Stress triaxiality and Lode angle along surfaces of elastoplastic structures

International Journal of Solids and Structures 67–68 (2015) 71–83

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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

Stress triaxiality and Lode angle along surfaces of elastoplastic structures Adrien Darlet a,b,⇑, Rodrigue Desmorat a a b

LMT (ENS Cachan, CNRS, Université Paris Saclay), 94235 Cachan, France Snecma Villaroche, Rond-point René Ravaud – Réau, 77550 Moissy-Cramayel, France

a r t i c l e

i n f o

Article history: Received 19 July 2013 Received in revised form 3 March 2015 Available online 25 March 2015 Keywords: Stress triaxiality Lode angle Free surfaces Stress concentration Neuber

a b s t r a c t Expressions for the stress triaxiality and the Lode angle along surfaces of elastoplastic structures are established in case of monotonic loading. The stress triaxiality is shown to be governed by the accumulated plastic strain when traction free boundary condition is considered. The exact expressions obtained are generalized to any loading thanks to the proposal of a multiaxiality rule or heuristics whose two parameters are determined from elastic computations of the structure considered: a first one with the elastic properties of the material, a second one quasi-incompressible. The multiaxiality rule proposed can then deal with both plane strain and plane stress conditions. The stress triaxiality at the surface is shown related to the Lode angle. The corresponding expressions are validated on different structures and loadings. Finally, two applications are presented: the enhancement of energetic methods for plasticity post-processing and the enhancement of homogenization localization laws. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The stress triaxiality defined as the ratio of mean stress or hydrostatic stress divided by von Mises equivalent stress and the Lode angle are a matter of interest in many mechanical fields as soon as the studied phenomena are influenced by the stress state. Ductile damage theories introduced by the early works of McClintock (1968), Rice and Tracey (1969) and then Gurson (1977) exhibit a void growth rate governed by the plastic strain rate but exponentially enhanced by the stress triaxiality (refer to Pineau and Pardoen (2007) for a review). Stress triaxiality is one of the main sensitive quantity for continuous damage and ductile failure (Lemaitre, 1971; Hayhurst and Leckie, 1973; Hult and Broberg, 1974; Hancock and Mackenzie, 1976; Murakami and Ohno, 1978; Beremin, 1981; Krajcinovic and Fonseka, 1981; Lemaitre and Chaboche, 1985; Johnson and Cook, 1985; Rousselier, 1987; Becker et al., 1988; Bernauer et al., 1999; Berdin et al., 2004; Lemaitre and Desmorat, 2005; François et al., 2012; Lemaitre et al., 2009). In recent works (Bao and Wierzbicki, 2004; Xue and Wierzbicki, 2008; Bai and Wierzbicki, 2008), the Lode angle is shown to play a major role on the fracture locus but also on void growth (Nahshon and Hutchinson, 2008) at low stress triaxiality. The stress triaxiality is a matter of interest in surface integrity as compressive residual stresses are sought to improve the fatigue life (Field and Kahles, 1964; Field and Kahles, 1971; Jawahir et al., ⇑ Corresponding author at: LMT (ENS Cachan, CNRS, Université Paris Saclay), 94235 Cachan, France. Tel.: +33 147 407 760; fax: +33 147 402 240. E-mail address: [email protected] (A. Darlet). http://dx.doi.org/10.1016/j.ijsolstr.2015.03.006 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.

2011). It is also a matter of interest in the study of diffusive phenomena in stressed solids, for instance in hydrogen embrittlement (Simpson, 1981; Huez et al., 1998). The solid diffusion is related to the atoms spacing and is obviously made easier in equi-biaxial tension than in uniaxial tension or in compression. Most of the problems involving these phenomena occur at surfaces and usually require elastoplastic computations when yielding occurs. It has been shown in a previous work (Desmorat, 2002) that the stress triaxiality at surfaces of structures subjected to monotonic loading is related to the accumulated plastic strain in plane strain condition. There is no systematic studies for more general multiaxial states, even at surfaces. Desmorat (2002) work can be extended to a wider range of stress state using different multiaxial constraints, i.e. different assumptions for the multiaxiality of the state of stresses or strains – or mixed quantities – at the surfaces. Such assumptions have mainly been developed in the attempts to extend the fast energetic methods, such as Neuber (1961) and Molski and Glinka (1981) methods, to 3D structural cases (Walker, 1977; Chaudonneret and Culie, 1985; Hoffmann and Seeger, 1985). One of them, giving good results for axisymmetric notched structures, is Hoffmann and Seeger (1985) assumption, that considers a constant strain ratio at the stress concentration point during loading, strain ratio determined from an elastic computation. However, this assumption does not apply to uniaxial stress states. None of the literature assumptions automatically deals with both plane stress and plane strain conditions. The aim here is to characterize surfaces stress/strain multiaxiality through the values of the stress triaxiality and of the lode

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angle. Lode angle and stress triaxiality are shown to be related on a free surface of an elasto-plastic structure, they are functions of the accumulated plastic strain in monotonic cases. First part of present work concerns the proposal of a novel assumption for the strain multiaxiality at surfaces, called next multiaxiality rule. Expressions for the stress triaxiality established by Desmorat (2002) are then extended to any stress/strain state along surfaces through the consideration of the proposed heuristics. These expressions are assessed on structural examples. Finally, two applications are presented. The first one is the enhancement of energetic methods for plasticity post-processing, the second one the enhancement of homogenization localization laws. 2. Existing stress/strain multiaxiality assumptions Three dimensional general stress states are usually difficult to handle in closed-form expressions as stress and strain components may evolve independently. This evolution is constrained by the geometry and/or the loading. For plane stress and for plane strain conditions relations in terms of stress or strain components exist and can be used to determine analytically the stress triaxiality (Walker, 1977; Desmorat, 2002). However, in the general case, the multiaxial constraint must be either numerically determined, or – as for fast energetic methods – it is hidden in the general set of equations (Neuber, 1961; Molski and Glinka, 1981; Hoffmann and Seeger, 1985; Moftakhar et al., 1995; Gallerneau, 2000; Chaboche, 2007; Herbland et al., 2007).

The stress triaxiality denoted T X in present work and the Lode angle H are both dimensionless invariants defined respectively with previous stress invariants or components as

TX ¼

rH 1 trr r2 þ r3 ¼ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi req 3 req 3 r2 þ r2  r2 r3 2

 cosð3HÞ ¼

r

3

req

¼

ð5Þ

3

ðr2  2r3 Þð2r2  r3 Þðr2 þ r3 Þ  3=2 2 r22 þ r23  r2 r3

ð6Þ

With these conventions, a stress triaxiality equal to 1=3 corresponds to uniaxial tension and 1=3 to uniaxial compression. Pure shear stress state presents a stress triaxiality equal to zero, in equibiaxial tension the triaxiality remains equal to 2=3 even after yielding. However, the stress triaxiality is in general not constant, even at surfaces, and evolves with respect to the loading. To describe constraint assumptions at surfaces, it is convenient to define the following stress and strain ratios in terms of the principal stress and strain components at surfaces:

k3 ¼

r3 1 3 and /1 ¼ ; /3 ¼ r2 2 2

ð7Þ

These ratios are defined with components obtained from an elastoplastic analysis (lowercase Greek letters). It is also possible to define similar ratios (still at the surface) but from the principal stresses and strains obtained from a linear elastic analysis of the same structure:

R3 R2

E1 ; E2

E3 E2

2.1. Stress state at surfaces

K3 ¼

Obviously, the stress state occurring along surfaces is either uniaxial or biaxial even if the rest of the structure presents a complex tridimensional stress state. In the principal coordinate system (Fig. A.1), the direction associated to principal stress r1 is considered normal to the free surface (free surface condition: r1 ¼ 0) and the convention for the principal stresses r2 P r3 is adopted. The free surface condition gives then r ¼ diag½0; r2 ; r3  so that the three stress invariants are defined respectively as the hydrostatic stress,

Capital notation means then, ‘‘components obtained from a linear elastic analysis’’, R and E being the stress and strain tensors obtained in elasticity.

1 3

1 3

rH ¼ trðrÞ ¼ ðr2 þ r3 Þ

ð1Þ

as von Mises equivalent stress,

req ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 r : r0 ¼ r22 þ r23  r2 r3 2

ð2Þ

and as the third invariant of the deviatoric stress, here made homogeneous to a stress,

r¼

 1=3  1=3 27 1 ¼ ðr2  2r3 Þð2r2  r3 Þðr2 þ r3 Þ detðr0 Þ 2 2

1 3

r0 ¼ r  tr r 1

ð4Þ

r3 R3  ¼ K3 r2 R2

ð9Þ

was first introduced by Walker (1977) who assumed a constant stress ratio in combination with Neuber’s rule to determinate the stress and the strain state at notch tip of different geometries. It corresponds to a constant stress triaxiality, i.e. a stress triaxiality identical in elasticity and in plasticity,

1 þ k3 1 þ K3 T X ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ T elasticity X 2 3 1  k3 þ ðk3 Þ 3 1  K3 þ ð K3 Þ 2

Fig. A.1. Notations for the coordinate system associated to the principal directions of the stress and strain tensors. Direction 1 is normal to the free surface.

ð8Þ

A wide range of assumptions has been proposed in the context of the extension of fast energetic methods to multiaxial stress state at stress concentration points such as notches (Walker, 1977; Chaudonneret and Culie, 1985; Hoffmann and Seeger, 1985; Moftakhar et al., 1995; Singh et al., 1996; Gallerneau, 2000; Knop et al., 2000; Buczynski and Glinka, 2003; Sethuraman and Viswanadha Gupta, 2004; Lim et al., 2005; Chaboche, 2007; Herbland et al., 2007; Ye et al., 2008). They equal elastic and elastoplastic stress, strain, or mixed ratios, i.e. quantities calculated in elasticity to the same quantities in elasto-plasticity. The assumption

k3 ¼

where r is deviatoric stress tensor,

U3 ¼

2.2. Existing multiaxiality assumptions linking elastic and elastoplastic quantities

ð3Þ

0

and U1 ¼

ð10Þ

As already mentioned, this holds for some special stress states only, as for shear, uniaxial tension and equi-biaxial plane stress tension/compression. However, this does not hold for plane strain conditions and many other intermediate states, even for small plastic strains.

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Hoffmann and Seeger (1985) have considered a constant strain ratio /3 to handle a wider range of problem including axisymmetric structures,

/3 ¼

3 E3  ¼ U3 2 E2

ð11Þ

This formulation can deal with many biaxial states but has been shown too rigid when reduced to a uniaxial stress state. Indeed setting e and p for elastic and plastic strains, the strain relation in uniaxial tension in direction 2 is given by 3 ¼ me2  12 p2 and it cannot be enforced from Eq. (11). Moftakhar et al. (1995) proposed the mixed assumption,

r2 2 R2 E2 ¼ r2 2 þ r3 3 R2 E2 þ R3 E3

ð12Þ

It was used with success for many geometry and type of loading (Singh et al., 1996; Ye et al., 2008). However and once again it does not discriminate plane stress and plane strain conditions. Note that other assumptions exist, such as (Gallerneau, 2000),

E

R

r2 ¼ br3 þ p3 b ¼ 2 m R3

ð13Þ

their projections on the principal basis of the total strain tensor are used in the heuristics. The difficulty in the determination of parameters a and b is that they likely evolve during complex loading. A first elastic calculation (p ¼ 0) with the real elastic parameters for the considered material leads to the determination of a from the stresses on the edge

a¼

R3  mR2 mR3  R2

ð21Þ

The determination of b is the key point of present work. It is possible to take b ¼ a, recovering then Hoffmann and Seeger’s assumption, but accepting then not to deal with uniaxial stress states. One prefers to consider that b reaches a value determined in generalized plasticity. In this limiting case, elasticity is neglected e  p . The multiaxiality rule becomes then:

_ p3  b_ p2

ð22Þ r0

The use of the normality rule _ p ¼ p_ 32 req (with p_ the accumulated plastic strain rate) altogether with the multiaxiality rule lead to the following expression of b:

r3  12 r2 r  r2 2 3

or, with GðpÞ a function of the accumulated plastic strain p (Chaboche, 2007),

b¼1

r 3 R3 ¼ GðpÞ r 2 R2

with ri the principal stresses obtained in generalized plasticity (lower case Greek letters). This expression is similar to the expression for a (Eq. (21)) when taking m ¼ 1=2. Therefore, one proposes to determine b from an incompressible elastic computation, setting

ð14Þ

or (Herbland, 2009),

r3 3 ¼ ð1 þ K 3 pÞR3 E3

ð15Þ

with either GðpÞ a function and K 3 a constant to be determined from plasticity finite element computations. Herbland (2009) has shown that the ‘‘constant’’ K 3 and also the function G in fact evolve with plasticity when the extension to cyclic loading is under consideraR _ where tion. The accumulated plastic strain p is defined as p ¼ pdt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p_ ¼ 2=3_ p : _ p . In most engineering problems, plane stress and plane strain conditions represent two usual design cases and often two bounding conditions that occur at notch roots. One proposes next a multiaxiality assumption that can handle both conditions in the same framework. 2.3. Multiaxiality rule Let us have then a close look at the strain state for classical loading cases encountered along surfaces. The idea is to propose an assumption, which will next be called multiaxiality rule, valid for both plane strain and plane stress conditions. Let us consider classical metal plasticity framework so that one has the strain partition  ¼ e þ p with  the total strain, e the elastic strain and p the plastic strain. Usual strain states along surfaces are cast as:

3 ¼ e2  p2

ðpure shearÞ

1 ðuniaxial tensionÞ 2 ¼ 0 ðplane strain tensionÞ

ð16Þ

3 ¼ me2  p2

ð17Þ

3 3 ¼ e2 þ p2

ð19Þ

ðequi-biaxial tensionÞ

ð18Þ

b¼

R3m¼1=2  12 R2m¼1=2 1 m¼1=2 R  R2m¼1=2 2 3

ð23Þ

ð24Þ

In practice the following calculations are performed with a quasiincompressible elastic behavior with m ¼ 0:499. To conclude, one proposes _ 3 ¼ a_ e2  b_ p2 as a novel assumption, called multiaxiality rule valid at surfaces (1 is the normal to the edge). For structures, the heuristics parameters a and b are determined from two elastic computations, one for a with the real Poisson’s ratio and one for b with m  0:5. It is important to notice that plane strain naturally corresponds to a ¼ b ¼ 0 and that uniaxial tension is naturally gained as a ¼ m and b ¼ 1=2 (from Eq. (21) and (24)). Both plane strain and plane stress conditions are handled with this single assumption. 3. Expressions for the stress triaxiality and Lode angle along surfaces One shows here that the multiaxiality rule (20) can be used to determine the stress triaxiality at surfaces by time integration of the plasticity constitutive equations. This generalizes the closedform expressions established by Desmorat (2002) for plane strain conditions. Indeed, plane strain is the particular case a ¼ b ¼ 0. One considers any stress state occurring along surfaces, from bicompression (triaxiality T X ¼ 2=3) to equibitension (triaxiality T X ¼ 2=3). Notations and conventions used in this section are those of Section 2. Monotonic loadings are considered.

where one recalls that 1 is the direction normal to the free surface. Therefore one proposes to generalize previous expressions as the so-called multiaxiality rule (or heuristics):

3.1. Relation between the stress triaxiality and Lode angle along surfaces

_ 3 ¼ a_ e2  b_ p2

In order to simplify the problem, a dummy variable u is introduced:

ð20Þ

where a and b are two heuristics parameters that one proposes to determine from elastic computations. Elastic and plastic strain tensors are implicitly assumed to be diagonal in the same basis. If not,

u¼

r2 req

ð25Þ

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No assumption about the loading proportionality is made as u may evolve during yielding. The free surface condition ensures r1 ¼ 0 (with 1 the direction normal to the free surface) and the von Mises equivalent stress is given by Eq. (2). Two expressions for r3 are obtained thanks to the roots of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ r2 =req ¼ r2 = r22 þ r23  r2 r3 and thus two expressions for the stress triaxiality and Lode angle H are obtained with respect to variable u:

8 > > > r2 ¼ ureq <

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

and r3 ¼ 12 u  e 4  3u2 req pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T X ¼ 12 u  16 e 4  3u2 > > ffi > : cos 3H ¼  1 e1  3u2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4  3u2 2 with the condition on

ð26Þ

e ¼ 1:

8 > < e ¼ 1 if

h h h h u 2 1; p2ffiffi3 and T X 2  23 ; p1ffiffi3 h i h i > : e ¼ 1 if u 2 1; p2ffiffi and T X 2 p1ffiffi ; 23 3 3

ð27Þ

The corresponding expressions are presented in the graph of Fig. A.2, in the triaxiality T X versus variable u plane. These results show that the stress triaxiality can be expressed with respect to the dummy variable u only. The definition domain h pffiffiffi pffiffiffii for the variable u is 2= 3; 2= 3 . The stress triaxiality is classically defined between the bounding values 2=3 and 2=3 on the free surface which match respectively with equi-biaxial compression and equi-biaxial tension stress states. An interesting remark comes from expression (26): the stress triaxiality and the Lode angle are bi-univoquely related along surfaces. This is consistent with the free surface condition written in Haigh–Westergaard representation. The principal stresses and thus the stress triaxiality are easily determined from Lode angle H as (Nayak and Zienkiewicz, 1972)

2 3

r1 ¼ req cos H1 þ rH ¼ 0; H1 ¼ H þ

2kp 3

k 2 f0; 1; 2g

ð28Þ

or

3.2. Closed-form expressions for stress triaxiality from the multiaxiality rule 3.2.1. Elasticity In elasticity (stress R, strain E), the multiaxiality rule writes:

E3 ¼ aE2

ð30Þ

with then

R3 ¼

ma R2 1  am

ð31Þ

Thus, the stress triaxiality only depends on the parameter a and Poisson’s ratio m as ma 1 þ 1a m T X ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ma þ ma 2 3 1  1a 1am m

This expression is valid for any geometry and any loading, but at surfaces. For proportional loading the parameter a is constant as the ratios of the stress components are fixed whereas for non proportional loadings the parameter a evolves during loading. 3.2.2. Plasticity with linear isotropic hardening For plasticity with linear isotropic hardening, the yield function (Lemaitre et al., 2009)

f ¼ req  RðpÞ  ry 6 0

ð29Þ

i.e. Lode angle H is related to stress triaxiality T X ¼ rH =req at surfaces.

ð33Þ

is considered with R ¼ Kp in case of linear hardening (K is then the plastic modulus). The evolution laws governing the accumulated plastic strain p and principal plastic strains pi are derived from the normality rule and from the consistency condition f ¼ 0 and f_ ¼ 0 during plastic flow. The boundary conditions (free surfaces of normal 1, r1 ¼ 0), the multiaxiality rule (Eq. (20)) and the elasto-plastic behavior considered altogether lead to:

_ 3 ¼ _ e3 þ _ p3 ¼ _ e2 ¼

r_ 2  mr_ 3 E

  1 1 p_ ðr_ 3  mr_ 2 Þ þ r3  r2 E 2 req   _ 1 p and _ p2 ¼ r2  r3 2 req

ð34Þ ð35Þ

Therefore one obtains the differential equation:

  1  am dr3 m  a dr2 dp ð2  bÞr3 þ ð2b  1Þr2 þ  1  am dt dt E dt 2req ¼0

3 cos H1 ¼  T X 2

ð32Þ

ð36Þ

To solve this equation, the dummy variable u ¼ r2 =req is used. The consistency condition (req ¼ RðpÞ þ ry and dreq =du ¼ R0 ðpÞdp=du) and the u-substitution applied in the differential Eq. (36) leads to the following equation:

dp F 1 ðuÞ½RðpÞ þ ry  ¼ 0 du R ðpÞF 3 ðuÞ þ F 2 ðuÞ

ð37Þ

with

ma 3u þ e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  am 4  3u2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii E 3bu þ ðb  2Þe 4  3u2 F 2 ðuÞ ¼ 2ð1  amÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ma

u  e 4  3u2 F 3 ðuÞ ¼ 1  2 1  am

F 1 ðuÞ ¼ 1  2

ð38Þ ð39Þ ð40Þ

This first-order differential equations is a separable variables equation if the isotropic hardening is linear (req ¼ Kp þ ry and dR=dp ¼ K). It can be integrated at a and b constant,  Z p Z u ma þ pffiffiffiffiffiffiffiffiffiffi 3u e 1  2 1a dp m 43u2

¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   du 2b b p0 Kp þ ry u0 K þ 2ð1a 4  3u2  e K ð1  2mH Þ þ 32 1a mÞ E mE u Fig. A.2. Expressions for the stress r3 and the stress triaxiality T X in the T X vs u plane. Classical triaxiality values are reported (black stars).

ð41Þ

A. Darlet, R. Desmorat / International Journal of Solids and Structures 67–68 (2015) 71–83

75

The initial state is defined at the onset of yielding:

1 p ¼ p0 ¼ 0 and u ¼ u0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ma 2 m a 1  1am þ 1a m

ð42Þ

Eq. (41) leads to the following parametric representation of the stress triaxiality as a function of the accumulated plastic strain  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (also of H ¼ HðpÞ using cos 3H ¼  2e 1  3u2 4  3u2 ):

8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e 2 > < T X ¼ 2 u  6 4  3u pffiffiffiffiffiffiffiffiffiffi  h pffiffi iu  2  r p43u r 3 > ffiffiffiffiffiffiffiffiffiffi022evu0  Ky exp e w arcsin u : p ¼ p0 þ Ky 2 43u 2evu u0

ð43Þ where the constants v; w and - only depend on the material parameters and on the heuristics parameters a and b:



maÞ 3 b K 1  2ð1a m þ 2 1am E

v¼ 2b 2 K þ 2ð1a mÞ E 1 pffiffiffi 0 2ðmaÞ 2 3K @ 1  1am v h ih iA w¼ 2b 3 þ 4v2 2 K þ 2b E K þ 2ð1a 2ð1amÞ mÞ E 0 2ðmaÞ K 4v2 K @ 1  1am iþ h ih -¼h 2 3 þ 4v 2 K þ 2b E K þ 2b E Kþ 2ð1amÞ

2ð1amÞ

ð44Þ

ð45Þ

v

1 iA

2b E 2ð1amÞ

ð46Þ To conclude those derivations in the monotonic loading case, several remarks may be formulated:  The stress triaxiality and the Lode angle are related along surfaces and then only depends on the accumulated plastic strain, the constitutive law parameters and the heuristics parameters a and b. Indirectly, they depend on the geometry and the loading through a and b which are determined from two elastic computations of the structure studied (the second one at Poisson’s ratio m  0:5).  The evolution of the stress triaxiality with respect to the accumulated plastic strain is monotonically increasing with yielding and reaches a saturation value given by the following expression:

3  2v T sat X ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 þ 4v 2

ð47Þ

 A special care has to be taken to describe continuously the evolution of the stress triaxiality T X if T X 0 < p1ffiffi3 < T sat X . Indeed, the variable e changes from a positive to a negative value.  For complex loading, a numerical time integration has to be performed considering that both parameters a and b are constant over a time increment. When isotropic hardening is non linear, the first-order differential equation (41) is not with separable variables anymore. In the numerical examples given next, the numerical integration is performed using Runge–Kutta scheme. 4. Validation example of a plate under plane stress condition and plane strain condition In order to study the validity of the expressions for the stress triaxiality and Lode angle presented in previous section, the classical problem of a plate with a circular hole is first discussed. Plane stress and plane strain conditions are considered. The plate geometry is presented in Fig. A.3 (dimensions L and R). Numerical

Fig. A.3. Geometry and loading of a plate with a circular hole under plane stress condition and plane strain condition.

simulations are carried out with the finite element code ABAQUS. The mesh of the plate consists of 2206 plane stress or plane strain biquadratic elements with reduced integration (CPS8R or CPE8R). A two dimensional model is considered. The plate is subjected to the uniaxial tension load F. One focuses on what happens at the notch root where stress concentration is maximum (marked as point of interest in the figure). One considers von Mises plasticity with  the total strain, e the elastic strain and p the plastic strain. The material behavior is elastoplastic here with linear isotropic hardening so that the plastic flow is given by:

_ ¼ p_ _ p ¼ pn

3 r0 2 req

ð48Þ

Material parameters are those of Moftakhar et al. (1995) who consider a linear isotropic hardening RðpÞ ¼ Kp with a Young’s modulus E ¼ 94400 MPa, a Poisson’s ration m ¼ 0:3, a yield stress ry ¼ 550 MPa, and a plastic modulus K ¼ 4720 MPa. Elastic and elasto-plastic calculations are performed using the finite element code ABAQUS. The stress triaxiality determined from a reference elastoplastic analysis is compared with the one obtained from present work parametric expressions (41) using one of the three assumptions:  Walker’s assumption (1977): constant stress triaxiality,  Hoffmann and Seeger’s assumption (1985): constant strain ratio /3 ¼ U3 (Eq. (11)) or b ¼ a from Eq. (21),  the proposed multiaxiality rule (Eq. (20)): a determined from an elastic computation (Eq. (21)) and b from a quasi-incompressible elastic computation (Eq. (24)). In order to perform an objective comparison, the accumulated plastic strain from the elastoplastic computation is considered as an input in the parametric expressions for the stress triaxiality T X ðpÞ. Walker’s assumption is naturally in perfect agreement with the elastoplastic solution in the plane stress condition but it cannot describe the stress triaxiality and the Lode angle evolution in plane strain condition (Fig. A.4). On the other hand, the Hoffmann and Seeger’s assumption matches perfectly with the elastoplastic solution in plane strain but its known rigidity in uniaxial tension is exhibited as it cannot handle the plane stress case (Fig. A.4). Fig. A.4 also shows that the proposed multiaxiality rule altogether with the constitutive equations (and their exact time integration on a free surface) handles both plane stress and plane strain conditions. 5. More complex structural examples In order to assess the accuracy of the different multiaxiality assumptions (Walker’s of constant stress triaxiality, Hoffman and

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Fig. A.4. Comparison of stress triaxiality expressions T X ðpÞ for a plate with a circular hole under plane stress condition and plane strain condition. Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

Seeger’s on strain ratios) and of the proposed multiaxiality rule in more complex structural examples, two structures subjected to different type of loading are considered.

5.1. Round bar with a circumferential notch under proportional and non proportional tension and torsion loading The first specimen is a round bar with a circumferential notch subjected to tension–torsion loadings (Moftakhar et al., 1995). The geometry of the bar and the mesh is presented in Fig. A.5. Numerical simulations are carried out with the finite element code ABAQUS.

The mesh consists of 11,042 quadrilateral axisymmetric stress elements with reduced integration (CGAX8R). These elements have an additional degree of freedom corresponding to the twist angle. A two dimensional axially symmetric model is considered. The loading applied on the structure is a combination of a tensile load F and a torque T. Different combinations are considered

to cover a wide range of stress triaxialities at the root of the notch. Indeed, the stress state for tension load T ¼ 0 locally almost matches with plane strain condition (T X  0:5 at notch root) and the stress state for torsion only (F ¼ 0) leads to local stress triaxiality equals to 0. The nominal tensile stress rnF and shear stress sn are determined using the net cross-sectional area,

rnF ¼

F

pðR  tÞ2

and

sn ¼

2T

pðR  tÞ3

ð49Þ

The material of the tube is SAE 1045 steel which is represented with an elastoplastic behavior with a non linear isotropic hardening, n

RðpÞ ¼ ry þ kp

ð50Þ

with parameters E ¼ 202 GPa, m ¼ 0:3; ry ¼ 202 MPa, n ¼ 0:208 and k ¼ 1258 MPa. The specimen is subjected to proportional and non proportional loading paths. During the proportional loading paths, the ratio of the nominal stresses rnF =sn remains constant. Two proportional loading paths are studied: ‘‘loading 1’’ with rnF =sn ¼ 0:5 and ‘‘loading 2’’ with rnF =sn ¼ 2 (see Table 1). As the torque is larger for loading 1 path the stress triaxiality is lower than for loading 2 path. Two non proportional loading paths are also studied denoted ‘‘tension–torsion’’ and ‘‘torsion–tension’’. During the ‘‘tension–torsion’’ loading, a uniaxial tensile load is applied and maintained. f =snf ¼ 0:84 (Fig. A.9). Torque is then applied until reaching rnF Alternatively, during the ‘‘torsion–tension’’ load, torque is applied and maintained. A uniaxial tensile load is then applied until reach-

Fig. A.5. Geometry and dimensions of the cylindrical specimen subjected to axial tension and torque studied by Moftakhar et al. (1995).

f =snf ¼ 1:92 (Fig. A.9). ing rnF The stress triaxiality as well as the Lode angle obtained from the different assumptions are compared to the finite element reference

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5.2. Hollow tube with a circumferential notch subjected to internal pressure and axial tensile load

Table 1 Combinations of tension–torsion loading applied to the round bar. Loading 1 Loading 2 Tension

rnF =sn ¼ 0:5 rnF =sn ¼ 2 sn ¼ 0

a ¼ 0:554 a ¼ 0:151 a ¼ 0:035

b ¼ 0:643 b ¼ 0:247 b ¼ 0:102

solution (round marks) in Figs. A.6, A.7, A.8. Hoffmann and Seeger’s assumption (black line) is accurate in small scale yielding but not anymore in generalized plasticity as it is seen in Fig. A.8. On the contrary, the multiaxiality rule is accurate in both small scale and large scale yielding (Fig. A.8), but a little less in the intermediary regime. Let us recall that a is determined at the onset of yielding and b in elasticity with a Poisson’s ratio m  1=2 representative of generalized plasticity. The error induced by the multiaxiality rule is lower than 10% in small scale yielding. For non proportional loading, the estimations obtained are quite in agreement with the elastoplastic reference computation. Indeed, the relative errors at the end of the loading using either Walker’s (dot line) and Hoffman or Seeger’s assumptions, or the multiaxiality rule are respectively:  17.0% for Walker’s assumption, 3.0% for Hoffman or Seeger’s assumption and 10.8% for the multiaxiality rule in the torsion–tension non proportional loading case (Fig. A.10),  4.7%, 12.7% and 0.3% in the tension–torsion loading non proportional loading case (Fig. A.10). Hoffmann and Seeger’s assumption and the multiaxiality rule give alternatively the best results for this axisymmetric geometry.

The second specimen is picked from Moftakhar et al. (1995) who studied a tube with a circumferential notch (geometry, dimensions and the mesh are given in Fig. A.11). Simulations were carried out with the finite element code ABAQUS. The mesh of the hollow tube consists of 4808 axisymmetric quadrilateral stress elements with reduced integration (CAX8R). Only a two dimensional axially symmetric model was considered. The tube is subjected to internal pressure p and axial tension F. The loading path is therefore defined by the combination of an axial nominal stress rnF and a hoop nominal stress rnp ,

F pRi

and rnp ¼ rnF ¼ R0  t  Ri p ðR0  tÞ2  R2i

ð51Þ

with R0 the outside radius of the pipe, Ri the internal radius of the pipe and t the wall thickness (see Fig. A.11). Proportional and non proportional loading paths are applied to the structure. The ratio of both nominal stresses remains constant during the proportional loading, i.e. rnF =rnp ¼ 0:79. For the non proportional loading, the ratio evolves following the path presented in the Fig. A.13 until reaching the final value rnF =rnp ¼ 0:79. The material of the round bar is still SAE 1045 steel. The material model and its parameters are those of Section 5.1 with non linear isotropic hardening. Elastic and elastoplastic calculations are performed using the finite element code ABAQUS. The stress state in the structure is multiaxial. At the notch root, the stress triaxiality is around 0.6 for the proportional loading

Fig. A.6. Comparison of stress triaxiality and Lode angle expressions (T X ðpÞ and HðpÞ) for the round bar with a circumferential notch subjected to proportional tension load. Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

Fig. A.7. Comparison of stress triaxiality and Lode angle expressions (T X ðpÞ and HðpÞ) for the round bar with a circumferential notch subjected to proportional tension– torsion load (rnF =sn ¼ 0:5). Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

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Fig. A.8. Comparison of stress triaxiality and Lode angle expressions (T X ðpÞ and HðpÞ) for the round bar with a circumferential notch subjected to proportional tension– torsion load (rnF =sn ¼ 2). Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

Fig. A.9. Non proportional loading paths for the round bar with a circumferential notch.

which is almost equivalent to the equi-biaxial tension state for which T X ¼ 2=3. For the proportional loading, the stress triaxiality at the root notch does not evolve much (T X  0:6). Both literature assumptions and the proposed multiaxiality rule all give an accurate estimation of the stress triaxiality and the Lode angle, Walker’s assumption (dotted line, constant stress triaxiality) being a little less accurate. The error at the end of the loading is less than 5% on the stress triaxiality (Fig. A.12). Fig. A.14 give the results for free surface stress triaxiality and Lode angle as function of accumulated plastic strain (T X ðpÞ and HðpÞ curves), still for the hollow tube but with the non

proportional loading of Fig. A.13. The Hoffman–Seeger’s assumption and the multiaxiality rule give almost the same results, in agreement with the elastoplastic reference computation, with an error is of about 3%. Walker’s assumption leads to larger errors, larger than 10%.

6. Application to the triaxiality enhancement of fast energetic methods Fast energetic methods were developed to estimate localized plasticity from an elastic computation, especially for stress

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79

Fig. A.10. Comparison of stress triaxiality and Lode angle expressions (T X ðpÞ and HðpÞ) for the round bar with a circumferential notch subjected to non proportional tension– torsion load. Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

Fig. A.11. Geometry and dimensions of the tube with a circumferential notch under internal pressure and axial tension load studied by Moftakhar et al. (1995).

Fig. A.13. Non proportional loading paths for the hollow tube with a circumferential notch.

Fig. A.12. Comparison of stress triaxiality and Lode angle expressions (T X ðpÞ and HðpÞ) for the hollow tube with a circumferential notch under internal pressure subjected to proportional load. Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

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Fig. A.14. Comparison of stress triaxiality and Lode angle expressions (T X ðpÞ and HðpÞ) for the hollow tube with a circumferential notch under internal pressure subjected to non proportional load. Stress triaxiality from: an elastoplastic analysis ( ), the multiaxiality rule ( ), Walker’s assumption ( ), Hoffmann and Seeger’s assumption ( ).

concentrations in notched bodies. Initially, Neuber (1961) established the well-known rule relating the elastic stress concentration factor to the elastoplastic stress and strain concentration factor for prismatic bodies subjected to shear. Then, Molski and Glinka (1981) proposed to use local equality between the Strain Energy R Density (SED) r : d obtained either from elastic or from elastoplastic calculations. Each rule is a scalar equation which allows in uniaxial for a closed form approximation of the scalar stress and plastic strain at notch concentration, simply from a specific time integration of elasto-plasticity constitutive equations. Such fast energetic methods are considered next, at surfaces, in multiaxial. Note that some authors (Moftakhar et al., 1995) propose to use Neuber’s and Glinka’s methods as bounds for the estimation of plasticity in small scale yielding.

Neuber rule is generalized in 3D as r :  ¼ ðr : Þelas (Lemaitre and Chaboche, 1985) where term ðr : Þelas is given from an elastic calculation. It can be rewritten

ðr : Þelas ¼

R2eq Rm ðT elas X Þ 2E

Rm ðT X Þ ¼

2 ð1 þ mÞ þ 3ð1  2mÞT 2X 3

ð52Þ

with R the stress tensor determined in elasticity, Rm the triaxiality ¼ trR=3Req the stress triaxiality in elasticity. function and T elas X Using constitutive equations in plasticity with isotropic hardening R ¼ RðpÞ ¼ req  ry gives at the stress concentration point

F ¼ r :   ðr : Þelas ¼

r2eq Rm ðT X Þ E

þ req gðreq Þ  ðr : Þelas ¼ 0 ð53Þ

Fig. A.15. Comparison with Finite Elements analysis (FEA) for the round bar with a circumferential notch subjected to non proportional tension–torsion load ( FEA, enhancement with proposed multiaxiality heuristics/rule, enhancement with Hoffmann and Seeger’s assumption, enhancement with Walker’s assumption).

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Fig. A.16. Comparison Finite Elements analysis (FEA) for the hollow tube subjected to non proportional loading path, (a) stress components, (b) strain components ( FEA, enhancement with proposed multiaxiality heuristics/rule, enhancement with Hoffmann and Seeger’s assumption, enhancement with Walker’s assumption).

e.g. a scalar equation for two unknowns in plasticity, von Mises stress req and stress triaxiality T X ¼ trr=3req . The accumulated plastic strain is p ¼ gðreq Þ ¼ R1 ðreq  ry Þ. R R SED rule r : d ¼ r : d elas leads in a similar manner to

F¼

r2eq Rm ðT X Þ E

þ2

Z req

req g 0 ðreq Þdreq  ðr : Þelas ¼ 0

ð54Þ

0

also a scalar equation for the two unknowns req and T X in plasticity. One has shown from Eq. (41) that the stress triaxiality on a free surface is a function of the accumulated plastic strain T X ¼ T X ðpÞ. Including this information within functions F allows to defined triaxiality enhanced fast energetic methods Neuber-T X and SEDT X as

F ðpÞ ¼ 0 ! p

req ¼ ry þ RðpÞ

ð55Þ

The principal stress components are finally gained from the dummy variable u ¼ r2 =req (Eq. 25) as  pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r2 ¼ 9T X þ 3 4  9T 2X eq r3 6   q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi req ¼ 9T X  3 4  9T 2X ð56Þ 6 The stress triaxiality enhancement allows to recover the uniaxial Neuber and SED rules encountered at surfaces in plane stress conditions (with then a – b). It includes Hoffmann and Seeger (1985) analysis when parameters a and b are equal. Two examples are given in Fig. A.15 and A.16 for the non proportional loading cases of structures of previous section. The same conclusion stand as for previous results on stress triaxiality estimation. The enhancement of SED method by the proposed multiaxiality rule gives good results in both cases (a) when the value of stress triaxiality remains close to the value obtained in elasticity (validity of Walker’s assumption), (b) when the strain ratio remains close in elasticity and in elasto-plasticity (validity of Hoffmann and Seeger’s assumption).

2 4  5m 15 1  m

ð58Þ

As the fields in the inclusion are not homogeneous in such a case, plasticity computations have been carried out to obtain numerical values for Mijkl coefficients. Using axisymmetry and the traction free condition (on the free surface of the inclusion of normal 1), non zero values were obtained only for parameters Z 2222 ¼ Z 3333 ¼ Z; Z 2233 ¼ Z 3322 ¼ zZ; Z 2323 ¼ Z 2323 ¼ Z 3223 ¼ Z 3232 ¼ Z 2332 ¼ 12 ð1  zÞZ. One has then r1i ¼ R1i ¼ 0 and trr – trR when p – 0. 7.1. Multiaxiality rule from scale transition law Let us assume proportional loading. If the localization law (58) is written in terms of principal components, one gets

(

R2 ¼ r2 þ 2Gð1  bÞp2 R3 ¼ r3 þ 2Gð1  bÞp3

ð59Þ

In a consistent manner with Eq. (21), let us calculate a ¼ ðR3  mR2 Þ=ðmR3  R2 Þ, expression which is expanded into

(

_ 3 ¼ a_ e2  b0 _ p2  b1 _ p3

ÞZ b0 ¼ 2Gð1b ½ðz  mÞ þ að1  mzÞ E ÞZ b1 ¼ 2Gð1b ½ð1  mzÞ þ aðz  mÞ  1 E

ð60Þ by use of strain partition

i ¼ 

e i

p i

þ  and elasticity law. Plasticity

0

r normality rule gives _ pi ¼ p_ reqi so that one finally recovers the mul-

_ 3 ¼ a_ e2  b_ p2

Localization laws allow for the closed form description of scale transition. When the microscale corresponds to a small sphere embedded in a large elastic medium Eshelby localization law is often used (Eshelby, 1957; Kroner, 1961),

b¼

r ¼ R  2Gð1  bÞZ : p

tiaxiality rule heuristically proposed in Section 5.1

7. Application to the triaxiality enhancement of a homogenization localization law

r ¼ R  2Gð1  bÞp

capital R for the stress tensor at Representative Volume Element (RVE) scale. When yielding occurs at microscale, the traction free condition R1i ¼ 0 (free surface of normal 1) is not conserved at microscale as then r1i – 0. From Eq. (57) still one has trr ¼ trR for incompressible plasticity. Localization laws have been extended to the case of hemispherical inclusions at surfaces (Sauzay and Gilormini, 2000; Lemaitre and Desmorat, 2005; Herbland, 2009; Herbland et al., 2007) as

b ¼ b0 þ b1

r03 r02

ð61Þ

Note that b is loading dependent but remains constant in case of proportional loading. 7.2. Stress triaxiality enhanced scale transition law

ð57Þ

here with elastic properties identical at both scale, where small letters stand for quantities at microscale (within the inclusion) and

In a consistent manner with the determination of b in Section 2.3, let us neglect the elastic strains in previous derivations. This allows to determine the localization parameter z as

82

z¼

A. Darlet, R. Desmorat / International Journal of Solids and Structures 67–68 (2015) 71–83

a  b  m þ abm ab þ mða  bÞ  1

ð62Þ

In other words, the localization law r ¼ R  2Gð1  bÞZ : p altogether with Z constant (equal to 1.79 in Sauzay and Gilormini (2000) work) and z loading dependent determined from Eq. (62), introduce the accurate estimation of the stress triaxiality of Section 3 in the localization law for hemispherical inclusion. 8. Conclusion A multiaxiality rule or heuristics has been proposed to describe the multiaxial constraint along surfaces of elastoplastic structures. It handles both plane stress and plane strain conditions. The validation structural examples studied shows that such a heuristics can be used for any stress state that occurs along surfaces, even when yielding does not remain confined. The influence of the geometry and the loading is taken into account through the two heuristics parameters a and b. The first parameter a is determined from an elastic analysis and b from a quasi-incompressible elastic analysis of the structure. The multiaxiality rule has been used to establish closed-form expressions for the stress triaxiality and for Lode angle at surfaces in case of monotonic loading with linear isotropic hardening. When the hardening is non linear, the numerical integration procedure is straightforward with Runge–Kutta scheme from differential Eq. (37). Along surfaces, the stress triaxiality and Lode angle are found related to each other. They depend only on the accumulated plastic strain and on the heuristics parameters a and b. The procedure has been validated on the classical example of the plate with a circular hole under both plane stress and plane strain conditions. Moreover, it is assessed on structures subjected to more complex proportional and non proportional loadings. The estimation of the stress triaxiality and Lode angle from the multiaxiality rule is found quite accurate, even if it is for rather small strain levels in non proportional cases. The proposed multiaxiality rule is the only assumption that can deal with both plane stress (therefore local tension) and plane strain conditions. It can be used to enhance fast energetic methods (such as Neuber or Molski and Glinka’s methods). It can be introduced within homogenization localization laws at surfaces. It can be extended to cyclic loading (see Appendix A). Acknowledgments The authors are particularly grateful to SNECMA SAFRAN group for its financial support. This work is conducted under the French program ‘‘PRC Structures Chaudes’’ involving Snecma-SAFRAN group, Turbomeca- SAFRAN group, ONERA and CNRS laboratories (Mines Paris Tech, Institut P-ENSMA, LMT-Cachan, LMS-X, CIRIMAT-ENSIACET and CEAT). Appendix A. Extension to cyclic loadings Monotonic applications have been addressed. For cyclic (fatigue) loadings, the same calculations apply, with the same results, the same figures, if one uses a cyclic plasticity law with then the stress r, the strain , the plastic strain p replaced by the stress amplitude Dr=2, the strain amplitude D=2, the plastic strain Dp =2. Note then the accumulated plastic strain, abscissa of many  1=2 figures, has to be replaced by 12 23 Dp : Dp . The stress triaxiality and Lode angle have to be defined in terms of amplitude

T cyclic ¼ X

1 trDr 3 ðDrÞeq

Hcyclic ¼

27 det Dr : 2 ðDrÞ3eq

ðA:1Þ

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