Mechanics of Materials considered inﬁnitesimal strain cycle. A transition between elastic unloading and plastic loading is a neutral loading, in which an inﬁnitesimal strain increment is tangential to the yield surface and represents pure elastic deformation. Thus, 1. Yield Surface 8 <> 40; for plastic loading Materials capable of plastic deformation usually have @g ’ an elastic range of purely elastic response. This range : E ¼ 0; for neutral loading ð3Þ @E :> is a closed domain in either stress or strain space o0; for elastic unloading whose boundary is called the yield surface. The shape of the yield surface depends on the entire deformation The gradient @[email protected] is codirectional with the path from the reference state. The yield surfaces for outward normal to a locally smooth yield surface actual materials are mainly smooth, but may have or g ¼ 0 at the state of strain E. For incrementally linear develop pointed pyramidal or conical vertices. response, all inﬁnitesimal increments dE with equal Physical theories of plasticity (Hill 1967) imply the projections on the normal @[email protected], produce equal formation of a corner or vertex at the loading point plastic increments of stress dpT, since the compo- on the yield surface. However, experimental evidence nents of dE obtained by projection on the plane suggests that, while relatively high curvature at the tangential to the yield surface represent elastic loading point is often observed, sharp corners are deformation only. seldom seen (Hecker 1976). Experiments also indicate that yield surfaces for metals are convex in Cauchy stress space, if elastic response within the yield 1.2 Yield Surface in Stress Space surface is linear and unaffected by plastic ﬂow. The yield surface in stress space is deﬁned by f ðT; HÞ¼0. The stress T is a work-conjugate to ’ 1.1 Yield Surface in Strain Space strain E, in the sense that T : E represents the rate of work per unit initial volume (Hill 1978). The function The yield surface in strain space is deﬁned by f is related to g by gðE; HÞ¼0, where E is the strain tensor, and H the pattern of internal rearrangements due to plastic f ½TðE; HÞ; H¼gðE; HÞ¼0 ð4Þ ﬂow, that is, the set of appropriate internal variables including the path history by which they are achieved provided that physically identical conditions of yield (Rice 1971). The shape of the yield surface speciﬁed are imposed in both spaces. If elastic response within by function g is different for different choices of E.If the yield surface is Green-elastic, associated with the elastic response within the yield surface is Green- complementary strain energy f ¼ fðT; HÞ per unit elastic, associated with the strain energy c ¼ reference volume, the corresponding strain is cðE; HÞ per unit reference volume, the correspond- E ¼ @[email protected] For material in the hardening range ing stress is T ¼ @[email protected] From the strain state on the relative to measures E and T, an increment of stress current yield surface, an increment of strain dE dT from the stress state on the yield surface, directed directed inside the yield surface constitutes an elastic inside the yield surface constitutes an elastic unload- unloading. The associated incremental elastic re- ing. The associated incremental elastic response is sponse is governed by governed by

2 ’ ’ @ c @2f T ¼ K : E; K ¼ ð1Þ E’ ¼ M : T’; M ¼ ð5Þ @E#@E @T#@T where K ¼ KðE; HÞ is a tensor of elastic moduli of The tensor M ¼ MðT; HÞ is a tensor of elastic the material at the considered state of strain and compliances of the material at the considered state of internal structure. An increment of strain directed stress and internal structure. An increment of stress outside the current yield surface constitutes plastic directed outside the current yield surface constitutes loading. The resulting increment of stress consists of plastic loading in the hardening range of material elastic and plastic parts, such that response. The resulting increment of strain consists of p elastic and plastic parts, such that T’ ¼ T’ e þ T’ ¼ K : E’ þ T’ p ð2Þ e p p E’ ¼ E’ þ E’ ¼ M : T’ þ E’ ð6Þ The elastic part of the stress rate T’ e gives a stress decrement deT associated with elastic removal of the During plastic loading of hardening material, the strain increment dE. The plastic part of the stress rate yield surface locally expands the stress state remain- T’ p gives a residual stress decrement dpT in a ing on the yield surface. The elastic part of the strain

1 Mechanics of Materials

e rate E’ gives an elastic increment of strain deE, which shown that (Hill and Rice 1973, Havner 1992) is recovered upon elastic unloading of the stress p increment dT. The plastic part of the strain rate E’ p d T : dEo0 ð9Þ gives a residual increment of strain dpE which is left upon removal of the stress increment dT. A transi- Since during plastic loading the strain increment dE is directed outward from the yield surface, and tion between elastic unloading and plastic loading is a p neutral loading, in which an inﬁnitesimal stress since the same d T is associated with a fan of increment is tangential to the yield surface and inﬁnitely many dE around the normal @[email protected], all having the same projection on that normal, the produces only elastic deformation. Thus, in the p hardening range inequality (9) requires that d T is codirectional with 8 the inward normal to a locally smooth yield surface <> 40; for plastic loading in strain space, @f : T’ ¼ 0; for neutral loading ð7Þ > p @g @T : d T ¼Àdg ð10Þ o0; for elastic unloading @E The scalar multiplier dg40 is called a loading The gradient @f [email protected] is codirectional with the p outward normal to a locally smooth yield surface index. At a vertex of the yield surface, d T must lie f ¼ 0 at the state of stress T. For incrementally linear within the cone of limiting inward normals. response, all inﬁnitesimal increments dT with equal The inequality (9) and the normality rule (10) hold projections on @f [email protected] produce equal plastic incre- for all pairs of conjugate stress and strain measures, ments of deformation dpE, since the components of irrespective of the nature of elastic changes caused by dT obtained by projection on the plane tangential to plastic deformation, or possible elastic nonlinearities the yield surface give rise to elastic deformation only. within the yield surface. Also, (10) applies regardless In a softening range of material response Eqn. (6) of whether the material is in a hardening or softening still holds, although the elastic and plastic parts of the range. strain rate have purely formal signiﬁcance, since in If elastic response within the yield surface is the softening range it is not physically possible to nonlinear, Ilyushin’s postulate does not imply that perform an inﬁnitesimal cycle of stress starting from a the yield surface is necessarily convex. For a linearly stress point on the yield surface. The hardening is, elastic response, however, it follows that however, a relative term: material that is in the E0 E : dpT40 11 hardening range relative to one pair of stress and ð À Þ ð Þ strain measures, may be in the softening range provided that there is no change of elastic stiffness relative to another pair. caused by plastic deformation ðdK ¼ 0Þ, or the change is such that dK is negative semi-deﬁnite. The strain E0 is an arbitrary strain state within the yield 2. Plasticity Postulates, Normality and p Convexity of the Yield Surface surface. Since d T is codirectional with the inward normal to a locally smooth yield surface in strain Several postulates in the form of constitutive inequal- space, (11) implies that the yield surface is convex. ities have been proposed for certain types of materials The convexity of the yield surface is not an invariant undergoing plastic deformation. The two most well property, because dK can be negative deﬁnite for known are by Drucker (1960) and Ilyushin (1961). some measures ðE; TÞ, but not for others. Plastic stress and strain rates are related by p ’ p T’ ¼ÀK : E , so that, to ﬁrst order, 2.1 Ilyushin’s Postulate dpT ¼ÀK : dpE ð12Þ According to Ilyushin’s postulate, the network in an isothermal cycle of strain must be positive Since for any elastic strain increment, dE, emanat- I ing from a point on the yield surface in strain space T : dE40 ð8Þ and directed inside of it, E dpT : dE40 ð13Þ if a cycle at some stage involves plastic deformation. The integral in (8) over an elastic strain cycle is equal substitution of (12) into (13) gives to zero. Since a cycle of strain that includes plastic deformation, in general, does not return the material dpE : dTo0 ð14Þ to its initial state, the inequality (8) is not a law of thermodynamics. For example, it does not apply to Here, dT ¼ K : dE is the stress increment from the materials which dissipate energy by friction. For point on the yield surface in stress space, directed materials obeying Ilyushin’s postulate it can be inside of it (elastic unloading increment associated

2 Mechanics of Materials with elastic strain increment dE). Inequality (14) outward normals. In the softening range, holds for any dT directed inside the yield surface. p Consequently, dpE must be codirectional with the d E : dTo0 ð19Þ outward normal to a locally smooth yield surface in stress T space, Since dT is now directed inside the current yield surface, (19) also requires that dpE is codirectional p @f with the outward normal to a locally smooth yield d E ¼ dg ; dg40 ð15Þ @T surface in stress T space, with the same generalization p at a vertex as in the case of hardening behavior. At a vertex of the yield surface, d E must lie within If elastic response is nonlinear, the yield surface in the cone of limiting outward normals. Inequality (14) stress space is not necessarily convex. A concavity of and the normality rule (15) hold for all pairs of the yield surface in the presence of nonlinear elasticity conjugate stress and strain measures. for a particular material model has been demon- If material is in a hardening range relative to E and strated by Palmer et al. (1967). For linear elastic T, the stress increment dT producing plastic defor- response, however, mation dpE is directed outside the yield surface, satisfying ðT À T0Þ : dpE40 ð20Þ p d E : dT40 ð16Þ provided that there is no change of elastic stiff- ness caused by plastic deformation ðdM ¼ 0Þ, or that If material is in the softening range, the stress p the change is such that dM is positive semi-deﬁnite. increment dT producing plastic deformation d E is The stress state T0 is an arbitrary stress state within directed inside the yield surface, satisfying the the yield surface. Since dpE is codirectional with the reversed inequality in (16). The normality rule (15) outward normal to a locally smooth yield surface in applies to both hardening and softening. Inequality strain T space, (20) implies that the yield surface in a (16) is not measure invariant, since material may be considered stress space is convex. Inequality (20) is in the hardening range relative to one pair of often referred to as the principle of maximum plastic conjugate stress and strain measures, and in the work (Hill 1950, Johnson and Mellor 1973, Lubliner softening range relative to another pair. 1990). If inequality is assumed at the outset, it by The normals to the yield surfaces in stress and itself assures both normality and convexity. strain space are related by @g @f ¼ K : ð17Þ 3. Constitutive Equations of Elastoplasticity @E @T 3.1 Strain Space Formulation This follows directly from Eqn. (4) by partial differentiation. The stress rate is a sum of elastic and plastic parts, such that

e p ’ @g 2.2 Drucker’s Postulate T’ ¼ T’ þ T’ ¼ K : E À g’ ð21Þ @E A noninvariant dual to (8) is I E : dTo0 ð18Þ For incrementally linear and continuous response T between loading and unloading, the loading index is requiring that the net complementary work (relative 1 @g @g g’ ¼ : E’ ; : E’40 ð22Þ to measures E and T) in an isothermal cycle of stress h @E @E must be negative, if the cycle at some stage involves plastic deformation. Inequality (18) is noninvariant where h40 is a scalar function of the plastic state on because the value of the integral in (18) depends on the yield surface in strain space, determined from the the selected measures E and T, and the reference state consistency condition g’ ¼ 0. Consequently, the con- with respect to which they are deﬁned. This is because stitutive equation for elastoplastic loading is T is introduced as a conjugate stress to E such that, 1 @g @g for the same geometry change, T : dE (and not T’ ¼ K À # : E’ ð23Þ E : dT) is measure invariant. If inequality (18) applies h @E @E to conjugate pair ðE; TÞ, it follows that in the hardening range (16) holds, and dpE is codirectional The fourth-order tensor (within the square brack- with the outward normal to a locally smooth yield ets) is the elastoplastic stiffness tensor associated with surface in stress T space, Eqn. (15). At a vertex of the the considered measure and reference state. Within the yield surface, dpE must lie within the cone of limiting framework based on Green-elasticity and normality

3 Mechanics of Materials rule, the elastoplastic stiffness tensor possesses reci- response, the plastic part of the strain rate is procal or self-adjoint symmetry (with respect to ﬁrst indeterminate to the extent of an arbitrary positive and second pair of indices) in addition to symmetries multiple, since g’ in Eqn. (27) is indeterminate. in the ﬁrst and last two indices associated with the symmetry of stress and strain tensors. 3.3 Yield Surface with a Vertex The inverted form of (23) is Physical theories of plasticity imply the formation of 1 @g @g a corner or vertex at the loading point on the yield E’ ¼ M þ M : # : M : T’ ð24Þ H @E @E surface. Suppose that the yield surface in stress space has a pyramidal vertex formed by n intersecting where segments f/iS ¼ 0, then near the vertex Yn @g @g H ¼ h À : M : ð25Þ f/iSðT; HÞ¼0; nX2 ð30Þ @E @E i¼1 It follows that "# 3.2 Stress Space Formulation Xn Xn ’ 1 @f/iS @f/jS E ¼ M þ HÀ # : T’ ð31Þ The strain rate is a sum of elastic and plastic parts, /ijS @T @T such that i¼1 j¼1 @f This is an extension of the constitutive structure E’ ¼ E’ e þ E’ p ¼ M : T’ þ g’ ð26Þ @T (28) for the smooth yield surface to the yield surface with a vertex. Elements of the matrix inverse to 1 The loading index is obtained from the consistency plastic moduli matrix H/ S are denoted by HÀ . ’ ij /ijS condition f ¼ 0, The papers by Koiter (1953), Hill (1978), and Asaro (1983) can be referred to for further analysis. 1 @f g’ ¼ : T’ ð27Þ H @T 4. Constitutive Models of Plastic Deformation where H is a scalar function of the plastic state on the yield surface in stress space. Thus, 4.1 Isotropic Hardening Experimental determination of the yield surface ’ 1 @f @f E ¼ M þ # : T’ ð28Þ shape is commonly done with respect to Cauchy H @T @T stress r. Suppose that this is given by f ðr; kÞ¼0, where f is an isotropic function of r and k ¼ kðWÞ is a The fourth-order tensor (within the square brack- scalar which deﬁnes the size of the yield surface. This ets) is the elastoplastic compliance tensor associated depends on the history parameter, such as the with the considered measure and reference state. effective plastic strain The scalar parameter H can be positive, negative, Z t or equal to zero. Three types of response can be p p 1=2 identiﬁed within this constitutive framework. These W ¼ ð2D : D Þ dt ð32Þ are (Hill 1978) 0 The hardening model in which the yield surface @f H40; : T’40 hardening expands during plastic deformation preserving its @T shape is known as the isotropic hardening model. @f ’ Since f is an isotropic function of stress, the material Ho0; : To0 softening ð29Þ @T is assumed to be isotropic. For nonporous metals, the @f onset of plastic deformation and plastic yielding is H ¼ 0; : T’ ¼ 0 ideally plastic @T unaffected by moderate superimposed pressure. The yield condition can consequently be written as an Starting from the current yield surface in stress isotropic function of the deviatoric part of the space, the yield point moves outward in the case of Cauchy stress, i.e., its second and third invariant, hardening, inward in the case of softening, and f(J2, J3, k) ¼ 0. The well-known examples are the tangentially to the yield surface in the case of ideally Tresca maximum shear stress criterion, or the von plastic response. In the case of softening, E’ is not Mises yield criterion. In the latter case, uniquely determined by prescribed stress rate T’, since either Eqn. (28), or the elastic unloading expression ’ ’ 2 1 0 0 E ¼ M : T, applies. In the case of ideally plastic f ¼ J2 À k ðWÞ¼0; J2 ¼ 2 r : r ð33Þ

4 Mechanics of Materials

The corresponding plasticity theory is referred to Thus, f ðr À a; kÞ¼0, where a represents the current as the J2 ﬂow theory of plasticity. The yield stress in center of the yield surface (back stress), and f is an simple shear is kp.Ifﬃﬃﬃ Y is the yield stress in uniaxial isotropic function of the stress difference r À a. The tension, k ¼ Y= 3. The consistency condition gives size of the yield surface is speciﬁed by the constant k. The evolution of the back stress is governed by 1 ’ 0 ( H k2hp g ¼ 2 p ðr : sÞ; ¼ 4 t ð34Þ p p p 1=2 4k ht a( ¼ cðaÞD þ CðaÞð2D : D Þ ð40Þ where the plastic tangent modulus in shear test is p where c and C are appropriate scalar and tensor ht ¼ dk=dW. The stress rate functions of a. This representation is in accord with 3 assumed time independence of plastic deformation, s ¼ r( þ r tr D; r( ¼ r’ À W Á r þ r Á W ð35Þ À which requires Eqn. (40) to be a homogeneous relation of degree one. represents the Jaumann rate of the Kirchhoff stress If C ¼ 0 and c is taken to be constant, the model s ¼ðdet FÞr, when the current state is taken as the corresponds to Prager’s linear kinematic hardening. reference ðdet F ¼ 0Þ. The deformation gradient is F, The plastic tangent modulus hp in shear test is in this t p and the material spin is W. The total rate of case constant and related to c by c ¼ 2ht . The deformation is therefore resulting constitutive structure is 0# 0 1 r r ( 1 ðr0 À aÞ#ðr0 À aÞ D ¼ M þ p : s ð36Þ D ¼ M þ : s( ð41Þ 2h r0 : r0 p 0 0 t 2ht ðr À aÞ : ðr À aÞ The elastic compliance tensor for inﬁnitesimal with the inverse elasticity is 0 # 0 ( 2m ðr À aÞ ðr À aÞ 1 l s ¼ K À p : D ð42Þ M ¼ I À d#d ð37Þ 1 þ h =m ðr0 À aÞ : ðr0 À aÞ 2m 2m þ 3l t If C in Eqn. (40) is taken to be proportional to a The Lame! elastic constants are l and m. The (i.e., C ¼Àc0 a; c0 ¼ constant), a nonlinear kine- second- and fourth-order unit tensors are designated matic hardening model of Armstrong and Frederick by d and I, respectively. The plastic deformation is in p (1966) is obtained. Details can be found in Khan and this case isochoric ðtr D ¼ 0Þ, and principal direc- Huang (1995). Ziegler (1959) used it as an evolution p pB 0 tions of D are parallel to those of r (D s ). The equation for the back stress inverted form of (36) is a( b’ r0 a 43 2m r0#r0 ¼ ð À ÞðÞ s( ¼ K À : D ð38Þ p 0 0 ’ 1 þ ht =m r : r The proportionality factor b is determined from the consistency condition in terms of r and a. where # K ¼ ld d þ 2mI ð39Þ 4.3 Combined Isotropic-kinematic Hardening is the elastic stiffness tensor. Constitutive structures In this hardening model, the yield surface expands (36) and (38) have been extensively used in analytical and translates during plastic deformation, so that and numerical studies of large plastic deformation problems (Neale 1981, Needleman 1982). Inﬁnitesi- f ðr À a; kÞ¼0; k ¼ kðWÞð44Þ mal strain formulation, derivation of classical Prandtl–Reuss equations for elastic-ideally plastic, The function kðWÞ, with W deﬁned by Eqn. (32), and Levy–Mises equations for rigid-ideally plastic speciﬁes expansion of the yield surface, while evolu- material models can be found in standard texts or tion Eqn. (40) speciﬁes its translation. review papers (Hill 1950, Naghdi 1960). 4.4 Multisurface Models 4.2 Kinematic Hardening Motivated by the need to better model nonlinearities To account for the Bauschinger effect and anisotropy in stress–strain loops, cyclic hardening or softening, of hardening, a simple model of kinematic hardening cyclic creep and stress relaxation, more involved was introduced by Prager (1956). According to this hardening models were suggested. Mroz! (1967) model, the initial yield surface does not change its size introduced a multiyield surface model in which there and shape during plastic deformation, but translates is a ﬁeld of hardening moduli, one for each yield in the stress space according to some prescribed rule. surface. Initially the yield surfaces are assumed to be

5 Mechanics of Materials concentric. When the stress point reaches the inner- 5.2 Gurson Yield Condition for Porous Metals most yield surface, the plastic deformation develops according to linear hardening model with a pre- Based on a rigid-perfectly plastic analysis of spheri- scribed plastic tangent modulus, until the active yield cally symmetric deformation around a spherical surface reaches the adjacent yield surface. Subsequent cavity, Gurson (1977) suggested a yield condition plastic deformation develops according to linear for porous metals in the form hardening model with another speciﬁed value of the 2 2 2 I1 2 Y0 plastic tangent modulus, until the next yield surface is f ¼ J2 þ vY0 cosh Àð1 þ v Þ ¼ 0 ð47Þ reached, etc. Dafalias and Popov (1975) and Krieg 3 2Y0 3 (1975) suggested a hardening model, which uses the yield (loading) surface and the limit (bounding) where v is the porosity (void/volume fraction), and surface. A smooth transition from elastic to plastic Y0 ¼ constant is the tensile yield stress of the matrix regions on loading is assured by introducing a material. Generalizations to include hardening matrix continuous variation of the plastic tangent modulus material were also made. The change in porosity between the two surfaces. during plastic deformation is given by the evolution equation ’ p 5. Pressure-dependent Plasticity v ¼ð1 À vÞ tr D ð48Þ For porous metals, concrete and geomaterials like Other evolution equations, which take into account soils and rocks, plastic deformation has its origin in nucleation and growth of voids, have been consid- pressure-dependent microscopic processes and the ered. To improve its predictions and agreement with yield condition for these materials, in addition to experimental data, Tvergaard (1982) introduced two deviatoric components, depends on hydrostatic com- additional material parameters in the structure of the ponent of stress, i.e., its ﬁrst invariant I1 ¼ tr r. Gurson yield criterion. Mear and Hutchinson (1985) incorporated the effects of anisotropic (kinematic) 0 5.1 Drucker–Prager Yield Condition for hardening by replacing J2 of r in Eqn. (47) with J2 of Geomaterials r0 À a, where a is the back stress. Drucker and Prager (1952) suggested that yielding of soil occurs when the shear stress on octahedral planes 5.3 Constitutive Equations of Pressure-dependent overcomes cohesive and frictional resistance to sliding Plasticity on those planes. The yield condition is consequently The two considered pressure-dependent yield condi- 1=2 1 tions are of the type f ¼ J2 þ 3 aI1 À k ¼ 0 ð45Þ where a is a frictional parameter. This geometrically f ðJ2; I1; HÞ¼0 ð49Þ represents a cone in the principal stress space with its For materials obeying Ilyushin’s postulate, the axis parallel to hydrostatic axis.pﬃﬃﬃ The radius of the circle in the deviatoric plane is 2k,wherek is the plastic part of the rate of deformation tensor is normal to the yield surface, so that yield stresspﬃﬃﬃ in simple shear. The angle of the cone is 1 tanÀ ð 2a=3Þ. The yield stresses in uniaxial tension @f @f @f @f Dp ¼ g’ ; ¼ r0 þ d ð50Þ and compression according to Eqn. (45) are @r @r @J @I pﬃﬃﬃ pﬃﬃﬃ 2 1 3k 3k Y þ ¼ pﬃﬃﬃ; Y À ¼ pﬃﬃﬃ ð46Þ The loading index is 1 þ a= 3 1 À a= 3 1 @f @f g’ ¼ r0 þ d : s( ð51Þ For the yield conditionp toﬃﬃﬃ be physically meaningful, H @J2 @I1 the restriction holds ao 3. If the compressive states of stress are considered positive (as commonly done in where H is an appropriate hardening modulus. geomechanics), the minus sign appears in front of the Therefore, second term in Eqn. (45). 1 @f @f @f @f When Drucker–Prager cone is applied to porous Dp ¼ r0 þ d # r0 þ d : s( ð52Þ rocks, it overestimates the yield stress at higher H @J2 @I1 @J2 @I1 pressures and inadequately predicts inelastic volume changes. To circumvent this, DiMaggio and Sandler The volumetric part of the plastic rate of deforma- (1971) introduced an ellipsoidal cap to close the cone tion is at certain level of pressure. Other shapes of the cap 3 @f @f @f were also used. Details can be found in Chen and tr Dp ¼ r0 þ d : s( ð53Þ Han (1988). H @I1 @J2 @I1

6 Mechanics of Materials

For the Drucker–Prager yield condition, Since paf , the plastic compliance tensor in Eqn. (61) does not possess a reciprocal symmetry. @f 1 À1=2 @f 1 Consider an inelastic behavior of geomaterials, ¼ J2 ; ¼ a ð54Þ whose yield is governed by the Drucker–Prager yield @J2 2 @I1 3 condition of Eqn. (45). A nonassociative ﬂow rule and can be used with the plastic potential

Z 1=2 t p J 1 bI k 0 62 dk 0 0 ¼ 2 þ 3 1 À ¼ ð Þ H ¼ ; W ¼ ð2Dp : Dp Þ1=2 dt ð55Þ dW 0 The material parameter b is in general different For the Gurson yield condition, from the frictional parameter a of Eqn. (45). The rate of plastic deformation is @p 1 1 @f @f 1 I1 p ’ ’ À1=2 0 ¼ 1; ¼ vY0 sinh ð56Þ D ¼ g ¼ g J2 r þ bd ð63Þ @J2 @I1 3 2Y0 @r 2 3 ’ and The consistency condition f ¼ 0 gives the loading index 2 I1 I1 3 1 1 À1=2 0 1 dk H ¼ vð1 À vÞY0 sinh v À cosh ð57Þ g’ ¼ J r þ ad : s(; H ¼ ð64Þ 3 2Y0 2Y0 H 2 2 3 dW

Consequently, 6. Nonassociative Plasticity p 1 1 À1=2 0 1 Constitutive equations, in which plastic part of the D ¼ J2 r þ bd rate of strain is normal to locally smooth yield H 2 3 surface f ¼ 0 in stress space, 1 À1=2 1 # J r0 þ ad : s( ð65Þ 2 2 3 ’ p @f E ¼ g’ ð58Þ @T The deviatoric and spherical parts are ! are often referred to as associative ﬂow rules. A 0 0 0 1 r r : s( 1 sufﬁcient condition for this constitutive structure to Dp ¼ þ a tr s( ð66Þ 2H 1=2 1=2 3 hold is that the material obeys the Ilyushin’s postulate. J2 2J2 However, many pressure-dependent dilatant materials with internal frictional effects are not well described by ! associative ﬂow rules. For example, associative ﬂow b r0 : s( 1 tr Dp ¼ þ a tr s( ð67Þ rules largely overestimate inelastic volume changes in H 2J1=2 3 geomaterials like rocks and soils (Rudnicki and Rice 2 1975), and in certain high-strength steels exhibiting the The parameter b can be expressed as strength-differential effect by which the yield strength is higher in compression than in tension (Spitzig et al. tr Dp b ¼ ð68Þ 1975). For such materials, plastic part of the rate of 0 0 1=2 ð2Dp : Dp Þ strain is taken to be normal to plastic potential surface p ¼ 0,whichisdistinctfromtheyieldsurface.The which shows that b is the ratio of the volumetric and resulting constitutive structure, shear part of the plastic strain rate, often called the p @p dilatancy factor (Rudnicki and Rice 1975). Frictional E’ ¼ g’ ð59Þ @T parameter and inelastic dilatancy of material actually change with progression of inelastic deformation. An is known as nonassociative ﬂow rule (Nemat-Nasser analysis which accounts for their variation is presented 1983). The consistency condition f’¼ 0gives by Nemat-Nasser and Shokooh (1980). Constitutive formulation of elastoplastic theory with evolving 1 @f elastic properties is studied by Lubarda and Krajci- g’ ¼ : T’ ð60Þ H @T novic (1995) and others (see also Lubarda 2002). so that 6.1 Yield Vertex Model for Fissured Rocks ’ p 1 @p @f E ¼ # : T’ ð61Þ In a brittle rock, modeled to contain a collection of H @T @T randomly oriented ﬁssures, inelastic deformation

7 Mechanics of Materials results from frictional sliding on the ﬁssure surfaces. is described macroscopically by constitutive structure Inelastic dilatancy under overall compressive loads is of nonlinear elasticity, where strain is a function of a consequence of opening the ﬁssures at asperities stress. The strain tensor is decomposed into elastic and local tensile fractures at some angle at the edges and plastic part, E ¼ E e þ E p. The elastic part is of ﬁssures. Individual yield surface may be associated expressed in terms of stress by generalized Hooke’s with each ﬁssure. The macroscopic yield surface is the law, and plastic part is assumed to be envelope of individual yield surfaces for ﬁssures of all orientations, similarly to slip models of metal E p ¼ jT0 ð70Þ plasticity (Rudnicki and Rice 1975, Rice 1976). Continued stressing in the same direction will cause where j is an appropriate scalar function. Suppose continuing sliding on (already activated) favorably that a nonlinear relationship t% ¼ t%ðg%pÞ is available oriented ﬁssures, and will initiate sliding for a from the elastoplastic shear test. Deﬁne the plastic p % %p p % %p progressively greater number of orientations. After secant and tangent moduli by hs ¼ t=g ; ht ¼ dt=dg , certain amount of inelastic deformation, the macro- and let scopic yield envelope develops a vertex at the loading 1=2 point. The stress increment normal to the original t% ¼ 1 T0 : T0 ; g%p ¼ð2E p : E pÞ1=2 ð71Þ stress direction will initiate or continue sliding of 2 ﬁssure surfaces for some ﬁssure orientations. In p The scalar function j is then j ¼ 1=2hs .Although isotropic hardening idealization with smooth yield deformation theory of plasticity is total strain theory, it surface, however, a stress increment tangential to the is useful to cast it in the rate-type form, particularly yield surface will cause only elastic deformation, when the considered boundary value problem needs to overestimating the stiffness of response. In order to be solved in an incremental manner. The resulting take into account the effect of the yield vertex in an expression for the plastic part of the total rate of approximate way, a second plastic modulus H1 is deformation is introduced, which governs the response to part of the stress increment directed tangentially to what is taken 0# 0 ( p 1 ( 1 1 ðs s Þ : s D ¼ s0 þ À ð72Þ to be the smooth yield surface through the same 2hp 2hp 2hp s0 : s0 stress point. Since no vertex formation is associated s t s with hydrostatic stress increments, tangential stress where s( is the Jaumann derivative of the Kirchhoff increments are taken to be deviatoric, and Eqn. (66) stress. is replaced with ! 0 0 0 1 r r : s( 1 D p ¼ þ a tr s( 7.1 Application of Deformation Theory Beyond 2H J1=2 2J1=2 3 Proportional Loading 2 2 0 ( 1 0 r : s Deformation theory agrees with ﬂow theory of þ s( À r0 ð69Þ 2H1 2J2 plasticity only under proportional loading, since then speciﬁcation of the ﬁnal state of stress also speciﬁes The dilation induced by the small tangential stress the stress history. For general (nonproportional) increment is assumed to be negligible, so that Eqn. loading, more accurate and physically appropriate (67) applies for tr Dp. The constitutive structure of is the ﬂow theory of plasticity, particularly with an Eqn. (69) is intended to model the response at a yield accurate modelling of the yield surface and hardening surface vertex for small deviations from proportional behavior. Budiansky (1959), however, indicated that (straight ahead) loading s(Br0. For the full range of deformation theory can be successfully used for directions of stress increment, the relationship certain nearly proportional loading paths, as well. between the rates of stress and plastic deformation The stress rate s(0 in Eqn. (72) does not then have to be is not expected to be necessarily linear, although it is codirectional with s0, and the plastic part of the rate homogeneous in these rates in the absence of time- of deformation depends on both components of the dependent creep effects. stress rate s(0, one in the direction of s0 and the other normal to it. In contrast, according to ﬂow theory with the von Mises smooth yield surface, the 0 0 7. Deformation Theory of Plasticity component of the stress rate s( normal to s (thus tangential to the yield surface) does not affect the Simple plasticity theory has been suggested for plastic part of the rate of deformation. Since the proportional loading and small deformation by structure of the deformation theory of plasticity Hencky (1924) and Ilyushin (1963). A large deforma- under proportional loading does not use any notion tion version of the theory can be formulated by using of the yield surface, Eqn. (72) can be used to the logarithmic strain and its conjugate stress. Since approximately describe the response when the yield stress proportionally increase, elastoplastic response surface develops a vertex. Rewriting Eqn. (72) in the

8 Mechanics of Materials form possible, 1 ðs0#s0Þ : s( f f D p s(0 @ ’ @ ’ ¼ p À 0 0 H40; : T þ y40 2hs s : s @T @y 1 ðs0#s0Þ : s( thermoplastic hardening þ ð73Þ 2hp s0 : s0 t @f ’ @f ’ Ho0; : T þ yo0 the ﬁrst term on the right-hand side gives the response @T @y to component of the stress increment normal to s0.The thermoplastic softening p associated plastic modulus is hs . The plastic modulus @f ’ @f ’ associated with the component of the stress increment H ¼ 0; : T þ y ¼ 0 0 p @T @y in the direction of s is ht . A corner theory that predicts continuous variation of the stiffness and ideally thermoplastic ð75Þ allows increasingly nonproportional increments of stress was formulated by Christoffersen and Hutch- This parallels the isothermal classiﬁcation of inson (1979). When applied to the analysis of necking Eqn. (29). in thin sheets under biaxial stretching, the results were Since rate-independence is assumed, the constitu- in better agreement with experiments than those tive relationship has to be homogeneous of degree obtained from the theory with smooth yield char- one in rates of stress, strain, and temperature. For acterization. Similar observations were long known in thermoplastic part of the rate of strain this is satisﬁed the ﬁeld of elastoplastic buckling. Deformation theory by the normality structure predicts buckling loads better than ﬂow theory with a smooth yield surface (Hutchinson 1974). p @f E’ ¼ g’ ð76Þ @T

8. Thermoplasticity which, in view of Eqn. (74), becomes Nonisothermal plasticity is considered here assuming p 1 @f @f @f that temperature is not too high, so that creep E’ ¼ : T’ þ y’ ð77Þ deformation can be neglected. The analysis may also H @T @y @T be adequate for certain applications under high stresses of short duration, where temperature increase The strain rate is a sum of thermoelastic and is more pronounced but viscous (creep) strains have thermoplastic parts. The thermoelastic part is no time to develop (Prager 1958, Kachanov 1971). 2 2 Thus, inﬁnitesimal changes of stress and temperature ’ e @ f @ f ’ E ¼ : T’ þ y ð78Þ applied to the material at a given state produce a @T#@T @[email protected] unique inﬁnitesimal change of strain that is indepen- dent of the speed with which these changes are made. For example, if Rate-dependent plasticity models will be presented in Sect. 9. 1 l The formulation of thermoplastic analysis under f ¼ tr T2 À tr2 T described conditions can proceed by introducing a 4m 3l þ 2m nonisothermal yield condition in either stress or þ aðyÞ tr T þ bðy; HÞð79Þ strain space. For example, the yield condition in stress space is f ðT; y; HÞ¼0. The response within there follows the yield surface is thermoelastic. If the Gibbs energy relative to selected stress and strain measures is f ¼ ’ e 1 l ’ 0 ’ fðT; y; Þ per unit reference volume, the strain is E ¼ I À d#d : T þ a ðyÞyd ð80Þ H 2m 2m þ 3l E ¼ @[email protected] Let the stress state T be on the current yield ! surface. The rates of stress and temperature asso- where l and b are the Lame-type elastic constants ciated with thermoplastic loading satisfy the consis- corresponding to selected measures, a and b are tency condition f’¼ 0, which gives appropriate functions of indicated arguments, and a0 ¼ da=dy. @f @f Suppose that nonisothermal yield condition in the : T’ þ y’ À g’H ¼ 0 ð74Þ @T @y Cauchy stress space is temperature-dependent von Mises condition The hardening parameter is H ¼ HðT; y; Þ, and H 1 0 0 2 the loading index is g’40. Three types of response are f ¼ 2 r : r À½jðyÞkðWÞ ¼ 0 ð81Þ

9 Mechanics of Materials

The thermoplastic part of the deformation rate is absolute temperature y. Geometrically, the plastic then part of the strain rate is normal to surfaces of constant ﬂow potential in stress space. There is no 1 r0#r0 j0 D p : s( r0 y’ 82 yield surface in the model and plastic deformation ¼ p 0 0 À ð Þ 2jht r : r j commences from the onset of loading. Time-inde- p 0 pendent behavior can be recovered under certain where ht ¼ dk=dW and j ¼ dj=dy. Combining with idealizations—neglecting creep rate effects, as an Eqn. (80), the total rate of deformation is appropriate limit (Rice 1970). 1 l 1 r0#r0 D I d#d : s( ¼ À þ p 0 0 2m2m þ 3l 2jht r : r 9.1 Power-law and Johnson–Cook Models 0 0 j 0 ’ þ a ðyÞd À 2 p r y ð83Þ The power-law representation of the ﬂow potential in 2j ht the Cauchy stress space is ! The inverse equation is 1=2 m 2g’0 J O ¼ 2 J1=2 2m r0#r0 m þ 1 k 2 s( ¼ ld#d þ 2mI À : D 1 jhp=m r0 : r0 þ t ð88Þ 1 j0 1 0 0 0 0 ’ J2 ¼ 2 r : r Àð3l þ 2mÞa d À p r y ð84Þ 1 þ jht =m j where k ¼ kðy; HÞ is the reference shear stress, g’0 is Inﬁnitesimal strain formulation for rigid-thermo- the reference shear strain rate to be selected for each plastic material was given by Prager (1958) (see also material, and m is the material parameter (of the order Lee 1969, Naghdi 1990). Experimental investigation of 100 for metals at room temperature and strain rates of nonisothermal yield surfaces was reported by below 104 sÀ1; Nemat-Nasser (1992). The correspond- Phillips (1982). ing plastic part of the rate of deformation is ! In the case of thermoplasticity with linear kine- m matic hardening ðc ¼ 2hpÞ, and the temperature- J1=2 r0 t Dp ¼ g’0 2 ð89Þ dependent yield surface k 1=2 J2 1 f ¼ ðr0 À aÞ : ðr0 À aÞÀ½jðyÞk2 ¼ 0; 2 The equivalent plastic strain is usually used as the only history parameter , and the reference shear k ¼ constant ð85Þ H stress depends on W and y according to there follows W a y À y k ¼ k0 1 þ exp Àb 0 ð90Þ 0 # 0 0 0 p 1 ðr À aÞ ðr À aÞ j ’ W ym À y0 D ¼ : s( À ðr0 À aÞy ð86Þ 2hp ðr0 À aÞ : ðr0 À aÞ j t Here, k0 and W0 are the normalizing stress and strain, y0 and ym are the room and melting temperatures, 9. Rate-dependent Plasticity and a and b are the material parameters. From the onset of loading, the deformation rate consists of This section is devoted to inelastic constitutive elastic and plastic constituents, although for large equations for metals in the strain rate sensitive range m the plastic contribution may be small if J2 is less of material response, where time effects play an than k. important role. There is an indication from the Another representation of the ﬂow potential, dislocation dynamics point of view (Johnston and constructed according to Johnson and Cook (1983) Gilman 1959) that plasticity caused by crystallo- model, is graphic slip in metals is inherently time dependent. "# ! 2g’0 J1=2 Once it is assumed that the rate of shearing on a given O ¼ k exp a 2 À 1 ð91Þ slip system depends on local stresses only through the a k resolved shear stress in slip direction, the plastic part of the rate of strain is derivable from a scalar ﬂow The reference shear stress is potential (Rice 1971) as "# c d 0 W y À y0 ’ p @OðT; y; HÞ k ¼ k 1 þ b 1 À ð92Þ E ¼ ð87Þ 0 y À y @T W m 0 The history of deformation is represented by the where a, b, c, and d are the material parameters. The pattern of internal rearrangements H, and the corresponding plastic part of the rate of deformation

10 Mechanics of Materials is, in this case, 10. Phenomenological Plasticity based on "# ! the Multiplicative Decomposition J1=2 r0 Dp ¼ g’0 exp a 2 À 1 ð93Þ k J1=2 In this section, a multiplicative decomposition of the 2 total deformation gradient into elastic and plastic parts is introduced to provide an additional frame- work for dealing with ﬁnite elastic and plastic 9.2 Viscoplasticity Models deformation. The decomposition in the speciﬁc context of a strain rate dependent, J2-ﬂow theory of For high-strain rate applications in dynamic plasti- plasticity is applied. Consider the current elastoplas- city (Cristescu 1967, Clifton 1983), the ﬂow potential tically deformed conﬁguration of the material sam- can be taken as ple. Let F be the deformation gradient that maps an inﬁnitesimal material element dX from initial conﬁg- 1 1=2 2 O ¼ ½J2 À ksðWÞ ð94Þ uration to dx in current conﬁguration, such that z dx ¼ F Á dX. An intermediate conﬁguration is intro- duced by elastically destressing the current conﬁg- where z is the viscosity coefﬁcient, and ksðWÞ represents the shear stress—plastic strain relationship uration to zero stress. Such conﬁguration differs from from the (quasi) static shear test. The positive the initial conﬁguration by a residual (plastic) 1=2 deformation, and from the current conﬁguration by difference J2 À ksðWÞ between the measure of the p current dynamic stress state and corresponding static a reversible (elastic) deformation. If dx is the stress state (at the given level of equivalent plastic material element in the intermediate conﬁguration, corresponding to dx in the current conﬁguration, strain W) is known as the overstress measure (Malvern e p e 1951). The plastic part of the rate of deformation is then dx ¼ F Á dx , where F represents a deforma- tion gradient associated with elastic loading from the 0 p 1 1=2 r intermediate to current conﬁguration. If the deforma- D ¼ ½J2 À ksðWÞ ð95Þ p z J1=2 tion gradient of plastic transformation is F , such 2 that dxp ¼ F p Á dX, the multiplicative decomposition The inverted form of Eqn. (95) is of the total deformation gradient into its elastic and plastic parts holds p 0 p D r ¼ zD þ 2ksðWÞ ð96Þ ð2Dp : DpÞ1=2 F ¼ F e Á F p ð100Þ which shows that the rate dependence in the model comes from the ﬁrst term on the right-hand side. In The decomposition was introduced in the phenom- quasi-static tests, viscosity z is taken to be equal to enological rate-independent theory of plasticity by zero, and Eqn. (96) reduces to time-independent von- Lee (1969). In the case when elastic destressing to Mises isotropic hardening plasticity. In this case, ﬂow zero stress is not physically achievable due to possible potential O is constant within the elastic range onset of reverse inelastic deformation before the state 1=2 of zero stress is reached, the intermediate conﬁgura- bounded by the yield surface J2 ¼ ksðWÞ. More general representation for O is possible by tion can be conceptually introduced by virtual using the Perzyna (1966) viscoplastic model. For destressing to zero stress, locking all inelastic example, one can take structural changes that would take place during the actual destressing. The deformation gradients F e and C mþ1 F p are not uniquely deﬁned because the intermediate O ¼ ½ f ðrÞÀksðWÞ ð97Þ m þ 1 unstressed conﬁguration is not unique. Arbitrary local material rotations can be superposed to the which yields intermediate conﬁguration, preserving it unstressed. @f In applications, however, the decomposition (100) Dp ¼ C½ f ðrÞÀk ðWÞm ð98Þ s @r can be made unique by additional speciﬁcations, dictated by the nature of the considered material 1=2 0 If f ¼ J2 , C ¼ 2=z, and ksðWÞ¼k ¼ constant, model. For example, for elastically isotropic materi- then Eqn. (98) gives als the elastic stress response depends only on the elastic stretch V e, and not on the rotation Re from 1 r0 e e e Dp ¼ ðJ1=2 À k0Þm ð99Þ the polar decomposition F ¼ V Á R . Conse- z 2 1=2 J2 quently, the intermediate conﬁguration can be speciﬁed uniquely by requiring that elastic unloading e e which is a nonlinear Bingham model. If ksðWÞ¼0, takes place without rotation ðF ¼ V Þ. An alter- 1=2 ’0 m f ¼ J2 , and C ¼ 2g =k , then Eqn. (98) reproduces native choice will be pursued in the constitutive the power-law J2 creep given by Eqn. (89). derivation presented here; see also Lubarda (2002).

11 Mechanics of Materials # The velocity gradient in the current conﬁguration The rectangular components of L are at time t is deﬁned by 2 e # e e @ C e e ’ À1 Lijkl ¼ FimFjn e e FkpFlq ð110Þ L ¼ F Á F ð101Þ @[email protected]

The superposed dot designates the material time Equation (109) can be equivalently written as derivative. By introducing the multiplicative decom- e position of deformation gradient (100), the velocity s’ ÀðF’ Á F eÀ1Þ Á s þ s ÁðF’ e Á F eÀ1Þ gradient becomes a a ’ e eÀ1 ¼ L : ðF Á F Þs ð111Þ L ¼ F’ e Á F eÀ1 þ F e ÁðF’ p Á F pÀ1ÞÁF eÀ1 ð102Þ The modiﬁed elastic moduli tensor L has the The rate of deformation D and the spin W are, components respectively, the symmetric and antisymmetric part of L, # 1 Lijkl ¼ Lijkl þ 2 ðtikdjl þ tjkdil þ tildjk þ tjldikÞð112Þ

’ e eÀ1 e ’ p pÀ1 eÀ1 D ¼ðF Á F Þs þ½F ÁðF Á F ÞÁF s ð103Þ In view of Eqn. (107), we can rewrite Eqn. (111) as

( ’ e eÀ1 s ¼ L : ðF Á F Þs ð113Þ ’ e eÀ1 e ’ p pÀ1 eÀ1 W ¼ðF Á F Þa þ½F ÁðF Á F ÞÁF a ð104Þ where Since Fe is speciﬁed up to an arbitrary rotation, and since the stress response of elastically isotropic s( ¼ s’ À W Á s þ s Á W ð114Þ materials does not depend on the rotation, an unloading program can be chosen, such that is the Jaumann rate of the Kirchhoff stress with respect to total spin. By inversion, Eqn. (113) gives e ’ p pÀ1 eÀ1 ½F ÁðF Á F ÞÁF a ¼ 0 ð105Þ the elastic rate of deformation as

e ’ e e 1 1 With this choice, therefore, the rate of deformation D ¼ðF Á F À Þ ¼ LÀ : s( ð115Þ and the spin tensors are s Physically, the strain increment De dt is a reversible ’ e eÀ1 e ’ p pÀ1 eÀ1 D ¼ðF Á F Þs þ F ÁðF Á F ÞÁF ð106Þ part of the total strain increment D dt, which is recovered upon loading–unloading cycle of the stress increment s( dt. The remaining part of the total rate of ’ e eÀ1 W ¼ðF Á F Þa ð107Þ deformation,

Dp ¼ D À De ð116Þ

10.1 Elastic and Plastic Constitutive Contributions is the plastic part, which gives a residual strain It is assumed that the material is elastically isotropic increment left upon the considered inﬁnitesimal cycle of stress. When the material obeys Ilyushin’s work in its initial undeformed state, and that plastic p deformation does not affect its elastic properties. postulate, the so-deﬁned plastic rate of deformation D The elastic response is then given by is codirectional with the outward normal to a locally smooth yield surface in the Cauchy stress space, i.e., @CeðE eÞ s ¼ F e Á Á F eT ð108Þ @f @E e Dpjj ð117Þ @r The elastic strain energy per unit unstressed volume, Ce, is an isotropic function of the Lagran- gian strain E e ¼ðF eT Á F e À IÞ=2. Plastic deforma- 10.2 Rate-dependent J2 Flow Theory tion is assumed to be incompressible e Classical J2 ﬂow theory uses the yield surface as ðdet F ¼ det FÞ, so that s ¼ðdet FÞr is the Kirchh- generated earlier as a ﬂow potential. Thus, the current off stress. By differentiating Eqn. (108), one obtains yield criteria s% ¼ k deﬁnes a series of yield surfaces in stress space, where k serves the role of a scaling e s’ ÀðF’ Á F eÀ1ÞÁs À s ÁðF’ e Á F eÀ1ÞT parameter. Here we rephrase the yield criterion in terms of the effective stress, s% ¼ð3=2s0 s0 Þ1=2; k is # ’ e eÀ1 ij ij ¼ L : ðF Á F Þs ð109Þ then the uniaxial yield stress. J2 ﬂow theory assumes

12 Mechanics of Materials

p 0 that D jjr . This amounts to taking Asaro R J 1983 Crystal plasticity. J. Appl. Mech. 50, 921–34 Budiansky B 1959 A reassessment of deformation theories of @s% plasticity. J. Appl. Mech. 26, 259–64 Dpjj ð118Þ @r0 Chen W F, Han D J 1988 Plasticity for Structural Engineers. Springer, New York or Christoffersen J, Hutchinson J W 1979 A class of phenomen-

0 ological corner theories of plasticity. J. Mech. Phys. Solids @s% 3 s 27, 465–87 Dpjj ¼ ij ð119Þ ij @s0 2 s% Clifton R J 1983 Dynamic plasticity. J. Appl. Mech. 50, ij 941–52 Thus one can write Cristescu N 1967 Dynamic Plasticity. North-Holland, Amsterdam 3 r0 Dafalias Y F, Popov E P 1975 A model of nonlinearly Dp ¼ e%’p ð120Þ hardening materials for complex loading. Acta Mech. 21, 2 s% 173–92 where e%’p is an effective plastic strain rate whose DiMaggio F L, Sandler I S 1971 Material model for granular speciﬁcation requires an additional model statement. soils. ASCE J. Eng. Mech. 97, 935–50 By incorporating Eqn. (120) one can write from Drucker D C 1960 Plasticity. In: Goodier J N, Hoff N J (eds.) Eqn. (113) Structural Mechanics – Proc. 1st Symp. Naval Struct. Mechanics. Pergamon, New York, pp. 407–55 Drucker D C, Prager W 1952 Soil mechanics and plastic s( ¼ L : D e ¼ L : ðD À D pÞ analysis or limit design. Q. Appl. Math. 10, 157–65 3 r0 Gurson A L 1977 Continuum theory of ductile rapture by void ¼ L : D À e%’p ð121Þ nucleation and growth – Part I: Yield criteria and ﬂow rules 2 s% for porous ductile media. J. Eng. Mater. Tech. 99, 2–15 By adopting a simple power law expression of the Havner K S 1992 Finite Plastic Deformation of Crystalline form Solids. Cambridge University Press, Cambridge Hecker S S 1976 Experimental studies of yield phenomena in 1=m biaxially loaded metals. In: Stricklin J A, Saczalski K J (eds.) s% Constitutive Equations in Viscoplasticity: Computational and e%’p ¼ e’ ð122Þ 0 g Engineering Aspects. ASME, New York, AMD Vol. 20, pp. 1–33. where e’0 is a reference strain rate and 1/m represents a Hencky H 1924 Zur Theorie plastischer Deformationen und der strain rate sensitivity coefﬁcient. For common metals, hierdurch im Material hervorgerufenen Nachspannungen. 50o1/mo200. For values of 1/mB100, or larger, the Z. Angew. Math. Mech. 4, 323–34 materials will display a very nearly rate independent Hill R 1950 The Mathematical Theory of Plasticity. Oxford response in the sense that s% will track g at nearly any University Press, London value of strain rate. Hill R 1967The essential structure of constitutive laws for Strain hardening is described as an evolution of the metal composites and polycrystals. J. Mech. Phys. Solids 15, 79–95 hardness function g. This is often taken to be Hill R 1978 Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1–75 %p n p e Hill R, Rice J R 1973 Elastic potentials and the structure of gðe% Þ¼s0 1 þ ð123Þ ey inelastic constitutive laws. SIAM J. Appl. Math. 25, 448–61 where Hutchinson J W 1974 Plastic buckling. Adv. Appl. Mech. 14, 67–144 Ilyushin A A 1961 On the postulate of plasticity. Prikl. Math. 2 1=2 e%’p ¼ Dp : Dp Mekh. 25, 503–7 3 Ilyushin A A 1963 Plasticity. Foundations of the General Z Mathematical Theory. Izd. Akad. Nauk SSSR, Moscow 1=2 ð124Þ t 2 (in Russian). e%p ¼ Dp : Dp dt 3 Johnson G R, Cook W H 1983 A constitutive model and data 0 for metals subjected to large strains, high strain rates, and high temperatures. Proceedings of the 7th International are the effective plastic strain rate and effective plastic Symposium on Ballistics. The Hague, The Netherlands, strain, respectively. The remaining parameters are the pp. 1–7. material parameters; the initial yield stress is s0, and n Johnson W, Mellor P B 1973 Engineering Plasticity. Van is the hardening exponent. Nostrand Reinhold, London Johnston W G, Gilman J J 1959 Dislocation velocities, dislocation densities, and plastic ﬂow in LiF crystals. Bibliography J. Appl. Phys. 30, 129–44 Kachanov L M 1971 Foundations of Theory of Plasticity. Armstrong P J, Frederick C O 1966 A Mathematical North-Holland, Amsterdam Representation of the Multiaxial Bauschinger Effect,G.E. Khan A S, Huang S 1995 ContinuumTheory of Plasticity . G. B. Report RD/B/N, 731. Wiley, New York

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Koiter W 1953 Stress–strain relations, uniqueness and varia- Nemat-Nasser S, Shokooh A 1980 On ﬁnite plastic ﬂows of tional theorems for elastic–plastic materials with a singular compressible materials with internal friction. Int. J. Solids yield surface. Q. Appl. Math. 11, 350–4 Struct. 16, 495–514 Krieg R D 1975 A practical two surface plasticity theory. Palmer A C, Maier G, Drucker D C 1967Normality relations J. Appl. Mech. 42, 641–6 and convexity of yield surfaces for unstable materials and Lee E H 1969 Elastic–plastic deformation at ﬁnite strains. structures. J. Appl. Mech. 34, 464–70 J. Appl. Mech. 36, 1–6 Perzyna P 1966 Fundamental problems in viscoplasticity. Adv. Lubarda V A 2002 Elastoplasticity Theory. CRC Press, Appl. Mech. 9, 243–377 Boca Raton Phillips A 1982 Combined stress experiments in plasticity – The Lubarda V A, Krajcinovic D 1995 Some fundamental issues effects of temperature and time. In: Lee E H, Mallett R L in rate theory of damage-elastoplasticity. Int. J. Plasticity (eds.) Plasticity of Metals at Finite Strain: Theory, Experi- 11, 763–97 ment and Computation. Div. Appl. Mechanics, Stanford Lubliner J 1990 Plasticity Theory. Macmillan Publishing, University, Stanford, pp. 230–52 New York Prager W 1956 A new method of analyzing stresses and Malvern L E 1951 The propagation of longitudinal waves of strains in work-hardening plastic solids. J. Appl. Mech. 23, plastic deformation in a bar of material exhibiting a strain- 493–6 rate effect. J. Appl. Mech. 18, 203–8 Prager W 1958 Non-isothermal plastic deformation. Konikl. Mear M E, Hutchinson J W 1985 Inﬂuence of yield surface Ned. Acad. Wetenschap. Proc. 61, 176–82 curvature on ﬂow localization in dilatant plasticity. Mech. Rice J R 1970 On the structure of stress–strain relations for Materials 4, 395–407 time-dependent plastic deformation in metals. J. Appl. Mech. Mroz! Z 1967On the description of anisotropic work-hardening. 37, 728–37 J. Mech. Phys. Solids 15, 163–75 Rice J R 1971 Inelastic constitutive relations for solids: an Naghdi P M 1960 Stress–strain relations in plasticity and internal variable theory and its application to metal thermoplasticity. In: Lee E H, Symonds P (eds.) Plasticity— plasticity. J. Mech. Phys. Solids 19, 433–55 Proc. 2nd Symp. Naval Struct. Mechanics, Pergamon, New Rice J R 1976 The localization of plastic deformation. In: York, pp. 121–67 Koiter W T (ed.) Theoretical and Applied Mechanics. North- Naghdi P M 1990 A critical review of the state of ﬁnite Holland, Amsterdam, pp. 207–20 plasticity. J. Appl. Math. Phys. 41, 315–94 Rudnicki J W, Rice J R 1975 Conditions for the localization of Neale K W 1981 Phenomenological constitutive laws in ﬁnite deformation in pressure-sensitive dilatant materials. J. Mech. plasticity. SM Archives 6, 79–128 Phys. Solids 23, 371–94 Needleman A 1982 Finite elements for ﬁnite strain plasticity Spitzig W A, Sober R J, Richmond O 1975 Pressure dependence problems. In: Lee E H, Mallett R L (eds.) Plasticity of Metals of yielding and associated volume expansion in tempered at Finite Strain: Theory, Experiment and Computation. Div. martensite. Acta Metall. 23, 885–93 Appl. Mechanics, Stanford University, Stanford, pp. 387–443 Tvergaard V 1982 On localization in ductile materials contain- Nemat-Nasser S 1983 On ﬁnite plastic ﬂow of crystalline solids ing spherical voids. Int. J. Fracture 18, 237–52 and geomaterials. J. Appl. Mech. 50, 1114–26 Ziegler H 1959 A modiﬁcation of Prager’s hardening rule. Nemat-Nasser S 1992 Phenomenological theories of elastoplas- Q. Appl. Math. 17, 55–65 ticity and strain localization at high strain rates. Appl. Mech. Rev. 45, S19–45 V. A. Lubarda

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