Math 272A: Continuum Mechanics

1 Introduction (1/4/10):

• Class structure: three HWs, equally weighted.

• Derivation of governing equations for materials using classical physics and continuum concep-tion of material.

• Materials modeled as a moving collection of particles, i.e. at any time material is associatedwith a subset of points in E3.

• Fluids: Navier Stokes equations, Stokes equations, compressible flow Euler equations, incom-pressible/inviscid Euler equations, non-Newtonian viscoelastic fluids.

• Solids: Linearly elastic materials, nonlinear Elasticity (e.g. Neo Hookean, Mooney Rivlin),anisotropy

1.1 Tensor algebra (chapter 1, Gonzalez and Stuart)

1.1.1 Points and vectors

• Points/particles: E3 = 3D Euclidean space = {x}.

• Vectors: quantity with direction and magnitude, also denoted with bold face symbols: v ∈ V.E.g. two points determine a vector.

• V has the structure of a real vector space:

u + v ∈ V ∀u,v ∈ V

αu ∈ V ∀α ∈ R,u ∈ V

1.1.2 Coordinate basis

• Basis: three mutually perpendicular unit vectors.

{i, j,k} = {e1, e2, e3} , or {ei}

|ei| = 1, ei · ej = 0 when i 6= j

• Right hand orientation:i× j = k, j× k = i, k× i = j

• Identifying a vector u with a given basis {ei}:

[u] =

u1

u2

u3

∈ R3 = R3×1, ui = u · ei

[u]T ∈ R1×3

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• Given a basis, we can uniquely identify every geometric vector in V with a numeric memberof R3. That’s the point of the basis, we’d rather work with numbers.

• Important to remember that we can only say we are working with R3 rather than V oncewe’ve introduced a basis.

1.1.3 Coordinate frame

• Will be convenient to also associate E3 with R3.

• Need a basis {ei} and a point 0 ∈ E3 (the origin) to do this.

• Given x ∈ E3, x− 0 ∈ V and [x− 0] ∈ R3

1.1.4 Index notation

• Convenient once we’ve introduced a basis or coordinate frame:

a,b ∈ V, {ei} →

a1

a2

a3

= [a] ,

b1b2b3

= [b]

a =3∑

i=1

aiei abbreviated as aiei.

• The summation is implied by the repeated index:

a · b = aibi

a2bjbjb3 = a2

(b21 + b22 + b23

)b3

aiajbjai =(a2

1 + a22 + a2

3

)(a1b1 + a2b2 + a3b3)

• Dummy index=any index in an expression that is repeated.

• Free index=any index in an expression that is not repeated.

• E.g.ai = cjbjbi = ckbkbi

i is a free index and j and k are dummies. They’re dummy indices because we can change it’sname without affecting the expression. Note that the above implies three equations becauseit implies it holds for any i ∈ {1, 2, 3}

• Each term in an equation should have the same free index. E.g.

Not allowed: ai = bj , aibj = cidjdj

Allowed: aibj = cickdkdj

• Kronecker Delta: given a basis {ei}, δij = ei · ej E.g. given v ∈ V and {ei}:

vi = δijvj

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• Permutation: εijk =

1, ijk = 123, 231, 312−1, ijk = 321, 213, 1320, repeats

Useful operator for expressions involving cross products. E.g.

ei × ej = εijkek, a× b = aibjεijkek, (a× b) · c = εijkaibjck

2 Higher order tensors, chapter 1 Gonzalez and Stuart (1/6/10):

• Since material will be associated with subsets of Euclidean space (E3), we will be interestedin functions like:

f : E3 → R, g : E3 → V

φ : E3 → E3, e.g. to describe how objects change shape over time.

• Differentiating these functions will be necessary in deriving the governing PDEs of motion forthe various materials.

• The derivatives are tensors: linear transformations over vector spaces.

E.g. vectors are first order tensors. They map V to R and vice versa (via dot productand scaling respectively).

• Example: force, velocity, acceleration are vectors or first order tensors. Other quantities suchas stress and strain are not well represented as vectors, but as higher order tensors.

• Example: stress σ is a second order tensor and relates direction with force per area. That is:

σ : V→ V, linear.

2.1 Second order tensors: V2

Linear maps S : V→ V. Formally

S(u + v) = S(u) + S(v), ∀u,v ∈ V

S(αv) = αS(v), ∀α ∈ R.

• Introduction of a basis {ei} allows us to associate S with a matrix [S] ∈ R3×3

[S]ij = Sij = ei · S(ej),

S11 S12 S13

S21 S22 S23

S31 S32 S33

∈ R3×3

• For example:

a,b ∈ V, {ei} →

a1

a2

a3

= [a] ,

b1b2b3

= [b]

If we have b = S(a), then b1b2b3

=

S11 S12 S13

S21 S22 S23

S31 S32 S33

a1

a2

a3

, or [b] = [S] [a] , or bi = Sijaj

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2.1.1 Second order dyadic product

Given a,b ∈ V, the second order dyadic product is denoted as a⊗ b ∈ V2 and is defined as:

a⊗ b : V→ V, a⊗ b(u) = (b · u)a.

In other words, this is just an outer product. That is, if we introduce a basis {ei}, then [a⊗ b]ij =aibj .

• Given a basis {ei}, S = Sijei ⊗ ej. I.e. you can think of {ei ⊗ ej} as the basis for V2 impliedby the basis {ei} for V

2.1.2 Special second order tensors

• Transpose of a tensor: Given S ∈ V2, ST ∈ V2 can be defined as the unique tensor such that

ST : V→ V, (S (u)) · v = u ·(ST (v)

), ∀u,v ∈ V

• The tensor is symmetric if ST = S, skew symmetric if ST = −S, orthogonal if STS = I. Herefunction composition defines the multiplication of tensors. Addition is also defined from afunction standpoint.

• Symmetric part:

E = sym (S) =12(ST + S

)• Skew symmetric part:

W = skew (S) =12(ST − S

), S = E + W

• Given S ∈ V2 with skew S, there exists a w ∈ V such that

S (u) = w × u, ∀u ∈ V

2.1.3 Change of basis

Given bases {ei} and {ei}:{ei} : V→ R3, {ei} : V→ R3

There is a tensor A for changing bases.

A = Aijei ⊗ ej, Aij = ei · ej

Can show A is orthogonal. By changing basis, we mean that for any u ∈ V, introduction of thebases {ei} and {ei} yield: u1

u2

u3

,

u1

u2

u3

, ui = u · ei, ui = u · ei

A is the tensor in V2 with the property that:

ui = Aij uj .

You can see this because u = ukek = uj ej and so ui = u · ei = uj ej · ei = Aij uj .

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2.1.4 Operations

• Trace: given S ∈ V2, Tr (S) = [S]ii

• Determinant: given S ∈ V2, det (S) = εijk [S]i1 [S]j2 [S]k3

Note that the definition of these operations assumes we have introduced a basis. This would implythat they are the same independent of the choice of basis. You can show that is the case.

2.1.5 Square root and polar decomposition

Given symmetric positive definite S ∈ V ∃!√

S ∈ V such that√

S2

= S.

• Let {λi} be the eigenvalues of S (thus λi > 0) and {ei} be the corresponding eigenvectors.Then {ei} form a basis for V and S = λiei and

√S =√λiei.

• Given F ∈ V, det(F) > 0, ∃U,V,R such that

F = RU = VR, VTV = UTU = I, R = RT

with det (R) , det (V) , det (U) > 0.

3 Fourth order tensors, chapter 1 Gonalez and Stuart (1/8/10)

The linear functions C : V2 → V2 are called fourth order tensors and are denoted as C ∈ V4.

3.1 Fourth order dyadic product

Just as with second order V2, there is a natural “outer product” like way of constructing a fourthorder tensor from vectors v ∈ V. Given a,b, c,d ∈ V, a⊗ b⊗ c⊗ d ∈ V4 is defined by its actionon S ∈ V2:

(a⊗ b⊗ c⊗ d) (S) = (c · Sd) a⊗ b

Given {ei}, the natural basis for V4 is {ei ⊗ ej ⊗ ek ⊗ el}.

• Introduction of {ei ⊗ ej ⊗ ek ⊗ el} associates C ∈ V4 with R3 × R3 × R3 × R3, specifically

Cijkl = ei ·C (ek ⊗ el) ej

• Given C ∈ V4 and {ei ⊗ ej ⊗ ek ⊗ el}, Cijkl ∈ R3 × R3 × R3 × R3 =

• [a⊗ b⊗ c⊗ d]ijkl = aibjckdl

3.2 Symmetries

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