[{"data":1,"prerenderedAt":33},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fcontact-stresses\u002Fsubsurface-contact-stress-state":3},{"topic":4,"trail":18,"children":28,"tasks":29,"alternates":30},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},131,"Stress State Beneath the Contact Surface","en","strength-of-materials\u002Fcontact-stresses\u002Fsubsurface-contact-stress-state","Stress State Beneath a Contact Surface","Subsurface contact stresses: normal, principal, and shear components, depth variation, friction effects, and identifying critical regions for strength.","This topic explains the three-dimensional stress state beneath a contact surface. It covers the localization and depth variation of normal, principal, shear, and equivalent stresses, the role of friction, and why subsurface stresses are important for rolling-contact fatigue and contact-strength assessment.","\u003Cp>\u003Cstrong>Beneath a contact surface\u003C\u002Fstrong>, a complex three-dimensional stress state develops. Surface contact pressure $p(x,y)$ is only a boundary condition; several normal and shear stress components arise inside the material.\u003C\u002Fp>\u003Ch2>Localization of the field\u003C\u002Fh2>\u003Cp>The highest contact-related stresses are concentrated in a volume whose characteristic dimensions are of the same order as the contact-patch radius $a$ or contact-strip half-width $b$. Stresses decrease rapidly with distance from the contact.\u003C\u002Fp>\u003Ch2>Normal stresses\u003C\u002Fh2>\u003Cp>Within the contact area, compressive normal pressure $p(x,y)$ acts on the surface. Beneath it, three-dimensional elastic interaction also creates normal stresses in other directions, so the state cannot be represented by a single value $-p$.\u003C\u002Fp>\u003Ch2>Shear and principal stresses\u003C\u002Fh2>\u003Cp>Even in frictionless normal contact, differences between principal normal stresses produce nonzero maximum shear stress:\u003C\u002Fp>\u003Cp>$$\\tau_{max}=\\frac{\\sigma_1-\\sigma_3}{2},$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$\\tau_{max}$\u003C\u002Fstrong> — maximum shear stress at the considered point;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\sigma_1$\u003C\u002Fstrong> — algebraically largest principal stress;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\sigma_3$\u003C\u002Fstrong> — algebraically smallest principal stress.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>In classical Hertz problems, $\\tau_{max}$ often occurs at a finite depth rather than directly at the surface. Its exact magnitude and position depend on contact type and Poisson's ratio $\\nu$, so a universal numerical value should not be used without specifying the problem.\u003C\u002Fp>\u003Ch2>Why subsurface stress matters\u003C\u002Fh2>\u003Cp>During repeated rolling, the subsurface region experiences a changing multiaxial stress state many times. This can promote fatigue-crack initiation below the surface and subsequent pitting or spalling.\u003C\u002Fp>\u003Ch2>Effect of friction\u003C\u002Fh2>\u003Cp>With tangential force or sliding, surface shear tractions are added to the normal Hertz problem. They change the principal-stress field and may move the critical region closer to the surface. A frictionless model should therefore not be applied automatically to contacts with substantial traction or sliding.\u003C\u002Fp>\u003Ch2>Assessment criteria\u003C\u002Fh2>\u003Cp>Depending on material and damage mechanism, engineers may examine maximum contact pressure $p_0$, principal stresses, $\\tau_{max}$, equivalent stress, or specialized contact-fatigue criteria. For ductile isotropic materials, local equivalent stress may be assessed with Tresca or von Mises criteria, but rolling-contact life requires a separate fatigue model.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>For ductile isotropic materials under a multiaxial stress state, the Tresca and von Mises criteria are widely used.\u003C\u002Fp>\u003Cp>\u003Cstrong>Tresca:\u003C\u002Fstrong>\u003C\u002Fp>\u003Cp>$$\\sigma_{\\mathrm{eq,T}}=\\max\\left(|\\sigma_1-\\sigma_2|,|\\sigma_2-\\sigma_3|,|\\sigma_3-\\sigma_1|\\right).$$\u003C\u002Fp>\u003Cp>\u003Cstrong>von Mises:\u003C\u002Fstrong>\u003C\u002Fp>\u003Cp>$$\\sigma_{\\mathrm{eq,VM}}=\\sqrt{\\frac{(\\sigma_1-\\sigma_2)^2+(\\sigma_2-\\sigma_3)^2+(\\sigma_3-\\sigma_1)^2}{2}}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$\\sigma_1$, $\\sigma_2$, $\\sigma_3$\u003C\u002Fstrong> — principal stresses;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\sigma_{\\mathrm{eq,T}}$\u003C\u002Fstrong> — Tresca equivalent stress;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\sigma_{\\mathrm{eq,VM}}$\u003C\u002Fstrong> — von Mises equivalent stress.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>In allowable-stress design, the corresponding equivalent stress is compared with an allowable value consistent with the material properties and the adopted design method.\u003C\u002Fp>\u003C\u002Fdiv>\u003Ch2>Practical procedure\u003C\u002Fh2>\u003Col>\u003Cli>Determine the contact-pressure distribution $p(x,y)$.\u003C\u002Fli>\u003Cli>Use the contact solution to obtain stress components in the subsurface region.\u003C\u002Fli>\u003Cli>Calculate principal, shear, or equivalent stresses.\u003C\u002Fli>\u003Cli>Locate the critical point.\u003C\u002Fli>\u003Cli>Relate the result to the expected damage mechanism and the appropriate strength or life criterion.\u003C\u002Fli>\u003C\u002Fol>",[14],{"id":15,"code":16,"type":17,"locale":7},78,"tresca-von-mises-criteria","formula",[19,23,27],{"id":20,"name":21,"path":22},45,"Strength of Materials","strength-of-materials",{"id":24,"name":25,"path":26},126,"Contact Stresses","strength-of-materials\u002Fcontact-stresses",{"id":5,"name":6,"path":8},[],[],{"en":31,"uk":32},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fcontact-stresses\u002Fsubsurface-contact-stress-state","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fkontaktni-napruzhennia\u002Fnapruzhenyi-stan-pid-kontaktnoiu-poverkhneiu",1787712541139]