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Mechanical

Third stress invariant (determinant)

Determinant of the stress tensor, equal to the product of the three principal stresses.

Ref: MEC-62
Rev: v1
Field: Mechanical
Result: I₃ [MPa³]

I_{3} = σ_{x} σ_{y} σ_{z} + 2 τ_{xy} τ_{yz} τ_{xz} - σ_{x} {τ_{yz}}^{2} - σ_{y} {τ_{xz}}^{2} - σ_{z} {τ_{xy}}^{2}

I₃ is the determinant of the stress tensor, σ_xσ_yσ_z + 2τ_xyτ_yzτ_xz − σ_xτ_yz² − σ_yτ_xz² − σ_zτ_xy². It equals the product σ₁σ₂σ₃ of the principal stresses and completes the characteristic equation λ³ − I₁λ² + I₂λ − I₃ = 0. It is zero whenever one principal stress is zero, which is always true for plane stress.

Determinant of the stress tensor, equal to the product of the three principal stresses. It completes the characteristic equation and is unchanged by any rotation of the axes.

Calculator

Solve for

I₃Third invariantComputed

MPa³

σ_xNormal stress in x

MPa

−200300

σ_yNormal stress in y

MPa

−200300

σ_zNormal stress in z

MPa

−200300

τ_xyShear stress xy

MPa

−150150

τ_yzShear stress yz

MPa

−150150

τ_xzShear stress xz

MPa

−150150

Product σ₁σ₂σ₃ of the principal stresses (should equal I₃)110,000 MPa³
First invariant I₁180 MPa
Second invariant I₂8,700 MPa²

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Third invariant (MPa³) current value 110,000 MPa³

σ_zNormal stress in zKey driver

−1,100,0001,650,000

τ_yzShear stress yz

−1,690,000

τ_xzShear stress xz

−1,690,000

σ_xNormal stress in x

−338,000462,000

σ_yNormal stress in y

−338,000462,000

τ_xyShear stress xy

−322,000128,000

The lecture state gives I₃ = 110,000 MPa³, which is exactly 110 × 50 × 20, the product of its principal stresses. Because I₃ is the product of the principal stresses, it is zero whenever any one of them is zero.
Setting σ_z = 0 with no out-of-plane shear gives I₃ = 0 MPa³: a plane stress state always has I₃ = 0, because one principal stress is zero.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa³)	Local step	Change in result
σ_z Normal stress in z	−200 to 300 MPa	−1,100,000	1,650,000	2,750,000	+10 MPa	+50 %
τ_yz Shear stress yz	−150 to 150 MPa	−1,690,000	110,000	1,800,000	+6 MPa	−2.62 %
τ_xz Shear stress xz	−150 to 150 MPa	−1,690,000	110,000	1,800,000	+6 MPa	−2.62 %
σ_x Normal stress in x	−200 to 300 MPa	−338,000	462,000	800,000	+10 MPa	+14.5 %
σ_y Normal stress in y	−200 to 300 MPa	−338,000	462,000	800,000	+10 MPa	+14.5 %
τ_xy Shear stress xy	−150 to 150 MPa	−322,000	128,000	450,000	+6 MPa	−7.2 %

Sweep of Normal stress in z

Normal stress in z (MPa)	Third invariant (MPa³)
−200	−1,100,000
−137.5	−756,300
−75	−412,500
−12.5	−68,750
50	275,000
112.5	618,800
175	962,500
237.5	1,306,000
300	1,650,000

Sweep one variable

Vary

Third invariant (MPa³)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

I₃

Third invariant

Result · MPa³

What the equation gives you. Everything else feeds it.

σ_x

Normal stress in x

MPa · typical −200 to 300

Normal stress on the x face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises I₃ by the change times (σ_yσ_z − τ_yz²). Raising it by 10 MPa (to 90 MPa) moves the result by +14.5 %, from 110,000 to 126,000 MPa³.

σ_y

Normal stress in y

MPa · typical −200 to 300

Normal stress on the y face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises I₃ by the change times (σ_xσ_z − τ_xz²). Raising it by 10 MPa (to 90 MPa) moves the result by +14.5 %, from 110,000 to 126,000 MPa³.

σ_z

Normal stress in z

MPa · typical −200 to 300

Normal stress on the z face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises I₃ by the change times (σ_xσ_y − τ_xy²). Raising it by 10 MPa (to 30 MPa) moves the result by +50 %, from 110,000 to 165,000 MPa³.

τ_xy

Shear stress xy

MPa · typical −150 to 150

Shear stress τ_xy, in MPa, acting on the x face in the +y direction. Shear is signed here, so the slider runs from −150 to +150 MPa. Changes I₃ through the term −σ_zτ_xy² and the triple product of the three shears. Raising it by 10 MPa (to 40 MPa) moves the result by −12.7 %, from 110,000 to 96,000 MPa³.

τ_yz

Shear stress yz

MPa · typical −150 to 150

Shear stress τ_yz, in MPa, acting on the y face in the +z direction. Shear is signed here, so the slider runs from −150 to +150 MPa. Changes I₃ through the term −σ_xτ_yz² and the triple product of the three shears. Raising it by 10 MPa (to 10 MPa) moves the result by −7.27 %, from 110,000 to 102,000 MPa³.

τ_xz

Shear stress xz

MPa · typical −150 to 150

Shear stress τ_xz, in MPa, acting on the x face in the +z direction. Shear is signed here, so the slider runs from −150 to +150 MPa. Changes I₃ through the term −σ_yτ_xz² and the triple product of the three shears. Raising it by 10 MPa (to 10 MPa) moves the result by −7.27 %, from 110,000 to 102,000 MPa³.

Worked example with the values above

Product of the normal stresses: 128,000 MPa³. Triple shear product: 0 MPa³.
Normal times shear squared: σ_xτ_yz² + σ_yτ_xz² + σ_zτ_xy² = 18,000 MPa³.
I₃ = 128,000 + 0 − 18,000 = 110,000 MPa³.

Assumptions and limits

A symmetric stress tensor with six independent components.
All in the same units: the result is in MPa³ and is very large for ordinary stresses.

Common mistakes

Forgetting the triple shear product 2τ_xyτ_yzτ_xz, which is zero only if one of the shears is zero.
Reading the sign of I₃ as an indication of tension or compression. It is the sign of the product of the principal stresses.

Statistics and uncertainty

I₃ is cubic in the stresses, so errors are strongly amplified: a 1 % error in every component gives about a 3 % error in I₃. It is best used as a check on principal stresses found another way, not as a direct measurement.

Where it is used

Completing the characteristic equation for the principal stresses.
Checking that a set of principal stresses multiplies to the determinant.

Maths you need

Matrix determinant: the 3 by 3 formula
Invariants: values that do not change with the axes

Related formulas

Practise this formula