Further Explanation of Strain Components

Further explanation of Strain components

Ansys LS-DYNA writes out two sorts of strain values:

Effective Plastic Strain (written at each integration point)	This is a scalar value which is the sum of all plastic strain increments in this element to date. It is directionless It is always +ve Its value will only ever increase It does not include any contributions from elastic strains It is a measure of the cumulative plastic deformation undergone by this element.
Strain Tensor (At all intg points in solids, but top and bottom intg points only in shell elements)	This is a full tensor [Exx, Eyy, Ezz, Exy, Eyz, Ezx] describing the current state of strain in the element. Being a fully populated tensor it contains directional information Values can be +ve or -ve Values can both increase and decrease It contains both elastic and plastic strain contributions. It is a measure of the current (instantaneous) state of strain in the element.

The question is often asked "how are these two values related?" Perhaps the easiest way to understand this is to consider a simple tensile test specimen involving an elastic/plastic material in which the specimen is first squashed into the plastic regime, then stretched again to its original shape.

The graph below shows curves of Effective Plastic strain (green) and von Mises strain (red) derived from the strain tensor for a typical element at the centre of the specimen:

Observe how in the squashing phase the two are identical, but that when stretching (implying a reversal of direction) takes place then:

• effective plastic strain continues to increase, since the material is still undergoing plastic deformation, albeit in the opposite direction.
• von Mises strain reduces by a similar amount, since the element's shape is reverting back towards its unsquashed condition.

(In an ideal world the von Mises strain curve in this analysis would drop back to the linear elastic value only as the specimen is pulled back out to its original length, but in practice the specimen in this analysis was allowed to bulge and hourglass a bit, meaning that it did not return to the original undeformed shape. This is why both strain curves level off at about 0.034s..)

Perhaps a good way to think of this is that the strain tensor contains information about the current element shape (or, more precisely, the distortions required to get from the original shape to the current one); and the effective plastic strain contains historical information about all the permanent deformation increments required to achieve that shape.

When loading is in a single direction then the effective plastic and von Mises strains will be more or less the same, as in the uniaxial "squash" phase above. (The von Mises strain contains elastic as well as plastic values, so it may be slightly greater.) However as soon as the loading direction, or more precisely the direction of deformation, changes then the two will diverge.

What is von Mises Strain, and why does it have a factor of 2/3?

Von Mises strain, sometimes referred to as "Equivalent strain", is the equivalent of von Mises stress in that it measures the deviatoric component of strain, and

VM stress / VM strain = 3G

and

G = E / 2(1 + υ)

where G is the shear modulus, E is the Young's modulus and υ is the Poisson's ratio.

Note that VM stress / VM strain does not equal the Young's modulus E except in the fully plastic state. To understand why consider the following:

Poisson's ratio (υ) only lies in the range 0.0 - 0.5 for elastic strain calculation, once the material goes plastic then the relationship between longitudinal and transverse strains is controlled by volume conservation as the material distorts permanently, and the plastic Poisson's ratio is required to be 0.5 in all cases if volume is to be preserved. Therefore for "conventional" materials that preserve volume, which implicitly excludes foams and other strange materials, we can eliminate the variability of Poisson's ratio from the problem allowing us to calculate a von Mises strain that has a fixed relationship with von Mises stress in the plastic regime.

In the elastic regime, where the Poisson's ratio is less than 0.5:

3G is less than E since E = 2(1 + υ) G therefore VM stress / VM strain gives some value < E

In the fully plastic regime where the Poisson's ratio = 0.5:

3G = E since E = 2(1 + 0.5) G therefore VM stress / VM strain should give - more or less - the current plastic modulus value in the element.

To see where factor of 2/3 comes from consider the following:

Perform the von Mises calculation for strain derived from stress with Poisson's ratio set to 0.5 and the strain comes out 1.5x larger. To demonstrate this consider the case of uniaxial stress in the plastic regime:

Let X stress σ_x = 1.0, all other stresses are zero.

Let the notional plastic modulus Ep be 1.0 for simplicity, then using 0.5 for Poisson's ratio we obtain the following strains. Stresses are shown for comparison.

Strain values		Stress values
X strain εx = 1.0 .		σx = 1.0
Y strain εy = -υ . σx / Ep = -0.5		σy = 0.0
Z strain εz = εy = -0.5		σz = 0.0

Performing the conventional von Mises calculation used for stress we obtain:

von Mises strain = 1/sqrt(2).((ε_x - ε_y)**2 + (ε_y - ε_z)**2 + (ε_z - ε_x)**2) = 1.5

Whereas we can see by inspection that the von Mises stress = 1.0

Therefore a factor of 2/3 factor is included in the formula for von Mises strain so that, during the plastic deformation phase of a uniaxial tensile test on a metal, VM strain will be the same as uniaxial strain, and therefore the plastic stress-strain curve from a tensile test can be used as input to material models expecting VM stress versus VM strain (after conversion from nominal stress/strain to true stress/strain).

But what does von Mises strain actually mean?

This section is being written because this question has been asked so many times. To be truthful it is not very useful!

• In predominantly plastic analyses it can be thought of as the quantity on the strain axis of the material's stress-strain input curve. When strains are occurring in two or three directions, the Von Mises strain gives the strain in a uniaxial tensile test that would work-harden the material to the same degree.
  
  
• In predominantly elastic analyses it has limited usefulness.

It is normally the case that Effective Plastic strain is far more useful in fully plastic analyses, and that the individual (directional) and principal components of the strain tensor are of more interest in elastic ones.