This chapter discusses nonlinear materials and their behavior. It covers the basics of linear and nonlinear materials, elasticity, plasticity, hyperelasticity, and the models available in Workbench. It then provides two step-by-step examples: a Belleville washer made of steel and subjected to compression and release, and a rubber seal between two steel plates, with the top plate displaced downward. Force-displacement curves are generated for each example and residual stresses examined.

1. Chapter 14 Nonlinear Materials 1
Chapter 14
Nonlinear Materials
14.1 Basics of Nonlinear Materials
14.2 Step-by-Step: Belleville Washer
14.3 Step-by-Step: Planar Seal
14.4 Review

2. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 2
Section 14.1
Basics 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

3. 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>. Stress(Force/Area)
Strain (Dimensionless)

4. 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.
Stress(Force/Area)
Strain (Dimensionless)
Stress(Force/Area)
Strain (Dimensionless)
[1] Elastic
material.
[2] Plastic
material.
[3] Plastic strain.

5. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 5
Stress
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.
Stress Strain

6. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 6
Stress(Force/Area)
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.

7. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 7
PLASTICITY
Idealized Stress-Strain Curve
Stress(Force/Area)
Strain (Dimensionless)
[1] Idealized
stress-strain
curve.
[2] Initial yield
point (or
elastic limit).
[3] The stress-
strain relation is
assumed linear
beforeYield
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.

8. 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 σy
′ , or
1
2
σ1 −σ2( )
2
+ σ2 −σ3( )
2
+ σ3 −σ1( )
2⎡
⎣
⎢
⎤
⎦
⎥ = σy
′
• The yielding initially occurs when σy
′ = σy , and the "current" uniaxial yield
strength σy
′ 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.

9. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 9
σ1
σ2
σ3
σ1
= σ2
= σ3
This is a von Mises yield surface, which
is a cylindrical surface aligned with the
axis σ1
= σ2
= σ3
and with a radius of
2σy
′ , where σy
′ is the current yield
strength.

10. 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.

11. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 11
σy
2σy
Stress
Strain
[1] Kinematic hardening
assumes that the difference
between tensile yield
strength and the
compressive yield strength
remains a constant of 2σy
.
σy
′
σy
′
Stress
Strain
[2] Isotropic hardening
assumes that the tensile
yield strength and the
compressive yield strength
remain equal in
magnitude.

12. 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.

13. 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.

14. 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.
= +

15. 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
Stress(psi)
Strain (Dimensionless)
[1] Uniaxial
test data.
[2]
Equibiaxial test
data.
[3] Shear test
data.

16. Chapter 14 Nonlinear Materials Section 14.1 Basics of Nonlinear Materials 16
Hyperelasticity
Models in
Workbench

17. Chapter 14 Nonlinear Materials Section 14.2 Belleville Washer 17
Section 14.2
Belleville Washer
Problem Description
250
260
270
280
0 0.001 0.002 0.003 0.004
Stress(MPa)
Plastic Strain (Dimensionless)
Stress-strain
curve of the
steel in this case.

18. 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.

19. 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
CompressiveForce(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.

20. 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.

21. Chapter 14 Nonlinear Materials Section 14.3 Planar Seal 21
Section 14.3
Planar Seal
Problem Description
0
40
80
120
160
200
0 0.1 0.2 0.3
Stress(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.

22. 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.

23. Chapter 14 Nonlinear Materials Section 14.3 Planar Seal 23
Results

24. 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.