MCE 526 Theory of Elasticity HW 03 Solutions · MCE 526 - Theory of Elasticity 2016-2017 Spring...

MCE 526 - Theory of Elasticity2016-2017 Spring Semester

Dr. M. SahinAnkara Yildirim Beyazit University

MCE 526 Theory of Elasticity HW 03 Solutions

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QUESTIONS

1. Derive stress-strain relation (Hooke’s law) for isotropic thermoelastic material in the form of

σij =E

(1 + ν) (1− 2ν)[(1− 2ν) eij + νekkδij]−

E

1− 2να (T − T0) δij

Hint; Use the strain-stress relation eij = 1+νEσij − ν

Eσkkδij + α (T − T0) δij

Solution: First find the ekk. Note that δkk = 3

ekk =1 + ν

Eσkk − 3

ν

Eσkk + 3α (T − T0) =

1− 2ν

Eσkk + 3α (T − T0)

then substitute σkk into the equation

σkk =E

1− 2ν(ekk − 3α (T − T0))

Algebraic operations after obtaining σkk

eij =1 + ν

Eσij −

ν

E

E

(1− 2ν)(ekk − 3α (T − T0)) δij + α (T − T0) δij

eij =1 + ν

Eσij −

ν

1− 2ν(ekk − 3α (T − T0)) δij + α (T − T0) δij

eij =1 + ν

Eσij −

ν

1− 2νekkδij +

ν

1− 2ν3α (T − T0) δij + α (T − T0) δij

eij =1 + ν

Eσij −

ν

1− 2νekkδij + α (T − T0) δij

(1 +

3ν

1− 2ν

)

eij =1 + ν

Eσij −

ν


(1 + ν

1− 2ν

)

eij =1 + ν

Eσij −

ν


(1 + ν

1− 2ν

)Finally, obtain the stress-strain relations

σij =E

1 + ν

(eij +

ν

1− 2νekkδij −

(1 + ν

1− 2ν

)α (T − T0) δij

)

σij =E

(1 + ν) (1− 2ν)[(1− 2ν) eij + νekkδij]−

E

(1− 2ν)α (T − T0) δij

Ankara Yıldırım Beyazıt University (AYBU) 1 2016-17 Spring Semester



2. Derive the Beltrami-Michell equations (Obtain six compatibility equations in terms of stresscomponents)

Solution: Write the compatibility equations

eij,kl + ekl,ij − eik,jl − ejl,ik = 0 (1)

There are 6 meaningful equations out of all possible equations when all indexed equations areconsideredThese 6 equations can be obtained by letting k = l

eij,kk + ekk,ij − eik,jk − ejk,ik = 0 (2)

Write the strains in terms of stresses by using Hooke’s law (constitutive equations)

eıj =1 + ν

Eσıj −

ν

Eσppδıj =

1

E((1 + ν)σıj − νσppδıj) (3)

Substitute the Eq. 3 into 2

1

E((1 + ν)σij,kk − νσpp,kkδij) +

1

E((1 + ν)σkk,ij − νσpp,ijδkk)−

1

E((1 + ν)σik,jk − νσpp,jkδik)−

1

E((1 + ν)σjk,ik − νσpp,ikδjk) = 0

Simplify the last equation

(σij,kk + σkk,ij − σik,jk − σjk,ik)−ν

(1 + ν)(σpp,kkδij + σpp,ijδkk − σpp,jkδik − σpp,ikδjk) = 0

Note that aqiδij = agj and a,pq = a,qp

=⇒ σpp,jkδik = σpp,ji = σpp,ij


(1 + ν)(σpp,kkδij + σpp,ijδkk − σpp,ij − σpp,ij) = 0

Recall that δkk = 3

σpp,ijδkk = 3σpp,ij


(1 + ν)(σpp,kkδij + 3σpp,ij − 2σpp,ij) = 0


(1 + ν)(σpp,kkδij + σpp,ij) = 0


(1 + ν)(σpp,kkδij + σkk,ij) = 0 (4)




Use the equlibrium equation

σij,j + fi = 0

σik,jk = σik,kj = −fi,jand

σjk,ik = σjk,ki = −fj,iSubstitute the last two equations into the Eq. 4

(σij,kk + σkk,ij + fi,j + fj,i)−ν

(1 + ν)(σpp,kkδij + σkk,ij) = 0

σij,kk + σkk,ij

(1− ν

(1 + ν)

)=

ν

(1 + ν)σpp,kkδij − fi,j − fj,i

σij,kk + σkk,ij1

(1 + ν)=

ν

(1 + ν)σpp,kkδij − fi,j − fj,i (5)

for i = j, and recall that δii = 3

σii,kk + σkk,ii1

(1 + ν)=

3ν

(1 + ν)σpp,kk − 2fi,i

rewrite the equation

σii,kk + σii,kk1

(1 + ν)=

3ν

(1 + ν)σii,kk − 2fi,i

σii,kk

(1 +

1

(1 + ν)

)=

3ν

(1 + ν)σii,kk − 2fi,i

σii,kk

(2 + ν

(1 + ν)− 3ν

(1 + ν)

)= −2fi,i

σii,kk = σpp,kk = −(1 + ν)

(1− ν)fi,i = −

(1 + ν)

(1− ν)fk,k

σii,kk = −1 + ν

1− νfi,i (6)

Substitute 6 into 5

σij,kk + σkk,ij1

(1 + ν)= − ν

(1 + ν)

(1 + ν)

(1− ν)fk,kδij − fi,j − fj,i

σij,kk + σkk,ij1

(1 + ν)= − ν

(1− ν)fk,kδij − fi,j − fj,i




3. For the general displacement formulation with no body forces, show that Navier’s equationsdepend on the Poisson’s ratio ν only.

Solution: The Naviers equations are

µuı,kk + (λ+ µ)uk,ki + Fi = 0

and wıthout body forces they reduce to

µuı,kk + (λ+ µ)uk,ki = 0

divide the equations by µ

uı,kk +

(λ

µ+ 1

)uk,ki = 0

and show that λµis function of Poisson’s ratio ν only.

Note thatλ =

Eν

(1− ν) (1 + 2ν)µ =

E

1 + ν

λ

µ=

Eν

(1 + ν) (1− 2ν) E2(1+ν)

=2ν

(1− 2ν)

Then, the Navier’s equations can be wriiten as

uı,kk +

(λ

µ+ 1

)uk,ki = uı,kk +

(2ν

(1− 2ν)+ 1

)uk,ki = uı,kk +

1

(1− 2ν)uk,ki = 0

which are functions of ν only


MCE 526 Theory of Elasticity HW 03 Solutions · MCE 526 - Theory of Elasticity 2016-2017 Spring...

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