1
Elastic Domains: Yield Conditions
1.1. Introductory remarks
Figure 1.1. Henri Tresca (1814–1885)1 honored on the North pillar of the Eiffel Tower
One hundred and fifty years ago, Henri Tresca (Figure 1.1), then a Professor of applied mechanics at the Conservatoire des arts et métiers in Paris, submitted a series of memoirs to The French Academy of Sciences (Tresca 1864a, 1867a, 1867c, 1869a, 1870). They were devoted to recording the extensive series of experiments he had carried out investigating punching, rolling, forging, stamping and extruding processes of metals, where he had definitely identified a phenomenon; heCOPYRIGHTED called it the “fluidity” of metals MATERIAL subjected to very high pressures (Figure 1.2).
1 Archives of The French Academy of Sciences, Paris. 2 Elastoplastic Modeling
Figure 1.2. “Flow of two iron slices, hot, under the action of a forging-hammer”
In addition to the precise description of the experiments that were initially performed on various lead specimens and, afterwards, on other metals (copper, iron, tin and zinc) or materials (ceramics and modeling wax), Tresca had proposed a mechanical theory for the observed phenomena with his “Fundamental assumption as to the resistance to fluidity”, which he stated as follows (Figure 1.3):
“Beyond this limit, we propose to consider, for the state of fluidity, that the forces that are developed are absolutely constant and entirely independent of the relative displacements, which amounts to admitting that they can always be evaluated by means of a resistance coefficient K per square meter or per square centimeter, this coefficient K remaining the same for the assessment of any molecular deformation developed during the fluidity period.” Elastic Domains: Yield Conditions 3
Figure 1.3. Excerpt from Tresca (1869a, p. 777)
As Tresca was not yet a member of the Academy, his memoirs had to be approved before abstracts, written by himself, could be published in the Comptes rendus (Tresca 1864b, 1867b, 1867d, 1869b, 1870). In 1870, Saint-Venant (1870b, 1871), who was one of the commissioners in charge of reviewing them, published a note whose rather long title may be considered as a perfect definition of what is now called the plastic behavior of materials: “Sur les équations des mouvements intérieurs opérés dans les solides ductiles au delà des limites où l’élasticité pourrait les ramener à leur premier état”. [On the equations of internal movements operated in ductile solids beyond the limits where elasticity could bring them back to their initial state]. In this note, considering the particular case of a two-dimensional problem, he derived five equations that governed what he called “hydrostéréodynamique” or “plasticodynamique”. They consisted of two differential equations of equilibrium, one equation expressing the fluidity condition2, another equation expressing incompressibility of matter and a last one expressing the coincidence of the principal directions of the two-dimensional strain-rate and stress tensors3.
2 Similar equations had been written by Lévy (1867) in his theory of earth pressure. 3 Original French wording: “afin d’obtenir la coïncidence des directions de plus grand glissement et de plus grande résistance au glissement.” 4 Elastoplastic Modeling
We may say that, for the first time, both questions that are the cornerstones of plastic modeling were thus evoked4: – When? i.e. the question of the yield condition; – What and How? i.e. the question of the plastic flow rule.
These two issues will be addressed successively in this chapter and the one following it.
1.2. An overview of the model
1.2.1. The infinitesimal transformation framework
Classical elastic–plastic modeling of ductile materials is expressed within the framework of infinitesimal transformation5. The model is well established by now and, although much work has been devoted to the formulation of finite transformation elastic–plastic theories, infinitesimal transformation is always the first framework – and often the only one – chosen when practical applications to structural analysis are concerned, which often proves sufficient. Moreover, it is clear that finite transformation theories call for the knowledge of infinitesimal models as they must match them under the infinitesimal assumption. Therefore, the presentation in this book focuses on infinitesimal plasticity within the framework of the small perturbation hypothesis (SPH) unless otherwise stated (see Chapter 4, section 4.3.7).
1.2.2. Time variable
In the same way as elastic modeling, the classical elastic–plastic model excludes any viscosity effect (no viscoplasticity) and does not take any ageing of the concerned material into account. It follows that the corresponding constitutive equations shall be invariant in any translation or positive rescaling of the time variable. Nevertheless, unlike the elastic case, the introduction of a time variable proves necessary due to the irreversibility of plastic deformation, which calls for an unambiguous chronology of loading and response histories. Such a time variable is just a “time-coding” parameter, monotonously increasing with the physical time. As it will be discussed in Chapter 2 (section 2.2.2), this status implies homogeneity conditions, with respect to time, to be satisfied by the constitutive equations.
4 “Tresca’s paper on the flow of solids may approximately be regarded as the birth of the mathematical theory of plasticity” (Koiter, 1960, p. 179). 5 As defined in Salençon (2001, p. 57). Elastic Domains: Yield Conditions 5
1.3. One-dimensional approach
1.3.1. Uniaxial tension test
1.3.1.1. Work-hardening material
Figure 1.4. Uniaxial tension test performed on a work-hardening specimen6
Figure 1.4 shows a typical test piece for a uniaxial tension–compression test, made from a homogeneous material and with specific geometrical features such that the stress field is assumed to be a simple homogeneous tension (or compression) field, parallel to the axis of the specimen7. In its simplest form, this experiment involves recording the change in the tensile force F or the ratio F S0 , where S0 is the initial value of the cross-section in the central part of the specimen, as a function of the relative extension Δ.
The experiment is carried out at a fixed deformation rate, slow enough to be considered as quasi-static, with temperature maintained constant. Figure 1.4a presents a classical diagram recorded in the case of a test piece made from stainless steel (Salençon and Gary 1999).
σ It shows first the reversibility of the material response up to a threshold 0 , σ marked by point A on the curve. After loading up to a level below 0 , total or
6 See: https://www.researchgate.net/publication/325757756. 7 Note that in the case of anisotropic materials, the simple off-axis tension experiment requires additional special precautions to be taken. 6 Elastoplastic Modeling
partial unloading causes the point (Δ ,)F S0 to run back down the same straight line in the opposite direction. The material behavior is linear elastic along OA .
σ When F S0 exceeds this threshold following the loading curve up to a value B along OAB , subsequent unloading shows that, when the load is made to decrease, point (Δ ,)F S0 does not regress along the loading curve BAO but moves down a straight line BC parallel to OA . In particular, when the load has been reduced to zero, the test piece still retains a certain degree of extension represented by OC , which is the plastic deformation of the specimen.
Then, a new loading process being carried out, starting from point C or any σ point on CB , it is observed that as long as F S0 remains below B , the material response is reversible and point (Δ ,)F S0 follows the same straight path CB ≤≤σ σ for any loading or unloading process with 0 FS0 B . When F S0 exceeds B , point (Δ ,)F S0 moves along a curve in the prolongation of the first loading σ curve OAB , which means that, for this new tension loading process, B stands as the new threshold.
These experimental observations are the basis of the stress–strain diagram shown in Figure 1.4b for modeling the elastic–plastic behavior in tension, where σ stands σ σσ=⊗ ε for the xx component of a uniaxial tension stress tensor xx eexx and ε εε=⊗8 represents the xx component of the linearized strain tensor ijeeij: σ – 0 is the initial elastic limit or initial plastic yield point for this experiment; σ – after loading has been carried out up to B , the current elastic limit or current σ plastic yield point is equal to B , this result sometimes being called “Coulomb’s principle”9;
– the permanent strain after complete unloading is the plastic strain ε p .
An example of the influence of the deformation rate ε on the experimental results regarding the stress–strain diagram, showing a viscosity effect, is shown in Figure 1.5 which refers to experiments on aluminum test pieces (Bui and Zarka 1972). This phenomenon is not taken into account in classical elastoplastic models10.
⊗ = 8 Symbol “ ” denoting the tensor product. eii ,1,2,3 unit vector. 9 As in Bouasse (1920). 10 See Bingham (1916), Hohenemser and Prager (1932), Oldroyd (1950) and Perzyna (1963, 1966, 1971, 2005). Elastic Domains: Yield Conditions 7
Figure 1.5. Uniaxial tension tests on aluminum specimens
No ageing effect being considered either, it follows that the response of the material in any uniaxial tension test is only characterized by the stress–strain diagram in Figure 1.4, which is independent of ε or, equivalently, σ 11.
The phenomenon observed in Figure 1.4, where the current plastic yield point actually depends on the plastic strain, characterizes work-hardening or strain- hardening of the material.
Finally, one important conclusion to be retained from this first approach to σ elastoplastic behavior is that, in the plastic range (i.e., once threshold 0 has been crossed), there is no longer a one-to-one relationship between stress σ and strain ε . The current value of ε is not determined by the current value of σ but depends on σ the whole stress history after 0 has been exceeded. Because of the independence of the stress–strain diagram, with respect to the strain rate and ageing not being taken into account, only the chronology of events in the histories of σ and ε matters. It follows that the time variable stands as a “time-coding” parameter, which can be subjected to any translation and positive rescaling.
1.3.1.2. Perfectly plastic material The work-hardening phenomenon, as observed in Figure 1.4 where the value of the plastic yield point is an increasing function of ε p , is sometimes called “positive hardening”. Although most commonly encountered, it cannot be considered as a
11 See Mase (1970). 8 Elastoplastic Modeling general feature of plastic behavior. Indeed some materials, such as mild steel, subjected to a uniaxial tension test reveal a force-extension diagram as shown in Figure 1.6a, with a plateau where F S0 remains constant while Δ −3 −2 increases in the range 10 to 10 (typical orders of magnitude). When F S0 has σ reached the plastic yield point 0 at point B , any unloading is reversible and follows the linear elastic path BC and further loading is elastic along CB, until σ F S0 reaches 0 , after which Δ resumes increasing along the plateau with = σ FS00.
Figure 1.6. Uniaxial tension test performed on a perfectly plastic specimen12
This behavior is modeled as shown in Figure 1.6b. It is said to be perfectly σ plastic. 0 is the plastic yield point in tension.
1.3.2. Uniaxial tension-compression test
Performed on the same test piece as shown in Figure 1.4a, starting from the initial unloaded state, a uniaxial compression test13 reveals the same type of behavior −σ as in tension, with an initial yield point in compression ( 0 ) and work-hardening as shown in Figure 1.7. We may then define the initial elastic domain in tension- −≤σσ ≤ compression as 000FS .
12 See: https://www.researchgate.net/publication/325757756. 13 The slenderness of the specimen must be controlled in order to avoid any buckling effect. Elastic Domains: Yield Conditions 9
Figure 1.7. Uniaxial compression test performed on a work-hardening specimen
Note that, in the case of a material identified as perfectly plastic in tension, perfect plasticity is also observed in compression with a plateau at the stress level of −σ the yield point in compression ( 0 ).
1.3.3. The Bauschinger effect
The question to be investigated now is whether the work-hardening phenomenon observed in tension has any influence on the current yield point in compression (and vice versa).
Figure 1.8. Uniaxial traction-compression test: Bauschinger effect
σ This is presented in Figure 1.8 where, after being loaded up to B in tension, the specimen is unloaded and reversely subjected to compression. It is observed that the behavior of the material along this compressive loading process remains elastic until σ F S0 is equal to D , a value that is usually algebraically superior to the initial 10 Elastoplastic Modeling elastic yield point in compression. From a practical viewpoint, this means that work- hardening in tension, besides causing the value of the plastic yield point in tension to increase, is at the origin of a decrease of the magnitude of the plastic yield point in compression. The current elastic domain in tension-compression, after the specimen = σ σσ≤≤ has been loaded up to FS0 B , is defined by D FS0 B . This experimental result is known as the Bauschinger effect (Bauschinger 1881).
Although by definition, no work-hardening phenomenon is observed with perfectly plastic materials, the question of the existence of some Bauschinger effect in the case of reversed loading processes may be investigated. A slight Bauschinger −σ effect has been experimentally reported (Figure 1.9a) with a fast return to the ( 0 ) plateau, which is not taken into account in the perfectly plastic model, as shown in Figure 1.9b where, for a perfectly plastic material, the elastic domain in tension- −≤σσ ≤ compression is invariable and just defined by 000FS .
Figure 1.9. Uniaxial traction-compression test on a perfectly plastic material
1.3.4. Other one-dimensional experiments
Figure 1.10. Torsion of a thin-walled tube Elastic Domains: Yield Conditions 11
Other experiments, such as the torsion of a thin-walled tube with circular cross- section, are also performed in order to investigate the elastic–plastic behavior of a material from a one-dimensional viewpoint.
In the example schematized in Figure 1.10, the geometry of the sample aims to attain, at each point of the test piece, a pure shear stress state, whose components σσ= θθzz in the orthonormal basis of cylindrical coordinates are constant and other σ = 14 σ ij 0 . Then, the diagrams relating θ z to the rotation of the top end-section with respect to the basis evidence the same characteristic features as their counterparts in Figures 1.4–1.9.
In fact, the diagrams that appear in Figures 1.4b, 1.6b, 1.8 and 1.9b provide the general form of one-dimensional elastoplastic modeling.
1.4. Multidimensional approach
1.4.1. A multidimensional experiment
The one-dimensional approaches presented in the preceding section made it relatively easy to introduce the basic concepts of the elastoplastic behavior of materials.
Shifting from the one-dimensional to the three-dimensional viewpoint, the force variable σ is substituted by the Cauchy stress tensor σ acting on the material σ 15 element, with components ij in an orthonormal basis . The natural initial state where the stress tensor is zero will be taken as the geometrical reference state (unless stated otherwise). The linearized strain tensor denoted by ε with ε components ij, counted from this reference state, is the geometrical variable associated with the stress tensor (SPH framework).
Figure 1.11 presents experimental results obtained by Bui (1970) in a typical multidimensional experiment, performed on a test piece made from copper, where a thin-walled tube with circular cross-section is subjected to traction-compression and torsion. Because of the geometrical specificities of the test piece and homogeneity
14 The analysis of this experiment must take material symmetries of the constituent material (isotropy, anisotropy) very carefully into account. 15 For basic notions regarding tensor calculus, the reader may refer to Appendix 1 in Salençon (2018a). 12 Elastoplastic Modeling and isotropy of the constituent material, it is considered that the components σσ= σ θθzz and zzof the stress field in the orthonormal basis of cylindrical σ = coordinates are constant and other ij 0 .
Starting from the natural initial state and following whatever loading path in the σσ (,zz zθ ) plane, the goal of the experiment is to determine the ultimate load that limits the elastic behavior of the material along this loading path. This leads to the σσ definition, in the (,zz zθ ) plane, of the initial elastic domain of the material in this experiment.
The difficulty of the experiment should not be overlooked. Indeed, delimiting the initial elastic domain requires checking at any point of the loading path – by means of incremental loading and unloading processes – whether irreversible deformation has been generated. These increments must be small enough not to alter the specimen significantly when the initial elastic limit is reached along a given loading path, so that a new investigation can be performed on the same specimen along a new loading path (see section 1.4.5).
Figure 1.11. Traction-compression and torsion of a thin-walled tube (Bui 1970)
1.4.2. Initial elastic domain
Such experiments confirm that the concept of an initial elastic domain of the material is realistic. Defined in the six-dimension space 6 of the Cauchy stress tensor σ , this domain, denoted by C , is such that the response of the material is Elastic Domains: Yield Conditions 13 purely elastic, i.e. reversible, along any loading path starting from O and running entirely within C . Referring to Figures 1.7 and 1.11, we see that both the segment −σσ σσ 00, in the first case and the domain delimited in the (,zz zθ ) plane in the second case are sections of C by the straight line or plane corresponding to the concerned loading process. It is worth noting that C is, by definition, star shaped with respect to O .
1.4.3. Work-hardening
In the general case of a work-hardening material, once the initial elastic limit has been reached by the loading point σ at a point A on a given loading path, this loading path can cross the initial elastic boundary. The loading process can still be carried on, with irreversible deformation being generated in the same way as in the one-dimensional case.
Let B be a loading point beyond the initial elastic limit on this loading path, as shown in Figure 1.12. The current elastic domain at point B after running along the loading path AB is generated by all the loading paths issued from B , such as BMP , along which the increment of strain of the material is elastic (i.e. reversible)16. Thus, it appears that the elastic domain is driven17 by the loading point when this point crosses the elastic boundary, with its shape usually being modified. This is the whole description of the work-hardening phenomenon.
Figure 1.12. Initial and current elastic domains in 6 for a work-hardening material
16 It is worth noting that it is therefore the current elastic domain for any loading point along the loading path BMP. 17 Moreau (1971) introduced the French terminology “rafle” (grabbing) to qualify this geometrical process in 6 . 14 Elastoplastic Modeling
The current elastic domain C ()E is a function of the whole time-oriented loading path followed to reach the current loading point, which characterizes the work-hardening state of the material that will be denoted symbolically by E .
An infinitesimal time-oriented arc of the loading path, such as PQ at point P , that crosses the elastic boundary outwards is said to be increasing: it generates an increment of plastic deformation and modifies the work-hardening state of the material. If the arc is tangent to the elastic boundary, it is said to be neutral and implies no additional plastic deformation and no change of the work-hardening state.
1.4.4. Perfectly plastic material
The elastic perfectly plastic model excludes any work-hardening and refers to a fixed elastic domain C . The loading point σ cannot cross this boundary. Plastic deformation can only be generated when σ stays on the elastic boundary or moves along a time-oriented loading path arc tangent to it (neutral loading arc) such as AB in Figure 1.13.
Figure 1.13. Elastic domain in 6 for a perfectly plastic material
1.4.5. Bui’s experimental results
As incidentally remarked in section 1.4.1, the determination of the initial elastic boundary requires detecting the first appearance of irreversible deformations along a given loading path. Then, in order to determine the current elastic boundary, it is necessary to test incremental unloading and loading arcs, in different directions
Elastic Domains: Yield Conditions 15 from the current position of the loading point, and look for the appearance of new irreversible deformations. Therefore, it is easily understood that the accuracy with which an increase in irreversible deformation can be appreciated is of paramount importance for the reliability and authenticity of the elastic boundaries so determined. In 1970, Bui published a series of results obtained through a tension- compression and torsion testing apparatus he had designed, which enabled him to − detect strain increments with a precision of 210× 5 . Some of these results will now be presented (Bui 1970).
Figure 1.14. Work-hardening in compression for an Al 99.5 specimen
Figure 1.14 presents the successive elastic boundaries for a specimen made from aluminum 99.5, where work-hardening is generated in compression. We can observe what Bui called an “expansion effect”, with the meaning that the first work- hardening steps induce an expansion of the current elastic boundary in all directions σσ of the (,zz zθ ) plane. The remarkable consequence is that both the elastic limits in pure shear and pure tension stress states are raised, in contradiction with a Bauschinger effect (section 1.3.3). Afterward, when work-hardening is sufficient – i.e. of the order of magnitude of current measurements – a Bauschinger effect is observed as expected. According to Bui, the expansion effect can only be observed with well-annealed metals.
Figure 1.15 provides experimental illustrations of the phenomenon described in Figure 1.12, where the elastic domain is driven by the loading point for two
16 Elastoplastic Modeling specimens made from ARMCO iron, subjected to different loading paths: (a) σ σ σ σ constant zz , with increasing zθ ; (b) radial loading path with zz and zθ increasing proportionally to each other. We note the appearance of a bump in the vicinity of the loading point, whose summit does not necessarily coincide with the loading point (Figure 1.15a).
Figure 1.15. Work-hardening of ARMCO iron specimens
1.5. Yield conditions
1.5.1. Initial yield condition and yield function
Practical use of the concept of initial elastic domain for any structural design requires this domain to be defined mathematically in 6 by an equation involving the components of the Cauchy stress tensor σ and physical constants characterizing the material under concern.
In view of the principle of material frame indifference18, this equation is an intrinsic property of the material and shall not depend on the spatial orientation of the considered material element. This means that the components of σ to be involved in the intrinsic form of this equation shall refer to an orthonormal basis physically significant for the element and attached to it. If this equation is written in an arbitrary orthonormal basis, the physical constants that are characteristic of the
18 See Šilhavý (1997). Elastic Domains: Yield Conditions 17 initial plastic behavior will include parameters defining the orientation of the element, with respect to that basis.
This remark is of no importance in the case of isotropic materials, which will be mainly considered in the following19, and enables us to write the (initial) yield function as f ()σ , a continuous function of σ ∈ 6 such that:
f (σ )<⇔ 0 interior of C f (σ )=⇔ 0 boundary of C [1.1] σσ>⇔∉ f () 0 C , where the second equation is also called the yield condition or yield criterion
f ()σ = 0. [1.2]
As remarked in section 1.4.2, the initial elastic domain is, by definition, star shaped with respect to O . Consequently, function f ()σ complies with:
∀<01,()0ασ ≤ff < ⇔ ()0 ασ < . [1.3]
σσ= In addition, function ff() (ij ), being a function of the symmetric tensor σ , only depends on six scalar components. Nevertheless, it is convenient to write it σ σσ≠ as a function of the nine components of , where components ijand ji (ij ) are considered distinct and play symmetric roles. Other mathematical properties of f ()σ will be stated later on.
We note also that defining f ()σ through [1.1] does not imply anything about its physical dimensions. In view of its physical significance as a measure of the intensity of the load applied to the material element, f ()σ will often be written with the dimension of a stress.
19 The case of some anisotropic materials will be briefly discussed in section 1.6.6. 18 Elastoplastic Modeling
1.5.2. Loading function and work-hardening
6 Figure 1.16. Work-hardening material: C ()E being driven by σ in
For a work-hardening material, the current elastic domain depends on the work- hardening state E of the material element, which is determined by its whole loading history. As already noted, only the time-ordered sequence of increasing loading arcs in this history shall be retained as relevant regarding work-hardening. As shown in Figure 1.16, the current elastic domain C (E ) is driven by the loading point σ along these arcs, while new plastic strain is generated according to the plastic flow rule, which will be the focus of Chapter 2. The evolution of the work-hardening state of the material is governed by the work-hardening rule.
The current elastic domain C (E ) can be described in the same way as C by means of a function f of σ and E , called the loading function in the current state, such that:
f (σ ,EE )<⇔ 0 interior of C ( ) f (σ ,EE )=⇔ 0 boundary of C() [1.4] σσ>⇔∉ f (,)EE 0 C ().
Expressing that, along an increasing loading arc, the loading point stays at the boundary of the current elastic domain, as shown in the box in Figure 1.16, yields the symbolical equation [1.6] where t is a monotonously increasing time-coding parameter:
with dσσ==> dttt , dEE d , d 0 [1.5] Elastic Domains: Yield Conditions 19 and the condition that dσ is an increasing or neutral loading arc20,
∂f (,)σ E f (,)σσ=≥ 0, : 0 E ∂σ [1.6] ∂∂σσ ff(,)EE (,) :0.σ += ∂∂σ E E
Equation [1.6] is called the consistency equation, which is symbolical in a sense that, under that form, it cannot be solved to derive the work-hardening rule. In fact, models have been developed, which aim to give an explicit expression of E in [1.4], using a few scalar or tensorial parameters, in order to make practical applications possible. Equation [1.6] can then be solved and the work-hardening parameter rates ∂f (,)σ E are obtained as proportional to :0σ > . ∂σ
It is worth noting that, in the case of a perfectly plastic material, which refers to a fixed elastic domain C defined by a loading function f ()σ , the consistency equation [1.6] reduces to:
∂f ()σ f ()σσ== 0, : 0, [1.7] ∂σ which is just the condition, already stated in section 1.4.4 (Figure 1.13), that plastic deformation can only be generated when σ stays on the elastic boundary or moves along a time-oriented loading path arc tangent to it (neutral loading arc).
1.5.3. Simple work-hardening models
1.5.3.1. Isotropic hardening model The isotropic hardening model is a simple model which depends on one scalar hardening parameter, to be denoted here by α . It was first introduced by Taylor and
20 It is assumed here that the loading function is continuously differentiable. A generalized expression of this equation will be given in Chapter 2 (section 2.2.6). The symbol ":" denotes the doubly contracted product of two second rank tensors.
∂∂ff(,)σσEE (,) ∂ f (,) σ E :σσ== : : σσ as is symmetric. ∂∂σσji ∂ σj j ij ij 20 Elastoplastic Modeling
Quinney (1934) and, according to Jirásek and Bažant (2002, p. 317), was also proposed by Odquist in 1933, with the difference that in Taylor and Quinney’s presentation, the hardening parameter α is the plastic work, while in Odquist’s approach it is the plastic strain (which explains the alternative terminology “strain- hardening”). It generates a family of elastic domains C (E ) , which follow the loading point along increasing loading arcs, while remaining similar to the initial elastic domain and homothetical with regard to the origin (Figure 1.17).
Figure 1.17. Isotropic hardening model
With f ()σ denoting the initial yield function, the loading function in the current state can be written as:
σ ff(,)σσαα== (,) f ( ), >1, [1.8] E α which makes it possible to solve the consistency equation [1.6] and obtain the work- hardening rule for α in the form:
d(f σ ) with = f ′(σ ) dσ [1.9] σσ σ αα =>ff′′(): σ (): σ if f ′ ():0. σ αα α Elastic Domains: Yield Conditions 21
1.5.3.2. Kinematic hardening model The isotropic hardening model amounts to an expansion of the elastic domain in all directions of the stress space 6 , following the loading point along an increasing loading path. This result can be accepted locally in the vicinity of the loading point but is not in accordance with experimental observations as presented in sections 1.3 and 1.4. In particular, it does not fit with the Bauschinger effect reported in section 1.3.3.
The kinematic hardening model, often credited to Prager (1955a, 1956), was also proposed by Melan (1938b). It introduces a tensorial hardening parameter, to be denoted here by α , usually called the back stress, which governs the translation of the current elastic domain, without any deformation, as it is driven by the loading point along an increasing loading arc. This parameter is null for the initial elastic domain and, with f ()σ denoting the initial yield function, the loading function in the current state can be written as:
ff(,)σσασαE ==− (,) f ( ), [1.10] which means that the back stress α defines the shift of the origin of the initial elastic boundary.
Figure 1.18. Kinematic hardening model 22 Elastoplastic Modeling
The hardening rule proceeds from the assumption by Melan (1938)21 that the hardening parameter rate α is proportional to the plastic strain rate, which, within the framework of associated plasticity (see Chapter 2, section 2.2.3), implies that it ∂f (,)σα is collinear with and lies along the outward normal to the current elastic ∂σ domain at the current loading point. It is then possible to solve the consistency equation [1.6] as:
d(f σ ) with = f ′(σ ) dσ [1.11] f ′():σασ− ασα =−ff′′() if ():0. σασ −> ff′′():()σα−− σα
1.5.3.3. Comments These two historical models do not account for all aspects of experimental results such as those already presented. However, they have been used in computational software, often in the form of an “isotropic-kinematic” mixed model, thus providing relevant results for practical applications. A large literature has been devoted to the development of work- or strain-hardening models, based upon numerous reliable experimental data and theoretical analyses, as required by industrial applications.
1.6. Yield criteria and loading functions
1.6.1. Convexity
Experimental results obtained from various multidimensional tests, such as those reported in Figures 1.14 and 1.15, support the general conclusion that the initial elastic domain and the current elastic domains as well, are convex in 6 . It may be worth noting that, for metals, this geometrical property of C and C (E ) can be derived theoretically (Hill 1956; Mandel 1966) from the convexity of the laws governing the plastic mechanisms that takes place at the elementary crystal level, such as Schmid’s shear-stress law (Schmid and Boas 1935), through micro–macro modeling processes (see Chapter 4).
21 See Jirásek and Bažant (2002, p. 324). Elastic Domains: Yield Conditions 23
From now on, C and C (E ) will be assumed to be convex and described by f ()σ and f (,)σ E , convex functions of σ
∀∀σσ′ ∀∈ λ[ ] ,, 0,1, [1.12] λσ+− λ σ′′ ≤ λ σ +− λ σ fff()(1 ) ,EE ( , ) (1 ) ( , E ) .
1.6.2. Isotropy
1.6.2.1. Isotropic yield functions As stated in section 1.5.1, we are concerned with materials that are isotropic in their initial natural state and we assume that isotropy is maintained throughout the elastic loading process. It follows that f ()σ , being only a scalar function of the symmetric tensor σ , can be written as a symmetric function of the sole principal stresses or, equivalently, a function of three invariants of that tensor (Wineman and Pipkin 1964) such as:
I ==trσσ 1 ii ==σσ σ σ I2 tr ( . ) 2ij ji 2 [1.13] ===σσσ σ σ σ σσ σ I3123tr ( . . ) 3 ij jk ki 3 also equivalent to a function of I1 and invariants J23and J of tensor s , the deviator of σ defined by:
=−σ sI1 1 3 ===−+−+−σσ222 σ σ σσ Jssss2122331tr ( . ) 2ij ji 2 [( ) ( ) ( ) ] 6 [1.14] Jsssssssss===tr ( . . ) 3 3 , 3123ij jk ki
δ where 1 denotes the unit isotropic tensor with components ij (Kronecker deltas) σσ σ and 123,, are the principal stresses.
In the case of work-hardening, if isotropy is retained throughout the loading process, beyond the initial elastic limit, f (,)σ E remains a symmetric function of the principal stresses. 24 Elastoplastic Modeling
1.6.2.2. Haigh–Westergaard stress space It follows that yield surfaces for isotropic materials can be represented in a three- dimensional space with coordinates 1,, 2 3 , where they admit the axis defined by
1 2 3 as the ternary axis of symmetry and the bisector planes
(1 2 ), ( 2 3 ) and ()31 as symmetry planes. As shown by Yang (1980a, 1980b), convexity of the yield surface with respect to is equivalent to the convexity of its geometrical representation in the (,,)1 2 3 space, known as the Haigh–Westergaard stress space (Mase 1970). Figure 1.19 shows the ternary symmetry axis ( 3 3, 3 3, 3 3) and the plane perpendicular to it passing through the origin, which is known as the Π-plane. Point corresponds to a typical stress state and vector O can be resolved into its component OP along the ternary symmetry axis and a component OS in the , with components I1 3 on each axis, represents the mean or “hydrostatic” stress
11I1m3 and, consequently, vector represents the deviator s.
Figure 1.19. Haigh–Westergaard stress space
1.6.3. The Tresca yield criterion
1.6.3.1. Maximum shear theory “La résistance au cisaillement est égale à la résistance de fluidité” [The shear strength is equal to the fluidity resistance]. This statement appears on page 827 of the memoir submitted by Tresca (1869a) on March 31, 1869 to The French Academy of Sciences as a result of the numerous experiments he had carried out since 1864; it is completed by, “ce qui est d’ailleurs une conséquence de l’hypothèse d’une force de cohésion constante” [which is, moreover, a consequence of the Elastic Domains: Yield Conditions 25 hypothesis of a constant cohesive force] in the summary written by Tresca (1869b). These statements are at the origin of the Tresca yield criterion for isotropic materials, based upon the concept of maximum shear stress, which can be written as:
f ()στ≤⇔∀ 0 nC, ≤, [1.15] where n , an arbitrary unit vector, is the outward normal to a facet at the considered point in the material and τ denotes the shear stress vector on this facet (Figure 1.20). C is a physical constant, which defines the maximum shear stress that limits the material elastic behavior. For this reason, the Tresca yield criterion is also called the maximum shear stress criterion.
Figure 1.20. Normal stress and shear stress on a facet with outward normal n
1.6.3.2. Geometrical representation In the Haigh–Westergaard stress space, the initial elastic boundary defined by [1.15] is represented by a cylindrical surface parallel to the ternary axis of symmetry, with a regular hexagonal cross-section in the Π-plane, as shown in Figure 1.21.
Figure 1.21. Tresca’s yield criterion in the Haigh–Westergaard stress space 26 Elastoplastic Modeling
Figure 1.22 recalls the principle of the Mohr representation of a stress state22. σσ σ σσ≥≥ σ With IIIIII,, , the principal stresses ordered according to IIIIII, the τσσ=− maximum value of the shear stress is just ()2I III and the Tresca yield criterion can be written as:
σσσσ≤⇔ − ≤= fC() 0 (IIII )2 0 2, [1.16]
σ with 0 denoting the initial yield stress in tension. In the form of a symmetric expression of non-ordered principal stresses, the criterion can be written as:
σσσσ=−−=≤ fi() Max{ ij0 ,j 1,2,3} 0. [1.17]
Figure 1.22. Mohr representation of the stress state (Salençon 2018b)23
It comes out from [1.17] and Figure 1.21 that f ()σ does not depend on the =σ σ “hydrostatic” stress 11I1m3 and only depends on the deviator s of . Also, as shown in Figure 1.21, the yield surface derived from [1.17] is not regular σσσσσ=−−== along six edges defined by ijik,00 , ijk,,1,2,3 and, for this
22 See Salençon (2018a). 23 See: https://www.researchgate.net/publication/325756864. Elastic Domains: Yield Conditions 27 reason, no closed form expression of f ()σ in terms of the invariants of s can be written.
1.6.4. The von Mises yield criterion
1.6.4.1. An assessment of the Tresca yield criterion
Figure 1.23. Dashed gray line: Tresca’s yield criterion compared with Bui’s experiments
Referring to Figure 1.11 we can try to assess the validity of the Tresca yield criterion to account for the results obtained by Bui on copper 99.5. In the case of a tension-compression and torsion test, equation [1.16] can be written in terms of σσ zz, zθ as:
1 σσ22+−≤2 σ (4zz zθ )0 0, [1.18] the equation of an ellipse we have marked as a dashed gray line in Figure 1.23. It obviously overestimates the initial elastic limit in pure tension if it is made to match the limit in a pure shear stress state. The von Mises yield criterion, which was drawn by Bui himself as a solid line ellipse in the same figure, provides a better agreement with experimental data. 28 Elastoplastic Modeling
1.6.4.2. Distortion energy theory As recalled in Christensen (2004), according to Timoshenko (1983), the origin of the criterion usually called the von Mises criterion seems to start with a letter written by Maxwell to Lord Kelvin in 1856 stating: “I have strong reasons for believing that when the strain energy of distortion reaches a certain limit then the element will begin to give way”. Beltrami’s (1885) contribution proposed that the total linear elastic energy be retained as a criterion, with the critical energy being estimated from a traction test. Huber (2004) observed that this criterion could not account for σ = the insensitivity of ductile materials to the hydrostatic stress m1I 3. This seems to have triggered von Mises in writing the criterion in terms of the deviatoric elastic energy, which for an isotropic material results in J2 2µ , where µ is the elastic shear modulus.
Finally, the Huber–Mises yield criterion can be written as:
σ ≤⇔ −≤ fJk() 0 2 0 [1.19] or, equivalently,
σσσσσσσ≤⇔ −222 + − + − −≤ fk() 0 [(12 ) ( 23 ) ( 31 )]6 0. [1.20]
The physical constant k that sets the limit to J2 can be determined by a pure shear stress test while, from [1.20], it comes out that the elastic limit in a simple tension or compression test is equal to k 3 . This fits correctly with the experimental results presented in Figure 1.23.
1.6.4.3. Equivalent stress, effective stress The concept of equivalent stress follows from this latter result. Within the framework of the von Mises yield criterion, the equivalent stress to a given stress σ σ state is defined as the magnitude eq of the pure tension stress that would give the same value to the loading function, i.e.
σσ⊗= feef()()eq xx , [1.21] from which we derive the Mises equivalent stress as:
σ = eq3 J 2 . [1.22] Elastic Domains: Yield Conditions 29
τ This concept should not be confused with the octahedral shear stress oct , defined from the stress vector exerted by σ on a facet with director cosines equal to 33 on the principal stress directions (Figure 1.24), whose normal and shear components are, respectively:
σστ== = octIJ 13, m oct 23 2 . [1.23]
Figure 1.24. Octahedral facet
The octahedral shear stress is sometimes called the effective stress for the von Mises criterion, meaning it is that shear stress, which is actually generated by the stress tensor σ , whose magnitude is limited by the criterion so that:
στ≤⇔ − ≤ fk() 0 oct 23 0. [1.24]
1.6.4.4. Geometrical representation
Figure 1.25. von Mises yield criterion in the Haigh–Westergaard stress space 30 Elastoplastic Modeling
In view of the discussion in section 1.6.2.2, it is clear that the initial elastic boundary defined by the von Mises criterion [1.19] is represented, in the Haigh– Westergaard stress space, by a cylindrical surface parallel to the ternary axis of symmetry with a circular cross-section in the Π-plane, as shown in Figure 1.25.
1.6.4.5. Further comments Although, as mentioned earlier, the von Mises criterion usually provides a better fitting with experimental results performed on metal specimens than the Tresca criterion, Hill (1948a) remarked that, in the case of “notably annealed steel,…when the specimen is traversed by Lüders’ bands” inducing a non-uniform deformation, “the maximum shear stress criterion is found to give a better approximation than that of von Mises”. This being due to the fact that “each Lüders’ band is a region where the deformation is a simple shear”.
1.6.5. Other yield criteria for metals
Besides these two “historical” criteria, more sophisticated forms have been proposed for the loading function f ()σ in order to reach better agreements with experimental results for various isotropic metals. Retaining as a well-established result that f ()σ must be independent of the hydrostatic stress, at least within the usual range of isotropic tensions or pressures, the concept of effective shear stress may be taken as a basis for writing such criteria. Indeed, as mentioned in section τ 1.6.4.2, the octahedral shear stress oct is the effective shear stress for the von Mises criterion and it is also obvious, from section 1.6.3.1, that the maximum shear stress τ max is the effective shear stress for the Tresca criterion.
Then, as another historical landmark, we may recall the criterion introduced by Drucker (1947) after the experiments carried out by Osgood (1949) on aluminum tubes, subjected to axial tension and internal pressure, where the effective shear stress, a function of the deviator s only, is defined as:
3 1 τ =−()JJ32()6 . [1.25] eff 22 3
Figure 1.26 presents a comparison of the three criteria we have just listed, in the σ = σ case of plane stress ( 3 0 ) where 0 , the elastic limit in uniaxial tension, is taken as the common experimental result so that they can be written, respectively, as: Elastic Domains: Yield Conditions 31
τσσσ=−=≤ Tresca: eff Max{ i j ij , 1,2,3} 0 1 τσσσσ=+−22 2 ≤ σ [1.26] von Mises: eff ( 1 2 1 2 ) 230 23 1 27(σσσσ22+− ) 3 6 1 1212 σ 1 τ =≤0 6 Drucker: eff 1 (26) . 33−−(2σσ )222 (2 σσ − ) ( σσ + ) 4 12 21 12
Figure 1.26. Comparing Tresca, Mises and Drucker criteria: (a) in plane stress (Drucker 1962); (b) cross-sections in the Π-plane
1.6.6. Yield criteria for anisotropic materials
1.6.6.1. Hill’s criterion The forming processes imposed on metal samples (such as rolling) may result in anisotropic behavior of the concerned material. As an example, Kelly (2019) reports tensile yield stress in the direction of rolling being typically 15% greater than in the transverse direction. In such a case, as explained in section 1.5.1, the yield function can no longer be written as a function of tensor σ only and must involve additional parameters to characterize material anisotropy. This is the case for instance when the material is orthotropic, which means that its behavior admits three mutually orthogonal symmetry planes. These planes define an orthogonal orientation basis R attached to the material element, whose spatial orientation angles are the additional parameters to be inserted in the yield function arguments written as
32 Elastoplastic Modeling
σσ= σ σ ff(,R )R (), where denotes the matrix of the components of in the basis R .
σσ= Retaining the experimental result that ff(,R )R () is independent of the hydrostatic stress, Hill (1948a, 1950) proposed a yield function for an orthotropic metal written in the form:
σσσσσσσ =⇔ −22 + − + − 2 fFGHR () 0 (22 33 ) ( 33 11 ) ( 11 22 ) [1.27] ++σσσ222 + −= 2LMN23 2 31 2 12 1 0, an expression that comes as a generalization of Mises’ criterion. Material constants F,GHLM , , , and N are determined after three uniaxial tension tests and three shear tests have been carried out24, yielding the elastic limits σσ στ τ τ (),(),(),(),()01 02 03 012 023 and ()031. They can be written as:
11 1 1 11 FL=+−=() σσσ222 τ 2 22()02 () 03 () 01 () 023 11 1 1 11 GM=+−=() [1.28] σσσ222 τ 2 22()03 () 01 () 02 () 031 11 1 1 11 HN=+−=(). σσσ222 τ 2 22()01 () 02 () 03 () 012
1.6.6.2. Yield criteria for anisotropic purely cohesive soils In soil mechanics, for classical stability analyses and bearing capacity problems, isotropic undrained saturated clay is usually modeled as a purely cohesive soil whose yield criterion, independent of the hydrostatic stress25, is written with a Tresca loading function:
σσσ=−−= fC() Max{ i ju 2i ,j 1,2,3} . [1.29]
The material constant Cu is the undrained cohesion of the material, which can be determined through what is known in soil mechanics as a classical “triaxial test” for instance. In such a test, a circular cylindrical sample is subjected to a confining pressure and an additional axial load, so that Cu can be estimated from the average value of the total normal stress along the sample axis and the confining stress taken
24 See Kelly (2019). 25 As an exception we may mention Boehler and Sawczuck (1977). Elastic Domains: Yield Conditions 33
σ σ σσ<<26 as the principal stresses, respectively, III and I , with III I 0 . Although most geotechnical materials prove to be slightly or strongly anisotropic (Hueckel and Nova 1983), it seems that only purely cohesive soil anisotropy has been taken in consideration in stability or bearing capacity analyses.
Classical triaxial tests have been carried out on various anisotropic purely cohesive soils after samples had been drilled out of the ground at various inclinations α about the vertical axis Oz (e.g. Duncan and Seed 1966; Lo and Milligan 1967)27. Because of the geotectonical origin of the anisotropy, soil materials turn out to be transversally isotropic about Oz . The value of the cohesion α ασσ=− determined for a given inclination is denoted by C() (IIII )2, with the = π = classical notations CC(0) v and CC(2) h . Results obtained that way are α presented in Figure 1.27 in the form of a polar diagram, where CC() v , which σσ= α 28 proves to be independent of III, is drawn as a function of .
Figure 1.27. Triaxial tests performed on anisotropic purely cohesive soils: Δ (Duncan and Seed 1966); □ (Bishop 1966); ○ (Lo and Milligan 1967). Experimental results and Bishop’s formula fitting
26 In soil mechanics, compressive normal stresses are most often counted positive but, for the notation consistency of this book, we stick to the continuum mechanics sign convention: tensile stresses are positive. 27 An important distinction was introduced in Casagrande and Carrillo (1944) between inherent and induced anisotropy, which points out that great care must be taken in terms of sampling and testing procedures. 28 See Tristán-López (1981) and Salençon (1984). 34 Elastoplastic Modeling
Other testing procedures are also referred to, among which the sophisticated triaxial test on thin-walled tubular samples (Broms and Casbarian 1965; Saada and Zamani 1969), where the stress field in the sample is governed by the inside and outside pressures, the axial load and the torque applied to the specimen (Figure 1.28). It operates on samples that are drilled out vertically and the variation of the acting stress tensor inclination about the vertical axis of the sample is obtained by varying the loading parameters (inside and outside pressures, axial load and torque). Two types of anisotropy can be identified in Figures 1.27 and 1.28: – polar diagrams with an “elliptic pattern” for “normally consolidated” soils; – polar diagrams that demonstrate a minimum cohesion for απ≅ 4 , significantly lower than Ch and Cv , for “highly over-consolidated” soils. It may be worth noting that, as established by de Buhan (1983) and reported in Salençon (1984), this same type of diagram, with a minimum cohesion for απ≅ 4 , is obtained as the homogenized yield criterion of a reinforced purely cohesive soil.
Figure 1.28. Thin-walled tubular testing sample. Experimental results (Broms and Casbarian 1965) and Bishop’s formula fitting
A first formula for describing C (α ) was proposed in Casagrande and Carrillo (1944) as:
ααα=+22 CC( )vh cos C sin , [1.30] Elastic Domains: Yield Conditions 35 which only accounts for the first type of anisotropy. Based upon his own triaxial test results on a London clay, Bishop (1966) proposed a more sophisticated expression:
C(4)π CC(αααα )=+ ( cos222 C sin )(cos 2 + 2 sin 2 2 α ) , [1.31] vh + CCvh
π =+ which reduces to [1.30] when CCC(4)(vh )2) and proves to fit many experimental data correctly, as shown in Figures 1.27 and 1.28.
On the basis of Bishop’s formula, a three-dimensional yield criterion for anisotropic purely cohesive soils was proposed in Tristán-López (1981), under the complementary assumption that it is independent of the intermediate principal stress σ II , in the form:
ασ= (,Oz III ) C(4)π [1.32] fCC(σσσ )=− ( ) 2 − ( cos222 α + sin α )(cos 2 α + 2 sin 2 2α ). III I v h + CCvh
Depending on the values of the anisotropy parameters CChv and π + CCC(4)(vh ), the yield boundaries corresponding to [1.32] may happen not to be convex.
1.6.7. Yield criteria for granular materials
1.6.7.1. The Coulomb yield criterion Coulomb’s (1773) yield criterion originates from the celebrated memoir presented by Coulomb in 1773 to the French Academy of Sciences29 where he exposed stability analyses of various structures in consideration of the influence of friction and cohesion (frottement et cohésion). Coulomb first wisely acknowledged that, contrary to gravity which is always acting, friction and cohesion are not active forces but coercive ones that can only be defined by the limits set to their resistance. He then referred to the previous work by Amontons (1699) stating that the frictional resisting force is proportional to the normal pressure exerted on the considered slip
29 “Le frottement et la cohésion ne sont point des forces actives comme la gravité, qui exerce toujours son effet en entier, mais seulement des forces coercitives ; l’on estime ces deux forces par les limites de leur résistance.” (“Friction and cohesion are not active forces such as gravity that always fully exerts its effect but only coercive forces; those two forces are defined through their limits of resistance”). 36 Elastoplastic Modeling plane, to which he added, for a complete analysis of the resistance of a masonry pillar subjected to a compressive external force, the cohesion of matter, which is independent of the exerted normal pressure.
It follows that the physical significance of Coulomb’s yield criterion, almost one century earlier than Tresca’s experiments, is quite similar to the statement we gave in section 1.6.3.1 for Tresca’s criterion. With the notations shown in Figure 1.20, where τ denotes the shear stress vector on a facet and σ is the normal stress counted positive in tension, the Coulomb yield criterion will now be written as:
fnC()στσ≤⇔∀ 0 , ≤− tanφ , [1.33] which implies that
∀≤nC, σ tanφ = H. [1.34]
This criterion is isotropic since C and φ are independent of the orientation of n . C denotes the cohesion of matter, while φ is the friction angle that defines the friction coefficient as tanφ (in Coulomb’s work the friction coefficient was denoted by 1 n ). Making φ = 0 , we retrieve Tresca’s criterion.
Looking for the expression of the criterion in terms of principal stresses, it is convenient to refer to the Mohr representation of a stress state (Figure 1.22). In Figure 1.29, where the stress vector T acting on a facet with normal n is represented by OT , we see that Coulomb’s yield criterion means that the radius σσ− 30 σσ+ ()2I III of the Mohr circle with center at point ()2IIII must not exceed ()σσ+ C cosφφ− IIII sin . Hence, in terms of principal stresses: 2
σσσσσ≤⇔ − + +φφ − ≤ fC( ) 0 (IIIIIIII ) ( )sin 2 cos 0 [1.35] or, in a symmetric form,
σσφσφφ=+−−−=≤ fC( ) Max{ ij (1 sin ) (1 sin ) 2 cosi ,j 1,2,3} 0 . [1.36]
σσ 30 The Mohr circle with diameter IIII is calledthe Mohr circle. Elastic Domains: Yield Conditions 37
Figure 1.29. Mohr representation of Coulomb’s and Tresca’s yield criteria
1.6.7.2. Geometrical representation In the Haigh–Westergaard stress space, the yield boundary defined by [1.36] is represented by a pyramidal surface with summit on the ternary axis of symmetry and =++=−σσσ a hexagonal cross-section in the Π-plane I11233P , as shown in Figure 1.3031. It clearly evidences the fact that Coulomb’s yield criterion depends on σ =− the hydrostatic stress m P .
Figure 1.30. Coulomb’s yield criterion in the Haigh–Westergaard stress space
σ =− 31 This plane is also called the deviatoric plane corresponding to m P . 38 Elastoplastic Modeling
1.6.7.3. Intrinsic curve yield criteria The Tresca and Coulomb criteria may be considered as landmark examples of the intrinsic curve yield criteria that were introduced by Mohr (1900), as recalled in Labuz and Zang (2012). This type of criterion, sometimes also credited to Caquot (1933), appears in the lecture notes of a course delivered at the French École nationale des ponts et chaussées, as shown in Figure 1.31.
Figure 1.31. The intrinsic curve as it appears in Caquot (1933) (tensile stresses are counted negative)
The concept of an intrinsic curve originates from experimental results obtained with granular materials and other materials commonly used in civil engineering, which indicate that those Mohr circles that correspond to yielding admit an envelope in the (στ , ) plane of the Mohr representation. This envelope is called the intrinsic curve. It is obviously symmetric with respect to the σ -axis . Assuming this result to be valid for all types of experiments performed on such a material means that, from a mathematical viewpoint, its yield criterion is governed by the values of the σ σ extremal principal stresses I and III only, independently of the intermediate σ σ σ principal stress II . Hence, it can be written as a relationship between I and III σσ− σσ+ or between ()2I III and ()2I III , respectively, the radius of the Mohr circle and the abscissa of its center, in the form:
σσ−+ σσ fg()σ ≤⇔ 0 I III − ( I III ) ≤ 0, [1.37] 22 which is the cornerstone of the intrinsic curve concept. Elastic Domains: Yield Conditions 39
From experimental results, it comes out that the intrinsic curve can usually be considered as unbounded when the material is subjected to an increasing compressive normal stress (0,)σσ< . Letting aside the specific case of a Tresca material, the intrinsic curve is bounded on the positive side of the σ -axis (tensile normal stress).
It follows that function g is a positive decreasing and concave32 function of its argument. In addition, it must be noted that the intrinsic curve being a real non- degenerated curve implies that g′ < 1 .
Experimental determination of the intrinsic curve usually proceeds from classical triaxial tests, as described in section 1.6.6.2, where the considered imposed stress σσ=> σ state is such that IIIIII. It is clear that such experiments make it possible to define a real envelope for the limit Mohr circles for these types of experiments but cannot decide about the influence of the intermediate principal stress. To cope with this difficulty, other types of tests have been carried out on granular media such as the “inverse triaxial” test, or the “true triaxial” test, where the three principal stresses can be varied independently from one another (Gudehus 1973; Lade and Duncan 1973).
1.6.7.4. The Drucker–Prager yield criterion In Drucker and Prager (1952), a yield criterion was proposed for granular materials that may be described as “affiliated” with Coulomb’s criterion in the same way as von Mises’ with Tresca’s criterion. Instead of the hexagonal pyramid of Coulomb’s criterion, the yield boundary is now a cone with the same ternary symmetry axis and a circular cross-section (Figure 1.32). The equation for this criterion can be simply written as:
σα≤⇔ − − ≤ ≤ f () 0 JHIIH211 ( 3)0, 3 , [1.38] with H the theoretical yield limit in an isotropic tension test, and α a scalar constant, 032<<α . This constant may be conveniently expressed as:
α =+<<3 sinφφφ 3 sin2 , 0π 2 , [1.39]
32 Which implies convexity of the intrinsic curve in the (,)στ plane of the Mohr representation and convexity of the yield boundary in the Haigh–Westergaard stress space. A rather extensive mathematical discussion about the shape and properties of the intrinsic curve may be found in Hill (1950) and Halphen and Salençon (1987). 40 Elastoplastic Modeling hence:
φ σ ≤⇔ −3sin − ≤ ≤ f () 0 JH21 (IIH 3)0, 13 . [1.40] 3sin+ 2 φ
Figure 1.32. Drucker–Prager’s yield criterion in the Haigh–Westergaard stress space
Equations [1.39] and [1.40] may look strangely sophisticated but they make it easy to compare Coulomb’s and Drucker–Prager’s criteria in the Haigh– Westergaard stress space. For the same values of φ and H , both surfaces have the same apex and the circular cross-section of Drucker–Prager’s boundary is internal and tangent to Coulomb’s boundary.
Figure 1.33. Yield surfaces of dense and loose sands (Lade and Duncan 1973) Elastic Domains: Yield Conditions 41
Experimental results do not bring much support in favor of this criterion, as shown in Figure 1.33, which reports the results of experiments carried out by Lade and Duncan (1973) on dense and loose sands. The shapes of the cross-sections of the yield surfaces in the Π-plane look more like a hexagonal pattern than a circular one (Figure 1.33a). Comparing these results with the cross sections of a Coulomb hexagon for various values of “phi”, as in Figure 1.33b, shows that it does not actually match with a description by a Coulomb criterion. In fact, the principal advantage in making use of the Drucker–Prager yield criterion lies in its continuous differentiability (see Chapter 2, section 2.6.4).
1.7. Final comments
Only most popular and important yield criteria have been presented here, in what may appear as a historical storytelling but provides an overview of the issue, with a rather complete review of the constraints they are subjected to and their most frequently encountered properties. Regarding the particular case of anisotropic materials, which has only been briefly evoked here, a detailed analysis can be found in Wang (1970), Boehler (1978), Hill (1979) and Hueckel and Nova (1983), for instance.
More will be said about plastic yield criteria in Chapter 2, in connection with the plastic flow rule, since the two topics are not independent from each other. Convexity of the yield criterion will thus be revisited within the framework of a general principle, with the particular cases of such criteria as Tresca’s or Coulomb’s, which are not continuously differentiable, being given special attention.
It must also be said that, in fact, although the concept of a yield criterion has been introduced from the analysis of the elastoplastic behavior, some of the criteria presented in the preceding sections will quite often only appear as yield criteria without any explicit reference to plastic behavior. This is the case in particular when they are considered from the yield design theory viewpoint, where they “set the rules of the game” by limiting the domain of resistance of the constituent material of a system (see Chapter 6).