1.2 The Strain-Displacement Relations

Transcription

1 1. The Strain-Displacement Relations The strain was introdced in ook I: 4. The concepts eamined there are now etended to the case of strains which var continosl throghot a material The Strain-Displacement Relations Normal Strain Consider a line element of length emanating from position (, ) and ling in the - direction, denoted b A in Fig After deformation the line element occpies A, having ndergone a translation, etension and rotation. (, ) (, ) A A Figre 1..1: deformation of a line element The particle that was originall at has ndergone a displacement (, ) and the other end of the line element has ndergone a displacement (, ). the definition of (small) normal strain, In the limit 0 one has A A (, ) (, ) (1..1) A (1..) This partial derivative is a displacement gradient, a measre of how rapid the displacement changes throgh the material, and is the strain at (, ). Phsicall, it represents the (approimate) nit change in length of a line element, as indicated in Fig Kell

2 A A Figre 1..: nit change in length of a line element Similarl, b considering a line element initiall ling in the direction, the strain in the direction can be epressed as Shear Strain (1..3) The particles A and in Fig also ndergo displacements in the direction and this is shown in Fig In this case, one has (1..4) (, ) A (, ) A Figre 1..3: deformation of a line element A similar relation can be derived b considering a line element initiall ling in the direction. A smmar is given in Fig From the figre, / tan 1 / provided that (i) is small and (ii) the displacement gradient / is small. A similar epression for the angle can be derived, and hence the shear strain can be written in terms of displacement gradients. 10 Kell

3 Figre 1..4: strains in terms of displacement gradients The Small-Strain Stress-Strain Relations In smmar, one has 1 -D Strain-Displacement relations (1..5) 1.. Geometrical Interpretation of Small Strain A geometric interpretation of the strain was given in ook I: This interpretation is repeated here, onl now in terms of displacement gradients. Positive Normal Strain Fig. 1..5a, Negative Normal Strain Fig 1..5b, 1 0, 0, 0 (1..6) 1 0, 0, 0 (1..7) 11 Kell

4 () ( ) () Figre 1..5: some simple deformations; (a) positive normal strain, (b) negative normal strain, (c) simple shear Simple Shear Fig. 1..5c, ( a) ( b) (c) 1 1 0, 0, (1..8) Pre Shear Fig 1..6a, 1 0, 0, (1..9) 1..3 The Rotation Consider an arbitrar deformation (omitting normal strains for ease of description), as shown in Fig As sal, the angles and are small, eqal to their tangents, and /, /. () () 1 Figre 1..6: arbitrar deformation (shear and rotation) 1 Kell

5 Now this arbitrar deformation can be decomposed into a pre shear and a rigid rotation 1 as depicted in Fig In the pre shear,. In the rotation, the 1 angle of rotation is then. arbitrar shear strain pre shear rotation (no strain) Figre 1..7: decomposition of a strain into a pre shear and a rotation This leads one to define the rotation of a material particle,, the signifing the ais abot which the element is rotating: 1 (1..10) The rotation will in general var throghot a material. When the rotation is everwhere ero, the material is said to be irrotational. For a pre rotation, note that 1 0, (1..11) 13 Kell

6 1..4 Fiing Displacements The strains give information abot the deformation of material particles bt, since the do not encompass translations and rotations, the do not give information abot the precise location in space of particles. To determine this, one mst specif three displacement components (in two-dimensional problems). Mathematicall, this is eqivalent to saing that one cannot niqel determine the displacements from the strain-displacement relations Eample Consider the strain field 0.01, 0. The displacements can be obtained b integrating the strain-displacement relations: d 0.01 d g( ) f ( ) (1..1) where f and g are nknown fnctions of and respectivel. Sbstitting the displacement epressions into the shear strain relation gives f ( ) g( ). (1..13) An epression of the form F( ) G( ) which holds for all and implies that F and G are constant 1. Since f, g are constant, one can integrate to get f ( ) A D, g( ) C. From 1..13, C D, and A C C (1..14) There are three arbitrar constants of integration, which can be determined b specifing three displacement components. For eample, sppose that it is known that ( 0,0) 0, (0,0) 0, (0, a) b. (1..15) In that case, A 0, 0, C b / a, and, finall, 0.01 ( b / a) ( b / a) (1..16) which corresponds to Fig. 1..8, with ( b / a) being the (tan of the small) angle b which the element has rotated. 1 since, if this was not so, a change in wold change the left hand side of this epression bt wold not change the right hand side and so the eqalit cannot hold 14 Kell

7 b a Figre 1..8: an element ndergoing a normal strain and a rotation In general, the displacement field will be of the form A C C (1..17) and indeed Eqn is of this form. Phsicall, A, and C represent the possible rigid bod motions of the material as a whole, since the are the same for all material particles. A corresponds to a translation in the direction, corresponds to a translation in the direction, and C corresponds to a positive (conterclockwise) rotation Three Dimensional Strain The three-dimensional stress-strain relations analogos to Eqns are, 1,, 1 1, 3-D Stress-Strain relations (1..18) The rotations are 1 1 1,, (1..19) 1..6 Problems 1. The displacement field in a material is given b A 3, A where A is a small constant. 15 Kell

8 (a) Evalate the strains. What is the rotation? Sketch the deformation and an rigid bod motions of a differential element at the point ( 1, 1) (b) Sketch the deformation and rigid bod motions at the point ( 0, ), b sing a pre shear strain sperimposed on the rotation.. The strains in a material are given b, 0, Evalate the displacements in terms of three arbitrar constants of integration, in the form of Eqn , A C C What is the rotation? 3. The strains in a material are given b A, A, A where A is a small constant. Evalate the displacements in terms of three arbitrar constants of integration. What is the rotation? 4. Show that, in a state of plane strain ( 0 ) with ero bod force, e where e is the volmetric strain (dilatation), the sm of the normal strains: e (see ook I, 4.3). 16 Kell