When analyzing the behavior of an engineering structure made of complex mixed materials, a three phase concept of the material profile can be introduced. The first phase of the profile is at a macro-continuum level of the structure. That is, the structure is built as an assembly of macro-continuum elements. The second phase is at a micro-continuum level of the structure. It should be noted that most macro-continuum materials, which seem to be homogeneous in the macro-continuum level, in fact comprise several constituent components when considered from the microscopic point of view. This kind of material is said to be micro-inhomogeneous. The third phase of the profile is at a molecular level. This phase is founded on the physical fact that each constituent material of the micro-continuum consists of a vast number of atoms and/or molecules.
As an example, a bridge is considered as a target structure that is made of granite. The granite is a macro-continuum material. Granite includes three major component minerals such as quartz, feldspar and mica in the micro-continuum level thereof. That is, the micro-structure of granite comprises three constituent minerals, each in turn including a set of atoms/molecules that constitutes a molecular system. Similarly, a concrete dam structure has the same phases. That is, the concrete is a macro-continuum material; gravel, sand and cement paste are micro-continuum materials; and each of the micro-continuum materials constitutes a molecular system.
When designing an engineering structure with plural member elements, it is known to follow a theory of macro-phenomenological mechanics, and to perform a series of experiments in order to determine the macro-mechanical properties of each macro- continuum element. It should be recalled that the material properties obtained in these experiments are averaged in the specimen. This procedure is applied in the same manner in many fields of mechanics such as mechanical engineering and civil engineering.
In order to overcome the theoretical insufficiency involved in the above described macro-phenomenological theory, it is commonly believed that for the micro-inhomogeneous material the size of experimental specimens must be more than ten times larger than the largest size of the constituent components. However, in fact it is not truly known if this belief is true. Furthermore, it is difficult for the macro-phenomenological theory to recognize what happens in the micro-continuum level, although the local phenomena are directly related to the global behavior of the structure.
It can thus be said that the conventional macro-phenomenological procedure is not appropriate for analyzing the behavior of the micro-inhomogeneous material, especially where such a material is used under very extreme engineering condition such as high pressure, high temperature and/or long elapsed time.
In analyzing the behavior of micro-inhomogeneous material two essential problems must be solved. Firstly, it is necessary to determine characteristics of constituent components of the micro-continuum which are directly affected by their molecular movement. Secondly, there must be developed an approach to relating the microscale characteristics to the macroscale behavior of the structure and the macro-continuum elements thereof.
The prior art has not yet succeeded in developing such a fully unified procedure to analyze the molecular movements of the constituent components of the micro-continuum with respect to the macroscale behavior of the structure, much less to design the micro-structure optimally based on such considerations.
MS is a known type of a computer simulation technique. In an MS computation, one gives a material system which consists of particles, atoms and/or molecules, and provides two physical laws, that is, the interatomic interaction potential and the equation of motion or equilibrium. Positions, velocities and/or accelerations of all particles are then calculated under the foregoing physical laws. A statistical thermodynamics procedure is applied to the simulated results, and one can estimate bulk-based physicochemical properties of the material (hereinafter called the bulk properties of the material) such as structure factor of the solid crystal. It is noted that the bulk properties represent thermodynamical averages of the ensemble of particles.
Three classes of MS methods are known:
1) the Monte Carlo method (hereafter abbreviated as MC), PA1 2) the Molecular Mechanics method (hereafter abbreviated as MM), and PA1 3) the Molecular Dynamics method (hereafter abbreviated as MD).
MC, developed by Metropolis et al, estimates the statistical equilibrium state of particles by generating their displacements randomly. Various thermomechanical properties are then calculated by averaging over the states in the Markov chain.
MM is applied for a molecular system which consists of a finite number of atoms, and determines the equilibrium state by optimizing the structure and potential energy. The bulk properties are calculated by using statistical thermodynamics procedure. MM is mainly used in the field of organic chemistry.
On the other hand, MD solves the equation of motion for a system of particles under a given interatomic interaction potential by using a time-discrete finite difference scheme, and the whole time trajectories of particles are specified. The bulk properties are calculated by using statistical thermodynamics procedure for the results.
As MC and MM provide no knowledge of chronological trajectories of particles, these techniques are incapable of considering quantities that are defined in terms of particle motion, such as diffusion. In this sense, except for computational efficiency, MD is more useful so the MD procedure is shown herein as a typical example of MS.
In the MD calculation, the law of conservation of linear momentum is applied for every particle to get the following equation of motion for the i-th component: ##EQU1##
where m.sub.i is the mass of the i-th particle, v.sub.i =dr.sub.i /dt is its velocity at the position r.sub.i, and the force F.sub.i is calculated from the potential function U.sub.ij between two particles by ##EQU2##
The MD system usually contains many particles of atoms and/or molecules in a basic cell, and (for simplicity of calculation) the method uses a three dimensional periodic lattice, which is repeated in each direction. Under these conditions one solves a time discrete form of the equation of motion, and the instantaneous position, velocity and acceleration of each particle are specified. Then, using these results and statistical thermodynamic theory, one computes for the material the bulk properties and their change with time, such as density, diffusivity of atoms, molecular vibrations, temperature- and/or pressure-dependent nonlinear elastic moduli, viscosity, heat capacity and heat conductivity.
The interatomic potential function for all atom-atom pairs plays an essential role for the MD calculation. Equation (3) presents a potential function developed by Kawamura so as to reproduce structural and physical properties of several oxide crystals such as quartz, corundum and feldspars properly.
2-body term: ##EQU3##
The right hand side of this equation shows the Coulomb, the short-range repulsion, the van der Waals and the Morse terms, respectively. These terms are selectively used for some materials due to the nature of the interaction. For water a 3-body term is also added to the H--O--H interaction because of its Sp.sup.3 hybrid orbital.
3-body term: ##EQU4##
Parameters {z, a, b, c} and {D, .beta., r*} for the 2-body term, and {f.sub.k, .theta..sub.0, g.sub.r, r.sub.m } for the 3-body term are specified by using experimental data of structural and physical properties of relevant materials.
Many types of simulation schemes have been developed in MD. For example, in the early stage of MD there was employed a scheme where the number of particles N, the volume V and the total energy E are constant (the constant-(NVE) ensemble), while in the 1980's there was developed a scheme in which the number of particles N, the pressure P and the temperature T are constant (the constant-(NPT) ensemble). MD generates information at the molecular level such as position, velocity and acceleration of each particle. Statistical thermodynamics provide averaged quantities of the system which are called the bulk properties. For example, one can calculate the temperature as ##EQU5##
where k.sub.B is Boltzmann's constant, and ##EQU6##
is the kinetic energy of the k-th particle with its mass m.sub.k and velocity v.sub.k. Note that &lt;A&gt;gives the time average of a quantity A(t) taken over a long time interval: ##EQU7##
The right hand side term is used for a discrete system with N-particles for N.sub.s -number of a time slice .DELTA..tau.. Then, by applying the virial theorem the pressure P is calculated as ##EQU8##
where r.sub.i is the position vector and F.sub.i the force acting on the i-th particle. Similar to this, one can calculate the stress tensor which is the force per unit area acting on three coordinate surfaces in the three dimensional case. On the other hand, if one uses the constant-(NPT) ensemble scheme for example, the normal strain .epsilon..sub.xx in the x-direction is calculated by ##EQU9##
where L.sub.x is the original size of the basic cell in the x-direction, and L.sup.c.sub.x is its size after relaxation by the MD calculation. Other components of strains including shear components can be calculated in a similar manner. If the pressure is changed, one gets a different value of strain. Plotting these values yields a stress-strain relation in the micro-continuum level. Thus, one gets a stress or strain dependent type of the Young's modulus E.
Though MD is quite powerful for simulating the true behavior of materials, it is impossible to use this method directly for designing an engineering structure on a human size scale, such as a car, an airplane and a bridge, because such structures involve extremely large numbers of molecules. Note that one mole of material (equivalent to 12 g of carbon) consists of 6.0221367.times.10.sup.23 molecules, and even if the fastest computer known at present were used, one can calculate a system with at most 10.sup.6 molecules. It is thus impossible to use MS in a simple manner for designing a practical engineering structure. This circumstance will not be changed in near future.
The HA method, which is based on a new type of perturbation theory, has been developed for micro-inhomogeneous media with a periodic microstructure. This method allows a determination of both macroscopic and microscopic distribution of field variables such as temperature, displacement, stress and strain. It is noted that the perturbation theory is a method mainly for solving nonlinear problems.
The simplest example of HA is shown herein for applying the one dimensional static equilibrium problem, which is described by the following equation. ##EQU10##
where u.sup..epsilon. (x) denotes the displacement which changes rapidly in the micro-continuum level, E Young's modulus, and f the body force. Note that E also varies in that micro-continuum level. Let the size of the microstructure be .epsilon.Y, and one introduces a local coordinate system y in the micro-continuum level. The global coordinate x is related to the local one y by ##EQU11##
By using this .epsilon., a perturbation expansion is introduced by EQU u.sup..epsilon. (x).congruent.u.sub.0 (x,y)+.epsilon.u.sub.1 (x,y)+.epsilon..sup.2 u.sub.1 (x,y)+.LAMBDA. (12)
where u.sub.0 (x,y), u.sub.1 (x,y), u.sub.2 (x,y), . . . are periodic functions satisfying the condition u.sub.i (x,y+Y)=u.sub.i (x,y) (i=0,1,2, . . .). If the two coordinates x and y are used, the differentiation is changed as ##EQU12##
On substituting the perturbed expansion (12) and the differentiation form (13) into the equilibrium equation (10), and setting each term of the series for .epsilon. to be zero, the following relations can be obtained: ##EQU13##
This implies that u.sub.0 is the function of only the global coordinate x, that is, u.sub.0 =u.sub.0 (x). ##EQU14##
This gives a differential equation to determine u.sub.1 (x,y) in y, and on introducing a separation of variable ##EQU15##
yields a differential equation for the characteristic function N(y): ##EQU16##
This equation is called the microscale equation. ##EQU17##
For this equation, the following average operation is introduced: ##EQU18##
then the macroscale equation is obtained for determining u.sub.0 as ##EQU19##
where ##EQU20##
The quantity E* is called the homogenized elastic coefficient. Thus, first the microscale equation is solved under the periodic boundary condition to get the characteristic function N(y). One substitutes it for the definition of the homogenized elastic coefficient, and solving the macroscale equation which is of the same form of the original equation of equilibrium yields u.sub.0. Now one gets u.sub.1 by using u.sub.0 and N(y), then u.sub.2 can be calculated by the differential equation of .epsilon..sup.0 -term. However since the value of terms .epsilon..sup.k u.sub.k (x,y)(k.gtoreq.2) is thought to be small, one sets EQU u.sup..epsilon. (x).congruent.u.sub.0 (x,y)+.epsilon.u.sub.1 (x,y). (22)
In this case, the strain can be obtained as ##EQU21##
and the stress is given by EQU .sigma..sup..epsilon. =Ee.sup..epsilon.. (24)
In practical engineering problems with two or three dimensions, a finite element approximation method is applied for solving the above mentioned microscale and macroscale equations.
The various methods of MS and HA have been developed in different fields, and no intercorrelation has been tried in these two fields. Both the MS methods and HA involve difficulties in attempting to apply them to simulate the behavior of a practical engineering structure made of a micro-inhomogeneous material: MS methods can only treat a system with far less particles than in a practical structure, while in HA it is difficult to find bulk properties of each constituent material and their interfaces. In view of the difficulties of the prior art, it is thus important to establish an accurate method for analyzing the behavior of a structure made of micro-inhomogeneous material and for designing the micro-structure of the micro-inhomogeneous material optimally in the case of manufacturing.
It is accordingly an object of the invention to overcome the difficulties of the prior art and to provide a novel method for analyzing and designing a structure made of micro-inhomogeneous material.