A number of failure theories have been proposed to explain the ways components manufactured from different types of materials fail that covers all types of loading. One complication is that some materials have a greater strength in compression than in tension. Although the main criteria are summarised below, only the following are widely used:
The Maximum - Normal - Stress Theory (Rankine)
Failure occurs when one of the three principal stresses equals the strength.
Assuming then failure occurs when or
where St is the tensile strength and Sc is the compressive strength.
This criteria is often used when designing with brittle materials such as concrete or some cast irons.
The Maximum Normal - Strain - Theory (Also called St Venant's theory)
Applies only in the elastic range. States yielding occurs when the largest of the 3 principal strains becomes equal to the strain corresponding to the yield strength. If it is assumed that the yield strength in tension and compression are equal, conditions for yielding are:
The Maximum - Shear - Stress Theory (Tresca)
States that yielding begins when the maximum shear stress becomes equal to the maximum shear stress in a tension test specimen of the same material when that specimen begins to yield.
This theory also predicts that the yield strength in shear is
For principal stresses:
Decompose the three normal principal stresses into the components -
etc so these equal stresses are called hydrostatic components.
If then the three shear stresses would all be zero and there could be no yielding - regardless of the hydrostatic stress. The magnitude of the hydrostatic stress has no effect on the size of the Mohr circle, but move it along the normal stress axis.
This criteria may be used when designing with ductile materials (most commonly used metals) but gives somewhat conservative designs for some combinations of loadings.
Maximum Strain Energy Theorem
Suggests failure by yielding occurs when the total strain energy per unit volume reaches or exceeds the strain energy in the same volume corresponding to the yield strength in tension or compression.
se / unit vol. when stressed uniaxially to yield: us = Sy2/2E, Energy in a unit vol. subject to combined stresses:
This is not much used presently.
The von Mises Theory (also known as the maximum distortion energy theory. This gives the same result as the von Mises-Hencky theory or the octahedral shear stress theory).
This states that yielding will occur when the distortion energy in a unit vol. equals the distortion energy in a unit vol. when uniaxially stressed to the yield strength. This was derived from the observation that yielding is not affected by a volume change caused by compression, so may be related to the angular distortion of a stressed element. With some algebra, the effective or von Mises stress is defined by:
and yielding occurs when vmsSy
This criterion is commonly used when designing with ductile metals - it gives a better fit to experimental data than the Tresca criterion.
Coulomb Mohr or Internal friction theory
This theory and variants try to cover materials whose yield stress in compression is not the same as their yield strength in tension.
David J Grieve, updated: 25th January 2014, 24th January 2010, 13th August 2004, original: 1st November 1999.
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