Plasticity Theory: Yield Criteria, Flow Rule, and Consistency Condition, Notas de aula de Análise Estrutural

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BASIC COMPUTATIONAL PLASTICITY

KRISTIAN KRABBENHØFT

Department of Civil Engineering Technical University of Denmark

June 2002

Contents

• 1 Introduction
  • 1.1 Plasticity
• 2 Plasticity Theory
  • 2.1 The yield criterion
  • 2.2 Specific yield criteria
    • 2.2.1 The Tresca criterion
    • 2.2.2 The von Mises criterion
    • 2.2.3 The Drucker-Prager criterion
  • 2.3 Loading/unloading conditions
  • 2.4 Plastic strains
  • 2.5 Incremental stress-strain relations
  • 2.6 Hardening
  • 2.7 Plane stress versus plane strain
• 3 Finite element implementation
  • 3.1 Integration of the constitutive relations
    • 3.1.1 Explicit integration
    • 3.1.2 Return algorithms
    • 3.1.3 Implicit methods
    • 3.1.4 Accuracy of return algorithms
  • 3.2 Solution of nonlinear finite element equations
    • 3.2.1 Newton’s method
    • 3.2.2 General solution procedures
  • 3.3 Consistent tangent matrix
• 4 Example
• 5 Solutions of nonlinear FE–equations
  • 5.1 Implementation of modified Newton-Raphson method for elasto-plasticity

(a) (b)

Figure 2: Material models: Linear elastic-perfectly plastic (a) and rigid-perfectly plastic (b).

1.1 Plasticity

Material nonlinearity itself may be subdivided into some fundamentally different cat- egories. In nonlinear elasticity the stress-strain relation is nonlinear but otherwise the behaviour follows that of linear elasticity, that is, no distinction is made between loading and unloading except for the sign. This is in contrast to what is that case with plastic or elasto-plastic materials, where irreversible strains occur. The difference between two such materials is illustrated in Figure 1. For low stress levels both materials follow a linear stress-strain relation. This is followed by a decrease in stiffness as the stress increases. If however, the stress is reduced the nonlinear elastic material will follow the same stress- strain curve as in loading, whereas unloading of the elasto-plastic material leads to a new branch on the σ − ε curve where the material is again elastic, often with a stiffness equal to the initial elastic stiffness. Furthermore, it is clear that when the material is completely unloaded, an irreversible plastic strain εp^ remains. The curved part above the elastic limit in Figure 1 (b) suggests that the stress-strain relation may be of a rather complicated nature and in general this may indeed be true. However, in practice, a number of approximations can be made. One of the most common

Figure 3: Hardening plasticity: Bauschinger effect.

approximations is to assume linear elasticity below some limit above which the mate- rial is perfectly plastic, i.e. the stress remains constant. This is shown in Figure 2 (a). Another common approximation is the rigid-perfectly plastic material model. Here the elastic strains are ignored altogether and deformations then take place only when the stress reaches a certain level, Figure 2 (b). Alternatively, a certain degree of hardening may be introduced such that yield stress σ 0 (α), now depending on one or more hardening parameters, is continuously increased as the plastic loading progresses. One of the most common types of hardening is described by the so-called Baushcinger effect which can be observed in metals. This model involves an increase in the yield stress under plastic loading while at the same time the negative limit is lifted such that the permissible stress range remains constant. This is illustrated in Figure 3. For a more complete discussion of the physical foundations of plasticity we refer to Chen and Han [1].

by a point either inside or on the yield surface. In plastic loading with perfect plasticity the state of stress can only be altered by a redistribution between the different stress components such that the stress point can be imagined as sliding along the yield surface. If hardening is present the stress point will still remain on the boundary defined by f = 0 but this boundary will be shifted according to the relevant hardening rules as the loading progresses. In unloading the stress point moves from the boundary of the yield surface to the inside and thus, immediately recovers its elastic properties.

2.2 Specific yield criteria

When defining the yield criterion care must be taken to ensure that a rotation of the coordinate system does not influence the conditions at which the material yields. For the uniaxial stress test a yield criterion may be formulated as

σx − σ 0 = 0 (3)

This criterion, however, only makes sense under the assumption that that the material is aligned in a coordinate system where all other stress components than σx are equal to zero. Therefore, it is convenient to define yield criteria in terms of certain invariants, that is, quantities which are not affected by a rotation of the coordinate system. These could be the magnitude of the principal stresses. Thus, rather than using (1) it is more convenient to write the yield criterion as

f (σ 1 , σ 2 , σ 3 , α 1 , ..., αn) = 0 (4)

where σ 1 , σ 2 , σ 3 are the principal stresses. In many cases, however, it turns out to be even more convenient to use another set of invariants, namely the invariants of the stress tensor I 1 , I 2 and I 3 and the invariants of the deviatoric stress tensor J 1 , J 2 and J 3 , see e.g. Chen and Han [1]. Furthermore, of these invariants it turns out that especially I 1 and J 2 are useful. The invariants are obtained by separating the state of stress into two components as

σij =

σ 1 − p 0 0 0 σ 1 − p 0 0 0 σ 1 − p

p 0 0 0 p 0 0 0 p

or σij = sij + pδij (6)

where the mean stress is defined as

p = 13 (σ 1 + σ 2 + σ 3 ) = 13 (σx + σy + σz ) = 13 I 1 (7)

The second invariant of sij is given by

J 2 = 16 [(σ 1 − σ 2 )^2 + (σ 2 − σ 3 )^2 + (σ 3 − σ 1 )^2 ] = 16 [(σx − σy)^2 + (σy − σz )^2 + (σz − σx)^2 ] + τ (^) xy^2 + τ (^) yz^2 + τ (^) zx^2

Figure 5: Mohr’s circles for triaxial state of stress.

Whereas pδij represents a state of hydrostatic stress the deviatoric part sij can be shown to represent a state of pure shear, and thus, the effects on yielding of shear and hydrostatic pressure can be effectively separated. Furthermore, the principal shear stresses are given by, see Figure 5,

τ 12 = 12 |σ 1 − σ 2 |, τ 23 = 12 |σ 2 − σ 3 |, τ 31 = 12 |σ 3 − σ 1 | (9)

which, except for the factor 12 are the terms contained in J 2. Thus, yield criteria making use if J 2 actually predict yielding to occur as a result of some combination of the principal shear stresses exceeding a certain limit. In the following, both examples of yield criteria developed directly from the principal stresses as well as from I 1 and J 2 are given.

2.2.1 The Tresca criterion

In 1864, after having subjected metal specimens to a combination of stresses, Tresca proposed that yielding occurs as a result of the maximal shear stress reaching a critical value. This was formulated as

τmax = max(^12 |σ 1 − σ 2 |, 12 |σ 2 − σ 3 |, 12 |σ 3 − σ 1 |) = max(τ 12 , τ 23 , τ 31 ) = k

where the material parameter k can be determined by the simple tension test as

k = 12 σ 0 (11)

In plane stress one of the principal stresses, say σ 3 , is equal to zero, and the yield surface may then be plotted in σ 1 − σ 2 space as shown in Figure 6. The yield surface is defined by six different expressions: When both σ 1 and σ 2 are greater than zero the limiting conditions are simply 1 2 |σ^1 |^ =^

1 2 σ^1 =^ k^ (line AB)^ (12)

and 1 2 |σ^2 |^ =^

1 2 σ^2 =^ k^ (line BC)^ (13)

Figure 7: von Mises criterion in principal stress space.

The similarities to the Tresca criterion are evident, and when plotted in plane stress, Figure 6, the difference is seen to be rather small. When the principal stress are identical or when one is equal to zero, the two criteria coincide. In contrast to the Tresca criterion, von Mises’ criterion predicts that the value of one principal stress may exceed σ 0 provided that the other principal stress is adjusted accordingly. As with the Tresca criterion, the von Mises criterion is independent of hydrostatic pres- sure. When plotted in three dimensional principal stress space the yield surface depicts a cylinder parallel to the hydrostatic axis σ 1 = σ 2 = σ 3 as shown in Figure 7.

2.2.3 The Drucker-Prager criterion

Whereas hydrostatic pressure independence is a realistic assumption for metals, it fails for other materials such as concrete and soils. Therefore Drucker and Prager formulated a modified von Mises criterion by adding a mean stress term

f (I 1 , J 2 ) =

J 2 + αI 1 − k (21)

or f (σ) = σe + ασm − σ 0 (22)

where the mean stress is given by

σm = I 1 = 13 (σ 1 + σ 2 + σ 3 ) = 13 (σx + σy + σz ) (23)

This criterion is illustrated in Figure 8. Compared to the von Mises criterion, there is a limit for positive (tensile) mean stresses whereas the material is strengthened by super- position of a negative (compressive) mean stress.

2.3 Loading/unloading conditions

As already touched upon earlier the yield condition defines not only the set of permissible stresses, but also the conditions for which plastic deformations can continue to occur.

Figure 8: Drucker-Prager criterion in principal stress space.

Whereas all elastic stress states are located inside the yield surface and defined uniquely by the elastic strain, plastic deformations can occur as long as the stress point is located on the yield surface. Thus, during loading in perfect plasticity, a stress point may remain in one fixed position on the yield surface or slide along it with redistribution of stresses among the different components. Mathematically, the conditions for plastic loading can be written as f (σ + dσ) = f (σ) + ∇f T^ dσ = 0 (24)

where ∇f = [∂f /∂σx, ..., ∂f /∂τzx]T^ (25)

is the normal to the yield surface and dσ is a stress increment, see Figure 9. Since f (σ) = 0

df = ∇f T^ dσ = 0 (26)

In other words, (26) states that during plastic loading the change in stress, if any, occurs tangential to the yield surface. This is the so-called consistency condition which, as shall be shown later, is a key ingredient in the general theory. In unloading the state of stress immediately becomes elastic which can be written as

df = ∇f T^ dσ < 0 (27)

Figure 9: Plastic loading and unloading.

guess at g it would be reasonable to assume that g = f. The plastic strain increment is then determined as

dεp^ = dλ

∂f ∂σ

= dλ∇f (30)

This is the so-called associated flow rule, in contrast to the nonassociated flow rule (29) where the strains are not connected with the yield criterion. Use of the yield criterion to derive the plastic strain increments is also refered to as the normality rule. The use of the associated flow rule is a cornerstone in the so-called mathematical theory of plastic- ity which was formulated around 1950 by among others Hill, Drucker and Prager. This mathematical theory of plasticity contains some very attractive results such as the neces- sity of a convex yield surface and the existence of the limit theorems which have been used extensively in engineering computations. Moreover, the physical interpretation of the results seem to comply so well with the basic thermodynamical requirement of energy conservation that the associated flow rule in some circles has almost the status of a fun- damental law of nature. It can be shown, however, that this is not the case, and although very attractive, associated plasticity often fails to describe the experimentally observed results for other materials than metals.

For the yield criteria discussed previously the plastic strain increments can now be de- termined using the associated flow rule. For the von Mises criterion the plastic strain increments are

dεp^ =

dεpx dεpy dεpz dγpxy dγpyz dγpzx

= dλ

2 σe

2 σx − σy − σz 2 σy − σz − σx 2 σz − σx − σy 6 τxy 6 τyz 6 τzx

where dλ again determines the magnitude of the increment. Given a particular state of strain the relative volume change can be determined as

ΔV V

= (1 + εx)(1 + εy)(1 + εz ) − 1  εx + εy + εz (32)

and thus, the associated von Mises flow rule predicts that no volumetric changes occur as a result of plastic straining,^2 which for metals is in good agreement with what can be observed experimentally. In contrast, soils, concrete and other granular materials do exhibit a volumetric dilatation during plastic flow. This is reflected in the Drucker-Prager criterion where the plastic

(^2) This is not to be confused with the total volumetric change. Usually the elastic contribution εex +εey +εez will be different from zero.

strain increments are given by

dεp^ =

dεpx dεpy dεpz dγxyp dγyzp dγzxp

= dλ

2 σe

2 σx − σy − σz 2 σy − σz − σx 2 σz − σx − σy 6 τxy 6 τyz 6 τzx

1 3 ασx 1 3 ασy 1 3 ασz 0 0 0

Here the relative change in volume is

ΔV V

= dεpx + dεpy + dεpz = dλ^13 α(σx + σy + σz ) (34)

which is not necessarily equal to zero. For soils, however, the volumetric dilatation pre- dicted by the associated Drucker-Prager flow rule is often somewhat larger than can be verified experimentally. Therefore, a nonassociated flow rule can be used, i.e. the elastic limit is still defined by the Drucker-Prager criterion whereas the flow rule is defined by some other function, e.g. f (σ) =

J 2 + αI 1 − k (35) g(σ) =

J 2 + βI 1 − k (36)

where β should be smaller than α. The situation is illustrated in Figure 11. Another

Figure 11: Drucker-Prager plasticity with nonassociated flow rule.

example of nonassociated plasticity can be illustrated by an example from elementary mechanics, namely the sliding on a rigid block on a rigid frictional surface as shown in Figure 12 (a). When the dragging force exceeds the friction force the block begins to slide. That is, at the instant of displacement the forces are related by

f = τ − μσ = 0 (37)

where μ is the coefficient of friction. This problem can be interpreted as a plasticity problem with the possibility of yielding in a thin layer in the interface between the block

where Dep^ is the elasto-plastic constitutive matrix. Such a relation was first derived and used in an finite element context by Zienkiewicz et al. [2]. The elasto-plastic constitutive matrix can be derived by considering the basic relations discussed in the foregoing. The total strain increment is given as the sum of the elastic strain increment and the plastic strain increment dε = dεe^ + dεp^ (39)

Hooke’s law gives the relation between stresses and elastic strains as

σ = Dεe^ = D(ε − εp) (40)

or in rate form as dσ = D(dε − dεp) (41)

where D is the elastic constitutive matrix. The plastic strain increment is determined by the flow rule (29) as

dεp^ = dλ

∂g ∂σ

Thus, the stress increment is given by

dσ = Ddε − dλD

∂g ∂σ

The expression for the stress increment is now substituted into the consistency condition (26) (^) ( ∂f ∂σ

)T (

Ddε − dλD

∂g ∂σ

Solving this equation for the scalar dλ one obtains

dλ =

∂f ∂σ

)T

D dε ( ∂f ∂σ

)T

D

∂g ∂σ

Finally, dλ is substituted back into (43) to yield

dσ = D

dε −

∂f ∂σ

)T

D dε

∂g ∂σ ( ∂f ∂σ

)T

D

∂g ∂σ

This can be rearranged to give the elasto-plastic constitutive relation

dσ =

D −

D

∂g ∂σ

∂f ∂σ

)T

D

∂f ∂σ

)T

D

∂g ∂σ

dε (47)

and thus, the elasto-plastic constitutive matrix introduced in (38) is

Dep^ = D −

D

∂g ∂σ

∂f ∂σ

)T

D

∂f ∂σ

)T

D

∂g ∂σ

The elasto-plastic constitutive relation defines the stress increment uniquely once the total strain increment and the current state of stress is known, whereas the a strain increment cannot be determined uniquely on the the basis of a stress increment, i.e. Dep^ is singu- lar. When used in finite element formulations (48) defines a nonlinear relation between stress and strain increments since the evaluation of the current stress must naturally be influenced by the magnitude of the stress increment. In this way the use of (48) leads to a classical type of finite element nonlinearity where the current state and an increment is known, but where the effect of the increment depends on the state that the increment gives rise to, i.e. an iterative procedure must be applied.

2.6 Hardening

As already discussed most materials exhibit some degree of hardening as an accompani- ment to plastic straining. In general this means that the shape and size of the yield surface changes during plastic loading. This change may be rather arbitrary and extremely diffi- cult to describe accurately. Therefore, hardening is often described by a combination of two specific types of hardening, namely isotropic hardening and kinematic hardening, see Figures 14 (a)-(b). In the following isotropic hardening related to the von Mises criterion is treated in some detail. For the von Mises criterion isotropic hardening implies an increase in the yield strength during plastic loading such that the yield criterion may be written as

f (σ) − σ 0 (α) = 0 (49)

(a) (b)

Figure 14: Isotropic (a) and kinematic (b) hardening.

By using the associated flow rule this equivalent plastic strain can be shown be be equal to the plastic multiplier dεpeq = dλ (58)

For the uniaxial tension test with dεp 2 = dε 3 = −^12 dεp 1 the equivalent plastic strain is then given by dεpeq = dεp 1 = dλ. The relation (56) can now be rewritten as

dσ 0 dεpeq

dσ 0 dλ

= H (59)

Integration of (59) yields

σ 0 =

Hdεpeq (60)

That is, at any given instant the yield strength is determined by the prior plastic strain history. This means that isotropic hardening is irreversible; once the material has experi- enced a certain degree of hardening the yield limit is shifted permanently. As this may not be in accordance with reality the isotropic hardening can be supplemented with kinematic hardening which is, however, a somewhat more demanding model to calibrate. Isotropic hardening is easily included into the elasto-plastic stress-strain relation (47) by considering the appropriate consistency condition. For (49) this is

∂f ∂σ

dσ −

∂σ 0 ∂α

dα = 0 (61)

or ∂f ∂σ

dσ − Hdλ = 0 (62)

This leads to the following elasto-plastic constitutive relation

dσ =

⎜⎝D^ −

D

∂g ∂σ

∂f ∂σ

)T

D

H +

∂f ∂σ

)T

D

∂g ∂σ

⎟⎠ dε^ (63)

Here it is clearly seen that for H tending to infinity the usual linear elastic constitutive matrix D is recovered while H = 0 of course corresponds to the elastic-perfectly plastic case (47). It should be noted that while for perfect plasticity Dep^ is singular, the introduc- tion of hardening results in a unique relationship between infinitesimal stress and strain increments such that this singularity is removed. The inverse of Dep^ can be found by use of the Sherman-Morrison formula which states that

(A − uvT^ )−^1 = A−^1 +

A−^1 uvT^ A−^1 1 − vT^ A−^1 u

where A is an n × n matrix and v and u are vectors of length n. The inverse of Dep^ can then be written as

(Dep)−^1 = D−^1 +

H

∂g ∂σ

∂f ∂σ

)T

which is clearly finite only for values of H different from zero.

2.7 Plane stress versus plane strain

The yield criteria discussed in the above were all formulated with reference to the most general triaxial stress state. Under certain circumstances, however, the number of variables can be reduced. This is for example the case in plane stress, plane strain and under axisymmetric conditions. In the following the two former states are treated. In plane stress only in-plane stresses are considered. This means that

σz = τyz = τzx = 0 (66)

The corresponding plane stress yield criterion is then obtained by simply deleting the above stress components from the general triaxial yield criterion. In this way von Mises’ yield criterion can be written as

f =

σ x^2 + σ y^2 − σxσy + 3τ (^) xy^2 − σ 0 (67)

Here it should be noted, however, that when computing the plastic strain increments reference must again be made to the general triaxial criterion, i.e. dεpz, dγyzp, dγzxp are not necessarily equal to zero, and the same is of course the case with respect to the elastic strains. In plane strain we have εz = γyz = γzx = 0 (68)

If this is to be valid for all strains it must hold that

dεez = dγeyz = dγzxe = 0 (69)

and dεpz = dγpyz = dγzxp = 0 (70)

In associated plasticity the relation between the plastic strain increments and the yield function now makes it possible to determine a two-dimensional yield criterion as in the case of plane stress. Again von Mises’ criterion is considered, and according to (31) the plastic strain increments are here given by

dεpz = dλ

2 σe

(2σz − σx − σy )

dγyzp = dλ

2 σe

6 τyz

dγzxp = dλ

2 σe

6 τzx

Whereas it follows directly from the plastic shear strain increments that the corresponding shear stresses should be deleted from the yield criterion, the expression for dεpz gives a condition for the normal stresses as

σz = 12 (σx + σy ) (72)