Failure criteria & Failure modes

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Strength of orthotropic lamina

The strength is defined by 5 quantitiesTh t th i t l ith th di tiThe strength varies strongly with the directionThe failure analysis is always done in the (L,T) frame

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1. Maximum stress theory

The failure occurs if one of the stresses in the naturalaxes (L,T) exceeds the corresponding allowable stress.To avoid failure, the material must satisfy the followingI litiInequalities:

In compression:

Assumes that the failure modes are independent !

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Example: Glass‐epoxy composite With the normalized properties:

Lscale!!

ogarithm

ic

L

LTLo

T

T

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2. Maximum strain theoryy

The failure occurs if one of the strains in the naturalaxes (L,T) exceeds the corresponding allowable strain.To avoid failure the material must satisfy the followingTo avoid failure, the material must satisfy the followingInequalities:

In compression

If the material is elasticlinear until failure, 

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Comparison Maximum stress theory – Maximum strain theory

Maximum stress criterion Maximum strain criterion

In stress space

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Expressing the maximum strain criterion in stress space:

7Maximum stress

criterion

Maximum strain theory:The failure occurs if one of the following inequalities holds:

Poisson effectPoisson effect

>

>

L

>

L

T

The maximum stress theoryand the maximum strain theoryyIgnore the interaction between

the failure modes.Tsai‐Hill criterion 8

Stress deviator tensor

Tsai‐Hill criterion, preliminary: Von Mises criterion

Stress deviator tensor

obtained by subtracting the hydrostatic stress from the stress tensorThe stress deviator has the same principaldirections as the stress tensor. The invariantsJ1 , J2 and J3 of the stress deviator are defined by

Because J1=0, the stress deviator tensoris in a state of pure shearis in a state of pure shear

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Because J1=0, the stress deviator tensor is in a state of pure shear.Von Mises criterion

1 , pvon Mises had the intuition that the yielding of materials begins when

the second deviatoric stress invariant J2 reaches a critical value. 

It is straightforward to determine the critical value from a uniaxial tension test:

In principal axes:

The criterion is also called the criterion of •Maximum distortion strain energy (Hencky) •Octahedral shear stressOctahedral shear stress

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Maximum distortion strain energy (Hencky) 

In principal axes

Stress‐strain relationship

In principal axes:

Hydrostatic stress state: uniform stress:

Proportional to J2

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Yielding occurs when:

In plane stresses: 

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For orthotropic materials, the criterion needs to be expressed in the material axes (L,T)von Mises in an arbitrary (non principal) frame:

Extended to anisotropic behaviour by Hill (1948).By analogy, Tsai‐Hill assume that failure occurs if the inequality is violated:

Plane stressesPlane stresses:

Finally:

Consistent with T !!! 13

LTsai‐Hill: example of glass‐epoxy composite

T

Both L and Tare in traction

•Accounts for the interaction between the failure modes

Tsai‐Hill

Accounts for the interaction between the failure modes•Conservative•The maximum difference occurs at the change of failure modes•One must transform the stress state in the (L,T) frame

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von Mises in principal plane stresses:

Tsai‐Hill in plane stresses (L,T)

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Importance of the sign of shear stress on the strength of composites

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Glass‐epoxy

Step 1:Transform in(L,T) frame:

Step 2:Tsai‐Hillcriterion

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Ultimate strength

xy = 5 MPaxy = 75.36 Mpa

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Failure modes

•Breaking of the fibersk f h•Microcracking of the matrix

•Debonding (separation of the interface between matrix and fibers)•Delamination (separation of laminae from each other)

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Microcracking in glass‐reinforced epoxy

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1. Failure under longitudinale tensile load

Individual fibers break in a random mannerat less than 50% of the ultimate load. Depending on the type of fibers and matrix and VDepending on the type of fibers and matrix and Vf

The following failure modes are observed:•Brittle fracture.•Brittle with fiber pullout (matrix breaking away from the fibers).•Shear failure of the matrix and debonding.

Glass fibers:   Vf

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2. Failure under longitudinale compression load

Failure modes:•Transverse tensile failure•Fiber microbuckling (extension mode or shear mode)•Shear failure Shear failure

Vf small Vf large

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Microbuckling in shear mode (large Vf)

Extension mode(low Vf)

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Based on the assumption of transverse tensile failure of the matrix,and on the empirical formula (3.43) (composite transverse breaking strain):

one can develop a model for the longitudinal compressive stress:

At failureAt failure:

Rule of mixtures:

Dominated by 

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ythe matrix !

3. Failure under transverse tensile loads

Failure modes:Fibers perpendicular to the loading produce stress concentrations at the interface and in the matrixconcentrations at the interface and in the matrix.The failure occurs because of the matrix or the interface  tensile failure (occasionally highly orientedfibers may also break in the transverse direction)

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4. Failure under transverse compression loads

Failure modes:•Matrix shear failure•Matrix shear failure plus debonding

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5. Failure under in‐plane shear loads

Failure modes:M i h f il•Matrix shear failure

•Constituent debonding

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