The stress-strain curve (diagram of the material)

1 The force-displacement curve for a traction loaded bar

Let consider a traction loaded bar:

We repeat the operation:

.

...

....................................................................

0121

0333123

022212

ruptureuntil

lllFFFF

lllFFF

lllFF

nnnnn −=⇒⇒>>>>

−=⇒⇒>>

−=⇒⇒>

− δ

δ

δ

In ( )δ−F coordinates, we can obtain two kinds of curves, for metallic materials:

F - The bar is subjected to an increasing load F

in a testing machine such that F and l can be

measured : for F1, we have the length l1⇒ the

elongation is: 011 ll −=δ

-

But, because ⇒

→

→

εδσF

We can obtain the material behavior curve.

( )εσ − curve is known as the traction diagram of the material (stress - strain curve).

2. Stress – strain curve. Hooke’s Law. Elastic constants.

a) The traction test

These kinds of tests are standardized. The specimen is a standard bar.

The stress :

where:

Fi – the applied force;

A0 – the initial section surface.

The strain :

δ

rupture

ere

F

rupture

ere

F

δ

a) Bar with low carbon

content b) Bar with high carbon

content

0A

F ii =σ

0

0

l

ll ii

−=ε

where li is the bar length for the Fi force and l0 is the initial bar length.

The ( )εσ − stress – strain curves have the same allure as the ( )δ−F force – displacement curves.

We can observe 4 important points:

1) pσ - proportional limit

- the point until the stress is proportional to strain by the relationship: εσ ⋅=E , Hooke’s law,

where

E = (longitudinal) modulus of elasticity or Young’s modulus,

2mm

NE

αtgE = ; E = the slope of the line OA;

- steel: 24 /1021 mmNE ⋅=

- wood : 24 /10 mmNE =

The zone of linearity is very important in most engineering applications. Strength calculations are

aimed at limiting values of stress, so this level does not exceed.

2) eσ - elastic limit

- the point until the material has a perfectly elastic behavior.

The elastic limit of a material is defined to be the maximum stress that can be developed in the

material without causing permanent set (residual deformation). It is found that for the most elastic

materials, the elastic limit has approximately the same value as the proportional limit of the material,

and in technical literature, the proportional limit is called (though incorrectly) the proportional elastic

limit.

Note: The deformation per unit of length (strain) retained by the bar after the load (and stress) has

been reduced to zero is called the permanent set or merely the set. For the bar subjected to the stress

σ, if the load is released gradually, the stress – strain curve will be represented by a straight line

parallel to the straight line portion of the curve below the proportional limit.

HeR

O

ε

σ

rmR σ=

α

eσ

F

E

C B

A

2.0PR

pσ

O

α

LeR

cσ

C’ D F

E C

B

A

ε

σ

rmR σ=

eσ

pσ

Elastic behavior ⇒ perfect elastic body, it doesn’t exist remanent or residual deformations

(permanent set).

Conventionally a material has an elastic behavior, if there are very small remanent deformations:

( ) erem σε ⇒= 0001.0%01.0 = elastic limit (point B).

- ep σσ ≅ ⇒ the error is not so great;

- liniarity ≠ elasticity;

3) cσ - yield strength (yield point – σy)

- there are certain materials, such as mild steel, that have a stress-strain diagram, characterized

by a flat, horizontal portion (C’D) that follows immediately after the elastic and proportional limit.

Thus, when this stress is reached, the strains continue to increase without any increase in stress. This

stress is called yield point – the minimum stress in the test specimen of the material at which the

material deforms appreciably without an increase in stress.

Usually, when a bar is a part of a machine or structure, the change in length (or shape) that occurs

while the stress is at the yield-point value is often sufficient to damage the load-resisting function of

the machine or structure, although perhaps a greater load than the yield-point load would be required

to cause collapse or fracture of the machine or structure.

The deformation that occurs when the yield-point stress is first reached starts as localized slip

bands visible as strain markings called Luder’s lines; the slip planes or bands are often approximately

in the direction of the maximum shearing stress. But the slip usually starts at a point of stress

concentration or at a defect or a weak point in the material.

The tension specimen becomes deformed and inelastic strains will occur. The names upper and

lower yield points are sometimes used to indicate the two yield points ( eHR , eLR ).

Usually, the stress on the landing C'D defines the apparent yield limit (yield strength).

remanentε

σ

ε α α

BC AC

NC

D

C

C

C’

MC

OC

!0

0

≠=

=

ONN

N

ε

σ

0A

FR c

ec ==σ

4) rσ - ultimate strength (σu)

- A brittle material breaks when stressed to the ultimate strength, whereas a ductile material

continues to stretch. The ultimate strength of a material, then, is defined to be the maximum stress

that can be developed in the material as determined from the original cross section of the bar or

specimen; the cross section of the bar of a ductile material decreases somewhat as the bar is stressed

above the yield strength and particularly as it is stressed beyond the ultimate strength .

The yield point and the ultimate strength are principal parameters to characterize a material.

- Strain hardening (DE)

- After the ultimate strength of a ductile material is developed, this begins to “neck down” (EF),

thereby rapidly reducing the area of cross section at the neck-down section and the load required to

cause the bar to continue to stretch decreases. The load on the bar at the instant of rupture is called

the breaking or rupture load.

- ultimate strength : o

mrA

FR max==σ

Another characteristics:

a) Ultimate strain :

[ ]%1000

0 ×−

=l

llurε

where :

ul - final length, measured after rupture.

b) Rupture neck-down:

[ ]%1000

0 ×−

=A

AAZ u

unde :

uA - neck-down section, at breaking.

a) rσ is a conventional value:

- ≠mF rupture load,

- =0A initial area, not neck-down section.

b) we are interested in the quality of the material!

For the materials without yield landing, we define the yield strength cσ as the maximum stress that

can be developed in a test specimen of the material without causing more than a specified permissible

set. A value of 0.2% (which means a set of 0.002 mm/mm) frequently is specified as the permissible

set for metals ( 2,0σ ).

c) γτ − curve (torsion loading)

d) Lateral Strain – Poisson’s Ratio

γτ ⋅= G

G – shearing modulus of elasticity 24 /104.8 mmNGsteel ⋅=

rτ

pτ

eτ cτ

τ

γ β

B

C A

C

D

C

C

E

C

O

C

2,02,0 pR≈σ

σ

%2,0=ε

N

C)

O

)

ε

The bar decreases in section as the load increases. The ratio of the lateral strain to the longitudinal

strain is a constant denoted by ν and is called Poisson’s ratio.

0

01

01

0

01

01

0

01

01

a

aabb

b

bbaa

l

llll

z

y

x

−=⇒<

−=⇒<

−=⇒>

ε

ε

ε

Experimentally, in the proportional zone of the diagram:

xz

xy

υεε

υεε

−=

−=

<

<

>

0

0

0

z

y

x

ε

ε

ε

υ = lateral strain coefficient = Poisson’s ratio

3.0=steelυ .

Note:

υ,,GE - characteristics of a material.

e) Isotropy

⇒−== xzy υεεε the material has the same behavior along the transversal axes.

Isotropic behavior: ( ) ctGE =υ,, , any loading direction.

Anisotropic materials are materials with preferential directions (crystals, laminated material).

y

z

a1

a0

l1

l0

b0

b1

F

x

II.3 Linear-elastic model in Deformable Body Mechanics

1) The hypothesis of continuous and homogeneous environment

The material is continuous and homogeneous – the material fills the all space bounded by its volume

and has the same physical properties at all points.

- The hypothesis is necessary to define the stress:

dA

dF

A

Fp

A=

∆∆

=→∆ 0

lim

2) The isotropy hypothesis

The material is isotropic when the same characteristics are any direction. It depends on the material

processing conditions.

3) The perfect elasticity hypothesis

After the removal of external forces the body returns to its original.

4) The hypothesis of linear deformation under the action of applied loads.

– The Hooke’s law is valid:

γτεσ⋅=

⋅=

G

E

5) The hypothesis of small deformations

– deformations are very small in relation to the body size.

We use the initial structure dimensions for calculation (called the first order calculation).

The second order calculation: the equations are written for the deformed state of the system.

- reality: L1<L

But we admit: L1 ≈L (there are infinitely small

deformations)

LFM ×=⇒ 1

⇒we admit to maintain the original size.

x

L

2 1

F

L1

L

x

2

1

F

6) Saint–Venant’s hypothesis

Regardless of the load application, away from the point of application, the material behaves the same.

It follows that the load can be replaced by an equivalent load and this change doesn’t influence far

away.

The bar as one-dimensional model

The hypotheses 1 - 6 + the Plane Sections Hypothesis (Bernoulli’s hypothesis)

Liniarity we can apply the principle of superposition.

- Partial effects can be added.

- The order of consideration of the tasks doesn’t matter, the final effect is the same.

- we can obtain information regarding the system of axes

⇒ variation of the stresses

⇒ maximum stress.

tiontheofsticcharacterilgeometrica

effortSTRESS

sec=