Elastic strain energy density decomposition in failure of ductile materials under combined torsion-tension
- Nikos P Andrianopoulos^{1}Email author and
- Vasileios M Manolopoulos^{1}
DOI: 10.1186/s40712-014-0016-5
© Andrianopoulos et al.; licensee Springer 2014
Received: 31 March 2014
Accepted: 21 August 2014
Published: 9 October 2014
Abstract
The constitutive behavior and failure of ductile materials are described in the present work for a general case of loading in terms of the secant moduli, which depend on the first (dilatational) and second (deviatoric) strain invariants. This approach exposes the distinct behavior of materials to the equivalent normal and shear stresses. The secant moduli enable the establishment of two (instead of one) constitutive equations necessary for the complete description of these materials. Emphasis is given in the accuracy of the resulting constitutive equations in terms of their predictions relative to actual experimental data for two materials systems. Failure predictions, according to T-criterion, are derived for two materials under combined torsion and tension, which are in good agreement with experimental data. Finally, the associated failure surfaces in a stress space are presented as well.
Keywords
Elastic strain energy density Nonlinear elastic deformations Combined stress paths Constitutive equations T-criterionBackground
The importance of strain energy density and in particular the distortional part of it relative to material failure was understood, conjectured and postulated, as early as in the era of Maxwell (1937) and others (Hencky 1924; Huber 1904). Strain energy, supplied to the material via some generalized loading, can be additively decomposed into two parts: the elastic and the inelastic one (for metals the plastic one); the elastic strain energy is recoverable upon removing the load (i.e., the corresponding strain at the unloaded state is zero) due to the fact that the materials remained elastic throughout its loading; the plastic strain energy is unrecoverable (and sometimes is called plastic work because it represents the energy lost to microstructural dissipative processes such as grain boundary slippage) that, when the load is removed, yields a permanent strain known as plastic strain.
Many failure criteria have been proposed and used to predict the initiation of macroscopic material failure under various loading conditions (Li 2001). In Christensen's failure theory (Christensen 2014), failure represents the termination of elastic behavior and not plastic behavior. In our failure theory, failure is defined as the loss of the ability of the material to store elastic strain energy. This may happen at the familiar yield point (elastic - perfectly plastic materials), or at maximum engineering loading point (hardening materials) or just at the moment of fracture (brittle materials). In this sense, the present definition of failure covers all the as above cases.
Phenomena are materialized in expenses of available resources. In case of failure of materials, the only available resource is elastic strain energy, stored into the material through elastic (linear or not) deformations. Elastic deformations exist from the very first load step until the last one. From this point of view, the amount of plastic work has no causative role in failure because it is the outcome of the action of available strain energy. Plastic work is a result of this action and cannot be the cause of other processes, including failure. It is a type of specific failure per se. This specific failure may be complete (plastic collapse), driving to the final loss of the ability of the material to store elastic strain energy when the material is elastic - perfectly plastic, implying saturation of the Bauschinger loop. Otherwise, in the so-called hardening materials (where both elastic and plastic strains coexist), the material may fail by either brittle or ductile fracture as it will be discussed in the sequel.
It is well known (Bao and Wierzbicki 2004) that plastic strains can depend on pressure. There are loading conditions (such as torsion) where in the absence of pressure plastic strains are dominant, or they are negligible when hydrostatic pressure prevails. Failure criteria based exclusively on stresses cannot capture the differentiation between elastic and plastic strains since there is no ‘elastic’ or ‘plastic’ differentiation for the stresses.
Concerning nonlinear elastic deformations, it is true that they, usually, can be neglected in engineering applications for practical reasons. However, all materials (especially brittle) show a degree of nonlinearity deserving consideration. Nevertheless, the strongest reason to incorporate nonlinear elastic deformations is the hardening behavior of materials, where both elastic and plastic strains coexist. In the hardening area, the nonlinearity of elastic deformations is certain.
A classical criterion marking pressure dependence is the Coulomb (Heyman 1997) criterion, which after Mohr (1914) states that a material fails when ‘a proper combination of shear and normal stresses is realized.’ This heuristic statement is logically perfect but lacks a quantitative description as far as it is based on experimental data interpolation and/or arbitrary assumptions for the shape of the failure envelope. In addition, it exhibits problematic behavior for tensile stresses (e.g., tension cut-off). Many failure criteria of this type exist in the open literature (Paul 1968; Stassi 1967; Schajer 1998; Bigoni and Piccolroaz 2004; Mahendra and Bhawani 2012; Drucker and Prager 1952), the simplest being the original Mohr-Coulomb criterion τ + a ⋅ σ = c.
- (a)
They are based on formulations involving a single function comprising the sum of two parts, one involving an expression of the shear stresses and the one involving the contribution of the normal stresses.
- (b)
They are based on stresses only and, especially, on those lying on the ‘critical’ plane of extreme stresses (σ_{1}, σ_{3}). It is difficult to accept that the intermediate stress σ_{2} plays no role, even in the critical plane.
An interesting exception is the criterion that was developed by Christensen (1997, 2004), where the dilatational and deviatoric (distortional) parts of the elastic stress tensor are introduced. This approach allows for a direct association of the decomposed parts of the stress tensor with the respective geometric changes activated during loading, i.e., volume (lengths) changes and shape (angles) changes, which are the only possible geometric changes in a deformable solid.
A criterion based on elastic strain energy density in case of linear elasticity was the first version of the so-called T-criterion which has been proved adequate to predict failure conditions for pre-cracked (Theocaris and Andrianopoulos 1982a, b) or uncracked geometries (Andrianopoulos 1993; Andrianopoulos and Boulougouris 1994). It was based on the von Mises's (1913) criterion and an addendum giving an answer to the question: ‘What happens with the, not covered by von Mises, dilatation of the material?’ However, this early version of the T-criterion was unable to give an answer to the phenomenon of ‘pressure dependence’ of failure.
In order to cover pressure dependence and nonlinearity of elastic strains, a generalization of the T-criterion was recently introduced (Andrianopoulos and Boulougouris 2004; Andrianopoulos et al. 2007, 2008; Andrianopoulos and Manolopoulos 2012). The general case of an isotropic material showing nonlinear elastic behavior was considered, and the total elastic strain energy density T was anticipated as the characteristic quantity for the respective conservative field. This quantity T is path independent and so, a relationship between dilatational T_{V} and distortional part T_{D} of T is obtained (Andrianopoulos and Manolopoulos 2010). This approach proved to be quite successful in predicting the failure behavior of metals under high levels of pressure (Andrianopoulos and Manolopoulos 2012) where for the first time - according to our best knowledge - the classical experiments of Bridgman (1952) were theoretically justified.
The goal of this paper is to give a different view of what a constitutive equation is and to emphasize the idea that the prediction of failure of a material is mainly equivalent to the problem of deriving the proper constitutive equations. Also, the thoughts, the assumptions and the considerations that made the writers to generalize T-criterion are presented in detail. The new generalized form of T-criterion, which is used in this work, was thoroughly explained and described in (Andrianopoulos and Manolopoulos 2012). Now the application of T-criterion is examined for combined loading paths of two ductile materials. Two sets of data with various combined tension-torsion loading paths are studied (Ali and Hashmi 1999; Marin 1948). The aim is to test T-criterion's forecast for these loading paths.
Methods
Constitutive equations
If one assumes that a constitutive relation of the form p = p(Θ) exists, then the area under the curve representing this constitutive relation represents the dilatational strain energy density stored into the material.
- 1.
By definition, strains in Figure 1a must be elastic as dictated by von Mises criterion (von Mises 1913). The stress-strain relationship is linear up to point A and generally nonlinear afterwards. Direct experimental data for p = p(Θ) curves are not available as triaxial tension is not easily accomplished. Usually, uniaxial tension data are used and an experimental or ‘graphical’ unloading is performed in order to find the elastic part of strain, i.e., the strain that can still be attributed to elastic response. Another option is from a graphical unloading of the curve p = p(Θ). These two alternatives are described thoroughly in Appendix 1.
- 2.
Plastic strains appear in Figure 1b after the end of both linear (point A) and nonlinear (point L) elasticity, although elastic strains continue developing up to the end (point H), in case of hardening materials.
- 3.
Both equations p = p(Θ) and σ_{eq} = σ_{eq}(ε_{eq}) are essential for the description of a material. They are considered as unique for each material and contain information about the energy balance and the type of failure. The traditional representation of materials through the second of them only (i.e., σ_{eq} = σ_{eq}(ε_{eq})) and von Mises failure are incomplete.
- 4.
In each curve of the above plots, a critical point can be located, marking the end of the development of elastic strains and, so, the collapse of the respective constitutive equation. The area of the graph under the loading part of the constitutive response line for any given level of strain on the horizontal access represents the value of the stored energy density. The elastic part of this energy density is represented by the area to the right of the unloading line when the stress returns to zero and the vertical from the strain reached for the terminal point on the curve. By definition, the terminal point in Figure 1a is the critical one when no plastic strains appear in this plot (see above remark 1). Consequently, the whole area from zero to terminal point in Figure 1a represents elastic strain energy density T_{V} stored for volume changes. Respectively, the unloading in Figure 1b from the terminal point H is a line (straight or not is unconcerned) ending at point \( \left({\varepsilon}_{\mathrm{eq}}^{\mathrm{pl}},0\right) \) whose abscissa represents the unrecoverable plastic strain. This line separates the whole area into available elastic strain energy density T_{D} and plastic work.
- 5.
The two branches of curve p = p(Θ) are not symmetric with respect to the origin of axes, especially in case of brittle materials. However, the area under each branch (T_{V}) is the same when the terminal point of each branch coincides with the critical one.
- 6.
The geometry of the specimen and the current stress state at any given point of the specimen affect the velocity of a ‘pointer’ running along these two curves. When normal stresses/strains prevail, the velocity of the pointer on p = p(Θ) is higher than that on σ_{eq} = σ_{eq}(ε_{eq}), the opposite being true when shear stresses/strains prevail. Then, according to the specimen geometry and the stress state, the pointer on one curve arrives at the respective critical point before its companion does on the other curve, marking the type of failure.
Equation (4) constitutes the necessary minimum set of constitutive equations for an isotropic material replacing Equations (1) and (2). Now, quantities T_{V} and T_{D} can be estimated as we are describing in the subsequent analysis. Referring to Figure 1a, the dashed green area in the first or third quadrant equals to dilatational strain energy density T_{V} and it is solely elastic. For the estimation of elastic distortional strain energy density T_{D}, the respective red dashed area in Figure 1b provides the necessary value while the remaining area under the curve represents plastic work. Depending on the type of failure, i.e., brittle fracture (cleavage) or plastic flow (slip), either T_{V} = T_{V,0} or T_{D} = T_{D,0} is satisfied, where T_{V,0}, T_{D,0} are the critical elastic strain energy densities.
Failure criterion
- (A)
Failure by cleavage (brittle fracture) occurs when T _{V} reaches a critical value T _{V,0}
- (B)
Failure by slip (plastic flow) occurs when T_{D}reaches a critical value T_{D,0}.
To quantify the failure behavior of an elastic material described by Equation (4), the evaluation of upper integration limits I_{1,0} and J_{2,0} for the quantities I_{1} and J_{2} appearing in Equation (5) is postulated.
Thus, Equations (5) and (6) constitute a system of three equations with four unknowns, K_{S}(I_{1}, J_{2}), G_{S}(I_{1}, J_{2}), I_{1,0}, and J_{2,0}. One of the functions K_{S} or G_{S} can be given experimentally, and subsequently the system can be solved for any prescribed loading path and the proposed criterion can be applied. Required constants, like (λ_{1}, λ_{2}) or \( \left({K}_{\mathrm{S}}^0,\ {G}_{\mathrm{S}}^0\right) \) can be obtained from Figure 1.
Results and discussion
Two sets of experimental data are used to verify the present theoretical predictions. The first one (Ali and Hashmi 1999) gives detailed description of the material (constitutive equations, type of failure, etc.) and permits a rigorous examination of its results. The second one is a classical series of data presented by Marin (1948).
Application to En8 steel
In this section we will be using the experimental results originally presented in (Ali and Hashmi 1999) for En8 (BS 970) steel. All the necessary information concerning material properties, loading paths, and failure conditions is given. The lack of hydrostatic tension or compression data is a barrier in defining the exact form of p = p(Θ) and the exact value of the critical dilatational strain energy density T_{V,0}. The required information was obtained under uniaxial tension conditions according to the procedure described in Appendix 1.
The experimental procedure, described in (Ali and Hashmi 1999) involved the application of combined torsion-tension loading under controlled conditions. Two types of loading paths were investigated:
The first loading path was torsion within the elastic range of the material and then axial tension beyond the uniaxial yield stress σ_{Y} (= 600 MPa) holding the initial angle of twist constant.
The second loading path was tension within the elastic range of the material and then torsion beyond the torsion yield stress τ_{Y} (= 350 MPa) holding the initial axial displacement constant.
In addition, distinct uniaxial tension and torsion experiments were performed to obtain the constitutive behavior of the material.
Constitutive equations
Properties of materials
Material | \( {\boldsymbol{K}}_{\mathbf{S}}^{\mathbf{0}} \)(GPa) | \( {\boldsymbol{G}}_{\mathbf{S}}^{\mathbf{0}} \)(GPa) | T_{V,0} (MPa) | T_{D,0}(MPa) | λ _{1} | λ _{2} | σ_{Y} (MPa) | τ_{Y}(MPa) | ν |
---|---|---|---|---|---|---|---|---|---|
Steel En8 | 193.7 | 80.4 | 0.16 | 1.24 | 0.0015 | 0.0020 | 600 | 350 | 0.3 |
Al 24S-T | 75.15 | 29.1 | 0.26 | 1.96 | 0.0019 | 0.0025 | 340 | 196 | 0.33 |
The failure boundary
Results for combined loading paths
Combined tension-torsion data for En8 steel
Path | Experiment | Maximum values at failure | |||
---|---|---|---|---|---|
σ_{eq}(MPa) | ε _{eq} | p(MPa) | Θ | ||
OD | Tension | 693 | 0.0060 | 231 | 0.0024 |
OBF | 50% σ_{Y} ➔ torsion | 725 | 0.0137 | 121 | 0.0008 |
OCE | 75% σ_{Y} ➔ torsion | 735 | 0.0136 | 181 | 0.0021 |
OL | Torsion | 745 | 0.0140 | 0 | 0 |
ONJ | 50% τ_{Y} ➔ tension | 702 | 0.0135 | 171 | 0.0012 |
OMK | 75% τ_{Y} ➔ tension | 710 | 0.0138 | 111 | 0.0005 |
The respective failure points are shown in Figure 5, along with the loading paths from Table 1. In the same figure, solid, red (T_{D} = T_{D,0}), and green (T_{V} = T_{V,0}) lines represent the bounds of the failure surface.
It is clear that the failure points L, K, F, J, and E belong to the curve T_{D} = T_{D,0} (red one), and point D belongs to the curve T_{V} = T_{V,0} (green one). The predictions of the T-criterion are quite satisfactory.
Application to Alcoa aluminum 24S-T alloy
The necessary parameters and constants are shown in Tables 1 and 2. The constitutive equations (Equation (11)) for Alcoa 24S-T are qualitatively similar to those in Figure 4 for En8 steel, but the sensitivity of K_{S} on J_{2} and G_{S} on I_{1} is much stronger than that in the case of steel. However, a critical difference in the behavior of the two materials is that in the case of Alcoa 24S-T, the Poisson ratio varied in order to satisfy the continuation of volume expansion Θ. Here, Poisson ratio took the values ν = 0.33 in the linear elastic area, ν_{L} = 0.34 in the nonlinear elastic area, and ν_{H} = 0.36 in the hardening area.
It is clear that the failure points B, C, D, E, and F belong to the curve T_{D} = T_{D,0} (red one) while the failure point A (uniaxial tension) belongs to the curve T_{V} = T_{V,0} (green one). The predictions of the T-criterion are quite satisfactory. All required parameters and constants for both materials are summarized in Tables 1 and 2.
Conclusions
In the present work, a careful analysis based on experimental data from Ali and Hashmi (1999) and Marin (1948) is performed by applying the T-criterion. The underlined physical principle is that a material fails because it cannot store more elastic strain energy for the formation of either new plastic or elastic strains. Consequently, it fails by slip or cleavage, respectively.
This binary alternative (slip-cleavage) necessitates the introduction of two, instead of one, constitutive equations for the complete description of materials; the first dealing with normal stresses/strains and the second one dealing with shear stresses/strains. In turn, two critical quantities are required, namely, T_{D,0} and T_{V,0}.
Even a couple of constitutive equations that predict accurately the experimental data, cannot guarantee acceptable predictions from any criterion as far as a clear distinction between ‘elastic energy’ and ‘plastic work’ is not considered. Plastic (or total) strains are results of loading and their magnitude depends on loading path (see Table 1 and for example, Bao and Wierzbicki (2004)). Proper functions of stresses and elastic strains separated into volume-shape changes give constant limits for failure.
Notes
Declarations
Authors’ Affiliations
References
- Ali, ARM, & Hashmi, MSJ. (1999). Theoretical and experimental results of the elastic-plastic response of a circular rod subjected to non-proportional combined torque and tension loadings. ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 1989-1996 (vols 203-210), 213(3), 251–261.Google Scholar
- Andrianopoulos, NP. (1993). Metalforming limit diagrams according to the T-criterion. Journal of Materials Processing and Technology, 39, 213–226.View ArticleGoogle Scholar
- Andrianopoulos, NP, & Boulougouris, VC. (1994). Failure by fracture or yielding in strain hardening materials according to the T-criterion. Engineering Fracture Mechanics, 47, 639–651.View ArticleGoogle Scholar
- Andrianopoulos, NP, & Boulougouris, VC. (2004). Loading path dependence of failure surfaces. Chania: 7th International Congress on Mechanics, HSTAM 2004.Google Scholar
- Andrianopoulos, NP, & Manolopoulos, VM. (2010). On a theoretical generalization of Coulomb criterion. Xanthi: First Greek-Ukrainian Conference on Fracture Mechanics of Materials and Structures.Google Scholar
- Andrianopoulos, NP, & Manolopoulos, VM. (2012). Can Coulomb criterion be generalized in case of ductile materials? An application to Bridgman experiments. International Journal of Mechanical Sciences, 54, 241–248.View ArticleGoogle Scholar
- Andrianopoulos, NP, Manolopoulos, VM, & Boulougouris, VC. (2007). Considerations on failure surfaces in brittle, non linear elastic materials. Patras: 8th Int. Congress on Mechanics, HSTAM 2007.Google Scholar
- Andrianopoulos, NP, Manolopoulos, VM, & Dernikas, IT. (2008). Coulomb failure surfaces in ductile nonlinear elastic materials. Patras: 1st International Conference of Engineering Against Fracture.Google Scholar
- Bao, Y, & Wierzbicki, T. (2004). On fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences, 46, 81–98.View ArticleGoogle Scholar
- Bigoni, D, & Piccolroaz, A. (2004). Yield criteria for quasibrittle and frictional materials. International journal of Sol and Structure, 41, 2855–2878.MATHMathSciNetView ArticleGoogle Scholar
- Bridgman, PW. (1952). Studies in large plastic flow and fracture (1st ed.). New York: McGraw-Hill.MATHGoogle Scholar
- Christensen, RM. (1997). Yield functions/failure criteria for isotropic materials. Proceedings of the Royal Society of London, 453, 1473–1491.MATHView ArticleGoogle Scholar
- Christensen, RM. (2004). A two-property yield, failure (fracture) criterion for homogeneous isotropic materials. Journal of Engineering Materials and Technology, 126, 45–52.View ArticleGoogle Scholar
- Christensen, RM. (2014). Failure mechanics - part I: the coordination between elasticity theory and failure theory for all isotropic materials. Journal of Applied Mechanics, 81(8), 1–7.View ArticleGoogle Scholar
- Drucker, DC, & Prager, W. (1952). Soil mechanics and plastic analysis for limit design. Quarterly of Applied Mathematics, 10, 157–165.MATHMathSciNetGoogle Scholar
- Hencky, H. (1924). Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Zeitschrift Angewandte Mathematics and Mechanics, 4, 323–334.View ArticleGoogle Scholar
- Heyman, J. (1997). Coulomb's memoir on statics, an essay in the history of civil engineering. London: Imperial College Press.View ArticleGoogle Scholar
- Huber, MT. (1904). Die spezifische formänderungsarbeit als mass der anstrengung eines materials. Czasopismo Techniczne, 22, 81–92.Google Scholar
- Li, QM. (2001). Strain energy density failure criterion. International Journal of Solids and Structures, 38, 6997–7013.MATHView ArticleGoogle Scholar
- Mahendra, S, & Bhawani, S. (2012). Modified Mohr-Coulomb criterion for non-linear triaxial and polyaxial strength of jointed rocks. International Journal of Rock Mechanics and Mining Science, 51, 43–52.View ArticleGoogle Scholar
- Manolopoulos, VM. (2009). Contribution to the study of failure criteria on non linear elastic materials, PhD Dissertation. In Greek: National Technical University of Athens.Google Scholar
- Marin, J. (1948). Stress-Strain relations in the plastic range for biaxial stresses (pp. 231–249). London: International Congress on Applied Mechanics.Google Scholar
- Maxwell, C. (1937). The origins of Clerk Maxwell's electric ideas, as described in familiar letters to W. Thomson. Mathematical Proceedings of the Cambridge Philosophical Society, 32, 695–748.View ArticleGoogle Scholar
- Mohr, O. (1914). Abhandlungen aus demGebiete der Technischen Mechanik (2nd ed.). Berlin: Ernst.Google Scholar
- Paul, B. (1968). In H Liebowitz (Ed.), Macroscopic criteria for plastic flow and brittle fracture in fracture (pp. 313–496). New York: Academic.Google Scholar
- Schajer, GS. (1998). Mohr-Coulomb failure criterion expressed in terms of stress invariants. Journal of Applied Mechanics, 65, 1066–1068.View ArticleGoogle Scholar
- Stassi, F. (1967). Flow and fracture of materials according to a new limiting condition of yielding. Meccanica, 3, 178–195.View ArticleGoogle Scholar
- Theocaris, PS, & Andrianopoulos, NP. (1982a). The Mises elastic-plastic boundary as the core region in fracture criteria. Engineering Fracture Mechanics, 16, 425–432.View ArticleGoogle Scholar
- Theocaris, PS, & Andrianopoulos, NP. (1982b). The T-criterion applied to ductile fracture. International Journal of Fracture, 20, R125–R130.View ArticleGoogle Scholar
- Theocaris, PS, & Koroneos, E. (1963). Stress-strain and contraction ratio curves for polycrystalline steel. Philosophical Magazine, 8(95), 1871–1893.View ArticleGoogle Scholar
- von Mises, R. (1913). Die mechanik der festen körper in plastisch deformierten zustand. Nachrichten Göttingen Akademie Wissenschaftliche, Math-physik, 1, 582–592.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.