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FAILURE CRITERIA

FOR COMPOSITES

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(tvid : 7a and 7b.)

7.1 Failure Criteria and Importance

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• Although increasing the size of the parts that make up a solid system allows us to obtain a safer structure in terms of strength, it will result in a more expensive system.
• Reducing the dimensions will allow us to obtain a more economical (cheaper) system, but this time it will risk durability.
• Therefore, obtaining an optimum design in terms of durability and economy and producing a problem-free system is only possible by calculating the strength limits, that is, the minimum dimensions of the parts. This is the fundamental subject of strength science and is a complete engineering study.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Calculation of these minimum dimensions is only possible by knowing and applying the failure (damage) criteria very well.
• However, when changing part dimensions, it should be ensured that there is no loss of functionality of the system.
• This is also related to deformation calculations, which are the subject of strength, and appears as a third criterion.

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• As the main failure criteria in isotropic materials, we can list Tresca and Von-Mises Yield Criteria, Rankine, Coulmb and Mohr Fracture criteria. These criteria are explained in «Strength of Materials» lessons.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Nano- or macro-sized particle reinforced, chopped fiber (Whiskers) reinforced or multi-directional continuous fiber reinforced composites show isotropic behavior and are evaluated according to the above-mentioned criteria.

• Failure criteria tell us whether yielding or fracture will occur at the examined point of the object, depending on the stress or strain values.

• Failure or damage can be defined as deformity and loss of functionality that occurs as a result of overload in a structure. Failure occurs when yielding occurs for ductile (formable) materials, and when rupture or fracture occurs for brittle materials.

• It is necessary to choose the appropriate criterion for damage detection according to the type of material and mechanical behavior of the object. It is extremely wrong to use a suitable damage criterion to determine the yielding of ductile materials in determining the fracture of a brittle material, or to use a criterion used for isotropic materials for a composite orthotropic material. Such mistakes also have negative effects on our engineering careers.

• For this reason, it is very important that we have full knowledge of the damage criteria.

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7.3 Main Failure Criteria in Orthotropic Composites

Unidirectional or bidirectional continuous fiber reinforced composites and sandwich composites show orthotropic properties and are examined according to the above criteria. These criteria are also used in damage detection of layered composites, which will be explained in the next topic. However, it is possible to find much different criteria developed specifically for composites in the literature.

1- Maximum Stress Criterion

2-Maximum Strain Criterion

3- Tsai – Hill Criterion

4- Modified Tsai – Hill Criterion

5- Tsai-Wu Criterion

6- Hoffman Criterion

7.2 Our aim in this chapter is to explain the main failure criteria used for composite materials with orthotropic properties. Criteria and examples will be explained according to the plane stress situation.

Now we will discuss the 6 criteria mentioned above one by one and understand with examples how damage (failure) detection is done for a single layer....>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

7. Failure Criteria for Composites

Note: The term Theory can also be used instead of Criterion

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

7.4. Let's remember the symbols of experimental material strengths according to Local Axes:

Direction 1

Direction 2

Tip 6: In order to detect failure (damage) at a point, the local stresses or local strains at that point must be known or calculated. Because all failure criteria have been developed according to these.

7. Failure Criteria for Composites

Experimental Strength Value

7.5.1 Maximum Stress Criterion

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According to this criterion, if all of the inequalities 7.1a-c shown on the side, for local stresses at a point of the composite are met, no damage will occur at that point.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In other words, if at least one of these three inequalities is not met according to this criterion, damage will occur at that point.

This criterion gives much better and more realistic results in detecting the fracture of brittle materials, especially at points where tensile stresses exist.

(7.1a-c)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

7.5.2 Maximum Strain Criterion

In other words, if at least one of these three inequalities is not met according to this criterion, damage will occur at that point.

This criterion gives results much closer to reality in detecting the fracture of brittle materials and especially at points where tensile stresses occur.

(7.2a-c)

Strains at the time of damage are calculated from Hooke's relations (equations 7.3.a-e).

(7.3a-e)

7. Failure Criteria for Composites

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

According to this criterion, if all of the inequalities 7.2a-c shown on the side for local strains at a point of the composite are met, no damage (failure) will occur at that point.

7.5.3 TSAI – HILL Criterion

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This criterion is an adaptation of the Von Mises distortion energy damage theory, which is valid for isotropic materials, to anisotropic materials.

According to this criterion, for damage to occur in a layer, the following condition 7.4 must be violated:

How to calculate the constants G₁, G₂, G₃, and G₆? …we will examine this now…>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Since σ₃ = τ₃₁ = τ₂₃ = 0 in the case of 2-Dimensional (Plane) stress, equation 7.4 turns into equation 7.5 below:

This criterion gives very good results

in damage detection in composites with the same tensile and compressive strengths,

in composites with ductile behavior and

in points subject to tensile stress.

In case of 3D Stress :

(7.4)

(7.5)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Final forms of Equation 7.5:

In case of plane stress;

7. Failure Criteria for Composites

(7.5)

Let's rewrite equation 75 on this page:..>>

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If these values are substituted into Equation 7.5:

According to the Tsai-Hill Damage Criterion, if inequality 7.6 is violated in the case of plane stress, damage occurs.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(7.6)

,

,

It is obtained from these four equations as

,

7. Failure Criteria for Composites

If you pay attention to inequality 7.6, only tension strengths are taken into account in the Tsai-Hill criterion. However, in brittle materials, the compressive strength may differ from the tensile strength. In this case, this criterion needs to be modified to suit brittle materials…>>

7.5.4 Modified TSAI – HILL Criterion

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This criterion is a modified version of the Tsai-Hill criterion, taking into account compressive strength.

​

According to this criterion, if inequality 7.7 on side is violated for the plane stress condition, damage (failure) occurs at the examined point.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(7.7)

This criterion is a general criterion and can be used not only for brittle composite materials, but also for all orthotropic material types and all point stress states.

7. Failure Criteria for Composites

The meaning of the values in the denominator :

7.5.5 TSAI – WU Failure Criterion

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According to this criterion, if the following inequality is violated, the material will be damaged:

The constants H₁, H₂, H₆, H₁₁, H₂₂, and H₆₆ will be found for 5 different strength values of the layer. H₁₂ can only be calculated experimentally.

This failure criterion is also a general criterion. It takes into account both the compressive and tensile strength of the materials. It can be used for all orthotropic material types and all point stress situations.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(7.8)

Now the determination of these constants will be explained…>>

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Equation 7.8 at the time of damage :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(7.9a)

(7.9b)

(7.9c)

(7.9d)

at the time of damage :

at the time of damage:

at the time of damage :

at the time of damage:

at the time of damage :

(7.9e)

(7.9f)

7. Failure Criteria for Composites

Let's write equation 7.8 here again:

(7.8)

Only the 𝐻₁₂ calculation remains..>>

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7.5.5.1) Methods of Calculating the H₁₂ constant

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

a.) 1st Experimental Method :

If there is a loading that will create equal tensile stress in the 1 and 2 directions at the examined point;

(σ₁ =σ₂=σ , τ₁₂= 0)

At the time of damage, equation 7.8:

(7.10a)

Or any of the following combinations can be used in this method:

There may be different alternatives for the experimental setup. The important thing is that only normal stresses occur at the same intensity in directions 1 and 2 at the examined point.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

b-) 2nd Experimental Method :

For the instant of damage, from the stress transformation equations (Equation 6.5a-c)

(Or these values can also be seen from the Mohr Circle on the side..)

At the time of damage, equation7.8:

(7.10b)

Mohr Circle

7. Failure Criteria for Composites

are found.

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Per Tsai-Hill failure Criterion:

Per Hoffman Criterion:

Per Mises-Hencky Criterion:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

c.-) Method 3: Empirical Equations

Equations that generalize experimental measurement results or observational data are called empirical equations.

(7.10c)

(7.10d)

(7.10e)

7. Failure Criteria for Composites

7.5.6 Hoffman Criterion�

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According to this criterion, if the following inequality is violated, the material will be damaged:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(7.11)

This criterion gives better results

• in composites with different tensile and compressive strengths,
• especially in composites that exhibit brittle behavior.

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Example 7.1

The plane stress state at one point of a composite layer reinforced with continuous fibers with an orientation angle of 45^o is as shown in the figure. The properties of the composite layer are given in the table below.

At this point, whether damage will occur, check according to

a-Maximum Stress Criterion,

b- Maximum Strain Criterion,

c- Tsai-Hill Criterion,

d- Modified Tsai-Hill Criterion.

​

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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a-)

According to Maximum Stress Theory (Criterion)

Local stresses and local deformations must be obtained to detect damage (Tip Point 6)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

c=cos45^o , s=sincos45^o

-200 < 5,5 < 100

-45 < -12,5 < 25

-18 < 16,5 < 18

No failure occurs.

Solution:

From equation 6.7,

Calculation of local stresses:

All 3 inequalities are satisfied.

(Equation 7.1 inequalities are checked. The values must be adjusted so that the left side of these equations must be negative..)

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b-)

According to Maximum Strain Criterion:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

-13,3x10^-4 < 0,617x10^-4 < 6,67x10^-4

-14,1x10^-4 < -4,01x10^-4 < 7,81x10^-4

-22,5x10^-4 < 20,63x10^-4 < 22,5x10^-4

Calculation of Local Strains from Equation 2.16b:

From Equation7.3a-e,

Strains at time of damage:

We check the inequalities in Equation 7.2. (The left side of these inequalities must be negative.)

7. Failure Criteria for Composites

No failure occurs.

All 3 inequalities are satisfied.

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c-) According to Tsai-Hill Criterion,

d-) According to Modifiye Tsai-Hill Criterion:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1,1 > 1

0,92 < 1

Damage occurs.

No Damage occurs.

The no-damage condition specified in Equation 7.6:

The no-damage condition specified in Equation 7.7:

(It has been found before.)

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Example 7.2

The forces shown in the figure act on a unidirectional and continuous fiber reinforced composite plate manufactured with a 60o fiber orientation angle and the plate remains in static balance. In the triple straingauge rosette attached to a Q point on the plate, a is placed horizontally, b is placed vertically, and c is placed at a 45^o angle with the –x axis. Values measured with strain gauges:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

7. Failure Criteria for Composites

-,

Obtain the results for the criteria listed below yourself with the calculations.

ε_a = -12.49x10^-4, ε_b = 6.747x10^-4 , ε_c = -1.111x10^-4

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In terms of all criteria: Local stresses or local strains must be obtained for damage detection. (Tip point 6)

Solution:

From Equation 5.1b, the strain (unit elongation) in the c strain-gauge is:

From Equation 6.11, local strains are calculated:

Looking at the directions of the Strain Gauges :

From this equation 𝛾_xy is found:

Local Strains

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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From equation 2.16a, local stresses are calculated:

We calculate each term of the [Q] matrix as follows:

From Equa. 3.13, Minor poisson ratio:

Local Stresses:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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e-) According to Tsai-Wu Criterion:

If we choose the Mises-Hencky empirical equation among the alternatives for H12 (Eq. 7.10 e);

Then, according to the modified Tsai-Wu criterion, no damage occurs at the Q point.

Let's rewrite the calculated local stresses:

Strength Values of Composite Material (Given in the Question)

From Equation 7.8, No Damage Condition:

Constants are found from Equation 7.9a-e:

If we substitute all numerical values into Equation 7.8:

(is zero in all cases)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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f-) According to Hoffman Criterion:

If numeric values are placed :

Therefore, no damage occurs at point Q according to the Hoffman criterion.

Local Stresses Found:

Strength Values of Composite Material (Given in the Question)

(No Damage Condition)

The inequality of Equation 7.11 is checked.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 7.3*

A composite layer with a fiber orientation angle of 0^o is placed in a rigid mold. The inner surfaces of the mold and the outer surfaces of the layer are in frictionless contact, and there is no compression between the surfaces. The required material properties of the layer are given in the table below. According to this,

a-) How much can the temperature of this layer be increased within the strength limits? Determine according to the Maximum Stress Criterion.

b-) Check for damage according to Tsai Hill and Modified Tsai-Hill criteria for fiber routing angle θ = 30⁰ and ΔT = 50 ⁰C

(This example includes formula inferences regarding thermal loads.)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Solution…>>

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Stresses at any temperature difference ΔT :

(7.12)

(7.13)

a-)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Solution:

There is no limitation in the thickness direction perpendicular to the plane (3 direction). The deformation is free and hence no stress occurs in this direction. Therefore, this problem is a plane problem.

If we first calculate the minor poisson ratio:

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If ΔT is found from equation 7.12 :

Review for direction 1:

(7.14)

If we examine it according to the Maximum Stress criterion:

Then, at the exact moment of damage :

No-damage condition for direction 1 (Eq. 7.1a):

Review for direction 2:

Similarly

Maximum temperature rise for direction 1, in the constrained layer in directions 1 and 2 :

Maximum temperature rise for direction 2, in the constrained layer in directions 1 and 2 :

Eq. 7.1.b:

at the time of damage:

From eq. 7.13 :

(7.15)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Since ΔT has no effect on shear stress and shear deformation, shear stress is zero and there is no need to examine inequality 7.1.c.

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When the given values ​​are substituted:

(Temperature difference that will cause yield in direction 1)

Or the condition that no damage occurs in this layer :

(Temperature difference that will cause yield in direction 2)

Therefore, the temperature increase limit value in this layer is :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

b-) Check for damage according to Tsai Hill and Modified Tsai-Hill criteria for fiber routing angle θ = 30⁰ and ΔT = 50 ⁰C

According to Modified Tsai-Hill

no damage occur

According to for Tsai-Hill

damage occur

Answers:

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Example 7.3 (2021 - 2nd Visa Question)

The global stresses at point a of a unidirectionally reinforced composite layer with an orientation angle of θ= 45⁰ were determined as follows:

σ_x = 27 MPa, σ_y = 26 MPa, τ_xy = 15MPa

A triple strain-gauge rosette with a 120^o angle between them is glued to point a.

Material Properties

At this point, determine whether damage will occur according to

a-) Tsai-Hill criterion and

b-) Maximum Strain Criterion.

c-) Calculate the unit elongation (strains ε ) values read from each of the strain-gauges.

Answers:

c-)

ε_d=3,38x10^-4

ε_c=4,14x10^-4

ε_b=1,79x10^-4

a-)

0,64<1

No damage ocur.

b-)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

7. Failure Criteria for Composites