Typical Solution Trails

As a general conclusion from the discussion so far, it may be stated that the governing equa-tions of elasticity for an anisotropic body may be expressed using two different approaches.

First, one may employ the approach that is generally referred to as “Differential Equilibrium”,

“Newton’s laws” or “Vector Mechanics”, which all stand for the creation of a set of equa-tions that will assure the fulfillment of the differential equilibrium equaequa-tions, the differential compatibility equations, and the local (natural and geometrical) boundary conditions. Alterna-tively, one may employ the variational analysis of energy based functionals discussed in S.1.5, S.1.6.4 to either derive the governing equations and boundary conditions, or to directly derive a specific solution.

Strictly speaking, the two methods described above are equivalent. One may claim that the former approach that utilizes equilibrium considerations is more “physical oriented”, while the concept of the latter is less intuitive in that sense. Generally speaking, while both approaches are suitable to accommodate closed-form or numerical solutions, variational analysis is some-times superior as it is capable of supplying, in addition to a specific solution, the governing equations, and sometimes part of the boundary conditions (see Remark 1.9), as well.

In what follows, we shall review the basic solution trails that are typically encountered within the theory of anisotropic elasticity.

1.6.5.1 Solution Trail “A”: Deformation Hypothesis

In view of S.1.6.3, Trail “A” consists of prescribing the expressions for the deformation com-ponents (or their measures) as presented by Fig. 1.15. Subsequently, the strain comcom-ponents

Figure 1.15:Solution Trail “A”.

are determined, see S.1.1, and the stress components are derived using the constitutive rela-tions thereafter, see S.2.1. The stress components enable the construction of the equilibrium equations, see S.1.3.2, and the natural boundary conditions, (1.185a), while the geometrical boundary conditions, (1.185b), are directly expressed in terms of the assumed deformation.

1.6.5.2 Solution Trail “B”: Stress/Strain Hypothesis

Trail “B” consists of prescribing the expressions for the stress or the strain components as presented by Fig. 1.16. Since the strain components may always be obtained by the stress components via the constitutive relations, see S.2.1, and vice versa, the following procedure is identical to both cases. Subsequently, the strain components enable us to construct the compati-bility equations, see S.1.2.1, while the stress components enable the construction of the equilib-rium equations, see S.1.3.2, and the natural boundary conditions (1.185a). The displacements

are then derived by strain integration (see S.1.2) and so, the geometrical boundary conditions (1.185b) may also be expressed.

Figure 1.16:Solution Trail “B”.

1.6.5.3 Solution Trail “C”: Stress Functions Hypothesis

Trail “C” consists of prescribing the stress functions expressions as presented by Fig. 1.17.

In this case, the stress components are determined according to a specific kind of the stress

Figure 1.17:Solution Trail “C”.

function/s employed and therefore satisfy equilibrium by definition. The strain components are then derived using the constitutive relations, see S.2.1. The strain components enable us to construct the compatibility equations, see S.1.2.1, while the stress components enable the construction of the natural boundary conditions (1.185a). In addition, the displacements are derived by strain integration (see S.1.2) so that the geometrical boundary conditions (1.185b) may also be expressed.

1.6.5.4 Solution Trails “D” and “E”: Energy Considerations

In this section, a generic methodology that employs energy theorems and provides the actual response without explicit formulation of Euler’s equations will be discussed.

In trail “D” we apply the Theorem of Minimum Potential Energy. In such a case, one is required to prescribe the displacements (in such a way that the geometrical boundary conditions are satisfied), see Fig. 1.18(a), and to minimize V of (1.124).

In trail “E” we apply the Theorem of Minimum Complementary Energy, and one is required to prescribe the stresses (in such a way that the equilibrium and the natural boundary conditions are satisfied), see Fig. 1.18(b), and to minimize V^∗of (1.128). In the latter case, displacements may be obtained by suitable integration, see S.1.2. The variational minimization process takes care of the missing part, i.e. the equilibrium and natural boundary conditions in the first case, and the compatibility conditions in the second case.

(a)Solution Trail “D”. (b)Solution Trail “E”.

Figure 1.18:Energy based solution trails “D” and “E” .

Within the methodologies in this class of trails, we shall provide some more attention to Ritz’s Method and confine the discussion to the one-dimensional case where we seek an ap-proximate solution of a governing equation that may be extracted from a functional, J.

To execute Ritz’s Method for a generic J> 0 functional, see e.g. (1.134), one first needs to adopt a relatively complete set of functionsηi (see Remark 1.10), each of which satisfies the boundary conditions (1.135). The approximate solution (the exact one is denoted by y^∗(x)) is therefore expressed as

y^∗_N(x) = y(x) +

∑

^Ni=1a_iηi (1.187) where y(x) is an auxiliary function and ai are unknown (real) coefficients (which actually depend on N) that should be determined by the method. One may therefore examine the value of J when N terms are employed in (1.187), which will be denoted JN. Subsequently, one may substitute y^∗_N(x) in the functional of (1.124) and express it as JN(a1,a2,...,aN) = J(y^∗N).

Minimization of JN with respect to{ai} yields the following linear system of N unknowns and equations:

∂JN

∂ai = 0, i= 1,...,N. (1.188)

Each approximation level includes the previous one, and a non-increasing series min JN ≤ min J_N−1≤ ··· ≤ min J1is obtained. Thus, this approximation yields an upper bound for min J.

Since F is a continuous function, the condition F(x,y^∗N,y^∗N,x) − F(x,y^∗,y^∗_,x) <ε1may be satisfied for any (small)ε1by a appropriate value of N, and since J=^x^x₀¹F dx, it will be also

bounded by (small) ε2= (x1− x0)ε1as

|JN^∗(y^∗N) − J(y)| =^{ x1}

x₀ |F^∗(x,y^∗N,y^∗_N,x) − F(x,y^∗,y^∗_,x)|dx <ε2. (1.189) In two (or more) dimensions we analogously write

u^∗(x,y) = u(x,y) +

∑

^Ni=1a_iηi(x,y), (1.190) where u(x,y) is an auxiliary function and aiare unknown (real) coefficients that should be de-termined by the method. In some, mainly “dynamics oriented” analyses, the coefficients aiare referred to as “Generalized Coordinates”. As shown in Example 1.8, when a time-dependent problem is under discussion, this concept assigns ai to be time-dependent while the “shape functions”ηiremain spatial functions only.

Remark 1.10 In the one-dimensional case, a set of functions{ηi(x)} on [x0,x1] is said to be relatively complete if for everyε> 0 and y(x) there are N and y^∗_N as defined by (1.187) such that|y^∗_N− y| <εandy^∗_N,x− y,x <εfor all x∈ [x0,x1]. In two dimensions, a set of functions {ηi(x,y)} onΩis said to be relatively complete if for everyε> 0 and u(x,y) there are N and u^∗_N as defined by (1.190) such that |u^∗_N− u| <ε,u^∗_N,x− u,x <εandu^∗_N,y− u,y <εfor all (x,y) ∈Ω. The extension of the above for higher dimensions is clear.

Example 1.7 Bending of a Beam.

We shall demonstrate here the usage of Ritz’s Method with the Theorem of Minimum Poten-tial Energy, i.e. J= V, see (1.124) for a “simply-supported” uniform isotropic beam of length l, and Young’s modulus E that undergoes a transverse loading fy(z) as shown in Fig. 1.19(a).

In the absence of body loads, (1.124) yields for this case (see also Example 1.2) V =^{ l}

0 [1 2EIx

v^∗

2

− fy(z)v^∗]dz. (1.191)

To employ Ritz’s method for determining v(z) we substitute v = v^∗in (1.191), where

v^∗=

∑

^Ni=1a_isin(iπz

l ), (1.192)

which clearly satisfies the geometrical boundary conditions v^∗(0) = v^∗(l) = 0 as required.

For uniform loading, i.e. fy= f0= const. and N = 3, one obtains

V= −2 l f0

π (a1+1

3a₃) +EIxπ⁴ l³ (1

4a²₁+ 4a²2+81

4 a²₃). (1.193) The solution of the three equations, V_,ai = 0, yields a1= _EI^{4 l4f0}

xπ⁵, a2= 0 and a3=_{243 EI}^{4 l4f0}

xπ⁵. Even with only the first term, the above approximation, i.e. v= 4_EI^l4f0

xπ⁵sin(πζ) whereζ=^z_l, is in a satisfactory agreement with the exact one given by v=_24EI^l4f0_xζ(ζ³− 2ζ²+ 1).

In addition, for linear loading distribution of fy= f0(z −₂^l) we obtain

V= a2

l²f0

2π +EIxπ⁴ l³ (1

4a²₁+ 4a²2+81

4 a²₃). (1.194)

The solution of the three equations, V_,ai= 0, in this case yields a1= a3= 0, a2= −_16EI^l5f0

xπ⁵.

When the above beam is “clamped-free” (cantilever) as shown in Fig. 1.19(b), one may alternatively use the approximation

v^∗=

∑

^Ni=2aizⁱ, (1.195)

which satisfies the geometrical boundary conditions v= v
= 0 at z = 0. For fy= f0= const.

such approximations yield the exact one, v=_24EI^l4f0_xζ²

ζ²− 4ζ+ 6

, for N= 4 (i.e. three terms).

Figure 1.19:Isotropic beam under transverse loading: (a) Simply-supported, (b) Cantilever.

Example 1.8 Lagrange’s Equations.

We adopt here the concept of expanding each displacement component as a relatively com-plete set of shape functions as in Example 1.7, but we consider a time-dependent problem by setting the generalized coordinates of the problem, a= (a1,...,a3N), to be functions of time, t. Hence, in the general case

u^∗_i(x,y,z,t) = ui(x,y,z,t) +

∑

^Nj=1am(t)ηj(x,y,z), i= 1,2,3, (1.196) where uiare auxiliary (known) functions and m= (i − 1)N + j. We now employ the Theorem of Minimum Potential Energy by writing the variation of V in (1.124) as

δV= (grad_aU− Q^S− Q^B) ·δa= 0 (1.197) where grad_a= (_∂a^∂₁,...,_∂a^∂_3N). In the above we have employed the notation

grad_aU= grada(



B

W), Q^S= (Q^S1,...,Q^SN), Q^Sj=



S

Fs· u,aj, Q^B= (Q^B1,...,Q^BN), Q^Bj =^

BFb· u,aj. (1.198) By assuming that in a time-dependent case, the dominant portion of the body force is Fb= −ρ¨u (whereρis the material density and a dot stands for a time derivative) we write

Q^B_j = −



Bρ¨u· u,aj = −∂

∂t(



Bρ˙u· u,aj) +



Bρ˙u· ˙u,aj. (1.199) We further note that the kinetic energy is given by T =¹₂^

B(ρ∑³_i=1u˙²_i), and therefore, the once underlined term in (1.199) is∂T/∂a˙j(note that∂u˙i/∂a˙j=∂Ui/∂aj in view of (1.196)),

and the twice underlined term is∂T/∂aj. Hence, (1.197) become

∂

∂t ∂T

∂a˙_j



−∂T

∂aj+∂U

∂aj− Q^Sj= 0, j= 1,...,3N, (1.200) which are well known as Lagrange’s equations.

1.6.5.5 Solution Trail “F”: Galerkin’s Method

Galerkin’s method is a general powerful method for solving boundary value problems, and there are many applications related to it in the theory of elasticity. It resembles Ritz’s method in the sense that it is also based on expanding an approximate solution into a series of rela-tively complete sets of “shape” functions, but it is founded on a different functional type. In Galerkin’s method, we select the functional to be the error L(u^∗), but instead of employing the operation

J=^

Ω|L(u^∗)|²→ min, (1.201)

we set the requirement

Ji=^

ΩL(u^∗)ηi= 0, (1.202)

by which we force the error function to be orthogonal to each one of the shape functions.

Since J is based on the differential operator and not on the potential energy, each of the shape functions must satisfy all boundary conditions (natural and geometrical). This may pose substantial constraints on the admissible families of shape functions. For example, the series used in (1.195) for a cantilever beam is not applicable here since theηifunctions do not satisfy the natural boundary conditions (one by one) at the free tip.

In document About the Authors Omri Rand Vladimir Rovenski (Page 59-65)