UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering,Department of Civil Engineering Mechanics and MaterialsFall 2005 Professor: S. GovindjeeHW 81. Consider a single crystal of Silicon in a homogeneous state of strain with cubic elasticconstants α = 166 GPa, λ = 64 GPa, and µ = 80 GPa. The lattice vector are knownto be a = e2, b = (−e1+ e3)/√2, and c = (e1+ e3)/√2. If the stress state has beenmeasured in the {ei}3i=1basis to beσ ∼0 10 5010 3 950 9 −3123MPa ,Find ε.2. Consider the strain energy of an isotropic linear elastic bodyEbody=ZΩµεijεij+λ2εiiεjjdV .Argue why the constant µ must necessarily be positive and that 3λ+2µ must necessarilybe positive. [Hint consider special states of strain and the meaning of Ebody.] Thisproblem “shows” that the strain energy density should be positive definite.3. In linear isotropic thermoelasticity we haveεij=1 + νEσij−νEσkkδij+ α∆T δij.Invert this relation to give σijas a function of strain and temperature change.4. The isotropic linear elastic moduli can be expressed asC = c1Idev,sym+ c21 ⊗ 1 ,where Idev,sym= Isym−131 ⊗ 1 is the “symmetric deviatoric projection operator”.(a) Justify the name symmetric deviatoric projection operator for Idev,symby consid-ering its action on an arbitrary 2nd order tensor.(b) Determine c1and c2in terms of the bulk modulus and the shear modulus.5. You have a microstructural model of a crystalline material which employs a unit cellas shown below. Assume that the diagonal atomic interactions have a potential Ur=12kr(r − a√3/3)2(shown in red), the outer horizontal and vertical atomic interactionshave a potential Um=12km(r − a)2(shown in magenta), and the inner horizontal1and vertical atomic interactions have a potential Ub=12kb(r − a/3)2(shown in blue).Determine the macroscopic (small strain) elastic moduli for this material. Only includethe atomic interactions that are shown by connecting lines. Assume the affine motionassumption for all atoms.Front ViewTop ViewSide Viewa/3a/3a/3a/3a/3 a/3aaaaaak_bk_rk_mx_1x_2x_2x_3x_1x_3Extra Credit: When you impose the deformation on the unit cell, assume that the 8outer atoms move in an affine manner but that the 8 inner atoms take up positionsthat minimize the stored energy of the cell. Find the macroscopic (small strain) elasticmoduli for this