Mechanics of materials Lecture 4 Strain and deformation

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1 Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206

2 of a continuum body 3 Definition of strain 4 Infinitesimal strain tensor Mechanics of materials R. Kouhia 2/6

3 Motion of a continuum body Continuum mechanics Stress Balance Force Constitutive equations Strains Displacements Kinematics B σ = f σ = Cε ε = Bu Reijo Kouhia 25/0/205 equilibrium constitutive model kinematical relation Mechanics of materials R. Kouhia 3/6

4 Description of motion A material point has coordinates X in the undeformed state. After deformation it is moved to the place x. A mapping χ is called the motion x = χ(x, t) = X + u(x, t), x i = χ i (X, t) = X i + u i (X, t), and u is the displacement vector. X are the material coordinates. Frequently used in solid mechanics. x are the spatial coordinates. Much used in fluid mechanics. Mechanics of materials R. Kouhia 4/6

5 of a continuum body 3 Definition of strain 4 Infinitesimal strain tensor Mechanics of materials R. Kouhia 5/6

6 Deformation gradient Deformation gradient F gives the change of an infinitesimal line element at P dx = F dx, F = χ X, Mechanics of materials R. Kouhia 6/6

7 Deformation gradient - cont d or in indicial notation dx i = F ij dx j, F ij = χ i X j = X i X j + u i X j = δ ij + u i X j. If there is no deformation, then F = I. It contains both strains and rigid body rotation and can be decomposed as (the polar decomposition) F = R U = V R, where R is orthogonal rotation tensor and U and V are the symmetric and positive definite right and left stretch tensors. Mechanics of materials R. Kouhia 7/6

8 of a continuum body 3 Definition of strain 4 Infinitesimal strain tensor Mechanics of materials R. Kouhia 8/6

9 Definition of strain Length of a line element PQ is ds = d X dx In deformed state pq = ds = d x dx 28 CHAPTER 4. KINEMATICAL RELATIONS q Q u Q ds P u P p du Figure 4.2: Relative displacement du of Q relative to P. 2 [(ds)2 (ds) 2 ] = 2 (d x dx d X dx) where the symmetric part ε is = the 2 d infinitesimal X (F T Fstrain I) tensor dx = d X E dx ε ε 2 ε 3 ε xx ε xy ε xz ε x γ where E is the Green-Lagrange strain tensor. 2 xy ε = ε 2 ε 22 ε 23 = ε yx ε yy ε yz = γ 2 yx ε y ε ε ε ε ε ε γ γ ε γ 2 xz Mechanics of materials R. Kouhia γ 2 yz, (4.30) 9/6

10 Green-Lagrange strain tensor E = 2 (F T F I) = 2 (C I), where C = F T F is the right Cauchy-Green deformation tensor. E = 0 for pure rigid body rotation. G-L in terms of displacement ( E = ( ) T u u 2 X + + u ( ) ) T u X X X If u/ X, then E ε = 2 ( u x + ( ) ) T u, x where ε is the infinitesimal strain tensor - notice x X. Mechanics of materials R. Kouhia 0/6

11 Other strain tensors A rather general strain definition can be stated as E (m) = m (U m I). m = 2 corresponds to the G-L strain tensor. The Hencky or logarithic strain tensor is obtained when m 0 + lim m 0 + E(m) = ln U. The Biot strain tensor for m = E () = U I. Mechanics of materials R. Kouhia /6

12 of a continuum body 3 Definition of strain 4 Infinitesimal strain tensor Mechanics of materials R. Kouhia 2/6

13 Infinitesimal strain tensor Also known as the small strain tensor ε = sym grad u in index notation ε ij = 2 ( ui + u ) j x j x i Von Kármán notation ε = ε x 2 γ xy 2 γ xy ε y 2 γ xz 2 γ xz 2 γ yz 2 γ yz ε z Mechanics of materials R. Kouhia 3/6

14 Strain in arbitrary direction Strain in direction n ( n = ) ε n = n ε n. Change in the angle between orthonormal vectors n and m γ nm = 2 n ε m. FIGURE 4.3 Mechanics of materials R. Kouhia 4/6

15 Principal strains Eigenvalues of the strain tensor εn = λn (ε λi)n = 0 Non-trivial solution for n if det(ε λi) = 0 Characteristic polynomial λ 3 + Iλ ε 2 + I2λ ε + I3 ε = 0 where I ε = trε = ε kk = ε + ε 22 + ε 33 I2 ε = 2 [tr(ε2 ) (trε) 2 ] I3 ε = det ε are called the principal invariants of the infinitesimal strain tensor. Mechanics of materials R. Kouhia 5/6

16 Volumetric - isochoric split The strain tensor can be split into volumetric and isochoric i.e. volume preserving parts ε = 3 (trε)i + e where trε = ε vol is the volumetric strain ε vol = V V 0 V 0. Mechanics of materials R. Kouhia 6/6

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