SME
Materials Science & Material Properties · Problem type
True stress and true strain
Use this for “At its UTS a bar carries 400 MPa at an engineering strain of 0.10. Find the true stress and the true strain.”
Diagram
The engineering curve: elastic, yielding, hardening to the UTS, necking, fracture (the knee is a sketch: the page gives no σY).
Dashed: the true curve, σT = σ(1 + ε) at εT = ln(1 + ε), to the UTS.
This is an example. Type your own values to start a new problem.
σT (MPa)
=
σ (MPa)
×
(1
+
ε
)
εT
=
ln(1
+
ε
)
Formulas
σT
=
σ
×
(1
+
ε
)
440 = 400 × (1 + 0.1)
εT
=
ln(1
+
ε
)
0.09531 = ln(1 + 0.1)
Assumptions
•
The point is at or before the UTS: past it the neck shrinks the area faster than this says.
•
The volume stays the same while the bar stretches, so A₀L₀ = AL.
•
Strains are plain numbers (mm/mm); 0.10 is a 10% stretch.
Step-by-step
WE KNOW
σ = 400 MPa, ε = 0.1
FIND
true stress (σT), true strain (εT)
Step 1 · Find true stress (σT)
Use σT = σ × (1 + ε)
The
true
stress
is
the
load
over
the
shrunk
area
A₀
1 +
ε
:
σ
times
1
+
ε
.
σT = 400 × (1 + 0.1)
σT = 400 × 1.1
σT = 440 MPa
Step 2 · Find true strain (εT)
Use εT = ln(1 + ε)
True
strain
adds
each
small
stretch
over
the
length
it
had
then:
the
integral
of
dL
L
is
ln(
L
L₀
).
εT = ln(1 + 0.1)
εT = ln(1.1)
εT = 0.09531
Check
Put the numbers back in. Both sides should match.
440 = 400 × (1 + 0.1) ✓
0.09531 = ln(1 + 0.1) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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