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2856x

002716

Dipl.-Ing. (FH) Robert Vogl

2023-12-15

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Strains

Specify in the Navigator which distortions are to be displayed on the surfaces. The table lists the strains of each surface according to the specifications set in the Result Table Manager .

Selecting Surface Strains in Navigator

Surface Strains in Table

The surface distortions are divided into the following categories:

Basic total strains: Distortions in the direction of the surface axes
Principal total strains: Distortions in the direction of the major axes
Maximum total strains: Extreme values of the distortions
Equivalent total strains: Distortions according to different equivalent stress hypotheses

Tip

Using the Layer side control field, you can quickly switch between the distortions present on the top and bottom sides of the surfaces (see image Displaying stresses of layer sides).

Basic Total Strains

The basic strains refer to the directions of the local surface axes. For curved surfaces, they refer to the local axes of the individual finite elements (see image Displaying FE axis systems ).

The basic strains mean in detail:

ε_{x, +} = \frac{\partial u}{\partial x} + \frac{d}{2} \frac{\partial φ_{y}}{\partial x}

d	Thickness of the surface

Strain ε_y,+ in Direction of Local y-Axis on Positive Surface Side

ε_{y, +} = \frac{\partial v}{\partial y} + \frac{d}{2} (- \frac{\partial φ_{x}}{\partial y})

Strain ε_y,+ in Direction of Local y-Axis on Positive Surface Side

γ_{x y, +} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{d}{2} (\frac{\partial φ_{y}}{\partial y} - \frac{\partial φ_{x}}{\partial x})

Related Rotation γ_xy,+ on Positive Surface Side

ε_{x, -} = \frac{\partial u}{\partial x} - \frac{d}{2} \frac{\partial φ_{y}}{\partial x}

Strain ε_x,- in Direction of x-Axis on Negative Surface Side

ε_{y, -} = \frac{\partial v}{\partial y} - \frac{d}{2} (- \frac{\partial φ_{x}}{\partial y})

Strain ε_y,- in Direction of y-Axis on Negative Surface Side

γ_{x y, -} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} - \frac{d}{2} (\frac{\partial φ_{y}}{\partial y} - \frac{\partial φ_{x}}{\partial x})

Related Rotation γ_xy,- on Negative Surface Side

Principal Total Strains

While the basic strains refer to the xyz coordinate system of a surface, the principal strains represent the extreme values of the distortions in a surface element. The major axes 1 (maximum value) and 2 (minimum value) are arranged orthogonally.

The principal strains are determined from the basic strains. They mean in detail:

ε_{1, +} = \frac{1}{2} (ε_{x, +} + ε_{y, +} + \sqrt{{(ε_{x, +} - ε_{y, +})}^{2} + {γ_{x y, +}}^{2}})

Strain ε_1,+ in Direction of Principal Axis 1 on Positive Surface Side

ε_{2, +} = \frac{1}{2} (ε_{x, +} + ε_{y, +} - \sqrt{{(ε_{x, +} - ε_{y, +})}^{2} + {γ_{x y, +}}^{2}})

Strain ε_2,+ in Direction of Principal Axis 2 on Positive Surface Side

α_{+} = \frac{1}{2} [\arctan (\frac{γ_{x y, +}}{ε_{x, +} - ε_{y, +}})]

Angle α₊ Between Local Axis x (or y) and Principal Axis 1 (or 2) for Strains on Positive Surface Side

ε_{1, -} = \frac{1}{2} (ε_{x, -} + ε_{y, -} + \sqrt{{(ε_{x, -} - ε_{y, -})}^{2} + {γ_{x y, -}}^{2}})

Strain ε_1,- in Direction of Principal Axis 1 on Negative Surface Side

ε_{2, -} = \frac{1}{2} (ε_{x, -} + ε_{y, -} - \sqrt{{(ε_{x, -} - ε_{y, -})}^{2} + {γ_{x y, -}}^{2}})

Strain ε_2,- in Direction of Principal Axis 2 on Negative Surface Side

α_{-} = \frac{1}{2} [\arctan (\frac{γ_{x y, -}}{ε_{x, -} - ε_{y, -}})]

Angle α_- Between Local Axis x (or y) and Principal Axis 1 (or 2) for Strains on Negative Surface Side

You can display the major axis directions α graphically as 'trajectories' (compare image Displaying trajectories of major axes ).

Maximum Total Strains

This category outputs the positive and negative extreme values of the distortions resulting from the principal total strains.

ε_max,+	Maximum value of strain on the positive surface side
ε_min,+	Minimum value of strain on the positive surface side
\|ε_max\|₊	Largest extreme value on the positive surface side
ε_max,-	Maximum value of strain on the negative surface side
ε_min,-	Minimum value of strain on the negative surface side
\|ε_max\|_-	Largest extreme value on the negative surface side
ε_max	Maximum value of strain on the positive or negative surface side
ε_min	Minimum value of strain on the positive or negative surface side
\|ε_max\|	Largest extreme value on the positive or negative surface side

Equivalent Total Strains

The Basic total strains are combined according to four equivalent stress hypotheses for the plane stress state.

Von Mises

The hypothesis according to von Mises is also known as the "distortion energy hypothesis". The distortion energy represents the energy that causes a distortion or deformation of the body.

The equivalent total strains according to von Mises mean:

ε_{v, Mises, +} = \frac{\sqrt{{(ε_{x, +} - ε_{y, +})}^{2} + {(\frac{ε_{x, +} + ν ε_{y, +}}{1 - ν})}^{2} + {(\frac{ν ε_{x, +} + ε_{y, +}}{1 - ν})}^{2} + \frac{3}{2} {γ_{x y, +}}^{2}}}{\sqrt{2} (1 + ν)}

Equivalent Strain ε_v,Mises,+ on Positive Surface Side

ε_{v, Mises, -} = \frac{\sqrt{{(ε_{x, -} - ε_{y, -})}^{2} + {(\frac{ε_{x, -} + ν ε_{y, -}}{1 - ν})}^{2} + {(\frac{ν ε_{x, -} + ε_{y, -}}{1 - ν})}^{2} + \frac{3}{2} {γ_{x y, -}}^{2}}}{\sqrt{2} (1 + ν)}

Equivalent Strain ε_v,Mises,- on Negative Surface Side

ε_{v, Mises} = \max (ε_{v, Mises, +}; ε_{v, Mises, -})

Maximum Equivalent Strain ε_v,Mises on Positive or Negative Surface Side

Tresca

The hypothesis according to Tresca assumes that failure is caused by the maximum shear stress.

The equivalent total strains according to Tresca mean:

ε_{v, Tresca, +} = \frac{\sqrt{{(ε_{x, +} - ε_{y, +})}^{2} + {γ_{x y, +}}^{2}}}{1 + ν}

Equivalent Strain ε_v,Tresca,+ on Positive Surface Side

ε_{v, Tresca, -} = \frac{\sqrt{{(ε_{x, -} - ε_{y, -})}^{2} + {γ_{x y, -}}^{2}}}{1 + ν}

Equivalent Strain ε_v,Tresca,- on Negative Surface Side

ε_{v, Tresca} = \max (ε_{v, Tresca, +}; ε_{v, Tresca, -})

Maximum Equivalent Strain ε_v,Tresca on Positive or Negative Surface Side

Rankine

The hypothesis according to Rankine assumes that the largest principal stress leads to failure.

The equivalent total strains according to Rankine mean:

ε_{v, Rankine, +} = \frac{1}{2} (\frac{|ε_{x, +} + ε_{y, +}|}{1 - ν} \frac{\sqrt{{(ε_{x, +} - ε_{y, +})}^{2} + {γ_{x y, +}}^{2}}}{1 + ν})

Equivalent Strain ε<sub>v,Rankine,-</sub> on Positive Surface Side

ε_{v, Rankine, -} = \frac{1}{2} (\frac{|ε_{x, -} + ε_{y, -}|}{1 - ν} \frac{\sqrt{{(ε_{x, -} - ε_{y, -})}^{2} + {γ_{x y, -}}^{2}}}{1 + ν})

Equivalent Strain ε_v,Rankine,- on Negative Surface Side

ε_{v, Rankine} = \max (ε_{v, Rankine, +}; ε_{v, Rankine, -})

Maximum Equivalent Strain ε_v,Rankine on Positive or Negative Surface Side

Bach

The hypothesis according to Bach assumes that failure occurs in the direction of the largest strain.

The equivalent total strains according to Bach mean:

ε_v,Bach,+	Largest absolute value of the principal strain ε_1,+ or ε_2,+ on the positive surface side
ε_v,Bach,-	Largest absolute value of the principal strain ε_1,- or ε_2,- on the negative surface side
\|ε_max\|	Largest equivalent strain ε_v,Bach on the positive or negative surface side

Parent Chapter

Results by Surface