Introduction to Composite Introduction
to Composite
Materials and Structures
Nachiketa Tiwari
Nachiketa
Tiwari
Indian Institute of Technology Kanpur
Lecture 32
Strength of a Composite Lamina
Failure Criteria for Isotropic Materials
Failure Criteria for Isotropic Materials
• Isotropic
Isotropic materials can fail in a variety of ways. Failure in such materials can fail in a variety of ways. Failure in such
materials can be predicted by using one of the following failure criteria.
– For brittle materials
• Maximum principal stress criterion
• Maximum principal strain criterion
• Coulomb‐Mohr criterion
– For ductile materials
• Maximum shear stress or Tresca criterion
• vonMises stress or distortion energy criterion
• To
To use these criteria, especially in multi
use these criteria, especially in multi‐axial
axial loading situation, the loading situation, the
concept of principal stress and principal strains is quite frequently invoked.
Failure Criteria for Isotropic Lamina
Failure Criteria for Isotropic Lamina
• However,
However, none of the failure criteria used for isotropic materials are none of the failure criteria used for isotropic materials are
of much use for predicting failure in composite lamina. • This so, because the planes along which the lamina may be possibly the weakest, may not be necessarily aligned with the direction of “principal”
principal stresses in a lamina. stresses in a lamina
Thus, for the same reason, the concept of principal stresses is of for the same reason the concept of principal stresses is of
• Thus
little use in case of composite materials.
• For this reason, several alternative failure theories have been developed, which may be used to predict failure of composite l i
lamina.
Different Strengths of a Composite Lamina
•
The strength of a unidirectional composite lamina may be characterized by five different material parameters. These are:
–
–
–
–
–
•
For a three‐dimensional analysis, i.e. for composite laminates, four additional strength parameters, are relevant. These are:
–
–
–
–
•
Longitudinal tensile strength, σu1.
Longitudinal compressive strength, σu1’
Transverse tensile strength, σu2.
Transverse compressive strength, σu2’
In‐plane shear strength, τu12.
Interlaminar tensile strength, σuI
Interlaminar compressive strength, σ
p
g
uI’
uI
Interlaminar shear strength, τu13, for the 1‐3 plane.
Interlaminar shear strength, τu23, for the 2‐3 plane.
For transversely isotropic materials, with 2‐3 as plane of isotropy:
– σuI = σu2 σuI’ = σu2’, and τu13 = τu23
•
Here, all material properties are measured in absolute numerical values.
Sign Convention for Shear Stresses
•
A positive shear stress applied in L‐T di ti
direction can also be expressed as a l b
d
specimen which experiences tensile stress at 45 degrees (anti‐clockwise turn w r t x‐axis) of value τ and a
turn w.r.t. x‐axis) of value τ, and a compressive stress applied in ‐45 degree direction (i.e. clockwise turn g
x‐axis).
)
of 45 degrees w.r.t
•
A negative shear stress applied in L‐
T direction can also be expressed as
T direction can also be expressed as a specimen which experiences compressive stress at 45 degrees (anti‐clockwise turn w.r.t. x‐axis) of value τ, and a tensile stress applied in ‐45 degree direction (i.e. clockwise turn of 45 degrees w.r.t x‐
axis).
i)
•
Fig. 32.1: Sign convention for positive and negative shear stresses.
Some Observations on Shear Strength
g
•
Figure 32.1 shows that when a unidirectional lamina is subjected to g
j
positive shear stress τ12, (i.e. shear stress referred to principal material planes), it is equivalent to application of normal tensile and compressive loads at 45⁰ to longitudinal (fiber) direction.
loads at 45
to longitudinal (fiber) direction
•
Given this, the strength of a lamina subjected to pure τ12 does not get affected by sense (positive or negative) of shear stress. However, the same is not true when shear stress is applied at an angle to principal material directions. •
Consider Fig. 32.2 as shown on the next slide.
Some Observations on Shear Strength
=
=
Fig. 32.2a: A Lamina Lamina
subjected to pure positive shear at 45 deg. h
t 45 d
to fiber direction.
Fig. 32.2b: A Lamina Lamina
subjected to pure negative shear at 45 deg. h
t 45 d
to fiber direction.
Some Observations on Shear Strength
•
In Fig. 32.2a, a lamina is subjected to pure positive shear at 45⁰ to fiber direction This stress‐state is equivalent to a tensile stress state in
direction. This stress‐state is equivalent to a tensile stress state in longitudinal direction, and compressive stress state in transverse direction.
•
In Fig. 32b, the same lamina is subjected to pure negative shear at 45⁰ to fiber direction. This stress‐state
fiber direction. This stress
state is equivalent to a compressive stress state is equivalent to a compressive stress state
in longitudinal direction, and tensile stress state in transverse direction.
•
Looking at Fig. 32a and Fig. 32b, and noting the fact that most composite lamina are weakest in transverse tension, we can make the case that the lamina would be weaker under negative shear, and stronger in positive shear.
Failure Theories for Composite Lamina
•
Failure theories for composite lamina can be classified into three distinct groups These are:
groups. These are:
– Non‐interactive or limit theories: Here, failure modes are predicted by comparing individual stresses or strains respective to their ultimate stresses and strains. However, such theories do not account for interplay between different stress components.
such theories do not account for interplay between different stress components. Examples of such theories are maximum stress criteria, and maximum strain criteria.
– Interactive theories: These theories go a step further than limit theories, and also g
p
,
account for interaction between various stress/strain components. Examples of such theories are those of Tsai‐Wu and Tsai‐Hill. They are able to predict overall failure, but cannot predict the exact failure mode.
– Failure mode based theories: These theories provide separate criteria for failure of matrix, fiber and interface. Examples of such theories are those of Puck, and Hashim‐
Rotem.
Rotem
•
Here, we discuss three theories in particular, maximum stress, maximum strain, and Tsai‐Hill.
i
d
i ill
Maximum Stress Theory
•
To use this theory:
– Calculate
Calculate stresses in material directions on a layer
stresses in material directions on a layer‐by‐layer
by layer basis.
basis
– Failure occurs, if at least one of the following conditions in any layer is satisfied.
– σ1 > σu1 if σ1 > 0.
– σ1 > σu1’ if σ1 < 0.
– σ2 > σu2 if σ2 > 0.
– σ2 > σu2’ if σ2 < 0.
– σ3 > σu3 if σ3 > 0.
– σ3 > σu3’ if σ3 < 0.
– |τ12| > τu12
– |τ13| > τu13
– |τ23| > τu23
•
These equations in red color are not relevant for two‐dimensional state of stress.
Maximum Strain Theory
•
To use this theory:
– Calculate
Calculate strains in material directions on a layer
strains in material directions on a layer‐by‐layer
by layer basis.
basis
– Failure occurs, if at least one of the following conditions in any layeris
satisfied.
– ε 1 > ε u1 if ε1 > 0.
– ε1 > εu1’ if ε1 < 0.
– ε2 > εu2 if ε2 > 0.
– ε2 > εu2’ if ε2 < 0.
– ε3 > εu3 if ε3 > 0.
– ε3 > εu3’ if ε3 < 0.
– |γ12| > γu12
– | γ13| > γu13
– | γ23| > γu23
•
These equations in red color are not relevant for two‐dimensional state of stress.
Tsai‐Hill Theory
•
To use this theory:
– Calculate stresses in material directions on a layer‐by‐layer basis.
Calculate stresses in material directions on a layer by layer basis
– Failure occurs, if the following condition is satisfied in any layer.
(σ1/F1)2 + (σ2/F2)2 + (τ12/F6)2 ‐ (σ1σ2/F1)2 > 1.
where,
F1 1 = σ
σu11 if σ
if σ1 1 > 0.
>0
F1 = σu1’ if σ1 < 0.
F2 = σu2 if σ2 > 0.
F2 = σu2’ if σ2 < 0.
F6 = τu12
• It
It may be noted that the above relation is valid only if the lamina may be noted that the above relation is valid only if the lamina
has a two‐dimensional stress state.
• This is an interactive theory, and is popularly used.
R f
References
1.
Analysis and Performance of Fiber Composites, Agarwal,
B D and
B.D.
d Broutman,
B t
L J.,
L.
J John
J h Wiley
Wil & Sons.
S
2
2.
Mechanics
M
h i off Composite
C
i Materials,
M
i l Jones,
J
R M.,
R.
M Mc‐Graw
M G
Hill.
3.
Engineering Mechanics of Composite Materials, Daniel, I.
M and Ishai,
M.
Ishai O.,
O Oxford University Press.
Press