9780203009512.ch4

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

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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

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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
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.

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

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

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