Mohr’s Circle for Strain Calculator (Plotter)

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Mohr’s Circle for Strain

Mohr’s Circle for Strain

Mohr’s Circle for Strain

This Mohr’s Circle plotter for strain is an interactive tool used to visualize the state of strain in a material. By inputting the components of the strain tensor—including normal strains (εxx, εyy, εzz) and shear strains (γxy, γxz, γyz)—the plotter calculates the principal strains and generates Mohr’s Circles. These circles graphically represent the transformation of strain components and allow engineers to assess the maximum and minimum normal and shear strains in different material orientations.

The strain tensor is a mathematical representation of strain within a material. It describes how the material deforms in each direction under applied loads. The normal components (εxx, εyy, εzz) indicate how much the material stretches or compresses along each axis, while the shear components (γxy, γxz, γyz) indicate the material’s deformation along planes between axes. Mohr’s Circle visualization provides insight into how these strains vary with orientation, helping in the design and analysis of materials under stress.

Please note that γxy = 2*εxy, γxz = 2*εxz, γyz = 2*εyz.

Other Strain Analysis Modules

Stress Invariants Calculator

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Stress Invariants Calculator

Stress Invariants Calculator

Input Stress Tensor Components

I1 (First Invariant): –
I2 (Second Invariant): –
I3 (Third Invariant): –

What is Stress Invariants

Stress invariants are fundamental quantities derived from the stress tensor that remain unchanged regardless of the coordinate system. They provide essential information about the internal state of stress in a material without depending on its orientation.

The three primary stress invariants are:

  1. First Invariant I1: This is the sum of the normal stresses along the principal axes, representing the trace of the stress tensor. It provides insight into the mean or hydrostatic stress within the material.
  2. Second Invariant I2​: This invariant combines both normal and shear stress components. It is related to the deviatoric (distortional) stress, which can cause shape changes in the material.
  3. Third Invariant I3​: Representing the determinant of the stress tensor, I3 indicates the volumetric effects of the stress state and is associated with pure shear states and material behavior under complex loading.

Together, these invariants are widely used in engineering and material science to analyze failure criteria, yielding, and material deformation, independent of orientation.

\[ \sigma_p^3 - I_1 \sigma_p^2 + I_2 \sigma_p - I_3 = 0 \]
\[ I_1 = \sigma_{xx} + \sigma_{yy} + \sigma_{zz} \]
\[ I_2 = \sigma_{xx}\sigma_{yy} + \sigma_{xx}\sigma_{zz} + \sigma_{yy}\sigma_{zz} - \sigma_{xy}^2 - \sigma_{xz}^2 - \sigma_{yz}^2 \]
\[ I_3 = \begin{vmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} \end{vmatrix} \]
Parent module

Courant Number Calculator for CFD

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Courant Number Calculator

Courant Number Calculator

Courant Number Formula

For a given grid cell, the Courant number CFL is defined as:

CFL=U⋅Δt/Δx​

where:

  • U is the fluid velocity (or characteristic speed of the flow).
  • Δt is the time step used in the simulation.
  • Δx is the spatial grid size or cell width.

The Courant number (often denoted as CFL for Courant-Friedrichs-Lewy number) is a crucial concept in computational fluid dynamics (CFD) and numerical simulations of partial differential equations. It helps in determining the stability and accuracy of numerical schemes used for solving fluid flow equations. Here’s an overview of the Courant number and its significance:

Definition

The Courant number is a dimensionless number that measures the ratio of the physical time step to the time required for a fluid particle to travel a distance equal to the grid spacing. It essentially evaluates how much of the physical domain a fluid particle can advance in one time step relative to the grid resolution.

Importance and Implications

  1. Stability:
    • The Courant number is used to ensure the stability of explicit time-stepping schemes in numerical methods. For many explicit methods (like Forward Euler), a CFL number greater than 1 can lead to numerical instability, causing the solution to diverge or become inaccurate.
    • In general, to ensure stability, the CFL number should be less than or equal to 1. However, in practice, a CFL number significantly smaller than 1 (e.g., 0.1 to 0.5) is often used to enhance stability and accuracy.
  2. Accuracy:
    • The choice of time step Δt and spatial grid size Δx affects the Courant number and, consequently, the accuracy of the simulation. A larger CFL number might lead to larger time steps but at the risk of reduced accuracy or stability.
  3. Adaptive Time Stepping:
    • In simulations with varying flow speeds or grid resolutions, adaptive time-stepping algorithms adjust Δt to keep the CFL number within acceptable limits, thus maintaining stability and accuracy throughout the simulation.
  4. Mesh and Solver Design:
    • When designing the mesh or choosing numerical solvers, understanding the relationship between velocity, grid spacing, and time step through the CFL number helps in balancing computational cost with the desired accuracy and stability.
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Other CFD Modules

Beam Structural Analysis Online Calculator (SFD, BMD, Deflection)

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SFD, BMD, TD, and Shaft Profile Calculator

Beam Dimensions

Supports Location

Forces

Distributed Loads

Material Properties

Beam Layout

Axial Force Diagram

Shear Forces Diagram

Bending Moment Diagram

Stress Tensor Components

Deflection (mm)

Stress-Strain Tensor Constitutive Equation Calculator (Isotropic Material)

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Stress-Strain Tensor Calculator

Strain Tensor Calculator

Stress Tensor [MPa], E (Young’s Modulus [GPa] ), and ν (Poisson’s Ratio):

σ Tensor:

E: ν:

Stress Tensor Calculator [MPa]

Strain Tensor, E (Young’s Modulus [GPa] ), and ν (Poisson’s Ratio):

ε Tensor:

E: ν:

Constitutive Equation for Isotropic Material (Hook’s Law)

Compute Strain Tensor

\[ \epsilon_{xx} = \frac{1}{E} \left( \sigma_{xx} - \nu \sigma_{yy} - \nu \sigma_{zz} \right) \]
\[ \epsilon_{yy} = \frac{1}{E} \left( \sigma_{yy} - \nu \sigma_{xx} - \nu \sigma_{zz} \right) \]
\[ \epsilon_{zz} = \frac{1}{E} \left( \sigma_{zz} - \nu \sigma_{xx} - \nu \sigma_{yy} \right) \]
\[ \epsilon_{xy} = \frac{1}{2G} \sigma_{xy} = \frac{1 + \nu}{E} \sigma_{xy} \]
\[ \epsilon_{xz} = \frac{1}{2G} \sigma_{xz} = \frac{1 + \nu}{E} \sigma_{xz} \]
\[ \epsilon_{yz} = \frac{1}{2G} \sigma_{yz} = \frac{1 + \nu}{E} \sigma_{yz} \]

Compute Stress Tensor

\[ \sigma_{xx} = \frac{E}{(1 + \nu)(1 - 2\nu)} \left[ (1 - \nu)\epsilon_{xx} + \nu (\epsilon_{yy} + \epsilon_{zz}) \right] \]
\[ \sigma_{yy} = \frac{E}{(1 + \nu)(1 - 2\nu)} \left[ (1 - \nu)\epsilon_{yy} + \nu (\epsilon_{xx} + \epsilon_{zz}) \right] \]
\[ \sigma_{zz} = \frac{E}{(1 + \nu)(1 - 2\nu)} \left[ (1 - \nu)\epsilon_{zz} + \nu (\epsilon_{xx} + \epsilon_{yy}) \right] \]
\[ \sigma_{xy} = \frac{E}{1 + \nu} \epsilon_{xy}, \quad \sigma_{xz} = \frac{E}{1 + \nu} \epsilon_{xz}, \quad \sigma_{yz} = \frac{E}{1 + \nu} \epsilon_{yz} \]

Normal and Unit Normal Vector from F(x,y,z) Calculator

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Normal Vector Calculator

Normal Vector Calculator

Point of Location (x, y, z)

Normal Vector: –
Unit Normal Vector: –

Equation Format

To input multiplication, use *

To input power use **

For example 2x+3yz-5x(y^2) –> 2*x + 3*x*z – 5*x*(y**2)

Back to Parent Module

Mechanical Shaft Design Calculator

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SFD, BMD, TD, and Shaft Profile Calculator

Shaft Segments

Bearing Locations

Forces

Torques

Material Properties

Critical Speed

N/A

Shaft Layout

Forces Diagram

Bending Moment Diagram

Torque Diagram

0°

Stress Tensor Components

Von Mises Stresses

Maximum Von Mises Stress

Midrange and Alternating Stress

Shaft Deflection

Angle of Twist

Stress Tensor (3D) Principal Stress

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Principal Stress Calculator

Principal Stress Calculator

Principal Stress 1: –
Principal Stress 2: –
Principal Stress 3: –
Principal Direction Calculator

Principal Direction Calculator

Input Stress Tensor Components

Input Principal Stress (σ_p)

Principal Direction (l, m, n):
l: –
m: –
n: –

What is Stress Tensor and It’s Principal Values?

A stress tensor is a mathematical representation of the internal forces within a material. It captures the intensity and direction of stress acting on different planes within a solid object. For a 3D object, the stress tensor is typically represented as a 3×3 matrix with normal stresses (σxx, σyy, σzz) along the diagonal and shear stresses (τxy, τxz, τyz) on the off-diagonals.

Due to the equilibrium, the stress tensor matrix is symmetry. If you put non-symmetry matrix in the calculator, the result will be unphysical!

Principal stresses are special values that represent the maximum and minimum normal stresses experienced by the material in certain directions. They are derived from the stress tensor and correspond to orientations where shear stresses vanish, meaning that the stress acts purely in the direction of the principal stresses. These stresses are crucial in engineering because they help identify critical points where materials are most likely to fail.

Stress Tensor
\[ \bar{\sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} \end{bmatrix} \]

Plane Stress (2D)

If you want to calcualte plane stress, just input zero values for Z components (σzz = τxz = τyz = 0)

Parent module

Vehicle Aerodynamics blockMesh and snappyHexMesh Calculator

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3D Model Viewer

Navigation

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How to use this Calculator?

You can import your model in the STL format, then determine the refinement box and blockMesh size using the percentage. Before you generate the mesh, please check your direction of the flow and the top direction, which will determine the size and orientation of the blockMesh.

Coordinates: X = Red, Y = Green, Z = Blue.

This calculator is designed for ground vehicle simulation, the ground coordinate will always the same as minimum object coordinate.

Click “Generate Mesh” to download your coordinates in the blockMeshDict and snappyHexMeshDict files.

Please note that this calculator only calculates the vertices’ coordinates, refinement region coordinates, and location in mesh coordinates; adjust the other parameters such as the boundary conditions, resolutions, object name, etc., based on your problem.

Download Simulation Template

This web calculator is part of tensorCFD project, which is development of OpenFOAM templates for industrial applications. You can download the full template to simulate the ground vehicle aerodynamics here:

tensorXFV
Parent Module

Turbulent Boundary And Initial Conditions Calculator

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Reynold Averaged Navier Stokes (RANS)

In some RANS models like κ-ω, κ-ε, and κ-ω SST, the iteration process can sometimes become sensitive to the input of boundary conditions or initial conditions for turbulent variables such as κ, ε, and ω. This is why differences in the initialization process can lead to distinct solutions.

Although in several commercial CFD software, these boundary conditions are usually computed automatically with inputs like velocity, characteristic length (hydraulic diameter), or turbulence intensity, it’s important to understand how these parameters are calculated.

For internal flows, these boundary conditions differ from external flows, thus requiring separate equations.









Generals

Internal Flow Results

Reynolds Number (Re)
Turbulence Intensity (I)
Beta (β)
Turbulent Kinetic Energy (κ)
Dissipation Rate (ε)
Specific Dissipation Rate (ω)

External Flow Results

Reynolds Number (Re)
Turbulence Intensity (I)
Beta (β)
Turbulent Kinetic Energy (κ)
Dissipation Rate (ε)
Specific Dissipation Rate (ω)
Turbulent Flow Boundary Conditions Calculator

Internal Flow Equation

For internal flow scenarios (e.g., flow in pipes, ducting, turbomachinery, etc.), the following equations are utilized:

External Flow Equation

Then, for external flows (e.g., aircraft aerodynamics, race car aerodynamics, wind around buildings, etc.), the κ equation takes the same form as in internal flows. However, for other variables, the relationships are as follows:

Meanwhile, the value of I can be calculated using the approximation:

With U being the reference velocity (usually the inlet velocity), C,miu is the coefficient in the turbulence model typically set to 0.09, L is the length scale usually set to 0.038 times the pipe diameter (for fully developed turbulent flow), and β has a value depending on the Reynolds number as follows:

ReIβ
less than 30000-1%0.1-0.2
3000-50001%11.6%-16.5%
5000-150001-5%16.5-26.7
15000–200005-20%26.7-34
more than 200005-20%100

It’s important to note that the calculations above serve as initial approximations to facilitate convergence, and eventually, they will be “washed out” as iterations progress. Therefore, extreme precision in these calculations isn’t necessary.

Parent Module