II. Fundamentals of Continuum Mechancis Supplimentary slides

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Division Mechanics of Functional Materials | Institute of Materials Science, TU Darmstadt | Xu | Introduction | 1

II. Fundamentals of Continuum Mechancis

Supplimentary slides

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2.1 Conventions and Theorem

Please refer to the mansucript for more details. Feel free to ask questions in Q & A

session in Zoom.

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2.2 Stress

Stress vector

Normal stress

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Shear stress

2.2 Stress

Stress vector

𝐹

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Stress vector at a material point:

Decomposition of the stress vector:

2.2 Stress

Stress vector

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Stress vectors on different cross sections at one point

2.2 Stress

Stress vector

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Stress vectors

2.2 Stress

Stress tensor

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Stress components in three perpenticular cross sections.

2.2 Stress

Stress tensor

Alternatively we can write all the components in one matrix

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2.2 Stress

Stress tensor

Comments

• Double indices notation: the first index describes the direction of the normal vector of the cross section, while the second indicates the direction of the stress component itself.

• Symmetry of the stress tensor, and thus also the symmetry of the

matrix: This is due to the momentum balance, the shear components

in two perpenticular sections.

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2.2 Stress

Stress tensor

• Cauchy‘s formula

• Transformation relation

where 𝑎 _𝑖𝑘 is the rotation tensor between the two coordinates.

𝑎 _𝑖𝑘 = 𝐞 _𝑖 ^′ ⋅ 𝐞 _𝑘 = cos(𝑥 _𝑖 ^′ , 𝑥 _𝑘 ) It appears twice in the

transformation relation. Thus the

stress tensor is of 2nd order.

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• Principal stress 𝜎 ₁ , 𝜎 ₂ , 𝜎 ₃ : Extreme values of the normal stress wrt. the rotation of coordinates. There exists always a special coordinate system, in which the shear stress components vanish and there left only with normal stress components. This coordinate system is called principle coordinates, and the related normal stress components are called principle stresses.

2.2 Stress

Stress tensor

Likewise there is a particular coordinate, in which the shear stress take extreme values, the so-called principle shear stress 𝜏 _𝑚𝑎𝑥 . The related normal stresses are in general not zero, but the mean of the principle stresses.

The related normal stresses are in general not zero, but the mean of the principle stresses.

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Invariants: quantities which do not change during rotation of coordinates

2.2 Stress

Stress tensor

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2.2 Stress

Stress tensor

• Decomposition of the stress state into the hydrostatic state and the deviatoric state

The second invariant of the stress deviator:

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• Alternative representation of stress components

o 𝑥 ₁ , 𝑥 ₂ , 𝑥 ₃ → 𝑥, 𝑦, 𝑧 ; Thus 𝜎 ₁₁ → 𝜎 _𝑥𝑥 , 𝜎 ₁₂ → 𝜎 _𝑥𝑦 , 𝜎 ₁₃ → 𝜎 _𝑥𝑧

o In this case, the normal stress 𝜎 _𝑥𝑥 , 𝜎 _𝑦𝑦 , 𝜎 _𝑧𝑧 are written in a short form of 𝜎 _𝑥 , 𝜎 _𝑦 , 𝜎 _𝑧 o 𝜎 ₁₂ , 𝜎 ₂₃ , 𝜎 ₃₁ → 𝜏 ₁₂ , 𝜏 ₂₃ , 𝜏 ₃₁

o Voigt Notation:

2.2 Stress

Stress tensor

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The stress tensor in 2D and its analysis:

2.2 Stress

Stress tensor

𝜎 _𝑖𝑗 = 𝜎 _𝑥 𝜏 _𝑥𝑦

𝜏 _𝑦𝑥 𝜎 _{𝑦 𝑥𝑦−𝑐𝑜.} = 𝜎 _𝜉 𝜏 _𝜉𝜂

𝜏 _𝜂𝜉 𝜎 _{𝜂 𝜉𝜂−𝑐𝑜.} = 𝜎 ₁ 0

0 𝜎 2 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑐𝑜.

in principle axis

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𝑎 _𝑖𝑘 = cos(𝐞 _𝑖 ^′ ⋅ 𝐞 _𝑘 )

= cos 𝜃 sin 𝜃

− sin 𝜃 cos 𝜃 The stress tensor in 2D and its analysis:

2.2 Stress

Stress tensor

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2.2 Stress

Stress analysis

Uniaxial tension test:

• Brittle materials fracture along vertical cross section

• Ductile materials fracture along 45 degree Torsion test

• Brittle materials fracture along 45 degree

• Ductile material fracture along vertical

cross section

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Overall strain:

Local strain:

P

P’

Q

Q’

Displacement field

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volume force

traction

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or

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II. Fundamentals of Continuum Mechancis Supplimentary slides

II. Fundamentals of Continuum Mechancis

Supplimentary slides

2.1 Conventions and Theorem

Please refer to the mansucript for more details. Feel free to ask questions in Q & A

session in Zoom.

2.2 Stress

Stress vector

Normal stress

Shear stress

2.2 Stress

Stress vector

𝐹

𝐹

Stress vector at a material point:

Decomposition of the stress vector:

2.2 Stress

Stress vector

Stress vectors on different cross sections at one point

2.2 Stress

Stress vector

Stress vectors

2.2 Stress

Stress tensor

Stress components in three perpenticular cross sections.

2.2 Stress

Stress tensor

Alternatively we can write all the components in one matrix

2.2 Stress

Stress tensor

Comments

• Double indices notation: the first index describes the direction of the normal vector of the cross section, while the second indicates the direction of the stress component itself.

• Symmetry of the stress tensor, and thus also the symmetry of the

matrix: This is due to the momentum balance, the shear components

in two perpenticular sections.

2.2 Stress

Stress tensor

• Cauchy‘s formula

• Transformation relation

where 𝑎 𝑖𝑘 is the rotation tensor between the two coordinates.

𝑎 𝑖𝑘 = 𝐞 𝑖 ′ ⋅ 𝐞 𝑘 = cos(𝑥 𝑖 ′ , 𝑥 𝑘 ) It appears twice in the

transformation relation. Thus the

stress tensor is of 2nd order.

2.2 Stress

Stress tensor

Likewise there is a particular coordinate, in which the shear stress take extreme values, the so-called principle shear stress 𝜏 𝑚𝑎𝑥 . The related normal stresses are in general not zero, but the mean of the principle stresses.

The related normal stresses are in general not zero, but the mean of the principle stresses.

Invariants: quantities which do not change during rotation of coordinates

2.2 Stress

Stress tensor

2.2 Stress

Stress tensor

• Decomposition of the stress state into the hydrostatic state and the deviatoric state

The second invariant of the stress deviator:

• Alternative representation of stress components

o 𝑥 1 , 𝑥 2 , 𝑥 3 → 𝑥, 𝑦, 𝑧 ; Thus 𝜎 11 → 𝜎 𝑥𝑥 , 𝜎 12 → 𝜎 𝑥𝑦 , 𝜎 13 → 𝜎 𝑥𝑧

o In this case, the normal stress 𝜎 𝑥𝑥 , 𝜎 𝑦𝑦 , 𝜎 𝑧𝑧 are written in a short form of 𝜎 𝑥 , 𝜎 𝑦 , 𝜎 𝑧 o 𝜎 12 , 𝜎 23 , 𝜎 31 → 𝜏 12 , 𝜏 23 , 𝜏 31

o Voigt Notation:

2.2 Stress

Stress tensor

The stress tensor in 2D and its analysis:

2.2 Stress

Stress tensor

𝜎 𝑖𝑗 = 𝜎 𝑥 𝜏 𝑥𝑦

𝜏 𝑦𝑥 𝜎 𝑦 𝑥𝑦−𝑐𝑜. = 𝜎 𝜉 𝜏 𝜉𝜂

𝜏 𝜂𝜉 𝜎 𝜂 𝜉𝜂−𝑐𝑜. = 𝜎 1 0

0 𝜎 2 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑐𝑜.

in principle axis

𝑎 𝑖𝑘 = cos(𝐞 𝑖 ′ ⋅ 𝐞 𝑘 )

= cos 𝜃 sin 𝜃

− sin 𝜃 cos 𝜃 The stress tensor in 2D and its analysis:

2.2 Stress

Stress tensor

2.2 Stress

Stress analysis

Uniaxial tension test:

• Brittle materials fracture along vertical cross section

• Ductile materials fracture along 45 degree Torsion test

• Brittle materials fracture along 45 degree

• Ductile material fracture along vertical

where 𝑎 _𝑖𝑘 is the rotation tensor between the two coordinates.

𝑎 _𝑖𝑘 = 𝐞 _𝑖 ^′ ⋅ 𝐞 _𝑘 = cos(𝑥 _𝑖 ^′ , 𝑥 _𝑘 ) It appears twice in the

Likewise there is a particular coordinate, in which the shear stress take extreme values, the so-called principle shear stress 𝜏 _𝑚𝑎𝑥 . The related normal stresses are in general not zero, but the mean of the principle stresses.

o 𝑥 ₁ , 𝑥 ₂ , 𝑥 ₃ → 𝑥, 𝑦, 𝑧 ; Thus 𝜎 ₁₁ → 𝜎 _𝑥𝑥 , 𝜎 ₁₂ → 𝜎 _𝑥𝑦 , 𝜎 ₁₃ → 𝜎 _𝑥𝑧

o In this case, the normal stress 𝜎 _𝑥𝑥 , 𝜎 _𝑦𝑦 , 𝜎 _𝑧𝑧 are written in a short form of 𝜎 _𝑥 , 𝜎 _𝑦 , 𝜎 _𝑧 o 𝜎 ₁₂ , 𝜎 ₂₃ , 𝜎 ₃₁ → 𝜏 ₁₂ , 𝜏 ₂₃ , 𝜏 ₃₁

𝜎 _𝑖𝑗 = 𝜎 _𝑥 𝜏 _𝑥𝑦

𝜏 _𝑦𝑥 𝜎 _{𝑦 𝑥𝑦−𝑐𝑜.} = 𝜎 _𝜉 𝜏 _𝜉𝜂

𝜏 _𝜂𝜉 𝜎 _{𝜂 𝜉𝜂−𝑐𝑜.} = 𝜎 ₁ 0

𝑎 _𝑖𝑘 = cos(𝐞 _𝑖 ^′ ⋅ 𝐞 _𝑘 )