On the equivalence of stress- and strain-based failure criteria in Elastic Media *
Edward Zywicz
The University of California, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550, USA (Received 19 October 1996; revised and accepted 14 August 1998)
Abstract– Conditions for which strain-based and stress-based failure criteria are mathematically equivalent in elastic media are explored by expressing the criteria in the spectral eigenspace of the elasticity tensor. For scalar-valued quadratic criteria that are homogeneous functions of degree one, stress-based and strain-stress-based criteria are most likely to be mathematically equivalent when the criteria’s operators are linear combinations of the elasticity’s spectral eigenbasis tensors.
For the commonly employed fiber-direction strain-based and fiber-direction stress-based “composite” failure criteria, the equivalence conditions are explicitly examined for orthotropic and transversely isotropic material symmetry. The difference between the two criteria decreases as the degree of extensional anisotropy increases but, in general, is unbounded for arbitrary deformation states. Over a moderate, but restricted, range of loading conditions, the difference between the criteria is small for high-modulus fiber-reinforced uni-directional lamina, often less than the uncertainties in either the elastic coefficients or failure values, and the two criteria appear interchangeable.Elsevier, Paris
failure criteria / fiber direction
1. Introduction
The prediction of failure in rate-independent elastic media typically involves either a strain-based or a stress-based failure criterion. The particular basis employed depends upon the material system and the physical phenomena involved. Sometimes, “complementary” and apparently similar strain-based and stress-based failure criteria coexist. For example, Swanson and Trask (1989) showed that the fiber-direction strain criterion and the direction stress criterion (see e.g., Jones, 1975) are both highly effective at predicting fiber-direction failure in uni-fiber-directional fiber-reinforced graphite/epoxy (Gr/Ep) laminates. The differences between complementary criteria are not always apparent, even when the criteria are expressed in the same basis, and can result in large and fundamental differences when used to derive damage evolution relationships, e.g., when the damage rate is assumed normal to the failure/damage surface. On the other hand, complementary strain-based and stress-based criteria may not differ significantly over a range of loading conditions, as is the case in the high-modulus fiber reinforced system examined by Swanson and Trask, and may be employed interchangeably to predict failure or selectively to derive damage evolution laws. Thus, it is important to understand the exact mathematical conditions when complementary strain-based and stress-based failure criteria are equivalent.
The differences between complementary failure criteria arise in part due to transformations induced from the constitutive relationship. Thus, as suggested by Schreyer and Zuo (1995) for the development of anisotropic yield surfaces, considerable insight may be gained by examining the directionality associated with the elasticity tensor. As shown by Sutcliffe (1992) and Elata and Rubin (1994), the positive definite elasticity tensor is expressible using alternative representations. Sutcliffe (1992) presents a spectral decomposition that involves
a summation of the eigenvalues and eigenvectors of the elasticity tensor, while Elata and Rubin (1994) show that the strain energy function is expressible in different directionally-dependent strain measures. Both works provide directional information regarding the stiffness tensor, and, as demonstrated by Christensen (1988), information contained in the elasticity tensor is indeed useful for developing and understanding damage criteria in composite materials.
This paper investigates the general mathematical conditions in which select complementary stress-based and strain-based failure criteria are equal for non-trivial deformation states. For “fiber-reinforced” orthotropic and transversely isotropic materials, it examines in detail the influence of the “fiber-direction” modulus upon the elastic spectral decomposition and complementary fiber-direction failure criteria.
The paper proceeds as follows. First, a family of restricted stress-based and strain-based failure criteria is expressed in the spectral eigenspace of the elasticity tensor. The equations are then specialized for the case of fiber-direction failure in materials with orthotropic and transversely isotropic symmetry. The equivalence conditions for this restricted characterization are extracted. Next, the characteristics of high-modulus fiber-reinforced material systems and their consequences on the differences between the stress-based and strain-based fiber-direction criteria are examined. Finally, an example is presented that quantitatively documents the numerical differences between the two complementary fiber-direction criteria in a common engineering composite.
2. Spectral representation of failure criteria
Consider a material whose response prior to failure is well represented by an elastic idealization. The fourth-order, positive definite, stiffness tensor, C, which characterizes the material’s elasticity is related to
the compliance tensorS in the usual manner, C=S−1. (Vectors and tensors are expressed with bold letters; fourth order tensors are denoted with bold calligraphic letters. The symbols : and ⊗ define inner and outer tensor products, respectively.) The constitutive relationship is given by
σ=C:ε, (1)
whereσ andεare the stress and strain tensors, respectively.
Sutcliffe (1992) showed that C can be spectrally decomposed in terms of its symmetric eigenvalues and eigenbasis tensors as
C= 6
X
i=1
λiOi. (2)
Hereλi andOi are thei-th eigenvalue and eigenbasis tensor, respectively. The eigenbasis tensor is assembled
from the set of mutually orthogonal unit eigenvectors ofC,Ni, such that
Oi =Ni⊗Ni. (3)
(Note,Nis actually a 3×3 tensor even though it is labeled here as an “eigenvector.”) Since, by definition, all
Oi are orthogonal to each other, it naturally follows that
S=
6
X
i=1
1
λi
Although the complete set of nine orthogonal eigenvectors contains six symmetric and three skew symmetric tensors, only the six symmetric ones are needed to define a symmetric positive definite elasticity tensor (Sutcliffe, 1992).
The set of symmetric stiffness eigenvectors span the entire space of symmetric 3×3 tensors and thus forms a complete basis. Because of this, ε, assumed symmetric, is expressible in terms of the stiffness tensor’s
eigenvectors as
whereαi represents the strain magnitude of thei-th mode.
Attention is now restricted to scalar-valued non-dimensional quadratic failure surfaces that are homogeneous functions of degree one. These surfaces use the rank-4 tensorsMandN to perform a weighted inner product onσ andε, respectively. The strain-based failure surface is given by
φstrain=ε : N : ε−1, (6)
while the stress-based failure surface is given by
φstress=σ : M: σ −1 (7)
or, in terms ofε,
φstress=ε : CMC : ε−1. (8)
The onset of failure is defined by the loci of points that satisfyφstrain=0 orφstress=0. Using the stiffness and
strain decompositions defined in Eqs (2), (3), and (5), along with the fact that the eigenvectors are mutually orthogonal, the two failure surfaces are expressible as
φstrain=
(no summation oniorj), the surfaces can be written simply as
φstrain=α : A: α−1 and φstress=α : B : α−1 (13)
When neitherAorBare rank deficient, the maximum difference between the two criteria can be found fromA
andBwith Rayleigh’s quotient (Strang, 1980).
Comparing Eqs (6) and (8), the two failure criteria are identical for arbitrary deformation states whenever
CMC=N, (14)
or, expressed in terms of the spectrally decomposed stiffness tensor, when
6
Clearly, Eqs (14) and (15) are most likely to be satisfied whenMandN are linear combinations ofOi.
3. Fiber-direction failure criteria for orthotropic and transversely isotropic media
The general equations are now specialized for the case of direction stress-based failure and fiber-direction strain-based failure in orthotropic materials. The primary goal is to identify the precise mathematical conditions in which these two commonly employed “one-dimensional” complementary criteria yield identical results for materials with orthotropic symmetry, i.e., exactly satisfy Eq. (15).
The stiffness tensorC is expressed in the material’s natural Cartesian coordinate system oriented such that
the “fiber” direction parallels the 1-axis. In compact notation,Ccontains the components
C=
When the material is transversely isotropic in the 2-3 plane, i.e., the plane perpendicular to the fiber,C13=C12,
C33=C22,C44=C66, andC55=(C22−C23)/2. The components ofC are related to the Young’s moduli (E11,
E22, and E33), Poisson’s ratios (ν12,ν23, and ν13), and shear moduli (µ12,µ23, and µ13) in the usual manner,
e.g., see Jones (1975).
The first three eigenvalues and eigenbasis tensors can be written in closed form. The symbolic expressions are lengthy and are omitted for brevity. However, the first three eigenbasis tensors are spanned by the basis matrices
but, in general, differ from Eq. (17).
The failure tensorsMand N, used in Eqs (6) and (8), for the direction strain criterion and the
fiber-direction stress criterion are defined by
M= 1
Hereεf denotes the fiber-direction strain at failure, whileσf defines the fiber-direction stress at failure. Since
both material properties are assumed to be measured from uniaxial tensile tests, Eq. (16) and the orthotropic idealization require thatσf =E11εf. With these specializations and becauseP is orthogonal to O4,O5,and O6, Eq. (15) can be given simply by
Fiber-direction failure criteria represent degenerate cases of the generic six-dimensional ellipsoidal failure surface. In strain (or stress) space, both failure surfaces define two parallel planes. One plane from each failure criteria passes through the point εf, where εf =εf(1,−ν12,−ν13,0,0,0)T, while the other plane
includes the point −εf. The normal vectors for the fiber-direction stress and strain surfaces are given by
nstress=(1, C12/C11, C13/C11,0,0,0)T and nstrain=(1,0,0,0,0,0)T, respectively. In general, because their
normals differ, the two complementary criteria define different surfaces.
or, alternatively, as
l=(ε22+νp12εf)C12/C11+(ε33+ν13εf)C13/C11 1+(C12/C11)2+(C13/C11)2
. (22)
When C12 and C13 are non-zero, the distance l between the two surfaces is unbounded for arbitrary ε22
and ε33. Only when ν12=ν13 =0, i.e., when C12=C13 =0 and λ1 =E11, are the surfaces coincidental.
Unfortunately, for many practical engineering materials and most uni-directional fiber reinforced composites, the complementary conditions are not identical. (Note, all of the above expressions hold for transverse isotropy as well.)
4. Characteristics of high-modulus “fiber-reinforced” material systems
High-modulus uni-directional fiber-reinforced materials commonly exhibit “one-dimensional” like charac-teristics in whichλ1≈E11 and O1≈P. This condition arises because typicallyν12 and ν13 are only≈1/3,
andE11, dominated by the fiber stiffness, is much greater than the matrix controlledE22andE33 moduli. This
characteristic is important since the differences between the fiber-direction failure criteria decrease asλ1→E11
andO1→P.
The migration to a “one-dimensional” like behavior and convergence of the fiber-direction criteria as the degree of extensional anisotropy increases is easily demonstrated for a transversely isotropic material. Hereλ1
is obtainable in closed form and is given by
λ1=
2E11
1+β− q
(1−β)2+8ν2 12
, (23)
whereβ =(1−ν23)E11/E22.Figure 1 shows the ratio of E11 and the eigenvalue λ1 as a function of β for
several values ofν12. Forν12=0.3, the ratio is in excess of 0.95 whenβ>4 or, alternatively,E11/E22>8 and
Table I.Elastic properties for a trans-versely isotropic AS4/3501-6 lamina.
E11GPa 144
Eq. (22) to definel, are expressible as
C12
C11
=C13 C11
=ν12/β. (24)
The same increases in β or decreases in ν12 that drive λ1 towards E11 simultaneously drive C12/C11 and,
consequentially, l toward zero. Clearly, as the fiber contribution dominates the extensional moduli, the one-dimensional like behavior arises and the differences between the fiber-direction criteria diminish.
5. Numerical example
The relevance and utility of the previous mathematical expressions can be best understood by examining a practical example. Consider the typical high-modulus Gr/Ep composite system AS4/3501-6. The transversely isotropic elastic properties, given by Kim et al. (1988), are listed intable I. The vector of eigenvalues for this system areλT =(148.6, 20.0, 6.35, 5.24, 3.17, 5.24)GPa and the first three eigenvectors, in compact notation,
are
NT1=(−0.9975,−0.0499,−0.0499,0,0,0),
NT2=(−0.0706,0.7053,0.7053,0,0,0),
NT3=(−1.3×10−18,−0.7071,0.7071,0,0,0).
The first eigenbasis tensor is
O1=
and the normalized difference between the two surfaces is given by
l
(This explains why the fiber-direction criteria yielded nearly identical agreement with the experimental data in Swanson and Trask (1989).) Given the inherent uncertainties in both the elastic coefficients and failure values, the error introduced by using one fiber-direction criterion instead of the other is insignificant in many practical engineering applications.
6. Summary and conclusion
In general, complementary stress-based and strain-based failure criteria do not satisfy Eq. (14) exactly, and thus do not yield mathematically identical results for non-trivial deformation states. However, complementary criteria are more likely to predict similar results when the failure surfaces are linear combinations of the stiffness’s eigenvectors.
For orthotropic and transversely isotropic materials that possess a high degree of extensional “anisotropy”, the differences between fiber-direction stress-based criterion and fiber-direction strain-based criterion diminish as the degree of extensional “anisotropy” increases. Specifically, for transversely isotropic uni-directional lamina manufactured from high modulus fibers, the difference between the complementary fiber-direction criteria decreases asE22/E11→0, and whenE11>10E22, the difference between the two criteria may be less
than the variation in the measured material properties. Under these conditions, it appears that fiber-direction criteria can be employed interchangeably over a range of loading conditions.
References
Christensen R.M., 1988. Tensor transformations and failure criteria for the analysis of fiber composite materials. J. Comput. Math. 22, 874–897. Elata D., Rubin M.B., 1994. Isotropy of strain energy functions which depend only on a finite number of directional strain measures. ASME J. Appl. Mech. 61, 284–289.
Jones R.M., 1975. Mechanics of Composite Materials. McGraw-Hill, New York, 71–83.
Kim R.Y., Abrams F., Knight M., 1988. Mechanical characterization of a thick composite laminate. In: Proc. Amer. Soc. for Comp., 3rd Tech. Conf., Technomic Publishing, Seattle, Washington, pp. 711–718.
Schreyer H.L., Zuo Q.H., 1995. Anisotropic yield surfaces based on elastic projection operators. ASME J. Appl. Mech. 62, 780–785. Strang G., 1980. Linear Algebra and Its Applications. Academic Press, New York, 266–273.
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