Exercises for continuum mechanics

Exercises of continuum mechanics course

HUANG Zaixing

College of Aerospace Engineering, NUAA

Chapter 1: Introduction to tensor

Exercises 1:

1. Show that

(1) ∈ijk ∈jki = 6

(2) ∈ijk A j A k = 0

(3) δij ∈ijk = 0

(4) ∈ijk ∈ist = δj s δk t ? δj t δk s

2. Show that ∈ijk σjk = 0 if σjk = σkj .

3. In an orthogonal Cartesian coordinate system, basic equations of elasticity may be written in the form of index notation:

)u u (2

1G 2u

f i ,j j ,i ij ij kk ij ij i i j ,ij +=εε+εμδ=σρ=ρ+σ&&

By using the equations above, prove the following equation:

i i kk ,i ki ,k u

f Gu u )G (&&ρ=ρ+++μ

Exercises 2:

1) Let ~~~w ,v ,u and ~x be arbitrary vectors, and ~A , ~B and ~

C be an arbitrary second-order tensor, show that a. )x u )(w v ()x w ()v u (~

~~~~~~~??=??? b. ~~~~~~v )u A ()v u (A ??=?? )v A (u A )v u (~T

~~~~~??=?? c. T ~T ~~~~~~~~~B :A B :A v A u )v u (:A =??=? ~T

~~~~T ~~~~B :)C A (C :)A B ()C B (:A ?=?=? d. ~T

~~~~~A tr A tr v u )v u (tr =?=?

~T ~~T ~~T ~~~T ~~T ~~A :B )A B tr()B A tr(A :B )A B tr()B A tr(=?=?==?=? e. n T ~T n ~

n 1~1n ~)A ()A ( )A ()A (==?? 2) Evaluate 2~A tr , 3~A tr if T

~~A A ?=.

3) Show that, if T

~~A A =, then ~~~~B sym :A B :A =.

Exercises 3 : With respect to the base vectors in the Cartesian coordinate system, the components of a tensor ~F are: 0F F F F 1F F F - 2F F 133132233322211211=====?====

Write out the polar decomposition of ~

F .

Exercises 4:

1) Show that ~

~~A A A I II and I I ,I are the invariants independent of the coordinate system.

2) Two symmetric second-order tensor ~A and ~B are coaxial, if they have the same eigenvectors ~

i n . Prove that the sufficient and necessary condition that ~A and ~

B are coaxial is ~

~~~A B B A ?=? 3) Give a counter-example to show that not all of the asymmetric second-order tensors have a principal representation.

Exercises 5: Suppose ~A is an antisymmetric tensor of second-order, then ~A

e may be represented as the form below:

~2A A ~A A ~A A II )II cos(1A II )II sin(I e ~~~~~?++=

Exercises 6:

1) Under the assumption T ~

~S S = calculate the partial derivative of the expression ~~S :S with respect to ~S . 2) Under the assumption T ~~S S = calculate the partial derivative of the expression ~~~n S n ?? with respect to ~

n , where ~

n is an unit vector.

3) Calculate the partial derivatives of three invariants of a second-order tensor.

Chapter 2: Finite deformation

1. A unit vector ~N is given at a point P 0 of a body in its undeformed state B 0. Express the unit vector ~

n which determines the direction of ~N in the actual deformed state in terms of the deformation gradient ~F and ~

N . 2. Evaluate the stretch in a given direction ~N of an undeformed body in terms of the deformation gradient ~

F and the left stretch tensor ~

v . 3. From the first and the second variation of the invariants of the right Cauchy-Green tensor ~

C . 4. Show that det 1b ~= if ~

3/2~

b J b ?=. 5. Show that ~

~~~T ~~n d n n L n ??=?? for an arbitray vector ~n . 6. Show that the real eigenvalue of an antisymmetric tensor is zero and that the corresponding eigenvector shows

in direction of the axial vector of this antisymmetric tensor.

7. )U (~Ψ and )C (~Ψ are tensor function of the right stretch tensor ~U and the right Cauchy-Green tensor ~

C , respectively, such that )U ()U ()C (~2~~Ψ=Ψ=Ψ. Establish the relation between the partial derivatives ~U ,Ψ and ~

C ,Ψ.

8. For the simple shear given by the equation ~

2~1~~g g I F ?γ+= find the directions in the deformed configuration in which no extension takes place. Find also the directions orthogonal to planes in which no change of area occurs.

9. A body is reinforced by embedding in it two families of inextensible fibres. The body has an undeformed configuration B 0 in which the fibres in each family are straight and parallel, and unit vectors defining their orientations have components (cos Θ, ±sin Θ, 0) (0<Θ<π/2) relative to an orthonormal basis ~

i g . The body is

subjected to a homogeneous triaxial extension in which stretches of amounts λ-1/2α, λ-1/2α-1, λ are applied in the directions defined by the base vectors ~

3~2~1g ,g ,g respectively. Obtain an equation connecting α, λ and Θ and deduce

from it that

(1). The extent to which the body can contract in the 3-direction is limited by the inequality Θ≥λ2sin .

(2). When sin2Θ<λ≠1, two deformed configurations are possible.

(3). When the maximum contraction in the 3-direction is achieved, the two families of fibres are orthogonal in the

deformed configuration.

Chapter 3: Kinematics of deformation

1. A motion of a continuum is given by the equations

?????++=++=++=2

2133

213222

3211t X t X X x t X t X X x t X t X X x .

(1). Find the velocity and acceleration of: (a) the particle which was at the point (1, 1, 1) at the reference time t = 0, and (b) the particle which occupies the point point (1, 1, 1) at time t = 0.

(2). Explain why this motion becomes physically unrealistic as t → 1.

(3). Find the components of the tensor L , d and w .

2. The velocity at a point x in space in a continuum is given by

322222121212222122212)

(2)()(e e x x x x a e x x x x a v γββ++++?=, where β, γ and a are constants.

(1). Show that div v = 0.

(2). Find the acceleration of the particle at x .

(3). Find the components of the tensor L , d and w .

Chapter 4: Stress

1. In a rectangular Cartesian coordinate system x 1, x 2 and x 3, the components of the stress tensor at a point P are given in appropriate units by

)(2)(224313311221332211e e e e e e e e e e e e e e ?+?+?+?+?+?+?=σ.

(1). Find the traction at P on a plane through P parallel to the plane x 1+2x 2+3x 3=1.

(2). Find the principal stress components at P .

(3). Find the principal directions of the stress at P . Verity the principal axes of the stress are mutually orthogonal.

(4). Let a new coordinate x’1, x’2, x’3 are related to x 1, x 2, x 3 by

?????????++?=++?=+?=)22(31')22(31')22(31'321332123211x x x x x x x x x x x x . Find the components of the stress tensor defined above in the new coordinate system. Use the answer to check the answers to (2) and (3) above.

2. Prove F C

&&:2:π=Σ, where π and Σ are first and second Piola-Kirchhoff stress tensor, respectively. x 1上标

3. Let n is an unit vector, p n the traction on the surface normal to n , and S the magnitude of the shear stress on this surface, so that S is the component of p n perpendicular to n . Prove that as n varies, S has stationary values when n is perpendicular to one of the principal axes of stress, and bisects the angle between the other two. Prove also that the maximum and minimum values of S are ±(T 1?T 3)/2.

Chapter 5: Balance equations

1. Determine the stress tensor conjugate to the Biot strain tensor.

2. Let σm be the average stress defined on ?, i.e.,

∫=Ω

dv V m σσ1, where V is the volume of ?. Prove the Signorini theorem: )(1∫∫?+?=

?ΩΩdv x f ds x p V n m σ, where p n be the traction on the surface and f the body force.

3 Derive the virtual power principle:

∫∫∫=?+??Ω

ΩΩdv d dv v f ds v p n δσδδ:, where p n be the traction on the surface, f the body force and v the velocity. d denotes the tensor of deformation rate.

Chapter 6: Constitutive equations

1. Calculate the stress components in a compressible elastic body subjected to the deformation

x 1 = X 1+kX 2, x 2 = X 2+kX 3, x 3 = X 3,

where k is a non-zero constant, and deduce that they satisfy the universal relations

σ22?σ11 = σ13 = (σ12?σ22)/k , σ22?σ33 = σ13+k σ23.

Show also that the deformation is isochoric.

虚功率原理