Stress Balance Principles 03 The Cauchy Stress （03柯西应力应力平衡原则）.pdfVIP

Section 3.3 3.3 The Cauchy Stress Tensor 3.3.1 The Traction Vector The traction vector was introduced in Part I, §3.3. To recall, it is the limiting value of the ratio of force over area; for Force ΔF acting on a surface element of area ΔS , it is ΔF t (n) lim (3.3.1) ΔS →0 ΔS and n denotes the normal to the surface element. An infinite number of traction vectors act at a point, each acting on different surfaces through the point, defined by different normals. 3.3.2 Cauchy’s Lemma Cauchy’s lemma states that traction vectors acting on opposite sides of a surface are equal and opposite1. This can be expressed in vector form: t (n) −t (−n) Cauchy’s Lemma (3.3.2) This can be proved by applying the principle of linear momentum to a collection of particles of mass Δm instantaneously occupying a small box with parallel surfaces of area Δs , thickness δ and volume Δv δΔs , Fig. 3.3.1. The resultant surface force acting on this matter is t (n) Δs +t (−n) Δs . t (n) n Δs thickness δ −n t (−n) Figure 3.3.1: traction acting on a small portion of material particles The total linear momentum of the matter is ∫ ρvdv ∫ vdm . By the mean value ΔV Δm theorem (see Appendix A to Chapter 1, §1.B.1), this equals vΔm , whe