Chapte 3 Waves n an Elastc Whole Space Equaton of Moton of a Sold Hopefully, many of the topcs n ths chapte ae evew. Howeve, I fnd t useful to dscuss some of the key chaactestcs of elastc contnuous meda. These concepts ae ctcal fo undestandng both sesmc waves n the Eath and also the esponse of engneeed stuctues (e.g. buldngs). I wll assume that you aleady know what stess and stan ae and I wll begn wth the equaton of moton. I wll use Ensten s summaton conventon that any epeated ndex sgnfes summaton ove thee spatal coodnates. In the fst two chaptes we consdeed dynamcs poblems n whch tme was the only dependent vaable. Howeve, n a contnuum, the moton s a functon of both tme and space. Consde an nfntesmally small cube of elastc sold shown n Fgue 3.. Although ths cube s suounded by a contnuous sold, we can ask about the net foces on the cube. Fgue 3.. Dstbuton of tactons on the faces of an nfntesmal cube of matte. th the vecto tacton (foce pe unt aea) on the face of the cube. We nque about the net foces on the cube whee T= σ j n j T s (3.) 3-
th and whee σ s stess n catesan codnates and n s unt nomal vecto to the j face j of the cube. We begn by assumng that thee s no net toque on the cube, othewse t would stat to spn. Ths condton s satsfed f and only f the stess tenso s symmetc; that s σ = σ (3.) j We next employ Newton s nd law to deve the ectlnea acceleaton of the mass, F = P mu (3.3) The cube s assumed to have a densty of ρ and dmensons of dx, dx, dx3. The th component of net foce on the cube s j j 3 F = d T dx dx + d T dx dx + d T dx dx (3.4) 3 3 Recognzng that dt σ j j = dxj (no summaton) (3.5) x j th we can ewte Newton s law (3.3) fo the component of net foce and acceleaton as σ j dxdxdx3 = ρu dxdxdx3 (summaton on j) (3.6) x j th o usng the notaton whee, sgnfes dffeentaton wth espect to coodnate, ths be wtten σ = ρu (3.7) j, j We can obtan a slghtly moe geneal expesson by allowng thee to be some extenal body foce f that s actng on the cube (e.g. gavty) and we then obtan σ + f = ρu (3.8) j, j Equaton (3.8) s the basc equaton of moton of a sold contnuum. Although we deved t fom Newton s law, t s fundamentally dffeent n that t contans a spatal devatve of foces as well as the tme devatve of lnea momentum. As we wll see, ths fundamentally changes the natue of the foces n the poblem. In patcula, t says that acceleaton at a pont s not elated to stess at that pont (foce pe unt aea), but to the spatal devatve of stess. Stan and Consttutve Laws In ode to actually solve elastcty poblems, we must have some elatonshp between the defomaton of the body and the ntenal stesses. If we consde ou nfntesmal cube as shown n Fgue 3., then we can descbe the moton of the cube as a combnaton of a gd body otaton and ntenal stan. We call the dagonal vecto of the unstaned element R and the dagonal of the element afte stanng s R. We defne the change n the dagonal element due as 3-
δ R= R R (3.9) If the moton of the nfntesmal cube s small, then n component fom δ R = u dx (3.0), j j whch can be ewtten n the fom of δ R = ω dx + ε dx (3.) j j j j whee ω j = ( u, j uj, ) (3.) and ε j = ( u, j + uj, ) (3.3) ωj epesents gd body otaton and t s ant-symmetc. ε j s the nfntesmal stan tenso and t s symmetc. Fgue 3.. Defomaton of an nfntesmal element. The elatonshp between stess and stan s called the consttutve elaton. Fo small stans, most mateals exhbt a lnea elatonshp between stess and stan that can be geneally wtten as σ = C ε (3.4) whee thee ae 8 elastc coeffcents C jkl j jkl kl. Howeve, due the symmety of the stess and stan tenso, and due to the equement fo a unque stan enegy, thee ae at most ndependent elastc coeffcents. If the mateal s sotopc (no ntnsc dectonalty to the popetes), then thee ae only ndependent elastc coeffcents. Table 3. povdes a handy conveson between seveal dffeent elastc coeffcents fo an sotopc sold. st nd Fo ou dscusson we wll use the and Lame constants λ and µ. In ths case (3.4) smplfes to σ = λε δ + µε (3.5) whee j kk j j 3-3
0 j δj = Konecke delta = j (3.6) Table 3.. Relatonshp between elastc constants fo an sotopc elastc medum 3-4
Nave s Equaton We ae now n a poston to wte the equaton of moton entely n tems of dsplacement of the medum. Combnng equatons (3.8), (3.3), and (3.5), we obtan ρ u = f + µ u + λ + µ u (3.7),j ( ), jj j Ths s Nave s equaton and t s such an mpotant equaton that t s woth wtng t out to see the tems moe explctly. 3 ρ u u u j f ( ) t µ j x λ µ = + + + (3.8) = j xj x In Nave s equaton nd devatves of dsplacements wth espect to tme ae lnealy elated to nd devatves of dsplacement wth espect to space. Eveythng that happens n an sotopc lnealy-elastc sold s a soluton to ths equaton. We can also wte Nave s equaton n vecto fom as µ u+ ( λ + µ ) u+ f = ρ u (3.9) Ths vecto fom of the equaton has the advantage that we can ewte t n any type of coodnate fame fo whch we know the Laplacan opeato, the gadent opeato, and the acceleaton vecto. In patcula, we can wte these opeatos fo Catesan coodnates u = u e (3.0) u =u, (3.) = e (3.) Cylndcal coodnates = xj xj x u u e (note the double sum on and j ) (3.3) u3 u u u 3 u u u= e+ e + e 3 (3.4) x x3 x3 x x x u e e e (3.5) = u + uθ θ + uz z uθ uz u = ( u ) + + θ z (3.6) = e + eθ + ez θ z (3.7) = + + θ z (3.8) uz uθ u uz u u= + θ + ( uθ ) z θ z e e z θ e (3.9) 3-5
Sphecal coodnates u = u e + u e + u e (3.30) θ θ ϕ ϕ uϕ u = ( u) + ( uθ snθ ) + snθ θ snθ ϕ (3.3) = e + eθ + ez θ snθ z (3.3) = snθ + + snθ θ θ sn θ ϕ (3.33) 3-6
u u u= snθ θ ϕ + snθ ϕ u + ( uθ) ϕ θ e θ ( u snθ) e snθ ( u ) ϕ ϕ θ e (3.34) Thee ae nfntely many solutons to Nave s equaton and the soluton to any ndvdual poblem s the one that has the coect ntal condtons and bounday condtons fo any patcula poblem. In geneal, t s not possble fo humans to analytcally solve 3.8 fo all classes of thee-dmensonal solutons to (3.8). Howeve, thee ae a numbe of analytc solutons to (3.8) f the poblem s assumed to be unfom n one decton (two-dmensonal). Ths s ultmately due to the fact that dvson s defned fo two dmensonal vectos (the same as dvson by complex numbes) but t cannot be defned fo hghe dmenson vectos. Theefoe, thee ae analytc (well mostly analytc) solutons to poblems n whch the elastc meda s descbed by a stack of hozontal plane laye, but entely numecal pocedues (fnte-element o fntedffeence) must be used to solve poblems n whch the stuctue s tuly thee dmensonal. The technques fo solvng geneal laye poblems often ely on expessng the dsplacement vecto feld as the sum of potentals (Helmholtz decomposton). That s, we can decompose the dsplacement as u = φ + ψ (3.35) whee ϕ and ψ ae scala and vecto functons of tme and space. If we make ths change of vaables, then Nave s equaton sepaates nto seveal wave equatons as follows. φ = φ (3.36) α ψ = ψ (3.37) β Of couse the bounday condtons must also be tansfomed nto potental fom. These potental foms can be used n any coodnate system as long as you know how to compute the Laplacan, the gadent and the cul. It s beyond the scope of ths class to demonstate geneal soluton technques fo Nave s equaton (see Achenbach fo a nce teatment), but we can demonstate seveal smple solutons whch have attbutes smla to those of solutons encounteed n the eal wold. Snce Nave s equaton s lnea, any soluton that s added to any othe soluton s also a soluton. Theefoe, we can often buld the appopate soluton by addng togethe known smple solutons n such a way that they poduce the desed stesses o dsplacements on the bounday of a doman; that s they match bounday condtons. When a doman contans layes, the solutons apply nsde the ndvdual laye and they ae constucted to poduce contnuous dsplacement at the boundaes and balanced tactons on the boundaes. Plane P-waves Suppose that we consde a moton defned by 3-7
and u x x x3 t f t (,,, ) = u u3 0 x α (3.38) = = (3.39) then t s a smple matte of substtutng (3.38) and (3.39) nto (3.8) to show that ths s a vald soluton fo any sngle vaable functon f, whch s twce dffeentable, and as long as α = λ + µ ρ (3.40) We could have altenatvely chosen the potentals, x φ = α f t α (3.4) ψ = 0 (3.4) It s a tval matte to show that ts gadent s the dsplacement feld gven by (3.38) and (3.39), and that t satsfes the wave equatons (3.36) and (3.37). Ths s the equaton of a plana P-wave tavelng at velocty α n the postve x decton. Snce the mateal s sotopc, ths decton s abtay and t could just as well be tavelng n the negatve x decton. Note that the shape of the wavefom s unchanged as t popagates though the medum. Ths popety s called nondspesve and t contasts wth some othe solutons that we wll exploe late whee the wave velocty depends on the fequency of the oscllaton. Snce the equaton s lnea, we could wte a moe geneal soluton that has dffeent P- waves tavelng n both postve and negatve dectons as x x u = f t + g t+ (3.43) α α whee g s some othe twce dffeentable functon. P-waves ae also called longtudnal waves snce the patcle moton s n the same decton as the wave popagates. They ae also called compessonal waves, although they have both compessonal and shea stesses as shown by the computng the stan and stess tenso fo (3.38) as follows. u ε = x x x f t f t α α = = α α x u t α = α (3.44) 3-8
and all othe stan components ae zeo. We see that the stan n ths wave s popotonal to the patcle velocty dvded by the wave speed. Ths wll be a ecung theme fo othe solutons of Nave s equaton. We can substtute (3.44) nto (3.5) to obtan the stess, whch gves ( ) ( λ µ ) ε σ = λ ε + ε + ε + µε 33 = + (3.45) and σ = σ33 = λε (3.46) σ = σ3 = σ3 = 0 (3.47) Substtutng (3.40) and(3.44) nto (3.45) and (3.46) we fnd that σ = ραu (3.48) and λ σ = σ33 = σ (3.49) λ + µ Equaton (3.48) tells us that the stess n ths wave s elated to the patcle velocty tmes the poduct of the densty and the wave speed. The ato of the stess to the patcle velocty s called the mechancal mpedance; t measues the stess that s needed to make a patcula gound moton. Notce that although thee ae no explct shea stesses n ths coodnate fame (whch s the pncpal coodnate fame fo ths poblem), thee ae shea stesses n othe coodnate fames. The maxmum shea stess s n the fame otated 45 degees fom the pncpal fame and n ths fame the maxmum shea stess s ( ) µ σ = σ σ = σ (3.50) λ + µ Theefoe thee ae shea stesses assocated wth these P-waves. We can also calculate the powe ( ) P x t assocated wth ths wave as the enegy flux n, the x decton. Ths enegy flux s the ate of wok pe unt aea done by the tacton vecto on a plane pependcula to the velocty of popagaton. Ths ate of wok pe unt aea s the stess tmes the patcle velocty, o P( x, t) = σu = ραu (3.5) The enegy pe unt volume ( ) dvded by the wave velocty, o E x t assocated wth the wave s just the enegy flux, (, ) ρu E x t = (3.5) As s the case fo all lnea dynamc systems, ths enegy s evenly dvded between knetc enegy and potental (stan) enegy f aveaged thoughout the system. 3-9
Fnally we can nque about the maxmum acceleatons that can occu n an elastc contnuum. We can dffeentate equaton (3.48) to obtan x σ t u( x, α t ) = (3.53) ρα That s the acceleaton of a pont scales lke the tme devatve of the compessve stess. If a fnte compessve stess wee suddenly appled to a suface then t would geneate a P-wave whose acceleaton would be descbed by a Dac-delta functon, whch has nfnte acceleaton. That s, f x σ = σ0h t α (3.54) whee H(t) s a Heavsde step functon, then x u = σδ 0 t α (3.55) Plane Shea Waves Anothe mpotant soluton to Nave s equaton can be expessed as x u = f t β (3.56) u = u3 = 0 (3.57) It s agan a smple matte to substtute (3.56) and (3.57) nto Nave s equaton (3.8) to fnd that ths s a soluton so long as β = µ ρ (3.58) As befoe, we could have used the dsplacement potentals φ = 0 (3.59) ψ = ψ = (3.60) 0 x ψ 3 = β f t β (3.6) whee the cul of ψ s the dsplacement and (3.6) solves the scala wave equaton (3.37). Ths s the descpton of a plana shea wave (S-wave) tavelng n the postve x decton wth velocty β. The patcle moton s n the x decton and t s paallel to the wave font and pependcula to the decton of moton. As was the case wth P- waves, f(t) s any functon wth a fnte nd devatve. Lke the plana P-wave, plana S- waves ae also nondspesve. Notce that the S-wave s slowe than the P-wave and that the aton of the veloctes s 3-0
α λ + µ = (3.6) β µ Ths can be expessed n tems of Posson s ato ν by usng Table 3.. In ths case, α ν = (3.63) β ν So the ato of P- to S-wave veloctes depends only on Posson s ato. Fo many solds, λ µ, o ν, n whch case we call the sold Possonan and α 3 =.77. 4 β We can also compute stan, stess, and enegy flux fo the S-wave wave as we dd fo the plana P-wave. In ths case, u ε = (3.64) β ε = ε = ε = ε = ε = (3.65) 33 3 3 0 σ = ρβu (3.66) σ = σ = σ = σ = σ = (3.67) 33 3 3 0 P( x, t) = σu = ρβu (3.68) Dagams of the moton of Plana P- and S-waves ae shown n Fgue 3.3. 3-
Hamonc Plane Waves x Whle plana P- and S-waves can be expessed fo any functon of the vaable, t c, whee c s the wave velocty, t s nstuctve to nvestgate the soluton f the functon s hamonc, a snusod o cosne. That s, thee ae many nstances n whch the supeposton of hamonc solutons can be used to constuct solutons to moe geneal poblems. To demonstate, let s consde the plana S-wave n the pevous secton, but we wll assume that ou functon s a cosne. That s, x u = cos ω t β (3.69) = cos( kx ωt) whee k s spatal wavenumbe gven by ω π k = = (3.70) β Λ and Λ s the wavelength. We can now consde what happens when two hamonc plane waves of dentcal stength and fequency, but tavelng n opposte dectons ae added togethe. We can use standad tgonometc denttes to easly show that. u = cos( kx ωt) + cos( kx+ ωt) (3.7) = cos kx cos ωt ( ) ( ) Equaton (3.7) s theefoe a standng wave wth the same fequency and wavenumbe as the two tavelng waves. Snce Nave s equaton s lnea, and snce the waves tavelng n each decton ae solutons, then the sum (the standng wave) s also a soluton of Nave s equaton. Obvously, standng wave solutons ae natual when dentcal waves ae tavelng n opposte dectons. Ths s a common occuence when hamonc waves ae eflected off of an nteface. It also happens n ou sphecal Eath when waves that tavel aound the Eath n opposte dectons meet. In ths case the ntefeence makes the fee oscllatons of the Eath. In a smla fashon, t s possble to add two hamonc standng waves togethe to poduce a sngle hamonc tavelng wave. Agan we can use standad tg denttes to show that u = cos( kx) cos( ωt) + sn ( kx) sn ( ωt) (3.7) = cos kx ωt ( ) We have shown that we can epesent any hamonc plane wave as ethe the sum of tavelng waves o the sum of standng waves. Obvously t woks fo P-waves too, snce we use the same tg denttes. As t tuns out, ths dualty of epesentatons s fa moe geneal and can be appled to a vaety of moe complex poblems. These two solutons ae sometmes efeed to as chaactestc solutons and mode solutons. Fgue 3.4 shows a schematc of how snusods tavelng n opposte dectons sum to make a standng wave. 3-
Fgue 3.4. Fom Vbaton and Waves by A. P. Fench. W.W. Noton and Co., 97. Sphecal Waves Many poblems that we encounte concen the adaton of waves fom a pont n the medum. These waves spead sphecally though the medum and the epesentaton wth Catesan coodnates s awkwad. In a homogeneous whole space t s usually most natual to solve these poblems n sphecal coodnates. Howeve, f thee ae layes n the medum, then t usually s moe convenent to solve these poblems n cylndcal coodnates. Geneal solutons fo these poblems ae qute complex and beyond the scope of ths class. Howeve, we can consde the followng potental n sphecal coodnates. Ths potental has adal symmety. ϕ ( t, ) = f t + g t+ (3.73) α α Ths solves the tansfomed fom of Nave s equaton gven by (3.33) and (3.36). When the poblem s adally symmetc, ths can be wtten as 3-3
ϕ = ϕ (3.74) α The dsplacement that esults fom ths s ϕ u = = f t g t f t g t + + + α α α α α (3.75) = f t + g t+ f t g t + α α α α α I have chosen a soluton wth waves that tavels both adally outwad (the f tems) and nwads (the g tems). Each of these has tems that decay wth dstance as both and ; these ae called fa-feld and nea-feld tems, espectvely. They ae both equed to solve Nave s equaton fo ths adal wave poblem. Notce that the fa-feld tem has a tme dependence that looks lke the tme devatve of the nea-feld tem. Also notce that the fa-feld tem s scaled by the factoα. We can exploe ths dffeence between nea-feld and fa-feld tems by nvestgatng the exact soluton to the poblem of a step change n pessue p 0 nsde a sphecal cavty of adus a. The devaton s somewhat lengthy and s gven by Achenbach. The answe fo a Posson sold s 3 ˆ ( ˆ a bt 0 ) sn ˆ u = p H t + ωt cosωˆ t e 4µ a (3.76) whee ˆ t t (3.77) α α ω = (3.78) 3a α b = (3.79) 3a At the suface of the cavty the dsplacement s a bt b bt u p0 e sn t e cos a 4 ω µ ω ω t = = + (3.80) Ths looks lke the pessue ate convolved wth the soluton of damped hamonc oscllato poblem subjected to a step n foce (see equaton.39). The peod of the undamped oscllato s gven by (.37), whch when combned wth (3.78) and (3.79) gves ω0 = ω + b = 3b (3.8) The facton of ctcal dampng of ths system s gven by (.5) and s equal to b ζ = = = 0.58 (3.8) ω0 3 So the suface of the cavty s a 58% damped oscllato that settles about ts new statc equlbum poston. Wth each hamonc swng, t adates wave enegy to the fa-feld tem, whch at lage become. 3-4
a u p e 4µ The dampng of the oscllatng hole s sometmes efeed to as adaton dampng and snce t s lnea and depends on the velocty at the souce, t s vey analogous to vscous dampng dscussed n the SDOF poblem of chapte. The concept of adaton dampng can become useful when nvestgatng the dampng of an oscllatng buldng that exctes sesmc waves as t oscllates. btˆ >> a 0 sn ˆ (3.83) Pont Foce The dsplacement n the decton fom a pont load n the a pont foce n the j decton wth tme hstoy f(t) was gven by Love (The mathematcal theoy of elastcty, Dove Pubs., 944) and s β δk u = τ f ( t τ) dτ + f t f t f t 4π x xk α + x xk α α β β β β (3.84) whee = xx (3.85) Ths s an mpotant buldng pont n sesmology, snce t allows us to calculate the wave feld that esults fom dstbutons of foces. Anelastc Attenuaton of a Tavelng Wave The solutons dscussed above ae fo an elastc medum. Howeve, t s useful to ntoduce the concept that the enegy slowly decays as they tavel due to some nelastc esponse of the medum. In addton, thee ae basc physcal consdeatons that eque that waves eventually attenuate. One convenent appoach to ths poblem s to beak a wavefom nto ts hamonc consttuent pats and to then ntoduce the followng defnton of Q whch s entely analogous to the one that we used n Chapte fo the SDOF poblem. Recall that fo a lghtly damped oscllato (equaton.30) E Q π (3.86) E whee E and E ae the total enegy and enegy lost pe cycle. We can also defne the logathmc decement of the ampltude lost pe cycle as ln A δ (3.87) A snce enegy s popotonal to the squae of ampltude, ln A = ln E (3.88) fom whch t follows that 3-5
Q π (3.89) δ We can now wte the expesson fo the ampltude A of a hamonc wave as a functon of dstance taveled as ( ) A 0 ( Qc) = Ae ω (3.90) whee c s the velocty of the wave. Sometmes the attenuaton s descbed by the paamete t whch s defned to be tavel tme t = = (3.9) cq qualty facto Homewok fo Chapte 3. Show that (3.38) and (3.56) ae solutons to Nave s equaton.. Show that (3.73) s a soluton to Nave s equaton. 3. If a plane hamonc wave wth a fequency of Hz and a popagaton velocty of 3 km/sec s ½ the ampltude afte tavelng 00 km though an attenuatng * medum, then what s the Q and t? 3-6