Mechanics and Design Chapter 2. Stresses and Strains Byeng D. Youn System Health & Risk Management Laboratory Department of Mechanical & Aerospace Engineering Seoul National University Seoul National University CONTENTS 1 Traction or Stress Vector 2 Coordinate Transformation of Stress Tensors 3 Principal Axis 4 Example 2019/1/4 Seoul National University - 2 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction Vector Consider a surface element, ∆ S , of either the bounding surface of the body or the fictitious internal surface of the body as shown in Fig. 2.1. Assume that ∆ S contains the point. The traction vector, t, is defined by Δf t = lim (2-1) ∆→S0∆S Fig. 2.1 Definition of surface traction 2019/1/4 Seoul National University - 3 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction Vector (Continued) It is assumed that Δ f and ∆ S approach zero but the fraction, in general, approaches a finite limit. An even stronger hypothesis is made about the limit approached at Q by the surface force per unit area. First, consider several different surfaces passing through Q all having the same normal n at Q as shown in Fig. 2.2. Fig. 2.2 Traction vector t and vectors at Q Then the tractions on S , S ′ and S ′′ are the same. That is, the traction is independent of the surface chosen so long as they all have the same normal. 2019/1/4 Seoul National University - 4 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Stress vectors on three coordinate plane Let the traction vectors on planes perpendicular to the coordinate axes be t(1), t(2), and t(3) as shown in Fig. 2.3. Then the stress vector at that point on any other plane inclined arbitrarily to the coordinate axes can be expressed in terms of t(1), t(2), and t(3). (1) Note that the vector t acts on the positive x1-side of the element. The stress vector on the negative side will be denoted by - t(1). Fig. 2.3 Traction vectors on three planes perpendicular to coordinate axes 2019/1/4 Seoul National University - 5 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Stress components The traction vectors on planes perpendicular to the coordinate axes, x 1 , x 2 and (1) (2) (3) x3 are t , t , and t . The three vectors can be decomposed into the directions of coordinate axes as (1) t=++TTT11 i 12 jk 13 (2) (2-2) t=++TTT21 i 22 jk 23 (3) t=++TTT31 i 32 jk 33 The nine rectangular components T ij are called the stress components. Here, the first subscript represents the “plane” and the second subscript represents the “direction”. Tij Direction Plane Fig. 2.4 Stress components 2019/1/4 Seoul National University - 6 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Sign convention A stress component is positive when it acts in the positive direction of the coordinate axes and on a plane whose outer normal points in one of the positive coordinate directions. Fig. 2.5 Sign convention of stress components Sign convention The stress state at a point Q is uniquely determined by the tensor T which is represented by TTT11 12 13 = T TTT21 22 23 (2-3) TTT31 32 33 2019/1/4 Seoul National University - 7 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction vector on an arbitrary plane: The Cauchy tetrahedron When the stress at a point O is given, then the traction on a surface passing the point Q is uniquely determined. Consider a tetrahedron as shown in Fig. 2.6. The orientation of the oblique plane ABC is arbitrary. Let the surface normal of ∆ ABC be n and the line ON is perpendicular to ∆ ABC . The components of the unit normal vector n are the direction cosine as n1 =cos( ∠ AON ) n2 =cos( ∠ BON ) (2-4) n3 =cos( ∠ CON ) Fig. 2.6 Geometry of tetrahedron 2019/1/4 Seoul National University - 8 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction vector on an arbitrary plane: The Cauchy tetrahedron (Conti.) If we let ON = h , then h = OA⋅ n1 = OB ⋅ n2 = OC ⋅ n3 (2-5) Let the area of ∆ ABC , ∆ OBC , ∆ OCA & ∆ OAB be ∆ S , ∆ S 1, ∆ S 2 & ∆ S 3 respectively. Then the volume of the tetrahedron, ∆ V , can be obtained by 1 1 1 1 ∆V = h ⋅ ∆S = OA⋅ ∆S = OB ⋅ ∆S = OC ⋅ ∆S (2-6) 3 3 1 3 2 3 3 From this we get, h ∆S =∆⋅ S =∆⋅ Sn 11OA h ∆S =∆⋅ S =∆⋅ Sn (2-7) 22OB h ∆S =∆⋅ S =∆⋅ Sn 33OC 2019/1/4 Seoul National University - 9 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction vector on an arbitrary plane: The Cauchy tetrahedron (Conti.) Now consider the balance of the force on OABC as shown in Fig. 2.7. The equation expressing the equilibrium for the tetrahedron becomes (n )* * * (1)* (2)* (3)* t∆+∆−∆−∆−∆=Sρ bt VS123 t S t S0 (2-8) Here the subscript * indicates the average quantity. Substituting for ∆ V , ∆ S 1 , ∆ S 2 and ∆ S 3 , and dividing through by ∆ S , we get 1 ∗ t(n )*+hρ * bt =++ (1)* nnn t (2)* t (3)* (2-9) 3 123 Fig. 2.7 Forces on tetrahedron 2019/1/4 Seoul National University - 10 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Traction vector on an arbitrary plane: The Cauchy tetrahedron (Conti.) Now let h approaches zero, then the term containing the body force approaches zero, while the vectors in the other terms approach the vectors at the point O. The result is in the limit (n ) (1) (2) (3) (k) tt=++=nnn123 t t t nk (2-10) The important equation permits us to determine the traction t(n) at a point acting on an arbitrary plane through the point, when we know the tractions on only three mutually perpendicular planes through the point. The equation (2-10) is a vector equation, and it can be rewritten by (n) (1) (2) (3) t1 = t1 n1 + t1 n2 + t1 n3 (n) (1) (2) (3) t2 = t2 n1 + t2 n2 + t2 n3 (2-11) (n) (1) (2) (3) t3 = t3 n1 + t3 n2 + t3 n3 Comparing these with eq. (2-2), we get (n) t1 = T11n1 + T21n2 + T31n3 = Tk1nk ()n =++= t2 Tn 12 1 Tn 22 2 Tn 32 3 Tnkk 2 (2-12) (n) t3 = T13n1 + T23n2 + T33n3 = Tk 3nk 2019/1/4 Seoul National University - 11 - Chapter 2 : Stresses and Strains Traction or Stress Vector; Stress Components Cauchy stress tensor Or for simplicity, we put (n) • in indicial notation ti = Tjin j • in matrix notation t(n)= Tn T (2-13) • in dyadic notation t()n =⋅= nT TT ⋅ n From the derivation of this section, it can be shown that the relation (2-13) also holds for fluid mechanics. Tij : Cauchy stress tensor. This stress tensor is the linear vector function which associates with n the traction vector t(n) 2019/1/4 Seoul National University - 12 - Chapter 2 : Stresses and Strains Coordinate Transformation of Stress Tensors Transformation matrix As we discussed in the previous chapter, stress tensor follows the tensor coordinate transformation rule. That is, let x and x be the two coordinate systems and A be a transformation matrix as v= Av or v= AvT Then the stress tensor T transforms to T as T = AT TA We may consider the stress tensor transformation in two dimensional case. Let the angle between x axis and x axis is θ . Then the transformation matrix A becomes aa11cosθθ− sin =12 = A 22 aa12sinθθ cos 2019/1/4 Seoul National University - 13 - Chapter 2 : Stresses and Strains Coordinate Transformation of Stress Tensors Transformation matrix (Continued) The stress T transforms to T according to the following T cosθθ sin TT11 12 cos θ− sin θ T= A TA = −sinθθ cos TT21 22 sin θ cos θ Evaluating the equation, we get 2 TT11=++ 11 cosθ 2 T21 sin θ cos θ T22 cos θθ sin 22 T12=−+−( TT 22 11 )sincosθθ T12 (cos θ sin θ ) 22 TT22=+− 11 sinθ T22 cos θ 2 T12 sincos θθ By using double angle trigonometry, we can get (TT+− )( TT ) TT, =±±11 22 11 22 cos2θθT sin 2 11 22 22 12 ()TT− TT=−+11 22 sin 2θθ cos2 12 2 12 2019/1/4 Seoul National University - 14 - Chapter 2 : Stresses and Strains Coordinate Transformation of Stress Tensors Transformation matrix (Continued) Now we recognize the last two equations are the same as the ones we derived for Mohr circle, (eq 4-25) of Crandall’s book. For the two dimensional stress state, Mohr circle may be convenient because we recognize the stress transformation more intuitively. However, for 3D stress state and computation, it is customary to use the tensor equation directly to calculate the stress components in transformed coordinate system. Fig. 2.8 Mohr’s circle 2019/1/4 Seoul National University - 15 - Chapter 2 : Stresses and Strains Principal Axes of Stress, Principal Stress, etc. Characteristics of the principal stress (1) When we consider the stress tensor T as a transformation, then there exist a line which is transformed onto itself by T. (2) There are three planes where the traction of the plane is in the direction of the normal vector, i.e. tn()n / / or tn()n = λ (2-14) Definitions • Principal axes • Principal plane • Principal stress 2019/1/4 Seoul National University - 16 - Chapter 2 : Stresses and Strains Principal Axes of Stress, Principal Stress, etc.