Elastic constants calculator
For steel, a Young's modulus of 200 GPa and a Poisson's ratio of 0.3 are enough to fix the shear and bulk moduli too. Enter any two of the four elastic constants to get the other two, checked against the stable range. Free in your browser.
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Run — free
Worked example
- Elastic modulus
- 200
- Poisson ratio
- 0.3
- Shear modulus
- 76.9230769231
- Bulk modulus
- 166.666666667
- Known
- elastic_modulus+poisson_ratio
Linear isotropic elasticity is completely described by any two independent elastic constants. Materials handbooks mix Young's modulus E, the shear modulus G, the bulk modulus K, and Poisson's ratio ν, while finite-element decks often need the pair the solver expects rather than the pair a coupon test reported.
What the four isotropic constants mean and why any two suffice
In linear isotropic elasticity a homogeneous solid has no preferred direction, so its stiffness tensor collapses to two independent scalars. Four popular names for those scalars appear in handbooks, finite-element cards, and laboratory reports: Young's modulus E, also called the elastic modulus; the shear modulus G; the bulk modulus K; and Poisson's ratio ν. They are not four independent material properties.
Any two determine the other two through algebraic identities that follow from Hooke's law in three dimensions: E equals two G times one plus ν, E equals three K times one minus two ν, and E equals nine K G divided by three K plus G. A tensile coupon often reports E and ν; an ultrasonic pulse often reports G and K; a compressibility study reports K; a torsion bar reports G.
Search queries such as elastic constants calculator, convert Young's modulus to shear modulus, bulk modulus from Poisson's ratio, and isotropic elastic constants converter all describe the same intent: given two numbers, recover the remaining pair without re-deriving continuum mechanics. This capability is that converter. It does not compute E from stress divided by strain, and it does not compute ν from lateral and axial strains; those are different search intents served by dedicated stress–strain tools.
Here the four symbols are treated as an isotropic set, and the job is to complete the set from whichever pair you already know.
Input contract, the six pairings, and the Poisson's ratio bound
Each modulus must be a strictly positive finite number in a consistent stress unit—pascals, gigapascals, or pounds per square inch are all acceptable as long as E, G, and K share that unit. Poisson's ratio is dimensionless. Numeric strings are accepted.
After the missing pair is computed, Poisson's ratio is checked against the thermodynamic window for a stable isotropic solid: it must lie strictly between minus one and one half. Values outside that range, including the incompressible limit ν equals 0.5 where the bulk modulus diverges and the auxetic bound ν equals minus one where the shear modulus diverges, are rejected rather than returning Infinity. Inconsistent pairs that imply a Poisson ratio outside the window—for example E greater than or equal to three G—are rejected with the same message.
Success returns all four constants, a known string naming the two inputs, and a formula string for the identities used. Results are rounded to twelve significant digits so goldens stay stable across JavaScript engines. Failed validation is not billed.
What you can do with it
Handbook E and ν into finite-element G and K
Take tensile Young's modulus and Poisson's ratio from a datasheet and obtain the shear and bulk moduli an isotropic elastic card expects.
Ultrasonic G and K checked against tensile E and ν
Convert laboratory shear and bulk moduli into E and Poisson's ratio and compare them with a coupon test on the same isotropic solid.
Teaching the isotropic identities with a bound on ν
Show students that any two of E, G, K, and ν determine the rest, and that Poisson's ratio outside minus one to one half is not a stable isotropic solid.
FAQ
What happens if Poisson's ratio is 0.5 or below −1?
For a stable isotropic solid Poisson's ratio must lie strictly between −1 and 0.5; at 0.5 the bulk modulus diverges (incompressible limit) and at −1 the shear modulus diverges.
What units should Young's, shear, and bulk moduli use?
Any consistent stress unit (Pa, GPa, psi, and so on). E, G, and K must share that unit; Poisson's ratio is dimensionless. The response uses the same unit you sent.
How are G and K computed from E and Poisson's ratio?
G = E / (2(1+ν)) and K = E / (3(1−2ν)). The other five pairings use the equivalent isotropic identities among E, G, K, and ν.
How do I get Young's modulus and Poisson's ratio from ultrasonic measurements?
An ultrasonic pulse test usually gives the shear and bulk moduli. Enter those two in the same stress unit and leave the others empty; the calculator returns Young's modulus and Poisson's ratio from the isotropic identities.
Why is my pair of values rejected as inconsistent?
Some pairs imply a Poisson's ratio outside the stable range between minus one and one half. A Young's modulus of three times the shear modulus or more, for example, cannot belong to a stable isotropic solid.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/elastic-constants \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"elastic_modulus":200,"E":1}'const res = await fetch("https://api.kit.forhosting.com/phys/elastic-constants", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"elastic_modulus": 200,
"E": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/elastic-constants",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"elastic_modulus": 200,
"E": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/elastic-constants", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"elastic_modulus":200,"E":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"elastic_modulus":200,"E":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/elastic-constants", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"elastic_modulus": 200,
"E": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.elastic_constants",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This elastic constants calculator accepts exactly two of elastic_modulus, shear_modulus, bulk_modulus, and poisson_ratio, applies the closed isotropic identities, and returns the remaining two together with the known pair and the formula used. Poisson's ratio must lie strictly between minus one and one half; values outside that thermodynamic window make the remaining moduli non-finite and fail with invalid_input.
Input contract, the six pairings, and the Poisson's ratio bound
Send a JSON object that contains exactly two of the four fields elastic_modulus, shear_modulus, bulk_modulus, and poisson_ratio. Aliases E, youngs_modulus, G, mu, K, kappa, nu, and poissons_ratio map onto the canonical names. Zero, one, three, or four provided constants fail with invalid_input because the isotropic problem is then under-specified or over-specified.
Where engineers use the conversion, limits, and what it costs
Typical flows include mapping handbook steel values such as E equals two hundred gigapascals and ν equals 0.3 into shear and bulk moduli, recovering E and ν from ultrasonic G and K, and rejecting rubber-like ν equals 0.5 so an incompressible material is not silently given a fake finite bulk modulus. Use the API when a materials database, a finite-element preprocessor, or a laboratory notebook must convert many pairs under a single contract. Each successful call is one billable item at $0.002; invalid input does not consume balance. Do not send mixed units, anisotropic stiffness matrices, temperature-dependent tables, or viscoelastic spectra—this endpoint is the closed isotropic conversion only. Pin the rounded JSON in tests, keep Poisson's ratio inside minus one to one half, and you get reproducible elastic constants for teaching, design notes, and automation without standing up a continuum-mechanics library of your own.
Technical questions
Which two elastic constants can I enter?
Any pair among elastic_modulus (E), shear_modulus (G), bulk_modulus (K), and poisson_ratio (ν). The calculator returns the other two. Providing fewer or more than two constants fails with invalid_input.
What does the elastic constants calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.