Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp between stress and stran: σ C :ε, σ (4.1.1) Cε where ε s the small stran tensor,.7. Stran Energy In ths purely mechancal theory of elastc materals, there s no dsspaton of energy all the energy of the loads s stored as elastc stran energy n the materal as t deforms, and can be recovered. For the lnear elastc materal, the rate of deformaton s equvalent to the rate of smallstran, d ε&, so the stran-energy functon can be wrtten as dw σ : dε (4.1.) and the total energy stored per unt volume over the complete hstory of stranng s nnd the stress can be wrtten as W σ : dε (4.1.3) σ W (4.1.4) Reducton n the number of Independent Elastc Constants Snce the stress and stran are symmetrc, σ σ and ε ε nm, the fourth-order elastcty tensor of stffness coeffcents contans the or symmetres 1.9.65, C C C (4.1.5) l l l and so the 81 coeffcents reduce to 36 ndependent coeffcents. Further, snce C s ndependent of the strans, 1 W σ : dε C : εdε e : C : e (4.1.6) Sold Mechancs Part III 353
Secton 4.1 and so C W (4.1.7) e e Now (4.1.8) and t follows that C possesses the maor symmetres: C C (4.1.9) Ths reduces the number of ndependent elastc constants from 36 to 1. Problems nvolvng Elastc Materals The sx consttutve equatons 4.1.1, together wth the equatons of moton and the 6 nematc relatons relatng the strans to the dsplacements, Eqn..7., ε ( u, + u, )/, gves a set of 15 equatons n the 15 unnowns: the sx stress components, the sx stran components and the three dsplacement components. To mantan a lnear theory, the acceleraton term n the equatons of moton must be lnear; ths s acheved by supposng the dsplacement gradents to be small: v du u u u d u u + v, (4.1.10) dt t t dt t When ths acceleraton term s ncluded, the problem s dynamc. When the equatons of equlbrum are used, the problem s statc. Compatblty An alternatve method of soluton s to remove the dsplacements from the above system and solve only for the stresses and strans. In ths case the stran-dsplacement relatons are replaced by three compatblty equatons, and there are then 1 equatons n 1 unnowns. Once the system s solved, the dsplacements can be obtaned from the strans by ntegraton. The Isotropc Lnear Elastc Sold When the materal s sotropc, the consttutve equaton holds n any coordnate system, σ (4.1.11) Cε Sold Mechancs Part III 354
Secton 4.1 and so the tensor of elastc constants s sotropc. The most general fourth-order sotropc tensor taes the form 1.10.7, ( I + I) C λδ δ + μ( δ δ δ δ ) C λ I I + μ, + (4.1.1) wth the fourth-order unt tensors gven by 1.9.60, m n n m I δ m I δ δ n δ e n m e m m n n (4.1.13) whch has only two ndependent materal constants. Snce the stran s symmetrc, one has σ C : ε ( I I + μ ( I + I ) λ : ε (4.1.14) ( λi I + μi) : ε and the consttutve equaton reduces to { Problem 1} and the two elastc constants Problems n Isotropc Elastcty ( tr ε) I + μ, σ λeδ + μe σ λ ε (4.1.15) λ, μ are called Lamé s constants. The 15 equatons mentoned earler can be reduced by elatng the strans from the consttutve equaton and the nematc equaton, and then substtutng the resultant expresson for stress nto the equatons of moton, gvng Naver s equatons u ( λ + μ ) grad( dvu ) + μ u + ρb ρu&, ( λ + μ ) + μ + ρb ρ Ths set of three partal dfferental equatons s approprate for problems nvolvng dsplacement boundary condtons. u u t (4.1.16) The Lame s constants and the Young s modulus and Posson s rato are related through μ(3λ + μ) λ E, ν λ + μ ( λ + μ) λ Eν, μ E ( 1+ ν )( 1 ν ) ( 1+ ν ) (4.1.17) The lnear elastc consttutve equatons n terms of the engneerng constants reads Sold Mechancs Part III 355
Secton 4.1 ν 1+ ν ε σ δ + σ E E E ν σ ε δ + ε 1+ ν 1 ν (4.1.18) The Bul Modulus The tensor of elastc constants can be wrtten n the alternatve forms C μ I + λi I ν μ I + I I 1 ν E ν (4.1.19) ( ) I + I I 1+ ν 1 ν 1 κ I I + μ I I I 3 where the new constant ntroduced s the bul modulus κ. Ths last expresson then leads to the alternatve form of the consttutve relatons, σ κ ( trε) I + μdevε (4.1.0) Ths expresson shows that the stress can be decomposed nto a sphercal component and a devatorc component. For a pure volume change, dev ε 0, and there are no shear stresses, σ ( trσ)i ; the bul modulus s thus a measure of the resstance of the materal to volume changes. 4.1. Geometrcally Non-Lnear Elastc Materals When the strans (dsplacement gradents) are not small, the behavour of the materal wll nevtably be non-lnear. Ths s due to the geometrc non-lnearty of the nematc stran-dsplacement relatons, for example usng the Green-Lagrange strans and the reference confguraton, E E T ( GradU + ( GradU) + ( GradU) GradU) 1 T 1 X + X + X X (4.1.1) The Krchhoff Materal The Krchhoff materal s an extenson of the lnear elastc model to the large stran range; E u : the consttutve relaton s a lnear tensor relaton, but non-lneartes enter through ( ) Sold Mechancs Part III 356
Secton 4.1 S C : E, S C E (4.1.) where S s the PK stress tensor and E s the Green-Lagrange stran. Snce both S and E are symmetrc, the fourth-order tensor C has the or symmetres, C C and C C nm, and so has 36 ndependent coeffcents. Followng the same arguments as before, one can defne a stran energy functon (per unt reference volume) dw S : de, dw S de (4.1.3) and the total energy stored per unt volume over the complete hstory of stranng s and the stress can be wrtten as 1 1 W SdE C E E E : C : E (4.1.4) S, S (4.1.5) E E Agan, the exstence of the stran energy functon mples that the matrx of elastc coeffcents only has 1 ndependent coeffcents. When the Krchhoff materal s sotropc, the consttutve relaton reduces further to ( tr E) I + μ, S λeδ + μe S λ E (4.1.6) As mentoned n.7., the lnear elastc model can not be used when there are large rgd body rotatons, even f the dsplacement gradents are not large. The Krchhoff model can be used n these cases. 4.1.3 Materally Non-Lnear Elastc Materals An elastc materal mght also exhbt materal non-lnearty through a non-lnear consttutve equaton, for example the Cauchy stress mght be some non-lnear functon of a stran measure, or of the deformaton gradent: σ f ( F( t)) (4.1.7) where f s some tensor functon of the deformaton gradent F. Ths consttutve equaton s called the Cauchy Elastc materal model. As can be seen, the stress s dependent on the current state only, and not on the path hstory, a requrement of elastcty. However, the stress n the case of a Cauchy elastc materal cannot n general be wrtten n terms of a stran-energy functon. In other words, the wor done mght be path-dependent. Sold Mechancs Part III 357
Secton 4.1 Obectvty Requrements The noton of obectvty was ntroduced n.8. When formulatng consttutve relatons for materals, one must ensure that the prncple of materal obectvty(or the prncple of materal frame ndfference) be satsfed. Ths prncple states that A consttutve law must be ndependent of the locaton of the observer (or frame of reference that s taen) Ths mples that two observers, even f n relatve moton wth respect to each other, observe the same stress n a gven body. Consder the Cauchy elastc materal 4.1.7. The Cauchy stress s an obectve tensor. Referrng to the example of Eqns..8.47-50 n.8.4, obectvty requres that the consttutve relaton be of the form T ( U) R σ Rf (4.1.8) The consttutve relaton can also be wrtten n terms of other stress measures. For T T T T example, usng J σ PF P( RU) PU R, one has -1 ( U) U P JRf (4.1.9) For the PK stress, one has 1 T S J F σf, so that S UU f -1 ( U) U 1 det (4.1.30) whch does not depend on the rotaton. Ths last relaton can also be wrtten n the form S det C 1/ C 1/ f ( 1/ C ) C 1/ g( C) (4.1.31) Ths s clearly obectve, snce S and C are unaffected by an observer transformaton, * * S S and C C. 4.1.4 Hypoelastc Materals A hypoelastc materal s one whose consttutve law relates the rate of stress to the rate of deformaton d. Ths can be wrtten n terms of the Cauchy stress as σ& f ( σ, d). Consder a smple one-dmensonal lnear model, σ& Ed (4.1.3) Snce, n one-dmenson, the stretch s λ dx / dx, the rate of deformaton s equvalent to the spatal velocty gradent l and the rate of change of a lne element dx s d ( dx) / dt ldx, dvdng through by dx gves d & λ / λ (see Eqn..5.10), so that & λ dλ & σ E σ dσ E E ln λ λ (4.1.33) λ Sold Mechancs Part III 358
Secton 4.1 Ths shows that the stress s clearly path-ndependent, dependng only on the current stretch. In fact, the stress can be wrtten as the dervatve of a stran energy functon accordng to σ dw / dλ, where W E / λ. In the three dmensonal case, however, the rate of deformaton can not n general be wrtten as the rate of change of some smple functon, d d ( ) / dt, and so the above calculaton cannot be done, mplyng that the hypoelastc materal cannot be wrtten n terms of a potental functon, and the wor done s path-dependent. The path-dependence s, however, small when the strans are small. 4.1.5 Hyperelastc Materals A hyperelastc materal (or Green elastc materal) s defned to be an elastc materal for whch a stran-energy functon W exsts, a scalar functon of one of the stran or deformaton tensors, whose dervatve wth respect to a stran component deteres the correspondng stress component. From the above, the lnear elastc model, the Krchhoff model and the one-dmensonal hypoelastc model are all examples of hyperelastc materals. The hyperelastc materal s a subset of the Cauchy-elastc materal. Hyperelastc materal models for components under large strans wll be the subect of the followng sectons. 4.1.6 Problems 1. Show that I I : ε ( trε) ( I + I) : ε ε I Sold Mechancs Part III 359