[{"data":1,"prerenderedAt":41},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fcontact-stresses\u002Fcontact-of-spheres-and-spherical-surfaces":3},{"topic":4,"trail":26,"children":36,"tasks":37,"alternates":38},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},129,"Contact of Spheres and Spherical Surfaces","en","strength-of-materials\u002Fcontact-stresses\u002Fcontact-of-spheres-and-spherical-surfaces","Elastic contact of spheres and spherical surfaces: contact radius, maximum Hertz pressure, reduced properties, examples, and applicability limits.","This topic develops Hertz contact for spheres and spherical surfaces. It covers reduced elastic modulus and curvature, circular contact-patch radius, maximum pressure, load scaling, a sphere-on-plane example, elliptical-contact limitations, and engineering applications.","\u003Cp>\u003Cstrong>Contact of spherical surfaces\u003C\u002Fstrong> is the classical example of initially point contact. Under a normal force $F$, elastic deformation creates a finite contact patch. For two spheres with an axisymmetric local geometry, the patch is circular.\u003C\u002Fp>\u003Ch2>Reduced properties\u003C\u002Fh2>\u003Cdiv class=\"formula-chunk\">\u003Cp>The reduced elastic modulus of two isotropic contacting bodies is:\u003C\u002Fp>\u003Cp>$$\\frac{1}{E^*}=\\frac{1-\\nu_1^2}{E_1}+\\frac{1-\\nu_2^2}{E_2}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$E^*$\u003C\u002Fstrong> — reduced elastic modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$E_1$, $E_2$\u003C\u002Fstrong> — Young's moduli of the contacting bodies;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\nu_1$, $\\nu_2$\u003C\u002Fstrong> — Poisson's ratios of the contacting bodies.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>Equivalently:\u003C\u002Fp>\u003Cp>$$E^*=\\left(\\frac{1-\\nu_1^2}{E_1}+\\frac{1-\\nu_2^2}{E_2}\\right)^{-1}.$$\u003C\u002Fp>\u003C\u002Fdiv>\u003Cdiv class=\"formula-chunk\">\u003Cp>For two convex spherical surfaces along a corresponding principal direction, the reduced radius can be written as:\u003C\u002Fp>\u003Cp>$$\\frac{1}{R^*}=\\frac{1}{R_1}+\\frac{1}{R_2}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$R^*$\u003C\u002Fstrong> — reduced radius of curvature;\u003C\u002Fli>\u003Cli>\u003Cstrong>$R_1$, $R_2$\u003C\u002Fstrong> — radii of curvature of the contacting surfaces.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>Equivalently, for two convex surfaces:\u003C\u002Fp>\u003Cp>$$R^*=\\frac{R_1R_2}{R_1+R_2}.$$\u003C\u002Fp>\u003Cp>For other combinations of curvature, the sign of an individual radius follows the adopted geometric convention. General three-dimensional contact requires the principal curvatures of both surfaces.\u003C\u002Fp>\u003Cp>The calculator below corresponds to two convex surfaces with positive radii.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>\u003Cp>For a sphere on a plane, one radius is infinite, so in the simple convex case $R^*=R$ of the sphere.\u003C\u002Fp>\u003Ch2>Contact radius and maximum pressure\u003C\u002Fh2>\u003Cdiv class=\"formula-chunk\">\u003Cp>For axisymmetric Hertz contact of two spherical surfaces, a normal force $F$ produces a circular contact area of radius $a$:\u003C\u002Fp>\u003Cp>$$a=\\left(\\frac{3FR^*}{4E^*}\\right)^{1\u002F3}.$$\u003C\u002Fp>\u003Cp>The maximum contact pressure is:\u003C\u002Fp>\u003Cp>$$p_0=\\frac{3F}{2\\pi a^2}.$$\u003C\u002Fp>\u003Cp>The pressure distribution is:\u003C\u002Fp>\u003Cp>$$p(r)=p_0\\sqrt{1-\\frac{r^2}{a^2}},\\qquad 0\\le r\\le a.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$F$\u003C\u002Fstrong> — normal force;\u003C\u002Fli>\u003Cli>\u003Cstrong>$a$\u003C\u002Fstrong> — contact radius;\u003C\u002Fli>\u003Cli>\u003Cstrong>$R^*$\u003C\u002Fstrong> — reduced radius of curvature;\u003C\u002Fli>\u003Cli>\u003Cstrong>$E^*$\u003C\u002Fstrong> — reduced elastic modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$p_0$\u003C\u002Fstrong> — maximum contact pressure;\u003C\u002Fli>\u003Cli>\u003Cstrong>$p(r)$\u003C\u002Fstrong> — contact pressure at radial coordinate $r$;\u003C\u002Fli>\u003Cli>\u003Cstrong>$r$\u003C\u002Fstrong> — radial coordinate within the circular contact patch.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>The formulas apply within the assumptions of classical elastic Hertz theory.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>\u003Cp>Increasing $F$ expands the patch as $F^{1\u002F3}$. The maximum pressure $p_0$ also increases as $F^{1\u002F3}$ because contact area does not grow rapidly enough to offset the increasing force completely.\u003C\u002Fp>\u003Ch2>Example: steel sphere on steel plane\u003C\u002Fh2>\u003Cp>Let $F=1000\\ \\text{N}$, $R=10\\ \\text{mm}$, $E_1=E_2=210000\\ \\text{MPa}$, and $\\nu_1=\\nu_2=0.30$. Then:\u003C\u002Fp>\u003Cp>$$E^*=\\left[2\\frac{1-0.3^2}{210000}\\right]^{-1}\\approx115385\\ \\text{MPa}.$$\u003C\u002Fp>\u003Cp>For the plane, $R^*=10\\ \\text{mm}$. The contact radius is:\u003C\u002Fp>\u003Cp>$$a=\\left(\\frac{3\\cdot1000\\cdot10}{4\\cdot115385}\\right)^{1\u002F3}\\approx0.402\\ \\text{mm}.$$\u003C\u002Fp>\u003Cp>The maximum pressure is:\u003C\u002Fp>\u003Cp>$$p_0=\\frac{3\\cdot1000}{2\\pi\\cdot0.402^2}\\approx2.96\\cdot10^3\\ \\text{MPa}.$$\u003C\u002Fp>\u003Cp>This high local value shows why contact strength cannot be assessed from average stress over the entire component. Before using the result in design, verify that the purely elastic contact assumptions remain acceptable.\u003C\u002Fp>\u003Ch2>Elliptical contact\u003C\u002Fh2>\u003Cp>If the principal curvatures differ in two perpendicular directions, the contact patch is generally elliptical. Its semi-axes require the full Hertz solution based on the principal curvatures; the circular-patch formula for $a$ is then not directly applicable.\u003C\u002Fp>\u003Ch2>Applications\u003C\u002Fh2>\u003Cp>Spherical contact is a basic model for ball bearings, spherical supports, ball mechanisms, and localized interaction of rounded components.\u003C\u002Fp>\u003Ch2>Procedure\u003C\u002Fh2>\u003Cdiv class=\"algorithm-chunk\">\u003Col>\u003Cli>Identify the initial contact type and local radii of curvature $R_1$ and $R_2$ of the surfaces.\u003C\u002Fli>\u003Cli>Check the assumptions of elastic Hertz theory: small contact area, small deformation, smooth surfaces, and no significant plasticity.\u003C\u002Fli>\u003Cli>Calculate the reduced elastic modulus $E^*$.\u003C\u002Fli>\u003Cli>Determine the reduced curvature or reduced radius $R^*$ for the relevant geometry.\u003C\u002Fli>\u003Cli>Use the normal force $F$, or load per unit length $F'=F\u002FL$ for line contact, to determine the dimensions of the contact area or strip.\u003C\u002Fli>\u003Cli>Calculate the maximum contact pressure $p_0$ and, where needed, the pressure distribution $p(r)$ or $p(x)$.\u003C\u002Fli>\u003Cli>Assess the subsurface stress state and the contact-strength criterion relevant to the component.\u003C\u002Fli>\u003Cli>Check cyclic loading, friction, lubrication, roughness, misalignment, and edge effects when they are significant.\u003C\u002Fli>\u003C\u002Fol>\u003C\u002Fdiv>",[13,17,20,23],{"id":14,"code":15,"type":16,"locale":7},124,"hertz-reduced-elastic-modulus","formula",{"id":18,"code":19,"type":16,"locale":7},125,"hertz-reduced-radius",{"id":21,"code":22,"type":16,"locale":7},127,"hertz-sphere-contact",{"id":5,"code":24,"type":25,"locale":7},"hertz-contact-calculation-algorithm","algorithm",[27,31,35],{"id":28,"name":29,"path":30},45,"Strength of Materials","strength-of-materials",{"id":32,"name":33,"path":34},126,"Contact Stresses","strength-of-materials\u002Fcontact-stresses",{"id":5,"name":6,"path":8},[],[],{"en":39,"uk":40},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fcontact-stresses\u002Fcontact-of-spheres-and-spherical-surfaces","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fkontaktni-napruzhennia\u002Fkontakt-kul-ta-sferychnykh-poverkhon",1787712541123]