Failure theories in brittle material.docx

8/13/2019 Failure theories in brittle material.docx

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Fracture Mechanics 2013

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Table of ContentsIntroduction ................................................................................................................................... 3

Tensile tests ............................................................................................................................... 5

Stress-Strain curve ..................................................................................................................... 6

Principal stress ........................................................................................................................... 6

Failure theories for brittle materials (franklin, 1968) .................................................................... 6

Brittle Materials exhibit the following characteristics .............................................................. 7

Maximum normal stress theory (even materials) ...................................................................... 9

Maximum normal stress theory (uneven materials) ................................................................ 11

Coulomb-Mohr or Internal friction (IFT) ................................................................................ 11

Mohrs Theory ......................................................................................................................... 12

Modified Mohr Theory ............................................................................................................ 13Maximum Normal-Strain theory. ............................................................................................ 16

Maximum Strain-Energy theory .............................................................................................. 16

Industrial application of failure theories ...................................................................................... 17

Conclusion ................................................................................................................................... 18

Works Cited ................................................................................................................................. 20


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Introduction In material engineering failure can be said to occur at a point when the material

loses its elastic property and starts behaving in an inelastic manner (A. R. Ingraffea,

2013). In science, materials or rather components fail simply because the materials

strength is lower as compared to the stresses applied on the component. For any load

combination, the shearing stress and normal stress combination in the material are the

two stresses responsible for any failure (S.A.F Murrell, 1972). From the above

discussions, failure can simply refer to fracture although in some components it can be

termed as yielding if and only this property of yielding distorts the material in that it can

no longer perform its intended function properly.

In engineering failure of materials can be classified as either brittle failure or

ductile failure. These materials exhibit different modes of failure which is further

dependent on the type of loading either static or dynamic (Science, 2013). For brittle

materials there is sudden failure once a material is subjected to a large component of the

load, thus there is no yielding before the material fails. Examples of brittle materials

include ceramics and some polymers. Unlike in ductile materials where failure occurs in

a systematic manner in that there is yielding whereby the material has to undergo plastic

deformation in preparation for failure to occur. Examples of ductile materials include

most metals. Usually for brittle materials where ductility is less than 5% failure islimited by the tensile strengths of the material (Chow, 2012).

Theoretical strength is the stress required to break the bond between atoms

(atomic bond) thus causing the separation in atoms (Anderson, 2006). The value of the

theoretical strength has been estimated to be approximately equal to a third of the

Youngs modulus (E/3). Despite this fact however, it can be seen that most materials

usually fail at a stress which is about one-thousandth or one-hundredth of the theoretical


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which when a material is deformed elastically it absorbs energy and then during

unloading the same energy is recovered.

Tensile testsThis is a standardized test used by researchers and allows them to share the results

amongst themselves in the form of the stress-stain curves (Dowling, 1993). In simple

tension test different materials are pulled at its from their two ends, the materials will

have to elongate in respect to the load and this values are recorded. From this

observation one can easily come up with the maximum strength of the materials. For

brittle materials this maximum strength is taken as the ultimate tensile of the material

while I ductile materials it called the upper yield point of the material (Theories of

Failure_Learn Engineering, 2012).The figure below called the test specimen is usually

chosen with standard dimensions as below. The values obtained are used to draw a

characteristic stress-stain curve. This curve gives the behavior of ductile and brittle

materials and further the need to have separate criteria for failure. But before the curve

true values of stress and strain are calculated using the cross-sectional area of the

material before loading.


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Figure: Test specimen

The curve below is a stress-strain curve of a material. At 0.2% it shows the property

of a brittle material where failure occurs and is indicated by the first dotted line.

Stress-Strain curve

Figure 1: a curve of stress against strain combination for both ductile and brittle

materials

Principal stress

The maximum normal stress that can occur at a given point is referred to as the

principal stress (Theories of Failure_Learn Engineering, 2012). This value can simple

be determined using the Mohrs circle and its analysis. The principal stresses are very

vital in the understanding of the failure theories

Failure theories for brittle materials (franklin, 1968)Currently, research is still being carried to try and explain and quantify the

strength of materials in terms of their properties and the atomic structure. These atomic

models are therefore not suitable when it comes to the design of structures and


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machines. Having this problem in mind, we then find a resolution in the failure theories

which are usually as a result of observations and testing for a long period of time. The

main aim of the failure theories is to give an extension of the values of strength that are

obtained uniaxial tests and states of stress in multi-axial form that practically exist in

nature. A good knowledge of the failure theories is very important in the design of

structures and mechanical equipment.

Considering the analogy of the weight lifter, the lift can be able to lift a

maximum of 100kg in a simple manner. The same weight can be lifted in a different

way. From this concept we can then assume that that the lifting ability or rather capacity

of the lift is about 100kg. If this case applies to both the two different scenarios, then it

forms a basis for failure theories. This concept of weight lifter can be applied to

different materials and has been the backbone of the failure theory in materials. For this

case the material can undergo simple test in tension. Or rather simple force test and thus

one can determine the maximum load capacity and capability a material has to be

subjected to. This understanding can be extrapolated to complex loading of the

materials. The assumption above has been the breaking point towards the understanding

of the failure theories. The two main prerequisites in the understanding of the theory of

failure are therefore, the principal stresses and the simple tension test concept (Theories

of Failure_Learn Engineering, 2012). These theories are explained in the foregoing

discussions.

Brittle Materials exhibit the following characteristicsFirst and foremost is that, the tensile strength of brittle material is less than its

compressive strength. The reason for this is that, when brittle material is loaded in

tension the fracture shown is a normal stress phenomenon. Secondly, brittle materials

do not have a definite or specific yield point, thus, they fail by what is called brittle

fracture. In this case the highest stress in compression and tension determines the


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ultimate strength. Third the material tests of tensile and compressive tests are critical in

determining the material behavior. And finally, brittle materials have an elongation of

less than five percent that is to say the strain is f 0.05 at failure.

Failure can be viewed as complete fracture or separation of a member. It may

also come as a result of excessive deformation either elastic or inelastic. There are

basically three modes of failure. These are: elastic deformation which is in excess and

this involves buckling, vibration, stretch, bending or twist, yielding which encompasses

creep at very high or elevated temperature, yield stress in design and at room

temperature plastic deformation. Finally, fracture which is a mode that involve fatigue

i.e. fracture in a progressive manner, at elevated temperature stress rupture and for

brittle material sudden fracture. Its important design factor is ultimate stress.

Basically, there are two theories that explain failure in brittle materials. One type

of the theory is the classical theories which only assume the materials in the question to

be uniform in structure. For non-uniform material structure for example large and thick

castings where large flaws appear, then fracture mechanics theory will be used to

determine failure with high accuracy. These theories are further classified into the

following to try and explain the failure mechanism in brittle materials.

a) Maximum normal stress theory(even materials)

b) Maximum normal stress theory(uneven materials)

c) Coulomb-Mohr theory

d) Modified-Mohr theory

These theories are developed both from experimental data and hypotheses in that,

tensile tests are first carried out and experimental data for failure obtained. Then Mohrs

circle plots are used to determine the correlation between experimental values to the

state of stress (Collins, 1993). A theory for failure is then developed from the concept


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responsible for the mechanism of failure. Finally, establishment of a design envelop is

done depending on the empirical and theoretical design equation. In static loading these

theories possess common features which include describing explicitly the mathematical

relationship between the external loading and the stress that occur at critical point in a

multi-axial stress state, basically they are based on measurable physical properties in the

materials and finally every theory has a relationship to a measurable failure criterion.

From observation and tests it can be concluded that all the theories of failure say the

same thing. That is to say, when the maximum magnitude of strain and stress in a multi-

axial state is more than or equal to the value of stress and strain that leads to failure in a

unit axis test of stress then the part in question fails (Fenster, 2007). Each of the above

theories is discussed below:

Maximum normal stress theory (even materials)In this theory, there is a limiting factor that determines failure. In that, a brittle

material will fracture or rather rupture when the maximum principal stress of the

material reaches a particular value called the limiting factor (Timoshenko, 1970). This

limiting value is always termed as the tensile strength which is normally obtained using

the normal uniaxial tensile test (Sriram, 2012).

In this case the ultimate tensile stress U is equal to the maximum principal stress 1

thus the equation:

It is important to calculate the principle stresses, 1 , 2 and 3 for any given state of

stress. Yield function in this case may be designed as follows

)

If f0

then the function is not defined. The surface for yielding is given as follows


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Fig2: six stress state

In a 2-dimensional case , 3=0, then the equations are modified to

Yield stress in tension is greater than yield stress in shear for experimental results

Figure 3: Failure envelope of the maximum principal stress theory

When one of the three important principal stresses reaches a strength that is permissible

then failure will have to occur (MA Meyers, 2009). According to this theory, if the

stresses i.e. principal fall within the quadrants then the part will resist failure(failure will

not occur)


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Maximum normal stress theory (uneven materials)This theory only holds in a case where the ultimate tensile strength of is less than the

maximum principal stress. That is, 1> U. In the diagram (b) below it can be seen that

the modulus of the maximum or rather critical principal stress is greater that the

modulus of the ultimate tensile stress (Caceres-Valencia, 2006).

Figure 4:

Coulomb-Mohr or Internal friction (IFT) If the maximum normal stress theory is modified in a manner such that the opposite

corners of the first and third quadrants of the failure envelop are connected. This

construction results in a failure envelope which is hexagonal. This theory only caters for

the uneven properties of brittle materials


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Figure 5: diagram showing how failure occurs using Coulomb-Mohr theory

Mohrs Theory This theory takes into consideration the following factors uni-axial ultimate stress in

compression, uni-axial ultimate stress in tension and pure shear. It states that a material

will fail if and only if there is a state of stress on the envelope which is tangential to the

three Mohrs circles that are correspondent to pure shear, uni-axial ultimate stress in

tension and uni-axial stress in compression (Dieter, 1961)

Figure 6 shows the behavior of a material in uni-axial ultimate stress in tension,

compression and pure shear.

T c

s


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Figure 7

Modified Mohr TheoryIs the most preferred theory for brittle materials. It comes as a result of the modification

of the Coulomb-Mohr theory. It can be shown diagramatically by the figure below.

Figure 8


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In this case the quadrants of interest are the first and third quadrant. And a derived

diagram is as given below

Figure9: Modified-Mohr Failure Theory for brittle materials.

Safety factor in zone 1, Modified Mohr theory is given as

where is the ultimate tensile of the material and is the maximum

principal normal stress.

Safety factor in zone 2, Modified Mohr theory is given as

)

Modified-Mohr theory applicable in thisarea


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The effective stress in this case therefore becomes;

Factor of safety (Measurement Group, 1993)

Modified Mohr theory. Effective stress; , where ultimate

tensile strength of the material and is the effective stress and is obtained as

) and if MAX


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Maximum Normal-Strain theory.This theory only takes place in the elastic range. It is also called Saint- Venants theory.

The theory assumes that inelastic behavior or rather fracture is controlled by specific

principal strain which is maximum. In this case failure will have to occur at a specific

part in the body if it is subjected to an independent state of strain when this principal

strain reaches the limit (Jayne, 1993). Failure is said to occur if either of normal strain

that comes as a result of the normal stresses is equal or more than the normal strain that

corresponds to the yield strength of the material loaded in the uni-axial compression or

tension (J.F. Young, 1998). The equation below show this conditions for failure to

occur:

Where S y is the yield strength for a biaxial state of stress.

Maximum Strain-Energy theoryWhen total strain energy in a given volume is more than or surpasses the strain

energy in the same volume which corresponds to the yield strength in either

compression or tension then failure is said to occur. Failure due to stain energy at a

given point in a body that is subjected to any state of stress will begin only when the

density of energy is equal to the energy absorbed by the material that has been subjected


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to elastic limit in a stress state in a uniaxial direction. This theory is applicable to many

tpe of engineering materials and failure will have to occur at a specific part in the body

if the total strain energy associated principal stresses reaches the limit (Jayne, 1993).

Failure is said to occur if either of normal strain that comes as a result of the normal

stresses is equal or more than the total strain energy that corresponds to the yield

strength of the material loaded in the uni-axial compression or tension (J.F. Young,

1998).

For uni-axial loading, the strain energy stored per unit volume (Us) is given as;

But in a biaxial state of stress, the strain energy are given as

Due to the invention of more simpler and suitable theories, this theory is no longer in

use.

Industrial application of failure theories

Currently, some FEA solvers are well designed to use failure theories. What one

requires is to specify the type of failure criterion in the solution method. It can be noted

that shear strain energy theory is the most commonly used method (Lecture Notes). The

software are integrated in such a way that they can give Von-mises stress within the

material which is normally based on the theory of shear strain energy. The user can


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simply verify whether this maximum induced stress in the body has surpassed the

allowable maximum stress value. In most design, the factor of safety is a very important

factor in that it allows the engineer or rather loader to take care of the loading scenarios

that are very bad (Theories of Failure_Learn Engineering, 2012).

ConclusionConclusively, it is important to remember that brittle material usually show a much

smaller ultimate strength in tension than in compression. The reason behind this is that,

unlike for yielding in ductile materials, fracture or failure of brittle materials loaded in

tension gives a phenomenon of normal stress. The material fails due to the fact that

normal tensile stresses separate or fracture the part in the normal direction to the plane

of the principal stress or the maximum normal stress. For compression, the literature is

quite distinct. If a brittle material is loaded in compression, the normal stress will not be

able to separate the part along the direction perpendicular to the plane of maximum

normal stress. If the separating normal stresses or tensile stresses are absent, then shear

stresses will have to come into play and work to separate or fracture the available

material in the direction of maximum shear stresses. For a pure compression, the

direction is at forty-five (45) degrees to the plane of loading. However, brittle materials

are usually very strong in shear a value that is almost of equal strength in shear as in

tension. The reason behind this is that it takes a great deal of compressive normal stress

for a shear stress to be created which is capable of creating a fracture in brittle material

loaded in compression.

The theories were discussed bearing in mind a 2-dimensional state of stress which is

similar to a three-dimensional but 3D is a little bit more abstract. Failure theories in

brittle fracture will divide the 1- 2 region into 4 quadrants. In the first quadrant, both

normal/principal stresses are always positive:


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If both 1 and 2 are positive values (tensile), then fracture is anticipated to occur when

either one of the principal stresses, 1 or 2 reaches S ut. If both 1 and 2 are negative

values (compressive), the fracture will occur when the value of one of the principal

stresses, - 1, - 2 reaches S uc. It has been discussed earlier that the value of S uc is

usually greater than S ut.

for the other two quadrants, where one of the principal stress is positive and the other is

negative, then Columb-Mohr theory of failure is the most applicable theory to predict

failure. The theory is also easy to learn and use. The Columb-Mohr theory failure line just connects the points of failure as represented in the figure with bold lines. In

quadrant IV and using the magnitudes of the stresses only:

S ut

S utS uc

S uc

III

III

IV


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From this formula ( 1, 2) is the load point (two principal stresses), and n is the factor

of safety associated with that load point. For Quadrant II, switch S ut and S uc.

Experimentally values of the theories of failure can be compared with the theoretical

values using the diagram below

Figure: Gray Cast Iron biaxial data as compared to various failure criteria (Dowling,

1993)

Works Cited

Theories of Failure_Learn Engineering. (2012, December). Retrieved December 10, 2013, fromwww.learnengineering.org: http://www.learnengineering.org/2012/12/teories-of-failure.html

A. R. Ingraffea, K. H. (2013). Engineering Fracture mechanics. An International Journal .

Anderson, T. (2006). Fracture Mechanics- Fundamentals and Applications, Third Edn. BocaRaton: CRC Press.

Caceres-Valencia, P. G. (2006, october 23). Theory of Failure in brittle materials. RetrievedDecember 08, 2013, from Failure Theories:www.bing.com/failuretheoriesinbrittlematerials/

nS S ucut

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Chow, P. C. (2012). Theory of failure ib brittle materials. International journal of Damagemechanics .

Collins, J. (1993). Failure of Materials in Mechanical Design: Analysis, Prediction, Prevention. John Wiley & sons.

Dieter, G. E. (1961). Mechanical Metallurgy. NewYork Toronto London: McGRAW-HILL.

Dowling, N. E. (1993). Mechanical behaviour of Materials. Englewood Cliffs: Prentice-Hall.

Fenster, A. C. (2007). Static Failure Theories. Advanced Strength and Applied Elasticity .

franklin, H. G. (1968). classic theories of failure of anisotropic materials. Retrieved December 9,2013, from www.sciencedirect.com:http://www.sciencedirect.com/science/article/pii/0015056868900043

J.F. Young, S. R. (1998). The science and techomlogy of civil engineering materials. PrenticeHall.

Jayne, J. B. (1993). Mechans of Wood and Wood composites. Krieger Publishing.

K.R.Y Simha, K. S. (2001). Mechanics of Fracture. Engineering Fracture Mechanics Journal , 53.

Lecture Notes. (n.d.). Retrieved December 10, 2013, from classes.mst.edu:http://classes.mst.edu/civeng120/lessons/failure/theories/index.html

MA Meyers, K. C. (2009). Mechanical Behavior of Materials. McGraw-Hill.

Measurement Group. (1993, December). Failure Prediction and Avoidance. Experimental stressanalysis Notebook (22), pp. 6-11.

S.A.F Murrell, P. D. (1972). The Thermodynamics of Brittle Fracture Initiation under triaxialstress conditions. international journal of Fracture Mechanics , 167-173.

Science, E. (2013). Fracture mechanics in solids. Engineering Fracture Mechanics Journal .

Sriram. (2012, september 18). Theories of Failure. Retrieved December 8, 2013, from Theoriesof failure: http://www.google.com/theories of failure

Timoshenko, S. a. (1970). Theory of Elasticity. New York: McGraw-Hill.

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Failure theories in brittle material.docx