5. Strain

Strain is deformation caused by an applied stress.

Types of Strain

• Elongation Strain ($\epsilon$): It measures how much a material stretches or squashes in a specific direction (change in length per unit length).
• Shear Strain ($\gamma$): It measures how much the angle between two originally perpendicular lines changes (distortion of the corner of a square).

$$\epsilon =\frac{\partial u}{\partial x}$$ $${\gamma }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}$$

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The Strain Tensor

Just like stress, strain is a tensor field. It varies point-by-point.

$$[\epsilon ]=\left[\begin{array}{ccc}{\epsilon }_x& {\epsilon }_{xy}& {\epsilon }_{xz}\\ {\epsilon }_{yx}& {\epsilon }_y& {\epsilon }_{yz}\\ {\epsilon }_{zx}& {\epsilon }_{zy}& {\epsilon }_z\end{array}\right]=\left[\begin{array}{ccc}{\epsilon }_x& \frac{1}{2}{\gamma }_{xy}& \frac{1}{2}{\gamma }_{xz}\\ \frac{1}{2}{\gamma }_{xy}& {\epsilon }_y& \frac{1}{2}{\gamma }_{yz}\\ \frac{1}{2}{\gamma }_{xz}& \frac{1}{2}{\gamma }_{yz}& {\epsilon }_z\end{array}\right]$$

Trap: Engineering vs. Tensor Shear

The tensor components (${\epsilon }_{ij}$) are NOT the same as the “Engineering Shear Strain” (${\gamma }_{xy}$) used in calculations.

• Tensor Shear: ${\epsilon }_{xy}=\frac{1}{2}{\gamma }_{xy}$
• Engineering Shear: ${\gamma }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}$

Why? The tensor must be symmetric. The factor of $1/2$ splits the total angle change ($\gamma$) equally between the two faces.

Volumetric Strain (${\epsilon }_v$)

The sum of the diagonal terms (the Trace) tells you how much the Volume changes (Dilation).

$${\epsilon }_v={\epsilon }_x+{\epsilon }_y+{\epsilon }_z=\text{Trace}([\epsilon ])=\frac{\Delta V}{V}$$

• ${\epsilon }_v>0$: Expansion (Dilation).
• ${\epsilon }_v<0$: Contraction (Compaction).
• ${\epsilon }_v=0$: Incompressible material (Volume is constant, only shape changes).

Principal Strains

Exactly like Stress, there is a specific orientation where Shear Strain is zero.

• These are the Principal Strains (${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$).
• You can find them using Mohr’s Circle for strain (identical process to stress, just plot $\epsilon$ on x-axis and $\gamma /2$ on y-axis).

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The Displacement Gradient ($\mathbf{J}$)

The derivative of displacement ($\nabla \mathbf{u}$) tells us how a point moves relative to its neighbours. It contains two things mixed together: Deformation and Rigid Rotation.

We mathematically split it to isolate the strain:

$$\mathbf{J}=\nabla \mathbf{u}=\underset{\text{Strain}}{\underbrace{\mathbf{\epsilon }}}+\underset{\text{Rotation}}{\underbrace{\mathbf{\Omega }}}$$

1. The Strain Tensor ($\mathbf{\epsilon }$)

• Maths: The Symmetric part of $\mathbf{J}$. $\mathbf{\epsilon }=\frac{1}{2}(\mathbf{J}+{\mathbf{J}}^T)$
• Physics: Represents pure stretching and shape change. Causes Stress.

2. The Spin Tensor ($\mathbf{\Omega }$)

• Maths: The Skew-Symmetric part of $\mathbf{J}$. $\mathbf{\Omega }=\frac{1}{2}(\mathbf{J}-{\mathbf{J}}^T)$
• Physics: Represents rigid body rotation. Causes NO Stress.

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$$\left[\begin{array}{ccc}{\epsilon }_x& {\epsilon }_{xy}& {\epsilon }_{xz}\\ {\epsilon }_{xy}& {\epsilon }_y& {\epsilon }_{yz}\\ {\epsilon }_{xz}& {\epsilon }_{yz}& {\epsilon }_z\end{array}\right]=\underset{\begin{array}{c}\text{Isotropic}\\ \text{(Volume Changes)}\end{array}}{\underbrace{\left[\begin{array}{ccc}{\epsilon }_m& 0& 0\\ 0& {\epsilon }_m& 0\\ 0& 0& {\epsilon }_m\end{array}\right]}}+\underset{\begin{array}{c}\text{Deviatoric}\\ \text{(Shape Changes)}\end{array}}{\underbrace{\left[\begin{array}{ccc}{\epsilon }_x-{\epsilon }_m& {\epsilon }_{xy}& {\epsilon }_{xz}\\ {\epsilon }_{xy}& {\epsilon }_y-{\epsilon }_m& {\epsilon }_{yz}\\ {\epsilon }_{xz}& {\epsilon }_{yz}& {\epsilon }_z-{\epsilon }_m\end{array}\right]}}$$ $${\epsilon }_m=\frac{{\epsilon }_x+{\epsilon }_y+{\epsilon }_z}{3}=\frac{1}{3}{\epsilon }_v(\text{Hydrostatic Strain})$$

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Constitutive Equations (Hooke’s Law)

Linear Elasticity (1D)

For a simple bar pulled in one direction (Uniaxial):

Hooke’s Law $\sigma =E\epsilon$

• $E$: Young’s Modulus (Stiffness). Higher $E$ = Stiffer material.

We can use Hooke’s law to get force and work done in two occasions:

$$F=k\Delta x$$

1. Instantaneous (Constant Force) Force $F$ is at 100% from start to finish.

• $W=F\Delta x=k\Delta x^2$
• Note: Contains 2x the energy of the progressive case.

2. Progressive (Linear Force) Force $F$ grows from 0 to max as displacement increases.

• $W={\int }_0^{\Delta x}k\tau d\tau =\frac{1}{2}k\Delta x^2$
• Note: Area of the triangle under the $F/\Delta x$ curve ($W=\frac{1}{2}F\Delta x$).

Poisson’s Ratio ($\nu$) When you pull a rubber band, it gets thinner. Poisson’s ratio measures this necking effect. $\nu =-\frac{{\epsilon }_{transverse}}{{\epsilon }_{longitudinal}}$

• Range: $0\le \nu \le 0.5$ (Most metals $\approx 0.3$).

Energy Density

$${\eta }_{\text{normal}}=\frac{1}{2}\sigma \epsilon$$ $${\eta }_{\text{shear}}=\frac{1}{2}\tau \gamma$$ $${\eta }_{\text{tot}}=\frac{1}{2}({\sigma }_{xx}{\epsilon }_{xx}+{\sigma }_{yy}{\epsilon }_{yy}+{\sigma }_{zz}{\epsilon }_{zz}+{\tau }_{xy}{\gamma }_{xy}+{\tau }_{xz}{\gamma }_{xz}+{\tau }_{yz}{\gamma }_{yz})$$

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Generalized Hooke’s Law (3D)

In the real world, stress in one direction ($x$) causes strain in all directions (due to Poisson). For an Isotropic material (same properties in all directions), the equations are:

Normal Strains

$$\{\begin{array}{l}{\epsilon }_x=\frac{1}{E}[{\sigma }_x-\nu ({\sigma }_y+{\sigma }_z)]\\ {\epsilon }_y=\frac{1}{E}[{\sigma }_y-\nu ({\sigma }_x+{\sigma }_z)]\\ {\epsilon }_z=\frac{1}{E}[{\sigma }_z-\nu ({\sigma }_x+{\sigma }_y)]\end{array}$$

Shear Strains Shear is decoupled from normal stress. It depends on the Shear Modulus ($G$). ${\gamma }_{xy}=\frac{{\tau }_{xy}}{G}{\gamma }_{yz}=\frac{{\tau }_{yz}}{G}{\gamma }_{xz}=\frac{{\tau }_{xz}}{G}$

Shear Modulus

$$G=\frac{E}{2(1+\nu )}$$

In matrix form:

$$\vec{\sigma }=[C]\vec{\epsilon }$$ $$\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\tau }_{yz}\\ {\tau }_{xz}\\ {\tau }_{xy}\end{array}\right]=\underset{\mathbf{C}\text{ (Stiffness Matrix)}}{\underbrace{\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& C_{15}& C_{16}\\ C_{21}& C_{22}& C_{23}& C_{24}& C_{25}& C_{26}\\ C_{31}& C_{32}& C_{33}& C_{34}& C_{35}& C_{36}\\ C_{41}& C_{42}& C_{43}& C_{44}& C_{45}& C_{46}\\ C_{51}& C_{52}& C_{53}& C_{54}& C_{55}& C_{56}\\ C_{61}& C_{62}& C_{63}& C_{64}& C_{65}& C_{66}\end{array}\right]}}\cdot \left[\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{zz}\\ {\gamma }_{yz}\\ {\gamma }_{xz}\\ {\gamma }_{xy}\end{array}\right]$$

• For Isotropic materials (standard metal), this giant matrix is mostly zeros. You only need two numbers to fill it: $E$ and $\nu$.
• The matrix is Symmetric ($C_{ij}=C_{ji}$), so you “only” need 21 constants for the worst-case scenario (like wood or composites), not 36.

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Thermal Strain

Temperature changes cause expansion/contraction without stress (unless constrained).

${\epsilon }_T=\alpha \cdot \Delta T$

• $\alpha$: Coefficient of thermal expansion.
• Total Strain: The sum of mechanical stress + thermal expansion. ${\epsilon }_{total}={\epsilon }_{mech}+{\epsilon }_{th}=\frac{\sigma }{E}+\alpha \Delta T$

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Some Properties

Metal	$E/Nm{m}^{-2}$	$\nu$
Steel	$2\times 10^5$	$0.3$
Cast Iron	$1\times 10^5$ to $1.8\times 10^5$	$0.27$
$Ti$	$1.2\times 10^5$	$0.3$
$Al$	$7\times 10^4$	$0.3$

${\sigma }_{\text{all}}\approx 1000Nm{m}^{-2}$ (max allowed stress) for steel, which corresponds to $\epsilon =0.005$.

Material (minimum values)	Type/Grade	${\sigma }_y$​ (MPa)	${\sigma }_{\text{ult}}$​ (MPa)	A%
STEEL - Structural	S 235	235	360	26
(UNI EN 10025)	S 275	275	430	22
	S 355	355	510	22
STEEL - annealed	C 30	400	600	18
(UNI EN 10083)	C 60	580	850	11
	41Cr4	800	1000	11
	36NiCrMo3	1050	1250	9
CAST IRON	G10	-	100	-
Gray	G20	-	200	-
	G30	-	290	-
CAST IRON	Gs370-17	230	370	17
Spheroidal	Gs500-7	320	500	7
	Gs700-2	420	700	2

NOTE

Values represent minimum thresholds. For Gray Cast Iron, yield strength (${\sigma }_y$) and elongation ($A\%$) are typically not defined due to the material’s brittle nature.