Yield strength, elastic limit, and ultimate strength

Definition and measurement.

Drilling down: yield, ultimate and elongation.

Why does a shear stress make a dislocation move?

Further reading.


Definition and measurement.  The yield strength  (or elastic limit ), (units: MPa or MN/m2) requires careful definition.  For metals, we often identify with the 0.2% offset yield strength, that is, the stress at which the stress-strain curve for axial loading deviates by a strain of 0.2% from the linear-elastic line as shown in Figure 1 (this 0.2% offset point is also associated with plastic strain).  can also be defined by the proportional limit.  For metals, it is often, but not always the same in tension and compression – notice for example that the wrought aluminum alloys datasheets show a tension/compression anisotropy.  For polymers,  is identified as the stress at which the gradient of the stress-strain graph is zero. When such a local maximum is not present, then it is defined as the stress at which the stress-strain curve becomes markedly non-linear: typically, a strain of 1% (Figure 2).  Polymers are a little stronger (≈ 20%) in compression than in tension.  The strength  of a composite is best defined by a set deviation from linear-elastic behavior: 0.5% is sometimes taken.  Composites that contain fibers (and this includes natural composites like wood) are a little weaker (up to 30%) in compression than tension because fibers buckle.  Strength, for ceramics and glasses, depends strongly on the mode of loading (Figure 3).  In tension, “strength” means the fracture strength – this value is taken as both the ultimate tensile and yield strength (elastic limit), , for ceramics.  In compression it means the crushing strength, which is much larger, by a factor of 10 to 15, than that in tension.

         The ultimate (tensile) strength    (units: MPa) is the maximum engineering stress (applied load divided by the original cross-sectional area of the specimen) in a uniaxial stress-strain test.  For non-deformable materials, it is the nominal stress at which a round bar of the material, loaded in tension, separates.  For deformable materials, it occurs at the onset of necking at strains preceding breakage (separation). For brittle solids – ceramics, glasses, and brittle polymers – it is the same as the failure strength in tension.  For metals and most composites, it is larger than the yield strength, , by a factor of between 1.1 and 5 because of work hardening or, in the case of composites, load transfer to the reinforcement.  The elongation  is the tensile strain at break, expressed as a percent.

         The compression strength is the yield or crushing strength in compression.  For ceramics this value is larger than the value stored under Yield strength (elastic limit), which is taken to be the same as the tensile fracture strength.

         Plastic work is the work done in deforming a material permanently by yield or crushing.  Its value, for a small permanent extension or compression  under a force , per unit volume , is


Thus the plastic work per unit volume at fracture, important in energy absorbing applications, is


and that is just the area under the stress-strain curve.  It can be estimated approximately in CES as


(See also Young’s modulus and Fracture toughness.)


Drilling down: yield, ultimate and elongation.  If the yield strength of a perfect crystal is computed from the known inter-atomic forces, the result, known as the “ideal strength” is very large – about  where  is the modulus.  In reality the strengths of engineering materials are nothing like this big; often they are barely 1% of it.  This was a mystery until half way through the last century (1950), when it was realized that a “dislocated” crystal could deform at stresses far below the ideal.  So what is a dislocation, and how does it do it?

         A dislocation can be made in the way shown in Figure 4(a).  The crystal is cut along an atomic plane up to the line shown as , the top part is slid across the bottom by one full atom spacing, and the atoms are reattached across the cut plane to give the configuration shown in 4(b).  There is now an extra half-plane of atoms with its lower edge along the   line, the dislocation line.  This particular configuration is called an edge dislocation because it is formed by the edge of the extra half plane, and it is written briefly as .

         We could, after making the cut in Figure 4(a), have displaced the upper part of the crystal parallel to the edge of the cut rather than normal to it.  That too creates a dislocation, but one with a rather different configuration of atoms along its line – one more like a cork-screw than like a squashed worm – and for this reason it is called a screw dislocation.  We don’t need the details of its structure; it is enough to know that its properties are like those of an edge dislocation except that when it sweeps through a crystal (moving normal to its line, of course), the lattice is displaced parallel to the dislocation line, not normal to it.  All dislocations are either edge or screw or mixed, meaning that they are made up of little steps of edge and screw.  The line of a mixed dislocation can be curved but every part of it has the same slip vector  because the dislocation line is just the boundary of a plane on which a fixed displacement  has occurred.

         Dislocation movement produces plastic strain.  Figure 5 shows how the atoms rearrange as the dislocation slides across the plane (called the slip plane), displacing the upper part of the crystal relative to the lower by the vector , the slip vector or Burgers vector.  It is far easier to move a dislocation through a crystal, breaking and remaking bonds only along its line as it moves, than it is to simultaneously break all the bonds in the plane before remaking them.  It is like moving a heavy carpet by pushing a ruck across it rather than sliding the whole thing at one go.  In real crystals it is easier to make and move dislocations on some planes than on others.  The preferred planes are called slip planes and the preferred directions of slip in these planes are called slip directions.



Why does a shear stress make a dislocation move?  Crystals resist the motion of dislocations with a friction-like resistance  per unit length – we will examine its origins in a moment.  For yielding to take place, the external stress must overcome the resistance  .  

Imagine that one dislocation moves right across a slip plane, traveling the distance  , as in Figure 6.  In doing so, it shifts the upper half of the crystal by a distance   relative to the lower half.  The shear stress acts on an area , giving a shear force on the surface of the block.  If the displacement parallel to the block is , the force does work


This work is done against the resistance  per unit length, or  on the length , and it does so over a displacement  (because the dislocation line moves this far against ) giving a total work against  of .  Equating this to the work done by the applied stress  gives


This result holds for any dislocation – edge, screw, or mixed.  So, provided the shear stress  exceeds the value    it will make dislocations move and cause the crystal to shear.



Further reading.




Ashby et al

Materials: Engineering, Science, Processing and Design

6, 7

Ashby & Jones

Engineering Materials Vol 1 & 2

Vol. 1, Chap. 8, 9


The Science and Engineering of Materials



Engineering Materials: Properties and Selection



Materials Science and Engineering: An Introduction


Callister & Rethwisch

Fundamentals of Materials Science and Engineering: An Integrated Approach



Introduction to Materials Science for Engineers


Further reference details