Ansys Workbench-Chapter14

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


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)


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)


Stre

ss (

Forc

e/A

rea)


[1] Elastic material.

[2] Plastic material.

[3] Plastic strain.


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


Stre

ss (

Forc

e/A

rea)


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.


PLASTICITY

Idealized Stress-Strain Curve

Stre

ss (

Forc

e/A

rea)


[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.


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.


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.


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.


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.


[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.


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.


• 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.

= +


0

60

120

180

240

300

0 0.2 0.5 0.7

Stre

ss (

psi)


[1] Uniaxial test data.

[2] Equibiaxial test

data.

[3] Shear test data.


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.


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.


-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.


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.


.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.


Results


A force-versus-displacement curve. Note that the force unit should

be read lbf/in instead of lbf.

Ansys Workbench-Chapter14

Engineering

Transcript of Ansys Workbench-Chapter14