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# Computer Art Design Model Based on Nonlinear Fractional Differential Equations

###### Accettato: 12 Jun 2022
Dettagli della rivista
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Introduction

Recently fractional calculus is being widely used in studying materials used in viscoelastic art design. It has been found that the fractional constitutive model has the advantages of few parameters and simple form. And the parameters in the model can refer to the mechanical properties of materials used in artistic design. This makes it possible to describe the mechanical properties quantitatively. However, the constant fractional order is invariable, and it does not consider the change of mechanical properties with time. In practice, the mechanical properties of materials used in artistic design change with time during the stress process [1]. The order should also change with time during the stress process.

The nonlinear fractional differential constitutive model has also begun to arouse the interest of many researchers. Nonlinear fractional differentiation is the latest development in fractional calculus. Its order is not only no longer an integer but can also vary in time or space. The so-called variational order model describes the evolution of mechanical properties during the deformation process through the change of order. Research in this area has just started in the international arena [2]. It has not attracted the attention of domestic scholars. In this paper, we hope to establish a variational fractional constitutive model.

This paper proposes a variable fractional constitutive model based on the constant fractional constitutive model. Theoretically, we can show the change in the mechanical properties of the materials used in art design by changing the order of variation. But whether the results are reliable is a question. We carried out constant strain rate loading tests on ductile metals copper, aluminum, and low carbon steel. We use the variational fractional constitutive model proposed in this paper for analysis. In this way, it is explored whether the evolution of its mechanical properties during the stress process is consistent with the actual situation [3]. In this way, it is verified whether the model can be used to describe the evolution of mechanical properties of materials used in artistic design.

The order studied in this paper varies from 0 to 1. The model can also be used to analyze the evolution of mechanical properties of materials used in other viscoelastic art designs in stress and deformation with time effects. For example, stress relaxation and creep of materials used in geotechnical and polymer art design, loading at equal strain rate, etc. Metals are chosen as research objects because the theory of solid mechanics is derived from plastic metals.

Introduction to Variational Order Differentiation

Nonlinear fractional calculus is developed based on constant fractional calculus. There are many definitions of variational order in academic circles, and order a(t, τ) generally has three forms: a(t, τ) = a(t), a(t, τ) = a(t), a(t, τ) = a(tτ). The first form has no order memory, the second form has a weaker order memory, and the third form has a strong order memory. The so-called order memory means that the calculation result of the order at the historical time point influences the calculation result of the current time, and the memory strength is determined by the influence of the change of the order. The third form is adopted in this paper. In this paper, the Caputo-type fractional differential theory operator is used. The a(t) order integral of a function f(t) at a(t, τ) = a(tτ) is defined as $aCDta(t)f(t)=∫at(t−τ)n−a(t−τ+a)−1Γ(n−a(t−τ+a))f(n)(τ)dτ, 0≤a(t)≤1$ _a^CD_t^{a\left(t \right)}f\left(t \right) = \int_a^t {{{{{\left({t - \tau} \right)}^{n - a\left({t - \tau + a} \right) - 1}}} \over {\Gamma \left({n - a\left({t - \tau + a} \right)} \right)}}{f^{\left(n \right)}}\left(\tau \right)d\tau,\,0 \le a\left(t \right) \le 1}

When f(t) = at is tτ = x in formula (1), then formula (1) becomes formula (2) $0CDta(t)f(t)=a∫0t−x−a(x)Γ(1−a(x))dx=g(t), 0≤a(t)≤1$ _0^CD_t^{a\left(t \right)}f\left(t \right) = a\int_0^t {{{- {x^{- a\left(x \right)}}} \over {\Gamma \left({1 - a\left(x \right)} \right)}}dx = g\left(t \right),\,0 \le a\left(t \right) \le 1}

In this paper, formula (2) is written in the form of superposition. Assuming $g(tk,ak)=0Dtkakf(t)$ g\left({{t_k},{a_k}} \right){= _0}D_{{t_k}}^{{a_k}}f\left(t \right) , then we can get $aCDta(t)f(t)=∑k=1ng(tk,ak)−g(tk−1,ak),(0=t0 \matrix{{_a^CD_t^{a\left(t \right)}f\left(t \right) = \sum\limits_{k = 1}^n {g\left({{t_k},{a_k}} \right) - g\left({{t_{k - 1}},{a_k}} \right),}} \hfill \cr {\left({0 = {t_0} < {t_1} < {t_2} \cdots < {t_n} = t} \right)} \hfill \cr}

a(t) in formula (3) is changed according to Table 1.

Values of order a(t)

Order Time zone
a1 0 < t < t1
a2 t1 < t < t2
an tn−1 < t < t2
The variational first-order constitutive model
Constant fractional constitutive model

The constitutive equations of materials used in artistic design between ideal solids and ideal fluids can be described by the “intermediate model” proposed by Smit and DeVries $σ=Eθadaεdta(0≤a≤1)$ \sigma = E{\theta ^a}{{{d^a}\varepsilon} \over {d{t^a}}}\left({0 \le a \le 1} \right)

E, θ, μ is the material constant used in art design. When a is a constant, the mechanical properties of the materials used in artistic design do not change, so this is called a constant fractional-order model. The fractional constitutive formula (4) model can treat the mechanical properties as a sequence. Ideal solids and Newtonian fluids lie at opposite ends of the sequence. Different a in a series of parameters a represent different mechanical properties of materials used in art design. It can find the exact location of the mechanical properties of materials used in artistic design between ideal solids and Newtonian fluids. When ɛ = at in formula (4), formula (4) can be transformed into $σ=E(aθ)aε1−aΓ(2−a)(0≤a≤1)$ \sigma = E{\left({a\theta} \right)^a}{{{\varepsilon ^{1 - a}}} \over {\Gamma \left({2 - a} \right)}}\left({0 \le a \le 1} \right)

The stress increases gradually during the constant strain rate loading process. The constant fractional-order model can describe the relationship between stress and strain during loading with a constant strain rate. But the order in the model proposed by formula (5) is constant. This means that the mechanical properties of the materials used in artistic design remain unchanged during constant strain rate loading. This does not match reality. Therefore, we must explore the use of nonlinear fractional differential theory.

Variational first-order constitutive model

In the constant fractional constitutive model, the order refers to the mechanical properties of materials used in artistic design [4]. Our understanding of the properties of materials used in art design changes from qualitative to quantitative. It still cannot quantitatively describe the changing process of mechanical properties. We consider establishing a variational fractional constitutive model [5]. Ingman and Samko have proposed $σ=Eda(t)εdta(t)(0≤a(t)≤1)$ \sigma = E{{{d^{a\left(t \right)}}\varepsilon} \over {d{t^{a\left(t \right)}}}}\left({0 \le a\left(t \right) \le 1} \right)

Because the E dimension in the formula is [stress] [time] a(t). So its physical meaning is uncertain. In addition, this variational order model still cannot reflect the evolution of mechanical properties of materials used in art design. This paper argues that the correct model should be $σ=Eθa(t)da(t)ε(t)dta(t)(0≤a(t)≤1)$ \sigma = E{\theta ^{a\left(t \right)}}{{{d^{a\left(t \right)}}\varepsilon \left(t \right)} \over {d{t^{a\left(t \right)}}}}\left({0 \le a\left(t \right) \le 1} \right)

Like formula (4), the mechanical parameters in formula (7) have precise physical meanings. $σ(t)=∑k=1nE(aθ)akεk1−ak−εk−11−akΓ(2−ak)$ \sigma \left(t \right) = \sum\limits_{k = 1}^n {E{{\left({a\theta} \right)}^{{a_k}}}{{\varepsilon _k^{1 - {a_k}} - \varepsilon _{k - 1}^{1 - {a_k}}} \over {\Gamma \left({2 - {a_k}} \right)}}}

a in the formula is the strain rate. Initial order a1 = 0. When the experimental curve reaches the horizontal state, the mechanical properties of the materials used in the artistic design are Newtonian fluids. The order is now 1. Therefore the parameter E can be obtained from the experimental points in the first few linear states [6]: $E=σ1ε1$ E = {{{\sigma _1}} \over {{\varepsilon _1}}}

σ1 and ɛ1 are the stress-strain values of the material used in artistic design in the elastic stage. The value of the parameter θ is $θ=σakEa$ \theta = {{{\sigma _{{a_k}}}} \over {Ea}}

σak is the stress when the curve is horizontal. a is the strain rate.

Mechanical properties evolution of plastic metal during tension
Introduction to ductile metal testing

Here, three kinds of metal copper, low carbon steel, and aluminum alloy were tested in constant strain rate tensile tests. The diameter of the test sample is 10mm, and the length is 50mm and 100mm, respectively. Tensile tests were divided into two groups—3 times per group. The strain rates during stretching were different for the two groups. Copper is 0.000133/s and 0.00133/s; mild steel is 0.0001/s and 0.001/s, aluminum alloy is 0.0002/s and 0.002/s.

Test data and analysis

Figure 1 shows the stress-strain results of the above-mentioned tensile test of the metal specimen. From these curves, it can be found that the specimen exhibits an elastic state in solid mechanics at the initial stage of loading. However, once the limit of this stage is exceeded, the properties of the materials used in artistic design undergo a sudden change. It was evident that the properties of the materials used in the artistic design are constantly changing during the whole nonlinear stage loading process. Figure 1 shows that the strain rate affects copper. The dependence on aluminum alloy and low carbon steel strain rate is relatively weak.

When the strain rate of copper is 0.000133/s, the strain-order curve is shown in Fig. 2. Since the order can describe the mechanical properties of the materials used in artistic design, we can describe the changing process of the properties of copper during the loading process according to Figure 2. It can be seen from Figure 2 that the entire changing process of mechanical properties can be divided into three stages: the first stage has an order of 0, which is a linear elastic stage. In the second stage, the order is abruptly changed. It corresponds to the lattice dislocation of materials used in artistic design. The properties of materials used in artistic design have undergone a sudden change. The order of the third stage changes uniformly in an approximately linear manner. At this time, the mechanical properties of the materials used in art design also changed uniformly. And we linearly fit the order-strain of the third stage and then back-substitute it into the above constitutive model. It can be found that the order change of the variational order model can describe the evolution process of the mechanical properties of the materials used in art design.

The above conclusion is only an analysis of the evolution of the mechanical properties of copper during the constant strain rate tensile process. Whether other materials used in metal art design will reach similar conclusions requires us to analyze the experimental data of materials used in other metal art designs [7]. Therefore, the calculation and comparison of low carbon steel and aluminum alloy are also made in this paper.

The strain rate of the test is 0.0001/s (low carbon steel) and 0.0002/s (aluminum alloy), respectively. The lattice dislocation of low carbon steel is evident in the test of low carbon steel. The mechanical properties are particularly evident in the second stage. The mutation of the order of the variational fractional model is also apparent at this stage, proving that the model is credible. Whether the evolution of mechanical properties of materials used in metal art design will be affected by the strain rate under different strain rate loading conditions also needs to be discussed. Here we compare and analyze the experimental data of materials used in the same metal art design at different strain rates. Variational fractional models can describe the evolution of properties of plastic metal mechanics at different strain rates [8]. And the evolution of mechanical properties of plastic metal under constant strain rate loading is not affected by the strain rate.

The variational fractional constitutive model can describe the evolution of mechanical properties quantitatively. And in the process of constant strain rate loading, we can think that the change in the properties of the materials used in metal art design can be divided into three stages. The first stage is the linear elastic stage. The second stage is the process of the material used in art design, from elastic to plastic. We call this the yield phase. The lattice dislocation of the metal at this stage leads to a sudden change in the mechanical properties. This agrees with the experiment. The variational fractional constitutive model can describe the evolution of mechanical properties quantitatively. In addition, the strain rate during loading has little effect on the evolution of mechanical properties of the materials used in the art design of the same metal.

Conclusion

Based on the constant fractional constitutive model, we propose a variational fractional constitutive model that can describe the evolution of mechanical properties. We verified it with material tests used in metal art design. It is found that the order of the model can quantitatively describe the evolution of the mechanical properties of plastic metals. In this paper, a variational fractional model is proposed. After processing the experimental data of copper, aluminum alloy, and low carbon steel under constant strain rate stretching, we found that the change of the order-time curve can be divided into three stages. The changes in mechanical properties can also be divided into three stages. The mechanical properties of the first stage remain unchanged. It is the linear elastic phase. The second stage corresponds to lattice dislocation during stretching. Its mechanical properties have changed abruptly. In the third stage, the mechanical properties of Wie are changed linearly. Under the condition of constant strain rate loading, the mechanical properties of metals are not affected by the strain rate.

#### Values of order a(t)

Order Time zone
a1 0 < t < t1
a2 t1 < t < t2
an tn−1 < t < t2

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