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Chapter 9
Modeling Material Failure
9.2 Stress and strain based fracture and fatigue criteria
Many of the most successful design procedures use simple, experimentally calibrated, functions of stress and strain to assess the likelihood of failure in a component.  Some examples of commonly used failure criteria are summarized in this section.
9.2.1 Stress based failure criteria for brittle solids and composites.
Experiments show that brittle solids (such as ceramics, glasses, and fiber-reinforced composites) tend to fail when the stress in the solid reaches a critical magnitude.  Materials such as ceramics and glasses can be idealized using an isotropic failure criterion.  Composite materials are stronger when loaded in some directions than others, and must be modeled using an anisotropic failure criterion.Â
The simplest brittle fracture criterion states that fracture is initiated when the greatest tensile principal stress in the solid reaches a critical magnitude,
(The subscript TS stands for tensile strength). To apply the criterion, you must 1. Measure (or look up) 2. Calculate the anticipated stress distribution in
your component or structure (e.g. using FEM).Â
Finally, you plot contours of principal stress, and find the maximum
value
More
sophisticated criteria must be used to model anisotropic materials
(especially composites). The criteria
must take account for the fact that the material is stronger in some directions
than others. For example, a fiber
reinforced composite is usually much stronger when loaded parallel to the
fiber direction than when loaded transverse to the fibers. There are many different ways to account
for this anisotropy
Orientation dependent fracture
strength. One approach is to make the tensile
strength of the solid orientation dependent.Â
For example, the tensile strength of a brittle, orthotropic solid
(with three distinct, mutually perpendicular characteristic material
directions) could be characterized by its tensile strengths
where
where
(i) Find the components of stress in the (ii) Maximize the function (iii) Check whether the maximum value of Â
Goldenblat-Kopnov failure criterion. A very general phenomenological failure criterion can be constructed by simply combining the stress components in a basis oriented with respect to material axes as polynomial function. The Goldenblat-Kopnov criterion is one example, which states that the critical stresses required to cause failure satisfy the equation
Here
A and B are material constants: A
is diagonal (
Tsai-Hill criterion:Â The
Tsai-Hill criterion is used to model damage in brittle laminated
fiber-reinforced composites and wood. A specimen of laminated composite
subjected to in-plane loading is sketched in the figure. The Tsai-Hill criterion assumes that a
plane stress state exists in the solid.Â
Let
at
failure, where 1. The laminate is loaded in uniaxial tension parallel
to the fibers. The material fails when 2. The laminate is loaded in uniaxial tension
perpendicular to the fibers. The
material fails when 3. In principle, the laminate could be loaded in shear
9.2.2 Probabilistic Design Methods for Brittle Fracture (Weibull Statistics)
The
fracture criterion
Weibull statistics refers to a technique used to predict the probability of failure in a brittle material. The following approach is used 1. Test a large number of samples with identical size and shape under uniform tensile stress, and determine their survival probability as a function of stress (survival probability is approximated by the fraction of specimens that survive a given stress level). 2. Fit the survival probability of these specimens
where 3. Given m,
To see this, note that the volume V can be
thought of as containing 4. More generally, the survival probability of a solid
subjected to an arbitrary stress distribution with principal values
where
This approach is quite successful in some applications: for example, it explains why brittle materials appear to be stronger in bending than in uniaxial tension. Like many statistical approaches it has some limitations as a design tool. The method can predict accurately the stress that gives 30% probability of failure. But who wants to buy a product that has a 30% probability of failure? For design applications we need to predict the probability of 1 failure in a million or so. It is very difficult to measure the tail of a statistical distribution accurately, and a distribution that was fit to predict 63% failure probability may be wildly inaccurate in the region of interest.
9.2.3 Static Fatigue Criterion for Brittle Materials
`Static fatigue’ refers to the progressive reduction in tensile strength of a stressed brittle material with time. The simplest way to model static fatigue is to make the tensile strength of the material a function of time and applied stress. The usual approach is to set
where
Since failure occurs when
so
that
Under multi-axial loading, the maximum principal tensile stress should
be used for
9.2.4 Constitutive laws for crushing failure of brittle materials
Brittle
materials are generally used in applications where they are subjected
primarily to compressive stress. Brittle materials are very strong in
compression, but they will fail if subjected to combined hydrostatic
compression and shear (e.g. by loading in uniaxial compression). Failure in compression is a consequence of distributed
microcracking in the solid
This type of crushing is often modeled using constitutive equations based on small-strain metal plasticity. The governing equations for a simple, small-strain, constitutive model of this form will be summarized briefly here. A more detailed discussion of plasticity theory is given in Section 3.6.
The material is characterized by the following properties:
The
constitutive equations specify a relationship between an increment in stress 1.
The strain is decomposed
into elastic and irreversible (damage) parts  as 2. The elastic part of the strain is related to the stress by the linear elastic constitutive equations
3. The critical stress that initiates crushing damage is given by a failure criterion (analogous to the yield criterion for a metal) of the form
where 4. The plastic strain components are determined using an associated flow rule
5. The magnitude of the plastic strain increment is related to the stress increment by
where
HEALTH WARNING: These constitutive equations should only be used in
regions where the hydrostatic stress is compressive
9.2.5Â Ductile Fracture Criteria
Strain to failure approach: Ductile fracture in tension occurs by the nucleation, growth and coalescence of voids in the material. A crude criterion for ductile failure could be based on the accumulated plastic strain, for example
at failure, where
Porous metal plasticity: Experiments show that the strain to cause ductile failure in a material depends on the hydrostatic component of tensile stress acting on the specimen, as shown in the figure. For example, the strain to failure under torsional loading (which subjects the material to shear with no hydrostatic stress) is much greater than under uniaxial tension. The critical strain is influenced by hydrostatic stress because ductile failure occurs as a result of the nucleation and growth of cavities in the solid. A hydrostatic stress greatly increases the rate of growth of the cavities. The simple strain-to-failure approach cannot account for this behavior.Porous metal
plasticity was developed to address this issue. The basic idea is simple: the solid is
idealized as a plastic matrix which contains a volume fraction
Both rate independent and viscoplastic versions of porous plasticity exist. The viscoplastic models have some advantages for finite element computations, because the rate dependence can stabilize the effects of strain softening. A simple small-strain viscoplastic constitutive law with power-law hardening and power-law rate dependence will be outlined here to illustrate the main features of these models. The constitutive law is known as the `Gurson model.’ Â
The material is characterized by the following properties:
The
constitutive equations specify a relationship between the stress 1.
The strain rate
is decomposed into elastic and plastic parts  as 2. The elastic part of the strain rate is related to the stress rate by the linear elastic constitutive equations
3. The magnitude of the plastic strain rate is determined by the following plastic flow potential
where 4. The components of the plastic strain rate tensor are computed from an associated flow law
5. Strain hardening in the matrix is modeled by
relating its flow stress
6. The effective plastic strain in the matrix is calculated from the condition that the plastic dissipation in the matrix must equal the rate of work done by stresses, which requires that
7. Finally, the model is completed by specifying the void volume fraction as a function of strain. The void volume fraction can increase due to growth of existing voids, or nucleation of new ones. To account for both effects, one can set
where the first term accounts for void growth, and the second accounts for strain controlled void nucleation. 9.2.6 Ductile failure by strain localization
If
you test a cylindrical specimen of a very ductile material in uniaxial
tension, it will initially deform uniformly, and remain cylindrical. At a
critical load (or strain) the specimen will start to neck, as shown in the
picture. Necking, once it starts, is
usually unstable
Neck formation is a consequence of geometric softening. A very simple model explains the concept of geometric softening. 1. Consider a cylindrical specimen with initial cross
sectional area 2. Assume that the material is perfectly plastic and
has a true stress-strain curve (Cauchy stress 3. The true strain in the specimen is related to its
length by 4. The force on the specimen is related to the Cauchy
stress by 5. At the point of maximum load 6. We can calculate
Notice
that 7. We can calculate
Notice
that 8. Finally, substituting the results of (6) and (7)
back into (5) and recalling thatÂ
9. Finally, note that by volume conservation the cross
sectional area is
It
turns out that the point of maximum load coincides with the condition for
unstable neck formation in the bar.Â
This is plausible
There are two important points to take away from this discussion.
Plastic localization can occur for many
reasons. There are two general classes
of localization
Examples of geometry induced localization are 1. Neck formation in a bar under uniaxial tension; 2. Shear band formation in torsional or shear loading at high strain rate due to thermal softening as a result of plastic heat generation
Examples of material induced localization are 1. Localization in a Gurson solid due to the softening effect of voids at large strains; 2. Localization in a single crystal due to the softening effect of lattice rotations; 3. Localization in a brittle microcracking material due to the increase in elastic compliance caused by the cracks.
Geometric
localization can be modeled quite easily, because it does not rely on any
empirical failure criteria. A
straightforward FEM computation, with an appropriate constitutive law and
proper consideration of finite strains, will predict localization if it is going
to occur
9.2.7 Criteria for failure by high cycle fatigue under constant amplitude cyclic loading
Empirical stress or strain based life
prediction methods are extensively used in design applications. The approach is straightforward
Here
we will review criteria that are used to predict fatigue life under
proportional cyclic loading. A typical stress cycle is parameterized by its
amplitude
For
tests run in the high cycle fatigue regime with any fixed value of mean
stress, the relationship between stress amplitude
where the exponent b is typically between 0.05 and 0.15. The constant C is a function of mean stress.
There
are two ways to account for the effects of mean stress. Both are based on the same idea: we know
that if the mean stress is equal to the tensile strength of the material
where
In practice, experimental data seem to lie between these two limits. Goodman’s rule gives a safe estimate.
These
criteria are intended to be used for components that are subjected to
uniaxial tensile stress. The criteria
can still be used if the loading is proportional
(i.e. with fixed directions of principal
stress). In this case, the maximum
principal stress should be used to calculate
9.2.8 Criteria for failure by low cycle fatigue
If a fatigue test is run with a high stress level (sufficient to cause plastic flow in a large section of the solid) the specimen fails very quickly (less than 10 000 cycles). This regime of behavior is known as `low cycle fatigue’. The fatigue life correlates best with the plastic strain amplitude rather than stress amplitude, and it is found that the Coffin Manson Law
gives a good fit to empirical data (the constants C and b do not have the same values as for Basquin’s law, of course)
9.2.9 Criteria for failure under variable amplitude cyclic loading
Fatigue tests are usually done at constant stress (or strain) amplitude. Service loading usually involves cycles with variable (and often random) amplitude. Fortunately, there’s a remarkably simple way to estimate fatigue life under variable loading using constant stress data.
Suppose
the load history is comprised of a set of
The Miner-Palmgren failure criterion assumes a linear summation of damage due to each set of load cycles, so that at failure
In terms of stress amplitude
The same approach works under low cycle fatigue conditions, in which case
The criterion is often used under random loading. To do so, we need
to find a way to estimate the number of cycles of load at a given stress
level. There are various ways to do
this
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(c) A.F. Bower, 2008 |