Lecture 6
Material Models
Introduction to ANSYS Explicit STR
Material 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 choose based on the material(s) present in your simulation
E Modeling Provided By Engineering Data
3
Material Deformation
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 Change in Shape
Principal Stresses
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.
Elastic Response
?For linear elasticity, stresses are given by Hooke’s law :
where 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 s i are the stress deviators
?Then :
f
Non‐linear Response
? 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
σi (max,min) =
Models Available for Explicit Dynamics
AUTODYN
Equation of State
Strength Model
Failure Model
Elastic Constants
Shear Modulus Young’s Modulus
Shear Modulus Poisson’s Ratio
Shear Modulus Bulk Modulus Young’s Modulus Poisson’s Ratio
Young’s Modulus Bulk Modulus Poisson’s Ratio Bulk Modulus Shear
Modulus G
E
2 (1+ n)
3EK
9K - E
3K (1 - 2n)
2 (1 + n)
Young’s
Modulus E
2G (1 + n)
9KG
3K + G
3K (1 - 2n)
Poisson’s
Ratio n
E - 2G
2G
3K - 2G
2 (3K + G)
3K - E
6K
Bulk
Modulus K
GE
3 (3G - E)
2G (1 + n)
3 (1 - 2n)
E
3 (1 - 2n)
Physical and Thermal Properties
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
Linear Elastic
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
Linear Elastic
Orthotropic Elasticity
?Used to define linear orthotropic elastic material behavior
– suitable for most orthotropic materials subjected to low compressions.
?Properties defined
–Young’s Modulii (E x, E y, E z)
–Poisson’s Ratios (νxy, νyz, νxz)
– Shear Modulii (G xy, G yz, G xz)
?Temperature dependence of the properties is not available for explicit dynamics
Linear 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, 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
Linear Elastic
Viscoelastic
ε= Constant
Stress Strain
Time
Stress Relaxation Creep
Time
Hyperelastic
E n g . S t r e s s (M P a )
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.
Tensile tests on vulcanised rubber
Mooney-Rivlin Arruda-Boyce O gden
T r eloar Experiments
1
2
3
4 5
6
7
8
Eng. Str a in
Need to verify the applicability of the model chosen
prior to use.
Currently hyperelastic materials may only be used for
solid elements
Hyperelastic
Examples of Hyperelasticity
Plasticity
)
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).
Plasticity
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
Plasticity
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, Y
–Models Bauschinger effect
–Often required to accurately predict response of thin structure (shells)
Plasticity
Isotropic vs Kinematic Hardening
σ
2
σ
2
Current Yield surface
σ
1
σ
1 Initial Yield surface
Isotropic Hardening (σ
3 = 0) Kinematic Hardening (σ
3
= 0)