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1762x

002716

2023-12-15

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Strains

In the navigator, define the strains 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 strains are subdivided into the following categories:

Basic Total Strains: strains in the direction of the surface axes
Principal Total Strains: strains in the direction of the principal axes
Maximum Total Strains: extreme values of strains
Equivalent Total Strains: strains analyzed according to different equivalent stress hypotheses

Basic total strains

The basic strains are related to the directions of the local surface axes. When analyzing curved surfaces, they refer to the local axes of finite elements (see image Displaying FE Axis Systems ).

The basic strains have the following meanings:

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 strains in a surface element. The principal axes 1 (maximum value) and 2 (minimum value) are arranged orthogonally.

The principal strains are determined from the basic strains. They have the following meanings:

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

It is possible to graphically display the principal axis directions α as 'Trajectories' (compare image Displaying Trajectories of Principal Axes ).

Maximum total strains

This category shows the strain's positive and negative extreme values resulting from the principal total strains.

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

Equivalent total strains

The Basic Strains are combined according to four equivalent stress hypotheses for the plane stress condition.

von Mises

The approach by von Mises is also called "shape modification hypothesis". This energy is the type of energy that causes a distortion or deformation of the object.

The equivalent total strains according to von Mises have the following meanings:

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

The equivalent total strains according to Tresca have the following meanings:

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 greatest principal stress leads to failure.

The equivalent total strains according to Rankine have the following meanings:

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

In the case of the Bach hypothesis, it is assumed that the failure occurs in the direction of the greatest strain.

The equivalent total strains according to Bach have the following meanings:

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

Parent Chapter

Results by Surface