{"id":85820,"date":"2025-06-16T14:43:53","date_gmt":"2025-06-16T12:43:53","guid":{"rendered":"https:\/\/www.linseis.com\/?p=85820"},"modified":"2025-07-31T07:26:09","modified_gmt":"2025-07-31T05:26:09","slug":"thermal-expansion","status":"publish","type":"post","link":"https:\/\/www.linseis.com\/en\/wiki\/thermal-expansion\/","title":{"rendered":"Thermal expansion"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"85820\" class=\"elementor elementor-85820\" data-elementor-post-type=\"post\">\n\t\t\t\t<div class=\"elementor-element elementor-element-20e7cce e-con-full e-flex e-con e-parent\" data-id=\"20e7cce\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div class=\"elementor-element elementor-element-330d9bc e-con-full e-flex e-con e-child\" data-id=\"330d9bc\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div class=\"elementor-element elementor-element-2fab8da e-con-full e-flex e-con e-child\" data-id=\"2fab8da\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div class=\"elementor-element elementor-element-756ba2d e-con-full e-flex e-con e-child\" data-id=\"756ba2d\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div class=\"elementor-element elementor-element-54ab761 e-con-full e-flex e-con e-child\" data-id=\"54ab761\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6204ea8 elementor-toc--minimized-on-tablet elementor-invisible elementor-widget elementor-widget-table-of-contents\" data-id=\"6204ea8\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;headings_by_tags&quot;:[&quot;h2&quot;],&quot;_animation&quot;:&quot;fadeInUp&quot;,&quot;no_headings_message&quot;:&quot;No headings were found on this page.&quot;,&quot;marker_view&quot;:&quot;numbers&quot;,&quot;minimize_box&quot;:&quot;yes&quot;,&quot;minimized_on&quot;:&quot;tablet&quot;,&quot;hierarchical_view&quot;:&quot;yes&quot;,&quot;min_height&quot;:{&quot;unit&quot;:&quot;px&quot;,&quot;size&quot;:&quot;&quot;,&quot;sizes&quot;:[]},&quot;min_height_tablet&quot;:{&quot;unit&quot;:&quot;px&quot;,&quot;size&quot;:&quot;&quot;,&quot;sizes&quot;:[]},&quot;min_height_mobile&quot;:{&quot;unit&quot;:&quot;px&quot;,&quot;size&quot;:&quot;&quot;,&quot;sizes&quot;:[]}}\" data-widget_type=\"table-of-contents.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-toc__header\">\n\t\t\t\t\t\t<h2 class=\"elementor-toc__header-title\">\n\t\t\t\tTable of Contents:\t\t\t<\/h2>\n\t\t\t\t\t\t\t\t\t\t<div class=\"elementor-toc__toggle-button elementor-toc__toggle-button--expand\" role=\"button\" tabindex=\"0\" aria-controls=\"elementor-toc__6204ea8\" aria-expanded=\"true\" aria-label=\"Open table of contents\"><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-down\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M207.029 381.476L12.686 187.132c-9.373-9.373-9.373-24.569 0-33.941l22.667-22.667c9.357-9.357 24.522-9.375 33.901-.04L224 284.505l154.745-154.021c9.379-9.335 24.544-9.317 33.901.04l22.667 22.667c9.373 9.373 9.373 24.569 0 33.941L240.971 381.476c-9.373 9.372-24.569 9.372-33.942 0z\"><\/path><\/svg><\/div>\n\t\t\t\t<div class=\"elementor-toc__toggle-button elementor-toc__toggle-button--collapse\" role=\"button\" tabindex=\"0\" aria-controls=\"elementor-toc__6204ea8\" aria-expanded=\"true\" aria-label=\"Close table of contents\"><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-up\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M240.971 130.524l194.343 194.343c9.373 9.373 9.373 24.569 0 33.941l-22.667 22.667c-9.357 9.357-24.522 9.375-33.901.04L224 227.495 69.255 381.516c-9.379 9.335-24.544 9.317-33.901-.04l-22.667-22.667c-9.373-9.373-9.373-24.569 0-33.941L207.03 130.525c9.372-9.373 24.568-9.373 33.941-.001z\"><\/path><\/svg><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<div id=\"elementor-toc__6204ea8\" class=\"elementor-toc__body\">\n\t\t\t<div class=\"elementor-toc__spinner-container\">\n\t\t\t\t<svg class=\"elementor-toc__spinner eicon-animation-spin e-font-icon-svg e-eicon-loading\" aria-hidden=\"true\" viewBox=\"0 0 1000 1000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M500 975V858C696 858 858 696 858 500S696 142 500 142 142 304 142 500H25C25 237 238 25 500 25S975 237 975 500 763 975 500 975Z\"><\/path><\/svg>\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b10c2f5 elementor-widget elementor-widget-spacer\" data-id=\"b10c2f5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8a71db9 elementor-widget elementor-widget-spacer\" data-id=\"8a71db9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5ef1fd4 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"5ef1fd4\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Thermal expansion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2a88229 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"2a88229\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Introduction to thermal expansion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-77833f4 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"77833f4\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Thermal expansion &#8211; also known as thermal expansion &#8211; is a physical phenomenon that is equally relevant in everyday life and technology. It describes the change in the geometric dimensions of a body when its temperature changes. Understanding and precisely measuring this property is particularly important in construction, mechanical engineering and materials research.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-50e1380 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"50e1380\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Fundamentals of thermal expansion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cffc1d6 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"cffc1d6\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">What is thermal expansion?<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d182312 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"d182312\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>An increase in temperature increases the molecular movement at the particle level of a material and consequently its space requirement also increases. The resulting effect is then an expansion of the material. On cooling, the space requirement decreases again and thermal contraction occurs accordingly.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0f43171 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"0f43171\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Causes and physical principle<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a261340 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"a261340\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul><li>The cause is the increase in the kinetic energy of the particles when heated.<\/li><li>The distance between the particles increases as a result.<\/li><li>Each material has a characteristic coefficient of expansion, which indicates how much it expands when the temperature changes, depending on the particles it contains and their degrees of freedom.<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-affb892 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"affb892\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Forms of thermal expansion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-397871d elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"397871d\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Thermal expansion can manifest itself in different dimensions depending on the shape of the body and the composition of the material: in length, area or volume.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-81c6ff9 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"81c6ff9\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Linear expansion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-521c097 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"521c097\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>With linear expansion, the length of a body changes along an axis, for example in the case of a metal rod, which visibly lengthens when heated. This form of thermal expansion is easy to measure and is the most common application.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-499af99 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"499af99\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Surface expansion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7083afa elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"7083afa\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Here, the surface area of a body increases &#8211; typically in the case of sheets or foils that expand evenly in two spatial directions.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8709bba elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"8709bba\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Volume expansion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1303a2e elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"1303a2e\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Volume expansion refers to the spatial expansion in all three directions. It mainly occurs in liquids and gases, but can also be observed in solids and calculated in part from the linear expansion of an axis.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5fe1e3d elementor-widget__width-initial elementor-invisible elementor-widget elementor-widget-image\" data-id=\"5fe1e3d\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"800\" height=\"800\" src=\"https:\/\/www.linseis.com\/wp-content\/uploads\/2025\/06\/Thermal-Expansion-1.png\" class=\"attachment-large size-large wp-image-85835\" alt=\"\" srcset=\"https:\/\/www.linseis.com\/wp-content\/uploads\/2025\/06\/Thermal-Expansion-1.png 1024w, https:\/\/www.linseis.com\/wp-content\/uploads\/2025\/06\/Thermal-Expansion-1-300x300.png 300w, https:\/\/www.linseis.com\/wp-content\/uploads\/2025\/06\/Thermal-Expansion-1-150x150.png 150w, https:\/\/www.linseis.com\/wp-content\/uploads\/2025\/06\/Thermal-Expansion-1-768x768.png 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\"><\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3120c46 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"3120c46\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Mathematical description<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aae0b1f elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"aae0b1f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Linear expansion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e48ceb0 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"e48ceb0\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\\(<br \/>\\Delta L = \\alpha \\cdot L_0 \\cdot \\Delta T<br \/>\\quad \\text{where} \\quad<br \/>\\begin{cases}<br \/>\\Delta L : \\text{Change in length (m)} \\\\<br \/>\\alpha : \\text{Linear expansion coefficient (1\/K)} \\\\<br \/>L_0 : \\text{Initial length (m)} \\\\<br \/>\\Delta T : \\text{Temperature change (K)}<br \/>\\end{cases}<br \/>\\)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fec8579 elementor-widget elementor-widget-spacer\" data-id=\"fec8579\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-117eb24 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"117eb24\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Volume expansion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1c7748e elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"1c7748e\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\\(<br \/>\\Delta V = \\gamma \\cdot V_0 \\cdot \\Delta T<br \/>\\quad \\text{where} \\quad<br \/>\\begin{cases}<br \/>\\Delta V : \\text{Change in volume (m\u00b3)} \\\\<br \/>\\gamma : \\text{Volumetric expansion coefficient (1\/K)} \\\\<br \/>V_0 : \\text{Initial volume (m\u00b3)} \\\\<br \/>\\Delta T : \\text{Temperature change (K)}<br \/>\\end{cases}<br \/>\\)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-10ceced elementor-widget elementor-widget-spacer\" data-id=\"10ceced\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eee1ba8 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"eee1ba8\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Measuring methods for determining thermal expansion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-727b42d elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"727b42d\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Dilatometer<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f231dcf elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"f231dcf\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>A dilatometer measures the change in length or volume of a sample of defined temperature and temperature change. The sample is heated or cooled and its change in length is transferred to a displacement transducer via a push rod. The inherent expansion of the measuring system is compensated for or corrected using reference material.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2cf348c elementor-widget elementor-widget-spacer\" data-id=\"2cf348c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0687a0f elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"0687a0f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><strong>Typical measured variables:<\/strong><\/p><ul><li>Linear coefficient of thermal expansion (CTE)<\/li><li>Sintering temperatures and stages<\/li><li>Glass transition temperatures (Tg)<\/li><li>Phase transitions<\/li><li>Volume and density changes<\/li><\/ul><p><strong>Areas of application:<\/strong><\/p><ul><li>Materials: Metals, ceramics, glasses, polymers, composites<\/li><li>Possible temperature range: -260 \u00b0C to +2800 \u00b0C (depending on device configuration)<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eed1324 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"eed1324\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Thermomechanical analysis (TMA)<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4864a43 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"4864a43\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Thermomechanical analysis (TMA) is a method for measuring dimensional changes under mechanical load at a defined temperature. A probe presses on the sample with a constant force while the temperature is slowly changed. A sensor, often a differential transformer or LVDT (Linear Variable Differential Transformer), measures the resulting changes in length in real time.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-29d7c21 elementor-widget elementor-widget-spacer\" data-id=\"29d7c21\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8812c3b elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"8812c3b\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><strong>Measured properties:<\/strong><\/p><ul><li>Linear expansion and expansion coefficient<\/li><li>Softening points<\/li><li>Shrinkage processes during sintering<\/li><li>Glass transition, melting point, crystallisation<\/li><li>Creep and relaxation (time-dependent deformation under load)<\/li><\/ul><p><strong>Typical applications:<\/strong><\/p><ul><li>Analysing solids, liquids and pasty materials<\/li><li>Suitable for: Plastics, elastomers, adhesives, metals, ceramics, composites<\/li><li>Complements other methods such as DSC, TGA and DMA<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d4cbe93 elementor-widget elementor-widget-spacer\" data-id=\"d4cbe93\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9d000c2 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"9d000c2\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Thanks to its high sensitivity, TMA not only allows the determination of expansion, but also the detection of weak physical transitions &#8211; such as changes in the crystal lattice, hardening or delamination. This makes it a versatile tool for research and industry.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f03bf48 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"f03bf48\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Use of dilatometers to analyse thermal expansion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c3b7d41 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"c3b7d41\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Functionality in detail<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-887d0a3 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"887d0a3\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Dilatometers are essential instruments for determining the thermal expansion of materials. The measurement is carried out either using mechanical components such as a push rod or contactless using optical sensors. The sample is placed in a heatable oven in which a defined temperature profile is run through. As the temperature changes, a high-precision displacement transducer measures the change in length of the sample material. This measurement allows the linear coefficient of thermal expansion (CTE) to be determined &#8211; a key parameter for describing the thermal behaviour of materials.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-67d18b5 elementor-widget elementor-widget-spacer\" data-id=\"67d18b5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f439301 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"f439301\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><strong>Key steps in the measurement process:<\/strong><\/p><ul><li>The sample is heated or cooled in a controlled manner.<\/li><li>The resulting change in length is recorded with high accuracy.<\/li><li>The CTE is calculated from the ratio of the change in length to the change in temperature.<\/li><\/ul><div>\u00a0<\/div><p>This method allows a resolution in the sub-micrometre range and is suitable for both routine quality controls and research applications.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-60aaa12 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"60aaa12\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Applications and typical measured variables<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-50bc3df elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"50bc3df\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Dilatometers provide a variety of measurable parameters that are of central importance for material development and industrial processes:<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7580681 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"7580681\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><strong>Typical applications:<\/strong><\/p><ul><li>Determination of glass transitions and phase transitions<\/li><li>Analysis of shrinkage behaviour in sintering processes<\/li><li>Measurement of volume and density changes<\/li><li>Use in quality assurance, material development and research<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4776162 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"4776162\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Differences between materials<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3cde593 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"3cde593\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Comparison of expansion coefficients<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eff1649 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"eff1649\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul>\n \t<li>Plastics: e.g. PE = 2,10<sup>-4<\/sup> K<sup>-1<\/sup> K<\/li>\n \t<li>Metals: aluminium = 23,8\u00b710<sup>-6<\/sup> K<sup>-1<\/sup>, Stahl = 11\u20131310<sup>-6<\/sup>  K<sup>-1<\/sup> <\/li>\n \t<li>Ceramics\/glass: Al\u2082O\u2083 = 7,5\u00b710<sup>-6<\/sup>  K<sup>-1<\/sup> <\/li>\n \t<li>Special materials: Invar = 0,55\u20131,2\u00b710<sup>-6<\/sup> K<sup>-1<\/sup> <\/li>\n<\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d11584f elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"d11584f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Significance for technical applications<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3334d97 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"3334d97\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul><li>Large differences can lead to stresses and material failure.<\/li><li>Precise material selection is crucial for durable designs.<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c6db477 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"c6db477\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Practical applications and challenges<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d04f6b1 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"d04f6b1\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Cross-industry significance<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aff03be elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"aff03be\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Thermal expansion is a key issue in numerous technical disciplines. Depending on the area of application, it can be utilised in a targeted manner or must be actively compensated for in order to avoid component damage or functional losses. Well thought-out design measures are essential, particularly in the case of permanent temperature changes, as occur in many technical environments.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5d66ef4 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"5d66ef4\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Typical areas of\napplication:<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bb1baa1 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"bb1baa1\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul><li>Construction: Expansion joints in bridges, carriageways or pipelines ensure that materials can expand or contract with temperature changes without causing cracks or structural damage.<\/li><li>Mechanical engineering: In machines and systems, components with different expansion behaviour often have to function together. If these differences are not taken into account, there is a risk of tension, leaks or premature wear.<\/li><li>Electronics: Printed circuit boards, semiconductors and housings are often made of different materials. An incorrect combination with different CTE values can lead to solder joint fractures or material cracks.<\/li><li>Aerospace: The requirements here are particularly high. Components must be able to withstand extreme cold in space as well as extreme heat when exposed to the atmosphere &#8211; without deforming or losing their function.<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8366138 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"8366138\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Examples from practice<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0f6c2cf elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"0f6c2cf\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul><li>Railway tracks are laid with gaps to prevent deformation due to thermal expansion.<\/li><li>Borosilicate glass in laboratory and cooking vessels withstands thermal shocks thanks to its low coefficient of expansion.<\/li><li>Water expands when it freezes &#8211; an exception that ensures that ice floats on water and insulates it.<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8836db2 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"8836db2\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-492d64f elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"492d64f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Thermal expansion is a central phenomenon in physics and technology. Its exact knowledge and precise measurement using dilatometers or TMA makes it possible to optimise materials for their application. This makes it possible to develop products that function reliably even under extreme conditions &#8211; whether in industry, research or everyday life.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-16379ac elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"16379ac\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Recommended sources for in depth study<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2049310 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"2049310\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInUp&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul><li><a href=\"https:\/\/www.leifiphysik.de\/waermelehre\/ausdehnung-bei-erwaermung\/grundwissen\/volumen-und-laengenaenderung-von-festkoerpern\" target=\"_blank\" rel=\"noopener\">LEIFIphysik: Volumen- und L\u00e4ngen\u00e4nderung von Festk\u00f6rpern<\/a><br \/>(Grundlagen der W\u00e4rmeausdehnung anschaulich erkl\u00e4rt)<\/li><li><a href=\"https:\/\/de.wikipedia.org\/wiki\/W%C3%A4rmeausdehnung\" target=\"_blank\" rel=\"noopener\">Wikipedia: W\u00e4rmeausdehnung<\/a><br \/>(\u00dcberblick \u00fcber Formen, Formeln und typische Werte)<\/li><li><a href=\"https:\/\/studyflix.de\/ingenieurwissenschaften\/waermeausdehnungskoeffizient-1091\" target=\"_blank\" rel=\"noopener\">Studyflix: W\u00e4rmeausdehnungskoeffizient<\/a><br \/>(Kompakte Erkl\u00e4rung mit Formeln und Beispielen)<\/li><li><a href=\"https:\/\/tu-dresden.de\/mn\/physik\/ressourcen\/dateien\/studium\/lehrveranstaltungen\/praktika\/pdf\/TA.pdf?lang=de\" target=\"_blank\" rel=\"noopener\">TU Dresden Praktikumsanleitung: Thermische Analyse<\/a><br \/>(Wissenschaftlich fundierte Messmethoden, inkl. TMA und Dilatometer)<\/li><li><a href=\"https:\/\/www.ptb.de\/cms\/fileadmin\/internet\/fachabteilungen\/abteilung_5\/5.3_koordinatenmesstechnik\/texte\/f_dilatometer.pdf\" target=\"_blank\" rel=\"noopener\">PTB \u2013 Physikalisch-Technische Bundesanstalt: Dilatometer<\/a><br \/>(Ausf\u00fchrliche technische Beschreibung von Dilatometern)<\/li><li><a href=\"https:\/\/wiki.polymerservice-merseburg.de\/index.php\/Thermomechanische_Analyse\" target=\"_blank\" rel=\"noopener\">Polymer Service Wiki: Thermomechanische Analyse<\/a><br \/>(Spezifische Anwendung und Funktionsweise von TMA)<\/li><li><a href=\"https:\/\/www.uni-due.de\/materials\/dilatometrie.shtml\" target=\"_blank\" rel=\"noopener\">Universit\u00e4t Duisburg-Essen: Dilatometrie<\/a><br \/>(Hintergrundinformationen zur thermischen Analyse fester Stoffe)<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Thermal expansion is the change in a material\u2019s dimensions due to temperature changes \u2013 a key factor in construction, engineering, and materials research.<\/p>\n","protected":false},"author":15,"featured_media":85821,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[106],"tags":[],"class_list":["post-85820","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-wiki"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/posts\/85820","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/comments?post=85820"}],"version-history":[{"count":0,"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/posts\/85820\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/media\/85821"}],"wp:attachment":[{"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/media?parent=85820"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/categories?post=85820"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.linseis.com\/en\/wp-json\/wp\/v2\/tags?post=85820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}