[{"data":1,"prerenderedAt":24},["ShallowReactive",2],{"formula-en-hertz-sphere-contact":3},{"formula":4,"related_topics":12,"alternates":21},{"id":5,"code":6,"slug":6,"title":7,"formula":8,"locale":9,"seo_title":10,"seo_description":10,"content_html":11},127,"hertz-sphere-contact","Hertz Contact of Spherical Surfaces","a = (3FR* \u002F 4E*)^(1\u002F3)","en",null,"\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>",[13,17],{"id":14,"name":15,"path":16},128,"Hertz Theory: Main Assumptions and Formulas","strength-of-materials\u002Fcontact-stresses\u002Fhertz-contact-theory",{"id":18,"name":19,"path":20},129,"Contact of Spheres and Spherical Surfaces","strength-of-materials\u002Fcontact-stresses\u002Fcontact-of-spheres-and-spherical-surfaces",{"en":22,"uk":23},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Fformulas\u002Fhertz-sphere-contact","https:\u002F\u002Fmechclassroom.com\u002Fformulas\u002Fhertz-sphere-contact",1787712544191]