DifferentialEquations_02_Strain_Disp_Eqns

DifferentialEquations_02_Strain_Disp_Eqns - Section 1.2 1.2...

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Section 1.2 Solid Mechanics Part II Kelly 9 1.2 The Strain-Displacement Relations The strain was introduced in Part I: §3.6. Expressions which relate the displacements of material particles to the strains for a continuously varying strain field are derived in what follows. 1.2.1 The Strain-Displacement Relations Normal Strain Consider a line element of length x Δ emanating from position ) , ( y x and lying in the x - direction, denoted by AB in Fig. 1.2.1. After deformation the line element occupies B A , having undergone a translation, extension and rotation. Figure 1.2.1: deformation of a line element The particle that was originally at x has undergone a displacement ) , ( y x u x and the other end of the line element has undergone a displacement ) , ( y x x u x Δ + . By the definition of (small) normal strain, x y x u y x x u AB AB B A x x xx Δ Δ + = = ) , ( ) , ( * ε (1.2.1) In the limit 0 Δ x one has x u x xx = ε (1.2.2) This partial derivative is a displacement gradient , a measure of how rapid the displacement changes through the material, and is the strain at ) , ( y x . Physically, it represents the (approximate) unit change in length of a line element, as indicated in Fig. 1.2.2. x A B A B ) , ( y x u x ) , ( y x x u x Δ + x x x Δ + y * B
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