- Using material symmetry through 180 rotations about each of the three coordinate axes, explicitly show the reduction of the elastic stiffness matrix to nine independent components for orthotropic
- A transversely isotropic material with an x3-axis of symmetry was specified by the elasticity matrix given in Eq. (11.2.12). Under an arbitrary q rotation about the x3-axis given by relation
- Determine the displacement field for the beam problem in Exercise 8.2. To determine the rigid body motion terms, choose fixity conditions:Note that with our approximate Saint-Veran solution, we
- The solution to the illustrated two-dimensional cantilever beam problem is proposed using the Airy stress function ∅ = C1x2 + C2x2y + C3y3 + C4y5 þ C5x2y3, where Ci are constants. First
- The following stress function: 13 xy = Cxy + C + C3= 6 + C4- 5 tys -+C5-9 + C6 20
- The cantilever beam shown in the figure is subjected to a distributed shear stress τ, ox/l on the upper face. The following Airy stress function is proposed to solve this problem:
- For the pure beam bending problem solved in Example 8.2, calculate and plot the in-plane displacement field given by relation (8.1.22)2. Use the vector distribution plotting scheme illustrated in
- For the beam problem in Example 8.3, the boundary conditions required that the resultant normal force vanish at each end (x =± l). Show, however, that the normal stress on each end is not zero, and
- Develop the general displacement solution (8.3.9) for the axisymmetric case.Equation 8.3.9 Ur = Ug = v) 1/2 [ - ( + 1 ) a E r + A sino + B cose 4r0 E -a + 2(1-v)a3r log r- (1+v)a3r +
- Consider the axisymmetric problem of an annular disk with a fixed inner radius and loaded with uniform shear stress τ, over the outer radius. Using the Airy stress function term a4θ, show that
- For the axisymmetric problem of Example 8.6, explicitly develop the displacement solution given by relation (8.4.5).Data from example 8.6Equation 8.4.5 The first example to be investigated involves a
- Consider the annular ring loaded with a sinusoidal distributed pressure as illustrated. Show that this problem can be solved using the Airy function = a log r+ ar + (a2r + a22r+ + a23r + a24) cos 20
- A long composite cylinder is subjected to the external pressure loading as shown in the following figure. Assuming idealized perfect bonding between the two materials, the normal stress and
- Numerically generate and plot the fields of stress (σr, σθ) and displacement (ur) within the composite cylinder of Exercise 8.20 for the specific case with material (1) = steel and material (2) =
- Resolve Exercise 8.20 for the case where material: (1) Is rigid and material. (2) Is elastic with modulus E and Poisson’s ratio v.Data from exercise 8.20A long composite cylinder is subjected to
- For the case of a thin-walled tube under internal pressure, verify that the general solution for the hoop stress (8.4.3)2 will reduce to the strength of materials relation (see Appendix D, Section
- Consider the cut-and-weld problem in which a small wedge of angle a is removed from an annular ring as shown in the figure. The ring is then to be joined back together (welded) at the cut section.
- Using superposition of the stress field (8.4.15), develop solution (8.4.18) for the equal but opposite biaxial loading on a stress-free hole shown in Fig. 8.15A. Also justify that this solution will
- An elastic circular plug of radius a with properties E1 and v1 is perfectly bonded and embedded in an infinite elastic medium with properties E2 and v2. The composite is loaded with a uniform
- An infinite elastic medium contains a perfectly bonded rigid plug and is loaded with uniform far-field biaxial stress T as shown. Using the results from Exercise 8.22, determine the stress and
- Show that the stress function:gives the solution to the problem of an elastic half-space loaded by a uniformly distributed shear over the free surface (x≤ 0), as shown in the figure. Identify
- Show that the Flamant solution given by Eqs. (8.4.31) and (8.4.32) can also be used to solve the more general wedge problem as shown.Equation 8.4.31Equation 8.4.32 X y Y 0 X
- The in-plane rotation in polar coordinates is given by relation Using this form with the Flamant displacement solution (8.4.43), show thatand thus conclude that ωz is an even function of θ that
- For the Flamant problem with only normal loading, explore the strain energy in a semicircular area of radius R centered at the loading point. Refer to the discussion in Section 5.8, and use relation
- Determine the stress field solution (8.4.47) for the problem of a half-space under a concentrated surface moment as shown in Fig. 8.23. It is recommended to use the superposition and limiting process
- Show that the problem of a half-space carrying a concentrated surface moment (see Fig. 8.23) can also be solved using the Airy function form∅ = a4θ+ b24 sin 2θ.Fig 8.23 d P M y
- Working in polar coordinates, show that an Airy stress function of the form ∅ = a2r2 + a21r2 cos 2θ (where a2 and a21 are constants) solves the illustrated problem of a shear loaded wedge
- For the problem of a half-space under uniform normal loading as shown in Fig. 8.24, show that the maximum shear stress can be expressed by:Plot the distribution of lines of constant maximum shear
- Generalize the integral superposition methods used in the examples shown in Section 8.4.9and Exercise 8.40. In particular, show that the stress solution for a half-space carrying general normal and
- Using the formulation and boundary condition results of the thin notch crack problem shown in Fig. 8.29, explicitly develop the stress components given by relations (8.4.56) and (8.4.57).Fig
- Consider the crack problem shown for the anti-plane strain case with u = v = 0, w = w(x,y). From Section 7.4, the governing equation for the unknown displacement component with zero body force was
- For the anti-plane stain crack problem solved in Exercise 8.44, plot contours of the displacement and stress fields.Data from exercise 8.44Consider the crack problem shown for the anti-plane strain
- For the mode III crack problem in Exercise 8.44, explore the strain energy in a circular area of radius R centered at the crack tip. Refer to the discussion in Section 5.8 and use relation (6.1.10)
- Using strength of materials theory (see Appendix D), the bending stress sq for curved beams is given by σθ =–M(r– B)/[rA(R– B)], where A = b –a, B = (b– a)/log(b/a), R = (a + b)/2. For
- For the disk under diametrical compression (Fig. 8.35), plot the distribution of the two normal stresses σx and σy along the horizontal diameter (y = 0, –R< x < R ).Fig 8.35 P 0 T 1 02 P X
- The behavior of granular materials has often been studied using photo elastic models of circular particles as shown in the following figure. This provides the full-field distribution of local contact
- Solve the rotating disk problem of Example 8.11 for the case of an annular disk with inner radius a and outer radius b being stress free. Explicitly show that for the case b >> a, the maximum stress
- Assume that we are dealing with a two-dimensional deformation problem in the x,y-plane where all field variables depend only on x and y. For the following strain fields (A and B are constants),
- Show that the general plane strain edge dislocation problem shown in Fig. 15.3 can be solved using methods of Chapter 10 with the two complex potentials:where b = bx + iby. In particular, verify the
- Justify that the edge dislocation solution (15.1.2) provides the required multivalued behavior for the displacement field. Explicitly develop the resulting stress fields given by (15.1.3) and
- Show that the screw dislocation displacement field (15.1.5) gives the stresses (15.1.6) and (15.1.7).Equation 15.1.5Equation 15.1.6Equation 15.1.7 u = v=0 b 2 W = tan -1 X
- For the edge dislocation model, consider a cylinder of finite radius with axis along the dislocation line (z-axis). Show that although the stress solution gives rise to tractions on this cylindrical
- The stress field (15.1.7) for the screw dislocation produces no tangential or normal forces on a cylinder of finite radius with axis along the dislocation line (z-axis). However, show that if the
- Show that the strain energy (per unit length) associated with the screw dislocation model of Example 15.2 is given by:where Ro is the outer radius of the crystal and Rc is the core radius of the
- Using similar notation as Exercise 15.6, show that the strain energy associated with the edge dislocation model of Example 15.1 can be expressed by:Note that this energy is larger than the value
- For the Kelvin state as considered in Example 15.4, explicitly justify the displacement and stress results given in relations (15.2.8) and (15.2.10).Data from example 15.4Equation 15.2.8Equation
- Verify that the displacements and stresses for the center of compression are given by (15.2.21)and (15.2.22).Equation 15.2.21Equation 15.2.22 u Xi 2R 1 (3xxj - dij R R
- A fiber discontinuity is to be modeled using a line of centers of dilatation along the x1-axis from 0 to a. Show that the displacement field for this problem is given by:Fig 5.11 where R = 1 - - -
- For the isotropic self-consistent crack distribution case in Example 15.12, show that for the case v = 0.5, relation (15.3.4)3 reduces to:Verify the total loss of moduli at ε = 9/16. Using these
- Develop the compatibility relations for couple-stress theory given by (15.4.12). Next, using the constitutive relations, eliminate the strains and rotations, and express these relations in terms of
- Explicitly justify that the stress-stress function relations (15.4.14) are a self-equilibrated form.Equation 15.4.14 Ox = Txy mxz 22 2 2 22 32 2 myz Tyx 22 22 + 2 22 32 + 20
- For the couple-stress theory, show that the two stress functions satisfy: = 0= +-zd'0=+
- Using the general stress relations (15.4.25) for the stress concentration problem of Example 15.13, show that the circumferential stress on the boundary of the hole is given by:Verify that this
- Starting with the general relations (15.5.6), verify that the two-dimensional plane stress constitutive equations for elastic materials with voids are given by (15.5.9).Equation 15.5.6Equation 15.5.9
- For elastic materials with voids, using the single strain-compatibility equation, develop the stress and stress function compatibility forms (15.5.10) and (15.5.11).Equation 15.5.10Equation 15.5.11
- Compare the hoop stress σθ(r,π/2) predictions from elasticity with voids given by relation (15.5.18) with the corresponding results from classical theory. Choosing N = 1/2 and L = 2, for the
- For the doublet mechanics Flamant solution in Example 15.15, develop contour plots (similar to Fig. 15.22) for the microstresses p1 and p2. Are there zones where these microstresses are tensile?Data
- Consider the gradient elasticity problem under a one-dimensional deformation field of the form u = u(x), v = w = 0. Using constitutive form (15.7.3), determine the stress components. Next show that
- Starting with the given displacement form (15.7.8), explicitly justify relations (15.7.9) – (15.7.12) in the gradient dislocation Example 15.16Equation 15.7.8Equation 15.7.9Equation 15.7.12Data
- Using integral tables verify that the gradient Flamant stress integral solution in Example 15.17 reduces to the classical elasticity form as per equations (15.7.20).Data from example 15.17Equation
- Starting with the general linear form (16.2.1), verify the interpolation relations (16.2.4) and (16.2.5).Equation 16.2.1Equation 16.2.4Equation 16.2.5 u(x,y) = C+Ccx + c3y
- For the constant strain triangular element, show that the stiffness matrix is given by (16.3.16).Equation 16.3.16 [K] = he 4Ae [CH+YC66 BIYI C12+B1Y1C66 YC2+ BC66 818 C11 +Y1Y2C66 B2Y1 C12 +1Y2C66
- For the case of element-wise constant body forces, verify that the body force vector is given by relation (16.3.17) for the linear triangular element.Equation 16.3.17 he Ae {F} = !e {F&Fy E FEFy}! 3
- Verify boundary relation (16.3.19) for the linear triangular element with constant boundary tractions Tnx and Tny.Equation 16.3.19 ~ LT {F} a 7 dS = he he T12 T12 VT VT VT VT 437" V3T y dS he L12 2
- For Example 16.1, show that the element stiffness equations for the isotropic case are given by relations (16.4.1) and (16.4.2).Data from example 16.1Equation 16.4.1Equation 16.4.2 Consider the plane
- Verify the nodal displacement solution given by (16.4.5) in Example 16.1.Equation 16.4.5Data from example 16.1 U V U3 V3 0.492 0.081 0.441 -0.030 Tx 10- m
- Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the stress concentration problem under biaxial loading given in Exercise 8.25. Compare the stress concentration factor from
- Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the stress concentration problem under shear loading given in Exercise 8.26. Compare the stress concentration factor from
- Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the curve beam problem shown in Fig. 8.32. At the fixed section, compare numerical stress results (σx)with analytical
- Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the torsion of a cylinder of circular section with circular keyway as shown in Exercise 9.23. Verify the result of Exercise
- Verify the traction relation (16.6.7) for the plane strain case.Equation 16.6.7 Pij = Tikink = 4n(1 - v)r [(1-2 -dij+rinj-rjni+2 r
- From strain energy arguments in Section 6.1, it was found that ∂σij/∂ekl = ∂σkl/σeij. Show that these results imply that Cij = Cji, therefore justifying that only 21 independent elastic
- Verify the inequality restrictions on the elastic moduli for orthotropic, transversely isotropic, cubic and isotropic materials given by Table 11.1.Table 11.1 Table 11.1 Positive definite
- For the orthotropic case, show that by using arguments of a positive definite strain energy function, v2ij i/Ej). Next, using typical values for E1 and E2 from Table 11.2, justify that this theory
- Similar to Example 11.1, consider the hydrostatic compression of a material with cubic symmetry. Determine the general expressions for the strains. We now specify that the material is aluminum with
- For the torsion of cylinders discussed in Section 11.4, show that with σx = σy = σz = τ xy = 0, the compatibility equations yield:Data from section 11.4As our first example, consider the torsion
- In terms of the stress function Ψ, the torsion problem was governed by Eq. (11.4.9):Equation 11.4.9 S44xx - 2S45xy + S554 yy = -2 Show that the homogeneous counterpart of this equation may be
- Explicitly justify relationships (11.5.3) between the compliances of the plane stress and plane strain theories.Equation 11.5.3 B11 B22 B66 = S11 S33-S2 $33 S22 S33-S23 S33 S66 S33 - S36 S33 B12 B16
- Investigate case 2 (μ1 = μ2) in Eq. (11.5.10) and determine the general form of the Airy stress function. Show that this case is actually an isotropic formulation.Equation 11.5.10 Case 1: = a +i
- Determine the roots of the characteristic equation (11.5.7) for S-Glass/Epoxy material with properties given in Table 11.2. Justify that they are purely imaginary.Equation 11.5.7Table 11.2 S112S16M+
- Recall that for the plane anisotropic problem, the Airy stress function was found to be: o=F1(21) + F1(21) + F2(22) + F2(22) where Zi = x + y and z2 = x + 2y. Explicitly show that the in-plane
- For the plane stress case, in terms of the two complex potentials ∅1 and ∅2, compute the two in-plane displacements u and v and thus justify relations (11.5.14).Equation 11.5.14 u(x,y) = 2Rep1
- Determine the polar coordinate stresses and displacements in terms of the complex potentials ∅1 and ∅2, as given by Eqs. (11.5.16) and (11.5.17).Equation 11.5.16Equation 11.5.17 2Re [(sinu cose)
- For the plane problem with an orthotropic material, show that the characteristic Eq. (11.5.7) reduces to the quadratic equation in μ2:Justify the isotropic case where β1,2 = 1. Finally, determine
- Consider an anisotropic monoclinic material symmetric about the x,y-plane (see Fig. 11.2) and subject to an anti-plane deformation specified by u = v = 0, w = w(x,y). Show that in the absence of body
- For Example, 11.5, consider the case of only a normal boundary load (X = 0), and assume that the material is orthotropic with μi = iβi (see Exercise 11.15). Show that the resulting stress field is
- Consider the case of the pressurized circular hole in an anisotropic sheet. Using orthotropic material properties given in Table 11.2 for Carbon/Epoxy, compute and plot the boundary hoop stress σθ
- Investigate the case of a circular hole of radius a in Example 11.7. Use orthotropic material properties given in Table 11.2 for Carbon/Epoxy with the 1-axis along the direction of loading. Compute
- Consider the elliptical hole problem in Example 11.7. By letting a→0, determine the stress field for the case where the hole reduces to a line crack of length 2b. Demonstrate the nature of the
- The potentials: were proposed to solve the plane extension of an anisotropic panel containing a crack of length 2a (see Fig. 11.12). Recall that the constants A1 and A2 correspond to the uniform
- Construct a contour plot of the crack tip stress component σy from solution (11.6.7). This result could be compared with the equivalent isotropic problem from Exercise 10.24.Equation 11.6.7Data
- Consider the case of a crack problem in an anisotropic monoclinic material under anti-plane deformation as described in Exercise 11.16. Following relation (11.6.6), choose the complex potential form
- Explicitly develop the governing Navier equation (11.7.5) for the polar orthotropic problem.Verify that its solution is given by (11.7.6) and show how this leads to the stress solution (11.7.7).
- For the spherically orthotropic problem, justify that Hooke’s law (11.7.11) can be inverted into form (11.7.13) under the relations (11.7.14).Equation 11.7.11Equation 11.7.13Equation 11.7.14 eR ed
- Under the stated symmetry conditions in Section 11.7.2, explicitly show that in the absence of body forces the general equilibrium equations reduce to forms (11.7.17) and (11.7.18).Verify the general
- For the rotating disk problem given previously in Example 8.11, the governing equilibrium equation was given by (8.4.74). Since this equation is also valid for anisotropic materials, consider the
- Using the results from Exercise 11.27, show that the stresses in a rotating solid circular polar-orthotropic disk of radius a with boundary condition σr (a) = 0 are given by:Data from exercise
- Consider the cantilever beam problem shown previously in Exercise 8.2 with no axial force (N =0). Assume a plane stress anisotropic model given by Hooke’s law (11.5.1) and governed by the Airy
- Using the assumption for isotropic materials that a temperature change produces isotropic thermal strains of the form α(T– To)δij, develop relations (12.1.6).Equation 12.1.6 ij = kkoij + 2e -
- For the general three-dimensional thermoelastic problem with no body forces, explicitly develop the Beltrami-Michell compatibility equations: Tij kk + 1 -kk,ij + (1 + v) Ea 1+v Tij 1+v + 1 + OUTk 1-v

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