1

1-D Elastic Models Theme 1 – linear and non-linear elasticity

Hyperplasticity model

Code 1-E

Description One dimensional linear elasticity

s

e

E

e

sE

1

Variables ,e s

Constants E Definitions

Functions 2

2

Ef

e

2

2g

E

s

Basic derivatives

fE

s e

e

g

E

se

s

Development

Notes The simplest of all elastic models and the basis for all further development of elasticity theories.

2

Hyperplasticity model

Code 1-N

Description One-dimensional non-linear elastic (power law)

s

e

e

s

Variables ,e s

Constants neo ,,s (strictly only two required, but dimensioning parameter os introduced for

convenience).

Definitions

Functions

2 11

2

n nof e ne n

s e

2

2 1

no

o

ge n n

s s

s

Basic derivatives

1 11n

of

e n

s s e e

11

1

n

o

g

e n

s

e s s

Development

11n

n n

o o

e e n e s s

e e e s s

so that the (variable) stiffness n

oo

E e s s

s e s

1n

o oe

s

e s s s

so that the (variable)

stiffness

n

oo

E e s s

s e s

Notes Based on 1-E, with introduction of nonlinear function.

Reduces to linear case 1-E with oE e s , 0n .

Origin for strain ( 0e ) is at 0s . See 1-N-alt for alternative strain origin.

3

Hyperplasticity model

Code 1-N-alt

Description One-dimensional nonlinear elastic (power law). Variant with origin for strain at

os s )

s

e

e

s

Variables ,e s

Constants neo ,,s (strictly only two required, but dimensioning parameter os introduced for

convenience).

Definitions

Functions

2 11 1 1

2

n nof e ne n

s e

2

112

1

no

o

o

g ne n

s s s s s

Basic derivatives

1 11 1n

of

e n

s s e e

11

11

n

o

g

e n

s e s s

Development

11 1n

n n

o o

e e n e s s

e e e s s

so that the (variable) stiffness n

oo

E e s s

s e s

1n

o oe

s

e s s s

so that the

(variable) stiffness

n

oo

E e s s

s e s

Notes Origin for strain ( 0e ) is at os s . See 1-N for alternative strain origin.

Approaches model 1-N-log for 1n .

4

Hyperplasticity model

Code 1-N-gen

Description One-dimensional nonlinear elastic (power law). General form that encompasses 1-N and 1_N-alt.

s

e

Variables ,e s

Constants neo ,,s (strictly only two required, but dimensioning parameter os introduced for

convenience), plus “switch parameter” N . Value 0N gives model 1-N (origin for

strain at 0s ), value 0N gives model 1-N-alt (origin for strain at os s ).

Definitions

Functions

2 11

2

n nof N e n Ne n

s e

2

12

1

no

o

o

Nng

e n

N

s s s s s

Basic derivatives

1 11n

of

N e n

s s e e

11

1

n

o

gN

e n

s e s s

Development 11

nn n

o o

e N e n e s s

e e e s s

so that the (variable) stiffness n

oo

E e s s

s e s

1n

o oe

s

e s s s

so that the

(variable) stiffness

n

oo

E e s s

s e s

Notes Reduces to model 1-E for 0N and 0n . Approaches model 1-N-log for 1N and 1n .

5

Hyperplasticity model

Code 1-N-log

Description One dimensional logarithmic elasticity (stiffness proportional to stress)

s

e

e

s

Variables ,e s

Constants os (arbitrary constant),

Definitions Note the definition of the special function ilog log 1x x x x , so that

ilog logd

x xdx

. The constant is introduced so that ilog 1 0 .

Functions exp 1of e

s

ilogoo

g s

s s

Basic derivatives expo

f e s s

e log

o

g se

s s

Development expo

s e s s e e

so that the

(variable) stiffness Es s

e

e s

s so that the (variable) stiffness

Es s

e

Notes Origin for strain ( 0e ) is at os s .

Limit of 1-N-alt as 1n but expressed in terms of flexibility parameter (by analogy with usage in Critical State Soil Mechanics) rather than stiffness parameter

1e .

NB Note that each of the non-linear models is written assuming that the stiffness is a power function of the positive stress. In practice these models may often be applied to cases where stiffness is a function of the compressive stress, in which case s must be replaced by s in all the expressions and e by e .

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Theme 2 – limits of perfectly flexible or rigid behaviour These models make use of the Indicator Functions that are employed in Convex Analysis. For a convex

set C of the variable x , the Indicator Function C xI is defined by:

0,

,Cx C

xx C

I

The sub-differential (in Convex Analysis the generalisation of a derivative) of an Indicator Function is given by the Normal Cone:

x C Cx x I =N

See Couse Material unit M4 – Convex Analysis. These models are largely of academic interest, but serve as a useful precursor to the unilateral models pursued in Theme 3.

Hyperplasticity model

Code 1-R

Description One dimensional rigid

s

e

e

s

Variables ,e s

Constants none

Definitions

Functions 0f eI

0g

Basic derivatives

0fes eN

0g

e s

Development

Notes The limit of 1-E as E .

7

Hyperplasticity model

Code 1-F

Description One dimensional perfectly flexible

s

e

e

s

Variables ,e s

Constants none

Definitions

Functions 0f

0g sI

Basic derivatives

0f

s e

0gse sN

Development

Notes The limit of 1-E when 0E .

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Theme 3 – bi-linear and unilateral response

Hyperplasticity model

Code 1-EB

Description One dimensional bi-linear elasticity

s

e

Ec

Et

e

s

Et

1

Ec1

Variables ,e s

Constants ,t cE E (elastic stiffnesses in tension and compression) Definitions

Uses Macaulay bracket 0, 0

, 0

xx

x x

Functions 2 2

2 2t cE Efe e

2 2

2 2t cg

E E

s s

Basic derivatives t c

fE E

s e e

e t c

g

E E

s se

s

Development

Notes Reduces to 1-E if t cE E E .

9

Hyperplasticity model

Code 1-ETR

Description One dimensional linear elastic, tension only (rigid in compression)

s

e

E

e

sE

1

Variables ,e s

Constants E Definitions

Functions

2

0, 2

Ef

e e I

2

2g

E

s

Basic derivatives

0,f Ee s e eN

g

E

se

s

Development

Notes The limit of 1-EB when cE .

10

Hyperplasticity model

Code 1-ECR

Description One dimensional linear elastic, compression only (rigid in tension)

s

e

E

e

s

E

1

Variables ,e s

Constants E Definitions

Functions

2

,0 2

Ef

e e I

2

2g

E

s

Basic derivatives

,0f Ee s e eN

g

E

se

s

Development

Notes The limit of 1-EB when tE .

11

Hyperplasticity model

Code 1-ET

Description One dimensional linear elastic, tension only

s

e

E

e

sE

1

Variables ,e s

Constants E Definitions

Functions 2

2

Ef

e

2

0, 2g

Es

s I

Basic derivatives

fE

s e

e 0,g Es

se s

N

Development

Notes The limit of 1-EB when 0cE .

This is the classical “light inextensible string”.

12

Hyperplasticity model

Code 1-EC

Description One dimensional linear elastic, compression only

s

e

E

e

s

E

1

Variables ,e s

Constants E Definitions

Functions 2

2

Ef

e

2

,0 2g

Es

s I

Basic derivatives

fE

s e

e ,0g Es

se s

N

Development

Notes The limit of 1-EB when 0tE .

13

Hyperplasticity model

Code 1-ET

Description One dimensional tension only

s

e

e

s

Variables ,e s

Constants -

Definitions

Functions ,0f eI

0,g sI

Basic derivatives

,0fe s eN

0,gs e sN

Development

Notes The limit of 1-EB when tE and 0cE .

14

Hyperplasticity model

Code 1-EC

Description One dimensional compression only

s

e

e

s

Variables ,e s

Constants -

Definitions

Functions 0,f eI

,0g sI

Basic derivatives

0,fe s eN

,0gs e sN

Development

Notes The limit of 1-EB when 0tE and cE .