A 3-dimensional singular kernel problem in viscoelasticity: an existence result

Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a singular kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional singular kernel initial boundary value problem.Comment: 15 page

KdV-type equations linked via Baecklund transformations: remarks and perspectives

Third order nonlinear evolution equations, that is the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered: they all are connected via Baecklund transformations. These links can be depicted in a wide Baecklund Chart} which further extends the previous one constructed in [22]. In particular, the Baecklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Baecklund Chart. That is, such a Baecklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30 is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Baecklund chart.Comment: 18 page

Some remarks on the model of rigid heat conductor with memory: unbounded heat relaxation function

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t=0$. That is, it is represented by a regular integrable function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions on $k$, whenever the heat flux is the same also the related thermal work is the same. Thus, also in the case under investigation, the notion of equivalence is introduced and its physical relevance is pointed out

Singular Kernel Problems in Materials with Memory

In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as well as the past history. Aiming to study new materials integro-differential problems admitting singular kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integro-differential equations. An initial boundary value problem with homogeneneous Neumann boundary conditions is studied.In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as wel

Some remarks on the model of rigid heat conductor with memory: unbounded heat relaxation function

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t=0$. That is, it is represented by a regular integrable function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions on $k$, whenever the heat flux is the same also the related thermal work is the same. Thus, also in the case under investigation, the notion of equivalence is introduced and its physical relevance is pointed out

Some remarks on materials with memory: heat conduction and viscoelasticity

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared

Ermakov-Pinney and Emden-Fowler equations: new solutions from novel B\"acklund transformations

The class of nonlinear ordinary differential equations $y^{\prime\prime}y = F(z,y^2)$, where F is a smooth function, is studied. Various nonlinear ordinary differential equations, whose applicative importance is well known, belong to such a class of nonlinear ordinary differential equations. Indeed, the Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov equations are among them. B\"acklund transformations and auto B\"acklund transformations are constructed: these last transformations induce the construction of a ladder of new solutions adimitted by the given differential equations starting from a trivial solutions. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficulty to apply

Materials with memory: Free energies & solution exponential decay

The model of a rigid linear heat conductor with memory is analyzed. Specifically, an evolution problem which describes the time evolution of the temperature distribution within a rigid heat conductor with memory is studied. The attention is focussed on the thermodynamical state of such a rigid heat conductor which, according to the adopted constitutive equations, depends on the history of the material; indeed, the dependence of the heat flux on the history of the temperature's gradient is modeled via an integral term. Thus, the evolution problem under investigation is an integro-differential one with assigned initial and boundary conditions. Crucial in the present study are suitable expressions of an appropriate free energy and thermal work, related one to the other, which allow to construct functional spaces which are meaningful both under the physical as well as the analytic viewpoint. On the basis of existence and uniqueness results previously obtained, exponential decay at infinity is proved

Abelian versus non-Abelian Baecklund Charts: some remarks

Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.Comment: 18 page