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. (A.19) and (A.21), λ˜ and κ˜ may be given by λ κ λ˜ = , κ˜ = (A.22) 1 + e0 1 + e0 which can be calculated from a plenty of data on λ and κ accumulated in the past. Then, the analysis would be improved over the finite deformation by using Eq. (11.4) with Eq. (A.22) instead of Eq. (11.15) or (11.17). Obviously, it is more appropriate to find the relations of λ˜ and κ˜ to λ and κ which are applicable over the whole range of pressure in relevant analysis. Needless to say, one has to determine the material parameters λ˜ and κ˜ directly from test data for soils without the data of λ and κ or for the case that an accurate formulation is required. Here, note that the curve fitting of lnv − ln p linear relation to test data is easier than the fitting of the e−ln p linear relation to test data because real soil behavior is far nearer to the former than the latter.

Appendix 7: Derivation of Eq. (11.22)

Differentiation of Eq. (11.18) under the condition f (σ ) = const. leads to

∂η ∂η (η ) + (η ) m + m σ  g m dp pg m ∂ dp ∂σ d P σ  1 = g(η )dp+ pg(η ) − dp+ dσ  = 0 m m p2M pM from it holds that σ    g(η ) − g (η ) dσ  m pM m g(η ) = = M m − η (A.23) dp 1 g(η ) m g(η ) m M m  Considering dσ /dp =0atηm = 1 in Eq. (A.23), one has Eq. (11.22). Appendixes 427

Appendix 8: Convexity of Two-Dimensional Curve

When the curve is described by the polar coordinates (r, θ) asshowninFig.A.2,the following relation holds rdθ tanα = (A.24) dr where α is the angle measured from the radius vector to the tangent line in the anti- clockwise direction. Eq. (A.24) is rewritten as

r cotα = (A.25) r where ( ) designates the first order differentiation with respect to θ. The equation of the tangent line at (r, θ) of the curve r = r(θ) is described by the following equation by using the current coordinates (R, Θ) on the tangent line.

Rcos[Θ −{θ − (π/2 − α)}]=rcos(π/2 − α) which is rewritten as 1 1 −Rsin(Θ − θ − α)=rsinα → = − Rsin(Θ − θ − α) rsinα

1 1 1 → = cos(Θ − θ) − cotα sin(Θ − θ) R r r Substituting Eq. (A.25) to this equation and noting (1/r) = −r/r2, one has the relation

Curve y α

r r R d d α rd θ r dθ T

a θ n d g r e n

t

l i α n − e π/2 θ Θ Θθπ−−{(/2)} −α 0 x

Fig. A.2 Curve in the polar coordinate (r, θ) 428 Appendixes

 1 1 1 = cos(Θ − θ)+ sin(Θ − θ), (A.26) R r r

Equation (A.26) is rewritten by applying the Taylor expansion to cosϑ and sinϑ as

 1 1 1 1 1 = cosϑ + sinϑ = 1 − ϑ 2 + ··· R(Θ) r(θ) r(θ) r(θ) 2  + 1 ϑ − 1ϑ 3 + ··· r(θ) 6  1 1 1 1 = + ϑ − ϑ 2 + ··· (A.27) r(θ) r(θ) 2 r(θ) where ϑ ≡ Θ − θ. On the other hand, the radius r(Θ) (Θ = θ + ϑ)ofthecurveis described by the Taylor expansion as follows:

 1 1 1  1 1 = +( ) ϑ + ϑ 2 + ··· (A.28) r(Θ) r(θ) r(θ) 2 r(θ)

Eqs. (A.27) and (A.28) lead to

 1 1 1 1 1 − = + ϑ 2 + ··· (A.29) r(Θ) R(Θ) 2 r(θ) r(θ)

In order that the curve is convex (r(Θ) ≤ R(Θ)), the following inequality must hold from Eq. (A.29).  1 1 + > 0 (A.30) r r

Appendix 9: Flow Rules with Plastic Spin in Multiplicative Hyperelasto-Plasticity

The flow rules with the plastic spin is shown below, which is referred to Yamakawa and Hashiguchi (2011) in which some numerical results are shown. p The stress which is work-conjugate with plastic velocity gradient L in the inter- mediate configuration K in Eq. (6.12) is given by the Mandel stress (Hashiguchi and Yamakawa, 2012): e M ≡ C S (A.31) where e C = FeT Fe (A.32)

S = Fe−1τ Fe−T = FpSFpT (A.33) Appendixes 429

Here, S is given by the hyperelasticity as

e ∂ψe(C ) S = 2 e (A.34) ∂C Ð The quantities with the over-bar ( ) are based in the intermediate configuration Kø . The plastic deformation gradient Fp is further decomposed multiplicatively into the p kinematic hardening-elastic deformation gradient Fe (energy storage part) and the kine- p matic hardening-dissipative deformation gradient Fd (energy dissipative part) as fol- lows: p = p p F Fe Fd (A.35) p Here, the configuration attained by extracting Fe from the intermediate configuration Kø is called the kinematic hardening-intermediate configuration and designated by K˜ (Lion, 2000). Then, the following storage variable induced in the kinematic hardening is incorporated. ˜ p = pT p Ce Fe Fe (A.36) The quantities with the over-tilde (˜) are based in the intermediate configuration K˜ for p the kinematic hardening. Further, the kinematic hardening variable S˜ e is also given by the hyperelastic form as follows: ∂ψp( ˜ p) ˜ p = e Ce Se 2 p (A.37) ∂C˜ e The Mandel stress-like variable for the kinematic hardening is given by ˜ p = ˜ p ˜ p Me Ce Se (A.38)

Further, we introduce the following velocity gradients for the deformation and the kinematic hardening. ⎫ • p − p p ⎪ L = FpFp 1 = D + W ⎬ (A.39) p p p p ⎭⎪ D = sym[L ], W = ant[L ]

• ⎫ ˜ p = p p−1 = ˜ p + ˜ p ⎬ Ld Fd Fd Dd Wd ⎭ (A.40) ˜ p = [ ˜ p], ˜ p = [ ˜ p] Dd sym Ld Wd ant Ld p where W and W˜ p are decomposed as follows: ⎫ • • • p − • − − ⎬ W = ant[F pFp 1]=ant[(RpUp) (RpUp) 1]=ant[Rp UpUp 1RpT ]+RpRpT • • • ˜ p = [ p p−1]= [( p p) • ( p p)−1]= [ p p p−1 pT ]+ p pT ⎭ Wd ant F dFd ant RdUd Rd Ud ant Rd Ud Ud R RdRd (A.41) 430 Appendixes

p = p p p = p p based on the polar decompositions F R U and Fd Rd Ud . The first and the second terms are called the plastic spin and the constitutive spin, respectively, in the plastic material spin W˜ p (Dafalias, 1988). Based on the isoclinic concept (Mandel, 1972), • • p pT = p pT = we assume that the constitutive spin is not induced, i.e. R R O and Rd Rd O, resulting in ⎫ • p − ⎬ W = ant[Rp UpUp 1RpT ] • (A.42) ˜ p = [ p p p−1 pT ] ⎭ Wd ant Rd UdUd Rd The plastic strain rate is induced by the variation of the plastic deformation gradi- ent tensor Fp one of the base vectors of which lives in the intermediate configuration. Therefore, the yield condition would have to be defined in the intermediate configura- tion as follows: f (Mˆ )=F(H) (A.43) where ˆ ≡ − p M M Me (A.44)

→ p =p ˜ p •Gø = p−T ˜ p pT Me e Me Gø Fe Me Fe (A.45) p p The symmetric part of the plastic velocity gradient L , i.e. the plastic strain rate D ˜ p ˜ p and the symmetric part of the velocity gradient Ld , i.e. the dissipative part Dd of the kinematic hardening are given as follows (Hashiguchi and Yamakawa, 2012):

• p D = λ Nˆ (A.46)

 • M˜ p D˜ p = bλ e (A.47) d c noting Eqs. (6.88) and (6.114), where ∂ f (Mˆ ) ∂ f (Mˆ ) Nˆ ≡ / (Nøˆ  = 1) (A.48) ∂M ∂Mø b and c are the material constants. • p ˜ p p = Further, noting that W and Wd are induced in the plastic deformation process U p O leading to D = O as shown in Eq. (A.42) and substituting Eqs. (A.46) and (A.47), they may be formulated as follows:

• p p p W = η p(M D − D M)=η p λ(M Nˆ − Nˆ M) (A.49) ˜ p = η p( ˜ p ˜ p − ˜ p ˜ p)=η p( ˜ p ˜ˆ p − ˜ˆ p ˜ p) Wd Me D D Me d Me Ne NeMe η p η p ˜ˆ p ˜ p where and d are the material parameters, and Ne and D are defined by

← ← ˜ N˜ˆ p =p N¯ˆ •G = FpT NFˆ p−T, D˜ p =p D p = FpTD¯ pFp (A.50) e e G˜ e e e G˜ G˜ e e References

Aifantis, E.C.: On the microstructural origin of certain inelastic models. J. Eng. Material Tech. (ASME) 106, 326–330 (1984) Alonso, E.E., Gens, A., Josa, A.: A constitutive model for partially saturated soils. Geotech- nique 40, 405–430 (1990) Amorosi, A., Boldini, D., Germano, V.: Implicit integration of a mixed isotropic–kinematic hard- ening plasticity model for structured clays. Int. J. Numer. Anal. Methods Geomech. 32, 1173– 1203 (2007) Anand, L.: A constitutive model for interface friction. Comput. Mech. 12, 197–213 (1993) Argyris, J.H.: Elasto-plastic matrix analysis of three dimensional continua. J. Roy. Aeronaut. Soc. 69, 231–262 (1965) Argyris, J.H., Faust, G., Szimma, J., Warnke, E.P., William, K.J.: Recent developments in the finite element analysis of PCRV. In: Proc. 2nd Int. Conf. SMIRT, Berlin (1973) Armstrong, P.J., Frederick, C.O.: A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N 731 (1966) (or in Materials at High Temperature 24, 1–26 (2007)) Asaoka, A., Nakano, M., Noda, T.: Soil-water coupled behaviour of heavily overconsolidated clay near/at critical state. Soils and Foundations 37(1), 13–28 (1997) Asaro, R., Lubarda, V.: Mechanics of Solids and Materials. Cambridge Univ. Press (2006) Batdorf, S.B., Budiansky, B.: A mathematical theory of plasticity based on the concept of slip. NACA TC1871, 1–31 (1949) Baumberger, T., Heslot, F., Perrin, B.: Crossover from creep to inertial motion in friction dynam- ics. Nature 30, 544–546 (1994) Bay, N., Wanheim, T.: Real area of contact and friction stresses at high pressure sliding contact. Wear 38, 201–209 (1976) Bazant, Z.P., Cedolin, L.: Stability of Structures. Oxford Univ. Press (1991) Becker, E., Burger, W.: Kontinuumsmechanik. B.G. Teubner, Stuttgart (1975) Belytschko, T., Liu, W.K., Moran, B.: Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons (2000) Bensson, J., Cailletaud, G., Chaboche, J.L., Forest, S., Bletry, M.: Non-linear Mechanics of Ma- terials. Springer (2001) Bertram, A.: Elasticity and Plasticity of Large Deformations. Springer (2008) Bingham, E.C.: Fluidity and Plasticity. McGraw-Hill, New York (1922) Biot, M.A.: Mechanics of incremental deformations. John Wiley & Sons, New York (1965) Bishop, A.W., Webb, D.L., Lewin, P.I.: Undisturbed samples of London clay from the Ashford Common shaft: strength-effective stress relationships. Geotechnique 15, 1–31 (1965) 432 References

Bland, D.R.: The associated flow rule of plasticity. J. Mech. Phys. Solids 6, 71–78 (1957) Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge Univ. Press (1997) Borja, R.I., Sama, K.M., Sanz, P.F.: On the numerical integration of three-invariant elastoplastic constitutive models. Comp. Meth. Appl. Mech. Eng. 192, 1227–1258 (2003) Borja, R.I., Tamagnini, C.: Cam-clay plasticity, Part III: Extension of the infinitesimal model to include finite strains. Comp. Meth. Appl. Mech. Eng. 155, 73–95 (1998) Bowden, F.P., Tabor, D.: The Friction and Lubrication of Solids. Clarendon Press (1958) Brockley, C.A., Davis, H.R.: The time-dependence of static friction. J. Lubr. Tech. (ASME) 90, 35–41 (1968) Bruhns, O.T.: Continuum Mechanics with Logarithmic Rate. Lecture Note in Kyushu University (2003) Budiansky, B.: A reassessment of deformation theories of plasticity. J. Appl. Mech. (ASME) 20, 259–264 (1959) Bureau, L., Baumberger, T., Caroli, C., Ronsin, O.: Low-velocity friction between macroscopic solids. C.R. Acad. Sci. Paris, Series IV, Different faces of Tribology 2, 699–707 (2001) Burland, J.B.: The yielding and dilatation of clay. Correspondence, Geotechnique 15, 211–214 (1965) Butterfield, R.: A natural compression law for soils (an advance on e-log p’). Geotechnique 29, 469–480 (1979) Callari, C., Auricchio, F., Sacco, E.: A finite-strain Cam-clay model in the framework of multi- plicative elasto-plasticity. Int. J. Plasticity 14, 1155–1187 (1998) Carlson, D.E., Hoger, A.: The derivative of a tensor-valued function of a tensor. Quart. Appl. Math. 406, 409–423 (1986) Castro, G.: Liquefaction of Sands. PH. D. Thesis, Harvard Soil Mech. Series 81 (1969) Chaboche, J.L.: Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plas- ticity 5, 247–302 (1989) Chaboche, J.L.: On some modifications of kinematic hardening to improve the description of ratcheting effects. Int. J. Plasticity 7, 661–678 (1991) Chaboche, J.L.: A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plas- ticity 24, 1642–1693 (2008) Chaboche, J.L., Dang-Van, K., Cordier, G.: Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. In: Trans. 5th Int. Conf. SMiRT, Berlin, Division L, Paper No. L. 11/3 (1979) Chaboche, J.L., Rousselier, G.: On the plastic and viscoplastic constitutive equations. Parts I and II, J. Pressure Vessel and Tech (ASME) 165, 153–164 (1983) Chadwick, P.: Continuum Mechanics. Goerge Allen & Unwin Ltd. (1976) Cheng, J.-H., Kikuchi, N.:: An incremental constitutive relation of uniaxial contact friction for large deformation analysis. J. Appl. Mech. (ASME) 52, 639–648 (1985) Christoffersen, J., Hutchinson, J.W.: A class of phenomenological corner theories of plasticity. J. Mech. Phys. Solids 27, 465–487 (1979) Chowdhury, E.Q., Nakai, T., Tawada, M., Yamada, S.: A model for clay using modified stress under various loading conditions with the application of subloading concept. Soils and Found. 39(6), 103–116 (1999) Coombs, W.M., Crouch, R.S.: Algorithmic issues for three-invariant hyperplastic Critical State models. Comp. Meth. Appl. Mech. Eng. 200, 2297–2318 (2011) Coombs, W.M., Crouch, R.S., Augarde, C.E.: A unique Critical State two-surface hyperplasticity model for fine-grained particulate media. J. Mech. Phys. Solis. 61, 175–189 (2013) Cosserat, E., Cosserat, F.: Theorie des Corps Deformation, Traite de Physique, transl. E. Davaux, ed. O.D. Chwolson, 2nd ed., 2, Paris, pp. 953–1173 (1909) References 433

Cotter, B.A., Rivlin, R.S.: Tensors associated with time-dependent stresses. Quart. Appl. Math. 13, 177–182 (1955) Courtney-Pratt, J.S., Eisner, E.: The effect of a tangential force on the contact metallic bodies. Proc. Roy. Soc. A 238, 529–550 (1957) Cundall, P., Board, M.: A microcomputer program for modeling large-strain plasticity problems. Prepare for the 6th Int. Conf. Numer. Meth. Geomech., Innsbruck, Austria, pp. 2101–2108 (1988) Curnier, A.: A theory of friction. Int. J. Solids Struct. 20, 637–647 (1984) Dafalias, Y.F., Herrmann, L.R.: A bounding surface soil plasticity model. In: Proc. Int. Symp. Soils Cyclic Trans. Load., Swansea, pp. 335–345 (1980) Dafalias, Y.F.: Corotational rates for kinematic hardening at large plastic deformations. J. Appl. Mech (ASME) 50, 561–565 (1983) Dafalias, Y.F.: The plastic spin concept and a simple illustration of its role in finite plastic trans- formation. Mech. Materials 3, 223–233 (1984) Dafalias, Y.F.: The plastic spin. J. Appl. Mech. (ASME) 52, 865–871 (1985a) Dafalias, Y.F.: A missing link in the macroscopic constitutive formulation of large plastic defor- mations. In: Sawczuk, A., Bianchi, G. (eds.) Int. Symp. Recent Trends and Results in Plastic- ity. Plasticity Today, pp. 135–151. Elsevier Publ. (1985b) Dafalias, Y.F.: Bounding surface plasticity. I: Mathematical foundation and hypoplasticity. J. Eng. Mech. (ASCE) 112, 966–987 (1986) Dafalias, Y.F.: Plastic spin: Necessity or redundancy? Int. J. Plasticity 14, 909–931 (1998) Dafalias, Y.F.: Finite elastic-plastic deformations: beyond the plastic spin. Theor. Appl. Mech. 38, 321–345 (2011) Dafalias, Y.F., Popov, E.P.: A model of nonlinearly hardening materials for complex loading. Acta Mech. 23, 173–192 (1975) Dafalias, Y.F., Popov, E.P.: Plastic internal variables formalism of cyclic plasticity. J. Appl. Mech. (ASME) 43, 645–651 (1976) Dafalias, Y.F., Popov, E.P.: Cyclic loading for materials with a vanishing elastic domain. Nucl. Eng. Design 41, 293–302 (1977) de Borst, R., Crisfield, M.A., Remmers, J.J.C., Verhoosed, C.V.: Nonlinear Finite Element Anal- ysis of Solids and Structures. Wiley Series in Computational Mechanics, 2nd edn. John-Wiley (2012) de Borst, R., Sluys, L.J., Muhlhaus, H.B., Pamin, J.: Fundamental issues in finite element analyses of localization of deformation. Eng. Comput. 10, 99–121 (1993) de Souza Neto, E.A., Peric, D., Owen, D.J.R.: Computational Methods for Plasticity. John Wiley and Sons (2008) Derjaguin, B.V., Push, V.E., Tolstoi, D.M.: A theory of stick-slipping of solids. In: Proc. Conf. Lubr. and Wear. Inst. Mech. Eng., London, pp. 257–268 (1957) Desai, C., Siriwardane, H.J.: Constitutive Laws for Engineering Materials with Emphasis on Ge- omatrials. Prentice-Hall, Inc. (1984) Dienes, J.K.: On the analysis of rotation and stress rate in deforming bodies. Acta Mech. 32, 217–232 (1979) Dokos, S.J.: Sliding friction under extreme pressure –I. Trans. ASME 68, A148–A156 (1946) Delobelle, P., Robinet, P., Bocher, L.: Experimental study and phenomenological modelization of ratchet under uniaxial and biaxial loading on austenitic stainless steel. Int. J. Plasticity 11, 295–330 (1995) Drucker, D.C.: A more fundamental approach to plastic stress-strain relations. In: Proc. 1st U.S. National Congr. Appl. Mech. (ASME), vol. 1, pp. 487–491 (1951) Drucker, D.C., Prager, W.: Soil mechanics and plastic analysis or limit design. Quart. Appl. Math. 10, 157–165 (1952) 434 References

Drucker, D.C.: Conventional and unconventional plastic response and representation. Appl. Meek. Rev (ASME) 41, 151–167 (1988) Dvorkin, E.N., Goldschmit, M.B.: Nonlinear Continua. Springer (2006) Eckart, G.: Theory of elasticity and inelasticity. Phys. Rev. 73, 373–380 (1948) Eidel, B., Gruttmann, F.: Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation. Compt. Materials Sci. 28, 732–742 (2003) Ellyin, F.: An anisotropic hardening rule for elastoplastic solids based on experimental observa- tions. J. Appl. Mech. (ASME) 56, 499–507 (1989) Ellyin, F.: Fracture Damage, Crack Growth and Life Prediction. Chapman & Hall (1997) Ellyin, F., Xia, Z.: A rate-independent constitutive model for transient non-proportional loading. J. Mech. Phys. Solids 37, 71–91 (1989) Eringen, A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York (1962) Eringen, A.C.: Mechanics of Continua. Rebert E. Krieger Publishing Co., New York (1967) Ferrero, J.F., Barrau, J.J.: Study of dry friction under small displacements and near-zero sliding velocity. Wear 209, 322–327 (1997) Flanagan, D.P., Taylor, L.M.: An accurate numerical algorithm for stress integration with finite rotations. Comput. Meth. Appl. Mech. Eng. 62, 305–320 (1987) Flugge, W.: Tensor Analysis and Continuum Mechanics. Springer (1972) Fredriksson, B.: Finite element solution of surface nonlinearities in structural mechanics with special emphasis to contact and fracture mechanics problems. Comput. Struct. 6, 281–290 (1976) Fukutake, K., Ohtsuki, M., Sato, M.: Analysis of saturated dense sand-structure system and com- parison with results from shaking table test. Earthquake Eng. Struct. Dynamics 19, 977–992 (1990) Fung, Y.C.: Foundations of Solid Mechanics. Prentice-Hall, Inc. (1965) Fung, Y.C.: A First Course in Continuum Mechanics. Prentice-Hall, Inc. (1969) Gearing, B.P., Moon, H.S., Anand, L.: A plasticity model for interface friction: application to sheet metal forming. Int. J. Plasticity 17, 237–271 (2001) Gotoh, M.: A class of plastic constitutive equations with vertex effect. Int. J. Solids Structures 21, 1101–1163 (1985) Goya, M., Ito, K.: An expression of elastic-plastic constitutive laws incorporating vertex formu- lation and kinematic hardening. J. Appl. Mech (ASME) 58, 617–622 (1991) Green, A.E., Naghdi, P.M.: A general theory of an elastic-plastic continuum. Arch. Ration. Mech. Anal. 18, 251–281 (1965) Gudehus, G.: A comparison of some constitutive laws for soils under radially symmetric loading and unloading. In: Wittke, W. (ed.) Proc. 3rd Int. Conf. Numer. Meth. Geomech., Aachen, pp. 1309–1323. Balkema Publ., Rotterdam (1979) Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, Inc. (1981) Gutierrez, M., Ishihara, K., Towhata, I.: Model for the deformation of sand during rotation of principal stress directions. Soils and Foundations 33(3), 105–117 (1993) Hashiguchi, K.: On a yielding of frictional materials – A hardening law. In: Proc. 27th Annual Meeting, JSCE, pp. 105–108 (1972) (in Japanese) Hashiguchi, K.: Isotropic hardening theory of granular media. Proc. Japan. Soc. Civil Eng. (227), 45–60 (1974) Hashiguchi, K.: An expression of anisotropy in a plastic constitutive equation of soils. In: Mu- rayama, S., Schofield, A.N. (eds.) Constitutive Equations of Soils, Proc. 9th Int. Conf. Soil Mech. Found. Eng., Spec. Ses. 9, Tokyo, pp. 302–305, JSSMFE (1977) Hashiguchi, K.: Plastic constitutive equations of granular materials. In: Cowin, S.C., Satake, M. (eds.) Proc. US-Japan Seminar on Continuum Mech. Stast. Appr. Mech. Granular Materials, Sendai, pp. 321–329 (1978) References 435

Hashiguchi, K.: Constitutive equations of elastoplastic materials with elastic-plastic transition. J. Appl. Mech. (ASME) 47, 266–272 (1980) Hashiguchi, K.: Constitutive equations of elastoplastic materials with anisotropic hardening and elastic-plastic transition. J. Appl. Mech. (ASME) 48, 297–301 (1981) Hashiguchi, K.: Macrometric approaches -static- intrinsically time-independent. In: Constitutive Laws of Soils, Proc. Dsicuss. Ses. 1A, 11th Int. Conf. Soil Mech. Found. Eng., San Francisco, pp. 25–65 (1985a) Hashiguchi, K.: Subloading surface model of plasticity. In: Constitutive Laws of Soils. Proc. Dsicuss. Ses. 1A, 11th Int. Conf. Soil Mech. Found. Eng., San Francisco, pp. 127–130 (1985b) Hashiguchi, K.: Elastoplastic constitutive model with a subloading surface. In: Proc. Int. Conf. Comput. pp. IV65–IV70 (1986) Hashiguchi, K.: A mathematical modification of two surface model formulation in plasticity. Int. J. Solids Structures 24, 987–1001 (1988) Hashiguchi, K.: Subloading surface model in unconventional plasticity. Int. J. Solids Struc- tures 25, 917–945 (1989) Hashiguchi, K.: Fundamental requirements and formulation of elastoplastic constitutive equa- tions with tangential plasticity. Int. J. Plasticity 9, 525–549 (1993a) Hashiguchi, K.: Mechanical requirements and structures of cyclic plasticity models. Int. J. Plas- ticity 9, 721–748 (1993b) Hashiguchi, K.: Loading criterion. Int. J. Plasticity 8, 871–878 (1994) Hashiguchi, K.: On the linear relations of V-lnp and lnv-lnp for isotropic consolidation of soils. Int. J. Numer. Anal. Meth. Geomech. 19, 367–376 (1995) Hashiguchi, K.: The extended flow rule in plasticity. Int. J. Plasticity 13, 37–58 (1997) Hashiguchi, K.: The tangential plasticity. Metals and Materials 4, 652–656 (1998) Hashiguchi, K.: Fundamentals in constitutive equation: continuity and smoothness conditions and loading criterion. Soils and Foundations 40(3), 155–161 (2000) Hashiguchi, K.: Description of inherent/induced anisotropy of soils: Rotational hardening rule with objectivity. Soils and Foundations 41(6), 139–145 (2001a) Hashiguchi, K.: On the thermomechanical approach to the formulation of plastic constitutive equations. Soils and Foundations 41(4), 89–94 (2001b) Hashiguchi, K.: A proposal of the simplest convex-conical surface for soils. Soils and Founda- tions 42(3), 107–113 (2002) Hashiguchi, K.: Subloading surface model with Tangential relaxation. In: Proc. Int. Symp. Plas- ticity 2005, pp. 259–261 (2005) Hashiguchi, K.: Constitutive model of friction with transition from static- to kinetic-friction – Time-dependent subloading-friction model – . In: Proc. Int. Symp. Plasticity 2006, pp. 178– 180 (2006) Hashiguchi, K.: General corotational rate tensor and replacement of material-time derivative to corotational derivative of yield function. Comput. Model. Eng. Sci. 17, 55–62 (2007a) Hashiguchi, K.: Anisotropic constitutive equation of friction with rotational hardening. In: Proc. 13th Int. Symp. Plasticity & its Current Appl., pp. 34–36 (2007b) Hashiguchi, K.: Extended overstress model for general rate of deformation including impact load. In: Proc. 13th Int. Symp. Plasticity & its Current Appl., pp. 37–39 (2007c) Hashiguchi, K.: Verification of compatibility of isotropic consolidation characteristics of soils to multiplicative decomposition of deformation gradient. Soils and Foundations 48, 597–602 (2008) Hashiguchi, K.: Elastoplasticity Theory, 1st edn. Springer (2009) Hashiguchi, K.: General interpretations and tensor symbols for pull-back, push-forward and con- vected derivative. In: Proc. JSME 24th Comp. Mech. Conf. (JSME), pp. 669–671 (2011) Hashiguchi, K., Chen, Z.-P.: Elastoplastic constitutive equations of soils with the subloading surface and the rotational hardening. Int. J. Numer. Anal. Meth. Geomech. 22, 197–227 (1998) 436 References

Hashiguchi, K., Kuwayama, T., Suzuki, N., Ogawa, S.: Time-dependent friction model – Subloading-overstress Model-. In: Proc. Compt. Eng. Conf. Japan, vol. 17, C-9-1 (2012b) Hashiguchi, K., Mase, T.: Extended yield condition of soils with tensile strength and rotational hardening. Int. J. Plasticity 23, 1939–1956 (2007) Hashiguchi, K., Mase, T.: Physical interpretation and quantitative prediction of cyclic mobility by the subloading surface model. Japanese Geotech. J. 6, 225–241 (2011) Hashiguchi, K., Okayasu, T., Saitoh, K.: Rate-dependent inelastic constitutive equation: The ex- tension of elastoplasticity. Int. J. Plasticity 21, 463–491 (2005a) Hashiguchi, K., Ozaki, S.: Constitutive equation of friction with rotational and orthotropic anisotropy. J. Appl. Mech (JSCE) 10, 383–389 (2007) Hashiguchi, K., Ozaki, S.: Constitutive equation for friction with transition from static to kinetic friction and recovery of static friction. Int. J. Plasticity 24, 2102–2124 (2008a) Hashiguchi, K., Ozaki, S.: Anisotropic constitutive equation for friction with transition from static to kinetic friction and vice versa. J. Appl. Mech (JSCE) 11, 271–282 (2008b) Hashiguchi, K., Ozaki, S., Okayasu, T.: Unconventional friction theory based on the subloading surface concept. Int. J. Solids Struct. 42, 1705–1727 (2005b) Hashiguchi, K., Protasov, A.: Localized necking analysis by the subloading surface model with tangential-strain rate and anisotropy. Int. J. Plasticity 20, 1909–1930 (2004) Hashiguchi, K., Saitoh, K., Okayasu, T., Tsutsumi, S.: Evaluation of typical conventional and un- conventional plasticity models for prediction of softening behavior of soils. Geotechnique 52, 561–573 (2002) Hashiguchi, K., Tsutsumi, S.: Elastoplastic constitutive equation with tangential stress rate effect. Int. J. Plasticity 17, 117–145 (2001) Hashiguchi, K., Tsutsumi, S.: Shear band formation analysis in soils by the subloading surface model with tangential stress rate effect. Int. J. Plasticity 19, 1651–1677 (2003) Hashiguchi, K., Tsutsumi, S.: Gradient plasticity with the tangential subloading surface model and the prediction of shear band thickness of granular materials. Int. J. Plasticity 22, 767–797 (2006) Hashiguchi, K., Ueno, M.: Elastoplastic constitutive laws of granular materials. In: Murayama, S., Schofield, A.N. (eds.) Constitutive Equations of Soils, Proc. 9th Int. Conf. Soil Mech. Found. Eng., Spec. Ses. 9, Tokyo, pp. 73–82. JSSMFE (1977) Hashiguchi, K., Ueno, M., Ozaki, T.: Elastoplastic model of metals with smooth elastic-plastic transition. Acta Mech. 223, 985–1013 (2012) Hashiguchi, K., Yamakawa, Y.: Introduction to Finite Strain Theory for Continuum Elasto- Plasticity. Wiley Series in Computational Mechanics. John-Wiley (2012) Hashiguchi, K., Yoshimaru, T.: A generalized formulation of the concept of nonhardening region. Int. J. Plasticity 11, 347–365 (1995) Hassan, S., Kyriakides, S.: Ratcheting in cyclic plasticity. Part I: Uniaxial behavior. J. Appl. Mech. (ASME) 8, 91–116 (1992) Hassan, T., Taleb, T., Krishna, S.: Influence of non-proportional loading on ratcheting responses and simulations by two recent cyclic plasticity models. Int. J. Plasticity 24, 1863–1889 (2008) Haupt, P.: Continuum Mechanics and Theory of Materials. Springer (2000) Hecker, S.S.: Experimental investigation of corners in yield surface. Acta Mech. 13, 69–86 (1972) Hencky, H.: Zur Theorie plastischer Deformationen und der hierdurch im Material herforgerufe- nen Nachspannungen. Z.A.M.M. 4, 323–334 (1924) Hill, R.: Theory of yielding and plastic flow of anisotropic metals. Proc. Royal Soc., London, A 193, 281–297 (1948) Hill, R.: Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950) References 437

Hill, R.: Some basic principles in the mechanics of solids without a natural time. J. Mech. Phys. Solids 7, 225–229 (1959) Hill, R.: Generalized constitutive relations for incremental deformation of metal crystals. J. Mech. Phys. Solids 14, 95–102 (1966) Hill, R.: On the classical constitutive relations for elastic/plastic solids. Recent Progress Appl. Mech., 241–249 (1967) Hill, R.: On the constitutive inequalities for simple materials –1. J. Mech. Phys. Solids 16, 229– 242 (1968) Hill, R.: Aspects of invariance in solid mechanics. Advances Appl. Mech. 18, 1–75 (1978) Hill, R.: On the intrinsic eigenstates in plasticity with generalized variables. Math. Proc. Cam- bridge. Phil. Soc. 93, 177–189 (1983) Hill, R.: Constitutive modeling of orthotropic plasticity in sheet metals. J. Mech. Phys. Solids 38, 241–249 (1990) Hinton, E., Owen, D.R.J.: Finite Elements in Plasticity: Theory and Practice. Pineridge Press, Swansea (1980) Hoger, A., Carlson, D.E.: On the derivative of the square root of a tensor and Guo’s theorem. J. Elasticity 14, 329–336 (1984) Hohenemser, K., Prager, W.: Uber die Ansatze der Mechanik isotroper Kontinua. Z.A.M.M. 12, 216–226 (1932) Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wi- ley & Sons, Ltd. (2000) Horowitz, F., Ruina, A.: Slip patterns in a spatially homogeneous fault model. J. Geophys. Re- search 94, 10279–10298 (1989) Houlsby, G.T.: The use of a variable shear modulus in elastic-plastic models for clays. Comput. Geotech. 1, 3–13 (1985) Houlsby, G.T., Amorosi, A., Rojas, E.: Elastic moduli of soils dependent on pressure: a hypere- lastic formulation. Geotechnique 55(5), 383–392 (2005) Houlsby, G.T., Puzrin, A.M.: Principles of Hyperelasticity; An Approach to Plasticity Theory Based on Thermodynamic Principles. Springer (2006) Howe, P.G., Benson, D.P., Puddington, I.E.: London-Van der Waals’ attractive forces between glass surface. Can. J. Chem. 33, 1375–1383 (1955) Hughes, T.J.R., Pister, K.S.: Consistent linearization in mechanics of solids and structures. Com- put. Struct. 9, 391–397 (1978) Hughes, T.J.R., Shakib, F.: Pseudo-corner theory: a simple enhancement of J2-flow theory for applications involving non-proportional loading. Eng. Comput. 3, 116–120 (1986) Huhges, T.J.R., Taylor, R.L.: Unconditionally stable algorithms for quasi-static elastoplastic finite element analysis. Comput. Struct. 8, 169–173 (1978) Hughes, T.J.R., Winget, J.: Finite rotation effects in numerical integration of rate consistent equa- tions arising in large-deformation analysis. Int. J. Numer. Meth. Eng. 15, 1862–1867 (1980) Iai, S., Ohtsuki, O.: Yield and cyclic behaviour of a strain space multiple mechanism model for granular materials. Int. J. Numer. Anal. Meth. Goemech. 29, 417–442 (2005) Ikegami, K.: Experimental plasticity on the anisotropy of metals. In: Proc. Euromech. Collo- quium, vol. 115, pp. 201–242 (1979) Ilyushin, A.A.: On the postulate of plasticity. Appl. Math, and Meek 25, 746–752 (1961); Trans- lation of Opostulate plastichnosti. Prikladnaya Mathematika i Mekkanika 25, 503–507 Ilyushin, A.A.: Plasticity – Foundation of the General Mathematical Theory. Izdatielistbo Akademii Nauk CCCR (Publisher of the Russian Academy of Sciences), Moscow (1963) Ishihara, K., Tatsuoka, F., Yasuda, S.: Undrained deformation and liquefaction of sand under cyclic stresses. Soils and Foundation 15, 29–44 (1975) Itasca Consulting Group: FLAC3D, Fast Lagrangian Analysis of Continua in 3 Dimensions, Min- neapolis, Minnesota, USA (2006) 438 References

Ito, K.: New flow rule for elastic-plastic solids based on KBW model with a view to lowering the buckling stress of plates and shells. Tech. Report Tohoku Univ. 44, 199–232 (1979) Iwan, W.D.: On a class of models for the yielding behavior of continuous and composite systems. J. Appl. Mech. (ASME) 34, 612–617 (1967) Jaumann, G.: Geschlossenes System physicalisher und chemischer Differentialgesetze. Sitzber. Akad. Wiss. Wien (IIa) 120, 385–530 (1911) Jaunzemis, W.: Continuum Mechanics. The Macmillan, New York (1967) Jiang, Y., Zhang, J.: Benchmark experiments and characteristic cyclic plasticity deformation. Int. J. Plasticity 24, 1481–1515 (2008) Kato, S., Sato, N., Matsubayashi, T.: Some considerations on characteristics of static friction of machine tool sideway. J. Lubr. Tech. (ASME) 94, 234–247 (1972) Khan, A.S., Huang, S.: Continuum Theory of Plasticity. John Wiley & Sons, Ltd. (1995) Khojastehpour, M., Hashiguchi, K.: The plane strain bifurcation analysis of soils by the tangential-subloading surface model. Int. J. Solids Struct. 41, 5541–5563 (2004a) Khojastehpour, M., Hashiguchi, K.: Axisymmetric bifurcation analysis in soils by the tangential- subloading surface model. J. Mech. Phys. Solids 52, 2235–2262 (2004b) Khojastehpour, M., Murakami, Y., Hashiguchi, K.: Antisymmetric bifurcation in a circular cylin- der with tangential plasticity. Mech. Materials 38, 1061–1071 (2006) Kikuchi, N., Oden, J.T.: Contact problem in elasticity: A study of variational inequalities and finite element methods. SIAM, Philadelphia (1988) Kiyota, T., Kozeki, J., Sato, T., Kuwano, S.: Aging effects on small strain shear moduli and liq- uefaction properties of in-situ frozen and reconstituted sandy soils. Soils and Foundations 49, 259–274 (2009a) Kiyota, T., Kozeki, J., Sato, T., Tsutsumi, Y.: Effects of sample disturbance on small strain charac- teristics and liquefaction properties of Holocene and pleistocene sandy soils. Soils and Foun- dations 49, 509–523 (2009b) Kleiber, M., Raniecki, B.: Elastic-plastic materials at finite strains. In: Sawczuk, A., Bianchi, G. (eds.) Plasticity Today, Modelling, Methods and Applications, pp. 3–46. Elsevier (1985) Kohgo, Y., Nakano, M., Miyazaki, T.: Verification of the generalized elastoplastic model for unsaturated soils. Soil and Foundations 33(4), 64–73 (1993) Koiter, W.T.: Stress-strain relations, uniqueness and variational theories for elastic-plastic mate- rials with a singular yield surface. Quart. Appl. Math. 11, 350–354 (1953) Kolymbas, D., Wu, W.: Introduction to plasticity. Modern Approaches to Plasticity, 213–224 (1993) Kratochvil, J.: Finite-strain theory of crystalline elastic-inelastic materials. J. Appl. Phys. 42, 1104–1108 (1971) Krieg, R.D.: A practical two surface plasticity theory. J. Appl. Mech. (ASME) 42, 641–646 (1975) Krieg, R.D., Key, S.W.: Implementation of a time dependent plasticity theory into structural com- puter programs. In: Strickin, J.A., Saczlski, K.J. (eds.) Constitutive Equations in Viscoplastic- ity: Computational and Engineering Aspects, AMD-20. ASME, New York (1976) Krieg, R.D., Krieg, D.B.: Accuracies of numerical solution methods for the elastic-perfectly plas- tic models. J. Pressure Vessel Tech. (ASME) 99, 510–515 (1977) Kroner, E.: Allgemeine Kontinuumstheoreie der Versetzungen und Eigenspannnungen. Arch. Ra- tion. Mech. Anal. 4, 273–334 (1960) Kuroda, M.: Roles of plastic spin in shear banding. Int. J. Plasticity 12, 671–694 (1996) Kuroda, M.: Interpretation of the behavior of metals under large plastic shear deformations: A macroscopic approach. Int. J. Plasticity 13, 359–383 (1997) Kuwayama, T., Suzuki, N., Ogawa, S., Hashiguchi, K.: Application of overstress-subloading fric- tion model to finite element analysis. In: Proc. 2012 Spring Conf. Tech. of Plasticity (Japan Soc. Tech. Plasticity), pp. 259–260 (2012) References 439

Lai, W.M., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics. Pergamon Press (1974) Lamaitre, J., Chaboche, J.-L.: Mechanics of Solid Materials. Cambridge Univ. Press (1990) Lee, E.H., Liu, D.T.: Finite-strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38, 19–27 (1967) Lee, E.H.: Elastic-plastic deformation at finite strain. J. Appl. Mech. (ASME) 36, 1–6 (1969) Leigh, D.C.: Nonlinear Continuum Mechanics: An Introduction to the Continuum Physics and Mechanical Theory of the Nonlinear Mechanical Behavior of Materials. McGraw-Hill, New York (1968) Lion, A.: Constitutive modeling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. Int. J. Plasticity 16, 469–494 (2000) Loret, B.: On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials. Mech. Materials 2, 287–304 (1983) Lubarda, V.A.: Elastoplasticity Theory. CRC Press (2002) Lubliner, J.: Plasticity Theory. Dover Publ., Inc., New York (1990) Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall (1969) Mandel, J.: Generalisation de la theorie de plasticite de W.T. Koiter. Int. J. Solids Struct. 1, 273– 295 (1965) Mandel, J.: Plastidite classique et viscoplasticite. Course & Lectures, No. 97, Int. Center Mech. Sci., Udine. Springer (1971) Mandel, J.: Director vectors and constitutive equations for plastic and viscoplastic media. In: Sawczuk, A. (ed.) Problems of Plasticity (Proc. Int. Symp. Foundation of Plasticity), Noord- hoff, pp. 135–141 (1972) Mandel, J.: Equations constitutives directeurs dans les milieux plastiques at viscoplastiques. Int. J. Solids Struct. 9, 725–740 (1973) Marsden, J.E., Hughes, T.J.R.: Mathematical Foundation of Elasticity. Prentice-Hall, Englewood Cliffs (1983) Mase, T., Hashiguchi, K.: Numerical analysis of footing settlement problem by subloading sur- face model. Soils and Foundations 49, 207–220 (2009) Masing, G.: Eigenspannungen und Verfestigung beim Messing. In: Proc. 2nd Int. Congr. Appl. Mech., Zurich, pp. 332–335 (1926) Matsuoka, H., Nakai, T.: Stress-deformation and strength characteristics of soil under three dif- ferent principal stress. Proc. Japan. Soc. Civil Eng. (232), 59–70 (1974) Matsuoka, H., Yao, Y.P., Sun, D.A.: The Cam-clay model revised by SMP criterion. Soils and Foundation 39(1), 81–95 (1999) Maugin, G.A.:: The Thermomechanics of Plasticity and Fracture. Cambridge Univ. Press (1992) McDowell, D.L.: An experimental study of the structure of constitutive equations for nonpropor- tional cyclic plasticity. J. Eng. Mater. Tech. (ASME) 107, 307–315 (1985) McDowell, D.L.: Evaluation of intersection conditions for two-surface plasticity theory. Int. J. Plasticity 5, 29–50 (1989) Menzel, A., Steinmann, P.: On the spatial formulation of anisotropic multiplicative elasto- plasticity. Compt. Meth. Appl. Mech. Eng. 192, 3431–3470 (2003b) Menzel, A., Ekh, M., Runesson, K., Steinmann, P.: A framework for multiplicative elastoplas- ticity with kinematic hardening coupled to anisotropic damage. Int. J. Plasticity 21, 397–434 (2005) Michalowski, R., Mroz, Z.: Associated and non-associated sliding rules in contact friction prob- lems. Archiv. Mech. 30, 259–276 (1978) Miehe, C.: Numerical computation of algorithmic (consistent) tangent moduli in large-strain com- putational inelasticity. Comput. Methods Appl. Mech. Eng. 134, 223–240 (1996) Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Experiment. Mech. 3, 1–7 (1963) 440 References

Mroz, Z.: On forms of constitutive laws for elastic-plastic solids. Arch. Mech. Stos. 18, 3–35 (1966) Mroz, Z.: On the description of anisotropic workhardening. J. Mech. Phys. Solids 15, 163–175 (1967) Mroz, Z., Norris, V.A., Zienkiewicz, O.C.: An anisotropic, critical state model for soils subject to cyclic loading. Geotechnique 31, 451–469 (1981) Mroz, Z., Stupkiewicz, S.: An anisotropic friction and wear model. Int. J. Solids Struct. 31, 1113– 1131 (1994) Muhlhaus, H.B., Vardoulakis, I.: The thickness of shear bands in granular materials. Geotech- nique 37, 271–283 (1987) Nakai, T., Hinokio, M.: A simple elastoplastic model for normally and over consolidated soils with unified material parameters. Soils and Foundations 44(2), 53–70 (2004) Nakai, T., Mihara, Y.: A new mechanical quantity for soils and its application to elastoplastic constitutive models. Soils and Foundations 24(2), 82–941 (1984) Nemat-Nasser, S.: Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Mate- rials. Cambridge Univ. Press (2004) Niemunis, A., Cudny, M.: On hyperelasticity for clays. Comput. Geotech. 23, 221–236 (1998) Norton, F.H.: Creep of Steel at High Temperature. McGraw-Hill, New York (1929) Nova, R.: On the hardening of soils. Arch. Mech. Stos. 29, 445–458 (1977) Oden, J.T.: An Introduction to Mathematical Modeling: A Course in Mechanics. Wiley Series in Computational Mechanics. John-Wiley (2011) Oden, J.T., Pires, E.B.: Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws. Comput. Struct. 19, 137–147 (1983a) Oden, J.T., Pires, E.B.: Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity. J. Appl. Mech. (ASME) 50, 67–76 (1983b) Oden, J.T., Martines, J.A.C.: Models and computational methods for dynamic friction phenom- ena. Comput. Meth. Appl. Mech. Eng. 52, 527–634 (1986) Odqvist, F.K.G., Hult, J.A.H.: Kriechfestigkeit Metallischer Werkstoffe. Springer, Berlin (1962) Odquivist, F.K.G.: Mathematical Theory of Creep and Creep Rupture. Oxford Univ. Press, Lon- don (1966) Ogden, R.W.: Non-linear Elastic Deformations. Dover Publ. Inc. (1984) Ohno, N.: A constitutive model of cyclic plasticity with a non-hardening strain region. J. Appl. Mech. (ASME) 49, 721–727 (1982) Ohno, N., Kachi, Y.: A constitutive model of cyclic plasticity for nonlinearly hardening materials. J. Appl. Mech. (ASME) 53, 395–403 (1986) Ohno, N., Wang, J.D.: Kinematic hardening rules with critical state of dynamic recovery, Parts I: Formulation and basic features for ratcheting behavior. Part II: Application to experiments of ratcheting behavior. Int. J. Plasticity 9, 375–403 (1993) Ohno, N., Tsuda, M., Kamei, T.: Elastoplastic implicit integration algorithm applicable to both plane stress and three-dimensional stress states. Finite Elements Anal. Design 66, 1–11 (2013) Oka, F., Yashima, A., Taguchi, A., Yamashita, S.: A cyclic elasto-plastic constitutive model for sand considering a plastic-strain dependence of the shear modulus. Geotechnique 49, 661–680 (1999) Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. Roy. Soc. London, Ser. A 200, 523–541 (1950) Ortiz, M., Popov, E.P.: Accuracy and stability of integration algorithms for elastoplastic constitu- tive relations. Int. J. Numer. Meth. Eng. 21, 1561–1576 (1985) Ortiz, M., Simo, J.C.: An analysis of a new class of integration algorithms for elastoplastic con- stitutive relations. Int. J. Numer. Meth. Eng. 23, 353–366 (1986) Ozaki, S., Hashiguchi, K.: Numerical analysis of stick-slip instability by a rate-dependent elasto- plastic formulation for friction. Tribology Int. 43, 2120–2133 (2010) References 441

Ozaki, S., Hikida, K., Hashiguchi, K.: Elastoplastic formulation for friction with orthotropic anisotropy and rotational hardening. Int. J. Solids Struct. 49, 648–657 (2012) P´erez-Foguet, A., Rodr´ıguez-Ferran, A., Huerta, A.: Numerical differentiation for non-trivial con- sistent tangent matrices: an application to the MRS-Lade model. Int. J. Numer. Meth. Eng. 48, 159–184 (2000a) P´erez-Foguet, A., Rodr´ıguez-Ferran, A., Huerta, A.: Numerical differentiation for local and global tangent operators in computational plasticity. Compt. Meth. Appl. Mech. Eng. 189, 277–296 (2000b) P´erez-Foguet, A., Rodr´ıguez-Ferran, A., Huerta, A.: Consistent tangent matrices for substepping schemes. Compt. Meth. Appl. Mech. Eng. 190, 4627–4647 (2001) Peric, D., Owen, R.J.: Computational model for 3-D contact problems with friction based on the penalty method. Int. J. Numer. Meth. Eng. 35, 1289–1309 (1992) Perzyna, P.: The constitutive equations for rate sensitive plastic materials. Quart. Appl. Math. 20, 321–332 (1963) Perzyna, P.: Fundamental problems in viscoplasticity. Advances Appl. Mech. 9, 243–377 (1966) Petryk, H.: On the second-order work in plasticity. Arch. Mech. 43, 377–397 (1991) Petryk, H.: Plastic instability: Criteria and computational approaches. Arch. Comput. Approach. Meth. Eng. 4, 111–151 (1997) Petryk, H., Thermann, K.: A yield-vertex modification of two-surface models of metal plasticity. Arch. Mech. 49, 847–863 (1997) Pietruszczak, S., Mroz, Z.: On hardening anisotropy of Ko-consolidated clays. Int. J. Numer. Anal. Meth. Geomech. 7, 19–38 (1983) Pietruszczak, S.T., Niu, X.: On the description of localized deformation. Int. J. Numer. Anal. Meth. Geomech. 17, 791–805 (1993) Pinsky, P.M., Ortiz, M., Pister, K.S.: Numerical integration of rate constitutive equations in finite deformation analysis. Comput. Meth. Appl. Mech. Eng. 193, 5223–5256 (1983) Prager, W.: Recent development in the mathematical theory of plasticity. J. Appl. Mech. (ASME) 20, 235–241 (1949) Prager, W.: A new methods of analyzing stresses and strains in work hardening plastic solids. J. Appl. Mech. (ASME) 23, 493–496 (1956) Prager, W.: Linearization in visco-plasticity. Ing. Archiv. 15, 152–157 (1961a) Prager, W.: Introduction to Mechanics of Continua. Ginn & Comp., Boston (1961b) Rabinowicz, E.: The nature of the static and kinetic coefficients of friction. J. Appl. Phys. 22, 1373–1379 (1951) Rabinowicz, E.: The intrinsic variables affecting the stick-slip process. Proc. Phys. Soc. 71, 668– 675 (1958) Raniecki, B.: Selected Fragments of Hill’s Course on Solid Mechanics. Lecture Note in Kyushu University (2004) Rice, J.R., Tracey, D.M.: Computational fracture mechanics. In: Feves, S.J. (ed.) Proc. Symp. Numer. Meth. Struct. Mech., Urbana, Illinois, p. 585. Academic Press (1973) Roscoe, K.H., Burland, J.B.: On the generalized stress-strain behaviour of ‘wet’ clay. In: Engi- neering Plasticity, 535-608. Cambridge Univ. Press (1968) Rudnicki, J.W., Rice, J.R.: Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371–394 (1975) Saada, A.S., Bianchini, G.: Proc. Int. Workshop on Constitutive Equations for Granular Non- cohesive Soils, Cleveland, Balkema (1989) Satake, M.: A proposal of new yield criterion for soils. Proc. Japan. Soc. Civil Eng. 189, 79–88 (1972) Schofield, A.N., Wroth, C.P.: Critical State Soil Mechanics. McGraw-Hill (1968) Seguchi, Y., Shindo, A., Tomita, Y., Sunohara, M.: Sliding rule of friction in plastic forming of metal. Compt. Meth. Nonlinear Mech., 683–692 (1974) 442 References

Sekiguchi, H., Ohta, H.: Induced anisotropy and its time dependence in clays. In: Constitutive Equations of Soils (Proc. Spec. Session 9, 9th ICSFME), Tokyo, pp. 229–238 (1977) Seth, B.R.: Generalized strain measure with applications to physical problems. In: Second-order Effects Inelasticity, Plasticity, and Fluid Dynamics. Pergamon, Oxford (1964) Sewell, M.J.: A yield-surface comer lowers the buckling stress of an elastic-plastic plate under compression. J. Mech. Phys. Solids 21, 19–45 (1973) Sewell, M.J.: A plastic flow at a yield vertex. J. Mech. Phys. Solids 22, 469–490 (1974) Siddiquee, M.S.A., Tanaka, T., Tatsuoka, F., Tani, K., Morimoto, T.: Numerical simulation of bearing capacity characteristics of strip footing on sand. Soils and Foundations 39(4), 93–109 (1999) Simo, J.C.: A J2-flow theory exhibiting a corner-like effect and suitable for large-scale computa- tion. Comput. Meth. Appl. Mech. Eng. 62, 169–194 (1987) Simo, J.C.: Numerical analysis and simulation of plasticity. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 6, Part 3. Elsevier (1998) Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer (1998) Simo, J.C., Ortiz, M.: A unified approach to finite deformation elastoplasticity based on the use of hyperelastic constitutive equations. Compt. Meth. Appl. Mech. Eng. 49, 221–245 (1985) Simo, J.C., Taylor, R.L.: Consistent tangent operators for rate-independent elastoplasticity. Com- put. Meth. Appl. Mech. Eng. 48, 101–118 (1985) Simo, J.C., Taylor, R.L.: A return mapping algorithm for plane stress elastoplasticity. Int. J. Nu- mer. Meth. Eng. 22, 649–670 (1986) Skempton, A.W., Brown, J.D.: A landslide in boulder clay at Selset. Yorkshire, Geotechnique 11, 280–293 (1961) Sloan, S.W., Randolph, M.F.: Numerical prediction of collapse loads using finite element meth- ods. Int. J. Numer. Anal. Meth. Geomech. 6, 47–76 (1982) Spencer, A.J.M.: In: Eringen, A.C. (ed.) Continuum Physics. Mathematics, vol. 1. Academic Press (1971) Stallebrass, S.E., Taylor, R.N.: The development and evaluation of a constitutive model for the prediction of ground movements in overconsolidated clay. Geotechnique 47, 235–253 (1997) Stark, T.D., Ebeling, R.M., Vettel, J.J.: Hyperbolic stress-strain parameters for silts. J. Geotech. Eng. (ASCE) 120, 420–441 (1994) Tamagnini, C., Castellanza, R., Nova, R.: A generalized backward Euler algorithm for the nu- merical integration of an isotropic hardening elastoplastic model for mechanical and chemi- cal degradation of bonded geomaterials. Int. J. Numer. Anal. Meth. Geomech. 26, 963–1004 (2002) Tanaka, T., Kawamoto, O.: Three dimensional finite element collapse analysis for foundations and slopes using dynamic relaxation. In: Proc. Numer. Meth. Geomech., Innsbruck, pp. 1213– 1218 (1988) Tanaka, T., Sakai, T.: Progressive failure effect of trap-door problems with granular materials. Soils and Foundations 33(1), 11–22 (1993) Tani, K.:: Mechanism of Bearing Capacity of Shallow Foundation. Master Thesis, Univ. Tokyo (1986) Tatsuoka, F., Ikuhara, O., Fukushima, S., Kawamura, T.: On the relation of bearing capacity of shallow footing on model sand ground and element test strength. In: Proc. Symp. Asses. Deform. Fail. Strength of Sandy Soils and Sand Grounds, Japan. Soc. Geotech. Eng., pp. 141– 148 (1984) Taylor, D.W.: Fundamentals of Soil Mechanics. John Wiley & Sons, Chichester (1948) Topolnicki, M.: An elasto-plastic suboading surface model for clay with isotropic and kinematic mixed hardening parameters. Soils and Foundations 30(2), 103–113 (1990) Truesdell, C.: Hypo-elasticity. J. Rational Mech. Anal. 4, 83–133 (1955) References 443

Truesdell, C. and Noll, W. (1965): The Nonlinear Field Theories of Mechanics. In: Flugge, S. (ed.) Encyclopedia of Physics, Vol. III/3. Springer, Heidelberg. Truesdell, C., Toupin, R.: The Classical Field Theories. In: Flugge, S. (ed.) Encyclopedia of Physics, vol. III/1. Springer, Heidelberg (1960) Truesdell, C.: A First Course in Rational Continuum Mechanics. General Concepts, vol. 1. Aca- demic Press (1977) Tsutsumi, S., Hashiguchi, K.: General non-proportional loading behavior of soils. Int. J. Plastic- ity 21, 1941–1969 (2005) Tsutsumi, S., Toyosada, M., Hashiguchi, K.: Extended subloading surface model incorporating elastic limit concept. In: Proc. Plasticity 2006, Halifax, pp. 217–219 (2006) Vardoulakis, I., Sulem, J.: Bifurcation Analysis in Geomechanics. Blackie Academic & Profess., London (1995) Vermeer, P.A.: A simple shear band analysis using compliances. In: Proc. IUTAM Symp. Defor- mation and Failure of Granular Materials, Balkema, pp. 493–499 (1982) Wang, Z.-L., Dafalias, Y.F., Shen, C.-K.: Bounding surface hypoplasticity model for sand. J. Eng. Mech. (ASCE) 116, 983–1001 (1990) Wesley, L.D.: Influence of structure and composition on residual soils. J. Geotech. Eng. (ASCE) 116, 589–603 (1990) Wilde, P.: Two invariants depending models of granular media. Arch. Mech. Stos. 29, 799–809 (1977) Wilkins, M.L.: Calculation of elastoplastic flow. In: Alder, B., et al. (eds.) Methods of Computa- tional Physics, vol. 3. Academic Press (1964) Wongsaroj, J., Soga, K., Mair, R.J.: Modeling of long-term ground response to tunneling under St James’ Park. Geotechnique 57, 75–90 (2007) Wriggers, P., Simo, J.C.: A general procedure for the direct computation of turning and bifurca- tion points. Int. J. Numer. Meth. Eng. 30, 155–176 (1990) Wriggers, P.: Computational Contact Mechanics. John Wiley & Sons, Ltd. (2003) Wriggers, P., Vu Van, T., Stein, E.: Finite element formulation of large deformation impact- contact problems with friction. Comput. Struct. 37, 319–331 (1990) Wu, H.-C.: Continuum Mechanics and Plasticity. Chapman & Hall/CRC (2004) Xia, Z., Ellyin, F.: Biaxial ratcheting under strain or stress-controlled axial cycling with constant hoop stress. J. Appl. Mech. (ASME) 61, 422–428 (1989) Yamada, S., Takamori, T., Sato, K.: Effects on reliquefaction resistance produced by changes in anisotropy during liquefaction. Soils and Foundations 50, 9–25 (2010) Yamada, Y., Yoshimura, N., Sakurai, T.: Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by finite element method. Int. J. Mech. Sci. 10, 343–354 (1968) Yamakawa, Y., Chida, M., Ikeda, K., Hashiguchi, K.: Finite strain elastoplastic model for fric- tional materials based on the multiplicative decomposition of plastic deformation gradient tensor. In: Proc. 62nd Natl. Cong. Theor. Appl. Mech., Japan (2013) Yamakawa, Y., Hashiguchi, K.: Elastoplasticity theory: Numerical methods for finite elastoplastic constitutive equation, Part 3. In: Science of Machine, vol. 63, pp. 251–254. Yokendo Publ. Co. (2011) Yamakawa, Y., Hashiguchi, K., Ikeda, K.: Implicit stress-update algorithm for isotropic Cam-clay model based on the subloading surface concept at finite strains. Int. J. Plasticity 26, 634–658 (2010a) Yamakawa, Y., Yamaguchi, Y., Hashiguchi, K., Ikeda, K.: Formulation and implicit stress-update algorithm of the extended subloading surface Cam-clay model with kinematic hardening for finite strains. J. Appl. Mech. (AICE) 13, 411–412 (2010b) Yamamoto, Y.: Evaluation of seismic behavior of clay and sand grounds. Ph.D. Thesis, Yam- aguchi University (1998) (in Japanese) 444 References

Yoshida, F., Uemori, T.: Elastic-plastic behavior of steel sheets under in-plane cyclic tension- compression at large strain. Int. J. Plasticity 18, 633–659 (2002a) Yoshida, F., Uemori, T.: A large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation. Int. J. Plasticity 18, 661–686 (2002b) Yoshida, F., Uemori, T.: A model of large-strain cyclic plasticity and its application to springback simulation. Int. J. Mech. Sci. 45, 1687–1702 (2003) Zaremba, S.: Su une forme perfectionnee de la theorie de la relaxation. Bull. Int. Acad. Sci., 594–614 (1903) Zbib, H.M., Aifantis, E.C.: On the concept of relative and plastic spins and its implications to large deformation theories. Part I: Hypoelasticity and vertex-type plasticity. Acta Mech. 75, 15–33 (1988) Zhang, F., Ye, B., Noda, T., Nakano, M., Nakai, K.: Explanation of cyclic mobility of soils: Approach by stress-induced anisotropy. Soils and Foundations 47, 635–648 (2007) Ziegler, H.: A modification of Prager’s. hardening rule. Quart. Appl. Phys. 17, 55–60 (1959) Zienkiewicz, O.C.: The Finite Element Method, 3rd edn. McGraw-Hill, London (1977) Index

A B

See Acoustic tensor 339 Back stress, Kinematic hardening Additive decomposition Backward-Euler method 384 strain rate 138 Bauschinger effect 153 continuum spin 138 Bingham model 309 Admissible field Biot See kinematically 85 strain tensor, Strain stress tensor, See Stress statically 85 Body force 78 Admissible transformation 50 Bounding surface 179 Algorithmic tangent modulus, See model 179 Consistent tangent modulus model with radial mapping 179 Almansi (Eulerian) strain, See Strain Bulk modulus 128 Alternating symbol, See Permutation symbol C Angular momentum 78 Anisotropy 152 Cam-clay model 251 kinematic hardening, See Kinematic modified 251 hardening original 251 orthotropic, See Orthotropic anisotropy Cap model 255 rotational hardening, See Rotational Cartesian hardening summation convention 1 Anisotropy for friction decomposition, See Tensor orthotropic 364 Cauchy rotational 364 ’s first law of motion See Equilibrium Anti(Skew)-symmetric tensor, See Tensor equation Associated flow rule (Associativity) ’s fundamental theorem (’s stress 140 principle) 79 Drucker’s interpretation 148 stress, See Stress second-order plastic relaxation work elastic material 126 rate 149 Cauchy-Green deformation tensor 57 Prager’s interpretation 147 Cayley-Hamilton theorem 31 Associative law of vector, See Vector Characteristic Axial vector, See Vector direction See Principal direction 446 Index

equation 23 tensor 424 length 334 vector 424 value See Eigenvalue Convected vector, See Eigenvector base vector 107 Circle of relative velocity 69 coordinate system 54, 106, 422 Closest point projection 384 time-derivative 106, 115 Closest point projection for friction stress rate 408 contravariant 115 Coaxial (Coaxiality) 31 covariant-contravariant 115 Cofactor 3 covariant 116 Commutative law of vector, See Vector corotational 116 Compliance method 341 tensor 109, 110 Component of vector 109, 110 tensor, See Tensor Convective, See Convected vector, See Vector Conventional plasticity model 131, Configuration 49 166 current (Eulerian) 49, 132 Conventional friction model 343 initial 49, 132 Convective term, See Steady term intermediate 132 Convexity condition of curve 427 reference (Lagrangian) 49 Convexity of yield surface 148 Conservation law of Coordinate transformation angular momentum 78 Cartesian 13 linear momentum 78 curvilinear 422 mass 77 Corner theory 161 Consistency condition Corotational rate 113, 117, 138, 139, conventional plasticity 140 317, 347 extended subloading surface model Green-Naghdi (Dienes) 114, 116, 204 319, 323 initial subloading surface model 173 Zaremba-Jaumann 114, 116, 317, Consistency condition for friction 349, 322, 347 352 Cosserat elastic material 121 Consistent tangent modulus 398 Cotter-Rivlin rate 113, 116 Consolidation of soils Coulomb sliding-yield condition 349, e − ln p linear relation 246 356 isotropic 243, 247 Couple stress 129 ln v − ln p linear relation 243, 425 Covariant and contravariant, See normal 243, 247 Contravariant and covariant swelling 243, 247 Covariant-contravariant convected stress, Constitutive spin, See Spin See Stress Contact elastic modulus for friction Creep model 310 348 Critical state 250, 291 Contact traction for friction 346 Cross product, See Vector normal and tangential 346 Curl of tensor field, See Tensor filed Continuity condition 167, 185 Current configuration, See Configuration Prager’s 169 Cutting plane projection 394 Continuity equation 77 Cutting plane projection for friction 411 Continuum spin, See Spin Cyclic loading Contraction of tensor, See Tensor elastoplastic deformation 166, 187 Contravariant and covariant friction 362 base vector 422 Cyclic mobility 288 Index 447

Cyclic plasticity model 187 Dilatancy locking 301 kinematic hardening type 188 Direct notation, See Tensor notation infinite surface model 191 Direction cosine 8 multi surface model (Mroz model) Discontinuity of velocity gradient 338 189 Dissipation energy 150 single surface model 194 Distributive law of vector 5 small single surface model 195 Divergence of tensor field, See Tensor Chaboche model 195 field Ohno-Wang model 196 Divergence theorem, See Gauss’ s two surface model 192 divergence theorem expansion of loading surface type 201 Drucker’s extended subloading surface model classification of plasticity model 167 201, 203 postulate for stress cycle 148, 151 Cyclic stagnation of isotropic hardening Drucker-Prager yield surface 257 215 Dummy index 1 Dyad, See Vector D Dynamic-loading subloading-overstress model Deformation gradient 54 ratio 313 elastic 131 surface 313 polar decomposition 135 Dynamic-loading for friction plastic 131 subloading-overstress polar decomposition 135 sliding ratio 376 polar decomposition 54 sliding surface 376 relative 58 Deformation theory E Hencky 162 J2- 162 Eddington’s epsilon, See Permutation Description symbol Eulerian 50 Eigen (principal) Lagrangian 50 direction, See Principal direction material, See, Lagrangian projection 29 relative 51 value 23, 28 spatial, See Eulerian value analysis 340 total Lagrangian 51 vector 23, 29 updated Lagrangian 51 Einstein’s summation convention, See Determinant 2 Summation convention product law 3 Elastic Vandermonde’s 35 bulk modulus 128 Deviatoric constitutive equation 123 principal invariant 27 deformation gradient 131 part 21 hyper- 123 plane 37 modulus 125, 127, 380 projection tensor (fourth-order) 23 predictor step, See trial step tangential stress rate 163, 183, 212 shear modulus 128 tangential projection tensor spin 137 (fourth-order) 163, 183, 212 strain energy function 123, 380, 413 Diagonal component, See Tensor strain energy function for kinematic Partial differential calculi 42 hardening 158, 429 448 Index

strain energy function for rotational Footing settlement analysis 301 hardening 271 Forward-Euler method 379 strain rate 137, 139 Friction 343 stress rate 141, 384 coefficient 344 tangent modulus 128 evolution rule 350 trial step 382, 408 kinetic 344, 350 volumetric strain 128, 134, 245 negative-rate sensitivity 376 Elastic-plastic transition 177 positive-rate sensitivity 376 Elastoplastic stiffness modulus tensor static 344, 350 142, 398 Functional determinant, See Jacobian Element test 333 G e − ln p linear relation, See Consolidation Embedded, See Convected Gauss’ s divergence theorem 46 Equilibrium equation 83 Gradient of tensor field, See Tensor field rate form 84 Gradient theory 334 moment 85 Green Equivalent elastic equation, See Hyperelastic plastic strain 142 equation stress 142 strain, See Strain viscoplastic strain 310 Green-Naghdi rate 114 Euler stress rate 116, 319, 323 ’s first law of motion 78 ’s second law of motion 78 H ’s theorem for homogeneous function 139, 424 Hamilton operator, See Nabra Eulerian Hardening configuration, See Configuration isotropic 139 description, See Description linear kinematic 155 spin tensor, See Spin nonlinear-kinematic 155 strain, See Strain rotational 268 tensor, See Tensor Hardening for friction 350 triad, See Triad Hencky Eulerian-Lagrangian two-point tensor deformation theory 162 54 strain, See Strain Explicit method, See Forward-Euler Hessian matrix 390 Extended subloading surface model, See Hessian matrix for friction 409 Subloading surface model Hooke’s law 126, 127, 380 Hyperelastic equation 123, 380 F metals 412 soils 413 Failure surface 258 Hyperelastic(-based) plasticity Finite strain theory 419, 429 infinitesimal 380, 418 First Piola-Kirchhoff stress, See Stress multiplicative 419 Flow rule Hypoelasticity 127, 138 associated 140, 147, 154, 175, 208, Hypoelastic-based 327 (Hypoelasto-)plasticity 138 non-associated 142 Hypoplasticity 129, 161 multiplicative plasticity 429 Hysteresis loop 191, 193, 201, 226, 227, Flow rule for friction 352 229 Index 449

I Prager, See linear rheological model 158 Identity tensor variable (back stress) 153, 154, 214 fourth-order 22 Ziegler 157 second-order 17 Kinematically admissible velocity field, Il-posedness of solution 334 See Admissible field Ilyushin’s Kinetic friction, See Friction isotropic stress space 241 Kirchhoff stress, See Stress postulate of strain cycle 149 Kronecker’ delta 2 Impact load 313 Implicit method, See Backward-Euler L method Indicial notation, See Tensor notation Lagrangian Infinite surface model, See Cyclic configuration, See Configuration plasticity model description, See Description Infinitesimal strain, See Strain spin tensor, See Description Initial configuration, See Configuration strain, See Strain Initial subloading surface model, See tensor, See Tensor Subloading surface model triad, See Triad Inner product, See Vector Lame constants 126, 127 Intermediate configuration, See Laplacian (Laplace operator) 44 Configuration Lee decomposition, See Multiplicative Internal variable 138, 167 decomposition Intersection of yield surfaces 161 Lie derivative 106 Invariant, See Principal invariant Limit dynamic loading surface 314 Inverse loading 201, 209 Limit dynamic loading surface for friction Inverse tensor, See Tensor 376 Isoclinic concept 137, 429 Linear kinematic hardening, See Isotropic Kinematic hardening tensor-valued tensor function 33 Linear transformation 14 hardening (variable) 139, 213, 266, ln v − ln p linear relation, 267, 293 See Consolidation definition of isotropic material 152 Liquefaction 288 scalar-valued tensor function 25, 117 Loading criterion tensor-valued tensor function 33 plastic sliding velocity 356 traverse 324 plastic strain rate 144 Local form 83 J Local theory 334 Local-time derivative, See Spatial-time Jacobian 50 derivative Jaumann, See Zaremba-Jaumann Local-time derivative term, See J2-deformation theory 162 Non-steady term Localization of deformation 333 K Lode angle 38, 253 Logarithmic Kinematic hardening strain 63, 72 linear 155 volumetric strain 63, 74, 134, 135, nonlinear 155, 158, 214 245 450 Index

M Nonlinear kinematic hardening, See Kinematic hardening Macauley bracket 172 Non-local theory 334 Mandel stress, See Stress Non-proportional loading 160, 200, Masing rule 190, 193 224, 231 Material Non-singular tensor, See Tensor description, See Lagrangian description Non-steady term 51 frame-indifference, See Objectivity Normal component 23, 41, 70 -time-derivative, See Time-derivative Normal isotropic hardening of volume integration 52 ratio 217 Maxwell model 308 surface 217 Mean part of tensor, See Tensor Normal stress rate 163 Mechanical ratcheting effect 190, 194, Normal-sliding for friction 201, 225 ratio 349 Mesh size dependence (sensitivity) 334 evolution rule 351 348 Metric tensor 107 surface , 354, 356, 365 Mises Normal-yield 171 ellipse 241 ratio , 203, 223, 281 173 yield condition 142, 213, 239 evolution rule , 211, 275, 294 170 plane strain 242 surface , 214, 250, 266, 268 See plane stress 240 Normality rule, Associated flow rule Modified Cam-clay model 251 Norton law 311 Mohr’s circle 40 Normalized orthonormal base 9 Momentum O linear 77 angular 78 Objective Motion 49 rate of tensor 113 Multi surface (Mroz) model, See Cyclic rate of vector 113 plasticity model time-derivative of scalar-valued tensor Multiplicative decomposition 131 function 117 Multiplicative hyperelastic-based time-integration of rate tensor 415 plasticity 419 stress rate tensor 115 tensor 13 N transformation 13 Objectivity 101 Nabra 44 Octahedral Nanson’s formula 75 plane, See deviatoric plane Natural strain, See Logarithmic strain shear stress 144 Navier’s equation 127 Oldroyd rate 112 Negative transformation 50 stress rate 115, 125 Nominal Original Cam-clay model 251 strain, See Strain Orthogonal stress, See Stress coordinate system 8 stress rate, See Stress rate tensor 15 stress vector, See Stress vector Orthotropic anisotropy 233 Nonassociated flow rule Orthotropic anisotropy for friction 364 (Non-associativity) 142, 259 Over stress 309, 376 Nonhardening, See Stagnation of Overstress model 309 isotropic hardening Bingham model 309 Index 451

Perzyna model 310 Positive proportionality factor 140, Prager model 309 141, 176, 209, 328, 335, 338, 391, return-mapping 404 396 subloading overstress model 313 Positive proportionality factor for friction Overstress friction model, See 352, 353, 356, 410, 412 Subloading-overstress friction Positive transformation 50 model Positive definite tensor 32 Prager’s P continuity condition 169 linear kinematic hardening rule 154 overstress model 309 Parallelogram law of vector 5 Prandtl plasticity model 309 Partial differential calculi 42 Prandtl-Reuss equation 143 Perfectly-plastic material 169 Primary Permutation symbol 2 base vector 107 Phase-transformation line 291 vector 7 Piola-Kirchhoff stress, See Stress Principal π-plane, See Deviatoric plane direction 23 Plastic invariant 26 corrector step 383, 384, 389, 392, time-derivative 120 394 space 36 deformation gradient, See Deformation stretch 55 gradient value, See Eigenvalue See material spin 429 vector, Eigenvector flow rule, See Flow rule Principle of See modulus 140, 154, 175, 183, 208, objectivity, see, Objectivity See 222, 275 material-frame indifference, multiplier, See Positive proportionality Objectivity factor maximum plastic work 151 potential 142, 152, 258 Product law of determinant 3 relaxation modulus tensor 141 Projection of area 421 Projection of tensor relaxation stress rate 141, 154, 384 deviatoric 23 shakedown, See Shakedown deviatoric-tangential 163, 183, 212 spin 135, 138, 324, 430 Proper value, See Eigenvalue strain rate 135, 138, 141, 176, 209, Pull-back operation 109, 415 328, 335, 384, 430 Pulsating loading 225 volumetric strain 134, 135, 245 Push-forward operation 109, 416 Plastic for friction corrector step 407, 408, 411 Q flow rule 352 modulus 353 Quasi-static deformation 312 multiplier 352 Quotient law 13 relaxation traction rate 407 sliding rate 345, 353, 409, 412 R Poisson’s ratio 129 Polar Ratcheting effect, See Mechanical decomposition 32, 54, 135 ratcheting effect spin, See Spin Rate of Positive definite tensor 32 elongation 70 452 Index

normal vector of surface 75, 76 triple product, See Vector shear strain 70 Second-order work rate 150 surface area 76 elastic stress 150 volume 75 plastic relaxation 150 Rate-type Second Piola-Kirchhoff stress, See Stress equilibrium equation 83 Shakedown 190 virtual work principle 86 Shear band 333 Reciprocal inception 338 base vector 107 band thickness 334 vector 7 -band embedded model 337 Reference configuration, See Shear modulus 128 Configuration Shear strain rate 70 Relative Similar tensor 25 deformation gradient 57 Similarity-center 203 description 51 enclosing condition 205 left and right Cauchy-Green translation rule 206 deformation tensor 58 surface 205 spin 65 yield ratio 205 Reloading behavior in subloading surface Similarity-ratio 171 model 209, 275 Simple shear 88 Rate of elongation 70 Single surface model, See Cyclic plasticity Representation theorem 36 model Return-mapping 379 Skew-symmetric tensor, See Tensor closest point 384 Sliding- cutting plane 394 hardening function 348 elastic trial (predictor) step 382 subloading surface 349 plastic corrector step 383 yield condition 348, 354, 356, 364 Return-mapping for overstress model Sliding velocity 345 404 elastic and plastic 345 Return-mapping for friction model 407 normal and tangential 345 Reynolds’ transportation theorem 52 Small single surface model, See Cyclic Rigid-body rotation 102 plasticity model Rigid-plastic material 321 Smeared crack model, See Shear-band Rotation embedded model of tensor field, See Tensor field Smoothness condition 169, 185 -free(insensitive) tensor 110, 415 Spatial description, See Eulerian of triad 55, 57 description rate tensor of material 113 Spatial-time derivative 51 rate of Lagrangian or Eulerian triad Spectral representation 24, 62, 413 65, 66 Spherical part of tensor, See Tensor Rotational anisotropy of friction 364 Spin Rotational hardening base vector 103 evolution rule 268 constitutive 429 rheological model 271 continuum 64, 114, 135 Rotational strain tensor 129 elastic 137 Eulerian 66 S Lagrangian 65 plastic 137, 138, 324, 430 Scalar plastic material 429 product, See Vector relative (polar) 65 Index 453

substructure 113 Stress vector 78 tensor 22 nominal 81 Stagnation of isotropic hardening 215, Stretch 229 left and right 55 Static friction, See Friction principal 55 Statically-admissible stress field, See Stretching, See Strain rate Admissible field Subloading surface model Steady term 51 extended 201, 203 Stick-sip phenomenon 371 initial 170 Strain 58 kinematic hardening 182 Almansi (Eulerian) 58 metals 213 Biot 61 normal-yield ratio 173, 203 energy function 123 return-mapping 384, 394 Green (Lagrangian) 58 stress-controlling function 178 Hencky 62 soils 254, 272 infinitesimal 59,72 subloading surface 171, 201, 203 logarithmic (natural) 63, 72 tangential-inelastic strain rate 183, volume 63, 74, 134, 135, 245 212 nominal 72 Subloading-friction model 348 volume 74, 248 normal-sliding ratio 349 Strain space theory 146 normal-sliding surface 448 Strain rate 64 return-mapping 407 elastic and plastic 135, 138 sliding-subloading surface 349 intermediate configuration 135, stress-controlling function 352 429 Subloading-overstress model 313 viscoplastic 310 dynamic-loading ratio 313 Stress-controlling function dynamic-loading surface 313 subloading surface model 178 subloading ratio 315 subloading-friction model 352 Subloading-overstress friction model Stress 78 376 Biot 121 Substructure spin, See Spin Cauchy 79, 104 Subyield state 171 covariant-contravariant convected 83 Summation convention 1 first Piola-Kirchhoff (nominal) 81, Superposition of rigid-body rotation 104, 123 102 Kirchhoff 81, 104, 121 Surface element 75 Mandel 428 Sylvester’s formula 30 Nominal 81 Symbolic notation, See Tensor notation second Piola-Kirchhoff stress 81, 104 Symmetric tensor, See Tensor pull-backed to intermediate SymmetryofCauchystress 79 configuration 428 Stress rate 115 T Cotter-Rivlin 115 Green-Naghdi 116, 319, 323 Tangent (stiffness) modulus 125, 128, nominal 116 398 Oldroyd 115, 125 Tangential Truesdell 115 inelastic strain rate 163, 183, 212, Zaremba-Jaumann 116, 125, 317, 316 322 stress rate 163, 183 Stress space theory 146 Tangential for friction 454 Index

associated flow rule of friction 352 skew-symmetric (anti-symmetric) 20 contact traction 346 skew-symmetrizing (fourth-order) 23 sliding velocity 345 similar 24 Tension and distortion behavior 97 spectral representation 24 Tension cut of yield surface 258 spherical part 21 Tensor spin 22 acoustic 339 strain, See Strain anti-symmetric, See skew-symmetric strain rate, See Strain rate Cartesian decomposition 21 stress, See Stress characteristic equation 26, 27 stress rate, See Stress rate coaxiality 31, 120 symbolic notation 15 component 17, 109 symmetric 20 contraction 14 symmetrizing (fourth-order) 23 definition 12 time-derivative, See Time derivative deviatoric part 21 trace 18 deviatoric projection (fourth-order) tracing (fourth-order) 22 23 transpose 19 diagonal component, See normal triple decomposition 21 component two-dimensional state 39 direct notation, See symbolic notation two-point 54, 107, 110 eigenprojection 29 Tensor field eigenvalue 23, 28 curl, See rotation eigenvector 23, 29 divergence 44 Eulerian 109 gradient 44 indicial notation 15 rotation (curl) 45 identity (second- and fourth-order) Tensor notation 17, 22 direct, See symbolic inverse 19 indicial 14 invertible (non-singular) 20 matrix 14 Lagrangian 109 symbolic 14 magnitude 19 Time derivative matrix notation 15 corotational 118 mean part, See spherical part local (spatial-time) 51 non-singular 20 material 51, 105, 117, 118 normal component 17, 23, 41, 70 moment of tensor 120 notation 14 non-steady (local time derivative) term objective (transformation) 13 51 orthogonal 15 principal invariant 120 partial derivative 42 scalar-valued tensor function 117 polar decomposition 32 steady (convective) term 51 positive definite 32 Time integration of objective rate tensor principal direction 23, 29 414 principal invariant, See Principal Total Lagrangian description, See invariant Description principal value, See Eigenvalue Trace 18 product, See Vector Transformation representation in principal space 36 objective 12 rotation-free(insensitive) 110, 415 negative 50 shear component 17 positive 50 similar, See Similar tensor tensor 12 Index 455

Transportation theorem, See Reynolds’ tensor product 17 transportation theorem Velocity gradient 63 Transpose 19 discontinuity 338 Traverse isotropy 324 Vertex theory, See Corner theory Traction, See Stress vector Virtual work principle 85 Triad Viscoelastic model 307 Eulerian 55 Viscoplastic model Lagrangian 55 creep model 310 Triple decomposition 21 ordinary overstress model 308, 312 Truesdell stress rate, See Stress rate subloading-overstress model 313 Two-dimensional state 39 Viscoplastic friction model Two-point identity tensor, See Tensor Subloading-overstress friction model Two-point tensor 54, 107, 110 376 Two surface model, See Cyclic plasticity Volume element 74 model Volumetric strain, See Strain Volumetric strain rate 74, 134, 135, U 245 elastic and plastic 134, 135, 249 Unconventional Vorticity 68 plasticity 167, 187 friction model 348 W Uniaxial loading behavior 87 Uniqueness of solution 169 Work Updated Lagrangian description, See conjugacy 120 Description conjugate pair 121 V hardening 143 rate (stress power) 121 Vandermonde’s determinant 35 Vector Y associative law 5 axial 21 Yield condition (surface) 139, 153 commutative law 5 Mises 142, 213 component description 8, 109 kinematic 213 cross product, See tensor product orthotropic 233 definition 5 Cam-clay 251 distributive law 5 rotational 268 dyad, See tensor product tensile strength 264 eigen 23, 29 Young’s modulus 129 equivalence 5 inner product, See scalar product Z magnitude 6, 11 parallelogram law 5 Zaremba-Jaumann rate 114 principal, See eigen Almansi strain 72 product 7 Cauchy stress 84, 116, 125, 317, 322 scalar product 6 Kirchhoff stress 116 scalar triple product 7 Ziegler’s kinematic hardening rule 156