Strength modules of FRC lamina

1. Strength of fiber reinforced composite lamina
1 Introduction
Fiber reinforced composite lamina can fail because of number of failure modes. This character-
istics of the FRC composite is further compounded due to the directional nature of its strength.
Typical failure mechanisms observed in FRC lamina are:
• Matrix cracks
• Fiber breakage
• Matrix-fiber debonding
• Fiber pullout
• Local fiber buckling
In this section, however, our focus is not to model or dwell in detail about the above mentioned
failure mechanisms but to make an assessment of the material at the macroscale. Homogenized
effective properties obtained/derived in earlier sections are used in the material constitutive
relation to determine the average stresses and strains in the lamina. These average quantities
are used to check the material for failure.
Effective strengths of the lamina may be defined as ultimate values of the volume-averaged
stresses that cause failure of the lamina under the simple states of stress like, longitudinal normal
stress, transverse stress and shear stress. The stress-strain curves in Fig. ?? show the graphical
interpretation of these simple states of stress, the effective strengths s+
L , s−
L , s+
T , s−
T , sLT and the
corresponding ultimate strains e+
L , e−
L , e+
T , e−
T , eLT . If we assume linear elastic behavior up to
failure, the ultimate stresses are related to the ultimate strains by
s+
L = E1e+
L s−
L = E1e−
L sLT = G12eLT
s+
T = E2e+
T s−
T = E2e−
T
Typical experimental values of the effective lamina strengths for selected composites are given
in Fig. 2. Note that the transverse tensile strength, s+
T , is the lowest of all the strengths. This is
because, when the loading is perpendicular to the fibers, stress and strain concentrations occur
in the matrix around the fibers and this reduces the tensile strength of the matrix material. This
condition is often responsible for what is typically known as the first-ply failure.

2. Figure 1: Stress-strain curves for uniaxial and shear loading showing lamina in-plane shear
strengths and ultimate strains.
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3. Figure 2: Typical values of Lamina strengths for several composites at room temperature.
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4. 2 Multiaxial strength criteria
In the cases of off-axis or multiaxial loading, we assume that lamina failure can be character-
ized by using a multiaxial strength criterion that incorporates the gross mechanical strengths
described above. The objective of such a theory is to provide the designer with the capability
to estimate quickly when lamina failure will occur under complex loading conditions other than
simple uniaxial or shear stresses. Most of the failure criteria for anisotropic composites are based
on generalizations of previously developed criteria for predicting the rtansition from elastic to
plastic behavior in isotropic metallic materials. As such, they make use of the concept of a f ailure
surface or f ailure envelope by ploaating stress components in stress space. The coordinate axes
for the stress space generall correspond to the stresses along the principal material axes. The
theory predicts that those combinations of stresses whose loci fall inside the the failure surface
will not cause failure, whereas those combinations of stresses whose loci fall on or outside the
surface will cause failure. Since we are only dealing with 2D state of stress, the failure surface
would be 2D.
2.1 Maximum stress criterion
This criterion for orthotropic lamina predicts failure when any principal material axis stress com-
ponent exceeds the corresponding strength. Thus to avoid failure, the following setof inequalities
must be satisfied.
−s−
L < σ1 < s+
L (1)
−s−
T < σ2 < s+
T (2)
|τ12| < sLT (3)
It is assumed that shear failure along the principal material axes is independent of the sign of the
shear stress τ12. Typical failure surface is as shown in the Fig. 3. In Fig. 4 comparison of theo-
retical failure surfaces with experimental biaxial failure data for a unidirectional graphite/epoxy
composite is shown. Since the strengths along the principal material directions provide the input
to the criterion, we would expect the agreement to be good when the applied stress is uniaxial
along those directions. Due to lack of stress interaction in the Maximum Stress Criterion, how-
ever, the agreement is not so good in biaxial stress situations. Scatter in composite strength
tests is typical of these materials.
Biaxial stress fields can be obtained indirectly by using off-axis uniaxial loading tests or off-
axis shear loading tests. The loading shown in Fig. 5 produces the following biaxial stress state
along the principal material axes.
σ1 = σxcos2
θ (4)
σ2 = σxsin2
θ (5)
τ12 = −σxsinθcosθ (6)
For the above described off-axis ply test, the predicted values of the failure stress σx as functions
of θ can be plotted. For a given θ, the corresponding value of the stress would be found by
drawing a vertical line at that angle and finding where it intersects with the curves.
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5. Figure 3: Maximum stress, strain and Tsai-Hill surface in stress space.
Figure 4: Comparison of predicted failure surfaces with experimental failure data for
graphite/epoxy.
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6. Figure 5: Off-axis uniaxial test of a unidirectional lamina specimen.
For off-axis shear test described in Fig. ??. The applied shear stress, τxy, generates the
following biaxial stress state along the principal material axes:
σ1 = 2τxysinθcosθ (7)
σ2 = −2τxysinθcosθ (8)
τ12 = τxy(cos2
θ − sin2
θ) (9)
The importance of the sign of the applied shear stress in the interpretation of test results is
described below. For example, if θ = 45◦, then the above relations reduces to σ1 = τxy, σ2 =
τxy, τ12 = 0. Thus, positive applied shear stress would produce longitudinal tension and rtans-
verse compression along the principal material axes. On the other hand, a negative applied shear
stress would produce longitudinal compression and transverse tension, as shown in Fig. ??a -
b. Given the fact that the transverse tensile strength is much lower than the other strengths,
the importance of the sign of the applied sheae stress is now obvious. It is easy to visualize a
situation where a negative shear stress of a certain magnitude could cause a transverse tensile
failure, whereas a positive shear stress of the same magnitude would not cause failure.
2.2 Maximum strain criterion
This criterion predicts failure when any principal material axis strain component exceeds the
corresponding ultimate strain. In order to avoid failure according to this criterion, the following
set of inequalities must be satisfied
−e−
L < ε1 < e+
L (10)
−e−
T < ε2 < e+
T (11)
|γ12| < eLT (12)
2.3 Quadratic interaction criteria
The so-called quadratic interaction criteria also evolved from early failure theories for isotropic
materials, but they differ from the Maximum stress and maximum strain criteria in that they
include terms to account for interaction between the stress components, and the quadratic forms
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7. of the equations for plane stress lead to elliptical failure surfaces. The most general form is given
by
A(σ2 − σ3)2
+ B(σ3 − σ1)2
+ C(σ1 − σ2)2
+ 2Dτ2
23 + 2Eτ2
31 + 2Fτ2
12 = 1 (13)
In order to avoid failure, the left hand side of the equation must be ¡ 1. This criteria is
extended for orthotropic, transversely isotropic lamina as below
σ2
1
s2
L
−
σ1σ2
s2
L
+
σ2
2
s2
T
+
τ2
12
s2
LT
= 1 (14)
In the above expression, the second term is negligible and hence usually neglected to give
the final form of the criteria as below
σ2
1
s2
L
+
σ2
2
s2
T
+
τ2
12
s2
LT
= 1 (15)
Failure is avoided if the left hand side of the equation is less than 1.
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