LMP D C LP;M C C MP;L C LM;P are Christhoffel symbols of the second kind

These 3⁴ D 81 equations (3.62) are not independent. In fact, due to the symmetries ofR_NMLP that can be deduced by inspection of (3.62),

R_LMNP D R_MLNP D R_LMPNI R_LMNP DR_NPLM;

there are only 6 distinct and nonvanishing componentsR_NMLP. To prove this, let a pair of indicesLP be fixed. SinceR_NMLP is skew-symmetric with respect to the pairNM, we conclude that indicesN andM can only assume the values12,13, and23. The same argument holds forLP if the pairNM is given. Finally, the 9 nonvanishing components ofR_NMLP

5The Riemann–Christoffel tensorRis also referred as the curvature tensor and the condition (3.62) as Riemann’s theorem.

100 3 Finite and Infinitesimal Deformations R1212; R1213; R1223;

R₁₃₁₂; R₁₃₁₃; R₁₃₂₃; R2312; R2313; R2323;

merge into the following 6 independent components, due to symmetry with respect to the first two indices:

R₁₂₁₂; R₁₂₁₃; R₁₂₂₃; R₁₃₁₃; R₁₃₂₃;

R₂₃₂₃: (3.63)

As a special case, consider the linearized theory of elasticity. SinceC ' IC 2E, (3.62) assumes the form

R_NMLP 'E_LM;NP CE_PN;LM E_NL;MP E_PM;LN D0; (3.64)

which is certainly true ifE.X/is a linear function. The equations of interest deduced from (3.64) are the following, known as the St.Venant compatibility conditions (St.Venant, 1864):

R₁₂₁₂'2E_12;12E_11;22E_22;11D0;

R₁₂₁₃'E_12;13CE_31;12E_11;23E_23;11D0;

R₁₂₂₃'E_22;13CE_31;22E_12;23E_23;21D0;

R₁₃₁₃'2E_13;13E_11;33E_33;11D0;

R1323'E13;23CE23;13E12;33E33;21D0;

R2323'2E23;23E22;33E33;22D0: (3.65)

and (3.65) can be summarized by the following relation:

QSRNMLEQN;SM D0, r r ED0: (3.66)

If the divergence operator is applied to (3.66), it follows that

QSRNMLEQN;SMRD0; (3.67)

which is identically satisfied by any second-order symmetric tensorE, since the permutation symbolQSRis skew-symmetric with respect to indicesRandS and EQN;SMRis symmetric with respect to the same indices.

WhenLD1; 2; 3, (3.67) gives the following three equations:

R1223;3CR1323;2R2323;1 D0;

R1213;3R1313;2CR1323;1 D0;

R1212;3CR1213;2R1223;1 D0: (3.68)

It is then possible to conclude that, if

R1213DR1223DR1323D0; (3.69)

it also holds that

R2323;1DR1313;2DR1212;3D0; (3.70)

i.e.,

R₂₃₂₃is independent onX¹; R1313is independent onX²; R₁₂₁₂is independent onX³:

(3.71)

It is convenient to summarize all the above considerations as follows:if Cis linearly simply connected,system(3.60)can be integrated if and only if the 6 distinct components of the curvature tensorR.C/vanish. In the linearized theory, the tensor Emust satisfy(3.65)or, equivalently, must satisfy(3.69)inCand (3.71)on the boundary@C.

We note that, when dealing with infinitesimal deformations, (3.64) can be obtained by observing that, given a symmetric tensor fieldE.X/, a displacement fieldu.X/has to be determined such that

u_L;M Cu_M;LD2E_LM:

By adding and subtractinguL;M, we can write the above system of 6 equations with 3 unknownsuL.X/as, see (3.36)

uL;M DELM CWLM: The 3 differential forms

duLDuL;MdXM D.ELM CWLM/ dXM

can be integrated if and only if

ELM;NCWLM;N DELN;M CWLN;M: (3.72)

102 3 Finite and Infinitesimal Deformations Cyclic permutation of indices in (3.72) gives two similar equations, and adding (3.72) to the first of these and subtracting the second one, leads to

ELM;NEMN;LDWLN;M; which is equivalent to the system of differential forms

d WLN D.ELM;NEMN;L/ dXM: These forms can be integrated if and only if (3.64) holds.

3.9 Curvilinear Coordinates

This section extends the description of the deformation to the case in which both the initial and current configurations are given in curvilinear coordinates.yⁱ/, which for sake of simplicity are still taken to be orthogonal (Chap.2).

In this case, we have to distinguish contravariant components, relative to the natural basis.e_i/, from covariant components, relative to the reciprocal or dual basis .eⁱ/, furthermore, for physical reasons it is convenient to express vectors and tensors with respect to an orthonormal basis.a_i/(see Chap.2, Sect.2.6).

If.yⁱ/are the coordinates of a point inC whose coordinates inCare.Y^L/, then the finite deformation fromC!C is given by

yⁱ Dyⁱ.Y^L/: (3.73)

Differentiation gives

dyⁱei D @yⁱ

@Y^L.Y/d Y^Lei; (3.74) where .e_i/ is the natural basis associated with .yⁱ/at the point y.Y/ 2 C. By recalling (2.59) and (2.61) and referring to the componentsd yⁱ in the basis of unit vectors.a_i/, we find that

d yⁱai DFⁱ_Ld Y^Lai; (3.75) where

Fⁱ_LD pgi i

pg_LLF_Lⁱ: (3.76)

The bar indicates that the quantities are evaluated in the basisa_i. It is useful to observe that the metric coefficient g_{i i} is computed at y.Y/ 2 C, while g_LL is

evaluated at Y 2 C. We can apply Cauchy’s theorem of polar decomposition in the form (3.7) to the matrix .Fⁱ_L/and, since.a_i/is an orthonormal basis, all considerations developed for Cartesian coordinates still apply to.Fⁱ_L/:

As an example, the components of tensorsCandBrelative to.a_i/are given by the elements of the matrices

CDF^TF; BDFF^T; and the eigenvalue equation is written as

CLMu^M DıLMu^M:

The displacement gradientruand tensorsEandWassume the expression given by the rules explained in Chap.2. The programDeformation, discussed at the end of this chapter, can be used to advantage in this respect.

3.10 Exercises

1. Describe the deformation process corresponding to a one-dimensional extension or compression.

The required deformation is characterized by two principal stretch ratios equal to unity, while the third one is different than unity. The vector function (3.1) is in this case equivalent to the following scalar functions:

x₁D˛X₁; x₂DX₂; x₃ DX₃; where˛ > 0if it is an extension and˛ < 0if it is a compression.

The deformation gradient, the right Cauchy–Green tensor, and the Green–

St.Venant tensor are expressed by

FD 0

@˛ 0 0 0 1 0 0 0 1

1 A;

CDF^TFD 0

@˛² 0 0 0 1 0 0 0 1

1 A;

GD 1

2.F^TFI/D 0 BB

@

˛²1 2 0 0 0 0 0 0 0 0

1 CC A:

104 3 Finite and Infinitesimal Deformations

Fig. 3.1 Plane shear deformation

2. Given the plane pure shear deformation

x₁DX₁; x₂DX₂CkX₃; x₃DX₃CkX₂;

determineF;C, andG, the stretch ratios along the diagonal directions, and the angle₂₃in the current configuration, as shown in Fig.3.1.

The required tensors are represented by the matrices

.F_iL/D

Given a line element parallel to the unit vectorNidentified in the reference configuration, the stretch ratioƒNis computed according to the relationship

ƒ²_NDNCN:

In the direction parallel to the diagonalOBwe obtain

ƒ²_OB D

and in the directionCAwe obtain

The angle₂₃in the current configuration is given by cos₂₃D p C23

C22

pC33

D 2k 1Ck²:

3. Referring to the pure shear deformation of Fig.3.1, apply the polar decomposi-tion to the tensorF.

Principal axesX₂⁰andX₃⁰are rotated45^ıaboutX1, so that the transformation matrix is

The Cauchy–Green tensorCin the basis of principal axes is:

C⁰DACA^T;

It is then proved, as stated by Theorem3.2, that the eigenvalues of the tensorC are the squares of the stretch ratios along the eigenvectors.

Moreover, the same Theorem3.2also states that bothUandChave the same eigenvectors, and the eigenvalues ofCare the squares of the eigenvalues ofU, so that

106 3 Finite and Infinitesimal Deformations

The matrixU¹in the initial frame is

U¹ DA^T.U⁰/¹A;

Finally, since we have proved that

RDFU¹;

Thus, the required decomposition is 0

and we observe that the matrixRrotates the principal axes ofCinX 2 Cin order to superimpose them on the principal axes ofB¹inx2C. As these axes coincide in this case,RDI.

4. Given the deformation

xi DXi CAijXj;

withAij constant, prove that plane sections and straight lines in the reference configuration correspond to plane sections and straight lines in the current configuration.

5. Verify that, if the deformation in Exercise 4 is infinitesimal, i.e., the quantities Aij are so small that their products are negligible, then the composition of two subsequent deformations can be regarded as their sum.

6. Given the deformation 8<

:

x1DX1CAX3; x₂DX₂AX₃; x3DX3AX1CAX2; findF;U;R;C;B, principal directions, and invariants.

7. Determine under which circumstances the infinitesimal displacement field u.X/Du1.X1/iCu2.X2/j

satisfies the compatibility conditions (3.66).

3.11 The Program Deformation Aim of the Program

Given a transformation fromorthogonal curvilinear coordinatesvarto Cartesian coordinates, the programDeformationallows the user to compute the tensorsF, C, andB, the eigenvalues and eigenvectors ofCandB, the deformation invariants, and the inverse ofB, as well as the tensorsUandR.

Furthermore, given two directions parallel to the unit vectorsvers1andvers2, the program defines the stretch ratios in these directions and the shearing angle, if vers1andvers2are distinct. We note that all tensor and vector components are relative to the basis of unit vectors associated with the holonomic basis of curvilinear coordinatesvar.

Dans le document Continuum Mechanics using Mathematica (Page 114-122)