The influence of dispersity in geometric structure on the stability of cellular solids

Mechanics of Materials 39 (2007) 183–198 www.elsevier.com/locate/mechmat

The influence of dispersity in geometric structure on the stability of cellular solids M.W. Schraad

*

Fluid Dynamics Group, Theoretical Division, Los Alamos National Laboratory, Mail Stop B216, Los Alamos, NM 87545, United States Received 25 October 2005

Abstract Geometric structures in cellular solids span the spectrum from perfectly periodic to strictly random. Depending on the degree of disorder at the cellular-scale, the corresponding continuum-scale mechanical response can admit instabilities or can remain stable for the entire range of compressive deformations. In the present work, the response of cellular materials to quasi-static uni-axial compression is investigated. The underlying geometric structures in these materials are allowed to range from highly ordered to highly disordered, and the corresponding transition from unstable to stable mechanical response is explored. A stochastic constitutive model is developed and used for this purpose. Model development begins with an established cellular-scale mechanical response description, but this cellular-scale model is generalized to accommodate finite strain. A continuum-scale constitutive model is established by averaging the cellular-scale model over an ensemble of foam cells, and stochastic variation in cellular-scale geometric structure and material properties is considered through the use of probability density functions for the associated model parameters. Results show that dispersity in geometric structure has little to no effect on the initial elastic properties of the cellular materials under investigation. For deformations occurring prior to any occurrence of instability, however, increasing dispersity is accompanied by decreasing stiffness, an increase in critical strain, and a decrease in the extent of localized deformation. Most notably, materials with the highest degrees of dispersity in their cellular structures exhibit mechanical response that remains stable for the entire range of compressive deformations, demonstrating a general stabilizing effect of dispersity in geometric structure on the continuum-scale mechanical response of cellular solids.  2006 Elsevier Ltd. All rights reserved. Keywords: Microstructures; Finite strain; Foam material; Probability and statistics; Stability and bifurcation

1. Introduction Failure and the often-associated instabilities in materials with microstructure are manifested in *

Tel.: +1 505 665 3946; fax: +1 505 665 5926. E-mail address: [email protected]

many different ways. The occurrence of an instability can be the precursor to shear band formation, pore and crack nucleation, and ultimately material failure, so an understanding of the relationships among microstructure, continuum-scale mechanical response, and stability is essential in material and engineering component design. Model materials

0167-6636/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2006.04.004

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with periodic microstructures inevitably succumb to bifurcation-induced instabilities as symmetries in the underlying idealized geometric structures are broken and more energetically favorable microstructural configurations emerge through deformation. Wavelengths of the corresponding mode shapes can be commensurate with the dimensions of the unit cell, or they can be infinite in extent; corresponding to either local or global modes of failure, respectively. Despite being modeled as perfectly periodic, the microstructures of genuine materials necessarily include natural material defects and geometric imperfections. Therefore, the actual materials being modeled possess, at best, nearly periodic, highly ordered microstructures. Such materials typically exhibit limit loads in their mechanical response, beyond which localized deformation occurs. The limit loads in the response of the materials with nearly periodic microstructures coincide with the occurrence of global instabilities and correspond to bifurcations occurring in the principle equilibrium solutions of the associated idealized model materials. The utility of the periodic models, therefore, lies in the fact that they provide an analytically tractable means of identifying critical points corresponding to bifurcations and provide upper limits on the maximum loads attainable in the actual materials being modeled. For materials with highly disordered microstructures, microscopic bifurcations do not occur, as by definition, symmetries in the underlying geometric structures no longer exist. Such materials, however, still can exhibit global instabilities, maximum loads, and localized deformation in their mechanical response, but there are no corresponding idealized periodic model materials, and therefore, the limit loads have no logical association with bifurcations at the microstructural scale. For additional details, the interested reader is referred to the significant body of work on this subject contributed by Triantafyllidis and co-workers (see Triantafyllidis and Maker, 1985; Geymonat et al., 1993; Triantafyllidis and Schnaidt, 1993; Triantafyllidis and Bardenhagen, 1996; Schraad and Triantafyllidis, 1997; Triantafyllidis and Schraad, 1998). Geometric structures in cellular solids also span the spectrum from perfectly periodic to strictly random. Two-dimensional honeycombs possess highly ordered, nearly periodic cellular structures, and thus, failure in these materials typically manifests itself as localized deformation; for example, the col-

lapse of rows of cells. The processing methods used in the manufacture of three-dimensional cellular solids, however, typically produce microstructures that are highly disordered. If the dispersity in the underlying cellular geometry is small, the mechanical response of three-dimensional foams can be quite similar to that of two-dimensional honeycombs. For materials with a high degree of dispersity, however, the stress can become a monotonically increasing function of strain, and thus, no critical point is reached, and the mechanical response remains stable through all regimes of uni-axial compressive deformation. A primary objective of this investigation is to study the behavior of cellular materials ranging over the spectrum of geometric structures from highly ordered to highly disordered, and to explore the corresponding transition from unstable to stable mechanical response. Synthetic cellular solids first appeared in the late 1930s in the form of latex rubber foams. Instabilities in cellular solids received little attention from early researchers, however, as predicting the initial mechanical properties of these new and unique materials provided sufficient challenges. Gent and Thomas (1959) were perhaps the first researchers to study the failure of cellular materials, focusing on tearing mechanisms under tensile loading. Shaw and Sata (1966) studied the plastic behavior of cellular materials under compressive loading and identified the onset of localized deformation, but failed to relate this failure mechanism to any instability mode. Few systematic investigations related to failure and the crucial role played by instabilities in cellular solids followed for the next two decades. Gibson and Ashby (1988) refocused the attention of the research community on this class of materials with a comprehensive study of the structure and related properties of both natural and synthetic cellular materials. Since the publication of this monograph, topics related to the failure of cellular solids and the associated instabilities have received significant attention in the literature. In addition to their now-classic text, Gibson, Ashby, and their co-workers (see Gibson et al., 1989; Triantafillou et al., 1989; Triantafillou and Gibson, 1990; Zhang and Ashby, 1992) developed analytical methods for constructing theoretical failure surfaces for cellular materials using idealized periodic models; comparing elastic buckling, plastic yielding, and brittle fracture mechanisms. Using similar techniques, Klintworth and Stronge (1988) proposed plastic yield limits for periodic hexagonal honeycombs

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subjected to various deformation modes, and more recently, Wang and McDowell (2005) derived initial yield surfaces for metallic honeycombs with various other periodic cellular configurations. Ohno et al. (2002), Okumura et al. (2004), and Ohno et al. (2004) established a theoretical framework to analyze the microscopic bifurcation of periodic honeycombs, and the stability of threedimensional periodic foams recently was studied by Gong et al. (2005,). A wealth of information regarding the biaxial compression and crushing of genuine aluminum and polycarbonate honeycombs is contained in the comprehensive experimental and analytical investigations carried out by Papka and Kyriakides (1994, 1998a,b, 1999a,b). The onset of failure in periodic honeycombs and the effects of microstructural imperfections on the stability of such materials was studied by Triantafyllidis and Schraad (1998), and an extension of this work to materials with three-dimensional cellular structures was provided by Laroussi et al. (2002). In all of these investigations, attention is focused on nearly periodic two-dimensional honeycombs and three-dimensional foams, or on periodic idealizations of these materials. Related work on cellular materials with higher degrees of geometric dispersity is less developed, as periodic idealizations are no longer useful and analytical models become intractable. Silva et al. (1995) used finite-element analysis to investigate the effect of non-periodic microstructures on the initial properties of two-dimensional ‘‘foams’’, Silva and Gibson (1997) extended this analysis to include the effects on the compressive strength of the same material, and Chen et al. (1999) conducted a similar investigation into the effects of geometric imperfections on plastic yielding. Huang (2003) established a theoretical approach for constructing failure surfaces for three-dimensional foams with moderately disordered cellular structures, while Gong et al. (2005,) and Gong and Kyriakides (2005) conducted a careful experimental investigation of the compressive response of polyester urethane foams with low degrees of non-uniformity, and a companion analytical study of the properties of periodic Kelvin-cell foams. Lastly, yield surfaces for genuine three-dimensional metallic foams with truly disordered cellular structures were investigated experimentally by Gioux et al., 2000. This volume of literature provides a clear picture of how idealized cellular materials with periodic microstructures succumb to bifurcation-induced

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instabilities and how nearly periodic honeycombs and three-dimensional cellular materials with morphological imperfections succumb to failure through plastic yielding and localizing deformation modes. Because global instabilities in cellular materials with nearly periodic geometric structures can be related to instabilities in the corresponding idealized, perfectly periodic, model material, unit cell models and periodic finite-element models serve useful purposes for understanding some of the key aspects of the mechanical behavior and onset of failure in such ordered cellular solids. Unfortunately, unit cell models do not accurately represent the geometric structure of most genuine cellular solids, and thus, there is a need for models that can be used to quantify the influence of cellular-scale geometric and material variability on the mechanical response of these materials. In the present work, a stochastic constitutive model is developed and used to investigate the stability of cellular solids. This constitutive model is based on an established cellular-scale mechanical model, but is generalized to accommodate finite strain. A continuum-scale mechanical response description is established by averaging the cellular-scale mechanical model over an ensemble of foam cells. Stochastic variation in cellular-scale geometric structure and material properties is considered through the use of probability density functions for the associated model parameters. Using this approach, ordered cellular structures are represented using monodisperse distributions, while disordered cellular structures with varying degrees of dispersity are represented using other appropriate distribution functions. Attention is restricted to the uni-axial compressive response of low-density, open-cell, polyurethane foams. Under uni-axial loading conditions, the loss of stability in the continuum-scale behavior of these materials coincides with a maximum load in the uni-axial stress–strain response. A similar investigation into the influence of dispersity on the stability of cellular solids under general threedimensional loading is the subject of a forthcoming article. Results demonstrate a general stabilizing effect of dispersity in geometric structure on the continuumscale mechanical response of cellular materials. Consistent with previous investigations (see, for example, Silva et al., 1995; Silva and Gibson, 1997; Chen et al., 1999; Fazekas et al., 2002), dispersity in geometric structure is shown to have no effect on the initial elastic properties of the cellular

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materials under investigation. For deformations occurring prior to any instability, however, increasing dispersity is accompanied by decreasing stiffness, and no additional information might lead one to believe that dispersity in geometric structure has an overall detrimental effect on the mechanical response of cellular materials. The results of the present investigation, however, show that, in general, as the dispersity increases, the critical strains increase, the extent of localized deformation diminishes, and the materials stiffen for deformations occurring after the critical load. Most notably, the mechanical response of the materials with the highest degrees of dispersity in their cellular structures remains stable for all compressive deformations through full densification. The results are consistent with trends shown in a wide range of analytical, numerical, and experimental studies of various cellular solids, suggesting that the stochastic constitutive model can be used to quantify the influence of cellular-scale geometric and material variability on the mechanical response, the stability, and the onset of failure in cellular materials. The constitutive model also provides useful information regarding the initial cellular-scale structural configurations in these materials, and the potential for these configurations to admit unstable behavior and eventually succumb to failure. Such information could prove valuable in tailoring these materials at the cellular scale for increased load-carrying capacity and reduced susceptibility to failure, possibly shifting the processing and manufacturing paradigm for this class of materials from properties by tentation to properties by design. 2. Stochastic constitutive model A scanning electron micrograph of a typical cellular material is shown in Fig. 1. This low-density, open-cell, polyurethane foam represents one of the lightest of the structural foams and is used extensively in cushioning, padding, and packaging applications. Traditional approaches for modeling the mechanical response of such materials most often focus on the development of expressions for the initial material properties in terms of the cellular-scale geometry, the properties of the parent solid, and the relative density, or solid volume fraction, of the material. In this section, the development of a continuum-scale constitutive model begins with an established expression for the initial tangent modu-

Fig. 1. A scanning electron micrograph of a low-density, opencell, polyurethane foam showing the intricate, disordered, geometric structure at the cellular scale. This material was provided courtesy of the Dow Chemical Company, and the micrograph was provided courtesy of Dr. D.J. Alexander of the Materials Science & Technology Division (MST-6) at Los Alamos National Laboratory.

lus of the material. Consideration is given, however, to the nonlinear response associated with large deformations, and the cellular-scale mechanical model is generalized to accommodate finite strain. Of primary interest to this presentation are the effects of dispersity in cellular-scale geometric structure on the mechanical response of cellular materials, so a constitutive model that includes a description of the stochastic nature of the cellular structure, and that is valid through the large deformations for which these materials are intended, is of particular interest. A stochastic constitutive model for cellular materials recently has been developed and is described by Schraad and Harlow (2006). In this section, an extension of this model is presented, allowing for a more general description of material response in the regime of the load plateau. 2.1. Cellular-scale constitutive model For cellular materials with open-cell structures, Gibson and Ashby (1988) show that, in general, the initial tangent modulus can be written as Ec0 ¼ A0 Es /20 ;

ð1Þ

in which the parameter, A0, is a geometric constant of proportionality, Es is the tangent modulus of the

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198

parent solid (assumed to be constant), and /0 is the initial solid volume fraction of the material. Subscripts denote initial material properties, which are relevant only for infinitesimal strain. This expression is derived by considering the mechanics of a single, idealized, foam cell or suitable representative structure and by assuming cell-wall bending is the dominant mechanism of deformation occurring at the cellular scale. Because cellular materials retain their cellular structure through all deformations up to full densification, and because cell-wall bending remains the dominant mode of deformation at the cellular scale for open-cell foams subjected to compressive deformations, it is postulated that the tangent modulus maintains this functional form through all deformations as well. When subjected to finite deformation, however, the cellular-scale geometry and solid volume fraction evolve with the strain, e. Therefore, the strain-dependent tangent modulus is written as 2

Ec ðeÞ ¼ AðeÞEs ½/ðeÞ ;

ð2Þ

in which A(0) = A0, /(0) = /0, and Ec ð0Þ ¼ Ec0 . The solid volume fraction of a particular foam cell is defined by the ratio of the strain-dependent density associated with the cell, qc(e), to the assumed strain-independent density of the parent solid, qs. That is /ðeÞ 

qc ðeÞ : qs

ð3Þ

Under uni-axial compressive deformations, it can be shown that /ðeÞ ¼

/0 ; 1þe

ð4Þ

in which /0 is the initial solid volume fraction of the particular foam cell under consideration. The effects of strain evolution on the geometric constant of proportionality are more difficult to quantify. A broad distribution of irregular polyhedra in liquid foams was characterized by Matzke (1946), so one can expect a broad distribution of values for the geometric constant as well. For any particular foam cell, the geometric ‘‘constant’’ of proportionality, A, (henceforth, referred to as the geometric stiffness) depends on cell shape, orientation, strut configuration, and strain. Analysis of any realistic geometric structure to determine all possible initial values of A for all possible cell shapes, orientations, and strut configurations, likely presents an intractable task, and additional analysis

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to account for the nonlinear effects associated with finite bending strains in the cell walls is virtually guaranteed to be impossible. Using physical behavior at the cellular scale as a guide, however, the following simple, yet useful, approximation is made: AðeÞ ¼ A0 H ðe  e1 Þ þ A1 H ðe1  eÞ þ A10 H ðe10  eÞ þ ðA2  A1  A10 ÞH ðe2  eÞ:

ð5Þ

The quantities, e1 and e2, are values of the strain corresponding to the transitions from initial elastic response to the load plateau, and from the load plateau to densification, respectively. Hence, e1 and e2 are referred to as the transition strains. A third transition strain, e10 , is incorporated into this extended model to allow for small variations in stress levels through the regime of deformation corresponding to the load plateau. The parameters, A0, A1, A10 , and A2, then are the values of the geometric stiffness evaluated at e = 0, e = e1, e ¼ e10 , and e = e2, respectively; and the function, H(x), is the Heaviside step function, defined by  1; x P 0; H ðxÞ ¼ ð6Þ 0; x < 0: The response of a cellular material approaching full densification suggests that A2 = 1. Substituting the relations provided in Eqs. (4) and (5) into Eq. (2) then provides the following expression for the tangent modulus of the foam cell: Ec ðeÞ ¼ ½A0 H ðe  e1 Þ þ A1 H ðe1  eÞ þ A10 H ðe10  eÞ þ ð1  A1  A10 ÞH ðe2  eÞEs

/20 ð1 þ eÞ

2

:

ð7Þ

In this way, the incremental form of the constitutive model for a particular foam cell can be written as Dr ¼ Ec ðeÞDe;

ð8Þ

in which De is the increment in strain for this particular cell and Dr is the corresponding increment in stress. Zhu et al. (1997) show that the mechanical response of a single foam cell can vary substantially when only the orientation to the direction of loading is changed (see Fig. 15). Of course, an infinite number of cell shapes, orientations, and solid volume fractions are possible. Simulated stress–strain response curves for three particular cell realizations are shown in Fig. 2. The results are obtained using the cellular-scale constitutive model and represent the quasi-static uni-axial compression of the foam cells. For these calculations, Es = 4.5 · 104 kPa,

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M.W. Schraad / Mechanics of Materials 39 (2007) 183–198 6.0 Cell 1 Cell 2 Cell 3

5.0

Stress, σ (kPa)

4.0

3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engineering Strain, ε

Fig. 2. Simulated cellular-scale stress–strain response curves for low-density, open-cell, polyurethane foam cells subjected to quasi-static uni-axial compression. These results were obtained using the cellular-scale constitutive model with different values of the geometric stiffness parameters, A1 and A10 , to generate material response with varying levels of stress in the regime of the load plateau.

/0 = 0.030, A0 = 0.32, e1 = 0.12, e10 ¼ 0:24, and e2 = 0.76 (these choices are explained in Section 2.3). Only the levels of stress in the regime of the load plateau were allowed to vary. For cell 1, A1 ¼ A10 ¼ 0; for cell 2, A1 = 0.038 and A10 ¼ 0:040; and for cell 3, A1 = 0.038 and A10 ¼ 0:070. Because only A1 and A10 are varied, the response curves are similar, but the results demonstrate that the cellular-scale constitutive model is capable of representing a broad distribution of mechanical response characteristics. Of interest, however, is the continuum-scale response for a cellular material comprised of a collection of many cells. The continuum-scale response is obtained by averaging the mechanical response description provided in Eq. (8) over an entire ensemble of cells, allowing for stochastic variations in the model parameters and material properties. 2.2. Continuum-scale constitutive model Eq. (8) provides a description of the mechanical response for any particular foam cell taken at random from a collection of foam cells comprising an ensemble of similar foam samples. Averaging over the ensemble provides Dr ¼ Ec ðeÞDe ¼ Ec ðeÞDe þ Ec 0 ðeÞDe0 ;

ð9Þ

in which a line over any quantity denotes that quantity’s properly weighted ensemble average. The

quantity, Ec ðeÞ, is the average tangent modulus of the cellular material, while Dr and De denote the average stress and average strain increments, respec0 tively. The quantities, Ec ðeÞ and De0 , are fluctuations in the cellular-scale modulus and strain increment (e.g., De ¼ De þ De0 ), respectively; and therefore, the term, Ec 0 ðeÞDe0 , represents an average of the cross-correlation between these two quantities. Schjødt-Thomsen and Pyrz (2004) used finiteelement analysis to show that cell-to-cell fluctuations in strain are small in cellular materials with random dispersions of cells, and suggested the Voigt approximation (see Voigt, 1889) provides reasonably accurate approximations to the actual strain response of disordered cellular materials. The experimental investigations of Wang and Cuitin˜o (2002) offer further validation of this result. The traditional approach offered by Voigt simplifies the form of the continuum-scale constitutive model by assuming the strain fluctuations are negligible. This is equivalent to assuming the fluctuations in the increments of strain are negligible, and therefore, consider De0 ¼ 0

)

De ¼ De:

ð10Þ

Because the fluctuations in the increments of strain are assumed to be zero, the cross-correlation term in Eq. (9) also is zero. The continuum-scale constitutive model then reduces to Dr ¼ Ec ðeÞDe:

ð11Þ

It remains then to determine the average tangent modulus, Ec ðeÞ, which depends on cell-to-cell fluctuations in the parameters describing the individual foam cell response. 2.3. Stochastic material representation At the cellular scale, the tangent modulus for an individual foam cell is given by Eq. (7). In general, the model parameters, A0, A1, A10 , e1, e10 , and e2, and the material properties, Es and /0, vary from cell to cell in an ensemble of foam cells (the Voigt approximation assumes e ¼ eÞ. Each of the model parameters and material properties, therefore, are treated as independent variables in the stochastic material representation. The average tangent modulus then is given by Z Z Z Z Z Z Z Z c E ðeÞ  Ec ðeÞ A0

A1

A 10

e1

e1 0

e2

Es

/0

 P ðA0 ; A1 ; . . . ; /0 Þd/0    dA1 dA0 ;

ð12Þ

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198

in which the quantity P(A0, A1, . . . , /0)d/0    dA1 dA0 is the probability that this particular foam cell is characterized by the parameters, A0, in the interval dA0; A1, in the interval dA1, . . .; and /0, in the interval d/0. The function P ðA0 ; A1 ; A10 ; e1 ; e10 ; e2 ; Es ; /0 Þ is the probability density function, which is decomposed into the following form: P ¼ P A0 ðA0 ÞP A1 ðA1 ÞP A10 ðA10 ÞP e1 ðe1 Þ  P e10 ðe10 ÞP e2 ðe2 ÞP Es ðEs ÞP /0 ð/0 Þ;

ð13Þ

in which P A0 ; P A1 ; . . . ; P /0 are the probability density functions associated with each of the respective independent variables in the stochastic material representation, under circumstances for which these variables are uncorrelated. Substitution of the relation given in Eq. (13) into Eq. (12) and integration provides the average tangent modulus, which can be written as h i2 ~ eÞ : e eÞEs /ð Ec ðeÞ ¼ Að ð14Þ In this simplified form, the effective geometric stiffness is defined by Z e e Þ  A0 Að H ðe  e1 ÞP e1 ðe1 Þde1 e1

þ A1

Z

H ðe1  eÞP e1 ðe1 Þde1 e1

þ A10

Z

e1 0

H ðe10  eÞP e10 ðe10 Þde10

þ ð1  A1  A10 Þ

Z

H ðe2  eÞP e2 ðe2 Þde2 ; ð15Þ

e2

and the effective solid volume fraction is defined by " #12 Z 1 2 ~ eÞ  /ð /0 P /0 ð/0 Þd/0 : ð16Þ 2 ð1 þ eÞ /0 For Gaussian distributions, the probability density functions for the transition strains are given by " # 2 1 ðe1  e1 Þ P e1 ðe1 Þ ¼ pffiffiffiffiffiffi exp ; ð17Þ 2r2e1 2pre1 " # 2 1 ðe10  e10 Þ P e10 ðe10 Þ ¼ pffiffiffiffiffiffi exp ; ð18Þ 2r2e10 2pre10 and

" # 2 1 ðe2  e2 Þ P e2 ðe2 Þ ¼ pffiffiffiffiffiffi exp ; 2r2e2 2pre2

ð19Þ

189

in which re1 , re10 , and re2 are the standard deviations in the distributions of e1, e10 , and e2, respectively, which provide measures of the dispersion of the distributions around the average values, e1 , e10 , and e2 . Upon substitution, the integrals in Eq. (15) can be evaluated and the effective geometric stiffness then is given by " !#   A  e e 0 1 e eÞ ¼ Að 1  erf pffiffiffi 2 2 r e1 " !# e  e1 A1 þ 1  erf pffiffiffi 2 2re1 " !# e  e10 A10 þ 1  erf pffiffiffi 2 2re 0 " 1 !# e  e2 ð1  A1  A10 Þ 1  erf pffiffiffi þ ; ð20Þ 2 2re2 in which the function, erf(x), is the error function (or probability integral), defined by Z x 2 2 erfðxÞ  pffiffiffi et dt: ð21Þ p 0 Similarly, for a Gaussian distribution, the probability density function for the initial solid volume fraction is given by " # 2 1 ð/0  /0 Þ P /0 ð/0 Þ ¼ pffiffiffiffiffiffi exp ; ð22Þ 2r2/ 2pr/ in which r/ is the standard deviation in the distribution of /0, which provides a measure of the dispersion of the distribution around the average value, /0 . Upon substitution, the integral in Eq. (16) can be evaluated and the effective solid volume fraction then is given by " #1 /20 þ r2/ 2 ~ /ðeÞ ¼ : ð23Þ 2 ð1 þ eÞ The relations for the effective geometric stiffness and the effective solid volume fraction provided in Eqs. (20) and (23), respectively, are substituted into the relation provided in Eq. (14) for the average tangent modulus of the material. The average tangent modulus then is used within the incremental form of the continuum-scale constitutive model provided in Eq. (11) to simulate the mechanical response of the cellular materials under investigation. Using the stochastic material representation, perfectly ordered cellular structures can be modeled with monodisperse

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distributions by setting re1 ¼ re10 ¼ re2 ¼ r/ ¼ 0, while a spectrum of dispersity in the underlying cellular structure and initial solid volume fractions can be considered using a range of values for these standard deviations. Experimental and simulated continuum-scale stress–strain response curves for a polyurethane foam material are shown in Fig. 3. This particular foam material is produced using a chemical reaction process. As a result, the parent solid does not exist in an unfoamed or bulk state, and the development of methods for obtaining the properties of the parent solid provides a research topic of its own. Therefore, for present purposes, the parent solid is assumed to have an average initial tangent modulus, Es ¼ 4:5  104 kPa, which is commensurate with the tangent modulus of bulk polyurethane materials. The average density of the polyurethane was taken 3 to be qs ¼ 1:2 g=cm , and the average initial density of the foam material was measured to be qc0 ¼ 0:036 g=cm3 . Therefore, the average initial solid volume fraction of this particular material is /0 ¼ qc0 =qs ¼ 0:030. Both the experimental and simulated stress– strain response curves represent the quasi-static uni-axial compression of the polyurethane foam material. Experimental results are plotted using solid data points and are shown for three different foam samples, each nominally possessing the same

6.0 Experiment 1 Experiment 2 Experiment 3 Stochastic Model

5.0

average material properties. Results obtained using the stochastic constitutive model are plotted using a solid line. For this calculation, A0 ¼ 0:32, A1 ¼ 0:038, A10 ¼ 0, e1 ¼ 0:12, e10 ¼ 0:24, e2 ¼ 0:76, re1 ¼ 0:030, re10 ¼ 0:060, re2 ¼ 0:060, and r/ = 0.009. Because the actual statistical details for this material are not yet known, the values for re1 ; re10 ; re2 , and r/ are chosen to provide the best fit to the experimental data. As a result, the simulated response reproduces the experimental stress– strain data quite well. Physical limits (e.g., /0 > 0) provide upper bounds on the standard deviations, and monodispersity provides lower bounds (e.g., r/ P 0). The values that best fit the experimental data reside very near median values between these two bounds. Values commensurate with actual statistical data, therefore, should provide simulated response very near what is presented in Fig. 3. The mechanical response of cellular materials with nearly periodic geometric structures typically exhibit instabilities. Notice, however, that the mechanical response of this particular material remains stable (the stress is a monotonically increasing function of strain) for the entire range of compressive deformations. It is postulated, at this point, that this stable response is due to the high degree of disorder in the cellular-scale geometric structure (see Fig. 1). As mentioned previously, the transition from unstable to stable mechanical response as cellular structures shift from order to disorder is of primary interest here. To investigate the influence of dispersity on the stability of cellular solids, however, an appropriate criterion must be defined.

Stress, σ (kPa)

4.0

2.4. Global instability and the onset of failure 3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engineering Strain, ε

Fig. 3. Continuum-scale stress–strain response curves for a lowdensity, open-cell, polyurethane foam material subjected to quasi-static uni-axial compression. Experimental results are plotted using solid data points, while simulated results obtained using the stochastic constitutive model are plotted using a solid line. The experimental data was provided courtesy of Dr. C. Liu of the Materials Science & Technology Divison (MST-8) at Los Alamos National Laboratory.

Defining failure for materials with microstructure can be an elaborate and formidable task. The complexity involved arises from the many potential failure mechanisms, some of which are present at negligible levels of deformation and loading, and some of which occur simultaneously with other modes of failure. The ultimate failure of materials with periodic microstructure can be related to the occurrence of a microscopic bifurcation instability. Microscopic bifurcations do not occur in cellular materials with disordered microstructures, however, as by definition, symmetries in the underlying geometric structures do not exist. Macroscopic or global instabilities, however, still can occur in the

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198

The failure criterion provided in Eq. (24) can be used in conjunction with the continuum-scale constitutive law to identify the first global instability in the material response, and thus, the onset of failure in the materials under investigation. 3. Results and discussion Of particular interest in the present work is the influence of dispersity in cellular-scale geometric structure on the continuum-scale stress–strain response of cellular materials. In this section, the stochastic constitutive model developed in Section 2 is used to simulate the finite-strain uni-axial compression of cellular materials to quasi-static loading conditions. The stability criterion defined in Section 2.4 is used to identify the first global instability in the continuum-scale mechanical response of the material. Model materials with varying amounts of dispersity in the underlying cellular-scale geometric structures are considered and the influence of this dispersity on the stability of these materials, as defined above, is investigated. If a thin sheet of a foam material is compressed quasi-statically between two rigid plates, and if the thickness of the foam sheet is small relative to the other two spatial dimensions, then a one-dimensional problem can be solved to determine the mechanical response of the cellular material. Because the loading is quasi-static, the continuum-

scale uni-axial strain and stress states in the cellular material are spatially uniform, and thus, the stochastic constitutive model can be used to determine the stress for any particular level of strain. Consider first an idealized, low-density, polyurethane, foam material with a highly ordered cellular structure; for example, a material with cells of the same shape, the same orientation to the direction of loading, and the same initial solid volume fraction. Under compressive loading conditions, such a material typically exhibits mechanical response with a prominant constant-stress load plateau (similar to cell 1 in Fig. 2). The load plateau emerges as the deformation localizes into bands of ‘‘failed’’ material oriented approximately perpendicular to the direction of loading, and the length of this load plateau determines the extent of localization that can occur for loads exceeding the critical stress. The influence of dispersity in geometric structure on the mechanical response of such a material is presented in Figs. 4 and 5. Simulated continuum-scale stress–strain response curves for this first class of polyurethane foam material are shown in Fig. 4. All of these curves correspond to material with identical average properties, but with varying degrees of geometric dispersity. The simulations represent the quasi-static uni-axial compression of such materials. For these calculations, A0 ¼ 0:32, A1 ¼ A10 ¼ 0, e1 ¼ 0:12,

6.0 m=0 m=1 m=2 m=3 m=4 m=5 m=6

5.0

4.0

Stress, σ (kPa)

continuum-scale mechanical response of such materials. The occurrence of a global instability in the response of a material with nearly periodic, disordered, or strictly random microstructure typically precedes the onset of localizing deformation, and can be the precursor to shear band formation, pore and crack nucleation, and ultimately failure of the material. Consequently, for the cellular materials considered here, the onset of failure is defined as the first loss of stability in the corresponding continuum-scale mechanical response, as predicted by the average elastic moduli. For the uni-axial compressive loading conditions under consideration, the loss of stability coincides with a maximum load in the continuum-scale uni-axial stress–strain response. The maximum load corresponds to the instant at which the material stiffness becomes zero. Therefore, the critical strain is defined by   drðeÞ c ec ¼ min root of E ðeÞ  ¼0 : ð24Þ de

191

3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engineering Strain, ε

Fig. 4. The influence of dispersity in geometric structure on the simulated continuum-scale mechanical response of a low-density, open-cell, polyurethane foam. For material class 1, model parameters were chosen to produce a constant-stress load plateau. Results for monodisperse distributions are plotted using a solid line, while the results obtained for values of the geometric dispersion parameter ranging from 1 to 6 are plotted using various dashed and dotted lines as indicated.

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198 0.35

0.70

0.30

0.60

0.25

0.50

0.20

Critical Strain, εc

0.40

0.15

Plateau Length, Δ ε

0.30

Plateau Length, Δε

Critical Strain, ε c

192

0.20

0.10 stable for m > 3.23

0.10

0.05

0.00

0.00 0

1

2

3

4

Geometric Dispersion Parameter, m

Fig. 5. The influence of dispersity in geometric structure on the critical strain and load-plateau length for material class 1. Notice that for m  3.23, the load plateau diminishes altogether, as the stress becomes a monotonically increasing function of strain. For m P 3.23, the mechanical response remains stable for all uniaxial compressive deformations.

e10 ¼ 0:24, e2 ¼ 0:76, re1 ¼ 0:010m, re10 ¼ 0:020m, re2 ¼ 0:020m, and r/ = 0. The parameter, m, provides a measure of the dispersion in the distributions for the transition strains around their respective average values and is referred to as the ‘‘geometric dispersion parameter’’. Results obtained using monodisperse distributions for e1, e10 , and e2 (i.e., for m = 0) are plotted using a solid line, while results obtained for values of the geometric dispersion parameter ranging from 1 to 6 are plotted using various dashed and dotted lines as indicated. The distribution of initial solid volume fractions is monodisperse for these simulations. For the material with monodisperse geometric structure, the stress–strain response exhibits a constant-stress load plateau, as dictated by the condition A1 ¼ A10 ¼ 0. Dispersity in geometric structure has no effect on the initial elastic properties of this material. As the geometric dispersity increases, however, the material softens for deformations occurring prior to the load plateau, the critical strain increases, and the load plateau contracts. Furthermore, the material stiffens dramatically for deformations approaching densification. Most notably, however, for m = 4, m = 5, and m = 6, the load plateau diminishes altogether, as the stress becomes a monotonically increasing function of strain. In other words, for the material with the most disperse geometric structures, the mechanical response no longer reaches a critical point, and thus, remains

stable for the entire range of uni-axial compressive deformations. These results demonstrate the stabilizing influence of increasing geometric dispersity. To more clearly illustrate the stabilizing influence of geometric dispersity, the critical strain, ec , and the load-plateau length, De, are plotted as functions of the geometric dispersion parameter, m, in Fig. 5. It is evident that the critical strain increases and the load plateau contracts as the geometric dispersity increases. For both quantities, there is an approximately linear dependence on the geometric dispersion parameter, m. By extrapolating these results, it is determined that the load plateau diminishes altogether (i.e., the stress becomes a monotonically increasing function of strain) for m  3.23, and thus, for m P 3.23, the mechanical response of the material remains stable for all uni-axial compressive deformations. Consider next a second polyurethane foam material, also with a highly ordered cellular structure. Under compressive loading conditions, this material also exhibits mechanical response with a load plateau. In this case, however, the load ‘‘plateau’’ does not occur at constant stress. The mechanical response for this material reaches a relatively sharp maximum load, followed by an unstable regime of behavior, over which the material stiffness is significantly reduced. Immediately after the maximum load, the average tangent modulus is negative, but re-stiffening occurs as the material approaches densification (similar to cell 2 in Fig. 2). The critical point coincides with the maximum load and corresponds to the onset of failure as deformation localizes. The ‘‘effective’’ load plateau in this case is defined from the critical point to the point at which an equivalent value of stress is realized upon re-stiffening. The influence of dispersity in geometric structure on the mechanical response of such a material is presented in Figs. 6 and 7. Simulated continuum-scale stress–strain response curves for this second class of polyurethane foam material are shown in Fig. 6. Again, all of these curves correspond to material with identical average properties, but with varying degrees of geometric dispersity. For these calculations, A0 ¼ 0:32, A1 ¼ 0:038, A10 ¼ 0:070, e1 ¼ 0:12, e10 ¼ 0:24, e2 ¼ 0:76, re1 ¼ 0:010m, re10 ¼ 0:020m, re2 ¼ 0:020m, and r/ = 0. Results obtained using monodisperse distributions for e1, e10 , and e2 (i.e., for m = 0) are plotted using a solid line, while results obtained for values of the geometric dispersion parameter ranging from 1 to 6 are plotted using

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198 6.0 m=0 m=1 m=2 m=3 m=4 m=5 m=6

5.0

Stress, σ (kPa)

4.0

3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engineering Strain, ε

Fig. 6. The influence of dispersity in geometric structure on the simulated continuum-scale mechanical response of a low-density, open-cell, polyurethane foam. For material class 2, model parameters are chosen to produce a relatively sharp maximum load. Results for monodisperse distributions are plotted using a solid line, while the results obtained for values of the geometric dispersion parameter ranging from 1 to 6 are plotted using various dashed and dotted lines as indicated.

0.24

0.22 Plateau Length, Δε De

0.20

0.16

0.18 stable for m = 6 0.16

0.12

0.14

0.08 Critical Strain, ε

c

0.12

0.10

0

1

2

3

Plateau Length, Δε

Critical Strain, εc

0.20

0.04

4

5

6

0.00

Geometric Dispersion Parameter, m

Fig. 7. The influence of dispersity in geometric structure on the critical strain and load-plateau length for material class 2. Notice that for m > 5, the load plateau diminishes altogether, as the stress becomes a monotonically increasing function of strain. For m = 6, the mechanical response remains stable for all uni-axial compressive deformations.

various dashed and dotted lines as indicated. Again, the distribution of initial solid volume fractions is monodisperse for these simulations. For the material with monodisperse geometric structure, the stress–strain response reaches a maximum load for e = 0.12, as dictated for A1 < 0; A10 > 0, and the condition e1 ¼ 0:12. Similar to the results obtained for the first class of material,

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dispersity in geometric structure has no effect on the initial elastic properties of this material. As the geometric dispersity increases, however, the material softens for deformations occurring prior to the maximum load, the critical strain increases, and the effective load plateau contracts. Also, the material stiffens dramatically for deformations approaching densification, and for m = 6, a maximum load no longer occurs, as the effective load plateau diminishes altogether, the stress becomes a monotonically increasing function of strain, and the mechanical response of the material remains stable for all uni-axial compressive deformations. The critical strain, ec , and the load-plateau length, De, are plotted as functions of the geometric dispersion parameter, m, in Fig. 7. Again, it is evident that the critical strain increases and the load plateau contracts as the geometric dispersity increases, but for both quantities, the dependence on the geometric dispersion parameter, m, is no longer linear. The mechanical response of the material becomes stable for a value of the geometric dispersion parameter between 5 and 6. Lastly, consider a third polyurethane foam material, once again with a highly ordered cellular structure. Under compressive loading conditions, this material also exhibits mechanical response with a load plateau. And again, in this case, the load ‘‘plateau’’ does not occur at constant stress. The mechanical response for this material reaches a relatively blunt maximum load, followed by an unstable regime of behavior, over which the material stiffness is significantly reduced. Immediately after the maximum load, the average tangent modulus is negative, but re-stiffening occurs as the material approaches densification (similar to cell 3 in Fig. 2). Again, the critical point coincides with the maximum load and corresponds to the onset of failure as deformation localizes, and the effective load plateau is defined from the critical point to the point at which an equivalent value of stress is realized upon re-stiffening. The influence of dispersity in geometric structure on the mechanical response of such a material is presented in Figs. 8 and 9. Simulated continuum-scale stress–strain response curves for this third class of polyurethane foam material are shown in Fig. 8. Once again, all of these curves correspond to material with identical average properties, but with varying degrees of geometric dispersity. For these calculations, A0 ¼ 0:32, A1 ¼ 0:038, A10 ¼ 0:040, e1 ¼ 0:12, e10 ¼ 0:24, e2 ¼ 0:76, re1 ¼ 0:010m, re10 ¼ 0:020m, re2 ¼

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M.W. Schraad / Mechanics of Materials 39 (2007) 183–198 6.0 m=0 m=1 m=2 m=3 m=4 m=5 m=6

Stress, σ (kPa)

5.0

4.0

3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engineering Strain, ε

Fig. 8. The influence of dispersity in geometric structure on the simulated continuum-scale mechanical response of a low-density, open-cell, polyurethane foam. For material class 3, model parameters are chosen to produce a relatively blunt maximum load. Results for monodisperse distributions are plotted using a solid line, while the results obtained for values of the geometric dispersion parameter ranging from 1 to 6 are plotted using various dashed and dotted lines as indicated.

0.50

0.60 Plateau Length, Δε

0.50

0.40

0.40

0.30

0.35 stable for m = 6

0.20

0.30 Critical Strain, εc

0.25

Plateau Length, Δε

Critical Strain, εc

0.45

For the material with monodisperse geometric structure, the stress–strain response reaches a maximum load for e = 0.24, as dictated for A1 > 0; A10 < 0, and the condition e10 ¼ 0:24. Similar to the results obtained for the first two classes of material, dispersity in geometric structure has no effect on the initial elastic properties of this material. Once again, however, as the geometric dispersity increases, the material softens for deformations occurring prior to the maximum load, the critical strain increases, and the effective load plateau contracts. And again, the material stiffens dramatically for deformations approaching densification, and for m = 6, a maximum load no longer occurs, as the effective load plateau diminishes altogether, the stress becomes a monotonically increasing function of strain, and the mechanical response of the material becomes stable for all uni-axial compressive deformations. The critical strain, ec , and the load-plateau length, De, are plotted as functions of the geometric dispersion parameter, m, in Fig. 9. Once again, it is evident that the critical strain increases and the load plateau contracts as the geometric dispersity increases, and the mechanical response of the material becomes stable for a value of the geometric dispersion parameter between 5 and 6. Qualitatively, the results for this class of material are quite similar to those obtained for the polyurethane foams from the second class of material. The critical stress, rc , is plotted as a function of the geometric dispersion parameter, m, in Fig. 10.

0.10 2.10

0

1

2

3

4

5

6

0.00

Geometric Dispersion Parameter, m

Fig. 9. The influence of dispersity in geometric structure on the critical strain and load-plateau length for material class 3. Notice that for m > 5, the load plateau diminishes altogether, as the stress becomes a monotonically increasing function of strain. For m = 6, the mechanical response remains stable for all uni-axial compressive deformations.

2.00

Critical Stress, σc (kPa)

0.20

Material Class 1 Material Class 2 Material Class 3

1.90

1.80

1.70

0:020m, and r/ = 0. Results obtained using monodisperse distributions for e1, e10 , and e2 (i.e., for m = 0) are plotted using a solid line, while results obtained for values of the geometric dispersion parameter, m, ranging from 1 to 6 are plotted using various dashed and dotted lines as indicated. And again, the distribution of initial solid volume fractions is monodisperse for these simulations.

1.60

0

1

2

3

4

5

6

Geometric Dispersion Parameter, m

Fig. 10. The influence of dispersity in geometric structure on the critical stress for material classes 1–3. The critical points from each of the preceding simulations are represented in this figure. Notice that, unlike the critical strain, the critical stress remains relatively constant as the geometric dispersity increases.

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198

The critical points from each of the preceding simulations are represented in this figure. Unlike the critical strain, the critical stress remains relatively constant as the geometric dispersity increases, even decreasing slightly depending on the material and the degree of dispersity. Analysis thus far has focused on variations in cellular-scale geometric structure and the corresponding effects on the continuum-scale mechanical response of cellular materials. In general, cellular materials also possess cell-to-cell variations in solid volume fraction. Simulated continuum-scale stress– strain response curves for the third class of polyurethane foam material are shown in Fig. 11. All of these curves correspond to material with identical average properties, but with varying degrees of dispersity in the initial solid volume fraction. For these calculations, A0 ¼ 0:32, A1 ¼ 0:038, A10 ¼ 0:040, e1 ¼ 0:12, e10 ¼ 0:24, e2 ¼ 0:76, re1 ¼ 0:020, re10 ¼ 0:040, re2 ¼ 0:040, and r/ = 0.003n. The parameter, n, provides a measure of the dispersion in the distributions for the initial solid volume fractions around the average value and is referred to as the ‘‘density dispersion parameter’’. Results obtained using monodisperse distributions for /0 (i.e., for n = 0) are plotted using a solid line, while results obtained for values of the density dispersion parameter ranging from 1 to 5 are plotted using various dashed and dotted lines as indicated. The distri-

6.0

2.6

2.5

Critical Stress, σc (kPa)

Stress, σ (kPa)

4.0

3.0

2.0

1.0

0.0 0.0

butions for the transition strains remained constant with m = 2 for these simulations. Unlike geometric dispersity, dispersity in initial solid volume fraction has a significant effect on the initial elastic properties of the material. For this material, as the density dispersion parameter, n, increases, the material stiffens, and this stiffening effect is present at all levels of compression. For each of these simulations, the stress–strain response reaches a maximum load, and the critical strain remains constant. The mechanical response for each of these simulations, therefore, eventually becomes unstable. Dispersity in the initial solid volume fraction for this class of materials, therefore, does not possess the same stabilizing influence as dispersity in the cellular-scale geometric structure. The critical stress, however, increases with increasing dispersity in the initial solid volume fractions, as shown in Fig. 12, which could be considered a stabilizing effect in its own right. Collectively, the results presented here demonstrate the significant influence dispersity at the microstructural scale can have on the general mechanical response of cellular solids. Dispersity in geometric structure is shown to have a stabilizing effect on the mechanical response, increasing the strain at which instabilities occur, decreasing the extent of localized deformation, and eventually leading to monotonicity in the stress–strain response and producing stable behavior for all uni-axial compressive deformations. Dispersity in initial solid volume fractions, on the other hand, does not produce the same stabilizing effects, but

n = 0 (m = 2) n=1 n=2 n=3 n=4 n=5

5.0

0.1

0.2

0.3

195

0.4

0.5

0.6

0.7

0.8

Engineering Strain, ε

Fig. 11. The influence of dispersity in initial solid volume fraction on the simulated continuum-scale mechanical response of a lowdensity, open-cell, polyurethane foam. Results for monodisperse distributions are plotted using a solid line, while the results obtained for values of the density dispersion parameter ranging from 1 to 5 are plotted using various dashed and dotted lines as indicated.

2.4

2.3

2.2

2.1

2.0

0

1

2

3

4

5

Density Dispersion Parameter, n

Fig. 12. The influence of dispersity in initial solid volume fraction on the critical stress. Dispersity in the initial solid volume fraction does not possess the same stabilizing influence as dispersity in the cellular-scale geometric structure.

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does increase stiffness, leading to an increased loadcarrying capacity in the material. The results point to the usefulness of any constitutive model that accounts for stochastic variations in material properties and model parameters in the construction of a material response description, and the conclusions offer potentially useful design information to be used in the fabrication of cellular materials for specific engineering purposes. 4. Concluding remarks Geometric structures in cellular solids span the spectrum from perfectly periodic to strictly random. As a result, failure and the associated instabilities in cellular materials are manifested in many different ways. Honeycombs and foams idealized as possessing periodic microstructures inevitably succumb to bifurcation-induced instabilities as symmetries in the underlying cellular structures are broken and more energetically favorable microstructural configurations emerge through deformation. The geometric structures of genuine three-dimensional cellular solids, however, necessarily include natural material defects and geometric imperfections, and most often are highly disordered due to the processing techniques used in their manufacture. The actual materials being modeled, therefore, possess nearly periodic microstructures at best, and at worst, may not resemble at all the periodic model materials at the cellular scale. This is important, because microscopic bifurcations do not occur in materials with disordered microstructures, as by definition, symmetries in the underlying geometric structures no longer exist. For such materials, however, global instabilities still can occur in the continuum-scale mechanical response. These disordered materials, therefore, typically exhibit limit loads in their mechanical response, beyond which localized deformation develops, followed ultimately by failure of the material. If the dispersity in cellular-scale structural geometry is small (i.e., for highly ordered cellular materials), the mechanical response can be quite similar to that of nearly periodic honeycombs and foams with only small morphological and material property imperfections. The mechanical response reaches a limit load and localized deformation dominates the high-strain response of the material. For materials with a high degree of dispersity (i.e., for highly disordered cellular materials), however, the stress can become a monotonically increasing function of

strain, and thus, no critical point is reached, and the mechanical response remains stable through all regimes of uni-axial compressive deformation. A primary objective of this investigation, therefore, is to study the behavior of cellular materials ranging over the spectrum of geometric structures from highly ordered to highly disordered, and to explore the corresponding transition from unstable to stable mechanical response. In the present work, the stability of cellular solids is studied through the development and use of a stochastic constitutive model. Model development begins with an established cellular-scale mechanical response description, but departs from traditional modeling approaches developed for cellular materials in the manner by which this cellular-scale model is generalized to accommodate finite strain. A continuum-scale constitutive model is established by averaging the cellular-scale mechanical model over an ensemble of foam cells or suitable representative cellular structures. Stochastic variation in cellular-scale geometric structure and material properties is considered through the use of probability density functions for the associated model parameters. In this way, ordered cellular structures are represented using monodisperse distributions for the relevant independent variables, while disordered cellular structures with varying degrees of dispersity are represented using other appropriate distribution functions. Attention is focused on the quasi-static, uni-axial, compressive response of low-density, open-cell, polyurethane foams. Under uni-axial loading conditions, the loss of stability in the continuum-scale mechanical response coincides with a maximum load, which immediately precedes the onset of localizing deformation. Results demonstrate a general stabilizing effect of dispersity in geometric structure on the continuum-scale mechanical response of cellular materials. Dispersity in geometric structure is shown to have little to no effect on the initial elastic properties of the cellular materials under investigation, and for deformations occurring prior to any occurrence of instability, increasing dispersity is accompanied by decreasing stiffness. These trends are consistent with the results of previous investigations, and provided with no additional information, one might be lead to believe that dispersity in geometric structure has an overall detrimental effect on the mechanical response of cellular materials. The results of the present investigation, however, show that as the dispersity increases, the critical strains increase, while the mechanical response of

M.W. Schraad / Mechanics of Materials 39 (2007) 183–198

the materials with the most disperse cellular structures becomes monotonic, and thus, these materials maintain stable mechanical response for all compressive deformations through full densification. Future research will focus on a more thorough representation of cellular-scale mechanics, and on developing a more general stochastic material representation for the cellular materials of interest through the correlation of independent variables and the use of appropriate joint or conditional probability density functions. In this way, the effect of dispersity in the tangent modulus of the parent solid and in the average geometric stiffness parameters also can be included in the investigation. More general expressions for the continuum-scale constitutive model also will be considered by allowing for fluctuations in the strain field and including the average of the cross-correlation between fluctuations in the cellular-scale tangent modulus and strain increment. And of course, the results presented herein will be extended to include the influence of dispersity on the stability of cellular solids under general three-dimensional loading. Qualitatively, the results presented herein are consistent with trends shown in a wide range of analytical, numerical, and experimental studies of cellular solids, suggesting the stochastic constitutive model not only reproduces the stress–strain response of the cellular materials under investigation, but also can be used to quantify the influence of cellular-scale geometric and material variability on the mechanical response, the stability, and the onset of failure in these materials. The results also provide useful design information, which could be used to tailor such materials at the cellular scale for increased load-carrying capacity and reduced susceptibility to failure, possibly shifting the processing and manufacturing paradigm for this class of materials to offer more predictive capabilities. Because the occurrence of an instability can be the precursor to shear band formation, and ultimately failure of the material, an understanding of the relationships among cellular-scale geometric structure, continuum-scale mechanical response, and stability is essential in the design of engineering components and the cellular materials from which they are made. Acknowledgements This work was performed under the auspices of the US Department of Energy, under contract

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