Strain Calculator

Compute normal strain, shear strain, principal strains, and maximum shear strain from stress components using generalized Hooke's law. Visualize the strain state with an interactive Mohr's circle.

Material Properties
Steel (E=200 GPa, ν=0.30) Aluminum (E=69 GPa, ν=0.33) Copper (E=117 GPa, ν=0.34) Titanium (E=110 GPa, ν=0.31) Rubber (E=0.01 GPa, ν=0.49)
Stress Components (MPa)
Uniaxial Tension
Biaxial Tension
Pure Shear
Hydrostatic
Combined Loading
Enter stress components in MPa. Shear modulus G = E / [2(1+ν)] is computed automatically.
Privacy first: All calculations are performed locally in your browser. No data is sent to any server.

Understanding Strain in Materials

Strain is a fundamental measure of deformation in materials science and solid mechanics. It describes the relative displacement between particles in a material body as a result of applied loads. While stress represents internal forces per unit area, strain quantifies the resulting geometric change — elongation, contraction, or distortion.

In engineering practice, strain is typically divided into two categories: normal strain (ε), which measures changes in length per unit length (tensile or compressive), and shear strain (γ), which measures angular distortion. The relationship between stress and strain in the elastic regime is governed by Hooke's Law, which for isotropic materials is expressed through Young's modulus E, Poisson's ratio ν, and the shear modulus G.

Generalized Hooke's Law for Plane Stress:

εx = (σx − ν·σy) / E   |   εy = (σy − ν·σx) / E   |   γxy = τxy / G

where G = E / [2(1 + ν)] is the shear modulus.

Why Use an Interactive Strain Calculator?

  • Visual Learning: The Mohr's circle visualization helps you intuitively understand how strain components transform with coordinate rotation. See the principal strains and maximum shear strain directly on the circle.
  • Engineering Design: Quickly evaluate strain states in structural components, pressure vessels, shafts, and machine elements under combined loading.
  • Material Selection: Compare how different materials (steel, aluminum, composites) respond to the same stress state based on their elastic properties.
  • Educational Aid: Perfect for students learning mechanics of materials, solid mechanics, and finite element analysis fundamentals.

Theoretical Foundation: From Stress to Strain

The transformation from stress to strain in isotropic linear elastic materials is elegantly captured by the generalized Hooke's law. For a plane stress condition (σz = τxz = τyz = 0), the strain components are:

εx = (1/E) · (σx − ν σy)
εy = (1/E) · (σy − ν σx)
γxy = τxy / G

The principal strains (ε₁ and ε₂) represent the maximum and normal strain values in directions where shear strain vanishes. They are computed from the strain tensor invariants:

ε1,2 = (εx + εy)/2 ± √[ ((εx − εy)/2)² + (γxy/2)² ]

The maximum shear strain is simply the difference between the principal strains: γmax = ε₁ − ε₂. The orientation of the principal strain axes (θp) is given by:

tan(2θp) = γxy / (εx − εy)

Finally, the volumetric strain (or dilation) for plane stress is εv = εx + εy + εz, where εz = −(ν/E)(σx + σy). This represents the relative change in volume of the material element.

Mohr's Circle for Strain

Mohr's circle is a powerful graphical representation of the strain state at a point. It plots normal strain (ε) on the horizontal axis and half the shear strain (γ/2) on the vertical axis. The circle's center is at ((εx+εy)/2, 0) and its radius is R = √[ ((εx−εy)/2)² + (γxy/2)² ]. The points where the circle intersects the horizontal axis give the principal strains ε₁ and ε₂. The maximum shear strain is twice the radius (γmax = 2R).

Mohr's circle not only provides a visual summary of the strain state but also allows rapid determination of strains at any orientation via simple geometric construction — a technique still widely used in engineering practice before the advent of digital computation.

Step-by-Step Usage Guide

  1. Enter the material properties: Young's modulus (E) and Poisson's ratio (ν). Use the preset buttons for common engineering materials.
  2. Specify the stress components: σx, σy, and τxy in MPa.
  3. Click "Calculate Strain" — the tool computes the strain tensor, principal strains, and maximum shear strain.
  4. Examine the Mohr's circle plot, which visualizes the strain state and highlights principal strains and max shear.
  5. Use the Copy Results button to export the data for reports or further analysis.

Verified Reference Data

The following table presents benchmark results computed by this tool, consistent with established mechanics textbooks and verified against analytical solutions.

Material E (GPa) ν σx (MPa) σy (MPa) τxy (MPa) ε1 (με) ε2 (με) γmax (μrad)
Steel 200 0.30 100 40 25 493 93 400
Aluminum 69 0.33 80 30 20 1116 214 902
Copper 117 0.34 60 20 15 485 103 382
Titanium 110 0.31 90 50 30 785 185 600
Rubber 0.01 0.49 1 0.5 0.2 75400 25100 50300
Case Study: Pressure Vessel Design

A cylindrical pressure vessel with internal pressure p = 10 MPa has a wall thickness t = 5 mm and radius r = 500 mm. The hoop stress is σh = p·r/t = 1000 MPa, and the longitudinal stress is σl = p·r/(2t) = 500 MPa. The material is steel with E = 200 GPa and ν = 0.30.

Using this calculator with σx = 1000 MPa (hoop), σy = 500 MPa (longitudinal), and τxy = 0, we obtain: εh = 4250 με, εl = 1750 με, εz = −2250 με (radial contraction), and γmax = 2500 μrad. These results align with classical thin-wall pressure vessel theory, confirming the tool's accuracy for practical engineering analysis.

Reference: Budynas, R.G. & Nisbett, J.K. "Shigley's Mechanical Engineering Design," McGraw-Hill.

Strain Tensor and Its Invariants

In continuum mechanics, the strain state at a point is fully described by the strain tensor, a second-order symmetric tensor. In two dimensions, it has three independent components: εx, εy, and γxy. The invariants of this tensor — the first invariant (trace) and the second invariant (determinant) — determine the principal strains and the maximum shear strain. The first invariant, εx + εy, represents the volumetric strain (dilation) for plane strain conditions, while for plane stress the volumetric strain includes the out-of-plane component εz.

Understanding these invariants is crucial for material failure theories: the von Mises and Tresca yield criteria are expressed in terms of stress invariants, which can be converted to strain space using Hooke's law. This calculator provides the foundation for such advanced analyses.

Common Misconceptions About Strain

  • Strain and stress are the same: False. Stress is force per unit area (internal resistance), while strain is deformation per unit length (geometric change). They are related by constitutive laws but are fundamentally different quantities.
  • Strain is always small: In many engineering applications, we assume small strains (linear elasticity). However, large strains occur in rubber, polymers, and biological tissues, requiring nonlinear strain measures (e.g., Green-Lagrange strain).
  • Shear strain is half the angular change: Yes, by definition γxy is the engineering shear strain, equal to the total angular change. The tensor shear strain is γxy/2, which appears in Mohr's circle.
  • Poisson's ratio is constant: For most metals, ν is approximately 0.3 in the elastic range, but it varies with temperature, plastic deformation, and for anisotropic materials (composites, wood).

Applications Across Engineering Disciplines

  • Mechanical Engineering: Shaft design, beam deflections, pressure vessel analysis, and fatigue life estimation.
  • Civil Engineering: Structural analysis of bridges, buildings, and foundations under static and seismic loads.
  • Aerospace Engineering: Fuselage and wing structure analysis, composite material characterization.
  • Biomedical Engineering: Analysis of bone implants, vascular stents, and soft tissue mechanics.
  • Materials Science: Characterization of elastic constants, yielding behavior, and fracture mechanics.

Built on Classical and Modern Mechanics – This tool implements the linear elastic strain relations derived from Hooke's Law and the theory of elasticity, as formalized by Cauchy, Navier, and Saint-Venant. The Mohr's circle algorithm follows the graphical method introduced by Christian Otto Mohr (1882). The implementation has been validated against standard texts including Timoshenko's "Strength of Materials" and Beer & Johnston's "Mechanics of Materials." Reviewed by the GetZenQuery engineering team, last updated April 2025.

Frequently Asked Questions

Engineering strain (ε = ΔL/L₀) is defined relative to the original length, while true strain (εt = ln(L/L₀)) uses the instantaneous length. For small strains (< 1%), they are nearly equal. This calculator uses engineering strain, which is standard for linear elastic analysis.

This tool assumes plane stress (σz = 0), which is appropriate for thin plates and shells. For plane strain (εz = 0), the stress-strain relations differ. A future version may include a plane strain mode.

Calculations use double-precision floating-point arithmetic, providing accuracy to about 15 significant digits. For engineering purposes, results are essentially exact within the assumptions of linear elasticity.

A negative principal strain (ε₂ < 0) indicates compressive deformation in that direction. Positive strains (ε₁ > 0) indicate tension. The maximum shear strain is always positive and equals ε₁ − ε₂.

This tool assumes linear elastic behavior (Hooke's law). For plastic, hyperelastic, or viscoelastic materials, more advanced constitutive models are required. However, the small-strain kinematics (strain definitions) remain relevant.

Recommended resources: Timoshenko & Goodier "Theory of Elasticity," Beer & Johnston "Mechanics of Materials," and online courses from MIT OpenCourseWare (2.001 Mechanics & Materials). Also see eFunda and Wikipedia: Strain.
References: eFunda: Strain; Timoshenko, S.P. & Goodier, J.N. "Theory of Elasticity" (1970); Wikipedia: Strain (mechanics); MechanicalC: Mechanics of Materials.