Localized Effects in Periodic Elastoplastic Composites

    

    

    Figure 1:  (a) A doubly periodic unidirectional fiber reinforced composite
with a localized damage, is subjected to a remote strain ${\overline{ϵ}}_{22}$
 at infinity. (b) A
rectangular domain 2H × 2L of the composite is divided into repeating cells.
These cells are labeled by (K2, K3) with -M2 = K2 = M2 and -M3 = K3 = M3,
and the size of every one of which is 2h × 2l (the figure is shown for M2 =
M3 = 2). (c) A characteristic cell (K2, K2) in which local coordinates $\left({x^{\prime }}_2,{x^{\prime }}_3\right)$

are introduced whose origin is located at the center. It is divided into Nβ × Nγ
subcells (the figure is shown for Nβ = Nγ = 10). (d) Subcell (βγ).
    
    

The response of the considered composite is determined by satisfying the equilibrium equations in every one of its constituents, namely:

∇ . σ = 0 (1)

in the absence of body forces, where σ is the stress tensor. In addition, the interfacial conditions that requires in the case of perfect bonding the continuity of displacements u and tractions t between the fibers (f) and matrix (m) constituents must be imposed:

u^f = u^m, t^f = t^m (2)

In the present investigation elastoplastic materials are considered whose constitutive equations are given (assuming isothermal conditions) by

σ = C: (Σ - Σ^P) (3)

where C is the stiffness tensor of the material, and Σand Σ^P are the strain and plastic strain tensors, respectively. In the framework of classical plasticity, the evolution law of the plastic strain Σ^P is governed by the Prandtl-Reuss equations, c.f. Mendelson [9]:

ΔΣ^P = Δλ s (4)

where Δλ is the Prandtl-Reuss proportionality function and s is the deviator tensor of σ.

In Aboudi and Ryvkin [1], the effect of localized damage in perfectly elastic composites was represented by eigenstresses as follows. The constitutive equations of elastic material in the presence of damage are determined by the principle of strain equivalence Lemaitre and Desmorat [10], according to which the strains in the damaged and effective configurations are equal. This implies that

σ = C: Σ - σ^e (5)

where σ^e is eigenstress which has the form

σ^e = DC: Σ (6)

where D is the damage variable which is equal to zero in the undamaged constituents, whereas D = 1 in cracks and cavity regions thus representing a complete damage. In addition to the contribution of Aboudi and Ryvkin [1] where this method was presented and applied to investigate localized damage in elastic composites, this approach has been also shown to be successful in the modeling of an H-crack (a transverse crack accompanied by two longitudinal ones) in ceramic matrix composites Ryvkin and Aboudi [5], and cracks, cavities and inclusions in electro-magneto-elastic composites Aboudi [6].

In the present paper where the constitutive equations of the phases incorporate inelastic effects we propose to continue representing the constitutive equations by Equation (5) but generalize Equation (6) as follows

σ^e = DC: Σ + (1-D)C: Σ^P (7)

Thus, the eigenstresses presently include the plasticity effects in addition to the damage. It should be noted that in the absence of damage (D = 0) the plasticity effects are included in the analysis in the form of eigenstresses and their effects extend over the entire region. In the presence of damage in certain localized regions (0 < D < 1) further terms are added to the latter. In regions with complete damage (D = 1) where cracks and cavities are simulated, the plasticity effects as expected are absent. It can be readily observed that with introduction of the damage parameter D, it is possible to model a region whose behavior is elastoplastic everywhere except for one or more localized domains where D = 1 which corresponds to cavities and cracks. Evolving damage, stiff and soft inclusions, where D takes suitable values between zero and one have been utilized in the modeling of thermoelastic, electro-magneto-thermo-elastic and electrostrictive composites, see Aboudi and Ryvkin [1], Aboudi [6] and Aboudi [7], respectively.

Finally, as it is discussed in the next section, the far-field boundary conditions that are applied on the composite should be incorporated.

Method of Solution

Far away from the region with localized damage, the periodic elastoplastic composite is governed at any instant of loading by its macroscopic (global) behavior. The constitutive equations that model this behavior can be micromechanically established by the HFGMC model, Aboudi et al. [4] chapter 6, and are given by

$\overline{\sigma }=C^{*}:\left(\overline{\in }-{\overline{\in }}^P\right)\text{ (8)}$

where C* is the effective stiffness tensor and , and $\overline{\sigma },\overline{\in }\text{ and }{\overline{\in }}^P$ are the global stress, total strain and plastic strain tensors, respectively.

Let us consider a rectangular domain -H = x₂ = H, -L = x₃ = L of the composite which includes the damaged region. Although this region includes the localized damage, it is assumed that it is extensive enough such that the inelastic stress, strain and displacement fields at its boundaries are not influenced by the damage existence and therefore, the macroscopic constitutive equations (8) are applicable. Consequently, the boundary conditions that are applied on x₂ = ± H and x₃ = ± L are referred to as the far-field boundary conditions. According to the representative cell method, Ryvkin and Nuller [2], this region is divided into (2M₂ + 1) × (2M₃ + 1) identical cells, see Figure 1(b) which is schematically shown for M₂ = M₃= 2 cells. Every cell is labeled by (K₂, K3) with K2 = -M₂, …, M₂ and K₃ = -M₃, …, M₃. In each cell, local coordinates (x’₂, x’₃) are introduced whose origins are located at its center, see Figure 1(c) which shows the representative cell that is schematically divided into N_β= 10 and N^γ = 10 subcells.

The equilibrium equation (1) of the materials within the cell (K₂, K₃) takes the form

$\nabla \cdot {\sigma }^{(k_2,k_3)}=0\text{ (9)}$

The constitutive equation in the cell, Equation (5), can be written as

${\sigma }^{(k_2,k_3)}=C:{\in }^{(k_2,k_3)}-{\sigma }^{e\left(k_2,k_3\right)}\text{ (10)}$

where the eigenstresses in cell (K₂, K₃) are given according to Equation (6) by

${\sigma }^{e(k_2,k_3)}=DC:{\in }^{(k_2,k_3)}+(1-D)C:{\in }^{P\left(k_2,k_3\right)}\text{ (11)}$

and

$D=\{\begin{array}{cc}\text{0}& \text{no damage}\\ 1& \text{full damage}\end{array}\text{ (12)}$

and 0 < D < 1 for a partial damage.

The continuity of displacements u(K₂,K₃) and tractions t(K₂,K₃) between adjacent cells should be imposed. Thus,

${\left[u\left(h,{x^{\prime }}_3\right)\right]}^{\left(k_2,k_3\right)}-{\left[u\left(-h,{x^{\prime }}_3\right)\right]}^{\left(k_2+1,k_3\right)}=0\text{ (13)}$

${\left[t^{(2)}\left(h,{x^{\prime }}_3\right)\right]}^{\left(k_2,k_3\right)}-{\left[t^{(2)}\left(-h,{x^{\prime }}_3\right)\right]}^{\left(k_2+1,k_3\right)}=0\text{ (14)}$

where K₂ = -M₂,…, M₂ - 1, K₃ = - M₃,…, M₃ - l = x’₃= l, and

${\left[u\left({x^{\prime }}_2,l\right)\right]}^{\left(k_2,k_3\right)}-{\left[u\left({x^{\prime }}_2,-l\right)\right]}^{\left(k_2,k_3+1\right)}=0△ABC\text{ (15)}$

${\left[t^{(3)}\left({x^{\prime }}_2,l\right)\right]}^{\left(k_2,k_3\right)}-{\left[t^{(3)}\left({x^{\prime }}_2,-l\right)\right]}^{\left(k_2,k_3+1\right)}=0\text{ (16)}$

where K₂ = -M₂, …, M₂, K₃ = -M₃, …, M₃ - 1, -h = x’₂ = h. Here t^(j) is the traction vector acting at a boundary perpendicular to the x_j -axis, j = 2, 3.

In the following, the appropriate form of the far-field boundary conditions that specify the tractions and displacements at the opposite sides x₂ = ±H, x₃ = ±L of the rectangle of Figure 1(b) are presented. The tractions at the opposite sides of the rectangular domain are equal:

${\left[t^{(2)}\right]}^{\left(M_2,K_3\right)}\left(h,{x^{\prime }}_3\right)-{\left[t^{\left(2\right)}\right]}^{\left(-M_2,K_3\right)}\left(-h,{x^{\prime }}_3\right)=0,K_3=-M_3,\mathrm{\dots },M_3\text{ -}l\le {x^{\prime }}_3\le l\text{ (17)}$

${\left[t^{(3)}\right]}^{\left(K_2,M_3\right)}\left({x^{\prime }}_2,l\right)-{\left[t^{\left(3\right)}\right]}^{\left(K_2,-M_3\right)}\left({x^{\prime }}_2,-l\right)=0,K_2=-M_2,\mathrm{\dots },M_2\text{ -}h\le {x^{\prime }}_2\le h\text{ (18)}$

The displacements at the opposite sides, on the other hand, differ by certain jumps as follows

$u^{\left(M_2,K_3\right)}\left(h,{x^{\prime }}_3\right)-u^{\left(-M_2,K_3\right)}\left(h,{x^{\prime }}_3\right)={\delta }_2,K_3=-M_3,\mathrm{\dots },M_3\text{ -}l\le {x^{\prime }}_3\le l\text{ (19)}$

$u^{\left(K_2,M_3\right)}\left({x^{\prime }}_2,l\right)-u^{\left(K_2,-M_3\right)}\left({x^{\prime }}_2,-l\right)={\delta }_3,K_2=-M_2,\mathrm{\dots },M_2\text{ -}h\le {x^{\prime }}_2\le h\text{ (20)}$

where σ₂ and σ₃ denote the vectors of the far-field displacement differences whose components are given by

${\delta }_{2j}=2H{\overline{\in }}_{2j},{\delta }_{3j}=2L{\overline{\in }}_{3j},\text{ j=1,2,3 (21)}$

and ${\overline{\in }}_{2j},{\overline{\in }}_{2j}$ are the macroscopic (global) strains of the unperturbed periodic composite which have to be determined from Equation (8) for a given type of applied loading. Suppose, for example, that the composite is subjected to a transverse strain loading ${\overline{\in }}_{22}\text{ with }{\overline{\sigma }}_{ij}=0$ in all other directions. The macroscopic equations (8) that describe the (undamaged) composite’s behavior are utilized to determine the remaining average strains ${\overline{\in }}_{ij},i=j\ne 2$ to be employed in Equation (21).

The double finite discrete Fourier transform of the displacement vector u^{(K₂ ,K₃ )} , for example, is defined by

$\widehat{u}\left({x^{\prime }}_2,{x^{\prime }}_3,{\varphi }_p,{\varphi }_q\right)={\displaystyle \sum _{K_2=-M_2}^{M_2}{\displaystyle \sum _{K_3=-M_3}^{M_3}{u}^{\left(K_2,K_3\right)}}\left({x^{\prime }}_2,{x^{\prime }}_3\right)\text{exp}\left[i\left(K_2{\varphi }_p+K_3{\varphi }_q\right)\right]}\text{ (22)}$

where

${\varphi }_p=\frac{2\pi p}{2M_2+1},p=0,\pm 1,\pm 2,\mathrm{\dots },\pm M_2,{\varphi }_q=\frac{2\pi q}{2M_3+1},q=0,\pm 1,\pm 2,\mathrm{\dots },\pm M_3,$

The application of this transform to the boundary value problem (9)-(20) for the rectangular domain -H < x₂ < H, -L < x₃ < L, divided into (2M₂ + 1) × (2M₃ + 1) cells, converts it to the problem for the single representative cell -h < x’₂ < h, -l < x’₃ < l with respect to the complex valued transforms. The field equations obtained from the equilibrium and constitutive equations have the form

$\nabla \cdot \widehat{\sigma }=0\text{ (23)}$

and

$\widehat{\sigma }=C\widehat{\in }-{\widehat{\sigma }}^e\text{ (24)}$

where the transformed eigenstress tensor is given by

${\widehat{\sigma }}^e=DC:\widehat{\in }+(1-D)C:{\widehat{\in }}^p\text{ (25)}$

The conditions relating the opposite boundaries of the representative cell in the elastic problem case were derived by Aboudi and Ryvkin [1]. Following their approach one obtains from (13)-(20)

$\widehat{u}(h,{x^{\prime }}_3)=\text{exp}(-i{\varphi }_p)\widehat{u}(-h,{x^{\prime }}_3)+{\delta }_{0,q}(2M_3+1){\delta }_2\text{exp}(i{\varphi }_p{M}_2),-l\le {x^{\prime }}_3\le l\text{ (26)}$

${\widehat{t}}^{(2)}(h,{x^{\prime }}_3)=\text{exp}(-i{\varphi }_p){\widehat{t}}^{(2)}(-h,{x^{\prime }}_3),-l\le {x^{\prime }}_3\le l\text{ (27)}$

$\widehat{u}({x^{\prime }}_2,l)=\text{exp}(-i{\varphi }_q)\widehat{u}({x^{\prime }}_2,-l)+{\delta }_{0,p}(2M_2+1){\delta }_3\text{exp}(i{\varphi }_q{M}_3),-h\le {x^{\prime }}_2\le h\text{ (28)}$

${\widehat{t}}^{(3)}({x^{\prime }}_2,l)=\text{exp}(-i{\varphi }_q){\widehat{t}}^{(3)}({x^{\prime }}_2,-l),-h\le {x^{\prime }}_2\le h\text{ (29)}$

where p = -M₂, …, M₂; q = -M₃, …, M₃. In these equations, σ_p,q denotes the Kronecker delta.

The representative cell boundary value problem (23)-(29) have been solved by employing the inelastic higher-order theory, Aboudi et al. (2013), chapter 11. According to this theory, the domain -h = x’₂ = h, -l = x’₃ = l (the representative cell) is divided into N_{β × Nγ} rectangular subcells β = 1, …, N_{β , γ = 1, …, Nγ ,} see Figure 1(c) where N_{β = Nγ = 10.} The transformed displacement vector is expanded into a second-order polynomial in the subcell (β,γ), Figure 1(d), and the equilibrium equations, interfacial and boundary conditions are imposed in the average (integral) sense. In order to model a (line) crack, a single row of subcells filled with a fully damaged (D = 1) elastoplastic material is introduced. Thus the value of the damage variable D is pre-determined in accordance with the crack configuration. If on the other hand a cavity is modeled, then its entire elastoplastic region of subcells is modeled with D = 1.

Once the solution at a current instant of loading in the transform domain has been established, the actual elastoplastic field can be readily determined at any point in the desired cell (K₂, K₃) of the considered rectangular region -H = x₂ = H, -L = x₃ = L by the inverse transform formula whose form for the displacements, for example, is:

$u^{\left(K_2,K_3\right)}\left({x^{\prime }}_2,{x^{\prime }}_3\right)=\frac{1}{(2M_2+1)(2M_3+1)}\text{ (30)}$

$\times {\displaystyle \sum _{p=M_2}^{M_2}{\displaystyle \sum _{q=M_3}^{M_3}\widehat{u}\left({x^{\prime }}_2,{x^{\prime }}_3,{\varphi }_p,{\varphi }_q\right)\exp \left[-i\left(K_2{\varphi }_p+K_3{\varphi }_q\right)\right]}}$

In the application of this theory, the eigenstress tensor${\widehat{\sigma }}^e$ to be used in Equation (24) is not known. Hence an iterative solution has to be employed as shown in the flow chart.

This procedure should be continued until a convergence to a desired degree of accuracy is achieved. Having established the solution at the current increment, one can proceed to the next increment by a slight change of the applied loading.

Verification

In all cases given in this paper, the computations of the elastoplastic field were carried out with a square representative cell h = l, Figure 1(c). Which is discretized into 2500 subcells with N_β = N_r = 50 subcells. The region -H = x₂ = H, -L = x₃ = L has been divided into 289 cells with M₂ = M₃ = 8. This choice was verified in providing a periodic (unperturbed) field far away from the localized damage which was always taken to exist in cell M₂ = M₃ = 0 only. Thus, the use of global constitutive relations, Equation (8), for the determination of the far-field inelastic boundary conditions is justified.

The verification of the present approach has been performed by comparing the resulting response to a remote biaxial loading of an elastoplastic material with an embedded circular cavity. This problem has been originally investigated by Budiansky and Mangasarian [11]. The material is described by the Ramberg-Osgood deformation theory of plasticity which for a uniaxial stress-strain case is given by

$\in =\frac{\sigma }{E}+\frac{\alpha Y}{E}{\left(\frac{\sigma }{Y}\right)}^n\text{ (31)}$

where E is the Young’s modulus of the material, Y is its yield stress and a and n are material parameters. For elastic strains which are negligible compared to the plastic strains, Budiansky and Mangasarian [11] provided a closed-form expression for the material response in this special case. An analytical solution for arbitrary values of the elastic strains has been derived by Ishikawa [8]. This solution is presently employed for comparison with the proposed analysis to verify our approach.

The uniaxial stress-strain Ramberg-Osgood relation (31) can be generalized to a multiaxial one yielding

${\in }_{ij}=\frac{1+v}{E}{\sigma }_{ij}-\frac{v}{E}{\sigma }_{kk}{\delta }_{i,j}+\frac{3\alpha }{2E}{\left(\frac{{\sigma }_{eq}}{Y}\right)}^{n-1}{s}_{ij}\text{ (32)}$

where ν is the Poisson’s ratio of the material and the equivalent stress is defined by ${\sigma }_{eq}=\sqrt{3s_{ij}{s}_{ij}/2}$ with s_ij being the deviators of the stresses σ_ij. This equation can be inverted (R. Haj-Ali, personal communication) to yield (see Appendix for a proof):

${\sigma }_{ij}=K{\in }_{kk}{\delta }_{i,j}+\frac{e_{ij}}{\frac{1}{2G}+\frac{3\alpha }{2E}{\left(\frac{{\sigma }_{eq}}{Y}\right)}^{n-1}}\text{ (33)}$

where K, G are the bulk and shear moduli and σ_eg is the root of the following nonlinear equation

$\frac{3\alpha Y}{2E}{\left(\frac{{\sigma }_{eq}}{Y}\right)}^n+\frac{Y}{2G}\left(\frac{{\sigma }_{eq}}{Y}\right)-e_{eq}=0\text{ (34)}$

In these equations, $e_{eq}=\sqrt{3e_{ij}{e}_{ij}/2}$ with e_ij being the deviators of the strain components Σ_ij. Consequently, in the present case Equation (4) should be replaced by

${\in }^p=\frac{3\alpha }{2E}{\left(\frac{{\sigma }_{eq}}{Y}\right)}^{n-1}s\text{ (35)}$

With a = 3/7, the analytical solution of Ishikawa [8], expressed in polar coordinates (r,θ) located at the cavity center whose radius is a, is based for plane stress conditions and remote biaxial loading on the following nonlinear equation

$\frac{3}{7}{\left[\frac{1}{4}{\left(\frac{a}{r}\right)}^2+\frac{1}{12}\right]}^{(n-1)/2}{\left(\sqrt{3\xi }\right)}^{n-1}{\left(\frac{2r^2}{r^2+a^2}{s}_{\theta \theta }\right)}^n+\frac{2r^2}{r^2+a^2}{s}_{\theta \theta }-2\left(1+\frac{3}{7}{\xi }^{n-1}\right)=0\text{ (36)}$

where$\xi ={\overline{\sigma }}_{rr}/Y$ being the applied remote radial stress ${\overline{\sigma }}_{rr}$ normalized with respect to the yield stress Y, and $s_{\theta \theta }={\sigma }_{\theta \theta }/{\overline{\sigma }}_{rr}$ Once the root s_θθ of this equation for an applied value of is ${\overline{\sigma }}_{rr}$ determined, the value of σ_rr is evaluated from

${\sigma }_{rr}=\frac{1-a^2}{1+a^2}{\sigma }_{\theta \theta }$

It should be noted that Ishikawa [8] solution is independent of the elastic parameters of the material.

The comparison between variations along x₃ at x₂ = 0 of σ₂₂ = σ_θθ and σ33 = σrr as predicted by the present approach and Ishikawa [8] analytical solution is shown in Figure 2 for a cavity radius of a/(2h) = 0.28. For a nonlinear epoxy, the material properties are given by Table 1 and the loading is given by${\overline{\in }}_{22}={\overline{\in }}_{33}$ which is incrementally increased up to the value of${\overline{\in }}_{22}={\overline{\in }}_{33}$ =1.8%. Plane stress condition is assumed in the present case such that σ₁₁ = 0 everywhere. This final value of applied strain corresponds to a far-field stress of${\overline{\sigma }}_{22}={\overline{\sigma }}_{33}=$ =142MPa (which is slightly below the yield stress Y = 157.6MPa). The figure shows comparisons between the analytical and the present solutions at far-field loading of${\overline{\in }}_{22}={\overline{\in }}_{33}$ =1.2% (which corresponds to ${\overline{\sigma }}_{22}={\overline{\sigma }}_{33}=$ = 95.6MPa ) and at${\overline{\in }}_{22}={\overline{\in }}_{33}$ =1.8%. Very good agreements between the two solutions can be observed (the radial stress variations coincide). As can be expected, the equivalent plastic strain ${\in }_{eq}^P$ attains its maximum value at the r = a and rapidly decays with increasing r. The maximum values of ${\in }_{eq}^P$ obtained from the applied far-field of 1.2% and 1.8% are 1.7% and 5.3%, respectively.

Figure 2: Elastoplastic material of the Ramberg-Osgood type with an embedded cavity, subjected to a remote biaxial strain loading. Comparison between the response along x₃ at x₂ = 0 between the present and Ishikawa (1975) analytical solution exhibiting σ₂₂ vs. x₃ and σ₃₃ vs. x₃. $(a){\overline{ϵ}}_{22}={\overline{ϵ}}_{33}=1.2\%\text{ and (b) }{\overline{ϵ}}_{22}={\overline{ϵ}}_{33}=1.8\%$, both with ${\overline{\sigma }}_{33}=0$ = 0.

    

    

    Figure 2:  Elastoplastic material of the Ramberg-Osgood type with an
embedded cavity, subjected to a remote biaxial strain loading. Comparison
between the response along x3 at x2 = 0 between the present and
Ishikawa (1975) analytical solution exhibiting σ22 vs. x3 and σ33 vs. x3.
$(a){\overline{ϵ}}_{22}={\overline{ϵ}}_{33}=1.2\%\text{ and (b) }{\overline{ϵ}}_{22}={\overline{ϵ}}_{33}=1.8\%$, both with ${\overline{\sigma }}_{33}=0$ = 0.
    
    

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Table 1: The parameters of the Ramberg-Osgood elastoplastic material.

  

    
E(GPa) 
    
ν 
    
Y (MPa) 
    
a 
    
n 
  
  

    
5.2 
    
0.35 
    
157.6 
    
3/7 
    
4
  
  

    
E, ν, Y , a and n denote the Young’s modulus,    Poisson’s ratio, yield stress and two parameters, respectively.
      
 
  

Table 1:  The parameters of the Ramberg-Osgood elastoplastic material.

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It should be noted that the selection of M₂ = M₃ = 8 as the number of cells results in a ratio of about 60 between the size of the considered region 2H = 2L and the cavity radius a which is sufficiently high.

Applications

In the following applications, results are shown in which an elastoplastic aluminum alloy whose behavior is governed by the incremental classical plasticity, Equation (4), is employed to form the inelastic phase. The parameters of this elastoplastic linear hardening material are given in Table 2.

Table 2: The parameters of the elastoplastic aluminum alloy with linear hardening material.

  

    
E(GPa) 
    
ν 
    
Y (MPa) 
    
H(GPa) 
  
  

    
68.3 
    
0.3 
    
371.5 
    
23 
  
  

    
E, ν, Y and H denote the Young’s modulus, Poisson’s ratio, yield stress and hardening, respectively. 
      
 
  

Table 2:  The parameters of the elastoplastic aluminum alloy with linear hardening
material.

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Figure 3 shows the response of an infinite elastoplastic aluminum alloy in which a cavity of radius a/(2h) = 0.28 is embedded. This system is subjected to a far-field uniaxial strain loading which is incrementally applied reaching the final value of${\overline{\in }}_{22}=0.02\text{ with }{\overline{\in }}_{11}=0$ For a transverse traction-free ${\overline{\sigma }}_{33}=0$ , the component ${\overline{\in }}_{33}$ is determined at any stage of loading by the HFGMC model for a homogeneous elastoplastic material. This figure shows the elastoplastic variations of the σ²² along x₃ at x₃ = 0 at the final stage of loading together with the equivalent plastic strain whose increment is given by

Figure 3: The response along x₃ at x₂ = 0 of elastoplastic material with an embedded cavity, subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$ .

    

    

    Figure 3:  The response along x3 at x2 = 0 of elastoplastic material
with an embedded cavity, subjected to uniaxial strain loading of
${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
 $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$
.
    
    

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$\Delta {\in }_{eq}^p=\sqrt{\frac{2}{3}\Delta {\in }^p:\Delta {\in }^p}\text{ (37)}$

This figure also shows, for a comparison, the corresponding response in the special case of perfectly elastic aluminum. Obviously, the cavity forms a stress concentrator which in the perfectly elastic case generates a stress concentration factor equal to 3 for normal stress in the loading direction. The ratio between σ₂₂ at the edge of the cavity and the far-field ${\overline{\sigma }}_{22}$ value in this latter case shows that the stress concentration factor is about 2.8 as compared to the theoretical value of 3. As expected, the elastoplastic results predict a far lower value of about 2. The graph of the equivalent plastic strain ${\in }_{eq}^P$ indicates that the intensity of the plastic field is concentrated near the edges of the cavity and decays rapidly away from it.

Similarly, the response to a biaxial strain loading of the infinite aluminum alloy with embedded cavity is shown in Figure 4. Here the far-field strain components${\overline{\in }}_{22}={\overline{\in }}_{33}$ are incrementally increased up to 0.02 together with${\overline{\in }}_{11}=0$ . The stress concentration factor in the elastic case provides the value of about 1.9 as compared to the theoretical value of 2. The corresponding elastoplastic value is seen to be about 1.6.

Figure 4: The response along x₃ at x₂ = 0 of elastoplastic material with an embedded cavity, subjected to biaxial strain loading of ${\overline{ϵ}}_{22}={\overline{ϵ}}_{33}=0.02\text{ with }{\overline{ϵ}}_{11}=0.$ with $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$.

    

    

    Figure 4:  The response along x3 at x2 = 0 of elastoplastic material with an
embedded cavity, subjected to biaxial strain loading of ${\overline{ϵ}}_{22}={\overline{ϵ}}_{33}=0.02\text{ with }{\overline{ϵ}}_{11}=0.$
 with $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$.
    
    

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Finally, the response of the infinite aluminum alloy with embedded cavity which is subjected to a biaxial loading of the form ${\overline{\in }}_{22}=-{\overline{\in }}_{33}=0.02\text{ with }{\overline{\in }}_{11}=0$ is shown in Figure 5. Here, the elastic and elastoplastic stress concentration factors are about 3.9 (theoretically 4) and 3.25, respectively.

Figure 5 : The response along x₃ at x₂ = 0 of elastoplastic material with an embedded cavity, subjected to biaxial strain loading of ${\overline{ϵ}}_{22}=-{\overline{ϵ}}_{33}=0.02\text{ with }{\overline{ϵ}}_{11}=0$ with $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$

    

    

    Figure 5:  The response along x3 at x2 = 0 of elastoplastic material with an
embedded cavity, subjected to biaxial strain loading of ${\overline{ϵ}}_{22}=-{\overline{ϵ}}_{33}=0.02\text{ with }{\overline{ϵ}}_{11}=0$
 with $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$ .
    
    

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The next two applications concern with a periodic unidirectional boron/aluminum metal matrix composite in which one of the elastic boron fibers is either completely lost (D = 1) thus forming a cavity, or his Young’s modulus deteriorated to one half of its original value (D = 0.5). These two situation might caused by a production defect or damage during service. The properties of the boron fibers are given in Table 3 and the fiber volume ratio is v_f = 0.25. Figure 6(a) shows the variations of σ₂₂ along x₃ at x₂ = 0 of the boron/aluminum composite with a missing filer at the final stage of uniaxial strain loading of ${\overline{\in }}_{22}=0.02\text{ with }{\overline{\in }}_{11}=0$. The other far-field strain component is determined by the HFGMC modeling of an unperturbed periodic unidirectional boron/aluminum metal matrix composite by imposing the condition that σ₃₃ = 0 . Figure 6(b) and the corresponding variations of the equivalent plastic strain ${\overline{\sigma }}_{33}=0$ reveal that the effect of missing fiber is confined to the vicinity of the cavity with a rapid decay away from the defect location. Figure 6(c) shows a similar behavior when the aluminum matrix is assumed to be perfectly elastic.

Figure 6: The response along x₃ at x₂ = 0 of boron/aluminum elastoplastic composite (vf = 0.25) with a missing fiber, subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ $\text{(a) }{\sigma }_{22}\text{ vs}\text{. }{x}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3,{\sigma }_{22}\text{ vs}\text{. }{x}_3$ assuming perfectly elastic aluminum.

    

    

    Figure 6:  The response along x3 at x2 = 0 of boron/aluminum elastoplastic
composite (vf = 0.25) with a missing fiber, subjected to uniaxial strain loading
of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
 $\text{(a) }{\sigma }_{22}\text{ vs}\text{. }{x}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3,{\sigma }_{22}\text{ vs}\text{. }{x}_3$
 assuming perfectly elastic
aluminum.
    
    

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Table 3: The elastic properties of the boron.

  

    
E(GPa) 
    
ν 
  
  

    
400 
    
0.2 
  
  

    
E and ν denote the Young’s modulus    and Poisson’s ratio,    respectively.
      
 
  

Table 3:  The elastic properties of the boron.

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The corresponding response of the boron/aluminum metal matrix composite in which the value of the Young’s modulus of a single boron fiber is reduced from 400GPa to 200GPa (e.g., by an imperfect bonding) is shown in Figure 7. The first two parts of this figure indicates that the effect of this defect is quite minor in the elastoplastic case, but Figure 7(c) in which the aluminum alloy is assumed to behave elastically a small stress deterioration can be detected across the defective fiber.

Figure 7: The response along x₃ at x₂ = 0 of boron/aluminum elastoplastic composite (ν_f = 0.25) with a half missing fiber, subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ . $\text{(a) }{\sigma }_{22}\text{ vs}\text{. }{x}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3,{\sigma }_{22}\text{ vs}\text{. }{x}_3$ assuming perfectly elastic aluminum.

    

    

    Figure 7:  The response along x3 at x2 = 0 of boron/aluminum elastoplastic
composite (νf = 0.25) with a half missing fiber, subjected to uniaxial strain
loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
 . $\text{(a) }{\sigma }_{22}\text{ vs}\text{. }{x}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3,{\sigma }_{22}\text{ vs}\text{. }{x}_3$

assuming perfectly elastic aluminum.
    
    

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Thus far, the effects of cavities, missing and partially missing fibers have been investigated. Presently, the effect of cracks in homogeneous, fiber reinforced and layered metal matrix composites are addressed.

Consider a single crack in the infinite aluminum alloy which is subjected to a far-field uniaxial strain${\overline{\in }}_{22}=0.02\text{ with }{\overline{\in }}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ . The length of the crack is 2a/(2l) = 0.5. The resulting σ₂₂ and ${\in }_{eq}^P$ variations along the crack line x₃ at x₂ = 0 are shown in Figure 8(a) and (b).

Figure 8: The response along x₃ at x₂ = 0 of elastoplastic material with an embedded crack, subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ . (a) Comparison with the perfectly elastic cases (present and analytical) of ${\sigma }_{22}\text{ vs}\text{. }{x}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$

    

    

    Figure 8:  The response along x3 at x2 = 0 of elastoplastic material
with an embedded crack, subjected to uniaxial strain loading of
${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
 . (a) Comparison with the perfectly elastic cases
(present and analytical) of ${\sigma }_{22}\text{ vs}\text{. }{x}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$.
    
    

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Also shown is the response in the special case when the aluminum is assumed to be perfectly elastic. Here an analytical solution is available, e.g. Sneddon [12], which is compared with the prediction of the present approach in this special elastic case. This comparison reveals an excellent agreement. The reduction of the magnitude of the stress in the vicinity of the crack’s tip in the elastoplastic case is well observed and the sharp high values of the plastic strains there is shown in Figure 8(b).

The next application concerns with the periodic boron/ aluminum metal matrix composite (ν_f = 0.25) with an embedded crack of length 2a/(2l) = 0.6 along the x3-axis within the aluminum matrix. The composite is subjected to a far-field uniaxial strain ${\overline{\in }}_{22}=0.02\text{ with }{\overline{\in }}_{11}=0$ . As discussed, the other strain component is determined by the HFGMC by imposing the condition σ₃₃ = 0 . The resulting variations of σ₂₂ and ${\in }_{eq}^P$ along the crack’s line are shown in Figure 9(a) and (b), respectively. Also shown in Figure 9(a) is thecorresponding variation when the aluminum is assumed to behave as a perfectly elastic material. The reduction of the magnitude of the stress in crack’s tip as compared to elastic case is noticeable. The equivalent plastic strain reaches the value of 0.22 (not shown in the figure) rendering the employed infinitesimal plasticity theory questionable there.

Figure 9: The response along x₃ at x₂ = 0 of boron/aluminum elastoplastic composite (ν_f = 0.25) with an embedded crack, subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ and $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$

    

    

    Figure 9:  The response along x3 at x2 = 0 of boron/aluminum elastoplastic
composite (νf = 0.25) with an embedded crack, subjected to uniaxial strain
loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
 and $\text{(a) }{\sigma }_{22}\text{ vs}{\text{. x}}_3,\text{ (b) }{ϵ}_{eq}^P\text{ vs}\text{. }{x}_3$
.
    
    

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It should be interesting to show the distribution of the equivalent stress σ_eq and plastic strain ${\in }_{eq}^P$ in the vicinity of the crack. These are shown in Figure 10(a) and (b), respectively, in the region -2 = x₂/ (2h) = 2, -2 = x₃/(2l) = 2. These two figures well display the resulting effect of the crack on its surrounding region of fibers and matrix. For comparison, Figure 10(c) shows the corresponding equivalent stress distribution when the aluminum matrix is assumed to behave elastically. Note that the scale of the plot is twice the scale of the elastoplastic case.

Figure 10: Field distribution of boron/aluminum elastoplastic composite (vf = 0.25) with an embedded crack in the region -2 = x₂/(2h) = 2, -2 = x₃/(2l) = 2. The composite is subjected to uniaxial strain loading of${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ . (a) The distribution of the equivalent stress σ_eq, (b) of the equivalent strain Σ_eq and (c) The distribution of the equivalent stress σ_eq assuming a perfectly elastic aluminum.

    

    

    Figure 10:  Field distribution of boron/aluminum elastoplastic composite (vf =
0.25) with an embedded crack in the region -2 = x2/(2h) = 2, -2 = x3/(2l) = 2. The
composite is subjected to uniaxial strain loading of${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
.
(a) The distribution of the equivalent stress σeq, (b) of the equivalent strain
Σeq and (c) The distribution of the equivalent stress σeq assuming a perfectly
elastic aluminum.
    
    

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Our last application of the present theory concerns with the effect of a broken ceramic layer in periodically ceramic/aluminum layered composite. The elastic Al₂O₃ ceramic layer is characterized in Table 4. The composite is subjected to a far-field uniaxial strain ${\overline{\in }}_{22}=0.02\text{ with }{\overline{\in }}_{11}=0$ and other applied strain component is determined by the HFGMC analysis by imposing the condition σ₃₃ = 0 . Comparisons between the variations of σ₂₂ along the crack’s line are shown in Figure 11 of the elastoplastic and elastic cases. In this figure, the widths of the ceramic and aluminum layer are equal so that the crack’s length is 2a/(2l) = 0.5.

Table 4: The elastic properties of the Al2O3 ceramic material

  

    
E(GPa) 
    
ν 
  
  

    
380 
    
0.2 
  
  

    
E and ν denote the Young’s modulus    and Poisson’s ratio,    respectively. 
      
 
  

Table 4:  The elastic properties of the Al2O3 ceramic material

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Figure 11: The response along x₃ at x₂ = 0 of periodically layered Al₂O₃/ aluminum elastoplastic composite (ν_f = 0.5) with an embedded crack in the ceramic phase. The composite is subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ . The figure shows a comparison of σ₂₂ vs. x₃ with the perfectly elastic case.

    

    

    Figure 11:  The response along x3 at x2 = 0 of periodically layered Al2O3/
aluminum elastoplastic composite (νf = 0.5) with an embedded crack in
the ceramic phase. The composite is subjected to uniaxial strain loading of
${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
 . The figure shows a comparison of σ22 vs. x3
with the perfectly elastic case.
    
    

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Let us consider a very thin aluminum layer such that the length of the crack in the broken ceramic layer is 2a/(2l) = 0.95. The resulting comparison between the variations of σ₂₂ along the crack’s line is shown in Figure 12(a) for the elastoplastic and elastic aluminum layer. More interesting is distribution of the equivalent plastic strain ${\in }_{eq}^P$ in the region -2 = x₂/(2h) = 2, -2 = x₃/(2l) = 2. This is shown in Figure 12(b) where ${\in }_{eq}^P$ reaches the value of 0.45 in the vicinity of the crack’s tip. In addition, the spread of plasticity over quite an extended region along the aluminum layer is notable.

Figure 12: a) The response along x₃ at x₂ = 0 of periodically layered Al₂O₃/ aluminum elastoplastic composite (ν_f = 0.95) with an embedded crack in the ceramic phase. The composite is subjected to uniaxial strain loading of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$ . The figure shows a comparison of σ₂₂ vs. x3 with the perfectly elastic case. (b) The resulting equivalent plastic strain distribution in the region -2 = x₂/(2h) = 2, -2 = x₃/(2l) = 2.

    

    

    Figure 12:  (a) The response along x3 at x2 = 0 of periodically layered Al2O3/
aluminum elastoplastic composite (νf = 0.95) with an embedded crack in
the ceramic phase. The composite is subjected to uniaxial strain loading
of ${\overline{ϵ}}_{22}=0.02\text{ with }{\overline{ϵ}}_{11}=0\text{ and }{\overline{\sigma }}_{33}=0$
. The figure shows a comparison of σ22 vs. x3
with the perfectly elastic case. (b) The resulting equivalent plastic strain
distribution in the region -2 = x2/(2h) = 2, -2 = x3/(2l) = 2.
    
    

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Asymptotic fracture analysis of the considered case of a very thin elastoplastic layers sandwiched between brittle elastic ones was performed by Chan et al. [13]. In this latter work a transverse crack in an elastic layer of thickness 2w was terminated at the plastic zone which was modeled by open or closed cracks of length 2d developed within elastoplastic layers, thus forming an H-crack configuration. It was found, in particular, that in the case of shear traction- free closed cracks in the plastic zone of the size characterized by the ratio d/w = 0.5, the stress concentration factor for the tensile stresses in the brittle layer adjacent to the cracked one is about 2. Inspection of the results in Figure 12 obtained in the present investigation shows that they confirm with Chan et al. [13] finding.

Conclusion

A method for the modeling of localized damage in elastoplastic composites with periodic microstructure has been presented. As a result of the localization effects (perturbations) the periodicity is lost and a repeating unit cell cannot be identified anymore. The method is based on the combination of three distinct types of analyses, namely the representative cell method, the higher-order theory for inelastic composites and the high-fidelity generalized method of cells micromechanical analysis which is needed for the determination of the far-field boundary conditions.

The representative cell method has been previously applied to solve linear and nonlinear problems, but the present method enables for the first time the analysis of elastoplastic materials. To this end, the plasticity effects are represented in the form of eigenstresses which are distributed over the entire region. In addition, these eigenstresses include the perturbation effects which are distributed over the damaged regions. The method has been applied in various cases for the prediction of the field distributions in elastoplastic materials and metal matrix composites with fiber loss and crack. Finally, the elastoplastic field in ceramic/aluminum layered composite in which a single ceramic layer is broken has been determined. Results confirm with a previously derived asymptotic analysis [13]. Presently, results for normal loading have been presented. Shear loading can be carried out in the same manner.

The present approach can be extended to the analysis of elasticviscoplastic composites with a localized damage by changing the flow rule of the elastoplastic matrix, Equation (4), to a viscoplastic one. In addition, although the present analysis has been confined to isothermal conditions, the inclusion of temperature effects in the eigenstresses can be easily performed, see Ryvkin and Aboudi [5] for the case of H-cracks in thermoelastic composites. Another possible general-ization is to allow the damage to evolve with the applied loading. Thus instead of applying an a priori a constant damage (e.g., D = 0.5 in the previously discussed case of a boron fiber with a reduced Young’s modulus), an evolution law can be adopted which according to Lemaitre and Desmorat [10] has the form

$\Delta D={\left(\frac{\text{y}}{S}\right)}^S\Delta {\in }_{eq}^P\text{ (38)}$

where y is the energy release, S and s are material parameters. Finally, extension to inelastic metal matrix composites with triply periodic microstructure (short-fiber composites) which include localized damage that appears in several locations is possible.

Acknowledgment

The first author is grateful for the support of the German-Israel Foundation (GIF) under contract 1166-163.

References

1. Aboudi J, Ryvkin M. The effect of localized damage on the behavior of composites with periodic microstructure. Int J Engng Sci. 2012; 52: 41-55.
2. Ryvkin M, Nuller B. Solution of quasi-periodic fracture problems by the representative cell method. Comp Mech. 1997; 20: 145-149.
3. Aboudi J, Ryvkin M. The analysis of localized effects in composites with periodic microstructure. Philos Trans A Math Phys Eng Sci. 2013; 371: 20120373.
4. Aboudi J, Arnold SM, Bednarcyk BA. Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach. Elsevier, Oxford, UK. 2013.
5. Ryvkin M, Aboudi J. Stress redistribution due to cracking in periodically layered composites. Eng Fract Mech. 2012; 93: 225-238.
6. Aboudi J. Field distributions in cracked periodically layered electro-magneto-thermo- elastic composites. J Intell Mater Sys Struct. 2012; 24: 381-398.
7. Aboudi J. The effect of localized internal defects on the field distributions of electrostrictive composites. Int J Engng Sci. 2014; 75: 135-153.
8. Ishikawa H. Stresses in the plastic range around a circular hole in an infinite sheet subjected to equal biaxial tension. J Appl Math Mech (ZAMM). 1975; 55: 171-178.
9. Mendelson A. Plasticity: Theory and Applications, MacMillan, New York. 1968.
10. Lemaitre J, Desmorat R. Engineering Damage Mechanics, Springer, Berlin. 2005.
11. Budiansky B, Mangasarian OL. Plastic stress concentration at a circular hole in an infinite sheet subjected to equal biaxial tension. J Appl Mech. 1960; 27: 59-64.
12. Sneddon IN. Fourier Transforms. McGraw Hill, New York. 1951.
13. Chan KS, He MY, Hutchinson JW. Cracking and stress redistribution in ceramic layered composites, Mater. Sci Eng. 1993; A167: 57-64.

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Citation: Aboudi J and Ryvkin M. Localized Effects in Periodic Elastoplastic Composites. Ann J Materials Sci Eng. 2015;2(1): 1017. ISSN:2471-0245

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