Question

(a) For a unit cube with sides along the coordinate axes, what is its deformed volume? What is the deformed area of the e, face of the cube? (b) If the Cauchy stress tensor is given by 712 = 721 = 100 MPa, and all other Tij = 0, calculate the first Piola-Kirchhoff stress tensor and the corresponding pseudo-stress vector for the plane whose undeformed plane is the e-plane and compare it with the Cauchy stress vector in the deformed state, (c) Calculate the second Piola-Kirchhoff tensor and the corresponding pseudo-stress vector for the plane whose undeformed plane is the e-plane. Also calculate the pseudo-differential force for the same plane. 4.46 The deformation of a body is described by x1 = 2X 13 = 2X2, x3 = 2X3. (a) For a unit cube with sides along the coordinate axes, what is its deformed volume? What is the deformed area of the e, face of the cube? (b) If the Cauchy stress tensor is given by 100001 0 100 0 Mpa, Lo 0100] calculate the first Piola-Kirchhoff stress tensor and the corresponding pseudo-stress vector for the plane whose undeformed plane is the e-plane and compare it with the Cauchy stress vector on its deformed plane. (c) Calculate the second Piola-Kirchhoff tensor and the corresponding pseudo-stress vector for the plane whose undeformed plane is the e-plane. Also calculate the pseudo-differential force for the same plane.

Answer 1

Rather simply using the formulas and finding answers, i would like to give an idea on how the formulas have been formed.

additionally as components of stress tensor have a physical interpretation like $\sigma$ ₁₁ is normal stress in x direction in x plane
,  $\sigma$ ₁₂ is shear stress is stress in y direction and in x plane and and so on , but all these components are in spatial coordinate system, and we need a stresses in material configuration system to perform operations like integration , this transformation leads to piola kirchoff stress tensors.

so these tensors do not have any physical significance but only are mathematical quantities.

hope you got a rough idea of what piola kirchoff tensors are.

now the first answer are attached in following pages
page 1:

Giuen mi : 2x ^,:2X, 243 there on deformation mappings from material - co-ordinate to spatial co-ordinat systems -- Deformation Heunds ! demo for mere om du and initial vol.is Let the final Volume be a good on to on a niken final volume. -> dus. Izdelf v final volume. , Saitine Volume : 1868131 Initial volume: 1x181=1 Initial volume a final volume : JX1 1 adet fx To z o le Det f=8 Deformed volume is 8 units. for area change, Nansons famula ginen Relation ie. - nda = JIT N da la face of cube is initial diree". he Noll,0,0] Ket final diveo be ? comide a live element /dXll of unit length & direi n a final diree" udall & divee" ? » f-day * r du = Fildall N = p o ol 110

page 2:

soldull= 2 & nato final diret" remain same but length double. 20=(1,0, ) & N:11,0, 0) Ming thin dA: 1x1=1 ? da = delf f - Nxi f 1 = 4 adj &. a Detf llu oon 1 % 40 01 lo 0 y lo o 42 * fides 82 ft : 1:)* 1000:10] comparing components da = 4 a) deformed area : 4 unit.. or simply fine final Vol. ic 8 unit and there is no - shape change a Side Length = 2 units & aua = 2X2 = Yunits (Am) (6) Balaning of linear momentum in spatial form Je lädo = fb do It da i t-traction. Jo s bubody force funnit voll converting them to spatial material ä acco the form I demity . a = 4(x,+) do J dv Drobe, J = det f. t = { from Cauchy's formule.

page 3:

hing there. tda - Enda - SB SB Now substifuting, Jedan proda: sept") da . from Nansoni famula. jäldo 15 d. J(5 9?") N da Bu w this eqr is in material form. Kirchoof eqa is P A JOAT 1 » 1st Piola L00 L 0001 X 8 de To 0 100 loy23 = 8x550 0 1 TO SO 0 lo o sol MOO لا 400 LoJo lo o yoo J Given that undeformed plane is e, » N=(1,0,011 * ! :PN - pano : 11:7-131 Pseudo stress vector lo As we found earlier , the undeformed plane diver" Ņ and deformed plane have same direi" n=11,0,0) > t = 9:11000 om 1000 Cauchy Stress vector lo o 100 T-4001 I 100 - DIE

page 4:

Aun) (e) Second Piola Kirchoff terror ? - first traction I = df force & ana in spatial system. - da adt = ţ da » df = &nda arJF-TN dA eNdA If we Pull back dlo to material system, dfo = f'df f'PNdA figÇEN da na dto. (J&*!")N da in material form vş Ş:JEET Or Y2 0 0 1 Oyo 07 110 120 100 100 142 0 100 6 0 Y2 0 To 0 Y2 JLO = x 420 150 001 o Y2 0 0 SO le 0 V2 JLO 0 0 -C = x 25 0 07 o 25 0 0 0 25 52000 01 planet To 0 200 corresponding Pseudo stress vector in undeformed Lowonose diveen is N:11,0,071 SN.200 0 1117 1200 200 0 200 JOJ . Vector & dfo = s NdA & N=11,0,0) de 18 ] where doo is force This only has i component sildfo 1 = 200 - If we comider unit area P dfo=df = 20 DAN

hope you understood my explanation and got your answer

hope that you liked my effort and give a positive review which will motivate me to write more.

thanking you.