S40 Solid Mechanics Fall 007 Stress field and momentum balance. Imagine the three-dimensional bod again. At time t, the material particle,, ) is under a state of stress ij,,, force per unit volume b b,,,. Denote the distributed eternal. An eample is the gravitational force, b g. The stress and the displacement are time-dependent fields. ach material particle has the acceleration vector u i / t. Cut a small differential element, of edges d, d and d. Let be the densit. The mass of the differential element is and we obtain that dd dd dd [ $ d,,, $,,, ] $, d,, $,,, [ ] ddd. Appl Newton s second law in the -direction, u t [ $,, d, $,,, ] bddd ddd Divide both sides of the above equation b ddd, and we obtain that u b. t 9/1/07 Linear lasticit-9
S40 Solid Mechanics Fall 007 9/1/07 Linear lasticit-10 This is the momentum balance equation in the -direction. Similarl, the momentum balance equations in the - and -direction are t u b t u b When the bod is in equilibrium, we drop the acceleration terms from the above equations. Using the summation convention, we write the three equations of momentum balance as t u b i j j ij.
S40 Solid Mechanics Fall 007 Hookes law. For an isotropic, homogeneous solid, onl two independent constants are needed to describe its elastic propert: Young s modulus and Poisson s ratio ν. In addition, a thermal epansion coefficient α characteries strains due to temperature change. When temperature changes b T, thermal epansion causes a strain T in all three directions. The combination of multi-aial stresses and a temperature change causes strains ) 1 % $ )*T ) 1 % $ )*T ) 1 % $ )*T The relations for shear are ) 1 ) 1 ) 1,, Recall the notation /, and we have. 1, 1, 1 Inde notation and summation convention. The si stress-strain relation ma be written as ij 1 ij $ kk% ij. The smbol ij stands for 0 when i j and for 1 when i j. We adopt the convention that a repeated inde implies a summation over 1, and 3. Thus, kk 11 33. Homogeneit. When talking about homogeneit, ou should think about at least two length scales: a large macro) length scale, and a small micro) length scale. A material is said to be homogeneous if the macro-scale of interest is much larger than the scale of microstructures. A fiber-reinforced material is regarded as homogeneous when used as a component of an airplane, but should be thought of as heterogeneous when its fracture mechanism is of interest. Steel is usuall thought of as a homogeneous material, but reall contains numerous voids, particles and grains. 9/1/07 Linear lasticit-11
S40 Solid Mechanics Fall 007 Isotrop. A material is isotropic when response in one direction is the same as in an other direction. Metals and ceramics in polcrstalline form are isotropic at macro-scale, even though their constituents grains of single crstals are anisotropic. Wood, single crstals, uniaial fiber reinforced composites are anisotropic materials. ample: a rubber laer pressed between two steel plates. A ver thin elastic laer, of Youngs modulus and Poissons ratio ν, is well bonded between two perfectl rigid plates. A thin rubber laer between two thick steel plates is a good approimation of the situation. The thin laer is compressed between the plates b a known normal stress σ. Calculate all the stress and strain components in the thin laer. Solution. The stress state at the edges of the elastic laer is complicated. We will neglect this edge effect, and focus on the field awa from the edges, where the field is uniform. This emphasis makes sense if we are interested in, for eample, the displacement of one plate relative to the other. Of course, this emphasis is misplaced if we are concerned of debonding of the laer from the plates, as debonding ma initiate from the edges, where stresses are high. B smmetr, the field has onl the normal components and has no shear components. Also b smmetr, we note that Because the elastic laer is bonded to the rigid plate, the two strain components vanish: 0. Using Hooke s law, we obtain that or $ ) 1 0 $,. 1 Using Hooke s law again, we obtain that 9/1/07 Linear lasticit-1
S40 Solid Mechanics Fall 007 1 ) 1 ) 1$ $ $ ). 1$ ) Consequentl, the elastic laer is in a state of uniaial strain, but all three stress components are nonero. When the elastic laer is incompressible, 0. 5, it cannot be strained in just one direction, and the stress state will be hdrostatic. 9/1/07 Linear lasticit-13
S40 Solid Mechanics Fall 007 Summar of lasticit Concepts Plaers: Fields Stress tensor: stress state must be represented b 6 components directional proper. Strain tensor: strain state must be represented b 6 components directional proper. Stress field: stress state varies from particle to particle positional proper. Stress field is represented b 6 functions,,, ;t ),,,;t ) Displacement field is represented b 3 functions, u,, ;,v,,;, w,, ;. Strain field is represented b 6 functions,,, ;t ),,, ;t ),... Rules: 3 elements of solid mechanics Momentum balance Deformation geometr Material law Complete equations of elasticit: Partial differential equations Boundar conditions - Prescribe displacement. - Prescribe traction. Initial conditions: For dnamic problems e.g., vibration and wave propagation), one also need prescribe initial displacement and velocit fields. Solving boundar value problems: OD and PD Idealiation, analtical solutions: e.g., S.P. Timoshenko and J.N. Goodier, Theor of lasticit, McGraw-Hill, New York Handbook solutions: R.. Peterson, Stress Concentration Factors, John Wile, New York, 1974. nd edition b W.D. Pilke, 1997 Brute force, numerical methods: finite element methods, boundar element methods 9/1/07 Linear lasticit-14
S40 Solid Mechanics Fall 007 3D lasticit: Collected quations Momentum balance b u t b v t b w t Strain-displacement relation Hookes Law u, 1 %v w $ v, 1 w u $ w, 1 % u v $ 1 [ $ ) ] %T, 1 $ 1 [ $ ) ] %T, 1 $ 1 $ ) Stress-traction relation [ ] %T, 1 $ t 1 t t 3 $ 11 1 13 $ 1 3 % 31 3 33 % n 1 n n 3 $ % 9/1/07 Linear lasticit-15
S40 Solid Mechanics Fall 007 3D lasticit: quations in other coordinates 1. Clindrical Coordinates r, θ, ) Momentum balance u, v, w are the displacement components in the radial, circumferential and aial directions, respectivel. Inertia and bod force terms are neglected. r r 1 r r r $ r 0 r r r 1 r r 0 r r r 1 r r r 0 Strain-displacement relation r u r, 1 $ v 1 w % r 1 $ v r % u, r 1 $ w r u % w, r 1 $ 1 u r v r v % r. Spherical Coordinates r, θ, φ) θ is measured from the positive -ais to a radius; φ is measured round the -ais in a righthanded sense. u, v, w are the displacements components in the r, θ, φ directions, respectivel. Inertia terms are neglected. 9/1/07 Linear lasticit-16
S40 Solid Mechanics Fall 007 Momentum balance r r 1 r r 1 r$ rsin $ 1 r % % % cot r $ r ) 0 r r 1 r 1 $ rsin $ 1 r % $ ) cot 3 r ) 0 r$ r 1 $ r 1 $ rsin $ 1 r 3 cot r$ $ ) 0 Strain-displacement relation r u r $ 1 r w % wcot 1 v ) sin $ * 1 v r u ) * $r 1 w % w r 1 u ) r sin $ * 1 w ) $ usin vcos r sin $ * 1 1 u r r v r % v ) r * 9/1/07 Linear lasticit-17