JN Reddy Nonlin Elast Plastcity: 1

Nonlinear Elasticity and Plasticity

• Nonlinear elasticity• Plasticity• Ideal plasticity and strain hardening plasticity• Stress-strain curves• Finite element models of nonlinear elasticity• Numerical examples• Small deformation theory of plasticity• Finite element formulation• Numerical examples

JN Reddy

Nonlinear Elasticity and Plasticity

Plasticity

Stress

Strain

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Plastic strain

Nonlinear Elasticity

Stress

Strain

Linear elastic

Non-linear elastic

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Nonlin Elast Plastcity: 2

JN Reddy

Ideal Plasticity and Strain Hardening Plasticity

Ideal (or Perfect) Plasticity

Stress

Strain

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Strain Hardening PlasticityStress

Strain

Plastic strain

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Nonlin Elast Plastcity: 3

JN Reddy

Ideal Plasticity and Strain Hardening Plasticity

Stress

Strain

dσp

EσY

dεdεp

dεe

Slope ET - Elastic-plastic tangent modulus

EffectiveStress, σ

Plastic Strain, εp

_Ep

σ = σ0 + Ep εp_

Nonlin Elast Plastcity: 4

JN Reddy

Stress-Strain Curves for Boron/Aluminum

Nonlin Elast Plastcity: 5

JN Reddy

Stress-Strain Curves for Graphite-Epoxy

Nonlin Elast Plastcity: 6

JN Reddy Nonlin Elast Plastcity: 7

Stress-Strain Curves for Boron-Epoxy

JN Reddy Nonlin Elast Plastcity: 8

Finite Element Models of Nonlinear Elasticity

Virtual Work Statement

Nonlinear Constitutive Equation

¾xx = E F ("xx)

0 =

Z

A

Z xb

xa

¾xx±"xx dxdA ¡Z xb

xa

f±u dx ¡ P e1 ±u(xa) ¡ P e

2 ±u(xb)

=

Z xb

xa

[EA F ("xx)±"xx ¡ f±u] dx ¡ P e1 ±u(xa) ¡ P e

2 ±u(xb)

JN Reddy Nonlin Elast Plastcity: 9

Finite Element Models of Nonlinear Elasticity

(continued)

Finite Element Model

Rei =

Z xb

xa

·EA F ("xx)

dÃei

dx¡ fÃi

¸dx ¡ P e

i

Keij =

@Rei

@uej

= EA

Z xb

xa

@F

@"xx

@"xx

@uej

dÃei

dxdx

= EA

Z xb

xa

µ@F

@"xx

¶dÃe

i

dx

dÃej

dxdx

Ramberg-Osgood Model

F ("xx) = ("xx)n

JN ReddyNonlin Elast Plastcity:

10

Numerical Examples

JN Reddy Nonlin Elast Plastcity: 11

Numerical Examples

JN Reddy Nonlin Elast Plastcity: 12

The theory of plasticity deals with an analytical description of the stress-strain relations of a deformed body after a part or all of the body has yielded.

The stress-strain relations must contain:

1. The elastic stress-strain relations.

2. The stress condition (or yield criterion) which indicates onset of yielding.

3. The stress-strain or stress-strain increment relations after the onset of plastic flow.

Small Deformation Theory of Plasticity

JN Reddy Nonlin Elast Plastcity: 13

Small Deformation Theory of Plasticity

Stress

Strain

dσp

EσY

dε

dεp

dεe

Slope ET - Elastic-plastic tangent modulus

¾ij < ¾Y linear elastic behavior

¾ij ¸ ¾Y plastic deformation (not recoverable)

JN Reddy Nonlin Elast Plastcity: 14

F (¾ij ; ·) = 0

F (J 02; J

03; ·) = 0; J 0

2 =1

2¾0

ij¾0ij ; J 0

3 =1

3¾0

ij¾0jk¾0

k`

The Tresca yield criterion:

F = 2¹¾ cos µ ¡ Y (·) = 0; ¹¾ =p

J 02

The Huber-von Mises yield criterion:

F =p

3¹¾ ¡ Y (·) = 0

Small Deformation Theory of Plasticity(continued)

General Yield Criterion

JN Reddy Nonlin Elast Plastcity: 15

Mathematical Models of the Strain Hardening Behavior

JN Reddy

Small Deformation Theory of Plasticity(continued)

Finite Element Formulation

d" = d"e + d"p; d"e =d¾

E;

d¾

d"= ET

H =d¾

d"p=

d¾d"

1 ¡ d"e

d"

=ET

1 ¡ ET

E

[Ke] =

Z xb

xa

[B]T [De][B]dx

du = hed"xx = he (d"e + d"p)

dF = Ad¾ = AeHd"p

Eep =dF

du=

AeHd"p

he (d"e + d"p)=

EAe

he

·1 ¡ E

(E + H)

¸

[Kep] =

Z xb

xa

[B]T [Dep][B]dxNonlin Elast Plastcity: 16

JN Reddy

Small Deformation Theory of Plasticity(continued)

Nonlin Elast Plastcity: 17

fd"g = fd"eg + fd"pg

fd"eg = [De]¡1fd¾g; fd"pg = d¸

½df

df¾g

¾

fd"eg = fd"g ¡ fd"pg

= fd"g ¡ d¸

½@f

@f¾g

¾

fd¾g = [De]

µfd"g ¡ d¸

½@f

@f¾g

¾¶

0 = f(f¾g; ·) ! 0 = df =

½@f

@f¾g

¾T

fd¾g ¡ Ad¸

0 =

½@f

@f¾g

¾T

[De]

µfd"g ¡ d¸

@f

@f¾g

¶¡ Ad¸

JN Reddy

Small Deformation Theory of Plasticity(continued)

Nonlin Elast Plastcity: 18

d¸ =f @f

@f¾ggT[De]fd"g

A + f @f@f¾gg[De]f @f

@f¾gg

A = H =d¹¾

d¹"p=

ET

1 ¡ ET =E

fd¾g = [Dep]fd"g

[Dep] = [De] ¡[De] @f

@f¾gf @f@f¾ggT [De]

H + f @f@f¾ggT [De]f @f

@f¾gg

JN Reddy Nonlin Elast Plastcity: 19

Numerical Examples

JN Reddy Nonlin Elast Plastcity: 20

Numerical Examples

JN Reddy Nonlin Elast Plastcity: 21

Numerical Examples

JN Reddy Introduction: 22

Simply Supported Isotropic Plate Under UDL(Combined Material and Geometric Nonlinearity)

E = 106 psi., G = 3.846 E, ν = 0.3, ρ = 0.000259 lb-s2/in4

σi = σ0i + Epi εp (i=1,2,4,5,6); σ01 = σ02 = σ04 = 3 × 104 psiσ04 = σ05 = σ06 = 1.372 × 104 psi, p0 = 300 psi

JN Reddy Nonlin Elast Plastcity: 23

Summary

The following topics were discussed:

• Nonlinear elasticity• Plasticity• Ideal plasticity and strain hardening plasticity• Stress-strain curves• Finite element models of nonlinear elasticity• Numerical examples• Small deformation theory of plasticity• Finite element formulation• Numerical examples