{
  "id": "youngs-modulus",
  "version": "25032d749ec9",
  "status": "published",
  "name": "Young's Modulus Calculator",
  "question": "What is Young's modulus?",
  "summary": "Young's modulus calculator: E = stress ÷ strain = F × L₀ ÷ (A × ΔL) from a load test, or the stretch or force for a material, with the stress and strain.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/youngs-modulus-calculator",
  "markdown": "https://www.acalculator.org/physics/youngs-modulus-calculator.md",
  "kind": "function",
  "method": "σ = F ÷ A; ε = ΔL ÷ L₀; E = σ ÷ ε = F × L₀ ÷ (A × ΔL); ΔL = F × L₀ ÷ (A × E); F = E × A × ΔL ÷ L₀; a round bar's A = π × d² ÷ 4.",
  "assumptions": [
    "The bar obeys Hooke’s law: the stress stays in the straight-line (elastic) part of the stress–strain curve, below the proportional limit.",
    "The force acts along the bar and spreads evenly over the cross-section; a push (compression) uses the same formulas with sizes, not signs."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "Which value to work out from E = F × L₀ ÷ (A × ΔL), or E = σ ÷ ε from the stress and strain.",
        "type": "string",
        "enum": [
          "modulus",
          "stretch",
          "force",
          "stress"
        ]
      },
      "F": {
        "title": "Force (F)",
        "description": "The pulling or pushing force along the bar, square to its cross-section.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "sec": {
        "title": "Cross-section",
        "description": "Type the cross-section area, or the diameter of a round bar or wire.",
        "type": "string",
        "enum": [
          "area",
          "round"
        ]
      },
      "A": {
        "title": "Cross-section area (A)",
        "description": "The area of the cut across the bar, square to the force.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "area",
        "minimum": 1e-12,
        "maximum": 1000000
      },
      "d": {
        "title": "Diameter (d)",
        "description": "The diameter of a round bar or wire; the area is π × d² ÷ 4.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.000001,
        "maximum": 1000
      },
      "L": {
        "title": "Original length (L₀)",
        "description": "The length of the bar before the force acts.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.000001,
        "maximum": 1000000
      },
      "dL": {
        "title": "Change in length (ΔL)",
        "description": "How much the bar stretches (or shortens under a push).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-12,
        "maximum": 1000000
      },
      "s": {
        "title": "Stress (σ)",
        "description": "The force per unit of area, in the unit chosen below.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "su": {
        "title": "Stress unit",
        "description": "The unit of the stress you type.",
        "type": "string",
        "enum": [
          "Pa",
          "kPa",
          "MPa",
          "GPa",
          "psi",
          "ksi",
          "Msi"
        ]
      },
      "e": {
        "title": "Strain (ε)",
        "description": "The change in length divided by the original length, a pure number (0.001 is 0.1%).",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 10
      },
      "mat": {
        "title": "Material",
        "description": "The material of the bar, which sets Young's modulus (OpenStax Table 12.1).",
        "type": "string",
        "enum": [
          "steel",
          "aluminum",
          "brass",
          "copper",
          "iron",
          "nickel",
          "lead",
          "glass",
          "granite",
          "marble",
          "concrete",
          "brick",
          "hardwood",
          "boneTension",
          "boneCompression",
          "polystyrene",
          "silk",
          "spider",
          "hair",
          "custom"
        ]
      },
      "E": {
        "title": "Young's modulus (E)",
        "description": "The material's Young's modulus, in the unit chosen below.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "Eu": {
        "title": "Modulus unit",
        "description": "The unit of the Young's modulus you type.",
        "type": "string",
        "enum": [
          "Pa",
          "kPa",
          "MPa",
          "GPa",
          "psi",
          "ksi",
          "Msi"
        ]
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The value worked out, to 6 significant figures, in SI and US units.",
      "format": "text"
    },
    "modulus": {
      "label": "Young's modulus (Pa)",
      "description": "E = stress ÷ strain, in pascals.",
      "format": "number"
    },
    "gpa": {
      "label": "Young's modulus (GPa)",
      "description": "E in gigapascals: 1 GPa = 10⁹ Pa.",
      "format": "number"
    },
    "psi": {
      "label": "Young's modulus (psi)",
      "description": "E in pounds-force per square inch.",
      "format": "number"
    },
    "stress": {
      "label": "Stress (MPa)",
      "description": "σ = F ÷ A, in megapascals.",
      "format": "number"
    },
    "stressPsi": {
      "label": "Stress (psi)",
      "description": "σ in pounds-force per square inch.",
      "format": "number"
    },
    "strain": {
      "label": "Strain",
      "description": "ε = ΔL ÷ L₀, a pure number.",
      "format": "number"
    },
    "strainPct": {
      "label": "Strain (%)",
      "description": "ε × 100.",
      "format": "number"
    },
    "stretch": {
      "label": "Change in length (mm)",
      "description": "ΔL in millimetres.",
      "format": "number"
    },
    "stretchIn": {
      "label": "Change in length (in)",
      "description": "ΔL in inches.",
      "format": "number"
    },
    "force": {
      "label": "Force (N)",
      "description": "F in newtons.",
      "format": "number"
    },
    "lbf": {
      "label": "Force (lbf)",
      "description": "F in pounds-force.",
      "format": "number"
    },
    "area": {
      "label": "Cross-section area (mm²)",
      "description": "A in square millimetres; π × d² ÷ 4 for a round bar.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "modulus",
      "F": "5390 N",
      "sec": "area",
      "A": "0.3 cm2",
      "d": "6 mm",
      "L": "2 m",
      "dL": "1.8 mm",
      "s": 180,
      "su": "MPa",
      "e": 0.0009,
      "mat": "steel",
      "E": 200,
      "Eu": "GPa"
    },
    "outputs": {
      "result": "E = 199.63 GPa (28,953,800 psi)",
      "modulus": 199629629629.62964,
      "gpa": 199.62962962962962,
      "psi": 28953829.865401026,
      "stress": 179.66666666666666,
      "stressPsi": 26058.446878860923,
      "strain": 0.0009,
      "strainPct": 0.09,
      "stretch": 1.8,
      "stretchIn": 0.07086614173228346,
      "force": 5390,
      "lbf": 1211.7202033074395,
      "area": 30
    },
    "text": "E = 199.63 GPa (28,953,800 psi), with a stress of 179.667 MPa and a strain of 0.0009."
  },
  "examples": [
    {
      "given": {
        "find": "stretch",
        "mat": "steel",
        "sec": "area",
        "F": 5390,
        "A": 0.00003,
        "L": 2
      },
      "expect": {
        "dL": 1.7966666666666666,
        "s": 179.66666666666666,
        "result": "ΔL = 1.79667 mm (0.0707349 in)"
      },
      "source": "OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19); hand calculation in content.mdx: ΔL = 5,390 × 2 ÷ (0.00003 × 2 × 10¹¹) = 0.00179667 m"
    },
    {
      "given": {
        "find": "modulus",
        "sec": "area",
        "F": 5390,
        "A": 0.00003,
        "L": 2,
        "dL": 0.0018
      },
      "expect": {
        "gpa": 199.62962962962962,
        "e": 0.0009,
        "result": "E = 199.63 GPa (28,953,800 psi)"
      },
      "source": "OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19); hand calculation in content.mdx: E = 5,390 × 2 ÷ (0.00003 × 0.0018) = 1.99630 × 10¹¹ Pa"
    },
    {
      "given": {
        "find": "stress",
        "s": 180,
        "su": "MPa",
        "e": 0.0009
      },
      "expect": {
        "gpa": 200,
        "E": 200000000000
      },
      "source": "OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19), Equation 12.36: E = σ ÷ ε = 180 MPa ÷ 0.0009 = 200 GPa, the steel value of Table 12.1"
    },
    {
      "given": {
        "find": "force",
        "mat": "steel",
        "sec": "area",
        "A": 0.00003,
        "L": 2,
        "dL": 0.0018
      },
      "expect": {
        "F": 5400,
        "s": 180,
        "result": "F = 5,400 N (1,213.97 lbf)"
      },
      "source": "OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19); hand calculation in content.mdx: F = 2 × 10¹¹ × 0.00003 × 0.0018 ÷ 2 = 5,400 N"
    },
    {
      "given": {
        "find": "stretch",
        "mat": "aluminum",
        "sec": "round",
        "d": 0.01,
        "L": 1,
        "F": 10000
      },
      "expect": {
        "dL": 1.8189136353359467,
        "A": 78.53981633974483
      },
      "source": "OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19), Table 12.1 (aluminum 7.0 × 10¹⁰ Pa); hand calculation in content.mdx: ΔL = 10,000 × 1 ÷ (π × 0.005² × 7 × 10¹⁰)"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus: Equation 12.34 (tensile stress F⊥ ÷ A), 12.35 (tensile strain ΔL ÷ L₀), 12.36 (Y = F⊥L₀ ÷ (AΔL)); Table 12.1 (approximate elastic moduli); Example 12.8 (a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus",
    "OpenStax, College Physics 2e, §5.3 Elasticity: Stress and Strain: Table 5.3, elastic moduli in 10⁹ N/m² (polystyrene 3, silk 6, spider thread 3), retrieved 2026-10-19. https://openstax.org/books/college-physics-2e/pages/5-3-elasticity-stress-and-strain",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 lbf = 4.4482216152605 N, 1 psi = 1 lbf/in², retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "force",
    "pressure",
    "moment-of-inertia"
  ],
  "changelog": []
}