9780203009512.ch4

9780203009512.ch4 - 0749_Frame_C04 Page 51 Wednesday,...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
51 Kinematics of Deformation The current chapter provides a review of the mathematics for describing deformation of continua. A more complete account is given, for example, in Chandrasekharaiah and Debnath (1994). 4.1 KINEMATICS 4.1.1 D ISPLACEMENT In finite-element analysis for finite deformation, it is necessary to carefully distin- guish between the current (or “deformed”) configuration (i.e., at the current time or load step) and a reference configuration, which is usually considered strain-free. Here, both configurations are referred to the same orthogonal coordinate system characterized by the base vectors e 1 , e 2 , e 3 (see Figure 1.1 in Chapter 1). Consider a body with volume V and surface S in the current configuration. The particle P occupies a position represented by the position vector x , and experiences (empirical) temperature T . In the corresponding undeformed configuration, the position of P is described by X , and the temperature has the value T 0 independent of X . It is now assumed that x is a function of X and t and that T is also a function of X and t . The relations are written as x ( X , t ) and T ( X , t ), and it is assumed that x and T are continuously differentiable in X and t through whatever order needed in the subse- quent development. FIGURE 4.1 Position vectors in deformed and undeformed configurations. 4 e 2 e 1 X x undeformed deformed © 2003 by CRC CRC Press LLC
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
52 Finite Element Analysis: Thermomechanics of Solids 4.1.2 D ISPLACEMENT V ECTOR The vector u ( X ) represents the displacement from position X to x : (4.1) Now consider two close points, P and Q, in the undeformed configuration. The vector difference X P X Q is represented as a differential d X with squared length dS 2 = d X T d X . The corresponding quantity in the deformed configuration is d x , with dS 2 = d x T d x . 4.1.3 D EFORMATION G RADIENT T ENSOR The deformation gradient tensor F is introduced as (4.2) F satisfies the polar-decomposition theorem: (4.3) in which U and V are orthogonal and Σ is a positive definite diagonal tensor whose FIGURE 4.2 Deformed and undeformed distances between adjacent points. ds Q' P' Q * P * dS u () . Xx X , t =− dd xFX F x X == FUV T , © 2003 by CRC CRC Press LLC
Background image of page 2
Kinematics of Deformation 53 entries λ j , the singular values of F , are called the principal stretches. (4.4) Based on Equation 4.3, F can be visualized as representing a rotation, followed by a stretch, followed by a second rotation. 4.2 STRAIN The deformation-induced change in squared length is given by (4.5) in which E denotes the Lagrangian strain tensor . Also of interest is the Right Cauchy- Green strain C = F T F = 2 E + I . Note that F = I + u / X . If quadratic terms in u / X are neglected, the linear-strain tensor E L is recovered as (4.6) Upon application of Equation 4.3, E is rewritten as (4.7) Under pure rotation x = QX , F = Q and E = [ Q T Q I ] = 0 . The case of pure rotation in small strain is considered in a subsequent section.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

Page1 / 21

9780203009512.ch4 - 0749_Frame_C04 Page 51 Wednesday,...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online