• Cowper Symonds Strength – Used to define the yield strength of isotropic strain hardening, strain rate dependant materials.

• Hardening term is same as that used in the Johnson Cook 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 frequency oscillations that are sometimes observed in the yield surface under high strain rates. A first order rate correction is applied by default. – Strain rate properties should be input assuming that the units of strain rate are 1/second. ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity • Steinberg Guinan Strength

Training Manual

– Computes the shear modulus and yield strength as functions of effective plastic strain, pressure and internal energy (temperature) – Fits experimental data on shock-induced free surface velocities – Yield Stress and Shear modulus increase with increasing pressure and decreases with increasing temperature – Yield stress reaches a maximum value which is subsequently strain rate independent.

subject to Y0 [1 + βε]n ≤ Ymax ε = effective plastic strain t = temperature (degrees K) η = compression = v0 / v Primed parameters (with subscripts P and τ) are derivatives with respect to pressure and temperature

– Constants for 14 metals in the library. ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Plasticity

Training Manual

• Zerilli Armstrong Strength – Used to model materials subjected to large strains, high strain rates and high temperatures. – Based on dislocation dynamics. • Applicable to a wide range of bcc (body centered cubic) and fcc (face centered cubic) metals.

bcc

fcc • 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Brittle / Granular

Training Manual

• 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 strengths in uniaxial compresion and tension

• Piecewise – Yield stress is a piecewise linear function of pressure


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Material Models

Brittle / Granular

Training Manual

• 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 (metallic plasticity)

– 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Brittle / Granular

Training Manual

• Johnson-Holmquist Strength Continuous (JH2) – Strength is modeled as smoothly varying functions of intact 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Brittle / Granular

Training Manual

• 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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Material Models

Brittle / Granular

Training Manual

• Johnson-Holmquist Strength Segmented – Example: Penetrator dwell


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Material Models

Brittle / Granular

Training Manual

• 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 such as rock and ceramic. – Combined plasticity and shear damage model in which the deviatoric stress in the material is limited by a generalised failure surface of the form:

– 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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Material Models

Brittle / Granular

Training Manual

• RHT Concrete Strength – Examples

Impact onto plain concrete Impact onto reinforced concrete ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Brittle / Granular

Training Manual

• MO Granular – Extension of the Drucker-Prager model • Takes into account effects associated with granular materials such as powders, soil, and sand. • In addition to pressure hardening, the model also represents density hardening and variations in the shear modulus with density.

– Yield stress has two components, one dependent on the density and one dependent of the pressure

Where σY , σp , and σρ denote the total yield stress, the pressure yield stress and the density yield stress respectively.

– The un-load / re-load slope is defined by the shear modulus which is defined as a function of the density of the material at zero pressure – The yield stress is defined by a yield stress – pressure and a yield stress – density curve with up to 10 points in each curve.

– The shear modulus is defined by a shear modulus – density curve with up to 10 points. • All three curves must be defined.


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Material Models

Equation of State

Training Manual

• 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 Modulus • 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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Material Models

Equation Of State

Training Manual

• Mei-Gruneisen form of Equation of State – Covers entire (p,v=1/ρ,e) space using a 1st order Taylor expansion from 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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Material Models

Equation of State

Training Manual

• Polynomial EOS – A Mie-Gruneisen form of equation of state that expresses pressure as a polynomial function of compression (density)

μ > 0 (compression):

μ < 0 (tension):

– Commonly found in early papers • Shock EOS is more commonly used today ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Equation of State

Training Manual

• 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 ρ (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 using

G = 2s1 - 1


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Material Models

Equation of State

Training Manual

• 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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Material Models

Porosity

Training Manual

• 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Porosity

Training Manual

• 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. – Young’s Modulus, E, is used for unloading / re-loading – Maximum Tensile Stress provides a tension cutoff


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Material Models

Porosity

Training Manual

• Compaction EOS Linear – Plastic compaction path is defined as a piecewise linear function of Pressure vs Density – The elastic unloading / reloading path is defined via a piecewise linear function of Sound 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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Material Models

Porosity

Training Manual

• Compaction EOS Non-Linear – Plastic compaction path is defined as a piecewise linear function of Pressure vs Density – Elastic unloading / reloading path is defined via a piecewise linear function of Bulk 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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Material Models

Porosity

Training Manual

• P-alpha EOS – Crushable Foam and Compaction EOS give good results for low stress levels and for materials with low initial porosities, but they may not do well for highly porous materials over a wide stress range – Herrmann’s P- alpha EOS is a phenomenological model which gives the correct behavior at high stresses but at the same time provides a reasonably detailed description of the compaction process 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 at identical conditions of pressure and temperature. • Solid EOS

• Porous EOS

where V is the specific volume of the porous material and Vs is the specific volume of the solid material

• α = g (p,e) ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

(fitted to experimental data)

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Material Models

Failure

Training Manual

• 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 strength characteristics 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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Material Models

Failure

Training Manual

• 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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Material Models

Failure

Training Manual

• 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Failure

Training Manual

• 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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Material Models

Failure

Training Manual

• Stochastic Failure – Example: Fragmenting Ring


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Material Models

Failure

Training Manual

• Tensile Pressure Failure – Used to represent dynamic spall (or cavitation) – Tensile pressure is limited by

If the pressure (P) becomes less than the Maximum Tensile Pressure (Pmin), failure occurs • Material instantaneously fails.

– If Material also uses damage evolution, the Maximum 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 to invoke fracture energy based softening ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Material Models

Failure

Training Manual

• Crack Softening Failure –

Fracture energy based damage model which provides a gradual reduction in the ability 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 possible principal 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 Kf2 = EGf

–

After failure initiation, a maximum principal tensile stress failure surface is defined to limit the maximum principal tensile stress in the cell and a Flow Rule is used to return 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 or brittle strength model.


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Material Models

Failure

Training Manual

• Example : Impact on Ceramic Target – 1449m/s impact of a 6.35mm diameter steel ball on a ceramic target – Johnson-Holmquist Strength model used in conjunction with Crack Softening

Simulation

Experiment (Hazell)


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Material Models

Failure

Training Manual

• Johnson Cook Failure – Used to model ductile failure of materials experiencing large pressures, strain rates and temperatures. – Consists of three independent terms that define the dynamic fracture strain (εf) as a function of pressure, strain rate and temperature:

– Can only be applied to solid bodies.


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Material Models

Failure

Training Manual

• 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 density c is the bulk sound speed Y is the yield stress ec is a Critical Strain Value

– If maximum principal tensile stress exceeds the critical spall stress (S), instantaneous failure of the element is initiated.

– 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 ANSYS, Inc. Proprietary © 2009 ANSYS, Inc. All rights reserved.

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Explicit Dynamics Chapter 9 Material Models

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