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Composite materials adhesion and cohesion properties research.

Name
Vladimir
Surname
Shorkin
Scientific organization
Orel State University named after I.S. TURGENEV
Academic degree
Doctor of Mathematics and Physics
Position
Professor on Physics department
Scientific discipline
Mathematics & Mechanics
Topic
Composite materials adhesion and cohesion properties research.
Abstract
The main property is the strength of the composite. It is caused by adhesion of its elements. It is proposed to solve the problem of selection of the elemental composition based on the idea of nonlocal potential many-particle interactions of its infinitesimal elements. Characteristically, the parameters of the potentials of interparticle interactions approximating functions are defined on the basis of the classical experiments to determine the characteristics of the linear elastic state of the composite elements.
Keywords
Mechanics of solids, solid state physics
Summary

1 Introduction

While consider adhesion of solid bodies based on non-traditional models of local Cosserat, Leroux, Tupin, Mindlin, Aero for the calculation of mechanical processes with specific constructional elastic materials, the problem of determining the elastic constants arise. The methods of solid-state physics for such materials are difficult or impossible to use. The mechanical processes in such materials are described on the basis of the phenomenological approach conveniently. This approach is based on the continuum thermodynamics. Therefore, methods for calculating the characteristics of the elastic state mediums with internal degrees of freedom should be based on the phenomenological of the macro experiment.

One of the phenomenon, whish described by the models of Cosserat, Leroux, Tupin, Mindlin, Aero, is the adhesion. The characteristics of adhesion are its energy and adhesive force. In this paper, the elastic constants are determined by calculation. It is based on the nonlocal elastic medium.

2 Theoretical positions

The interaction of bodies \(B \equiv B_{(k)} \) (\(k=1,2,...\) – number of body) is considered. These bodies are bounded by smooth surfaces \(A \equiv A_{(k)}\). Surfaces have the outward unit normal \(\overrightarrow{n} \equiv \overrightarrow{n_{(k)}} \). The bodies \(B \equiv B_{(k)}\)composed of homogeneous, isotropic linear elastic materials. Each of them is considered dedicated from the infinite medium \(\Omega \equiv \Omega_{(k)}\). The material \(\Omega \equiv \Omega_{(k)}\) and \(B \equiv B_{(k)}\) is the same. This assumption excludes the influence of the material properties of the body \(B \equiv B_{(k)}\) by its boundary region. The state \(B \equiv B_{(k)}\) inside \(\Omega \equiv \Omega_{(k)}\) is the reference. It corresponds to time \(t=0\).

Every body \(B \equiv B_{(k)}\)can be presented as a union of non-intersecting parts \(\Delta B_{n}\equiv\Delta B_{(k)n}\). The density of materials \(\rho \equiv \rho_{(k)}\) and their temperatures \(T\equiv T_{(k)}\) are distributed uniformly and do not change over time. Thus: \(T_{(1)}=T_{(2)}\).

The particles – material points. 

In the reference configuration arbitrary body B occupies the region V and center of inertia of its particles have a dB radius vectors \(\overrightarrow r\in V\). The position of an arbitrary particle \(dB_{2}\) relative to other arbitrary particle \(dB_{1}\equiv dB\) is defined relative radius vector \(\overrightarrow {l_{12}}=\overrightarrow {r_{2}}-\overrightarrow {r_{1}}\) with length \(l_{12}=|\overrightarrow{l_{12}}|=|\overrightarrow{r_{2}}-\overrightarrow {r_{1}}|\).

Under the influence of external mechanical impacts, including the allocation of \(B \equiv B_{(k)} \) from \(\Omega \equiv \Omega_{(k)}\), particles \(dB \equiv dB_{(k)}\) acquire new provisions, which are characterized by the radius vectors \(\overrightarrow R\equiv \overrightarrow{R_{(k)}}\in V_{t(k)}\) and the displacement vector \(\overrightarrow u(\overrightarrow r,t)=\overrightarrow R(\overrightarrow r,t)-\overrightarrow r\). Area \(V_{t(k)}\) – is the area, which occupied by the body \(B \equiv B_{(k)} \) in the current configuration. The position of a particle \(dB_{2}\) relative particle \(dB_{1}\equiv dB\) will change and will be determined by the radius vector \(\overrightarrow {L_{12}}=\overrightarrow{R_{2}}-\overrightarrow {R_{1}}=(\overrightarrow {r_{2}}-\overrightarrow {r_{1}})+(\overrightarrow{u_{2}}-\overrightarrow {u_{1}})=\overrightarrow {l_{12}}+\Delta \overrightarrow {u_{12}}\) and length \(L_{12}=|\overrightarrow {L_{12}}|=|\overrightarrow{R_{2}}-\overrightarrow{R_{1}}|\). If \(B \equiv B_{(k)} \) body is deformed, then \(L_{12}\neq l_{12}\).

It is assumed that the deformations are small: \(|L-l|/l<<1\). Therefore, the density of the material equality and volumes of elementary particles in the reference and current status are equal.

Vectors \(\Delta \overrightarrow{u_{1,j}}\) can be represented as a series in exterior powers \(\overrightarrow{l_{1,j}}\).

                                                                              \(n \ times\)                                      \(n \ times\)

                                   \(\Delta \overrightarrow{u_{1,j}}=\displaystyle\sum_{n=1}^\infty\frac{1}{n!}\big( \nabla_{1,j}^{n}\overrightarrow u\big)\cdot \widehat{...} \cdot \overrightarrow{l_{1,j}^n}=\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{n!}\big( \nabla^{n}\overrightarrow u\big)\cdot \widehat{...} \cdot (\overrightarrow{l_{1,j}})^n,\ \ j=2,3,\dots,\)           (1)

where \(\nabla=d\dots /d\overrightarrow r\) – differential del operator on a vector \(\overrightarrow r\), and \(\nabla_{1,j}=d\dots /d\overrightarrow{l_{1,j}}\) – on a vector \(\overrightarrow{l_{1,j}}\).

It is believed that the vector \(\overrightarrow{r_{(1)}}\) receives the increment \(d\overrightarrow{r_{(1)}}\), then the vector \(\overrightarrow{l_{1,j}}\) receives the increment \(d\overrightarrow{l_{1,j}}=-d\overrightarrow r\). It means: \(\nabla_{1,j}^{n}=(-1)^n\nabla^n\).

The distortion tensor is the gradient of the curvature tensor. It is expressed by the second gradient of the displacement vector.

                                              \(\nabla \nabla \overrightarrow u=\nabla^2\overrightarrow u=u_{i,jn}\overrightarrow {e_i}\overrightarrow {e_j}\overrightarrow {e_n}=D_{ijn}\overrightarrow {e_i}\overrightarrow {e_j}\overrightarrow {e_n}=d_{ij,n}\overrightarrow {e_i}\overrightarrow {e_j}\overrightarrow {e_n}\)                                (2)

If the characteristics of the kinematics of the continuum is necessary to use a second gradient of displacement, then it may appear dislocation.

In known scientific works is formulated kinematic sign of adhesion of the two bodies. Any material fiber, which intersects the contact surface \(A_{(12)}\), should preserve the smoothness of the distribution of their deformation. It is suggested that this characteristic may be made by using the curvature tensor for a description deformations (2). This means that for the adhesion of elastic materials must arise field dislocations. When there is adhesion of two metals, this feature is enough.

If there is metal adhesion, then allowed, that the total potential energy of the combined body \(B=B_{(1)}\cup B_{(2)}\) is the sum of the potential energies of many-particle interactions inside each of the bodies \(B_{(1)}\) and \(B_{(2)}\), and between them. The quantities \(\Phi_{(kp)}^{(2)}(\overrightarrow{R_{(k)}},\overrightarrow {R_{(p)}})dV_{(k)}dV_{(p)}\), \(\Phi_{(kpq)}^{(2)}(\overrightarrow{R_{(k)}},\overrightarrow {R_{(p)}},\overrightarrow{R_{(q)}})dV_{(k)}dV_{(p)}dV_{(q)}\), … are the potentials of pair, triple, etc. interactions of particles pair \(dB_{(k)},dB_{(p)},dB_{(q)}\) bodies \(B_{(k)},B_{(p)},B_{(q)} \ \ k,p,q=1,2\). In this case, \(dV_{(k)},dV_{(p)},dV_{(q)}\) – the volume of the interacting particles in the reference state. Functions (the potential)  \(\Phi_{(kp)}^{(2)}(\overrightarrow{R_{(k)}},\overrightarrow {R_{(p)}}),\Phi_{(kpq)}^{(2)}(\overrightarrow{R_{(k)}},\overrightarrow {R_{(p)}},\overrightarrow{R_{(q)}})\) for a homogeneous isotropic material depend only on the distance between the interacting particles in the current state.

The energy \(dW_{(1)}(\overrightarrow{R_{(1)}})=w_{(1)}(\overrightarrow{R_{(1)}})dV_{(1)}\) of infinitesimal particle, e.g., \(dB_{(1)}\) with the volume \(dV_{(1)}\) and center of inertia \(\overrightarrow{R_{(1)}}\) is presented in the form of

\(w_{(1)}(\overrightarrow{R_{(1)}})dV_{(1)}=(w_{(11)}+w_{(12)})dV_{(1)}=\Bigg[\Phi_{(11)}^{(2)}dV_{(1)}+\frac{1}{2!}\int\limits_{V_{(1)}}\int\limits_{V_{(1)}}\Phi_{(111)}^{(3)}dV_{(1)}dV_{(1)}+]\dots\Bigg]dV_{(1)}+\Bigg[\int\limits_{V_{(2)}}\Phi_{(12)}^{(2)}dV_{(2)}+\frac{1}{2!}\sum\limits_{k=1}^{2}\int\limits_{V_{(2)}}\int\limits_{V_{(k)}}\Phi_{(12k)}^{(3)}dV_{(k)}+\dots\Bigg]dV_{(1)}\)

In this equality \(w_{(11)}\) – the cubic density of the potential energy, which arose due to the interaction of the particles of the body \(B_{(1)}\) among themselves; \(w_{(12)}\) – addition to the quantity of \(w_{(11)}\), which arises from the interaction particles of the body \(B_{(1)}\) with the particles of the body \(B_{(2)}\). Each particle \(dB_{(1)}\subset B_{(1)}\) is affected by the forces from the other particles \(dB_{(1)}\) of the same body \(B_{(1)}\), particles \(dB_{(2)}\) body \(B_{(2)}\) and the medium, which surrounds both the body. The first forces are called forces of cohesive interaction parts of the body. Their cubic density is:

\(\overrightarrow{f_{(11)}}=-\nabla w_{(11)}\).

The second forces are adhesive forces. Their cubic density is:

                                                                    \(\overrightarrow{f_{(12)}}=-\nabla w_{(12)}=-\overrightarrow{f_{(21)}}\).                                               (3)

During the deformation of the material interacting particles \(dB_{1(k)}\) and \(dB_{j(p)}\) experienced relative displacements \(\Delta \overrightarrow{u_{1j}}\). For the particles decomposition (1) is valid. At the same time, pair, triple, etc. potential interaction is permissible to submit second-order polynomials relatively \(\Delta \overrightarrow{u_{1j}}\). Absolute term of the polynomial and its coefficients are expressed in the potentials of many-particle interactions in the reference state.

Changing the cubic density \(\Delta w_{(11)}\)of the potential energy body \(B_{(1)}\) is a function of the sequence \(\{\nabla^n \overrightarrow{u}\}\) displacement gradients. If we differentiate dependence \(\Delta w_{(11)}(\nabla \overrightarrow{u},\nabla^2 \overrightarrow{u},\dots)\)by the gradients \(\nabla^n \overrightarrow u\), we obtain the expression for the stress tensor, which develop in the material body \(B_{(1)}\).                                                                                            \(m \ times\) 

                                               \(P^{(n)}=\frac{\partial\Delta w_{(11)}}{\partial(\nabla^n \overrightarrow u)}=P^{0(n)}+\sum\limits_{m=1}^{\infty}\big(\nabla^m \overrightarrow u\big)\cdot\widehat{...}\cdot C^{(m,n)}\),                                   (4)

where \(P^{0(n)}\) – tensor of initial stress; \(C^{(m,n)}\) – tensors, which characterize the mechanical properties of the material. Taking into account only pair and triple interactions, the defining relations have the form:

                                      \(P^{0(m)}=\frac{1}{2!}\int\limits_{V}\frac{1}{m!}\big(\nabla_{12}\Phi_{(11)}^{(2)}\big)\overrightarrow{l_{12}^{m}}dV_{2}+\frac{1}{3!}\sum\limits_{j=2}^{3}\int\limits_{V}\Bigg[\int\limits_{V}\frac{1}{m!}\big(\nabla_{1j}\Phi_{(111)}^{(3)}\big)\overrightarrow{l_{1j}^{m}}dV_{2}\Bigg]dV_{3}\),                (5)

                                \(C^{(n,m)}=\frac{1}{2!}\int\limits_V\frac{1}{m!n!}\overrightarrow{l_{12}^n}\big(\nabla_1^2\Phi_{(11)}^{(2)}\big)\overrightarrow{l_{12}^m}dV_2+\frac{1}{3!}\sum\limits_{p,q=2}^{3}\int\limits_V\Bigg[\int\limits_V\frac{1}{m!n!}\overrightarrow{l_{1p}^n}\big(\nabla_p \nabla_q\Phi_{(111)}^{(3)}\big)\overrightarrow{l_{1q}^m}dV_2\Bigg]dV_3\).       (6)

Jump to a specific local model is the replacement of (1) the sum of one, two, etc. terms. Herewith, sequence \(\{P^{(n)}\}\) stored a corresponding number of cells.

The equation of motion for interacting bodies \(B_{(1)}\) and \(B_{(2)}\) in stresses for the local model has the form:

                                        \(\rho_{(k)}=\frac{\partial^2 \overrightarrow{u_{(k)}}}{\partial t^2}=\nabla \cdot\big(P_{(k)}^{(1)}-\nabla \cdot\big(P_{(k)}^{(2)}-\nabla \cdot\big(P_{(k)}^{(3)}-\dots \big) \big) \big)+\overrightarrow{f_{(kp)}}+\psi_{(k)}\),                (7)

                                                                              \(k,p=1,2; \ k\neq p\)

where \(P_{(k)}^{(m)}, \ m=1,2,3,\dots\)– internal stress tensor.

The field of vectors \(\overrightarrow{\psi_{(k)}}=\overrightarrow{\psi_{(k)}}(\overrightarrow{r_{(k)}})\) are defined. The fields \(\overrightarrow{f_{(kp)}}=\overrightarrow{f_{(kp)}}(\overrightarrow{r_{(k)}},\overrightarrow{r_{(p)}})\) are defined by (3). The value \(w_{(12)}\) is calculated through the interaction potentials of particles in the assumption of the absence in these strains. The interaction potentials must be known.

At time \(t=t_{0(k)}\) We set the initial conditions of the displacements distribution and velocities of the particles of the body \(B_{(k)}\), which occupied an area of \(V_{(k)}\).

Therefore, the use of expressions (7) and (4) – (6) makes a conjugate problem of the contact interaction of elastic bodies with regard to their adhesion. In the reference state, the potentials of all the many-particle interactions should be known.

3 The results of calculation

Expressions (5) and (6) show, that the characteristics of the elastic state of the material are calculated by the potentials of nonlocal interaction of its particles. The feature of metals – pressure of the electron gas is taken into account in known scientific works. Nonlocal interaction potentials-material point are requested to identify with a nonlinear dispersion law. The dispersion law – the dependence of \(w^2=f(K^2)\) is determined experimentally. It is approximated by a polynomial of degree n. The value of the degree n is determined by the condition of current task. Geometrical conditions of adhesion is the continuity and smoothness of field variations for those displacements of contacting bodies, which they obtained by adhesion. For performing the conditions is sufficient to apply only the first two displacement gradients in the description of the deformations, which occur in the adhesion of the two bodies. It's enough to take \(n=2\).

In this case, depending on the potential pair and triple interactions are approximated by functions

                                                          \(\Phi _{(kp)}^{(2)}=\Phi _{0(kp)}^{(2)}\Big(e^{-2\beta_{(kp)}l_{12(kp)}}-2e^{-\beta_{(kp)}l_{12(kp)}}\Big)\),

                                        \(\Phi_{(k,p,q)}^{(3)}=\Phi_{0(k,p,q)}^{(3)}\Big(e^{-2\beta_{(kp)}l_{12(kp)}}-2e^{-\beta_{(kp)}l_{12(kp)}}\Big)\Big(e^{-2\beta_{(kq)}l_{13(kq)}}-2e^{-\beta_{(kq)}l_{13(kq)}}\Big)\).               (8)

These functions are equal to zero at an infinite distance. These particles may belong to the body \(B_{(k)} \ (k=p=q)\) or another \((k\neq p \lor k\neq q)\).

In the first case \((k=p=q)\), for the parameters \(\Phi_{0(kp)}^{(2)}, \ \Phi_{0(kpq)}^{(3)}, \ \beta_{(kp)}\) were obtained calculating formulas.

                                                           \(\Bigg(\frac{\Phi_{0(k)}^{(2)}}{\beta_{(k)}^{3}}\Bigg)=\frac{4\mu_{(k)}+45\lambda_{(k)}}{294\pi}=\frac{E_{(k)}}{294\pi}\frac{2+41\nu_{(k)}}{(1+\nu_{(k)})(1-2\nu_{(k)})}\)                                          (9)

                                                          \(\Bigg(\frac{\Phi_{0(k)}^{(3)}}{\beta_{(k)}^{6}}\Bigg)=\frac{27(\lambda_{(k)}-\mu_{(k)})}{1764\pi}=\frac{27E_{(k)}}{1764\pi}\frac{4\nu_{(k)}-1}{2(1+\nu_{(k)})(1-2\nu_{(k)})}\)                                        (10)

                                                     \(\beta_{(kk)}=\frac{1}{2}\sqrt{3\pi\Bigg(\frac{f_{0(kk)}}{f_{1(kk)}}\Bigg)\frac{15\Bigg(\frac{\Phi_{0(kk)}^{(2)}}{\beta_{(kk)}^3}\Bigg)+\Big(\frac{1563\pi}{4}\Big)\Bigg(\frac{\Phi_{0(kk)}^{(3)}}{\beta_{(kk)}^6}\Bigg)}{2\mu_{(kk)}+\lambda_{(kk)}}}\)                                (11)

Formulas (9) and (10) are the result of the comparison Voigt notation for tensor traditional characteristics of the elastic state of the material with the first term of the polynomial \(w^2=f(K^2)\). Equations (9) and (10) are constructed with help (5) and (8).

Equation (11) is obtained by comparing the first and second terms the submission of the dispersion law \(w^2=f(K^2)\) for of plane longitudinal waves in the form of a polynomial of the second degree.

In the second case \((k\neq p \lor k\neq q)\), when the bodies of different materials interact, for determining the parameters \(\Phi_{0(kp)}^{(2)}, \ \Phi_{0(kpq)}^{(3)}, \ \beta_{(kp)}\) are used depending on the characteristics of elastic state two-component solid solutions from the concentration of their components.

The methodology, which is proposed, allowed to calculate the interaction potential semi-infinite bodies \(B_{(1)}\) and \(B_{(2)}\) (fig. 1, a), the force of attraction of р (fig. 1, b), which depends on the distance between the h units of area boundary planes \(A_{(1)}\) and \(A_{(2)}\). The results of calculation are compared with existing works. Conformity is satisfactory. 

                                       

               a)                                                                                           b)

Fig. 1. The dependence of the potential Ea and the force of attraction р from the distance \(\eta\) for combinations \(Cu-Al\) – curves 1 and \(Al-Al\) – curves 2.