ASEN 3112 - Structures
5
Stress-Strain
Material Laws
ASEN 3112 Lecture 5 – Slide 1
ASEN 3112 - Structures
Strains and Stresses are Connected
by Material Properties of the Body (Structure)
Recall the connections displayed in previous lecture:
MP
internal forces ⇒ stresses ⇒ strains ⇒ displacements ⇒ size&shape changes
MP
displacements ⇒ strains ⇒ stresses ⇒ internal forces
The linkage between stresses and strains is done through
material properties, as shown by symbol MP over red arrow
Those are mathematically expressed as constitutive equations
Historically the first C.E. was Hooke's elasticity law, stated in 1660 as "ut tensio sic vis"
Since then recast in terms of stresses and strains, which are more modern concepts.
ASEN 3112 Lecture 5 – Slide 2
ASEN 3112 - Structures
Assumptions Used In This Course As Regards
Material Properties & Constitutive Equations
Macromodel
material is modeled as a continuum body; finer
scale levels (crystals, molecules, atoms) are ignored
Elasticity
stress-strain response is reversible and has a preferred
natural state, which is unstressed & undeformed
Linearity
relationship beteen strains and stresses is linear
Isotropy
properties of material are independent of direction
Small strains
deformations are so small that changes of geometry are
neglected as loads (or temperature changes) are applied
For additional explanations see Lecture notes.
ASEN 3112 Lecture 5 – Slide 3
ASEN 3112 - Structures
The Tension Test Revisited:
Response Regions for Mild Steel
Nominal stress
σ = P/A 0
Max
nominal
stress
Strain
hardening
Localization
Yield
Elastic
limit
Linear elastic
behavior
(Hooke's law is
valid over this
response region)
Undeformed state
Nominal
failure
stress
Mild Steel
Tension Test
P
L0
gage length
Nominal strain ε = ∆L /L0
ASEN 3112 Lecture 5 – Slide 4
P
ASEN 3112 - Structures
Other Tension Test Response Flavors
Brittle
(glass, ceramics,
concrete in tension)
Moderately
ductile
(Al alloy)
ASEN 3112 Lecture 5 – Slide 5
Nonlinear
from start
(rubber,
polymers)
ASEN 3112 - Structures
Tension Test Responses of Different Steel Grades
Nominal stress
σ = P/A0
Tool steel
Note similar
elastic modulus
High strength steel
Mild steel
(highly ductile)
Conspicuous yield
Nominal strain ε = ∆ L /L 0
ASEN 3112 Lecture 5 – Slide 6
ASEN 3112 - Structures
Material Properties For A
Linearly Elastic Isotropic Body
E
Elastic modulus, a.k.a. Young's modulus
Physical dimension: stress=force/area (e.g. ksi)
ν
Poisson's ratio
Physical dimension: dimensionless (just a number)
G
Shear modulus, a.k.a. modulus of rigidity
Physical dimension: stress=force/area (e.g. MPa)
α
Coefficient of thermal expansion
Physical dimension: 1/degree (e.g., 1/ C)
E, ν and G are not independent. They are linked by
E = 2G (1+ν),
G = E/(2(1+ν)),
ν= E /(2G)−1
ASEN 3112 Lecture 5 – Slide 7
ASEN 3112 - Structures
State of Stress and Strain In Tension Test
Cross section (often circular) of area A
(a)
P
P
gaged length
σ xx = P/A (uniform over cross section)
y
z
(b)
P
Stress
state
σxx
0
0
0
0
0
0
0
0
Strain
state
x Cartesian axes
εxx
0
0
at all points in the gaged region
For isotropic material, εyy = εzz
0
εyy
0
0
0
εzz
is called the lateral strain
ASEN 3112 Lecture 5 – Slide 8
ASEN 3112 - Structures
Defining Elastic Modulus and Poisson's Ratio
Isotropic material properties E and ν are obtained from the linear elastic
response region of the uniaxial tension test (last slide). For simplicity call
σ = σ xx = axial stress, ε = ε xx = axial strain,
εyy = ε zz = lateral strain
The elastic modulus E is defined as the ratio of axial stress to axial strain:
def
E =
σ
ε
whence σ = E ε, ε =
σ
E
Poisson's ratio ν is defined as the ratio of lateral strain to axial strain:
def
ν =
lateral strain
axial strain
=−
lateral strain
axial strain
The − sign in the definition of ν is introduced so that it comes out positive.
For structural materials ν lies in the range [0,1/2). For most metals (and their
alloys) ν is in the range 0.25 to 0.35. For concrete and ceramics, ν ≈ 0.10.
For cork ν ≈ 0. For rubber ν ≈ 0.5 to 3 places. A material for which ν = 0.5 is
called incompressible. If ν is very close to 0.5, it is called nearly incompressible.
ASEN 3112 Lecture 5 – Slide 9
ASEN 3112 - Structures
;
;
Which material would work best
for capping a wine bottle?
Rubber
Cork
ASEN 3112 Lecture 5 – Slide 10
ASEN 3112 - Structures
Which material would work
best for a shock absorber?
Rubber
Foam
ASEN 3112 Lecture 5 – Slide 11
;yy;
y;
ASEN 3112 - Structures
State of Stress and Strain In Torsion Test
Circular cross section
(a) T
T
gaged length
For distribution of shear stresses and
strains over the cross section, cf. Lecture 7
y
z
x Cartesian axes
(b) T
Stress
state
0
τyx
τxy
0
0
0
0
0
Strain
state
0
γyx
γxy
0
0
0
0
0
0
0
at all points in the gaged region. Both the shear stress τ yx = τ xy as well
as the shear strain γ xy = γyx vary linearly as per distance from the
cross section center (Lecture 7). They attain maximum values on the
max
specimen surface. For simplicity, call those values τ = τ max
xy and γ = γ xy
ASEN 3112 Lecture 5 – Slide 12
ASEN 3112 - Structures
Defining Shear Modulus Of An Isotropic
Linearly Elastic Material
Isotropic material property G (the shear modulus, also called modulus
of rigidity) is obtained from the linear elastic response region of the
torsion test of a circular cross section specimen (last slide).
For simplicity call
max
τ = τ xy
= max shear stress on specimen surface over gauged region
max
γ = γ xy = max shear strain on specimen surface over gauged region
The shear modulus G is defined as the ratio of the foregoing
shear stress and strain:
def
G =
τ
γ
whence τ = G γ, γ =
τ
G
ASEN 3112 Lecture 5 – Slide 13
ASEN 3112 - Structures
Defining The Coefficient of Thermal
Expansion Of An Isotropic Material
Take a standard uniaxial test specimen:
x
gaged length
At the reference temperature T0 (usually the room temperature) the
gaged length is L 0 . Heat the unloaded specimen by ∆T while allowing it
to expand freely in all directions. The gaged length changes to
L = L 0 + ∆L. The coefficient of thermal expansion is defined as
def
α =
∆L
L0 ∆T
whence ∆L = α L 0 ∆T
T
The ratio ε T = εxx
= ∆L /L 0 = α ∆T is called the thermal strain in the
axial (x) direction. For an isotropic material, the material expands equally in
T
all directions: ε xx
= εTyy = εzzT , whereas the thermal shear strains are zero.
ASEN 3112 Lecture 5 – Slide 14
ASEN 3112 - Structures
1D Hooke's Law Including Thermal Effects
Stress To Strain:
ε=
σ
+ α ∆T = εM + εT
E
expresses that total strain = mechanical strain +
thermal strain: the strain superposition principle
Strain To Stress:
σ = E ( ε − α ∆T )
A problem in Recitation 3 uses this form
ASEN 3112 Lecture 5 – Slide 15
ASEN 3112 - Structures
3D Generalized Hooke's Law (1)
Stresses To Strains (Omitting Thermal Effects)
 1
E


x x
 ν
−
 yy   E

  ν
 zz   − E

=
 γx y   0

 
γ yz

 0
γ
zx
0
ν
−E
1
E
ν
−E
ν
−E
ν
−E
1
E
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
G
0
0
0
0
1
G
0
0
1
G
For derivation using the strain superposition principle,
as well as inclusion of thermal effects, see Lecture notes
ASEN 3112 Lecture 5 – Slide 16



 σx x

  σ yy 


  σzz 


  τx y 


 τ yz

τzx
.
ASEN 3112 - Structures
3D Generalized Hooke's Law (2)
Strains To Stresses (Omitting Thermal Effects)

 
σx x
Ê (1 − ν)
 σ yy   Ê ν

 
 σzz   Ê ν

=
0
 τx y  

 
0
τ yz
0
τzx
in which
Ê ν
Ê (1 − ν)
Ê ν
0
0
0
Ê =
Ê ν
Ê ν
Ê (1 − ν)
0
0
0
0
0
0
G
0
0
0
0
0
0
G
0


x x
0
0   yy 


0   zz 


0   γx y 


0
γ yz
G
γzx
E
(1 − 2ν)(1 + ν)
This is derived by inverting the matrix of previous slide. For the
inclusion of thermal effects, see Lecture notes
ASEN 3112 Lecture 5 – Slide 17
ASEN 3112 - Structures
2D Plane Stress Specialization
Stresses
σx x
τ yx
0
τx y
σ yy
0
Strains
0
0
0
x x
γ yx
0
ASEN 3112 Lecture 5 – Slide 18
γx y
yy
0
0
0
zz
ASEN 3112 - Structures
2D Plane Stress Generalized Hooke's Law
Strains To Stresses (Omitting Thermal Effects)
 1
E
x x
 ν
 yy   − E

= ν
zz
−
E
γx y
0


ν
−E
1
E
ν
−E
0
0

 σx x
0 
 σ yy
0  τ
xy
1
G
Stresses To Strains (Omitting Thermal Effects)
in which
σx x
σ yy
τx y
=
Ẽ
Ẽ ν
0
Ẽ =
Ẽ ν
Ẽ
0
0
0
G
x x
yy
γx y
E
1 − ν2
Used in Ex. 3.1 of HW #3. For inclusion of thermal effects as well as the
plane strain case (which is less important in Aerospace) see Lecture Notes
ASEN 3112 Lecture 5 – Slide 19