# A Continuum Damage Mechanics Model for the Static and Cyclic Fatigue of Cellular Composites

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Properties and Fatigue Behavior

#### 2.1. Constituent Materials

#### 2.2. Quasi-Static Mechanical Properties

_{y}is defined as stress at 0.1% plastic strain.

#### 2.3. Static and Cyclic Fatigue Behavior

_{m}≠ 0 [24].

_{max}/σ

_{y,T}≤ 1.4 in tension and 0.66 < |σ|

_{max}/σ

_{y,C}≤ 0.91 in compression. For the cyclic damage, the stress ranges 0.4 < σ

_{max}/σ

_{y,T}≤ 0.91 in tension and 0.66 < |σ|

_{max}/σ

_{y,C}≤ 0.91 in compression were used. The stress ratios R and the frequencies were also varied. It was found that the damage process under cyclic loading is an interaction between the static and the cyclic damage.

_{max}/σ

_{y,T}= 0.5 and R = 0.1. A crack passing through the grain can be seen. At low load levels, only a few cracks were found in single grains of the tested specimens. The cracks are oriented primarily perpendicular to the loading direction.

_{max}/σ

_{y,T}= 0.82. It was found that, in this case, almost all of the grains have several cracks. Small cracks were also observed in the matrix walls between two adjacent grains.

_{max}/σ

_{y,T}= 0.7 are shown in Figure 5a. In general, a permanent decrease in stiffness was observed with ongoing damage in all specimens.

_{0}is the initial Young’s modulus of the undamaged specimen, which is defined as the initial stiffness of the first unloading path of the initial σ-ε hysteresis loop at low stresses. Figure 5b shows a schematic of a typical damage evolution curve. This kind of damage evolution behavior is typical for static and for cyclic loading. In the first stage, the damage rate is high. An initial damage D

_{0}may occur immediately depending on the applied load level. During Stage II, which is the longest stage, the damage rate is low. In the final Stage III, the damage rate increases again and a macrocrack develops in the specimen, leading to the final failure of the specimen. The critical value of damage prior to fracture is called D

_{c}(see e.g., [30]).

_{0}and D

_{c}. In case of compression, D

_{0}and D

_{c}are nearly independent of the load level and can be set as constant to 0.05 and 0.4, respectively (see Figure 6a). When D

_{c}is higher than 0.4, the material is considered to be completely damaged and not suitable for load carrying applications anymore. Such behavior in compression is different to that in tension as shown in Figure 6. Figure 7 shows the values of D

_{0}and D

_{c}as function of the applied load level determined during static tensile tests and cyclic tensile tests at different frequencies.

_{0}as well as of D

_{c}when they are described as a function of maximum stress for the static and cyclic tests. When the stress is sufficiently low, no initial damage develops. This behavior can be explained by the existence of a threshold stress intensity factor K

_{I0}in static tests and ΔK

_{I0}in cyclic tests. The threshold values K

_{I0}and ΔK

_{I0}depend on the crack length and density in the glass foam grains. These cracks are formed during cooling phases in the manufacturing process. When the applied stress intensity factor is lower than the threshold value, no cracks will develop in the glass foam. The critical damage D

_{c}also depends on the load level, and can reach a maximum value of about 0.75 (see Figure 7b). This is a stage when all grains are damaged and the load is carried only by the matrix.

_{0}. In the range 0.0 < σ

_{max}/σ

_{y,T}≤ 1.0, D

_{0}can be described by

_{0}is not exactly zero below σ

_{max}/σ

_{y,T}= 0.5. However, the calculated damage is very low for these load levels so that this has no significant effect on the results of the damage model.

_{c}is approximated by the quadratic polynomial

_{max}/σ

_{y,T}≤ 1.0. The specific numbers in Equations (2) and (3) result from a fitting process. For the derivation of Equation (3), only results at a 20 Hz test frequency are considered (see Figure 6b), which leads to a conservative fatigue life estimation. At load levels σ

_{max}/σ

_{y,T}≤ 0.5, no initial damage develops, and there was also no fracture observed. This can be explained by the threshold stress intensity factor, which is higher than the applied stress intensity factor in the specimen, which prohibits crack growth.

## 3. Damage Modeling and Parameter Identification

#### 3.1. Static Damage Model

_{max}/σ

_{y,T}≤ 0.91 for tensile damage and 0 ≤ |σ|

_{max}/σ

_{y,C}≤ 0.91 for compressive damage. The parameters for the damage model are identified on the basis of the experimentally determined damage evolution in pure static damage tests.

_{0}and the fracture condition D(t = t

_{f}) = D

_{c}, the time to fracture t

_{f}can be calculated by the integration of Equation (4) and results in

_{f}depends on the initial damage D

_{0}, the critical damage D

_{c}, and the damage exponent p. The identification of D

_{0}and D

_{c}is explained in Section 2.3. The identification of the damage exponent p is carried out in Section 3.1.1.

#### 3.1.1. Damage Exponent, p

_{f}depends on the applied stress level (see Figure 8b). This means that there is nonlinear damage accumulation, which leads to sequence effects. Negative damage exponents describe disproportionately high damage growth at the beginning of the lifetime, whereas positive damage exponents describe highly progressive damage growth at the end of the lifetime.

_{c}= 0.5 is chosen alternatively in order to identify the damage exponent p. This critical damage is reached for all tested stress levels, and the damage gradient becomes very small then (see Figure 8b). In the case of compressive loading, a critical damage D

_{c}= 0.4 is chosen because the damage rate becomes very high when this value is reached, which leads to the very fast failure of the specimens.

#### 3.1.2. Stress Exponent, m

_{a}, the stress exponent β, the damage exponent α, and the mean stress parameter M, by

#### 3.1.3. Material Parameter, A

#### 3.1.4. Static Damage Model Parameters

#### 3.1.5. Validation of the Static Damage Model

#### 3.2. Cyclic Damage Model

_{a}, the stress exponent β, the damage exponent α, and the parameter M for the mean stress dependency, by

_{0}at N = 0 yields the damage D as a function of the number of cycles

_{c}at N = N

_{f}, the fatigue life N

_{f}can be calculated as

_{f}as

_{0}, the critical damage D

_{c}, and the damage exponent α.

#### 3.2.1. Damage Exponent, α

_{0}and D

_{c}are known from experiments (see Figure 7), α remains the only unknown parameter. It can be identified directly by a parameter variation when modeling damage evolution curves with the help of Equation (14) and a comparison with damage evolution curves obtained by experiments. Since α has to be a function of applied stress level because of sequence effects, an appropriate relationship has to be found. It was found in the experiments that there is a good correlation between the damage rate and the maximum stress magnitude |σ| applied in the cyclic tests [13]. The parameter identification is performed using damage evolution curves from compressive loading at R = 10 and from tensile loading at R = 0.1.

_{0}= 0.05 and a critical damage D

_{c}= 0.4 are used for all load levels in the compressive loading tests, which results in a good correlation between the experiments and the damage model. For tensile loading, the initial damage D

_{0}and the critical damage D

_{c}depend on the load level (see Table 3). Also, in the case of cyclic tensile loading, a good correlation between the experiments and the damage model is achieved (see Figure 10).

#### 3.2.2. Stress Exponent, β

#### 3.2.3. Mean Stress Parameter, M

_{0}is a material parameter, and σ

_{A}is the fatigue limit, which is determined experimentally. σ

_{A}is approximated using the Gaussian distribution

_{a}(σ

_{m}) for N

_{f}= 5 × 10

^{6}). The Gaussian distribution is commonly used for the description of the fatigue limit in the case of fiber reinforced plastics; see e.g., [35]. The material parameter M

_{0}is identified together with the stress exponent β with the help of a parameter variation.

#### 3.2.4. Cyclic Damage Model Parameters

#### 3.2.5. Validation of the Cyclic Damage Model

#### 3.3. Damage Interaction Model

#### 3.4. Numerical Integration

_{c}/500, the cycle jumping will be activated. This threshold value represents a good agreement between accuracy and computation costs. The number of omitted cycles is computed by

_{jump}and N + ΔN

_{jump}. A linear damage growth is supposed during cycle-jumping. In the case when the damage rate is high and the damage growth in a single loading cycle is higher than the threshold value, the damage computation is continued by the next cycle.

## 4. Damage Model Application

#### 4.1. Mean Stress Effect in Pure Cyclic Damage

_{m}≤ −5 MPa are calculated with the model parameters for compressive damage, and those in the region of σ

_{m}≥ −4 MPa are calculated with the model parameters for tensile damage. The values of the region −5 MPa < σ

_{m}< −4 MPa are linearly interpolated (see Figure 12). This avoids the discontinuity which would otherwise occur. In addition, the experimentally determined results for constant R-values are drawn in Figure 12.

_{f}= 5 × 10

^{6}cycles is represented by the damage model with high precision except for a small range at about σ

_{m}= −4 MPa. In this range, the model for tensile damage results in slightly higher endurable amplitudes compared to the experimental values. The finite fatigue life strength values for R = 10 and R = 0.1 coincide with the experimental data, since the parametrization of the damage model is based on these R-values. For reversed loading (R = −1), the endurable amplitudes at high lifetimes are approximated by the model very accurately. At lower lifetimes (N

_{f}≤ 10

^{5}), the model provides higher endurable amplitudes in comparison to the experimental results. The reason is the fact that the damage model was developed and validated for cyclic tensile (R = 0.1) and compressive (R = 10) loading. To increase the accuracy of the model, more R-values could be taken into account for the parameterization of the damage model. Within the model, either tensile or compression damage is considered. Especially in the range of about σ

_{m}= −4.5 MPa, both tensile and compressive damage may appear as shown in [13]. Thus, a higher total damage within the cellular composite occurs, which is not included in the model. Further research work is needed to adequately describe the damage evolution in this region.

#### 4.2. Frequency Effect in Static and Cyclic Damage Interaction

_{f}and the time to failure t

_{f}, which is obtained by dividing the number of cycles N

_{f}by the test frequency f. The results from the damage model agree well with the experimental results.

_{f}decrease accordingly. It can be seen in Figure 13a that the cycles to failure N

_{f}decrease continuously by reducing the test frequency, because the static damage gains more influence. In contrast, the fatigue life for f = 100 Hz is only slightly higher than that for f = 20 Hz. This means that the influence of static damage becomes lower at higher frequencies.

_{f,static}for the amount of pure static damage at cyclic loading can be calculated, when setting the term for the cyclic damage to zero. This results in a time to failure t

_{f,static}which is for the same waveform independent of the frequency, and represents an upper limit for the accessible time to failure t

_{f}. Figure 13b shows that almost pure static damage is present for f = 0.1 Hz. The corresponding Wöhler line coincides with the Wöhler line for pure static damage, which is represented by the time to failure t

_{f,static}. A further reduction of the frequency does not lead to an increasing time to fracture t

_{f}. Therefore, the frequency f = 0.1 Hz represents the lower frequency limit, where static damage is dominant. The time to failure at higher frequencies is shorter than that of lower frequencies, because more cyclic damage is accumulated.

_{f}for different frequencies and waveforms are calculated via Equation (17) and normalized by the cycles to fracture for pure static damage N

_{f,static}or the cycles to fracture for pure cyclic damage N

_{f,cyclic}, and are drawn as an interaction diagram in Figure 14. This diagram illustrates the portions of static damage or cyclic damage of the total damage. The cycles to fracture for pure cyclic damage N

_{f,cyclic}are calculated with Equation (13). The cycles to fracture for pure static damage N

_{f,static}are determined using t

_{f,static}and the frequency f, with

_{f,static}= t

_{f,static}f.

_{max}/σ

_{y,C}= 0.70…0.91 in five equidistant steps for all selected waveforms and frequencies. The interaction diagram shows that the cyclic damage is dominant for the triangular waveform with f = 30 Hz. The amount of static damage is, on average, below 3%. Therefore, there is nearly pure cyclic damage present. In the case of the sinusoidal signal with f = 0.1 Hz, static damage dominates.

#### 4.3. Influence of the Waveform

## 5. Conclusions

_{f}does not decrease anymore. On the other hand, there is an upper frequency limit, above which the number of cycles to fracture N

_{f}does not depend on the frequency. Between the lower and the upper frequency limit, there is always an interaction between the static and the cyclic damage. In addition, it was shown that the damage interaction between the static and cyclic damage is less pronounced, and that there is almost a linear damage accumulation present.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Sample of a cellular composite with 1–2 mm glass foam granules as place holders and epoxy resin matrix; (

**b**) SEM image of a glass foam grain; (

**c**) SEM image of the fracture surface of a cellular composite with 1–2 mm glass foam granules, taken from a tensile test (from [9] by permission of Springer Science + Business Media).

**Figure 4.**SEM image of the lateral surface of a cyclic damaged specimen after fracture (loading in vertical direction) for (

**a**) cyclic tensile loading at σ

_{max}/σ

_{y,T}= 0.5 and R = 0.1, and (

**b**) static tensile loading at σ

_{max}/σ

_{y,T}= 0.82.

**Figure 5.**(

**a**) Typical stress-strain hysteresis loops obtained from a cyclic tensile test (R = 0.1) at 20 Hz and σ

_{max}/σ

_{y,T}= 0.7; (

**b**) Schematic of a typical damage development during static and cyclic tests.

**Figure 6.**Experimental results of typical damage evolution curves for pure cyclic damage: (

**a**) Compressive loading (R = 10, 30 Hz, triangular waveform); (

**b**) Tensile loading (R = 0.1, 20 Hz, sinusoidal waveform).

**Figure 7.**Values of initial damage D

_{0}(

**a**) and critical damage D

_{c}(

**b**) during static and cyclic tensile tests (R = 0.1) for different load levels and frequencies.

**Figure 8.**Influence of damage exponent p on the damage evolution curves: (

**a**) damage development for different values of p, calculated by Equation (6) (D

_{0}= 0.05, D

_{c}= 0.4); (

**b**) comparison of damage evolution curves for tensile and compressive loading (experiment and damage model).

**Figure 9.**(

**a**) Comparison of creep strength lines for static loading (damage model, Equation (5) and experiment); (

**b**) Comparison of numerically (damage model, Equation (5)) and experimentally determined lifetimes at the marked load levels in (

**a**).

**Figure 10.**Comparison of damage evolution curves for tensile and compressive loading and pure cyclic damage (experiment and damage model): (

**a**) compressive loading (R = 10); (

**b**) tensile loading (R = 0.1).

**Figure 11.**(

**a**) Comparison of fatigue lives for cyclic tensile (R = 0.1, f = 20 Hz, sinusoidal waveform) and compressive (R = 10, f = 30 Hz, triangular waveform) loading (experiment and damage model, Equation (13)); (

**b**) Comparison of numerically (damage model, Equation (13)) and experimentally determined fatigue lives at the marked load levels in (

**a**).

**Figure 13.**Effect of loading frequency in the case of compressive cyclic loading (R = 10) in terms of (

**a**) cycles to failure N

_{f}; and (

**b**) time to failure t

_{f}.

**Figure 14.**Calculated portions of static and cyclic damage with different waveforms and frequencies in the case of compressive cyclic loading (R = 10) for different load levels.

**Figure 15.**Influence of the waveform and frequency on the cycles to fracture N

_{f}for compressive cyclic loading (R = 10).

**Table 1.**Mechanical properties of the cellular composite with 1–2 mm glass foam granules (data from [13]).

Density ρ (g/cm^{3}) | Young’s Modulus E (MPa) | Poisson’s Ratio ν (–) | Tensile Yield Strength σ_{y,T} (MPa) | Tensile Strength σ_{u,T} (MPa) | Compressive Yield Strength σ_{y,C} (MPa) | Compressive Strength σ_{u,C} (MPa) |
---|---|---|---|---|---|---|

0.72 | 3098 | 0.31 | 5.8 | 9.1 | 17.1 | 18.3 |

Loading | A | m | p | D_{0} | D_{c} |
---|---|---|---|---|---|

compression | 4.5 × 10^{−39} | 30 | −1.8 |σ|/(MPa) + 27.5 | 0.05 | 0.4 |

tension | 7.0 × 10^{−29} | 40 | −4.5 σ/(MPa) + 8.0 | Equation (2) | 0.5 |

Damage in | α | β | M_{0} | D_{0} | D_{c} |
---|---|---|---|---|---|

compressive region | −0.43 |σ|_{max}/(MPa) + 7.3 | 4.75 | 12.7 | 0.05 | 0.4 |

tensile region | −3.2 σ_{max}/(MPa) + 8.8 | 4.5 | 90.9 | Equation (2) | Equation (3) |

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**MDPI and ACS Style**

Diel, S.; Huber, O.
A Continuum Damage Mechanics Model for the Static and Cyclic Fatigue of Cellular Composites. *Materials* **2017**, *10*, 951.
https://doi.org/10.3390/ma10080951

**AMA Style**

Diel S, Huber O.
A Continuum Damage Mechanics Model for the Static and Cyclic Fatigue of Cellular Composites. *Materials*. 2017; 10(8):951.
https://doi.org/10.3390/ma10080951

**Chicago/Turabian Style**

Diel, Sergej, and Otto Huber.
2017. "A Continuum Damage Mechanics Model for the Static and Cyclic Fatigue of Cellular Composites" *Materials* 10, no. 8: 951.
https://doi.org/10.3390/ma10080951