Tham khảo tài liệu 'finite element analysis - thermomechanics of solids part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5 Mechanical Equilibrium and the Principle of Virtual Work TRACTION AND STRESS Cauchy Stress Consider a differential tetrahedron enclosing the point x in the deformed configuration. The area of the inclined face is dS and dSị is the area of the face whose exterior normal vector is -ei. Simple vector analysis serves to derive that n dSjdS see Exercise 1 in Chapter 1 . Next let dP denote the force on a surface element dS and let dP i denote the force on area dS. The traction vector is introduced by T dP dS. As the tetrahedron shrinks to a point the contribution of volume forces such as inertia decays faster than surface forces. Balance of forces on the tetrahedron now requires that dP y dp. J J i The traction vector acting on the inclined face is defined by dP T dS from which v dP T J x ds i dP dSt dSi dS . i i -Ỵ i 73 2003 by CRC CRC Press LLC 74 Finite Element Analysis Thermomechanics of Solids FIGURE Equilibrium of a tetrahedron. with T ij dP i dsi It is readily seen that Tj can be interpreted as the intensity of the force acting in the j direction on the facet pointing in the -i direction and is recognized as the if entry of the Cauchy stress T. In matrix-vector notation the stress-traction relation is written as T T Tn. The next section will show that T is symmetric by virtue of the balance of angular momentum. Equation implies that TT is a tensor thus it follows that T is a tensor. To visualize T consider a differential cube. Positive stresses are shown on faces pointing in positive directions as shown in Figure . In traditional depictions the stresses on the back faces are represented by arrows pointing in negative directions. However this depiction can be confusing the arrows actually represent the directions of the traction components. Consider the one-dimensional bar in Figure . The traction vector te1 acts at x L while the traction vector -te1 acts at x 0. At x L the corresponding stress is t11 te1 -e1 t. At x
