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LESSON 31: THEORIES OF FAILURE Introduction Failure: Every material has certain strength, expressed in terms of stress or strain, beyond which it fractures or fails to carry the load.

application of the theories. This has been explained in the table below.

Failure Criterion: A criterion used to hypothesize the failure. Failure Theory: A Theory behind a failure criterion.

Why Need Failure Theories?

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To design structural components and calculate margin of safety. To guide in materials development.

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To determine weak and strong directions.

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Failure is generally perceived to be fracture or complete separation of a member. However, failure may also occur due to excessive deformation (elastic or inelastic).

Failure Modes Excessive elastic deformation 1. Stretch, twist, or bending 2. Buckling 3. Vibration

Failure Theories

Ductile

Maximum shear stress criterion, von Mises criterion

Brittle

Maximum normal stress criterion, Mohr's theory

A brief summary of the common theories used to predict yielding of ductile materials follows: Stress Theories •

Maximum Principal Stress Theory (Rankine, Lamé) Maximum Octahedral Shearing Stress Theory

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Maximum Shear Stress Theory (Tresca, Guest, Coulomb)

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Yielding • • •

Fracture •

Plastic deformation at room temperature Creep at elevated temperatures Yield stress is the important design factor

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During the latter part of the 19th century and continuing up to the present, a number of basic failure theories were proposed and tested on a few materials. Most of the theories were based on the assumption that failure occurs when some physical variable such as stress, strain, or energy reaches a limiting value. When a component is subject to increasing loads it eventually fails. It is comparatively easy to determine the point of failure of a component subject to a single tensile force. The strength data on the material identifies this strength. However when the material is subject to a number of loads in different directions some of which are tensile and some of which are shear, then the determination of the point of failure is more complicated. Several theories of failure have been proposed, each of which gives good results for some materials under some stress states. Unfortunately, none of the theories gives uniformly good results when applied to a large variety of materials and loading conditions. Thus different materials require different theories for their analysis. Materials being brittle and ductile have led to the division of theories into two groups based on the field of

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Material Type

Sudden fracture of brittle materials Fatigue (progressive fracture) Stress rupture at elevated temperatures Ultimate stress is the important design factor

Strain Theories •

Maximum Strain Theory (Saint-Venant)

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Total Strain Energy Theory (Beltrami-Haigh)

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Maximum Distortion Energy Theory (Huber-Henky-von Mises)

The Maximum Principle Stress Theory The theory associated with Rankine. This theory is approximately correct for cast iron and brittle materials generally. According to this theory failure will occur when the maximum principal stress in a system reaches the value of the maximum stress at elastic limit in simple tension. For the two-dimensional stress case this is obtained from the formula below (ref page on Mohr’s circle).

The design Factor of Safety for the two dimensional case=FoS = Elastic Limit from tensile test / highest principle stress.

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This theory is appealing since for some ductile materials (e.g. hot-rolled carbon steel) we can observe slip occurring at orientations, which appear to agree with the maximum shear planes. Recall the orientation of the slip planes from your tensile and torsion tests of hot rolled carbon steel. This theory is quite simple to apply and gives reasonable results when applied to many ductile materials subjected to fairly simple loading states.

The Maximum Shear Stress Theory The theory associated with Tresca and Guest. This is very relevant to ductile metals. It is conservative and relatively easy to apply. It assumes that failure occurs when maximum shear strength attains a certain value. This value being the value of shear strength at failure in the tensile test. In this instance it is appropriate to choose the yield point as practical failure. If the yield point = σ y and this is obtained from a tensile test and thus is the sole principal stress then the maximum shear stress σ sy is easily identified as σ y /2. (ref to notes on Mohrs circle). σ sy = σ y /2 In the context of a complicated stress system the initial step would be to determine the principle stress i.e. σ 1, σ 2 & σ 3 in order of magnitude σ 1 > σ 2 > σ 3. Then the maximum shear stress would be determined from Maximum Shear Stress = σ max The factor of safety selected would be FoS = σ y / (2. σ max ) = σ y / (σ 1 - σ 3 ) The theory is conservative especially if the yield strength is more then 50% of the tensile strength. For the simple case of a tensile stress σ x combined with a shear stress σ xy. The design FOS + FoS = σ y / (σ x 2 + 4. τxy 2 )1/2 For a case of a component with σ 1 > σ 2 both positive (tensile) and with σ 3 = 0 then the maximum shear stress = (σ x - 0 ) / 2 Independent of the complexity of the stress state, yielding is assumed to occur when the maximum shearing stress in the material reaches a value equal to the maximum shearing stress for the material as determined from a tensile test at yield:

Examination of experimental results shows that the shearing stress at yielding as determined from a torsion test is slightly higher than that determined from a tensile test.

Shear Strain Energy Theory This theory is also known as the Von Mises-Hencky theory Detailed studies have indicated that yielding is related to the shear energy rather than the maximum shear stress. Strain energy is energy stored in the material due to elastic deformation. The energy of strain is similar to the energy stored in a spring. Upon close examination, the strain energy is seen to be of two kinds: One part results from changes in mutually perpendicular dimensions, and hence in volume, with no change angular changes: the other arises from angular distortion without volume change. The latter is termed as the shear strain energy, which has been shown to be a primary cause of elastic failure. It can be shown by strain energy analysis that the shear strain energy associated with the principal stresses σ 1, σ 1 & σ 3 at elastic failure, is the same as than in the tensile test causing yield at direct stress σ y when: (σ 1 - σ 2) 2 + (σ 2 - σ 3) 2 + (σ 1 - σ 3) 2 > = 2 σ y2 In terms of 3 dimensional stresses using Cartesian co-ordinates

For a plane stress state where the two in-plane principal stresses are of opposite sign the maximum shear stress is given by:

(σ x - σ y) 2 + (σ y - σ z) 2 + (σ z - σ x) 2 + 6. (σ xy2 + σ yz2 + σ zx2) >= 2 σ y2 In terms of plane stress this reduces to. (σ x2 - σ x. σ y + σ y2 + 3 σ xy2) >= σ y2 In terms of simple linear stress combined with shear stress.

If the in-plane principal stresses are of the same sign then we must consider the third principal stress. The third principal stress may be the maximum, minimum or intermediate 7.153

Factor of Safety FOS = σ y / (σ x2 + 3 τ xy2) ½

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principal stress. In a thin-walled pressure vessel for example, the in-plane principal stresses are both positive and the minimum normal stress acts normal to the surface of the pressure vessel.

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two Mohr’s circles for uniaxial tensile strength and uniaxial compression strength. This envelope is shown in the figure below,

The Maximum Strain Energy Theory The theory associated with Haigh. This theory is based on the assumption that strains are recoverable up to the elastic limit, and the energy absorbed by the material at failure up to this point is a single valued function independent of the stress system causing it. The strain energy per unit volume causing failure is equal to the strain energy at the elastic limit in simple tension.. The following relationship can be derived from this theory. (σ y is the yield point in simple shear and n = poissons ratio.) (σ 1 - σ 2) 2 + (σ 2 - σ 3) 2 + (σ 1 - σ 3) 2 + 2 n. (σ 1. σ 2 + σ 2. σ 3 + σ . σ 3) > = σ y. 1 Maximum Octahedral Shear Stress Theory of Failure Independent of the complexity of the stress state, yielding is assumed to occur when the octahedral shearing stress in the material reaches a value equal to the octahedral shearing stress for the material as determined from a tensile test at yielding. The octahedral planes make equal angles with the three principal axes. The Octahedral shearing stress for a plane stress state can be shown to be:

The left circle is for uniaxial compression at the limiting compression stress ? c of the material. Likewise, the right circle is for uniaxial tension at the limiting tension stress ? t. The middle Mohr’s Circle on the figure (dash-dot-dash line) represents the maximum allowable stress for an intermediate stress state. All intermediate stress states fall into one of the four categories in the following table. Each case defines the maximum allowable values for the two principal stresses to avoid failure. Case

Principal Stresses

Criterion Requirements

1

Both in tension

σ1 > 0, σ2 > 0

σ 1 < σt, σ2 < σ t

2

Both in compression

σ1 < 0, σ2 < 0

σ1 > -σ c, σ2 > -σc

3

σ 1 in tension, σ2 in compression

σ1 > 0, σ2 < 0

4

σ 1 in compression, σ2 in tension

σ1 < 0, σ2 > 0

Graphically, Mohr’s theory requires that the two principal stresses lie within the green zone depicted below,

In a Uniaxial Tensile test this Reduces to

Maximum Distortion Energy Failure Theory Strain energy can be separated into energy, which is associated with volume change, and energy, which causes distortion of the element. The maximum distortion energy failure theory predicts yielding when the distortion energy reaches a critical value. This theory of failure can be shown to be equivalent to the maximum octahedral shear stress theory of failure. Mohr’s Theory The Mohr Theory of Failure, also known as the CoulombMohr criterion or internal-friction theory, is based on the famous Mohr’s Circle. Mohr’s theory is often used in predicting the failure of brittle materials, and is applied to cases of 2D stress. Mohr’s theory suggests that failure occurs when Mohr’s Circle at a point in the body exceeds the envelope created by the

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Also shown on the figure is the maximum stress criterion (dashed line). This theory is less conservative than Mohr’s theory since it lies outside Mohr’s boundary. Summary Below is a summary of the theories of failure applied to a simple uniaxial stress state and to a pure shear stress state.

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Failure Criteria Theory \ Loading

Uniaxial

Pure Shear

Relationship

Max. Shear Stress Theory

Oct. Shear Stress Theory

Where ó yp is the tensile (or compressive) yield point determined for uniaxial loading and ô yp is the shearing yield point as determined from a pure shear (e.g. torsion) test. Failure Criteria Theory \ Loading Uniaxial Pure Shear Relationship Maximum principal σmax = σYP σmax = τYP τYP = σYP stress Maximum principal εmax = σYP / E εmax = 5τYP / 4E τYP = 0.8 σYP strain Maximum shear τmax = σYP / 2 τmax = τYP τYP = 0.5 σYP stress Maximum octahedral shear stress

τYP = 0.577 σYP

Maximum distortional energy density

τYP = 0.577 σYP

Notes

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