II Ii il ~~~~~~~~~~~~~~~~~~Part III ANALY~hSIS OF A MEM~ER ELEIENTGoverning Equations for a Deformable Solid 10-1. GENERAL The formulation of the governing equations for the behavior of a deformable solid involves the following three steps: 1. Study of deformation. We analyze the change in shape of a differential volume element due to displacement of the body. The quantities re-quired to specify the deformation (change in shape) are conventionally called strains. This step leads to a set of equations relating the strains and derivatives of the displacement components at a point. Note that the analysis of strain is purely a geomnetricul problem. 2. Study of forces. We visualize the body to consist of a set of differential volume elements. The forces due to the interactions of adjacent volume elements are called internal forces. Also, the internal force per unit area acting on a differential area, say dAj, is defined as the stress vector, 6j. In this step, we analyze the state of stress at a point, that is, we investigate how the stress vector varies with orientation of the area element. We also apply the conditions of static equilibrium to the volume elements. This leads to a set of differential equations (called stress equilibrium equations) which must be satisfied at each point in the interior of the body and a set of algebraic equations (called stress boundary conditions) which must be satisfied at each point on the surface of the body. Note that the study of forces is purely an equilibrium problem. 3. Relate forces and displacements. In this step, we first relate the stress and strain components at a point. The form of these equations depends on the material behavior (linear elastic, nonlinear elastic, inelastic, etc.). Substitution of the strain-displacement relations in the stress-strain relations leads to a set of equations relating the stress componentfs and derivatives of the displacement components. We refer to this system as the stress-displacement relations. 229231 230 GOVERNING EQUATIONS FOR A DEFORMABLE SOLID CHAP. 10 The governing equations for a deformable solid consist of the stress equilib-rium equations, stress-displacement relations, and the stress and displacement boundary conditions. In this chapter, we develop the governing equations for a linearly elastic solid following the steps outlined above. We also extend the variational principles developed in Chapter 7 for an ideal truss to a three-dimensional solid. In Chapter 11, we present St. Venant's theory of torsion-flexure of prismatic members and apply the theory to some simple cross sections. St. Venant's theory provides us with considerable insight as to the nature of the behavior and also as to how we can simplify the corresponding mathematical problem by introducing certain assumptions. The conventional engineering theory of prismatic members is developed in Chapter 12 and a more refined theory for thin walled prismatic members which includes the effect of warping of the cross section is discussed in Chapter 13. In Chapter 14, we develop the engi-neering theory for an arbitrary planar member. Finally, in Chapter 15, we present the engineering theory for an arbitrary space member. 10-2. SUMMATION CONVENTION; CARTESIAN TENSORS Let a and b.represent nth-order column matrices: a -{al, a2 ., an} ti b = {b, b2,..., b} ti-A1) Their scalar (inner) product is defined as aTb = bTa = alb + a2b2 + + a,,b = aibi (a) i=1 To avoid having to write the summation sign, we introduce the convention that when an index is repeated in a term, it is understood the term is summed over the range of the index. According to this convention aibi = aibi (i = 1,2.., n) (10-2) i=I and we write the scalar product as aTb = aibi (10-3) The summation convention allows us to represent operations on multi-dimensional arrays in compact form. It is particularly convenient for formu-lation, i.e., establishing the governing equations. We illustrate its application below. Example 10-1 1. Consider the product of a rectangular matrix, a, and a column vector, x. c = ax a is 11 x n (a) SEC. 10-2. SUMMATION CONVENTION; CARTESIAN TENSORS The typical term is Ci = E ijxj axj (b) j=1 2. Let a, b be square matrices, x a column vector, and f, g scalars defined by f = xTax g = xTbx (c) The matrix form of the product, fg, is fg =(xax)(xTbx) (d) One could expand (d) but it is more convenient to utilize (b) and write (c) as f = aijxixj g = bk(-xkxt (e) Then, fg = aijbkrtxfxjXkX( = Dijk(XiXjXkX-(f) 3. We return to part 1. The inner product of c is a scalar, IH, II = xr(aTa)x (g) Using (b), H = CiCi = aikaitxkxf (h) The outer product is a second-order array, d, d = CCT _aX aT (i) and can be expressed as dij = CiCj = aikajtXkXt (j)= Aijk(XkX( According to the summation convention, dii = dl + d22 + = trace ofd (k) Then, we can write (h) as H = dii = Aiik(XkX( (1) 4. Let ij, eij represent square second-order arrays. The inner product is defined as the sum of the products of corresponding elements: Inner product (aij, eij) = oijeij i j 1let1 + 2 2e2 2 + + 1l2e12 + U2 1e21 +... ajjej j (m)= (TIn order to represent this product as a matrix product, we must convert aij, eij over to one-dimensional arrays. Let b), b ), b ) represent a one-dimensional set of elements associated with an orthogonal reference frame having directions X1 ), X(2), X( ). If the232 233 GOVERNING EQUATIONS FOR A DEFORMABLE SOLID CHAP. 10 corresponding set for a second reference frame is related to the first set by b( 2) = .k bW) (10-4)Ojk = COS (X(2) X))j, k = 1,2,3 we say that the elements of b comprise a first-order cartesian tensor. Noting (5-5), we can write (10-4) as b(2 ) = Rt2b(1) (10-5) and it follows that the set of orthogonal components of a vector are a first-order cartesian tensor. We know that the magnitude of a vector is invariant. Then, the sum of the squares of the elements of a first-order tensor is invariant. bi)bj)-b2b2 (10-6) A second-order cartesian tensor is defined as a set of doubly subscripted elements which transform according to jk j,nJk T (10-7) j. k. ,m. = 1.2,3 An alternate form is b(2) = R 2b(1)(R 12)T (10-8) The transformation (10-8) is orthogonal and the trace, sum of the principal second-order minors, and the determinant are invariant.t 3(t2 = (Il) p(22 = pi) (10-9) fliZ = fly) where P = hjj l3 = ibl A2 = bl + b23 + bil b13 2= b2 l b1 2 ±b3 b22 b3333b22 2 1 Ib3 1 In the cases we encounter, b will be symmetrical. 10-3. ANALYSIS OF