{"id":848,"date":"2026-10-04T11:26:17","date_gmt":"2026-10-04T10:26:17","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-modulu-younga\/"},"modified":"2026-10-04T11:26:17","modified_gmt":"2026-10-04T10:26:17","slug":"kalkulator-modulu-younga","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-modulu-younga\/","title":{"rendered":"Kalkulator modu\u0142u Younga"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Oblicz modu\u0142 Younga na podstawie napr\u0119\u017cenia i odkszta\u0142cenia<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"848\" data-favorite-title=\"Modu\u0142 Younga\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-modulu-younga\/\" data-favorite-excerpt=\"Oblicz modu\u0142 spr\u0119\u017cysto\u015bci (modu\u0142 Younga) na podstawie napr\u0119\u017cenia i odkszta\u0142cenia albo z si\u0142y, pola przekroju i zmiany d\u0142ugo\u015bci. Wyniki w GPa, MPa, Pa i psi.\" aria-label=\"Dodaj do ulubionych\" data-favorite-icon=\"icon icon-physic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-modulu-younga\/\" method=\"POST\" class=\"calculator\" id=\"calculator-848\" data-post=\"848\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Metoda oblicze\u0144<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"metoda_oblicze\"><option value=\"ss\" selected>Z napr\u0119\u017cenia i odkszta\u0142cenia<\/option><option value=\"fal\">Z si\u0142y, pola i d\u0142ugo\u015bci<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field is-single has-term\" id=\"input-2\"><label for=\"field-2\">Napr\u0119\u017cenie (\u03c3)<\/label><div class=\"term\">\u03c3<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"naprenie_\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 200\" min=\"0\" value=\"200\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"Pa\">Pa<\/option><option value=\"kPa\">kPa<\/option><option value=\"MPa\" selected>MPa<\/option><option value=\"GPa\">GPa<\/option><option value=\"psi\">psi<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-2\">Si\u0142a dzia\u0142aj\u0105ca na jednostk\u0119 powierzchni (\u03c3 = F\/A). Pozostaw puste, je\u015bli chcesz u\u017cy\u0107 poni\u017cszej metody opartej na sile.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-3\"><label for=\"field-4\">Odkszta\u0142cenie (\u03b5)<\/label><div class=\"term\">\u03b5<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"odksztacenie_\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.001\" min=\"0\" value=\"0.001\"\/><small id=\"calc-info-4\">Odkszta\u0142cenie wzgl\u0119dne \u03b5 = \u0394L\/L\u2080 (bezjednostkowe). 0,001 = wyd\u0142u\u017cenie o 0,1%.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-4\" hidden><label for=\"field-5\">Si\u0142a (F)<\/label><div class=\"term\">F<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"sia_f\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 2000\" min=\"0\" value=\"2000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"N\" selected>N<\/option><option value=\"kN\">kN<\/option><option value=\"lbf\">lbf<\/option><option value=\"kgf\">kgf<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-5\">Si\u0142a rozci\u0105gaj\u0105ca lub \u015bciskaj\u0105ca dzia\u0142aj\u0105ca na pr\u00f3bk\u0119.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-5\" hidden><label for=\"field-7\">Pole przekroju poprzecznego (A)<\/label><div class=\"term\">A<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"pole_przekroju_poprzecznego_a\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 100\" min=\"0\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"mm2\" selected>mm\u00b2<\/option><option value=\"cm2\">cm\u00b2<\/option><option value=\"m2\">m\u00b2<\/option><option value=\"in2\">in\u00b2<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-7\">Pierwotny przekr\u00f3j pr\u00f3bki, prostopad\u0142y do kierunku si\u0142y. Pr\u0119t o wymiarach 10 mm \u00d7 10 mm ma A = 100 mm\u00b2.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-6\" hidden><label for=\"field-9\">D\u0142ugo\u015b\u0107 pocz\u0105tkowa (L\u2080)<\/label><div class=\"term\">L\u2080<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"dugo_pocztkowa_l\" id=\"field-9\" aria-describedby=\"calc-info-9\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 1000\" min=\"0\" value=\"1000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-9\">D\u0142ugo\u015b\u0107 pr\u00f3bki przed przy\u0142o\u017ceniem si\u0142y (baza pomiarowa).<\/small><\/div><div class=\"field is-single has-term\" id=\"input-7\" hidden><label for=\"field-11\">Zmiana d\u0142ugo\u015bci (\u0394L)<\/label><div class=\"term\">\u0394L<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"zmiana_dugoci_l\" id=\"field-11\" aria-describedby=\"calc-info-11\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.1\" min=\"0\" value=\"0.1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-12\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-11\">Wyd\u0142u\u017cenie (dodatnie) lub skr\u00f3cenie pr\u00f3bki pod wp\u0142ywem si\u0142y. Nie mo\u017ce by\u0107 zerowa.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"848\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Wyniki modu\u0142u Younga<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Modu\u0142 spr\u0119\u017cysto\u015bci Younga<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Wz\u00f3r, podsumowanie napr\u0119\u017cenia i odkszta\u0142cenia oraz uwagi<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Kopiuj link do tego obliczenia<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Zostaw komentarz<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"848\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Lubi\u0119 to <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Pomocne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Niejasne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Rozczarowanie <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Niedok\u0142adne <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Wz\u00f3r na modu\u0142 Younga<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ E = \\frac{\\sigma}{\\varepsilon} = \\frac{F \/ A}{\\Delta L \/ L_0} = \\frac{F \\, L_0}{A \\, \\Delta L} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(E\\) \u2014 modu\u0142 Younga (modu\u0142 spr\u0119\u017cysto\u015bci pod\u0142u\u017cnej) (Pa)<\/li>\n<li>\\(\\sigma\\) \u2014 napr\u0119\u017cenie, czyli si\u0142a dzia\u0142aj\u0105ca na jednostk\u0119 powierzchni \\(\\sigma = F\/A\\) (Pa)<\/li>\n<li>\\(\\varepsilon\\) \u2014 odkszta\u0142cenie wzgl\u0119dne \\(\\varepsilon = \\Delta L \/ L_0\\) (bezwymiarowe)<\/li>\n<li>\\(F\\) \u2014 przy\u0142o\u017cona si\u0142a rozci\u0105gaj\u0105ca lub \u015bciskaj\u0105ca (N)<\/li>\n<li>\\(A\\) \u2014 pierwotne pole przekroju pr\u00f3bki (m\u00b2)<\/li>\n<li>\\(L_0\\) \u2014 pierwotna d\u0142ugo\u015b\u0107 pr\u00f3bki (m)<\/li>\n<li>\\(\\Delta L\\) \u2014 zmiana d\u0142ugo\u015bci pod obci\u0105\u017ceniem (m)<\/li>\n<\/ul>\n<p>Wz\u00f3r jest zapisem prawa Hooke'a w postaci ilorazowej: modu\u0142 Younga to sta\u0142y stosunek napr\u0119\u017cenia do odkszta\u0142cenia w zakresie spr\u0119\u017cystym, czyli nachylenie prostoliniowego odcinka wykresu napr\u0119\u017cenie\u2013odkszta\u0142cenie.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Modu\u0142 Younga \u2014 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> pr\u0119t stalowy obci\u0105\u017cono tak, \u017ce napr\u0119\u017cenie w nim wynosi 200 MPa, a jego wyd\u0142u\u017cenie stanowi 0,1% pierwotnej d\u0142ugo\u015bci.<\/p>\n<p>\\[ \\varepsilon = 0.1\\% = 0.001 \\]<\/p>\n<p>\\[ E = \\frac{\\sigma}{\\varepsilon} = \\frac{200\\,000\\,000\\ \\text{Pa}}{0.001} = 200\\,000\\,000\\,000\\ \\text{Pa} \\]<\/p>\n<p><strong>Wynik:<\/strong> \\(E = 200\\ \\text{GPa}\\) \u2014 typowa warto\u015b\u0107 dla stali konstrukcyjnej.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Otw\u00f3rz list\u0119 rozwijan\u0105 <strong>Metoda oblicze\u0144<\/strong> i wybierz opcj\u0119 <em>Z napr\u0119\u017cenia i odkszta\u0142cenia<\/em>, je\u015bli znasz ju\u017c te dwie warto\u015bci, albo <em>Z si\u0142y, pola przekroju i d\u0142ugo\u015bci<\/em>, je\u015bli chcesz, aby kalkulator wyznaczy\u0142 napr\u0119\u017cenie i odkszta\u0142cenie za Ciebie.<\/li>\n<li>W pierwszej metodzie podaj <strong>Napr\u0119\u017cenie (\u03c3)<\/strong> wraz z jednostk\u0105 (Pa, kPa, MPa, GPa lub psi) oraz <strong>Odkszta\u0142cenie (\u03b5)<\/strong> jako u\u0142amek dziesi\u0119tny \u2014 0,001 oznacza wyd\u0142u\u017cenie o 0,1%.<\/li>\n<li>W drugiej metodzie podaj <strong>Si\u0142\u0119 (F)<\/strong> (N, kN, lbf lub kgf), <strong>Pole przekroju (A)<\/strong> (mm\u00b2, cm\u00b2, m\u00b2 lub in\u00b2), <strong>Pierwotn\u0105 d\u0142ugo\u015b\u0107 (L\u2080)<\/strong> oraz <strong>Zmian\u0119 d\u0142ugo\u015bci (\u0394L)<\/strong> w sp\u00f3jnych jednostkach d\u0142ugo\u015bci.<\/li>\n<li>Odczytaj wynik: modu\u0142 Younga jest wy\u015bwietlany w GPa, MPa, Pa, psi, ksi i psf, wraz z podsumowaniem napr\u0119\u017cenia i odkszta\u0142cenia u\u017cytych w obliczeniu.<\/li>\n<li>Zapoznaj si\u0119 z not\u0105 interpretacyjn\u0105, kt\u00f3ra por\u00f3wnuje Tw\u00f3j wynik z typowymi warto\u015bciami dla popularnych materia\u0142\u00f3w, takich jak stal, aluminium, beton i guma.<\/li>\n<\/ol>\n<h2>Czym jest modu\u0142 Younga?<\/h2>\n<p>Modu\u0142 Younga (nazywany tak\u017ce modu\u0142em spr\u0119\u017cysto\u015bci) to w\u0142a\u015bciwo\u015b\u0107 materia\u0142u opisuj\u0105ca jego sztywno\u015b\u0107 \u2014 jak bardzo opiera si\u0119 on rozci\u0105ganiu lub \u015bciskaniu. Jest definiowany jako stosunek napr\u0119\u017cenia do odkszta\u0142cenia w spr\u0119\u017cystym zakresie deformacji, w kt\u00f3rym materia\u0142 powraca do pierwotnego kszta\u0142tu po zdj\u0119ciu obci\u0105\u017cenia. Im wy\u017cszy modu\u0142, tym wi\u0119ksza si\u0142a jest potrzebna do wywo\u0142ania tego samego odkszta\u0142cenia.<\/p>\n<p>Napr\u0119\u017cenie \\(\\sigma = F\/A\\) mierzy intensywno\u015b\u0107 obci\u0105\u017cenia \u2014 si\u0142\u0119 roz\u0142o\u017con\u0105 na pole przekroju \u2014 natomiast odkszta\u0142cenie \\(\\varepsilon = \\Delta L \/ L_0\\) mierzy reakcj\u0119 \u2014 wyd\u0142u\u017cenie lub skr\u00f3cenie wzgl\u0119dem pierwotnej d\u0142ugo\u015bci. Poniewa\u017c odkszta\u0142cenie jest bezwymiarowe, modu\u0142 Younga ma te same jednostki co napr\u0119\u017cenie: w uk\u0142adzie metrycznym zwykle gigapaskale (GPa), a w jednostkach imperialnych funty na cal kwadratowy (psi).<\/p>\n<p>Wynik m\u00f3wi, jakie jest nachylenie liniowej cz\u0119\u015bci wykresu napr\u0119\u017cenie\u2013odkszta\u0142cenie. Warto\u015b\u0107 200 GPa oznacza, \u017ce pr\u0119t o d\u0142ugo\u015bci 1 metra wyd\u0142u\u017ca si\u0119 jedynie o 5 mikrometr\u00f3w przy ka\u017cdym megapaskalu przy\u0142o\u017conego napr\u0119\u017cenia. Typowe warto\u015bci wahaj\u0105 si\u0119 od oko\u0142o 0,01 GPa dla mi\u0119kkiej gumy do oko\u0142o 1100 GPa dla diamentu; stal konstrukcyjna mie\u015bci si\u0119 w okolicach 200 GPa, stopy aluminium oko\u0142o 70 GPa, a beton oko\u0142o 30 GPa. Znajomo\u015b\u0107 modu\u0142u materia\u0142u jest niezb\u0119dna w in\u017cynierii konstrukcyjnej do przewidywania ugi\u0119\u0107, w in\u017cynierii materia\u0142owej do charakteryzacji nowych materia\u0142\u00f3w oraz w ka\u017cdej sytuacji, w kt\u00f3rej odkszta\u0142cenie spr\u0119\u017cyste ma znaczenie \u2014 od most\u00f3w po spr\u0119\u017cyny.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki praktyczne<\/h2>\n<ul>\n<li><strong>Tylko liniowo-spr\u0119\u017cysty zakres.<\/strong> Wz\u00f3r \\(E = \\sigma\/\\varepsilon\\) jest wa\u017cny jedynie poni\u017cej granicy proporcjonalno\u015bci, gdzie napr\u0119\u017cenie jest proporcjonalne do odkszta\u0142cenia. Powy\u017cej niej materia\u0142 p\u0142ynie lub odkszta\u0142ca si\u0119 nieliniowo i pojedynczy modu\u0142 nie opisuje ju\u017c jego zachowania.<\/li>\n<li><strong>Materia\u0142 jednorodny i izotropowy.<\/strong> Obliczenie zak\u0142ada, \u017ce sztywno\u015b\u0107 jest taka sama w ca\u0142ym materiale i w ka\u017cdym kierunku. Drewno, kompozyty i materia\u0142y wzmocnione w\u0142\u00f3knami s\u0105 anizotropowe \u2014 ich modu\u0142 wzd\u0142u\u017c w\u0142\u00f3kien mo\u017ce by\u0107 kilkukrotnie wy\u017cszy ni\u017c w poprzek nich.<\/li>\n<li><strong>Ma\u0142e odkszta\u0142cenia.<\/strong> U\u017cywaj pierwotnego pola przekroju \\(A\\) i pierwotnej d\u0142ugo\u015bci \\(L_0\\), a nie warto\u015bci zmierzonych pod obci\u0105\u017ceniem. Przy du\u017cych odkszta\u0142ceniach in\u017cynierska definicja napr\u0119\u017cenia i odkszta\u0142cenia traci dok\u0142adno\u015b\u0107.<\/li>\n<li><strong>Warto\u015bci dodatnie.<\/strong> Podaj warto\u015b\u0107 bezwzgl\u0119dn\u0105 wyd\u0142u\u017cenia lub skr\u00f3cenia; kalkulator odrzuca warto\u015bci zerowe i ujemne. W praktyce modu\u0142 przy rozci\u0105ganiu i \u015bciskaniu tego samego materia\u0142u jest zwykle zbli\u017cony, ale niekt\u00f3re materia\u0142y (np. beton) zachowuj\u0105 si\u0119 zupe\u0142nie inaczej na \u015bciskanie ni\u017c na rozci\u0105ganie.<\/li>\n<li><strong>Wp\u0142yw temperatury i szybko\u015bci obci\u0105\u017cania.<\/strong> Modu\u0142 Younga maleje ze wzrostem temperatury i mo\u017ce zale\u017ce\u0107 od szybko\u015bci przy\u0142o\u017cenia obci\u0105\u017cenia, zw\u0142aszcza w polimerach. Warto\u015bci w tablicach referencyjnych s\u0105 zwykle mierzone w temperaturze pokojowej przy powolnym obci\u0105\u017caniu.<\/li>\n<\/ul>\n<h2>Typowe warto\u015bci modu\u0142u Younga<\/h2>\n<table>\n<thead>\n<tr>\n<th>Materia\u0142<\/th>\n<th>Modu\u0142 Younga (GPa)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Guma (naturalna)<\/td>\n<td>0,01\u20130,1<\/td>\n<\/tr>\n<tr>\n<td>Nylon<\/td>\n<td>2\u20134<\/td>\n<\/tr>\n<tr>\n<td>Drewno (wzd\u0142u\u017c w\u0142\u00f3kien)<\/td>\n<td>9\u201316<\/td>\n<\/tr>\n<tr>\n<td>Beton<\/td>\n<td>30\u201340<\/td>\n<\/tr>\n<tr>\n<td>Stop aluminium<\/td>\n<td>69\u201379<\/td>\n<\/tr>\n<tr>\n<td>Mosi\u0105d\u017a<\/td>\n<td>100\u2013110<\/td>\n<\/tr>\n<tr>\n<td>Mied\u017a<\/td>\n<td>110\u2013130<\/td>\n<\/tr>\n<tr>\n<td>Stal (konstrukcyjna)<\/td>\n<td>190\u2013210<\/td>\n<\/tr>\n<tr>\n<td>Wolfram<\/td>\n<td>400\u2013411<\/td>\n<\/tr>\n<tr>\n<td>Diament<\/td>\n<td>1050\u20131200<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Je\u015bli analizujesz si\u0142y w obci\u0105\u017conej konstrukcji, przydatne mog\u0105 by\u0107 te\u017c nasz <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-naciagu-lin\/\">kalkulator naci\u0105gu lin<\/a> oraz <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-sily-tarcia\/\">kalkulator si\u0142y tarcia<\/a>. Dla energii zgromadzonej w odkszta\u0142conym ciele spr\u0119\u017cystym zobacz <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-energii-sprezystosci\/\">kalkulator energii potencjalnej spr\u0119\u017cysto\u015bci<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Oblicz modu\u0142 spr\u0119\u017cysto\u015bci (modu\u0142 Younga) na podstawie napr\u0119\u017cenia i odkszta\u0142cenia albo z si\u0142y, pola przekroju i zmiany d\u0142ugo\u015bci. Wyniki w GPa, MPa, Pa i psi.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[54],"class_list":["post-848","post","type-post","status-publish","format-standard","hentry","category-fizyka","tag-mechanika"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/848","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=848"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/848\/revisions"}],"predecessor-version":[{"id":1779,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/848\/revisions\/1779"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=848"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=848"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=848"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}