The area under each stress-strain curve represents the maximum energy that can be absorbed and is a measure of toughness.

Ductile materials undergo observable plastic deformation and absorb significant energy before fracture. Brittle fracture is characterized by very low plastic deformation and low energy absorption prior to breaking.

Toughness is the ability of a material to absorb energy and plastically deform without fracturing. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. This measure of toughness is different from that used for fracture toughness, which describes load bearing capabilities of materials with flaws.

Toughness requires a balance of strength and ductility. A material should withstand both high stresses and high strains in order to be tough. Generally, strength indicates how much force the material can support, while toughness indicates how much energy a material can absorb before rupturing.

Toughness (or, deformation energy, UT) is measured in units of joule per cubic metre (J·m−3) in the SI system and inch-pound-force per cubic inch (in·lbf·in−3) in US customary units.

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Stress Strain Test Data Reference

As an example, consider that a plate made from A286 Stainless Steel must withstand shock loading defined by the following acceleration shock response spectrum (SRS) and its corresponding pseudo velocity response spectrum. Note that A286 is a ductile material with excellent toughness.

The conversion formula is: PV SRS (f) = scale*[Accel SRS (f) / (2 π f) ]

For English units: scale = 386 in/sec^2 / G

The natural frequency of the plate is unknown. So assume the peak pseudo velocity of 250 in/sec, which is the plateau.

The strain-velocity relationship for a plate is:

ε ~ 2 * ( PV / c ) = 2 * PV *sqrt( ρ / E )

c = sqrt ( E / ρ )

ε=strain

PV = pseudo velocity

c = speed of sound in the material

E = elastic modulus

ρ = mass/volume

Note that the elastic modulus is a function of strain, and thus the speed of sound is also.

The velocity-strain graph is difficult to derive given the need to take the slope of the plotted nonlinear stress-strain curve. The velocity-strain graph is plotted up to 0.2 strain because this is the point at which the elastic modulus becomes approximately zero.

The resulting strain for 250 in/sec is per the graph is ε = 0.0037, or it could jump to ε = 0.021 depending on the response time history.

Both strain estimates show that the material will yield but is well within its fracture limit per the previous stress-strain curve. The engineering fracture strain threshold is above 0.2 for A286 stainless steel. The yield point for engineering material in general is defined as 0.002 strain.

Note that this approach is a rough approximation given that the shock response spectra curves assumed a linear model. Further research is needed.

Stress Strain Velocity Relationship Paper

Stress Velocity Relationship

– Tom Irvine

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