Department of Structural Mechanics

Faculty of Civil Engineering, VSB - Technical University Ostrava

Elasticity and Plasticity

Strain and Deformation

• Deformations and Displacements of 3D solid

• Physical Relations between Strains and Deformations• Hook’s Law, Physical Constants and Stress-Strain

Diagram of Construction Materials• Deformation by the Temperature Changes

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Deformations and displacements

Deformations and Displacements of 3D solid

3D bodies are deformated by influence of loading or

temperature changes,it is possible to define relative deformations or displacement components.

Relative deformations:

- length ε (relative elongation or contraction)

- angular γ (cross-section tapering)

Small-deformation theory: 1

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Geometrical equations

x

y

M A

B C

C’

B’

A’

M’

dy

dx

dx’

v

u

x

u

x

xuxx

uux

x

xxx

∂

∂=

−

−

∂

∂++

=−′

=d

ddd

d

ddε

xx

uu d

∂

∂+

Deformations and Displacements of 3D solid 6 / 33

Geometrical equations

x

y

M A

B C

C’

B’

A’

M’

dy

dx

dx’

dy’

v

u

xx

uu d

∂

∂+

xx

vd

∂

∂

yy

ud

∂

∂

β

α

y

u

x

v

y

yy

u

x

xx

v

xy∂

∂+

∂

∂=

∂

∂

+∂

∂

=+=d

d

d

d

βαγ

Deformations and Displacements of 3D solid

7 / 33

Geometrical equations

x

ux

∂

∂=ε

y

vy

∂

∂=ε

z

wz

∂

∂=ε

y

u

x

vxy

∂

∂+

∂

∂=γ

z

v

y

wyz

∂

∂+

∂

∂=γ

x

w

z

uzx

∂

∂+

∂

∂=γ

Deformations and Displacements of 3D solid

Relations between components of relative deformationin 3D body and components of displacements of any points of 3D body are expressed by geometrical equations.

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Stress-strain diagram of construction materials

Physical Relations between Strains and Deformations

ε

σ

Relations between strains and deformations are expressed by stress-strain diagram. It depends on physical and mechanical properties of construction materials.

Tension

A

N

Ar

r

∆

∆=

→∆ 0limσ

A

N=σ

l

l∆=ε

Dir

ect

str

ess

Relative deformation

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N

N

Physical Relations between Strains and Deformations

Tensile stress

Tensile test of steel

Stress-strain diagram of construction materials

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N

N

Physical Relations between Strains and Deformations

Tensile stress

Tensile test of steel

Stress-strain diagram of construction materials

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N

N

Physical Relations between Strains and Deformations

Tensile stress

Tensile test of steel

Stress-strain diagram of construction materials

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Broken steel

specimen after tensile test

Physical Relations between Strains and Deformations

Tensile stress

Stress-strain diagram of construction materials

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Tensile stress

Physical Relations between Strains and Deformations

Broken steel

specimen after tensile test

Stress-strain diagram of construction materials

14 / 33Physical Relations between Strains and Deformations

Tensile test of steel,

stress-strain diagram

Stress-strain diagram of construction materials

15 / 33Physical Relations between Strains and Deformations

Tensile test of steel,

stress-strain diagram

Stress-strain diagram of construction materials

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Tensile test of steel, stress-strain diagram

Physical Relations between Strains and Deformations

Stress-strain diagram of construction materials

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Tensile test of steel, stress-strain diagram

ε

σ

Dir

ect

str

ess

Relative deformation

Linear-elastic matherial

Physical Relations between Strains and Deformations

Stress-strain diagram of construction materials

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ε

σ

Plastic behaviour of matherial

Physical Relations between Strains and Deformations

Tensile test of steel, stress-strain diagram

Dir

ect

str

ess

Relative deformation

Stress-strain diagram of construction materials

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ε

σ

Permanent deformation

ach

ievem

ent in

tensile

test

Physical Relations between Strains and Deformations

Tensile test of steel, stress-strain diagram

Dir

ect

str

ess

Relative deformation

Stress-strain diagram of construction materials

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Linear-elastic matherial, Hook’s law

εx

Tension

EE xx

x

x σεε

σα =→== tan

A

Nx =σ

l

lx

∆=ε

α = arctan E

E

A

N

l

l=

∆

σx ... direct stress [Pa]

εx ... relative deformation [-]

AE

lNl

.

.=∆→

Hook’s law

σ

E ... modulus of elasticity in tension andcompression (Young’s modulus) [Pa]

Physical Relations between Strains and Deformations

Dir

ect

str

ess

Relative deformation

21 / 33

Linear-elastic matherial, Hook’s law

υ ... Poisson's ratio [-]

dx

σx σx

dy

dx’

dy’

after deformation

E

xxzy

συευεε .. −=−==

5,0≤υ

In simultaneously operation

of σx, σy and σz

( )[ ]zyxzyxxEEEE

σσυσσ

υσ

υσ

ε +−=−−= ..1

..

likewise ( )[ ]zxyyE

σσυσε +−= ..1 ( )[ ]

yxzzE

σσυσε +−= ..1

Physical equation – 1st part

Physical Relations between Strains and Deformations 22 / 33

was a British natural philosopher, architect and polymath who played an important role in the Scientific Revolution, through both experimental and theoretical work. In 1660, Hooke discovered the law of elasticity which bears his name and which describes the linear variation of tension with extension in an elastic spring. He first described this discovery in the anagram "ceiiinosssttuv", whose solution he published in 1678 as "Ut tensio, sic vis" meaning "As the extension, so the force.”

Historical persons

Robert Hooke

Thomas Young

Siméon-Denis Poisson

was an English genius and polymath. He is famous with the public for having partly deciphered Egyptian hieroglyphs before Champollion did. Young made notable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony and Egyptology. Young described the characterization of elasticity that came to be known as Young's modulus, denoted as E, in 1807, and further described it in his subsequent works such as his 1845 Course

of Lectures on Natural Philosophy and the Mechanical Arts.

was a French mathematician, geometer, and physicist. After him is named

Poisson's ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).

Physical Relations between Strains and Deformations

(18 July 1635 – 3 March 1703)

(13 June 1773 – 10 May 1829)

(21 June 1781 – 25 April 1840)

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Shear, shear stresses

x

y

dy

dx

βafter d

eformation

τxyτxy

τyx

τyx

Physical Relations between Strains and Deformations 24 / 33

Linear-elastic matherial, Hook’s law in shear

γxy

τxy = τyx

α = arctan G

τxy ... shear stress [Pa]

γxy ... tapering cross-section [-]

G ... modulus of elasticity in shear [Pa]

Hook’s law in shear

GG

xy

xy

xy

xy τγγ

τα =→== tan

likewise

G

yz

yz

τγ =

G

zxzx

τγ =

Physical equation – 2nd part

Physical Relations between Strains and Deformations

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Physical equations

Physical equations expressing relations between components of relative deformations and components of strains in 3D body.

G

yz

yz

τγ =

G

zxzx

τγ =

G

xy

xy

τγ =

( )[ ]zxyy

Eσσυσε +−= ..

1

( )[ ]yxzz

Eσσυσε +−= ..

1

( )[ ]zyxx

Eσσυσε +−= ..

1

Physical Relations between Strains and Deformations 26 / 33

Physical constants

In case of isotropic substance is not E, G and υ mutual indipendence.

5,00 ≤≤ υ( )υ+= 1.2G

E→

23

EG

E≤≤

Benchmark values of physical constants of some matherials:

E G υ

Steel 210 000 MPa 81 000 MPa 0,3

Glass 70 000 MPa 28 000 MPa 0,25

Granite 12 000 to 50 000 MPa - 0,2

SoftwoodE|| = 10 000 MPa

E⊥ = 300 MPa600 MPa -

Physical Relations between Strains and Deformations

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Stress-strain diagram of concrete in compression

Physical Relations between Strains and Deformations 28 / 33

Physical constants of concrete

Grade of concreteEcm

modulus of elasticityG υ

C12 26 000 MPa

0,42.E 0,2

C16 27 500 MPa

C20/25 29 000 MPa

C25/30 30 500 MPa

C30/35 32 000 MPa

C35/45 33 500 MPa

C40/50 35 000 MPa

C45/55 36 000 MPa

C50/60 37 000 MPa

Physical Relations between Strains and Deformations

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Design stress-strain diagram of concrete in compression

Parabolic-rectangular

Idealized diagram

Design diagram

Physical Relations between Strains and Deformations 30 / 33

Bilinear

Idealized diagram

Design diagram

Physical Relations between Strains and Deformations

Design stress-strain diagram of concrete in compression

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Stress-strain diagram of steel

Plasticity: matherial ability to permanent deformation without fracture.

fe … limit of elasticity

fy ... yield stressfu ... ultimate strength

Steel ductility: plastic elongation of broken bar, steel 15%.

Strain energy

Physical Relations between Strains and Deformations 32 / 33

Ideal elasto-plastic matherial

εx

σx

Compression

α = arctan E

Tension

fy

0

Y

Y’

A,C

B

section

Y-Y’ Hook’s law

Y-A Plastic state – free increase of

deformations

A-B Unloading

B-C Re-increasing of strain

εp εe

εp … plastic (permanent) deformationεe … elastic deformation

-fy

Physical Relations between Strains and Deformations

Stress-strain diagram

33 / 33

Deformation by Temperature Changes

Deformation by Temperature Changes

dy

dx

( )CT o∆

dx’

dy’

TTTzTyTx ∆=== .,,, αεεε 0=== zxyzxy γγγ

αt … coefficient of thermal expansivity [oC-1]

Steel αt=12.10-6 oC-1 Timber α

t=3.10-6 oC-1

Concrete αt=10.10-6 oC-1 Masonry α

t=5.10-6 oC-1