Section 1.2 Solid Mechanics Part II Kelly 91.2 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 xΔemanating from position ),(yxand lying in the x- direction, denoted by ABin Fig. 1.2.1. After deformation the line element occupies BA′′, having undergone a translation, extension and rotation. Figure 1.2.1: deformation of a line element The particle that was originally at xhas undergone a displacement ),(yxuxand the other end of the line element has undergone a displacement ),(yxxuxΔ+. By the definition of (small) normal strain, xyxuyxxuABABBAxxxxΔ−Δ+=−′=),(),(*ε(1.2.1) In the limit 0→Δxone has xuxxx∂∂=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),(yx. Physically, it represents the (approximate) unit change in length of a line element, as indicated in Fig. 1.2.2. x••A••BA′B′),(yxux),(yxxuxΔ+xxxΔ+y*B
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