9.1 Elastic Material
Recap Kinematic relation – vector format:
𝜀𝑥𝑥
𝜀𝑦𝑦
𝜀𝑧𝑧
Using the strain vector: 𝜺 = 𝛾 , and the
𝑥𝑦
𝛾𝑦𝑧
𝛾𝑧𝑥
𝑢
displacement vector: 𝐮 = 𝑣
𝑤
the strain–displacement relation is written as:
𝜺 = ℒ𝐮,
𝜕
𝜕𝑥
0
0
0
𝜕
𝜕𝑦
0
0
0
ℒ= 𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝜕
𝜕𝑥
𝜕
𝜕𝑧
0
𝜕
𝜕𝑧
0
0
𝜕
𝜕𝑦
𝜕
𝜕𝑥
9.1 Elastic Material
Recap - Equilibrium equations
𝜎𝑥𝑥
𝜎𝑦𝑦
𝑓𝑥
𝜎𝑧𝑧
Using the stress vector: 𝝈 = 𝜏
,and the body force vector:𝐟 = 𝑓𝑦
𝑥𝑦
𝑓𝑧
𝜏𝑦𝑧
𝜏𝑧𝑥
the equilibrium equations can be written in vector form as:
𝜕𝝈 + 𝐟 = 𝟎
where:
∂
∂
∂
0
0
0
∂𝑥
∂𝑦
∂𝑧
𝑇
∂
∂
∂
𝜕=ℒ = 0
0
0
∂𝑦
∂𝑥 ∂𝑧
∂
∂
∂
0
0
0
∂𝑧
∂𝑦 ∂𝑥
9.1 Elastic Material
Tonti Diagram – 3D elasticity
𝜕𝑫ℒ𝒖 + 𝐟 = 𝟎
𝒇
𝒖
𝝏𝝈 + 𝒇 = 𝟎
𝜺 = ℒ𝒖
𝝈
𝝈 = 𝑫𝜺
𝜺
9.2 Mean and deviator components
Mean and Deviator Components
Idea: Decompose stresses (and strains) into:
• Mean (hydrostatic) component
• Deviatoric component
𝝈 = 𝝈𝑚 + 𝒔
where:
𝝈𝑚 = 𝜎𝑚 𝐈 mean stress tensor
𝒔 deviatoric stress tensor
Why: Provides the foundation for some yield criteria and failure theories
9.2 Mean and deviator components
From Tensor Form to Principal Stress Space (Haig-Westergaard Space)
The stress state in the generic 𝑥, 𝑦, 𝑧 coordinate is described by the tensor
𝜎𝑥𝑥 𝜏𝑥𝑦 𝜏𝑥𝑧
𝝈 = 𝜏𝑦𝑥 𝜎𝑦𝑦 𝜏𝑦𝑧
𝜏𝑧𝑥 𝜏𝑧𝑦 𝜎𝑧𝑧
By rotating to the principal directions, the tensor becomes diagonal:
𝜎1 0 0
𝝈 = 0 𝜎2 0
0 0 𝜎3
In this basis, the stress tensor is completely defined by the three principal
stresses. Therefore, the tensor can be represented geometrically by the
vector:
𝜎1
𝝈 = 𝜎2
𝜎3
in a three-dimensional space called principal stress space.
9.2 Mean and deviator components
Mean stress as Hydrostatic Stress State
The mean stress corresponds to the hydrostatic stress state, namely a
stress tensor with
𝜎𝑚 0
0
𝝈 = 𝜎𝑚 𝐈 = 0 𝜎𝑚 0
0
0 𝜎𝑚
where
1
𝜎𝑚 = 𝜎1 + 𝜎2 + 𝜎3
3
is the mean stress.
9.2 Mean and deviator components
In the principal direction, the deviatoric stress tensor becomes
𝜎1 − 𝜎𝑚
0
𝐬 = 𝝈 − 𝜎𝑚 𝐈 =
0
0
𝜎2 − 𝜎𝑚
0
0
𝑠1
0
= 0
0
𝜎3 − 𝜎𝑚
0
𝑠2
0
The principal deviatoric stresses in the principal vector space are:
𝑠1
1 2𝜎1 − 𝜎2 − 𝜎3
𝑠2 = 2𝜎2 − 𝜎3 − 𝜎1
3 2𝜎 − 𝜎 − 𝜎
𝑠3
3
1
2
The sum of the deviatoric stresses is:
𝑠1 + 𝑠2 + 𝑠3 = 0
the deviatoric stress contains no hydrostatic contribution
0
0
𝑠3
9.2 Mean and deviator components
Deviator Plane
The scalar product of the deviatoric stress vector 𝐬 with
the vector
1
1
1
1
namely the direction associated with the mean stress,
𝜎𝑚 1
gives
1
𝑠1
1 1 1 𝑠2 = 𝑠1 + 𝑠2 + 𝑠3 = 0
𝑠3
𝜎1
Hence, the deviatoric stress vector is orthogonal to the
mean stress direction. Therefore, all deviatoric stress
states lie in the plane normal to the hydrostatic direction
1
1
𝜎𝑚 1
𝜎𝑚 1
1
1
called the deviator plane.
𝜎3
𝐬
𝜎2
𝜎3
𝐬
9.2 Mean and deviator components
Mean and Deviator Decomposition
Remark:
1
The mean stress 𝜎𝑚 1 acts equally in all directions:
1
• It produces uniform expansion or compression, associated
with volume change
𝐬 has zero mean value, 𝑠1 +𝑠2 + 𝑠3 = 0
• There is net volumetric contribution. It is associated with
distortional (shape-changing) deformation without volume
change.
9.2 Mean and deviator components
Mean and Deviator Decomposition
Remark:
The magnitude of the deviatoric stress is:
∥ 𝐬 ∥2 = 𝜎1 − 𝜎𝑚 2 + 𝜎2 − 𝜎𝑚 2 + 𝜎3 − 𝜎𝑚 2
or equivalently,
2
∥ 𝐬 ∥2 = 𝜎12 + 𝜎22 + 𝜎32 − 3𝜎𝑚
9.2 Mean and deviator components
Yield Conditions for Metals
Metals are ductile materials. They yield when the stress state
exceeds a characteristic stress level.
For a uniaxial stress state, the limit stress is:
𝜎𝑥 = 𝜎𝑌
where
𝜎𝑌 : yield stress
Yield conditions define the boundary between:
• elastic behavior
• plastic deformation.
Different criteria exist to define the yielding condition of metals
9.2 Mean and deviator components
Von Mises Yield Criterion
Experimental evidence shows that yielding in metals is nearly
independent of the mean stress:
1
𝜎𝑚 = 𝜎1 + 𝜎2 + 𝜎3
3
Therefore, yielding depends only on the deviatoric stress state.
For isotropic materials, the criterion must also be invariant with respect
to the coordinate system.
In principal stress space, the von Mises criterion is:
or equivalently,
𝑠1 2 + 𝑠2 2 + 𝑠3 2 = const.
𝜎1 − 𝜎2 2 + 𝜎1 − 𝜎3 2 + 𝜎2 − 𝜎3 2 = 2𝜎𝑌2
9.2 Mean and deviator components
For a uniaxial stress state:
The material reaches yield for:
𝜎1 0 0
𝜎1 = ±𝜎𝑌
Equivalent Stress
For a general stress state, the von Mises equivalent stress is defined as:
1
𝜎𝑒 =
𝜎1 − 𝜎2 2 + 𝜎1 − 𝜎3 2 + 𝜎2 − 𝜎3 2
2
The yield condition becomes simply
𝜎𝑒 = 𝜎𝑌
where
𝜎𝑌 : material property
𝜎𝑒 : measure of the current stress state. The equivalent stress is an
invariant quantity.
9.2 Mean and deviator components
General Form in Cartesian Coordinates
Using general stress components:
𝜎𝑒 =
1
2
𝜎𝑥𝑥 − 𝜎𝑦𝑦
2
+ 𝜎𝑥𝑥 − 𝜎𝑧𝑧
2+
𝜎𝑦𝑦 − 𝜎𝑧𝑧
2
2
2
2
+ 6 𝜏𝑦𝑧
+ 𝜏𝑧𝑥
+ 𝜏𝑥𝑦
This extends the von Mises criterion to arbitrary stress states.
The criterion depends only on stress differences and shear stresses,
confirming that yielding is governed by the deviatoric stress state.
9.2 Mean and deviator components
Geometric Interpretation of von Mises Yield Criterion
The von Mises yield condition defines a cylinder in
principal stress space. The cylinder axis is aligned with
the hydrostatic direction because the von Mises criterion
is independent of the mean stress:
𝜎3
𝜎1 + 𝜎2 + 𝜎3
𝜎𝑚 =
3
Thus, hydrostatic loading changes volume but does not
contribute to yielding.
The yielding depends only on the magnitude of the
deviatoric stress. The intersection of the yield surface
with the deviator plane is a circle:
𝜎𝑒 = 𝜎𝑌
𝜎2
𝜎1
9.2 Mean and deviator components
Geometric Interpretation of von Mises
Yield Criterion
Stress states:
𝜎3
• inside the cylinder:⇒elastic behavior
• on the cylinder:⇒onset of yielding
• outside the cylinder:⇒inadmissible for
perfectly plastic materials.
The von Mises criterion agrees well with
experimental observations for metals.
𝜎2
𝜎1
9.2 Mean and deviator components
von Mises Criterion for Plane Stress
For a plane stress state:
𝜎𝑧𝑧 = 𝜏𝑥𝑧 = 𝜏𝑦𝑧 = 0
The von Mises equivalent stress becomes
𝜎𝑒 =
2
2
2
𝜎𝑥𝑥
+ 𝜎𝑦𝑦
− 𝜎𝑥𝑥 𝜎𝑦𝑦 + 3𝜏𝑥𝑦
9.2 Mean and deviator components
von Mises Criterion for a beam under in plane load
For a beam the stress state:
𝜎𝑧𝑧 = 𝜎𝑦𝑦 = 𝜏𝑥𝑧 = 𝜏𝑦𝑧 = 0
The von Mises equivalent stress becomes
𝜎𝑒 =
2
2
𝜎𝑥𝑥
+ +3𝜏𝑥𝑦
This form is widely used in the design of steel beam.
9.2 Mean and deviator components
von Mises Criterion for Plane Stress
In principal stress space, plane stress corresponds to
stress states lying in the
𝜎1 − 𝜎2
plane.
For principal stresses, the yield condition becomes:
𝜎12 + 𝜎22 − 𝜎1 𝜎2 = 𝜎𝑌2
This equation represents an ellipse in the 𝜎1 𝜎2 plane.
major axis aligned with
𝜎1 = 𝜎2
• inside the ellipse:⇒elastic behavior
• on the ellipse:⇒yielding.
9.2 Mean and deviator components
Rankine Failure Criterion for Fragile Materials
Fragile (brittle) materials fail primarily by cracking when one of the
principal stresses reaches the corresponding material strength limit.
Unlike ductile metals, brittle materials generally exhibit different
strengths in tension and compression:
𝜎𝐿𝑇 ≠∣ 𝜎𝐿𝐶 ∣
where
𝜎𝐿𝑇 :tensile strength limit
𝜎𝐿𝐶 :compressive strength limit.
Typical examples include: concrete, ceramics, rocks, glass.
9.2 Mean and deviator components
Rankine Failure Criterion for Fragile Materials
The Rankine criterion assumes that failure is governed
directly by the principal stresses.
Tensile failure condition
Failure occurs when the maximum principal stress
reaches the tensile strength:
max 𝜎1 𝜎2 ≤ 𝜎𝐿𝑇
Compressive failure condition
Failure occurs when the minimum principal stress
reaches the compressive strength:
min 𝜎1 𝜎2 ≥ 𝜎𝐿𝐶
Thus, the material remains safe only while all principal
stresses stay within the admissible interval.