# Energies of Mechanical Fractional-Order Elements: Causal Concept and Kernel Effects
## 5 More General Formulations of the Causal Force-Displacement Relationship
### Fractional Element with Prabhakar Memory
#### 5.3.3 An Example of the Use of the Complete Prabhakar Function  
Let us now turn to the complete definition of the three-parameter Mittag-Leffler function, which for clarity of the following calculations can be presented as  
\[ \begin{split} E_{\lambda,\beta}^{\gamma}\Big{(}-t^{\lambda} \Big{)}=\frac{1}{\Gamma(\gamma)}\sum\limits_{k=0}^{\infty}{(-1)}^{k}\frac{ \Gamma(\gamma+k)}{k!\Gamma(\lambda k+\beta)}t^{k}=\sum\limits_{k=0}^{\infty}{ p_{k}t^{k}},\\ p_{k}=(-1)^{k}\frac{1}{\Gamma(\gamma)}\frac{\Gamma(\gamma+k)}{k! \Gamma(\alpha k+\beta)}.\end{split}  \tag{105}\]  
Then, the Prabhakar function (A26) can be presented as  
\[e_{\lambda,\beta}^{\gamma}(t)=t^{\beta-1}\sum\limits_{k=0}^{\infty}{p_{k}t^{ k}}=\sum\limits_{k=0}^{\infty}{p_{k}t^{k+\beta-1}}, \tag{106}\]  
while the displacement can be defined in general as  
\[x_{\text{PrabG}}(t)=\frac{1}{k_{\lambda}}\int\limits_{0}^{t}e_{\lambda,\beta} ^{\gamma}(t-\tau)F(\tau)d\tau=\frac{1}{k_{\lambda}}\int\limits_{0}^{t}\sum \limits_{k=0}^{\infty}{p_{k}(t-\tau)^{k+\beta-1}F(\tau)d\tau}. \tag{107}\]  
In the case of a unite step function \(F(t)=1\), the integration by parts in (107) yields  
\[x_{\text{PrabG-1}}(t)=\frac{1}{k_{\lambda}}\sum\limits_{k=0}^{\infty}{\frac{p _{k}}{k+\beta}t^{k+\beta}}\Rightarrow\frac{d\,x_{\text{PrabG-1}}(t)}{dt}=\frac {1}{k_{\lambda}}\sum\limits_{k=0}^{\infty}{p_{k}t^{k+\beta-1}}. \tag{108}\]  
Thus, the energy of the fractional element is  
\[W_{\text{PrabG-1}}(t)=\frac{1}{k_{\lambda}}\int\limits_{0}^{t}\sum\limits_{k=0 }^{\infty}{p_{k}t^{k+\beta-1}dt}=\frac{1}{k_{\lambda}}\sum\limits_{k=0}^{\infty }{\frac{t^{k+\beta}}{k+\beta}}. \tag{109}\]  
For the linear ramp force \(F(t)=t\), skipping the intermediate calculations, we obtain  
\[x_{\text{PrabG-Linear\ Ramp}}(t)=\frac{1}{k_{\lambda}}\int\limits_{0}^{t}\sum \limits_{k=0}^{\infty}{p_{k}(t-\tau)^{k+\beta-1}\tau d\tau}=\frac{1}{k_{\lambda }}\sum\limits_{k=0}^{\infty}{\frac{p_{k}}{(k+\beta)(k+\beta+1)}t^{k+\beta+1}} \tag{110}\]  
\[\frac{d\,x_{\text{PrabG-Linear\ Ramp}}(t)}{dt}=\frac{1}{k_{\lambda}}\sum \limits_{k=0}^{\infty}{\frac{p_{k}}{k+\beta}t^{k+\beta}} \tag{111}\]\[W_{\text{PrabG-Linear\,Ramp}}(t)=\frac{1}{k_{\lambda}}\sum_{k=0}^{\infty}\frac{p_{k }}{(k+\beta)(k+\beta+1)}t^{k+\beta+1}. \tag{112}\]  
Outcomes from the Examples Solved through Convolution Integrals with Memories Different from the Power Law  
In general, the extension of the modeling approach applying more complex memory kernels can be considered a step towards generalization of the force-displacement relationships stemming from the causal formulation. This is a natural process in the modeling of nonlocal force-displacement relationships; the first step was carried out by [PERSON] and [PERSON], with a simple example based on the Maxwell spring-dashpot model. Looking more deeply into the physics that this simple mechanical construction can model, it is apparent that the damping represented by the dashpot should strongly depend on the relaxation function of the medium on which the external force is exerted. Moreover, taking into account the huge variety of material behaviors in the area of linear viscoelasticity (the [PERSON] model belongs to this group) which do not obey the requirement of relaxation following the power law (modeled by the singular Abel kernel), it is reasonable to check whether other functions can model them [8]. If these functions are completely monotone (CM), then there is no harm in applying them in the nonlocal force-displacement relationship. The examples developed in this section of the article have especially applied both new and old memory kernels (relaxation functions) known from the field of fractional calculus. This is a step beyond the level attained by [PERSON] and [PERSON] [6], and allows us to see how this can be done.