## From engineering to true strain, true stress

First of all, you may check that your experimental data from a uniaxial tension test is expressed in terms of true stress vs. true strain, not engineering stress or strain.

True strain = ln(1 + engineering strain) where ln designates the natural log

True stress = (engineering stress) * exp(true strain) = (engineering stress) * (1 + engineering strain) where exp(true strain) is 2.71 raised to the power of (true strain).

Be aware that experimental data always includes some degree of error and thus tends to be somewhat noisy or erratic. When using *MAT_24 , one should input a smoothed stress-strain curve utilizing a minimal number of points. Input of noisy experimental data may cause spurious behavior, particularly in the case of the default, 3-iteration plane stress plasticity algorithm for shells. Full iterative plasticity can be invoked for shells, at greater expense, for material models 3, 18, 19, and 24 by setting MITER=2 in *CONTROL_SHELL .

The effective plastic strain values input in defining a stress vs. effective plastic strain curve in a LS-DYNA plasticity model should be the residual true strains after unloading elastically. True stress is input directly for the stress values.

Using experimental data from a true stress vs. true strain curve .

effective plastic strain (input value) = total true strain – true stress/E

Note that as the stress value increases, the recoverable strain ( true stress/E ) increases as well. For metals, E is very large compared to the yield stress so it’s fairly common practice in the case of metals to just subtract off a constant value equal to the strain at initial yield from all subsequent strain values. For plastics/polymers, you probably should consider the increase in recoverable strain as stresses increase (since the elastic component of strain may be quite large). In any case, the first plastic strain value should be input as zero and the first stress value should be the initial yield stress.

In the case where the user elects to input only an initial yield stress SIGY and the tangent modulus Etan in lieu of a true stress vs. effective plastic strain curve (in *MAT_PIECEWISE_LINEAR_PLASTICITY ),

Etan = (Eh * E)/(Eh + E) where Eh = (true stress – SIGY)/(true strain – true stress/E)

Eh = (Etan * E)/(E – Etan) if E > Etan

E should not be less than Etan where Etan is computed from E and Ep, where Ep is the initial slope of the piecewise linear stress vs. epspl curve (presumably this is the steepest portion of the curve). In *MAT_24 , this is exactly the input check that is made if LCSS=0 and cards 3 and 4 are blank (E must be greater than ETAN or else you get a fatal error).

Actually, this condition of E > Etan is ALWAYS met if a stress vs. epspl curve is given. For example, if Ep = 3253 and E were set to an extremely low value, say 10, Etan is then equal to Ep*E/(Ep + E) = 9.97 . If cards 3 and 4 are used to define the curve, the job will stop due to an improper though conservative check of E against Ep. You can always bypass this check by using LCSS instead of cards 3 and 4.

From engineering to true strain, true stress First of all, you may check that your experimental data from a uniaxial tension test is expressed in terms of true stress vs. true strain, not

## Engineering Stress/Strain vs True Stress/Strain

In engineering and materials science, stress–strain curve for a material gives the relationship between stress and strain. That is obtained by gradually applying load to a test coupon and measuring the deformation from tensile testing, which the stress and strain can be determined. These curves reveal many of properties of materials, such as the Young’s modulus, the yield strength, the ultimate tensile strength and so on.

Stress-strain curve for material is plotted by elongating the sample and recording the stress variation with strain until the sample fractures. The strain is set to horizontal axis and stress is set to vertical axis. It is often assumed that the cross-section area of the material does not change during the whole deformation process. This is not true since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on the original cross-section and gauge length is called the *engineering stress-strain curve*, while the curve based on the instantaneous cross-section area and length is called the *true stress-strain curve*.

For engineering stress, we assume the length and diameter of the sample remain constant throughout the whole experiment.

Engineering stress is calculated by:

Engineering strain is calculated by:

True stress is the applied load divided by the actual cross-sectional area (the changing area with time) of material. Engineering stress is the applied load divided by the original cross-sectional area of material. Also known as nominal stress.

This shows the cross-section of the specimen has changed during the experiment process.

The cross-section does not remain constantly and will be different from the given value of diameter. This stress is called True Stress. Applied force is divided by the area of the section at that instant.

Before examine thoroughly true stress and strain, let’s reminisce about tensile testing (tension test).

**Tensile Testing**

Tensile testing, also known as tension testing, is a fundamental materials science and engineering test in which a sample is subjected to a controlled tension until failure.

Properties that are directly measured via a tensile test are ultimate tensile strength, breaking strength, maximum elongation and reduction in area. From these measurements some properties can also be determined: Young’s modulus, Poisson’s ratio, yield strength, and strain-hardening characteristics. Uniaxial tensile testing is the most commonly used for obtaining the mechanical characteristics of isotropic materials. For Some materials, biaxial tensile testing is used. The main difference between these testing machines being how load is applied on the materials.

**Fracture Behaviour**

Fracture behavior is considered under two main material behaviours which are called Ductile and Brittle materials.

**Ductile material:**

Significant plastic deformation and energy absorption (toughness) reveals before fracture. Characteristic feature of ductile material is necking before material failure.

**Brittle material**:

Little plastic deformation or energy absorption reveals before fracture. Characteristic feature of brittle materials is different compare to ductile materials. Brittle materials fracture without any necking.

Different materials exhibit different behaviours/trends under the same loading condition.

More traditional engineering materials such as concrete under tension, glass metals and alloys exhibit adequately linear stress-strain relations until the onset of yield point.

Axial tensile test and bending test for two different materials:

True stress (*σ _{t}*) and true strain (

*ε*) are used for accurate definition of plastic behaviour of ductile materials by considering the actual dimensions.

_{t}Brittle materials usually fracture(fail) shortly after yielding or even at yield points whereas alloys and many steels can extensively deform plastically before failure. The characteristics of each material should be chosen based on the application and design requirements.

**True Stress and Strain**

True stress and strain are different from engineering stress and strain.

In a tensile test, true stress is larger than engineering stress and true strain is less than engineering strain. The difference between the true and engineering stresses and strains will increase with plastic deformation. At low strains (in elastic region), the differences between the two are negligible.

True stress is the stress determined by the instantaneous load acting on the instantaneous cross-sectional area.

True strain is logarithmic and engineering strain is linear. However it appears to be almost same for small deformation owing to small values in Taylor expansion.

The true stress and strain can be expressed by engineering stress and strain.

For true stress:

For true strain:

Integrate both sides and apply the boundary condition,

The stress and strain at the necking can be expressed as:

**In Conclusion**

**Engineering stress** is the applied load divided by the original cross-sectional area of a material. Also known as nominal stress.**True stress** is the applied load divided by the actual cross-sectional area (the changing area with respect to time) of the specimen at that load**Engineering strain** is the amount that a material deforms per unit length in a tensile test. Also known as nominal strain.**True strain** equals the natural log of the quotient of current length over the original length.

There is no decrease in true stress during the necking phase. Also, the results achieved from tensile and compressive tests will produce essentially the same plot when true stress and true strain are used. Engineers will produce an acceptable stress and an acceptable deformation in a given member and they want to use a diagram based on the engineering stress and the engineering strain with the cross-sectional area *A*_{0} and the length *L*_{0} of the member in its undeformed state.

The engineering stress is obtained by dividing *F* by the cross-sectional area * A_{0}* of the deformed specimen. Engineering stress becomes apparent in ductile materials after yield has started directly proportional to the force (

*F)*decreases during the necking phase.

The true stress (*σ _{t)}*, which is proportional to

*F*and inversely proportional to

*A*, is observed to keep increasing until rupture of the specimen occurs.

Dividing each increment Δ*L* of the distance between the gage marks, by the corresponding value of *L*, the elementary strain is obtained:

Adding the values of Δ*ε**ε _{t}* = ∑ Δ

*ε*= ∑

*Δ*

*L*/LWith summary by an integral, the true strain can also be expressed as:

Engineering Stress/Strain vs True Stress/Strain In engineering and materials science, stress–strain curve for a material gives the relationship between stress and strain. That is obtained by