MOM-II Theories of Failure

Theories of Failure

Mechanics of Materials II

Contents

• Introduction to failure

• Failure of various types of materials

• Theories of failure

• Comparison of failure theories

INTRODUCTION

Theories of Failure

What is Failure?

Anything that might cause a component to lose its structural tolerances, preventing it from serving its intended purpose

• This means

– Fracture

– or plastic deformation

– or excessive elastic deformation

• For this course, plastic deformation is considered as failure

Failure and fracture are not the same

Introduction

• When designing using a specific material, an upper limit on the state of stress needs to be defined

• Ductile Material

– Failure is usually specified by the initiation of yielding

• Brittle Material

– Failure is usually specified by fracture

Why Failure Theories are Required?

• Modes of failure are readily defined if the member is subjected to a uniaxial state of stress e.g. simple tension

• In case of biaxial or triaxial stress, the criterion for failure becomes more difficult to establish

• Various failure theories are used to predict the failure of a material subjected to a multiaxial state of stress

Why Failure Theories are Required?

• No single theory of failure can be applied to a specific material at all times

• A material may behave in ductile or brittle manner depending on the temperature, rate of loading, chemical environment, etc.

• When using a particular theory of failure, it is first necessary to calculate the maximum normal and maximum shear stress

• Principal stresses at these critical points are then determined

FAILURE IN DUCTILE MATERIALS

Theories of Failure

Failure in Ductile Materials

• Most common type of yielding of a ductile material such as steel is caused by slipping

• Slipping occurs along the contact planes of randomly ordered crystals

• A highly polished thin strip, subjected to a simple tension test, can show slipping

• Slipping causes the material to yield

• The edges of the planes of slipping are referred to as Lüder’s lines

• These lines clearly indicate the slip planes in the strip, which occur at approximately 45° with the axis of the strip

Failure in Ductile Materials

• The slipping is caused by shear stress

• Consider an element of the material taken from a tension specimen when subjected to yield stress

• The maximum shear stress can be determined by drawing Mohr’s circle for the element

• The results indicate that

𝜏𝑚𝑎𝑥 =𝜎𝑌2

• Shear stress acts on planes that are 45° from the planes of principal stress

Maximum Shear Stress Theory

• In 1868, Henri Tresca proposed the maximum-shear-stress theory or Tresca yield criterion

• Can be used to predict the failure stress of a ductile material subjected to any type of loading

• The theory states that

“yielding of the material begins when the absolute maximum shear stress in the material reaches the shear stress that causes the same

material to yield when it is subjected only to axial tension”

• To avoid failure, it is required that 𝜏𝑚𝑎𝑥𝑎𝑏𝑠 in the material must be less

than or equal to 𝜎𝑌 2

• 𝜎𝑌 is determined from a simple tension test


• For application, express the absolute maximum shear stress in terms of the principal stresses

• Consider the plane stress condition

• If the two in-plane principal stresses have the same sign

– i.e., they are both tensile or both compressive, then

– failure will occur out of the plane, and from

𝜏𝑚𝑎𝑥𝑎𝑏𝑠 =

𝜎12

• If the in-plane principal stresses are of opposite signs, then failure occurs in the plane, and from

𝜏𝑚𝑎𝑥𝑎𝑏𝑠 =

𝜎1 − 𝜎22


• Maximum-shear-stress theory for plane stress for any two in-plane principal stresses 𝜎1 and 𝜎2 by the following criteria

𝜎1 = 𝜎𝑌

𝜎2 = 𝜎𝑌𝜎1, 𝜎2 have same signs

𝜎1 − 𝜎2 = 𝜎𝑌 𝜎1, 𝜎2 have opposite signs


𝜎1

𝜎2

(𝜎𝑌𝑆, 𝑂)(−𝜎𝑌𝑆, 𝑂) 𝑂

(𝑂, 𝜎𝑌𝑆)

(𝑂,−𝜎𝑌𝑆)Tresca

Maximum Distortion Energy Theory

• An external loading will deform a material, causing it to store energy internally throughout its volume

• The energy per unit volume of material is called the strain-energy density

• If the material is subjected to a uniaxial stress the strain-energy density

𝑢 =1

2𝜎𝜖


• Under triaxial stress, each principal stress contributes to the total strain-energy density

𝜖𝑥 =1

𝐸𝜎𝑥 − 𝜈𝜎𝑦 − 𝜈𝜎𝑧

𝜖𝑦 =1

𝐸𝜎𝑦 − 𝜈𝜎𝑥 − 𝜈𝜎𝑧

𝜖𝑧 =1

𝐸𝜎𝑧 − 𝜈𝜎𝑥 − 𝜈𝜎𝑦

𝑢 =1

2𝜎1𝜖1 +

1

2𝜎2𝜖2 +

1

2𝜎3𝜖3

• For linear elastic materials, Hooke’s law applies


• Putting strains and simplifying

𝑢 =1

2𝐸[𝜎1

2 + 𝜎22 + 𝜎3

2 − 2𝜈 𝜎1𝜎2 + 𝜎1𝜎3 + 𝜎3𝜎2 ]

• Strain-energy density is sum of two parts

𝜎𝑎𝑣𝑔 =𝜎1 + 𝜎2 + 𝜎3

3 𝜎1 − 𝜎𝑎𝑣𝑔 , 𝜎2 − 𝜎𝑎𝑣𝑔 , (𝜎3 − 𝜎𝑎𝑣𝑔)Energy needed to cause a volume change with no change in shape Energy needed to distort the element


• The maximum-distortion-energy theory

“Yielding in a ductile material occurs when the distortion energy per unit volume of the material equals or exceeds the distortion energy per unit volume of the same material when it is subjected to yielding in a simple tension test”

• Also known as von Mises theory, as was presented by Richard von Mises

Richard von Mises


• Distortion energy per unit volume is obtain by substituting 𝜎1, 𝜎2,

𝜎3 with 𝜎1 − 𝜎𝑎𝑣𝑔 , 𝜎2 − 𝜎𝑎𝑣𝑔 , (𝜎3 − 𝜎𝑎𝑣𝑔)

𝑢𝑑 =1 + 𝜈

6𝐸[ 𝜎1 − 𝜎2

2 + 𝜎2 − 𝜎32 + 𝜎3 − 𝜎1

2]

• For case of plane stress, 𝜎3 = 0

𝑢𝑑 =1 + 𝜈

3𝐸[𝜎1

2 − 𝜎1𝜎2 + 𝜎22]

• For uniaxial tension test, 𝜎1 = 𝜎𝑌, 𝜎2 = 𝜎3 = 0

𝑢𝑑 =1 + 𝜈

3𝐸𝜎𝑌2


𝜎1

𝜎2



(𝑂,−𝜎𝑌𝑆)

von Mises

𝜎12 − 𝜎1𝜎2 + 𝜎2

2 = 𝜎𝑌2For plane or biaxial stress state

FAILURE IN BRITTLE MATERIALS

Theories of Failure

Failure in Brittle Materials

• Brittle materials, such as gray cast iron, tend to fail suddenly by fracture with no apparent yielding

• In a tension test, the fracture occurs when the normal stress reaches the ultimate stress

• Brittle fracture occurs in a torsion test due to tension since the plane of fracture for an element is at 45° to the shear direction

Maximum Normal Stress Theory

“A brittle material will fail when the maximum tensile stress, , in the material reaches a value that is equal to the ultimate normal stress the material can sustain when it is subjected to simple tension”

• If material is subjected to plane stress

• Also known as Rankine theory as was presented by W. Rankine in mid-1800s

𝜎1 = 𝜎𝑢𝑙𝑡

𝜎2 = 𝜎𝑢𝑙𝑡

Maximum Normal Stress Theory

𝜎1

𝜎2



(𝑂,−𝜎𝑌𝑆)

Rankine

𝜎1

𝜎2



(𝑂,−𝜎𝑌𝑆)Tresca

Rankine

von Mises

Comparison of Yield Criteria

• Biaxial state of stress

• Tresca is more conservative than von Mises

• The two criteria are only equivalent for uniaxial loading (𝜎1, 𝜎2 >0 𝑤𝑖𝑡ℎ 𝜎2, 𝜎1 = 𝜎3 = 0) or for balanced biaxial loading (𝜎1 = 𝜎2, 𝜎3 = 0)

Reference

• 10.7 Material Property Relationships, Mechanics of Materials, R. C. Hibbeler, 8th Ed

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MOM-II Theories of Failure