Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

Of course this simple type of damage model has limited use in practice~ but describes both the dissipation mechanisms and the irreversible rearrangements of the strucll.lre .. To refine the model more general (tensorial) forms are needed, especially to describe anisotropic damage.

In order to obtain the stress relation we differentiate first (6.274) with respect to time. Using the chain rule we find that

and therefore the second Piola-Kirchoff stress tensor S and the non-negative internal dissipation Vint. are

In constitutive equation (6.277) the quantity Su denotes the effective second Piola- Kirchhoff stress tensor. The dissipation inequality (6.278) clearly shows that damage is a dissipative process, the quantity f therein denotes the thermodynamic force which governs the damage evolution. In continuum -damage mechanics the thermodynamic quantity f has the meaning of the effective strain energy \JI0 released per unit reference volu·me.

see relation (6.274). Therefore, instead of controlling the damage process by the i.nterM nal variable ( we can equivalently use its conjugate quantity, i..e. the effective strain energy ·wo.

The evolution off is given by

recognizing that .f characterizes the effective stress power per unit reference volume

300 6 Hypere:Jastic Materials

according to (4J 13).

As noted above we assume that the .total damage accumulation is based on Mullins type (stress) softening of the ·material. For a strain-controlled cyclic loading process with a fixed strain amplitude this type of phenomenological damage occurs only within the first eye.le. We adopt the smooth function ( =-((a:) as the damage variable, with conditions ((0) = 0 and ({oo) E [O, l]. The .phenomenological variable a describes

the discontinuous damage. A constitutive particularization of the damage variable ( may, according to MIEHE [l 995a], .be given by

where ( 00 describes the dimensionless maxi-mum {possible) da.rnage and,, is referred to as the damage satu-ration parameter.

We now aim to determine the discontinuous damage variable a over the past history

up to the current time_, i.e. the history time interval To control a discontinuous drunage process we use the evolution of the effective strain-energy function '110 . Within the closed time interval (0, t] we take the phenomenological variable a to be related to the maximum value of lJl0 and write

mum the-rmodynamic force with the same dimension as the effective strain energy p.er unit reference volume. Definition (6.282) was employed by, for example~ DE SOUZA NETO et al. [1994] and by MIEHE [l 995a], while the rate-independent damage model of SIMO [1987] invoked the principle ofstrain-equivalence (see the chapter on damage mechanics in the book by LEMAITRE and CHABOCHE [1.990, Chapter 7]).

Other strain-based models for the finite strain domain are due to GOVINDJEE and SIMO

[1991, l992a, bJ, JOHNSON and BEATTY [1993a, b] among others.

Re.Jation (6.282) _generalizes the one-dimensional damage model (for small strains) proposed by GURTIN and FRANCIS [1981b]. It uses the hypothesis that the current state of damage is characterized by the maximum axial strain attained in the history of-defonnation. Here we do not recall the many important works in the area of small strain damage mechanics.

A straightforward refinement of the isotropic damage model may be obtained by including damage effects governed by continuous damage accumulation. In particular, the work of MIEHE [l 995~], which is exclusively presented in an Eulerian setting, describes continuous damage accumulation. lt is based on the arc-length of the effective strain-energy function.

A fairly simple and efficient energy-based damage model to describe the main features of the Mullins effect in filled rubber was proposed by OGDEN and ROXBURGH

II99.9a]. The material model is .composed of a (classical) strain-energy function describing the primary loading path from the unstressed virgin .state and an additive dam.- age function responsible for the unloading path which .is initiated from any point on the primary loading path. The formulation is based on the concept of

in which the material is treated as one elastic material in loading and another elastic material .in unloading. This idea was used by, for example, FUNG et al. [1979] within the context of modelling arterial walls. It has the significant advantage of convenient and simple description of the stress-strain relationships in cyclic loading and their numerical (finite element) realization.

Damage criterion and damage evolution. We now define a damage criterion in the strain space at any time of the loading process in the form

with the damage function </J. For <P < 0 no evolution of damage occurs. The second possible situation is </> = 0 which characterizes the so-called damage surface with normal

For a fixed a, the damage surface delimits the strain space in which the behavior of the material is considered to be fully elastic (.no damage accumulation occurs). A representation of the damage surface with .the associated normal in the principal strain space is shown in Figure 6.8.

.a= 1, 2, 3

.. ········.··-.... Damage surface

c/J(Ca,.a) = 0

Figure 6.·s Illustration of the damage criterion in the principal strain space.

Hence, a double contraction of the two tensors N and Cgives, using (6.284h, the scalar N : C= 8f (C)/oC : C= j(C). Borrowing the terminology .from strain space

302 6 Hyperelastic Materials

plasticity {see NAGHDI and TRAPP [1975] and SIMO and HUGHES [1998, p. 84J), at

</> = 0 we must distinguish between

.

Advancement of damage only occurs for the cas·e of loading, and the .initial condition for n is zero. The evolution equation (6.286), which dearly shows the discontinuous property of this damage model, corresponds to those given in GURTIN and FRANCIS [198lb], S·tMO ["1987] and ·MIEHE [1995a].

Substituting (6288) into (6.287), we obtain, using (6.286), the ·evolution of the stress tensor in the farm

with the effective elasticity tensor C 0 in the material description. For an undamaged material, C0 is defined to be

(6.153") we find that the terms within the bracket -char- acterize the elasticity tensor C in the -material description. The second term in .the rate equation (6.289)i governs the damage that causes stress softening of the material.

Dam.age -model in decoupled materi.al descr.i_ption. To complete our consid- erations of finite strain elasticity with isotropic damage in the sense of decoupled vo"Jumetric-isochoric response (introduced due to a multiplicative split of the deforma-

Here, W\'ol is a strictly convex function (with the minimum at J = 1) which describes the volwnetric elastic response. The second function '\JI 0 iso denotes the isochoric cf...

:fective strain energy of the undamaged material, which describes the isoc/wric elastic response. Hence_, the damage phenomenon is assumed to affect -only the isochoric

.part of the deformation, as proposed by, for example, SlMO [I 987]. We require that \Jivol ( J) = 0 and WcHso(C) = 0 hold ifand only if J = l and C =I, respectively.

Consider the structure (6.291), the purely volumetric contribution to the stress and the elasticity tensor are pres·en.ted by eqs. (6,.89) and (6.166), respectively. The isoclwric contribution to the stress is, by analogy with (6.277), given by

The isochoric contribution to the elasticity tensor includes damage and is~ by anal-

ogy with (6289), given by

.

with the isochoric part C0 iso of the e.ffectiv.e elasticity tensor in the material description. For an undamaged material, C 0 iso is defined to be

.Explicit forms of (6.292.h and (6.294) are given by (6.90) and (6.168), respectively.

By analogy with eq. (6.278"h the thermodynamic force f has here the meaning of the isochoric effective strain energy W0 ;80 ( C) of the undamaged material. Within the decoupled framework of volumetric-isochoric response, eqs. (6.278)2 and (6.280)2 take on the forms

The applicability of the constitutive damage model thus described is limited to sufficiently slow ·processes (viscous effects are .not considered). However, the damage model may easily be combined with the viscoelastic model as proposed in the last section. A suitable decoupled free energy for characterizing finite-strain viscoelastic damage mechanisms might be given, with reference to (6.236) and (6.291), as w(C, (, ri, ... ,I'm) = W~i(J)+(l-()[WOfso{C)+L::=l Too(C, r n)]. Here, \llOiso(C) and L~~1 denote, respectively, the strain energy and the configurational free energy (per unit reference volume) for the hyperelastic undamaged material. Both functions are associated with the isochoric response.

EXERCISES

1.Pure shear with isotropic damage. Consider a thin sheet of (incompressible) hy- perelastic material which is subjected to a homogeneous pure shear deformation

with the kinematic relation ,\1 = A, A = = 1/;\ (compare with Exer~

2

cise 1(b) on p. 227). The stress state of this mode of deformation is characterized by a2 and a3 = 0 (recall eqs. (6.76), (6.77)). The material is supposed to undergo stress softening of Mullins type.

(a)Based on the isotropic damage model introduced, compute the loading path up to A. = 2 followed by the unloading path back to A = 1.. Apply

the constitutive particularization of the phenomenological damage variable

(=({a), as given in eq. (6.281), and use the strain-energy function of the Mooney-Rivlin form, Le. W = cr(I1-- 3) + c2(/2 - 3), with the ratio of the material constants c1/ c2 = 7.

(b) Plot the loading and unloading path in the form of the two functions £T1 =

2. Equibif1xial deformation with isotropic damage. Consider an equibiaxial deformation of an incompressible material, which may be modeled by the strain

energy due to Ogden. Recall the stress strain relation derived in Example 6.6 on p. 239, i.e. a = z:,:=1 µp(,.\'!11t - ,.\-20J• ). During a load cycle,,\= 1 -->- 3 -+ 1 -t

5 --+ 1 .the material accumulates damage of Mullins type.

Plot the load cycle a= a(A.) with the typical values of the constants O'p, µ.P, p =

1,,. .. , 3, for Ogden's model given by (6.121). Assume the maximum damage ( 00 = 0.8 and the damage saturation _parameter .1. = 1.0 · .106 N/rn2 •

(a)

I

I

I

rI

'\

\

\.

''...........

........

........-.....__

.....

Tic point

(b)

Figure 7.1 Single po'lymerchain lying between two tic .points with various distances r. Two possible conformations are shown in (a). The number of possible conformations decreases with

.increasing end-to-end dislancc (b). Fully extended chain showing the limiting case wilh only one poss.ible conformation (c).

.Statistical concept. Based on several physical techniques we know that amorphous

polymers are composed of bundles of long-chain molecules (which may .be imagined

.as strings) having a high degree of ·flexibility. Figure 7.1 shows a model for a single polymer chain lyin.g .between successive points of cross-linkage, which we call tie points. The sing'le .polymer chain forms a typical segment in the coherent and threedimensional network of rubber. This model is due to GUTH and MARK [1935] and

KUHN fl 938, I946l The distance between the tie points of the chain molecule, denoted by r, we call subsequently end...to-end distance (or separation). The distance r is a parameter that characterizes a molecular conformation. The name conforma- tion comes from chemistry and refers to different shapes (arrangements) ·of .a chain molecule. The most powerful physical technique now available for determining .conformations of ·chain molecu'les is small-angle neutron scattering (for more details see, for example, SPERLING [1992, Section 5.2]).

308 7 Thermodynamics of Materials

Figure 7. l(a) shows a polymer chain with the end-to-end distance r 1 of the tie points, which are assum·ed to be fixed in .space. The distance r 1 is ·much smaller than the contour length L which is the length of the fully -extended chain. Consequently, the chain may take on an enormous range of possible conformations, two of which are shown in Figure 7.l(a). Obviously, the number of possible conformations decreases for a larger -end-to-end distance, and in the limit the number -of possible conformations diminishes to only one if the chain is in its most extended state, as illustrated in Figure 7.1 (c). Then the value of the end-to-end distance reaches its max.i.mum1 i.e. r 3 , and equals the contour length L, and the chain .is straight. The most crumpled conformation occurs when r tends to zero, and the tie points coincide. Clearly the end-to..-end

of a chain characterizes the molecular conformation.

The statistical theory of rubber elasticity (see, for example, the notable works by TRELOAR [1943a, b], JAMES and GUTH [1943~ 1949] and FLORY [1953, Section XT-

3, and references therein]), which is basically set up on these concepts makes use of the idea to express the number of conformations th.at a chain molecule can .assume as an 'en.tropic effect'. The chains occur in random.Jy coiled conformations in the unstretched state, as seen in Figure 7. l(a), and as the chains are -extended the number of conformations and the entropy decrease.

We proceed now to model this characteristic behavior and to analyze the conformations of a chain molecule. However, just for clarity, consider first the problem in one dimension and project the conformation on one coordinate axis, say the :I:i.-axis (the chain may be imagined as being constrained .artificially so that the tie points lie on the ~7-:1-axis). The conformations of the individual chains are distributed in a random manner. The _probability p(~z:-1 )dx 1 that the end-to-end distance of a chain ties in the interval between :i:1 and :r1 + d.ri1 is expressed by the Gaussian distribution function

where v(:c 1) is the probability density (per unit length) and b is a parameter of the model. This parameter is a measure of a representative length, as we see later in Example 7.1, p. 313. For an explicit derivation of the Gaussian distribution function (7 .. :I) for a chain in one dimension the interested reader is referred to the classical book by FLORY [195.3, Appendix A of Chapter Xl

The Gaussian function is a bell-shaped curve, which provides numerous applications in engineering practice and statistics. The most probable value of :ri, LeA the

maximum of the Gaussian function, may be found by differentiating eq. (7. l) with respect to ~l;1 and occurs at :r 1 = 0. The probability decreases monotonically as :z:i increases. Note that the entropy .may be interpreted as a quantitative measure of probability (microscopic random.ness and disorder) by using a fundamental finding which is

due to Boltzmann and Planck. This will be made clear in Section 7.2.