6.1 The Linear Elastic Model

Transcription

1 Linear lasticit The simplest constitutive law or solid materials is the linear elastic law, which assumes a linear relationship between stress and engineering strain. This assumption turns out to be an ecellent predictor o the response o components which undergo small deormations, or eample steel and concrete structures under large loads, and also works well or practicall an material at a suicientl small load. The linear elastic model is discussed in this chapter and some elementar problems involving elastic materials are solved. Anisotropic elasticit is discussed in ection..

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3 ection.. The Linear lastic Model.. The Linear lastic Model Repeating some o what was said in ection.: the Linear lastic model is used to describe materials which respond as ollows: (i) the strains in the material are small (linear) (ii) the stress is proportional to the strain, (linear) (iii) the material returns to its original shape when the loads are removed, and the unloading path is the same as the loading path (elastic) (iv) there is no dependence on the rate o loading or straining (elastic) From the discussion in the previous chapter, this model well represents the engineering materials up to their elastic limit. It also models well almost an material provided the stresses are suicientl small. The stress-strain (loading and unloading) curve or the Linear lastic solid is shown in Fig...a. Other possible responses are shown in Figs...b,c. Fig...b shows the tpical response o a rubber-tpe material and man biological tissues; these are nonlinear elastic materials. Fig...c shows the tpical response o viscoelastic materials (see hapter ) and that o man plasticall and viscoplasticall deorming materials (see hapters and ). hsteresis loop load unload ( a) ( b) (c) Figure..: Dierent stress-strain relationships; (a) linear elastic, (b) non-linear elastic, (c) viscoelastic/plastic/viscoplastic It will be assumed at irst that the material is isotropic and homogeneous. The case o an anisotropic elastic material is discussed in ection.. i the small-strain approimation is not made, the stress-strain relationship will be inherentl non-linear; the actual strain, qn...7, involves (non-linear) squares and square-roots o lengths

4 ection... tress-train Law onsider a cube o material subjected to a uniaial tensile stress, Fig...a. One would epect it to respond b etending in the direction,, and to contract laterall, so, these last two being equal because o the isotrop o the material. With stress proportional to strain, one can write, (..) (a) (b) Figure..: an element o material subjected to a uniaial stress; (a) normal strain, (b) shear strain The constant o proportionalit between the normal stress and strain is the Young s Modulus, qn..., the measure o the stiness o the material. The material parameter is the Poisson s ratio, qn.... ince, it is a measure o the contraction relative to the normal etension. Because o the isotrop/smmetr o the material, the shear strains are zero, and so the deormation o Fig...b, which shows a non-zero, is not possible shear strain can arise i the material is not isotropic. One can write down similar epressions or the strains which result rom a uniaial tensile stress and a uniaial stress:,, (..) imilar arguments can be used to write down the shear strains which result rom the application o a shear stress:,, (..)

5 ection. The constant o proportionalit here is the hear Modulus, qn...8, the measure o the resistance to shear deormation (the letter was used in qn...8 both and are used to denote the hear Modulus, the latter in more mathematical and advanced discussions). The strain which results rom a combination o all si stresses is simpl the sum o the strains which result rom each :,,,, (..) These equations involve three material parameters. It will be proved in. that an isotropic linear elastic material can have onl two independent material parameters and that, in act, This relation will be veriied in the ollowing eample. ample: Veriication o qn.... (..) onsider the simple shear deormation shown in Fig..., with and all other strains zero. With the material linear elastic, the onl non-zero stress is. Figure..: a simple shear deormation this is called the principle o linear superposition: the "eect" o a sum o "causes" is equal to the sum o the individual "eects" o each "cause". For a linear relation, e.g., the eects o two causes, are and, and the eect o the sum o the causes is indeed equal to the sum o the individual eects: ( ). This is not true o a non-linear relation, e.g., since ( ) 7

6 ection. Using the strain transormation equations, qns..., the onl non-zero strains in a o second coordinate sstem, with at rom the ais (see Fig...), are and. Because the material is isotropic, qns.. hold also in this second coordinate sstem and so the stresses in the new coordinate sstem can be determined b solving the equations,,,, (..) which results in, v v (..7) But the stress transormation equations, qns...8, with, give and and so qn... is veriied. Relation.. allows the Linear lastic olid stress-strain law, qn..., to be written as tress-train Relations (..8) This is known as Hooke s Law. These equations can be solved or the stresses to get 8

7 ection. ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) tress-train Relations (..9) Values o and or a number o materials are given in Table.. below (see also Table..). Material (Pa) re ast Iron.9 A tainless teel 9. A Aluminium 8. Bronze. Pleiglass.9. Rubber oncrete -. ranite -.7 Wood (pinewood) ibre direction transverse direction Table..: Young s Modulus and Poisson s Ratio ν or a selection o materials at o Volume hange Recall that the volume change in a material undergoing small strains is given b the sum o the normal strains (see ection.). From Hooke s law, normal stresses cause normal strain and shear stresses cause shear strain. It ollows that normal stresses produce volume changes and shear stresses produce distortion (change in shape), but no volume change... Two Dimensional lasticit The above three-dimensional stress-strain relations reduce in the case o a twodimensional stress state or a two-dimensional strain state. 9

8 ection. Plane tress In plane stress (see ection.),, Fig..., so the stress-strain relations reduce to tress-train Relations (Plane tress) (..) with, (..) z Figure..: Plane stress Note that the strain is not zero. Phsicall, corresponds to a change in thickness o the material perpendicular to the direction o loading. Plane train In plane strain (see ection.),, Fig..., and the stress-strain relations reduce to

9 ection. ( ) ( ) ( ) ( )( ) ( ) ( )( ) tress-train Relations (Plane train) (..) with, (..) Again, note here that the stress component is not zero. Phsicall, this stress corresponds to the orces preventing movement in the z direction. Figure.. Plane strain - a thick component constrained in one direction imilar olutions The epressions or plane stress and plane strain are ver similar. For eample, the plane strain constitutive law.. can be derived rom the corresponding plane stress epressions.. b making the substitutions, (..) in.. and then dropping the primes. The plane stress epressions can be derived rom the plane strain epressions b making the substitutions, (..)

10 ection. in.. and then dropping the primes. Thus, i one solves a plane stress problem, one has automaticall solved the corresponding plane strain problem, and vice versa... Problems. A strain gauge at a certain point on the surace o a thin A Aluminium component (loaded in-plane) records strains o μm, μm, μm. Determine the principal stresses. (ee Table.. or the material properties.). Use the stress-strain relations to prove that, or a linear elastic solid, and, indeed,, Note: rom qns... and.., these show that the principal aes o stress and strain coincide or an isotropic elastic material. onsider the case o hdrostatic pressure in a linearl elastic solid: p ij p p as might occur, or eample, when a spherical component is surrounded b a luid under high pressure, as illustrated in the igure below. how that the volumetric strain (qn...) is equal to p, so that the Bulk Modulus, qn...9, is K. onsider again Problem rom..7. (a) Assuming the material to be linearl elastic, what are the strains? Draw a second material element (superimposed on the one shown below) to show the deormed shape o the square element assume the displacement o the bo-centre to be zero and that there is no rotation. Note how the ree surace moves, even though there is no stress acting on it.

11 ection. (b) What are the principal strains and? You will see that the principal directions o stress and strain coincide (see Problem ) the largest normal stress and strain occur in the same direction.. onsider a ver thin sheet o material subjected to a normal pressure p on one o its large suraces. It is ied along its edges. This is an eample o a plate problem, an important branch o elasticit with applications to boat hulls, aircrat uselage, etc. (a) write out the complete three dimensional stress-strain relations (both cases, strain in terms o stress, qns...8, stress in terms o strain, qns...9). Following the discussion on thin plates in section.., the shear stresses,, can be taken to be zero throughout the plate. impli the relations using this act, the pressure boundar condition on the large ace and the coordinate sstem shown. (b) assuming that the through thickness change in the sheet can be neglected, show that p p z. A thin linear elastic rectangular plate with width a and height b is subjected to a uniorm compressive stress as shown below. how that the slope o the plate diagonal shown ater deormation is given b b / tan a / What is the magnitude o or a steel plate ( Pa,. ) o dimensions cm with MPa?

12 ection.. Homogeneous Problems in Linear lasticit A homogeneous stress (strain) ield is one where the stress (strain) is the same at all points in the material. Homogeneous conditions will arise when the geometr is simple and the loading is simple... lastic Rectangular uboids Hooke s Law, qns...8 or..9, can be used to solve problems involving homogeneous stress and deormation. Hoooke s law is equations in unknowns ( stresses and strains). I some o these unknowns are given, the rest can be ound rom the relations. ample onsider the block o linear elastic material shown in Fig.... It is subjected to an equi-biaial stress o. ince this is an isotropic elastic material, the shears stresses and strains will be all zero or such a loading. One thus need onl consider the three normal stresses and strains. There are now equations (the irst o qns...8 or..9) in unknowns. One thus needs to know three o the normal stresses and/or strains to ind a solution. From the loading, one knows that and. The third piece o inormation comes rom noting that the suraces parallel to the plane are ree suraces (no orces acting on them) and so. From qn...8 then, the strains are ( ),, As epected, and. z Figure..: A block o linear elastic material subjected to an equi-biaial stress

13 ection... Problems. A block o isotropic linear elastic material is subjected to a compressive normal stress o over two opposing aces. The material is constrained (prevented rom moving) in one o the direction normal to these aces. The other aces are ree. (a) What are the stresses and strains in the block, in terms o o,,? (b) alculate three maimum shear stresses, one or each plane (parallel to the aces o the block). Which o these is the overall maimum shear stress acting in the block?. Repeat problem a, onl with the ree aces now ied also.

14 ection.. Anisotropic lasticit There are man materials which, although well modelled using the linear elastic model, are not nearl isotropic. amples are wood, composite materials and man biological materials. The mechanical properties o these materials dier in dierent directions. Materials with this direction dependence are called anisotropic (see ection..7)... Material onstants The most general orm o Hooke s law, the generalised Hooke s Law, or a linear elastic material is (..) where each stress component depends (linearl) on all strain components. This new notation, with onl one subscript or the stress and strain, numbered rom, is helpul as it allows the equations o anisotropic elasticit to be written in matri orm. The ij 's are material constants called the stinesses, and in principle are to be obtained rom eperiment. The matri o stinesses is called the stiness matri. Note that these equations impl that a normal stress will induce a material element to not onl stretch in the direction and contract laterall, but to undergo shear strain too, as illustrated schematicall in Fig.... Figure..: an element undergoing shear strain when subjected to a normal stress onl In section 8.., when discussing the strain energ in an elastic material, it will be shown that it is necessar or the stiness matri to be smmetric and so there are onl independent elastic constants in the most general case o anisotropic elasticit. qns... can be inverted so that the strains are given eplicitl in terms o the stresses:

15 ection. (..) The s here are called compliances, and the matri o compliances is called the ' ij compliance matri. The bottom hal o the compliance matri has been omitted since it too is smmetric. It is diicult to model ull anisotropic materials due to the great number o elastic constants. Fortunatel man materials which are not ull isotropic still have certain material smmetries which simpli the above equations. These material tpes are considered net... Orthotropic Linear lasticit An orthotropic material is one which has three orthogonal planes o microstructural smmetr. An eample is shown in Fig...a, which shows a glass-ibre composite material. The material consists o thousands o ver slender, long, glass ibres bound together in bundles with oval cross-sections. These bundles are then surrounded b a plastic binder material. The continuum model o this composite material is shown in Fig...b wherein the ine microstructural details o the bundles and surrounding matri are smeared out and averaged. Three mutuall perpendicular planes o smmetr can be passed through each point in the continuum model. The,, z aes orming these planes are called the material directions. z ibre binder bundles ( a ) ( b ) z Figure..: an orthotropic material; (a) microstructural detail, (b) continuum model 7

16 ection. The material smmetr inherent in the orthotropic material reduces the number o independent elastic constants. To see this, consider an element o orthotropic material subjected to a shear strain, as in Fig.... and also a strain z Figure..: an element o orthotropic material undergoing shear strain From qns..., the stresses induced b a strain onl are,,,, (..) The stresses induced b a strain onl are (the prime is added to distinguish these stresses rom those o qn...),,,, (..) These stresses, together with the strain, are shown in Fig... (the microstructure is also indicated) ( a) (b) Figure..: an element o orthotropic material undergoing shear strain; (a) positive strain, (b) negative strain Because o the smmetr o the material (print this page out, turn it over, and Fig...a viewed rom the other side o the page is the same as Fig...b on this side o the page), one would epect the normal stresses in Fig... to be the same,, 8

17 ection. 9, but the shear stresses to be o opposite sign,. qns...- then impl that (..) imilar conclusions ollow rom considering shear strains in the other two planes: : : (..) The stiness matri is thus reduced, and there are onl nine independent elastic constants: (..7) These equations can be inverted to get, introducing elastic constants, and in place o the s ij ' : (..8) The nine independent constants here have the ollowing meanings: i is the Young s modulus (stiness) o the material in direction,, i ; or eample, or uniaial tension in the direction. ij is the Poisson s ratio representing the ratio o a transverse strain to the applied strain in uniaial tension; or eample, / or uniaial tension in the direction.

18 ection. ij are the shear moduli representing the shear stiness in the corresponding plane; or eample, is the shear stiness or shearing in the - plane. I the -ais has long ibres along that direction, it is usual to call and the aial shear moduli and the transverse (out-o-plane) shear modulus. Note that, rom smmetr o the stiness matri,, (..9), An important eature o the orthotropic material is that there is no shear coupling with respect to the material aes. In other words, normal stresses result in normal strains onl and shear stresses result in shear strains onl. Note that there will in general be shear coupling when the reerence aes used,,, z, are not aligned with the material directions,,. For eample, suppose that the aes were oriented to the material aes as shown in Fig.... Assuming that the material constants were known, the stresses and strains in the constitutive equations..8 can be transormed into,, etc. and,, etc. using the strain and stress transormation equations. The resulting matri equations relating the strains, to the stresses, will then not contain zero entries in the stiness matri, and normal stresses, e.g., will induce shear strain, e.g., and shear stress will induce normal strain. Figure..: reerence aes not aligned with the material directions.. Transversel Isotropic Linear lasticit A transversel isotropic material is one which has a single material direction and whose response in the plane orthogonal to this direction is isotropic. An eample is shown in Fig..., which again shows a glass-ibre composite material with aligned ibres, onl now the cross-sectional shapes o the ibres are circular. The characteristic material direction is z and the material is isotropic in an plane parallel to the plane. The material properties are the same in all directions transverse to the ibre direction.

19 ection. z Figure..: a transversel isotropic material This etra smmetr over that inherent in the orthotropic material reduces the number o independent elastic constants urther. To see this, consider an element o transversel isotropic material subjected to a normal strain onl o magnitude, Fig...7a, and also a normal strain o the same magnitude,, Fig...7b. The plane is the plane o isotrop. ( a) (b) Figure..7: elements o a transversel isotropic material undergoing normal strain in the plane o isotrop From qns...7, the stresses induced b a strain onl are,,,, (..) The stresses induced b the strain onl are (the prime is added to distinguish these stresses rom those o qn...),,,, (..) Because o the isotrop, the ( ) due to the should be the same as the ( ) due to the, and it ollows that. Further, the ( ) should be the same or both, and so.

20 ection. Further simpliications arise rom consideration o shear deormations, and rotations about the material ais, and one inds that and. The stiness matri is thus reduced, and there are onl ive independent elastic constants: (..) with being the material direction. These equations can be inverted to get, introducing elastic constants, and in place o the ij ' s. One again gets qn...8, but now,,,, (..) so (..) with, due to smmetr, / / (..) qns...- seem to impl that there are independent constants; however, the transverse modulus is related to the transverse Poisson ratio and the transverse stiness through (see qn..., and.. below, or the isotropic version o this relation) ( ) (..)

21 ection. These equations are oten epressed in terms o a or ibre (or a or aial) and t or transverse: t t t t t t t (..7).. Isotropic Linear lasticit An isotropic material is one or which the material response is independent o orientation. The smmetr here urther reduces the number o elastic constants to two, and the stiness matri reads (..8) These equations can be inverted to get, introducing elastic constants, and,

22 ection. (..9) with (..) which are qns...8 and... qns...8 can also be written concisel in terms o the engineering constants, and with the help o the Lamé constants, and : (..) with, ( ) (..).. Problems. A piece o orthotropic material is loaded b a uniaial stress (aligned with the material direction ). What are the strains in the material, in terms o the engineering constants?. A specimen o bone in the shape o a cube is ied and loaded b a compressive stress MPa as shown below. The bone can be considered to be orthotropic, with material properties

23 ection..9pa,.pa,., 8.Pa,.Pa,.,. 8.Pa.9Pa What are the stresses and strains which arise rom the test according to this model (the bone is compressed along the direction)? suraces are ied. onsider a block o transversel isotropic material subjected to a compressive stress p (perpendicular to the material direction) and constrained rom moving in the other two perpendicular directions (as in Problem ). valuate the stresses and in terms o the engineering constants t, and t,.. A strip o skin is tested in biaial tension as shown below. The measured stresses and strains are as given in the igure. The orientation o the ibres in the material is later o measured to be. Pa Pa Fibre orientation.7.7. (a) alculate the normal stresses along and transverse to the ibres, and the corresponding shear stress. (Hint: use the stress transormation equations.) (b) alculate the normal strains along and transverse to the ibres, and the corresponding shear strain. (Hint: use the strain transormation equations.) (c) Assuming the material to be orthotropic, determine the elastic constants o the material (assume the stiness in the ibre direction to be ive times greater than the stiness in the transverse direction). Note: because the material is thin, one can take. (d) alculate the magnitude and orientations o the principal normal stresses and strains. (Hint: the principal directions o stress are where there is zero shear stress.) (e) Do the principal directions o stress and strain coincide?

24 ection.. A biaial test is perormed on a roughl planar section o skin (thickness mm) rom the back o a test-animal. The test aes ( and ) are aligned such that deormation is induced in the skin along the spinal direction and transverse to this direction, under the assumption that the ibres are oriented principall in these directions. However, it is ound during the eperiment that shear stresses are necessar to maintain a biaial deormation state. Measured stresses are kpa, kpa, kpa Determine the in-plane orientation o the ibres given the data kpa, kpa, kpa,.. [Hint: derive an epression or involving onl, where is the inclination o the material aes rom the aes]