# Failure Theories

##### SEPTEMBER 2018

The successful application of materials in engineering structures and mechanical systems depends on good design and the efficient use of materials properties in relation to their strength and yield properties. The common approach is to use results from a standard tensile test, and design to values of measured yield stress and yield strain, but this effectively assumes that there is a single principal stress and does not accurately account for complex stress fields or shear stresses.

In the general case, structural stresses are typically three dimensional and fully varying through tension to compression, which in turn is best expressed by principal stresses. The criteria for failure, however, also need to take account of these principal stresses and reflect how the material actually fails under such loading conditions, and what the relevant criterion in the mode of failure is. A normal tensile test does not easily represent these modes of failure, and so-called ‘theories of failure’ have been developed to address these different criteria, which are surprisingly difficult to measure and observe experimentally. A number of theories have been proposed in an attempt to predict elastic failure in actual components - these include the:

__a. Maximum principal stress theory (Rankine)__

__b. Maximum shear stress theory (Guest –Tresca)__

__c. Maximum principal strain (St Venant)__

__d. Total strain energy, per unit volume (Haigh)__

__e. Shear strain energy, per unit volume (von Mises)__

Of these, (a), (b) and (e) are the most well-known/applicable, while (c) and (d) are hardly used nowadays. The applicability of the various failure theories is dependent on whether the material behaves in a ductile or brittle manner and each is discussed in more detail below. For clarity in the following the principal stresses are defined as σ_{1},σ_{2 }andσ_{3}where σ_{1}> σ_{2}> σ_{3}and σ_{y}is the yield stress of the material.

__a. Maximum principal stress theory__

The theory postulates that failure occurs when the principal stress reaches or exceeds the yield strength of the material,σ_{1}≥σ_{y}(in tension) or σ_{3}≥σ_{y }(in compression) with the principal stress in the third plane not being significant. Although the theory is applicable for brittle materials it is **not **applicable to ductile materials (in compression) where higher (compressive) stresses than are predicted can be sustained without failure. Even in pure tension in a tensile test failure in ductile materials, failure does not occur due to direct stresses but occurs in shear. The yield surface diagram (plot that defines the limiting values of stress on a three dimensional plot of σ_{1,}σ_{2,}and σ_{3}, but is often simplified to a two dimensional plot of σ_{1}, and σ_{2} as per examples below) for the maximum principal stress theory is a simple square defined by compressive and tensile yield stresses

__b. Maximum shear stress theory __

This theory assumes that failure occurs when the maximum shear stress in a complex system becomes equal to or exceeds yield in a simple tensile test specimen, i.e. (σ_{1}– σ_{3}) ≥σ_{y }This so-called Tresca criterion of failure correlates well with experiments for ductile materials, and for biaxial and tri-axial applications is slightly conservative. The yield surface diagram is the well-known truncated hexagonal form in the 2D stress field as shown in the figures below.

__c. Maximum principal strain theory __

In the maximum principal strain theory, failure is assumed to occur when the maximum **strain** reaches or exceeds the yield point in a tensile test, i.e. σ_{1}– νσ_{2}– νσ_{3}≥σ_{y}. This theory is now **contradicted** by results of experiments (of flat (ductile) steel plate, but is acceptable for cast iron). In 2D the stress limits, or failure surface, predicted by the theory is the typical elongated and inclined rhomboid shape, whose size is recognisably too large. The theory it is now not widely applied.

__d. Maximum total strain energy theory __

The theory predicts failure occurs when total strain energy in the complex stress field is equal to or exceeds the yield point of a tensile test. This can be written as σ_{1}^{2}+ σ_{2}^{2}+ σ_{3}^{2}- 2𝜈(σ_{1}σ_{2}+ σ_{2}σ_{3}+ σ_{3}σ_{1}) ≥σ_{y}^{2}. This theory gives fairly good results for ductile materials but is seldom used in favour of the more well-known Maximum Shear Strain Energy theory/von Mises criterion, below.

__e. Maximum shear strain energy theory (von Mises)__

The strain energy of a stressed component can be divided into two parts, the volumetric strain energy (which causes volume change with no distortion) and shear strain energy (which causes distortion). The theory predicts that failure occurs when the maximum shear strain energy in the complex stress system is equal to or exceeds the yield point in the tensile test. The criterion may be written as (σ_{1}– σ_{2})^{2}+ (σ_{2}–σ_{3})^{2}+ (σ_{3}– σ_{1})^{2}≥2 σ_{y}^{2}

In 2D this results in the familiar elliptical failure surface through the critical nodes and is often regarded as the **best **theory to predict failure in ductile materials. Even though this von Mises stress is difficult to ‘visualise’ or identify as a more conventional ‘stress’ with which engineers are familiar, it is widely favoured by the finite element analysis community for its applicability to ductile materials.

It is also appropriate to mention other failure theories, namely Mohr’s theory (which is applicable to ‘internal friction’ materials such as brittle materials, compacted soil or concrete, and which often have different yield criteria in tension and compression), and which are addressed (unsurprisingly) using Mohr circle approaches, but are outside the scope of the above conventional Failure Theories for metallic materials.

Failure of materials due to **fracture **(as opposed to yield) of fast crack propagation from pre-existing flaws can be addressed by Fracture Mechanics and characterised (elastically at least) in terms of stress intensity K = Yσ√πa are also very relevant and has been addressed in other tech tips. Linear elastic fracture mechanics predicts that failure will occur when the stress intensity is equal to or exceeds the material fracture toughness K_{Ic}.

The preceding discussion highlights that failure cannot be characterised by a single parameter and it is important to understand whether the material will behave in a ductile or brittle manner. It is therefore important to understand both the 3D stress state (which often can be reduced to a simple 2D case), as well as the potential modes of failure, and not simply assume that any of the individual components of the stress, as predicted using FEA or other techniques, being below yield is a sufficient criterion to preclude failure.

*Published in Technical Tips by Origen Engineering Solutions on 1 September 2018*