ansys workbench-chapter14
TRANSCRIPT
Chapter 14 Nonlinear Materials 1
Chapter 14Nonlinear Materials14.1 Basics of Nonlinear Materials
14.2 Step-by-Step: Belleville Washer
14.3 Step-by-Step: Planar Seal
14.4 Review
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 2
Section 14.1Basics of Nonlinear Materials
Key Concepts
• Linear versus Nonlinear Materials
• Elasticity
• Linear Elasticity
• Hyperelasticity
• Plasticity
• Plasticity
• Yield Criteria
• Hardening Rules
• Plasticity Models
• Hyperelasticity
• Required Test Data
• Strain Energy Functions
• Hyperelasticity Models
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 3
Linear/Nonlinear Materials
• When the stress-stain relation of a material
is linear, it is called a linear material,
otherwise the material is called a nonlinear
material.
• For a linear material, the stress-strain
relation is expressed by Hooke's law, in
which two independent material parameters
are needed to completely define the
material.
• Orthotropic and anisotropic linear elasticity
are also available in <Workbench>.St
ress
(Fo
rce/
Are
a)
Strain (Dimensionless)
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 4
Elastic/Plastic Materials
• If the strain is totally recovered after
release of the stress, the behavior is
called elasticity.
• On the other hand, if the strain is not
totally recoverable (i.e., there is no
residual strain after release of the
stress), the behavior is called plasticity
and the residual strain is called the
plastic strain.
Stre
ss (
Forc
e/A
rea)
Strain (Dimensionless)
Stre
ss (
Forc
e/A
rea)
Strain (Dimensionless)
[1] Elastic material.
[2] Plastic material.
[3] Plastic strain.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 5
Stre
ss
Strain
Hysteresis
• The term hysteresis is used for the energy
loss in a material during stressing and
unstressing.
• Most of materials have more-or-less hysteresis
behavior. However, as long as it is small
enough, we may neglect the hysteresis
behavior.
Stre
ssStrain
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 6
Stre
ss (
Forc
e/A
rea)
Strain (Dimensionless)
Hyperelastic material.
Hyperelasticity
• Nonlinear non-hysteresis elasticity are characterized
by that the stressing curve and the unstressing curve
are coincident: the energy is conserved in the cycles.
• Challenge of implementing nonlinear elastic material
models comes from that the strain may be as large
as 100% or even 200%, such as rubber under
stretching or compression.
• Additional consideration is that, under such large
strains, the stretching and compression behaviors
may not be described by the same parameters.
• This kind of super-large deformation elasticity is
given a special name: hyperelasticity.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 7
PLASTICITY
Idealized Stress-Strain Curve
Stre
ss (
Forc
e/A
rea)
Strain (Dimensionless)
[1] Idealized stress-strain
curve.
[2] Initial yield point (or
elastic limit).
[3] The stress-strain relation is assumed linear
before Yield point, and the
initial slope is the Young's modulus.
[4] When the stress is released,
the strain decreases with a
slope equal to the Young's modulus.
• Plasticity behavior typically occurs in ductile
metals subject to large deformation. Plastic strain
results from slips between planes of atoms due to
shear stresses. This dislocation deformation is a
rearrangement of atoms in the crystal structure.
• A stress-strain curve is not sufficient to fully
define a plasticity behavior. There are two
additional characteristics that must be described: a
yield criterion and a hardening rule.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 8
Yield Criteria
• <Workbench> uses von Mises criterion as the yield criterion, that is, a stress state reaches yield state when the von Mises stress
σe is equal to the current
uniaxial yield strength σ y′ , or
12
σ1−σ2( )2 + σ2 −σ3( )2 + σ3 −σ1( )2⎡⎣⎢
⎤⎦⎥=σ y
′
• The yielding initially occurs when σ y′ =σ y , and the "current" uniaxial yield
strength σ y′ may change subsequently.
• If the stress state is inside the cylinder, no yielding occurs. If the stress state is on
the surface, yielding occurs. No stress state can exist outside the yield surface.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 9
σ
1
σ
2
σ
3
σ
1= σ
2= σ
3
This is a von Mises yield surface, which
is a cylindrical surface aligned with the
axis σ
1= σ
2= σ
3 and with a radius of
2σ
y′ , where
σ
y′ is the current yield
strength.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 10
Hardening Rules
• If the stress state is on the yield surface and the stress state continues to "push" the
yield surface outward, the size (radius) or the location of the yield surface will
change. The rule that describes how the yield surface changes its size or location is
called a hardening rule.
• Kinematic hardening assumes that, when a stress state continues to "push" a yield
surface outward, the yield surface will change its location, according to the "push
direction," but preserve the size of the yield surface.
• Isotropic hardening assumes that, when a stress state continues to "push" a yield
surface, the yield surface will expand its size, but preserve the axis of the yield
surface.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 11
σ
y 2σ
y
Stre
ss
Strain
[1] Kinematic hardening assumes that the difference
between tensile yield strength and the
compressive yield strength
remains a constant of 2σ
y.
σ
y′
σ
y′St
ress
Strain
[2] Isotropic hardening assumes that the tensile yield strength and the
compressive yield strength remain equal in
magnitude.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 12
[2] To complete a description of plasticity
model, you must include its linear elastic properties.
Plasticity Models in Workbench
[1] Currently, <Workbench>
provides six plasticity models.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 13
HYPERELASTICITY
Test Data Needed for Hyperelasticity
• In plasticity or linear elasticity, we use a stress-strain curve to describe its
behavior, and the stress-strain curve is usually obtained by a tensile test. Since only
tension behavior is investigated, other behaviors (compressive, shearing, and
volumetric) must be drawn from the tensile test data.
• When the strain is large, all the moduli (tensile, compressive, shear, and bulk) can
not assume simple relations.
• Therefore, to describe hyperelasticity behavior, we need following test data: (a) a
set of uniaxial tensile test data, (b) a set of uniaxial compressive test data, (c) a set
of shear test data, and (d) a set of volumetric test data if the material is
compressible.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 14
• It is possible that a set of test data is obtained by superposing two sets of other test
data. For example, the set of uniaxial compressive test data can be obtained by adding a
set of hydrostatic compressive test data to a set of equibiaxial tensile test data.
[1] Uniaxial compressive test.
[2] Equibiaxial tensile test.
[3] Hydrostatic compressive test.
= +
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 15
0
60
120
180
240
300
0 0.2 0.5 0.7
Stre
ss (
psi)
Strain (Dimensionless)
[1] Uniaxial test data.
[2] Equibiaxial test
data.
[3] Shear test data.
Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 16
Hyperelasticity Models in Workbench
Chapter 14 Nonlinear Materials Section 14.2 Belleville Washer 17
Section 14.2Belleville Washer
Problem Description
250
260
270
280
0 0.001 0.002 0.003 0.004
Stre
ss (
MPa
)
Plastic Strain (Dimensionless)
Stress-strain curve of the
steel in this case.
Chapter 14 Nonlinear Materials Section 14.2 Belleville Washer 18
40 mm
22 mm
The Belleville washer is made of steel, with
thickness of 1.0 mm.
• We will compress the Belleville
spring by 1.0 mm and then
release it completely.
• A force-displacement curve will
also be plotted.
• We will examine the residual
stress after the spring is
completely released.
Chapter 14 Nonlinear Materials Section 14.2 Belleville Washer 19
-80
-60
-40
-20
0
20
40
60
80
0 0.2 0.4 0.6 0.8 1.0
Com
pres
sive
For
ce (
N)
Displacement (mm)
Force-versus-Displacement Curve
[1] The curve is quite different
between loading and unloading.[3] Let's explore the
residual stress at this point when the external
force is completely released.
[2] There is no practice use of this section. It is the force required to pull the spring back to its original position.
Chapter 14 Nonlinear Materials Section 14.2 Belleville Washer 20
Residual Stress
[1] Residual equivalent stress.
[2] Residual hoop stress. Note that the top surface is
dominated by tension, while the bottom surface is
dominated by compression.
Chapter 14 Nonlinear Materials Section 14.3 Planar Seal 21
Section 14.3Planar Seal
Problem Description
0
40
80
120
160
200
0 0.1 0.2 0.3
Stre
ss (
psi)
Engineering Strain (Dimensionless)
[1] Uniaxial test.
[2] Biaxial test.
[3] Shear test.• The seal is used in the door of a
refrigerator. The seal is a long
strip, and we will model it as a
plane strain problem.
Chapter 14 Nonlinear Materials Section 14.3 Planar Seal 22
.800
1.100
.333 .500
.133
.133
.867 R.150
R.150
R.200
R.200
R.050
R.050
Unit: in.
[2] Steel plate.
[1] Rubber seal.
[3] Steel plate.
[4] The upper plate is displaced 0.85" downward.
Chapter 14 Nonlinear Materials Section 14.3 Planar Seal 23
Results
Chapter 14 Nonlinear Materials Section 14.3 Planar Seal 24
A force-versus-displacement curve. Note that the force unit should
be read lbf/in instead of lbf.