Lecture 2: Stress. 1. Forces Surface Forces and Body Forces

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1 Lectue : Stess Geophysicists study pheomea such as seismicity, plate tectoics, ad the slow flow of ocks ad mieals called ceep. Oe way they study these pheomea is by ivestigatig the defomatio ad flow of Eath mateials. he sciece of the defomatio ad flow of mateials is called heology. he fudametal cocepts of heology ae stess ad stai. Stess is a dymaic quatity that expesses the magitude of foces actig withi o at the suface of cotiuous media ad stai is a kiematic quatity that expesses defomatio of the media. his lectue itoduces the basic cocepts of stess. It explais how to completely descibe a object s state of stess i a D coodiate system ad how to chage the coodiate system fo coveiece. It also descibes how to fid the stess o a abitay plae withi a object ad how to detemie if the stess o that plae will cause the object to beak.. Foces Suface Foces ad Body Foces Suface foces ae diffeet fom body foces. Fo example, coside pushig a ock off the edge of a cliff. Pushig the ock ivolves a suface foce that acts o a aea boudig pat of the ock. Doppig it off the cliff ivolves gavity, a body foce that acts thoughout the volume of the ock ad depeds o its mass.. actios Motivatio If you push a balloo with you foot, it will slide acoss the goud. If you push a balloo with the same foce usig a eedle, it will pop. he diffeece esults fom the fact that a eedle ad a foot apply foce ove a diffeet suface aea of the balloo. he eedle applies foce ove a vey small aea ad poduces a lage tactio. he foot applies foce ove a much bigge aea ad poduces a smalle tactio. Defiitio A suface foce F v that acts uifomly ove plaa suface of aea A esults i a tactio v of magitude v v F = A he tactio is a vecto that has the same diectio as F v ad has uits of pessue. I SI uits, tactios ae measued i Pascals (Pa). F kg m N = = = = Pa A m s m

2 Sice tactios ae vectos, they ca be decomposed ito omal ad tagetial vecto compoets. σ ad deote the magitude of these compoets. Notes o the Oietatio of Sufaces = σ = = ta getial omal ta getial actio depeds o the aea of the suface ove which it acts. Sice aea ofte chages with the oietatio of the suface, the values of σ ad do ot behave like vecto compoets whe the suface oietatio chages. Coside the followig figue: + omal W F ' θ = 0 o A' F s ' ' σ ' σ s L/cosθ θ L L Compae how the omal ad tagetial foces chage with the oietatio of the suface with how the omal ad tagetial tactios chage: F F t = W cosθ = W si θ σ cos = W θ = W siθ cosθ Figue. Figue by MI OCW. he decompositio of tactios ito omal ad shea compoets is useful i witig Admoto s Law i tems of tactios. his ew fom of Admoto s law is the lik betwee stess ad failue acoss a plae.

3 Admoto s Law Admoto s law elates a omal foce ad a tagetial foce by a popotioality costat called the coefficiet of static fictio f s : F ta getial = f s F omal Dividig both sides of the equatio by the aea A ove which the foces act leads to Admoto s law i tems of tactios: = f s σ σ Figue. Figue by MI OCW. As show i the pictue, Admoto s law expesses a failue citeio. Whe the tagetial foce is lage tha the poduct of the omal foce ad f s, the block i the pictue will slide dow the iclie. his failue citeio will help solve the poblem of houses slidig ito the ocea i Southe Califoia. he poblem i Southe Califoia, though, is moe difficult tha the block poblem because it is uclea which plae below the house has σ ad that will cause failue. he est of this lectue ad the ext oe will addess this difficulty. It will exploe how to completely epeset the state of stess below the house at ay poit, how to fid the tactio o a abitay plae below the house, ad how to evaluate which oe of those plaes will fail. Motivatio. actios at a poit actios at a poit ae difficult to coceptualize because the aea at a poit ove which a foce acts is ifiitesimal. Despite this difficulty, the cocept of poit tactios is extemely impotat because it allows oe to fid the tactio o a abitay plae.

4 Defiitio Coside the fee body i the pictue below. Call the boudig suface S, the volume V, ad a cuttig plae though the body C. he omal to plae C is ). P is a poit o C ad is suouded by a small volume of C called C. Pat of the body exets iteal foces o Pat ad gives ise to a foce o C called f. ( ^ x P Pat Pat o C x Figue. Figue by MI OCW. x P i Pat f i o C x Figue.4 Figue by MI OCW.

5 he Cauchy stess piciple states that as the aea aoud P shiks to zeo, the followig limit holds lim C 0 f C = ( ˆ) () is called the tactio o stess vecto at poit P. he (ˆ) is a emide that this stess vecto is defied oly fo a paticula plae though P with omal vecto (ˆ). Newto s Laws ad the actio at a Poit Newto s thid law says that the foce exeted by Pat of the body o Pat is equal ad opposite to the foce exeted by Pat o Pat. What about the poit tactio (-) exeted by Pat o Pat? x () Pat - Pat (-) o x Figue.5 Figue by MI OCW. Applyig Newto s secod law to both pats of the body ad the body as a whole eveals that (-) is equal to (). Fo a complete teatmet of the poblem see pages 50-5 i Cotiuum Mechaics fo Egiees by Mase ad Mase. 4. he Complete Repesetatio of Stess at a Poit Motivatio he last sectio povided the fist step towad fidig the stess o a abitay plae i cotiuous media. he ext step is descibig the stess teso. he stess teso is a epesetatio of stess o thee mutually pepedicula plaes i a coodiatio system. It specifies the complete state of stess.

6 Defiitio esos Most physical quatities that ae impotat i cotiuum mechaics like tempeatue, foce, ad stess ca be epeseted by a teso. empeatue ca be specified by statig a sigle umeical value called a scala ad is called a zeoth-ode teso. A foce, howeve, must be specified by statig both a magitude ad diectio. It is a example of a fist-ode teso. Specifyig a stess is eve moe complicated ad equies statig a magitude ad two diectios the diectio of a foce vecto ad the diectio of the omal vecto to the plae o which the foce acts. Stesses ae epeseted by secodode tesos. he Stess eso Repesetig a foce i thee dimesios equies thee umbes, each efeeced to a coodiate axis. Repesetig the state of stess i thee dimesios equies ie umbes, each efeeced to a coodiate axis ad a plae pepedicula to the coodiate axes. o udestad what each of the ie umbes meas, it is helpful to visualize a ifiitesimally small cube i a cotiuous medium, oieted i a D coodiate system. x x x x Figue.6 Figue by MI OCW. he coodiate system has axes x,, x omal to the faces of the cube. he faces of ) ) ) the cube ae defied by uit omal vectos x, x, x that ae positive whe poitig out fom the cube.

7 If thee ae uifom foces actig o each of the faces of the cube, the tactio vectos ) ) ) ca be defied o each face. Let the tactios o faces x, x, x be called,, espectively. hese tactios ae ot ecessaily omal to the faces. x x x x Figue.7 Figue by MI OCW. actios,, ae each defied by thee compoets oe omal tactio ad two shea tactios. hese compoets ae labeled with two subscipts, as i ij. he fist subscipt i deotes the face of the cube o which the tactio acts ad the secod subscipt j deotes the diectio of the tactio.

8 x dx dx d x Figue.8 Figue by MI OCW. I vecto otatio, the tactios o the faces of the cube ae witte: =,, =,, =,, I matix otatio, the tactios ae witte: ij = his matix is geeally efeed to as the stess teso. It is the complete epesetatio of stess at a poit. Featues of the Stess eso he stess teso is a symmetic teso, meaig that ij = ji. As a esult, the etie teso may be specified with oly six umbes istead of ie. Fo a poof, see Chapte.4 i Cotiuum Mechaics fo Egiees by Mase ad Mase.

9 5. he Cauchy etahedo ad actio o Abitay Plaes Applicatio his sectio shows that the tactio vecto at a poit o a abitaily oieted plae ca be foud if,, at that poit ae kow. Agumet Apply Newto s secod law to a fee body i the shape of a tetahedo ad let the height of the tetahedo shik to zeo. Figues ad Defiitios Coside the tetahedo below. he poit O is the oigi ad the apices ae labeled A, B, ad C. x C o h N B A S x Figue.9 Figue by MI OCW. he elevat quatities ae defied as follows: ρ = desity F v = body foce pe uit mass i the i diectio i

10 a i = acceleatio i the i diectio h = height of the tetahedo, measued to ABC δs = aea of the oblique suface ABC = the compoet of the tactio vecto o the oblique suface i the i diectio i he mass of the tetahedo is Foce Balace ρhδs. he aea of a face pepedicula to xi is i δ S. Coside the foce balace i the i= diectio. Ovebas deote values aveaged ove a suface o volume. v F = ma F ρhδs + δs ( δs ) ( δs ) ( δs ) = ρhδs a Divide both sides by δs. F ρh + ( ) ( ) ( ) = ρh a Allow h to appoach zeo i such a way that the sufaces ad volume of the tetahedo appoach zeo while the sufaces peseve thei oietatio. he body foce ad the mass both appoach zeo. = Pefomig the same foce balace i the othe two coodiate diectios leads to expessios fo the thee tactio compoets o a abitay plae. = = = he set of these thee equatios is called Cauchy s fomula.

11 Diffeet Notatios. A geeal equatio fo the explicit expessios above is give by: = i j=. Summatio otatio is a way of witig summatios without the summatio sig Σ. o use it, simply dop the Σ ad sum ove epeated idices. he equatio i summatio otatio is give by: ji j i = ji j. he equatio i matix fom is give by =