HALLFASTHETSLARA, LTH

Examination in computational materials modeling

TID: 2015-01-10, kl 8.00-13.00

Maximalt 60 poang kan erhallas pa denna tenta. For godkant kravs 30 poang.

Tillatet hjalpmedel: raknare

Uppgift nr 1 2 3 4 5 6Besvarad(satt x)Poang

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PROBLEM 1 (10p.)

For a linear elastic material the stress response ij = sij +13ijkk can be

described by the relations

kk = 3Kkk

sij = 2Geij

where the total strain is kl = ekl +13klpp and

G =E

2(1 + ), K =

E

3(1 2)a) Assume that a strain energy function W (I1, J2, J3) exists, where

I1 = kk, J2 =1

2eijeji, J3 =

1

3eikekleli

Derive the most general stress strain response from

ij =W

ij

b) Based on the result in a) identify the specific format for the strain energyconsidered in this example.

c) Assuming that the strain energy is positive, i.e. W > 0, identity theconstraints on E and .

d) For the linear elastic material derive the material stiffness Dijkl definedby the relation ij = Dijklkl.

PROBLEM 2 (10p.)

Derive the incremental stress strain relation in elasto-plasticity. Make use ofthat the strain can be decomposed as

ij = eij +

pij

where superscipt e denotes the elastic part and superscipt p the plastic partof the strains. The Hookes law is given as

ij = Dijklekl (1)

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The evolution laws are defined as

pij = f

ij, = k, 0

where f(ij , K) is the yield surface, K = K() is the hardening function andk = k(ij , K) is related to the hardening of the material.

a) Assume plastic loading, use the consistency condition to derive

f

ijDijklkl A = 0

and identify A.

b) Use the consistency condition in the above relations to derive the straindriven incremental law

ij = Depijklkl

Name one situation where the above relation breaks down.

PROBLEM 3 (10p.)

For materials with a special type of tetragonal crystal structure only 7 ma-terial parameters are needed to describe the linear relation between stressesand strains.

To derive the constitutive relation for tetragonal symmetry we will con-sider a rotation of the coordinate system in the x1 x2 plane by /2 whichis described by the transformation matrix

[Aij ] =

0 1 01 0 00 0 1

Recall that the transformation of second-order tensors, i.e. stresses andstrains are given by

ij = AikklAjl ij = AikklAjl

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The general stress- strain relation is given by112233121323

=

D11 D12 D13 D14 D15 D16D21 D22 D23 D24 D25 D26D31 D32 D33 D34 D35 D36D41 D42 D43 D44 D45 D46D51 D52 D53 D54 D55 D56D61 D62 D63 D64 D65 D66

112233212213223

a) Derive the transformation of the different stress and strain components

if transformation matrix Aij is used.

b) Use the arguments about elastic symmetry planes and the transformationmatrix stated above show how the D-matrix reduces. Only the two firststresses 11 and 22 need to be considered.

PROBLEM 4 (10p.)

For uniaxial experimental tests, i.e. uniaxial loading

[ij ] =

0 00 0 00 0 0

on a metal it was found that a good fitting could be obtained by using apower law format for the stress strain relation, given by

=

{E when yo

E

yo + kyo(p)n when yo

E

where , and p are the uniaxial stress, strain and plastic strain, respectively.Moreover, E is Youngs modulus and yo is the initial yield stress, and finallyn and k are material parameters.

Assuming that the material can be described by a kinematic hardeningvon Mises model during plastic loading

f =

(3

2sij sij

)1/2 yo = 0

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where sij = sij ij is the reduced deviatoric stress tensor, ij is a devia-toric back-stress tensor and sij = ij 13ijkk is the deviatoric stress tensorAssociated plasticity is assumed, i.e

pij = f

ij, 0

For evolution of the back-stress it is assumed that Melan-pragers evolutionlaw is valid

ij = cpij

where c is a positive material parameter that may depend on the load history.

a) Calculations the plastic strain rate for the above model.

b) For a general load case and the choice strain hardening, i.e.

= eff =

(2

3pij

pij

)1/2identify the rate of internal variable by use of the flow rule for plasticstrain rates.

c) Considering uniaxial loading identify the choice of internal variable (i.e.).

d) For uniaxial loading identify the form of ij

e) Given that uniaxial loading, given that the experimental data can befitted to the power law function, identify c().

PROBLEM 5 (10p.)

Let failure of a material be defined by the Drucker-Prager criterion

F (ij) =

3J2 + I1 = 0 (1)

where

J2 =1

2sijsij , I1 = kk, sij = ij 1

3ijkk

Moreover, and are material parameters.

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a) Illustrate (1) in the deviatoric plane and in the meridian plane, i.e. theI1

J2 plane. (Note that I1 = kk). A principal sketch is sufficient.

b) Derive the uniaxial tension and uniaxial compression paths in terms ofI1 and

J2. Draw the stress paths in the deviatoric plane and in the

meridian plane.

c) Reduce the Drucker-Prager criterion to plane stress conditions, i.e. 3 =0.

d) Determine the uniaxial tensile strength t, uniaxial compression strengthc, biaxial tensile strength bt and the biaxial compressive strength bcin terms of and .

e) Draw the shape of the Drucker-Prager criterion in the 1 2 plane, andmark the locations of the stresses in problem d).

PROBLEM 6 (10p.)

In a certain experiment it is found that yielding of a material occurs underthe following states of principal stresses

(1, 2, 3) = (20, 0, 5)

(1, 2, 3) = (1, 0, 18)Assuming that the material is isotropic, that the hydrostatic stress does notaffect the yielding and the yield stress is the same in tension and compression.

Plot as many points as you can derive from these observations in the1 2 space, i.e. biaxial loading, where 3 = 0 (plane stress).

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