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Elastoplasticity Theory (Backmatter Pages)

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Elastoplasticity Theory (Backmatter Pages) Appendixes Appendix 1: Projection of Area Consider the projection of the area having the unit normal vector n onto the surface having the normal vector m in Fig. A.1. Now, suppose the plane (abcd in Fig. A.1) which contains the unit normal vectors m and n. Then, consider the line ef obtained by cutting the area having the unit normal vector n by this plane. Further, divide the area having the unit normal vector n to the narrow bands perpendicular to this line and their projections onto the surface having the normal vector m. The lengths of projected bands are same as the those of the original bands but the projected width db are obtained by multiplying the scalar product of the unit normal vectors, i.e. m • n to the original widths db . Eventually, the projected area da is related to the original area da as follows: da = m • nda (A.1) a e n b db f m d db= nm• db c Fig. A.1 Projection of area 422 Appendixes Appendix 2: Proof of ∂(FjA/J)/∂x j = 0 ∂ FjA 1 ∂(∂x /∂X ) ∂x ∂J J = j A J − j ∂x J2 ∂x ∂X ∂x j ⎧ j A j ⎫ ∂ ∂ ∂ ⎨ ∂ε x1 x2 x3 ⎬ ∂(∂x /∂X ) ∂ ∂ ∂ ∂x PQR ∂X ∂X ∂X = 1 j A ε x1 x2 x3 − j P Q R 2 PQR J ⎩ ∂x j ∂XP ∂XQ ∂XR ∂XA ∂x j ⎭ ∂ 2 ∂ 2 ∂ 2 ∂ ∂ ∂ = 1 x1 + x2 + x3 ε x1 x2 x3 J2 ∂X ∂x ∂X ∂x ∂X ∂x PQR ∂X ∂X ∂X A 1 A 2 A 3 P Q R ∂ ∂ 2 ∂ ∂ ∂ ∂ ∂ 2 ∂ − ε x1 x1 x2 x3 + x2 x1 x2 x3 PQR ∂X ∂X ∂x ∂X ∂X ∂X ∂X ∂X ∂x ∂X A P 1 Q R A P Q 2 R ∂ ∂ ∂ ∂ 2 + x3 x1 x2 x3 ∂X ∂X ∂X ∂X ∂x A P Q R 3 ∂ 2 ∂ ∂ ∂ ∂ ∂ 2 ∂ ∂ = 1 ε x1 x1 x2 x3 + x1 x2 x2 x3 J2 PQR ∂X ∂x ∂X ∂X ∂X ∂X ∂X ∂x ∂X ∂X A 1 P Q R P A 2 Q R ∂ ∂ ∂ 2 ∂ + x1 x2 x3 x3 ∂X ∂X ∂X ∂x ∂X P Q A 3 R ∂ 2 ∂ ∂ ∂ ∂ ∂ 2 ∂ ∂ − x1 x1 x2 x3 + x2 x2 x1 x3 ∂X ∂x ∂X ∂X ∂X ∂X ∂X ∂x ∂X ∂X P 1 A Q R A Q 2 P R ∂ ∂ ∂ 2 ∂ + x1 x2 x3 x3 ∂X ∂X ∂X ∂x ∂X P Q R 3 A ∂ 2 ∂ ∂ ∂ ∂ 2 ∂ ∂ ∂ ∂ 2 = 1 ε x1 x2 x3 + x1 x2 x3 + x1 x2 x3 J2 PQR ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X ∂X A P Q R P A Q R P Q A R ∂ 2x ∂x ∂x ∂x ∂ 2x ∂x ∂x ∂x ∂ 2x − 1 2 3 + 1 2 3 + 1 2 3 = 0 ∂XP∂XA ∂XQ ∂XR ∂XP ∂XQ∂XA ∂XR ∂XP ∂XQ ∂XR∂XA (A.2) Appendix 3: Covariant and Contravariant Base Vectors and Components Consider the general curvilinear coordinate system (θ 1,θ 2,θ 3) with the primary base i {ai} and the locally defined coordinate system (θ1,θ2,θ3) with the reciprocal base {a }. The infinitesimal line-element dx is described from Eq. (1.35) in these coordinate sys- tems as follows: ⎧ ⎪ ∂x ⎨ dθ i = dθ ia ∂θi i dx = (A.3) ⎩⎪ i dθ ai Appendixes 423 with i i dθ = dx • a , dθi = dx • ai (A.4) ∂x ∂θi a = , ai = (A.5) i ∂θi ∂x which satisfies Eq. (1.34), i.e. j ∂x ∂θ j a • a j = • = δ (A.6) i ∂θi ∂x i ∗ ∗ ∗ Now, consider the another curvilinear coordinate system (θ 1,θ 2,θ 3) with the ∗ ∗ ∗ ∗ base {a } and the locally defined coordinate system (θ ,θ ,θ ) with the reciprocal ∗i 1 2 3 base {a i}. The following coordinate transformation rules hold for the base vectors by the chain rule of differentiation. ∗ ∗ ∗ ∂x ∂x ∂θ j ∂θ j ∂θ j ∗ ∂θ j ∂θr a = = = a , a = a = a = δ ra (A.7) i ∂θ∗i ∂θ j ∂θ∗i ∂θ∗i j i ∂θi j ∂θi ∂θ∗ j r i r ∗ ∗ ∗ ∂θ i ∂θi ∗ ∂θi ∂θ j a i = a j, ai = a j = ar = δ iar (A.8) ∂θ j ∂θ∗ j ∂θ∗ j ∂θr r The vector v is described by ⎧ j ⎪ ∗ ∗ ∗i ∂θ ⎪ via = v ia = v a ⎨ i i ∂θ∗i j v = (A.9) ⎪ ∗ ⎩⎪ ∗ ∗ ∗ ∂θ i v ai = v a i = v a j i i i ∂θ j from which one has the transformation rules: ⎫ i ∗i ∂θ ∗ ∗ ∂θ ⎪ vi = v j, v i = v j ⎪ ∂θ∗ j ∂θ j ⎬ (A.10) ∗ j j ⎪ ∂θ ∗ ∗ ∂θ ⎪ v = v , v = v ⎭ i ∂θi j i ∂θ∗i j In the analogous way, one has the following transformation rules for the second-order tensor. ⎫ ∗i ∗ j i j ∗ ∂θ ∂θ ∂θ ∂θ ∗rs ⎪ t ij = trs, tij = t ⎪ ∂θr ∂θs ∂θ∗r ∂θ∗s ⎪ ⎪ ⎪ ∗i s i ∗s ⎪ ∗ ∂θ ∂θ ∂θ ∂θ ∗ ⎪ t i = tr , ti = t r ⎪ • j ∂θr ∂θ∗ j •s • j ∂θ∗r ∂θ j •s ⎬ (A.11) r ∗ j ∗r j ⎪ ∗• j ∂θ ∂θ • • j ∂θ ∂θ ∗ • s ⎪ i = s, = r ⎪ t ∗i s Tr ti i ∗s t ⎪ ∂θ ∂θ ∂θ ∂θ ⎪ ⎪ ∗ ∗ ⎪ ∗ ∂θr ∂θs ∂θ r ∂θ s ∗ ⎭⎪ t = t , t = t ij ∂θ∗i ∂θ∗ j rs ij ∂θi ∂θ j rs 424 Appendixes It can be recognized from Eqs. (A.7), (A.8), (A.10) and (A.11) that the transforma- tion rules of the base vectors and the components of vector and tensor are classified into the type with the subscript and the other type with the superscript. The partial-derivative ∗ operator ∂θi or ∂θ i is placed in the denominator and the numerator for the base vec- i tor or component denoted by ()i and () , respectively, in their transformation rules. The former and the latter are referred to as the “covariant”andthe“contravariant”, respectively, base vector, component and description of vector and tensor. Appendix 4: Euler’s Theorem for Homogeneous Function The homogeneous function of degree n is defined to fulfill the relation n f (ax1, ax2, ···, axm)=a f (x1, x2, ···, xm) (A.12) for the variables x1, x2,···, xm, letting a denote an arbitrary scalar constant. Then, con- sider the homogeneous function given by the polynomial expression: s ni ni i ( , , ···, )= 1 2 ··· nm f x1 x2 xm ∑ ci x1 x2 xm (A.13) i=1 where s is the number of terms of polynomial expression and ci are constants, provided to fulfill m i = ∑ n j n for each i (A.14) j=1 Eq. (A.13) leads to m s m s ∂ f (x , x , ···, x ) ni ni ni −1 i ni ni i 1 2 m x = c ni x 1 x 2 ···x j ···xnm x = n c x 1 x 2 ···xnm ∑ ∂ j ∑ ∑ i j 1 2 j m j ∑ i 1 2 m j=1 x j i=1 j=1 i=1 Then, it holds that m ∂ ( , , ···, ) f x1 x2 xm = ( , , ···, ) ∑ ∂ x j nf x1 x2 xm (A.15) j=1 x j which is called the Euler’s theorem for homogeneous function. For the simple example (m =3,n =4,s =3): f (x, y, z)=αx4 + βx3y + γx2yz Eq. (A.15) is confirmed as follows: ∂ f ∂ f ∂ f x + y + z =(4αx3 + 3βx2y + 2γxyz)x +(βx3 + γx2z)y + γx2y • z = 4 f ∂x ∂y ∂z Eq. (A.15) yields Eq. (6.33) for the yield function (n =1). Appendixes 425 Appendix 5: Normal Vector of Surface The quantity (∂ f (t)/∂t): dt is regarded as the scalar product of the vectors ∂ f (t)/∂t and dt in the nine-dimensional space (t11,t12,t33,···,t31,t13) . Here, it holds that ⎧ ⎨ > 0: dt is directed outward-normal to surface ∂ f (t) : dt = 0: dt is directed tangential to surface (A.16) ∂t ⎩ < 0: dt is directed inward-normal to surface Therefore, ∂ f (t)/∂t is interpreted to be the vector designating the outward-normal of the surface. This fact holds also for the yield surface f (σ )=F. Appendix 6: Relationships of Material Constants in lnv − ln p and e − ln p Linear Relations The following relation holds from Eqs. (2.131), (11.3) and (11.14) for pe = 0, provided εe = ( + εe) that Eq. (2.131) holds for elastic volumetric strain, i.e. v ln 1 v . p κ p −κ˜ ln = ln 1 − ln (A.17) p0 1 + e0 p0 from which one has κ ( − p ) ln 1 + ln κ = 1 e0 p0 ˜ p (A.18) −ln p0 ∼ It follows from Eq. (A.18) for infinitesimal deformation under p = p0 that κ − 1 1 + e0 p κ p κ p ln(1 − ln ) 1 − ln 1 + e p 1 + e p κ lim κ˜ = lim 0 0 = lim 0 0 = (A.19) → → p → + p p0 p p0 −ln p p0 − 1 1 e0 p0 p resulting in κ κ˜ =∼ (A.20) 1 + e0 Further, substituting Eqs. (11.3) and (11.4) into Eq. (2.131), i.e. εv = ln(1 + εv),itfol- lows that p p κ p κ p λ p −κ˜ ln − (λ˜ − κ˜)ln y = ln 1 − ln + ln y − ln y p0 py0 1 + e0 p0 1 + e0 py0 1 + e0 py0 426 Appendixes i.e. p p κ p p λ p κ˜ y − − + y − − y ln ln ln 1 + ln ln + ln λ˜ = py0 p0 1 e0 py0 p0 1 e0 py0 p ln y py0 from which one has λ λ˜ = lim + (A.21) p → p0 1 e0 py → py0 Based on Eqs.
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