I I I i il Part III ANALY hSIS MEM ER OF A ELEIENT Governing 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 229 230 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 SEC 10 2 231 SUMMATION CONVENTION CARTESIAN TENSORS The typical term is Ci E ijxj b axj j 1 2 Let a b be square matrices x a column vector and f g scalars defined by f xTax g xTbx c fg xax xTbx d The matrix form of the product fg is One could expand d but it is more convenient to utilize b and write c as f aijxixj e g bk xkxt Then fg aijbkrtxfxjXkX f Dijk XiXjXkX 3 We return to part 1 The inner product of c is a scalar IH 10 2 SUMMATION CONVENTION CARTESIAN TENSORS a al a2 II xr aTa x g H CiCi aikaitxkxf h Using b Let a and b represent nth order column matrices an ti ti A1 b b b2 b The outer product is a second order array d i aT d CCT aX Their scalar inner product is defined as and can be expressed as dij CiCj aikajtXkXt aTb bTa alb a2 b2 aibi a b a j Aijk XkX 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 According to the summation convention dii dl d22 k trace ofd Then we can write h as H 1 dii Aiik XkX 10 2 i I 4 Let ij eij represent square second order arrays The inner product is defined as the sum of the products of corresponding elements 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 oijeij Inner product aij eij i j T 1 let 1 2 2 e2 2 1 l2 e12 U2 1 e21 ajjej j m In order to represent this product as a matrix product we must convert aij eij over to one dimensional arrays 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 Let b b b represent a one dimensional set of elements associated with an orthogonal reference frame having directions X1 X 2 X If the 232 GOVERNING EQUATIONS FOR A DEFORMABLE SOLID CHAP 10 corresponding set for a second reference frame is related to the first set by b 2 bW k 10 4 COS X 2 X Ojk SEC 10 3 ANALYSIS OF DEFORMATION CARTESIAN STRAINS 233 Excluding rigid body motion the displacement from the initial undeformed position will be small for a solid and it is reasonable to take the initial cartesian coordinates xj as the independent variables This is known as the Lagrange j k 1 2 3 X3 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 Undeformed dp 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 P Deformed 10 6 bi bj b2b2 A second order cartesian tensor is defined as a set of doubly subscripted elements which transform according to jk j nJk j k m 1 2 3 173 13 10 7 T 7 I An alternate form is b 2 R o 2b 1 R 12 T I 771 10 9 approach Also to simplify the derivation we work with cartesian components for ft Then i l xj lujYj 10 11 P xj l3 ibl b2 l b1 2 b22 b b3 22 2 b23 b3 31 bil Ib 3 1 b1 3 33 We consider a differential line element at P represented …