EP1490804A1 - Method and device for determining deformations - Google Patents

Description

Method and device for determining deformations

Description:

The invention relates to a method for determining deformations, these being recorded and investigating how a supply of energy, caused by a load, deforms a body.

The invention further relates to a device suitable for carrying out the method.

Various methods are known with which the deformation of a body is determined under the action of forces.

The known methods provide for the representation of the body by a grid model and for each individual grid point to determine an expected change in the relative position under the action of a force.

The known methods are very complex, especially in the event that the body has a complex shape.

For example, determining the deformation of a vehicle in a traffic accident requires a very high amount of calculation.

Although the mathematical models included in the known methods are simple, the underlying mathematical models are essentially based on work

Eulers from the 18th century - so far no processes are known which differ in principle from the processes based on Euler. The inventor has further developed theoretical approaches to deformation theory and presented them in a series of freely accessible articles.

The first of these articles, “An approach to deformation theory based on Boyle's law. I. The concept, and kinematics of pure shear "deals with a concept of kinematics of solid state deformation on the basis of an adaptation of the ideal state equation pV = nRT. This theory covers both isotropic and anisotropic materials and stresses.

Further developments to the kinematics are in the article “An approach to deformation theory based on Boyle's law. II. Kinematics of simple shear, and some energetic considerations ".

The requirement to consider a comprehensive consideration of geometrically very simple deformations in all three spatial dimensions was presented in the article: "An approach to deformation theory based on Boyle's law. III. Three-dimensional properties of plane-strain deformation: origin of conjugate joint sets, sheath folds in plastic shear zones, and turbulence in viscous flow ".

The invention has for its object a generic

Process so evolve that deal with it

Have deformations determined as quickly and reliably as possible.

According to the invention, this object is achieved in that any point Q in a solid can be thought of as a reference point, which is surrounded by correspondence points P in all directions at a standard unit distance. This Correspondence points P serve as directional representatives from the point of view of Q and, in their entirety, form a surface of ideal, spherical shape. The change in location of all P relative to as a function of the acting forces describes the deformation at point Q. In the conventional method, a point grid is predefined, and all points serve simultaneously as a reference point and point of correspondence. As a result, the accuracy of the method depends on the number of pairs of points, which is very high, but finite and necessarily incomplete. In contrast, in the new method, each point can serve as a reference point Qi, but the corresponding surface A of all assigned P for this Qi is thus determined; that a point Pi can function as a point Q ₂ is irrelevant from the point of view of Q _x . In this way there is an infinitely large number of P for each Q, and an infinitely large number of Q within the scope of the approach, which makes it possible to present it as a homogeneous differential approach.

The distance between two The distance Q → P is knot is finite. Independent of scale.

Any two nodes in the network. The route r ₀ is used as the starting and connecting line between Q and end point of a given P, but not between two routes. Q. There is no relationship between the surface point Pi of a standard volume assigned to Qi and another Q ₂ (even if Pi is a P ₂ at the same time) unless a Pi accidentally falls on a Q ₂ .

Deformation is modeled by changing this help from r relative only to distance in length and / or its own reference point Q, direction relative to one and only indirectly with reference to or to the other, and an external reference point. then relative to an external reference point.

The deformation modeling The deformation modeling refers to the x happens homogeneously discrete connecting lines continuously over all P of all point pairs. relative to their Q, and homogeneously continuous over all Q relative to an external reference point.

Deformation is called deformation is called mathematical transformation energetic state change

("mapping") of mass points. Two different ones from the undeformed in points P and Q cannot deformed coordinates coincide because they understand. To prevent the work equation logarithmic that two mass points are at and that an infinite amount of transformation on each other would be necessary. The fall is the apparatus the apparatus of compatibility equations compatibility equations is required. not mandatory.

Symbolteπtiinologie

T, M, F tensors f,, r, s, v vectors

I identity tensor n, t

unit vectors

TY _j , F _1D tensor components r_

vector components

P, V, H, U scalars, state f, r

vector quantities

Functions P, Q points α, γ, θ angles

The forces that shift the surface of all P are made up of the sum of external forces f _ext , the forces exerted by the system on the environment m _syst , and the surface binding forces m _A that do no work themselves (in the sense of a Force), but expansion components of f _{ext /} and the shear components m _a ( _Syst ) and f _S ( _e χ _t ) make it possible to do work:

-ext + m, syst + m _A = 0 (G1.01)

What is new about this method is the consideration of stress and deformation the introduction of a defined area (the

System), which is mentally separated from the environment and interacts with it mechanically, and its physical properties regardless of scale

(regardless of the size of the system),

the explicit separation of forces f _ext / which act on the system from the outside in a controlled manner from the surroundings, and forces m _syst which the system exerts on its surroundings,

the inclusion of the existence of binding forces m _A (permanent in the solid, non-permanent in a liquid), which considerably limit the freedom of the system's reactions to external forces and

the recording of the energetic effect of shear forces

(the work done by shear forces) and the

Conversion of the shear components of the fields f _ext and m _syst into normal components f _S (eχt) ^and dm _S (sys) parallel to

Main stretch direction.

The invention thus provides for a method for determining deformations of a body, in which it is investigated how the action of forces deforms the body, in such a way that for a part of the body or its entirety, as a function of material properties, it is determined Boundary conditions and the existence of binding forces between the system and the environment, a balance of forces between internal and external forces is achieved. The material properties must be known as a function of a location Q. The external boundary conditions (properties of the external force field) must also be known. All the work done by an external vector field is done by dividing the work done by the normal and shear components of a vector or vector field according to the equation

r * f = ^ | rxf | ² + | r - f | ² = | r || f | = const (Eq. 18)

determined.

The radius r is a mechanical lever. Within a period of time that is short compared to the material-dependent diffusion rate or relaxation time, the method can also be applied to liquids and gases.

The total shift is determined according to a logarithmic working equation which is to be derived from the state equation corresponding to the material; in the simplest case, the work equation

which can be derived from PV = nRT.

The invention is explained below using two-dimensional example cases. An expansion to three dimensions is possible without difficulty.

Further advantages, special features and expedient developments of the invention result from the subclaims and the following illustration of preferred exemplary embodiments with reference to the drawings:

From the drawings show: Fig.01 Above: Previous method (finite element method): on

Network of nodes is predefined and the deformation on

Point Q (in the center) is determined by changing the orientation and length of all routes, one end of which is Q. The respective outer points are then correspondence points P. The finite

The element method is called this because the distance between two nodes is finite. Bottom: Any point can be used as a reference point

Q are viewed, all Q assigned correspondence points then form a closed area around Q, and all relevant

Variables - material properties, boundary conditions, deformation can be represented as differentiable functions of the location f (Q).

Fig.02 Compressibility data for the alkali elements. External stress in multiples of internal pressure. Data converted from Bridgman (1958, p.180).

Fig.03 Relationship of f _op to fdev Vertical axis: vector size, horizontal axis: angular distance. The left and right sides represent a contracting and an extending direction (c and e) of an anisotropic field. Only in an orthogonal field are the eigendirections orthogonal and at the same time parallel to the main axes of an ellipsoid. Fig.03 shows the relationship between the minimum and maximum external forces relative to the operative field. Gravity is not taken into account. The ambient pressure and the operative field are isotropic-hydrostatic.

Fig.04 (a) Deviator force field with pure shear. fi and f ₂ are perpendicular to each other and parallel to the coordinates and to the intrinsic directions of the field. The same figure represents the displacement field or flow field when the arrows represent displacements, (b) orthorhombic entire force field, schematic. m: material vector, f _op : hydrostatic operative pressure, f: deviator force field.

Fig.05 Force sizes along a unit body with pure shear. Solid line: f _n , dashed line: f _s . Their total is 1 at all points.

Fig.06 Dilatation effect of a shear force. Curved line: surface of the thermodynamic system. Tangential force vector (dashed) will shift material point P from P to P '; the distance from the origin increases. Point coordinate Xi is lengthened from f to x _± 'by the horizontal component, x ₂ is shortened from f to x ₂ ' by the vertical component. X: external coordinates parallel to the natural directions.

Fig.07 Outer thick line: surface of a unit body before loading. Inner thin line: unit body after hydrostatic loading due to an operative field; Volume shrinkage. Inner elliptical line: shape change of the stressed body only due to the normal component of the deviatoric field; Constant volume. Outer elliptical line: finite deformation after inclusion of the work done by shear forces. Area between the elliptical lines: volume increase due to the dilatation effect. X: outer coordinates parallel to the natural directions.

Fig.08 External force field for simple shear with unit body. (a) The force field consists only of forces parallel to X. (b) The unit body is fixed in space and cannot rotate. P: starting point of a vector, O:

Coordinate origin, θ: angle between the position vector of P and x_.

Fig.09 Forces on the surface of a solid in subjected to a continuum and a force field as in Fig. 08. Average force f _av at its starting point P _κ on the surface of a unit body with simple shear. f _av with size sin K breaks down into normal component f _n and shear component f _s . The latter is balanced by surface binding forces. f _{s (K} ) is subtracted from all shear forces at all points, which results in sinistral shear forces between P _κ and 0. λ is the angle of a coordinate transformation that occurs due to the imbalance of the shear forces. The equilibrium condition J fxr dθ = 0 is met.

Fig.10 Thermodynamic system or volume element with unit mass and elliptical shape, which is in equilibrium with a monoclinic external force field. An acute angle between the directions c and e = 2κ.

Fig.11 Vector sizes with simple shear. (a) in the untransformed state, (b) in the transformed state. Solid line: normal component f _n (-i _e v) (Eq. 44, long dashed line: | f _de v ^x r _e ιι I (G1.42), dash line: f _s (G1.45, see text). -. R, JP: Orientations perpendicular to the R and P surfaces, see Fig.12.

Fig.12 Simple shear geometry. (a) Calculated geometric properties: Xi, X ^ -. Coordinates, c: contracting direction, e: extending direction, _LR: direction from the origin to the R point and perpendicular to the R (iedel) surface; ± P: Direction from the origin to the P point, perpendicular to the P surface. (b) Observed geometric properties (Weber & Juckenack 1990). The only major difference between (a) and (b) is the orientation of the grain shape film. However, it is believed that the extending self-direction must be correlated with the structure orientation, which is generally somewhat lower.

Fig.13 (a) Displacement field for simple shear. Although the external force field has no component parallel to x ₂ , there is such a component in the effective field fdev and the displacement field. Own directions and coordinates see above. (b) Natural SC structure: Mylonite in lower green slate facies from the Insubrische Linie, Sesia zone, Val Strona, Italy. The long dimension of the photo is parallel to the overall foliation, which serves as an external reference system. The shear surfaces falling moderately to the right are the C surfaces. The material layer structure falling slightly to the left is the S surface. Significant sinistral total body rotation between C-surface discontinuities, which have even dextral shear sense. The entire sense of shear is dextral. (c) Viscose simple shear in subrecent Obsidian River, Lipari, Italy. Upper layer was softer, made of black glass with bubbles and shows drag. The lower layer was stiff, consists of partially crystallized material and reacted fragile. Drag in the upper layer and gap orientation in the lower layer indicate dextral shear.

Fig.14. Above: Rectangular solid between two stamps with the locations for which the boundary conditions are shown below. Middle: boundary conditions as a function of the location in the body. Circles stand for (from the inside out): operational field (isotropic); Normal component of the deviatoric field; system-contributed shear-stretching effect, - volume effect due to the external boundary conditions. The latter is directed inward in (1) because of the friction effect on the workpiece-die contact, in (2) outward due to the shear-stretching effect contributed by the environment, and zero in (3) for points on the interface between solid and free space , Below: representation of the expected effect; white circles: unloaded, black ellipses: loaded.

Fig.15. Dimensions 10x2 distance units. Wavy structure along the horizontal edges in the lower image are mathematical artifacts due to summation over non-infinite coefficients n.

Fig.16. Dimensions 10x5 distance units. Wavy structure along the horizontal edges in the lower image are mathematical artifacts due to summation over non-infinite coefficients n.

Fig.17 Dimensions 10x10 distance units.

Fig.18 Dimensions 10x30 distance units.

For the sake of simplicity, model calculations are limited to two-dimensional examples. In the present

As a rule, the equations are given in vector notation and in algebraic form, e.g.

I f x r dθ = I sinθ cosθ dθ

The invention comprises in particular the following four parts: the definition of a scale-independent unit volume with a shape corresponding to the equilibrium conditions, the definition of two force fields which are in equilibrium with one another, the establishment of a material-independent

State calibration for solids, the transformation of the state calibration from scalar to vector form.

Definition of the system

The purpose of the system is to provide a reference body in the unloaded state, which represents the material and other mechanical properties. Work is carried out on this system under load, which is independent of scale in the sense that, for example, the same amount of work is done on a small and a large system with the same deformation relative to its mass. This ensures that a finite volume or radius can be expected, but the actual size of the system is not relevant.

Any mass is associated with a potential. In the case considered here, the internal energy of a thermodynamic system is considered as its potential. In a given energetic state it is proportional to the mass, du ∞ dn, where n is the number of moles (the thermodynamic mass is dimensionless). The energy density per mass is then dU / dV = P, in an integrated form U / V = P, the mass being proportional to both U and V. U / V = P is the energy density, or the thermodynamic potential per unit mass. The latter is then finite.

In the standard state U ₀ , PV = nRT applies. An infinitesimal

Change of state requires the additional energy du. At constant n, this dU is by nature quite different from dU Q dn above. Thus a system of finite mass and finite volume, associated with its own potential, is one Prerequisite to consider infinitesimal changes in state.

In order to be able to define work acting on a point P ₀ , a reference point Q ≠ P ₀ must be selected. The choice is arbitrary and may arise from the nature of the problem (Kellogg 1929, p.53, 63; Boas 1983). The basic distance Q → P ₀ is the zero potential distance r and is interpreted here as the radius of the thermodynamic system, in the anisotropic case: as the distance of each point on the surface of the thermodynamic system to its center of mass. Since the mass is evenly distributed in the region occupied by the thermodynamic system, it represents a distributed source in the sense of potential theory (Kellogg 1929, p.156). The potentials of distributed sources are generally logarithmic. The zero potential distance can then have any finite value (except zero), but is often set to 1 by convention.

In thermodynamics, the properties of the materials are given in a standard state (P *, V *, T *), which can then also be assumed to be a zero potential state. A pressure change to absolutely 0 or oo then requires an infinite amount of work. The distance term r is interpreted here as the radius r _{0 of} the thermodynamic system in the standard state.

It is the one-dimensional equivalent of the volume of mass n in the equation PV = nRT. For solids that have a finite molar volume and a density> 0 in vacuum, the zero potential state is then defined as P * = _pinn . A change in the size of r ₀ then requires one

Change of state. In a first approximation, r _{0 is} initially considered to be invariant with respect to the direction, ie it is assumed that the thermodynamic system has a spherical shape and the surface per volume has a minimum value. The Consideration of other boundary conditions such as material anisotropy can lead to the fact that a different shape must be selected.

In the isotropic case only normal forces need to be considered, so the divergence theorem

fully describes the situation, n is the unit vector perpendicular to A. Force is interpreted as energy flow f = & idU / dx _± . A is a closed surface that contains the volume element JdV; the forces are exerted by the mass in the system on the environment, and vice versa, ie there are two force fields that interact with each other and each fulfill Eq. 02 (where K has the opposite sign). Under the same external conditions, a small and a large mass must do the same work on its surroundings relative to the mass under consideration, so the consideration must be independent of scale. The thermodynamic system therefore represents a source of forces. The divergence of the force field exerted by the system (and therefore also the divergence of the field exerted on the system from outside) is proportional to the mass, so the RHS is a linear function of V; therefore div f is a constant. From this it follows that given external conditions, div f is independent of scale. If the scale under consideration, i.e. V, is changed, since the relation of V oc r ³ to A ∞ r ^{2 is} not linear, the ratio of A to | f | not be constant; f | | must necessarily be a linear function of the scale, | f | oc r. div f is the trace of the tensor F defined below. Both follows from Eq. 02 and G1.06

f | | - = div f = const (Eq. 03)

| R | under constant external conditions, where r is the radius of the Systems and r is a measure of the scale under consideration.

These conclusions are not changed by further developments in the divergence concept, which can only be explained below if the context is clearer. Eq.03 is compatible with the properties of a thermodynamic continuum, with the spatial properties of a thermodynamic system that interacts with its surroundings, and with the fact that the system represents a distributed potential (Kellogg 1929, p.156).

A thermodynamic system is characterized by a given amount

Mass defined. Its location in space is due to its mass center

(or its focus) given in an external coordinate system Xi. The surface points of the system are determined with the help of an inscribed, internal coordinate system Xi, the origin of which coincides with the mass center of the selected system. The physical conditions at the points P in X are functions of Q in Xi in the sense of a vector field, T (Q) v = x, where T is a tensor and a location function of Q; v is the location vector of a nearby point P relative to Q, and x is a vector at location P as a function of T and v.

The relationship between a surface point P and a certain direction in xi is clear. Another vector through P can only be part of another system with its own origin and coordinates, and other boundary conditions may apply if the gradients in Xi are not zero. It is therefore of no interest at first.

Derivation of the external and the material force field

A definition of a tensor is the derivation of one Vector field after the coordinates, or the second derivative of a scalar field after the coordinates (Malvern 1969, p.57). Energy or work are such scalars. It is assumed that an external potential U _ext exists and the material enthalpy is E then

- ^ei = f - = m, (G1. 04) dx, dx, d ² UV, d ² E dm, -. ,, e, F - e, = - = e, M - e _/ (Eq. 05) dx _t dX _j dX _j dx.dX _j dX _j

, , <*. rlγ - fc f - '-dx, = Mp = Hl (G1.06)

^■ *. dX _j * J dX _j where r is the radius or the location vector of a point on the surface of the system, f the external force field, m the material force field of the system (simply: material field), and F and M are the property tensors of the two vector fields f and m. F represents the external boundary conditions, and M represents the material properties, which can therefore also be understood as a kind of internal boundary condition. Forces acting on the system are understood as an energy flow that changes the internal energy state, but not as an external acceleration of a point.

Since extremely unbalanced forces cannot cause deformation, they are ignored. In addition, mass forces are ignored that interact with the inertial mass, but not with the thermodynamic mass. The inner equilibrium condition is then given by f + m = 0 (G1.07) for each point P on the surface of the system, or if f and m are understood as functions of the direction θ,

as a sum over the surface of the system in 2D. The external rotational equilibrium is maintained by definition, since the system and the environment are firmly connected; the case of imbalance cannot occur in an elastic medium as long as no bonds are broken. The condition

Jfxr θ = 0 (G1.09) is therefore not an equilibrium condition, but merely the determination of the orthogonality in the event that the rotational equilibrium is maintained without additional help from binding forces m _s . The complete condition for the rotational balance is

J (f xr - m _s r) dQ = 0 (G1.10) where m _s _L r, its size is invariant, δlmJ

^^ = 0, (Gl.ll) ae and m _s xr has the same sign at all surface points P. Binding forces are neither external forces nor material forces; they exist regardless of the material properties. They do not appear to interact with inward (compressing) normal components of f, but only outward

(dilating) components and generally with shear forces; their existence follows from the precondition that balance must exist and - if necessary - balance the torque of f. The topic will be illustrated in more detail later.

Eq. 07 and G1.08 are essential for the process because they clearly distinguish external and material forces, i.e. Recognize material forces as independent forces. They are exerted by the system on the environment due to a change in state in the system due to the influence of external factors

Forces. The system therefore represents a potential. Balance between system and environment means under isotropic conditions that divf + divm = 0 _; (Gl.12)

G1.12 therefore consists of two Poisson equations.

Tension in the sense of this application is a force vector field ftotai / which consists of the interaction of two independent force fields f and m, the outer and the material force field, their respective boundary conditions in the state of equilibrium, plus the binding forces if necessary (cf.Gl.01), f _tota , = / (f, m, m _s ). (G1.13)

Establishment of a material-independent condition calibration for solids

The ideal gas law, PV = nRT, ignores the atomistic structure of matter and only looks at macroscopic overall behavior. Boyle's law can be viewed as a material law for an ideal gas, since it predicts a certain behavior of the gas due to a change in state. It relates the internal energy of a system, its mass and its volume to each other and therefore completely describes its own energetic state. In principle, such a state equation must also exist for solids. It must therefore be possible to define an ideal solid.

P ^k V = z (G1.14) is chosen as the most general form of state calibration for a solid, wöbei

In ^solld K k = ° -. (G1.15) 'mol

The formula says that all solids are the same Have compressibility dV / dP when taking the molar volume into account, z = f (P) is a number that is characteristic of a given state; the function itself is not known, but is successfully modeled using the Birch-Murnaghan equation. This is used in the modeling of material behavior under high pressures, eg the prediction of the elastic properties of the earth's core (Jeanloz & Knittle 1986).

It is known to compress alkali metals to 100 kb. In a first approximation, they all follow the same function if the externally applied load is expressed in multiples of the internal pressure of the respective material (Fig. 02). It is therefore justified to generally use internal pressure as the standard for material behavior. Eq. 14 is isotropic, therefore k is independent of the direction and is also only important for modeling real solids. It is sufficient if k = 1 is set; thus z = const. Thus, the thermodynamic properties of the ideal solid are identical to those of an ideal gas, and Boyle's law is applicable.

Transformation of the state equation from scalar to vector form

That is the fundamental law of materials in thermodynamics

Boyle-Mariottesche law,

PV = const (G1.16)

Both sizes on the left can be reinterpreted without the units themselves changing (the joule remains); the

The result is a vector form of Boyle's law, r * f = const (G1.17) where r is the location vector of a point P on the surface of the

Systems is relative to its mass center. The star product is interpreted as r * f = -J | rxf | ² + | r • f | ² = | r | f | = const; (G1. 18) r * f describes the work done by the normal and the shear components of an external force f at point P, which has the position vector r and is part of the surface of a thermodynamic system. Eq. 18 is a known identity. If r is a unit vector, then r * f = | f |; but the ratio of | r | to | f | now has the freedom to be variable as a function of the direction, even if Eq. 18 is adhered to, namely if | f | and | r | behave inversely to each other.

The force field f _tota i _/ which develops during the transition from the unloaded to the loaded state can be broken down into an isotropic component, the operative force field f _op , and the deviating component f _dev , where f _{op represents} the change in state. Both for fr and | fxr | integrated over the surface of the system add up to zero. A reference state is defined by f _op , which serves to assign signs to the directions of the deviatoric field (Fig. 03)

In contrast to gases and liquids, solids can withstand shear forces, and the energetic state of a system is influenced by shear forces and normal forces alike. A normal force will cause the system to dilate or contract, depending on the sign. The

The sign of a shear force, on the other hand, only shows its direction of rotation, but it has no information about a volume effect. The effect of the shear forces on the volume can only be dilating. Example: a deviator field f _dev can be further divided into a field consisting only of normal components f _n and a field consisting only of shear components f _s . A spherical volume element is subjected to the shear forces f _{a of} a force field with orthogonal directions; the The origin of the coordinate system is at the mass center. On the system surface, the f _{g start} at the points P with the position vector r and move them parallel to the direction of f _s . You will have positions P 'with position vector r'. Since r and v = P - • P 'are perpendicular to each other, | r' | is always > | r |. Therefore, the work done by a shear force has a volume effect that is always dilating. The dilational component adds up from zero at the starting point of the contracting self-direction towards the starting point of the extending self-direction, where it reaches its maximum. This effect applies both to shear forces that are exerted on the system by the environment and, conversely, to the forces that are exerted by the system on the environment. The common effect of the shear forces results in two additional normal-stretching force field components f _S (ext) and πis (sys) parallel to the extending direction. In the end result, due to the existence of the shear forces, a dilating (i.e. normal) effect is achieved parallel to the extending direction, so that f _total = f _op + f _dev + f _{s (ext)} + m _{s (syst)} (Eq. 19) where f _ex t = fo + fdev / F _op = cl, and det F _dev = + 1. An example is given below. The tensors are field property tensors as in the definition of a vector field Tv = x in linear algebra.

If the entire force field f is _completely known, the work can be calculated analogously to the PdV work,

J / ö? R = cJ— = clnr (G1.20)

This equation, integrated over the surface of a thermodynamic system, gives the same result as the simple calculation of the PdV work when the load is isotropic and compressive. The relationship of a material distance, i.e. r ₀ , to a colinear force is given by differentiation of Eq. 18, r ₀ df + f ₀ dr = 0 (Eq. 21)

If Eq. 21 is divided by r ₀ , re-integrated and divided by f ₀ , it follows

in complete analogy to In (Vι / V ₀ ) = - ΔP / P ₀ in isotropic thermodynamics. G1.22 represents the equation that connects cause and effect; With their help, the displacement field Δr = s is calculated from the force field ftotai, which therefore have identical geometric properties.

A solid has its ideal volume V ₀ with a unit radius r ₀ in a vacuum. The internal pressure Pi _{nt of} a solid is around 100 MPa. Analogous to the internal pressure in vector form, the internal or material force is m ₀ , which is a force with size> 0, but internally balanced in the unloaded state. It is set to one size below for modeling purposes. The size of the external effective force must be standardized with reference to the size of the material force m ₀ , from which Eq. 07 reads completely:

-Δm = Δf _ext (Eq. 23) where the first term on the right is the magnitude of f _ext in multiples of the internal pressure of the material, the second is a unit vector with the direction of f _ext . This standardization is always applied in thought; In the following, unit sizes are used for the sake of simplicity, and Δf is referred to as f.

The equilibrium condition for the rotational moment depends on the properties of the force field and the shape of the System. The balance is always maintained because the system and the environment are firmly connected. The shape of the system is determined by the properties of the force field, the material properties and the condition that the A / V ratio should be minimized for a given mass.

If the effect of a progressive deformation is to be shown in 2D, a circle is usually transformed into an ellipse by a displacement field. The unit circle in the undeformed state is certainly an adequate concept for the demonstration; but the shape of the thermodynamic system (or volume element) must adhere to the equilibrium conditions. In particular, the system in the undeformed state can have an elliptical shape under a certain set of boundary conditions, in another state it can be a circle. The two concepts - a circular drawing in the undeformed state versus the mechanically active form of the system - must not be confused.

The strain tensor is a function of the displacement field. If the main axes of the strain ellipsoid do not rotate during a progressive deformation, they must coincide with the characteristic directions of the displacement field (and therefore also the force field). It can be concluded from this that the natural directions of the force field and therefore also the displacement field are orthogonal. From a mathematical point of view, however, this is a very special case and is by no means a prerequisite. The strain tensor is therefore not helpful in understanding deformation. It is therefore necessary to (a) deduce the form of the system from the equilibrium conditions, and (b) find the proper directions of the force field. They are defined as the directions along which the stress field f has no shear components. Shape of the volume element

Eq. 18 provides boundary conditions for the shape of the system. The simplest form is the sphere - it is isotropic, minimizes the A / V ratio, and for each point source at the center of mass, its surface is an equipotential surface. For isotropic external conditions and an isotropic material, the spherical shape is the most natural choice for the system because there are no shear forces. For other types of deformation, Eq. 18 stipulates that the radius and normal force along the contracting eigen direction c and along the extending eigen direction e are of identical size. The swirl of the deviatoric field for pure shear (Fig.04)

.___. 0 x. -1 0

^F dev = (G1.24)

0 ___. 0 1 dx. is zero; therefore the directions of the resulting field coincide with those of F _dev . The conditions tr T = 0 and det T = ± 1 are both conservation conditions and must be observed together, i.e. I En I = IF ₂₂ 1 = 1. Since F _{dev is} orthogonal, there are no restrictions from the equilibrium conditions for the shape of the System, which is therefore still spherical.

The sizes of the normal and shear components of the deviating forces are shown in Fig.05. Under the given boundary conditions, all deviating forces on the surface have the components f = [cosθ, -sinθ], where θ is the angle between the orientation of r and the xi coordinate. The external equilibrium condition for the torque for the whole body is then ^* f xr Θ = 2 ^ ^π cosθ sinθ-iθ = 0, (Eq. 25) and the surface binding forces m _s are not activated.

Pure shear kinematics

m ₀ is the material force that is in equilibrium with a vacuum. For a body with a unit radius in an orthogonal deviator field, the components of the position vector r of a surface point P are given by [r _x r ₂ ] = [cosθ sinθ]; the operative force vector is a normal force with the components f _op proportional to [-cosθ -sinθ]; the deviating force f _dev is proportional to [cosθ -sinθ]; and the radius-normal tangent unit vector t = [sinθ -cosθ].

External forces acting on P are the isotropic operative forces f _op and the deviating component f _de vf _o p consists only of normal forces. f _dev can be further divided into a normal component f _n (dev) and a tangential component f _S ( _de v) • The sum of all normal components is Σf _n = f _op + (rf _dev ) r = l + (cos ² θ-sin ² θ) (G1.26)

The corresponding inward displacement vector s _n parallel to r can be found by Eq. 22.

The cross product | rxf _dev | = | f _{s (dev)} | = 2cosθsinθ (Eq. 27) is equivalent to f ^■ t. A shear force causes P to be pushed away from the contracting direction and towards the extending direction (Fig. 06). The components of f _s ( _de v) at P cause material forces of the same size and direction but opposite orientation in the system, so that the torque at P is balanced. The effect of f _s is that P is shifted to P '(Fig.06); however, since f _{s is} perpendicular to r, the distance r 'from the origin to P' is always longer than the distance from the origin to P (Fig.06). So shear will always cause local elongation proportional to its own size. This effect is additive from point to point. The jeo-baαhte e dilation at P 'is therefore the sum of all dilations from the contracting eigen direction to P, j ( ^f _dev ^• t) r dθ = 2 sinθcosθ / cos ² θ + sin ² θ dθ (G1.28) where α is the angular distance from the contracting eigen direction c at x ₂ to P. For α = ^π / ₂ , Jf-t dθ = 1. There is of course no dilatory effect at the lower integration limit; the integral reaches its highest value at the extending eigen direction e, although there are no shear forces at this point.

In a continuum connected on all sides, the result of Eq. 28 counts double t. Assuming that a body is hydrostatically loaded, so that the operative force field (which causes the change in state) has a size of -1 and the deviatoric field has a size of zero. The radius is shortened along x _x and x ₂ . Then the boundary conditions along Xi are relaxed; the environment is believed to be immobile. In an attempt to reduce the stored energy, the material expands by itself parallel to x _lt but nothing happens in parallel to x ₂ . The size of the operative field is reduced to -%, and a deviating field with the size ±% is created. Since the expansion is not isotropic, the body exerts shear forces on its surroundings; to the right of the contracting self-direction, a left-turning pair of forces is observed at the interface between the system and the environment above the extending self-direction (in the 1st quadrant), but below it a clockwise rotation (in the 2nd quadrant). So the system affects its surroundings, reducing the energetic state of the system from -1 to -% until the system has reached its energetic minimum state. However, too note that the sign of the shear force pairs is opposite to what you would normally expect with pure shear in the continuum. However, if the environment is deformed together with the system, it will first expand with the system in Xi and then exert additional shear forces on the system, which causes a right-handed pair of forces in the 1st quadrant and a left-handed pair of forces in the 2nd quadrant, which in turn causes further expansion in Xi. This shearing work is done by the environment and leads to an increase in the energetic state from -% to -1. So part of the expansion is due to the shear effect of the system, the second part is due to the shear effect of the environment, and both are proportional to the size of the deviating field. In the end, a dilatory force of twice the size of the integrated external shear forces is applied to the surface point under the extending self-direction, which is therefore pushed away from the system's center of mass.

Volume effect of the deviatoric load

A numerical example based on G1.22 is given. r ₀ = 1, fO = 1 / and volume V ₀ / π = 1. After charging to the operative field strength -1/5, the quantities along x _{2 are} : Δf = f _op + fn (ev) = -2/5, r ₂ = 0.670, along Xi: f _op + f _n (dev> = -1/5 + 1/5 = 0, Zi = 1,000. However, f _S (i _nt ) + f _B {eχt) = 2/5, therefore r_ = 1.492. The volume of the resulting ellipse is V / π = rr ₂ = 1, ie the volume is the same as in the unloaded state (Fig.07).

Model calculation for simple shear

The own directions The position of the body's mass center is given in an external coordinate system Xi. A second coordinate network x ^ is connected to the body, so that its surface points are given in the second system.

For a dextral shear, the deviatorial external force field has the form F _ex r = f _ex t / whereby

(Abb.08a). This field interacts with a unitary body (the thermodynamic system) with isotropic material properties (Fig.08b). Therefore, in a continuum, an infinite number of force vectors start from an infinite number of points on the system surface. It is therefore necessary to find the average starting point of the average force vector f _av on the body. Just like that

The problem in Fig.08 is f ∞ sinθ, where θ is the angle between r and the coordinate Xi. Then the average for a quadrant is

π fsinθ-θ ₂

^ = -. (G1.30) π π

2 The average force is a vector of the form f _av = [2 / π,

0], the position vector of its starting point P _κ forms the angle K = ± 39.54 ° with Xx (Fig.09). If the body is held in space and exposed to a force field as in Eq. 29, the normal force has the form f _n = (f - n) n = cos θ sin θ, _Q -, ₃₁ \ while the shear force by f = (f • t) t = sin ² θ. (G1.32) given is. The latter is therefore at P _κ

\ ² f ^{s (κ)} = -; = 0.405. (G1.33)

If the body is allowed to react to the applied force field, f _{B (κ} ) is balanced: since there is coherence between system and environment, an external imbalance is impossible. Therefore, the system cannot rotate freely, but the surface binding forces m _s appear, which prevent the rotation. Therefore | f _S ( _K ) l of all | f _s | subtracted from all points, reversing the sign of the effective f ₃ in some areas. The effective shear force size at all points P is therefore

so the total effective force on P (θ) is.

At the beginning there were two points without a force component perpendicular to the radius, at θ = 0 and θ = π (Fig.08b). Subtracting f _S ( _K ) instead creates two new ones with angular spacing θ = + K on both sides of the X coordinate. Since only normal components are active along these directions, they are interpreted as intrinsic directions (contracting: c, extending: e) that are no longer orthogonal. Together they enclose the angles π - 2κ = 100.92 ° and 2κ = 79.08 °. However, the unit vectors n and t J. n were used in Eq. 31 and G1.32 without further explanation; however, this implicitly assumed a spherical shape for the volume element, since there is no difference between n and r. However, it is still not possible to compensate the torque for a spherical body since J _2π f _S ( _eff ) ^xr dθ ≠ 0. It is therefore assumed that the spherical body is assumed is incorrect. The angular relationship between the eigendirections suggests that both the effective force field f _eff for simple shear and the shape of the system have elliptical properties. It is therefore necessary to calculate the shape of the ellipse, which is in equilibrium with f _eff with the eigendirections as above, and which fulfills the conditions of Eq. 18.

For-τ? of force field and volume element

In the example of pure shear (RS), the field property tensor has the properties tr F _RS = 0 and det F _RS = ± 1. Eq. 29 applied to simple shear (ES) differs from this. However, the germ in Eq.33 is not part of Eq.29. It is therefore assumed that the field matrix for simple shear after the subtraction has the property det F _ES = ± 1, which indicates that all flows represented by the deviating field and entering the system also leave it. Therefore, the force field for the conditions of elastic simple shear and the shape properties of the system are modeled here by using an orthogonal field of shape

is assumed to be in equilibrium with a spherical system. The main directions should coincide with the points below the coordinates, which is why the field is rotated 45 ° sinistral. Both the force field and the radius field are then converted using a transformation matrix T, so that they obtain elliptical properties, where p are the position vectors of the surface points in the non-transformed state: cotκ 0

T = (G1.37) 0 camouflage

Tp = r _ell = [E _n cos θ E ₂₂ sin θ], (G1.39) and

F _dv P = f _dev = [^, sinθ E ₂₂ cosθ]. (G1.40)

The normal directions of the elliptical radius field r _e ιι are therefore orthogonal and identical to the main axes of the resulting ellipse, which has cross-sectional area A / π = T- - _{I 22} = 1, while the eigen directions v _e and v _{c of} the force field are not orthogonal as desired are (Fig.10),

1

^V e, c = (G1.41)

± tani

Simple shear kinematics

It is convenient that all equations relating to the equilibrium conditions are only the transformed equations of the pure shear example. The cross product simplifies itself

^< jl ^f dev ^{X r} ell | ^dQ = ^ Ä <f ( ^{COs2 θ ~ sin2} θ) -iθ = 0, (Eq. 42 ⁾ ie the elliptical properties of the radius field and force field shorten each other. The linear moment is

<jf _dev τ _ell dθ = (T, + T? ₂ ) jcosQsmθ dθ = 0 (G1.43)

E ^ + E, ² > 2, is the maximum size of the dot product along the perpendicular directions> 0. This is considered an artifact of the elliptical shape, because the radii along the normal directions are not of unit length due to the transformation by T. If, on the other hand, the scalar product of the inverse position vector r _e ιι ^{" 1} = T ^_1 r is carried out,

<jl ^f „ _(de . ₎ | ^ θ = ^ r ^{"1 •} f _dev -iθ = <J2 cos θ sin θ dQ = 0 (Eq. 44) the equilibrium condition is met. The elliptical properties of the force field and the radius field cancel each other out, which results in the reference to a normal deviator force f ( _n ) _de v per unit radius, the maximum size of which is 1. Eq.44 shows the normalized relationship of | f _n (dev) | to | r _e ιι | ; this normalization of f _n with reference to r is necessary (see G1.23). f _p (see Eq. 19-Eq. 23) is therefore also a unit vector. Eq.42 and Eq.44 together hold Eq.18. Some vector sizes are shown in Fig.11.

If t ± n, t is not a tangent vector for the ellipse, but is defined as a unit vector perpendicular to the radius and independent of the orientation of the ellipse surface.

The size of the shear force component is through

| f _{S (d} - _v) | = f _dev - t = sin ² Θ - E ₂₂ cos ² θ (G1. 45). If G1.53 is integrated for separate sectors (e.g. from the contracting in direction in one from both sides to the next extending Ε direction, or from 0 to K and from K to ^π / ₂ ), it shows that there is an imbalance between dextral and sinistral Shear forces on the system surface. However, G1.45 is not an equilibrium condition and the shear forces alone do not have to be balanced. Since the forces for the ellipse are balanced (Eq. 42), an imbalance for f _s ( _d ev) does not mean free rotation. The imbalance is therefore interpreted in such a way that the force field is permanently reoriented by transforming the inner coordinates x ^ against the outer Xi by an angle λ,

where the vector sizes under the root stand for the integrated sums over neighboring sectors. The root is drawn because all angular terms in Eq. 45 are square terms. There the directions are real, the angle λ is stable.

It is advisable to interpret the left side in G1.46 as a tangent term because there is no reason why the term should be limited to values <1. If LHS - »∞, λ - 90 °, which would be the condition for external rotation. At the same moment, the characteristic equation should degenerate, i.e. have only one solution, whereupon the directions of their own become imaginary.)

The invention involves a shear force moving a point P to P 'away from c and towards e. Shear will always cause local elongation proportional to its size. This effect is additive from point to point. The same effect applies to simple elastic shear, except that the eigendirections are no longer orthogonal. However, the result is similar to that of Eq. 28 (which resulted in an integration over one quadrant 1), except that the surface over which integration takes place is larger than a corresponding spherical segment. The result per quadrant is therefore compared with the ball (by a factor) TχχT _22/2 = 1.018. As explained in the example of pure shear (according to Eq. 28), this result counts twice, i.e. w _s = 2.036.

w _β is the dilating effect of the shear forces on the radius - r_ parallel to e. The reduction factor for the radius r _c along c is given by Eq. 22. If v _c and v _{e are} normalized eigenvectors, they can be understood as the radii of a unit circle that is exposed to the monoclinic field Fr = f, and the displacement field can be calculated. A numerical example using Eq. 22: r ₀ = 1, fo = 1, and the volume V ₀ / π = 1. After loading on the operative field strength -1/5, the sizes are along c Δf = f _op + n (dev) = -2/5, r _c = | v _c | = 0.670. Along e they are f _op + fn.dev) = -1/5 + 1/5 = 0, r _e = 1,000; but w _s = 2.036 / 5 = 0.407, therefore r _e = I v _e l = 1.503. If S is the displacement field property tensor,

| v _c | v _β | = detS = 1.0155 (Eq .47) which means that in this example the volume experiences a real expansion of 0.75%.

Geo-net-risαe properties of deformation after simple shear

The coordinates Xi are rotated by 1 in the shear direction compared to Xi. In the outer frame Xi, the following picture emerges (Fig.12): the extending direction e is 10.71 °, the contracting direction c is 111.63 °. The bisectors of the angles enclosed by c and e are parallel to the xi coordinates and mark the directions of maximum shear. The orientation vector, which halves the larger sector at 61.18 °, marks a shear direction perpendicular to itself, the R surface is therefore at -28.83 °; the orientation vector that divides the small sector at 28.83 ° marks the P-shear area at 61.18 ° (Fig.12).

An important part of the invention is the determination of the force field that causes the deformation and the

Displacement field. However, regardless of the type of deformation - elastic or plastic - the displacement field must reflect the properties of the force field, even if the resulting phenomena differ significantly in their nature. Therefore, it is justified that

Properties of the calculated force field with

To compare structural properties that in plastically deformed

Rocks can be observed. X is the shear zone boundary

(Fig.12a). From the proper directions, e becomes with the S-surface in SC structure correlates (Berthe et al. 1979) and with the direction parallel to which the main anisotropy of the crystals aligns. The predicted angle of approx. 11 ° is in good agreement with the skewness of structural diagrams for monomineral shear zones with minerals with only a well-developed shear direction (ice, mica, olivine).

The contracting intrinsic direction c is not well developed in high temperature tectonites. The direction can be recognized by the absence of pressure shadows or by minimal mica alignment along the surface of feldspar porphyroclasts. However, when the Mylonites are lifted out, they develop frequent fissures that cut the layer structure at an angle of approx. 70-80 ° against the shear direction. They are apparently caused by the elastic energy that was stored in the rock during the plastic deformation and which is released when the surrounding pressure is no longer sufficient to hold the rock together. The procedure presented here allows the conclusion that cracks open parallel to c.

The shear surface at θ = -28.83 ° is the Riede surface R (Riedel 1929) or the C surface in the SC structure (Berthe et al. 1979). The P-surface is mostly suppressed in the case of natural plastic deformation, but was observed in experiments (e.g. Merzer & Freund 1975) and sometimes also forms a poorly developed shear direction (Fig.12b). It becomes more apparent when the material behavior is close to the plastic-breakable limit. The R surface (synthetic, here dextral) and the P surface (antithetical, here sinistral) are not mechanically equivalent. The R-surface is probably the first to start flowing when plastic behavior begins. Fig.13 shows the flow field. The shear sense on the R surface is synthetic to the overall shear direction, but the surface itself rotates antithetically to e; In this process it is progressively stretched until it disintegrates. R and P surfaces are stretch faults.

Deformation energetics with elastic and plastic, pure and simple shear

Elastic deformation

The state in which the system is located is determined by the operational force field, which serves as a measure of the field strength. It can be used to compare the types of deformation with each other. In the elastic range, the operative and deviatoric fields are proportional to each other during the progressive loading; they therefore both need work that represents the elastic potential. Since the sum of f _n <dev) from 0 to π is zero, the work performed by normal forces is only that of f _op ,

^ f _op dθ = -2π (G1.48) for r ₀ = o = Δf ^" _op = 1 (cf. Eq. 22). In the isotropic case, this would be the only term that would have to be taken into account in 2D; in absolute numbers this is the case of the greatest work of all possibilities. This work manifests itself in isotropic contraction. Work done by shear forces causes directed dilation. Eq. 28 integrated from 0 to p / 2 for a quadrant, times 4 for all quadrants, gives 8,000 ; the total work for elastic pure shear is therefore

In simple elastic shear, the size of f _{op is the} same as above. The shear work (G1.47) must be integrated for a sector from c to e and multiplied by 4 in order to obtain the sum over all sectors. The total work with simple elastic shear is therefore w _SS ( _e i) = -2π + 8.144 = 1.861. With the same operative field strength, elastic simple shear therefore requires 8.4% more energy than pure shear.

Plastic deformation

The mechanism by which plastic deformation is brought about is not the subject of the invention. It is assumed that the material remains continuous and that the field f remains in a homogeneous state. The transition from elastic to plastic deformation (the pour point) is characterized by a change from reversible to irreversible behavior. The operational field therefore achieves a stable field strength and requires no further work.

All further work, the plastic deformation energy, is therefore done only by shear forces. Since the dilation they cause is an elastic phenomenon, the structure of which stops at the pour point, the energy is not stored in the volume of the system, but dissipates along its surface. Since the system therefore behaves passively, the principle of double work (see discussion after Eq. 28) no longer applies; so the shear energy required is only half that for elastic deformation. Dissipated energy is lost energy and does not have to be balanced; however, since a free rotation of the entire system is not possible in a field with real directions of its own - which still exist due to the continued existence of the elastically tensioned state - the rotational equilibrium must still be balanced.

For simple shear, the field f is orthorhombic, and sinistral and dextral shear equalize, so external rotation of the system is not possible. In the When considering the work, the sinistral and dextral shear work are to be summed up, therefore the work performed with plastic pure shear is half as large as with elastic loading, i.e. w _P s ( _P i) = 4.

With simple shear, the conditions are somewhat more complex. Eq.45 from c to e integrated for both sectors gives different values: 1.229 dextral shear work in the large sectors, and 0.733 sinistral work in the small sectors. The imbalance for the entire system is 1.202; the total shear work done on the system is 4,144. Because the major sectors are overweight, the imbalance triggers a plastic dextral overall flow. Triggering a dextral imbalance leads to torque imbalance for the entire system. Since the shape of the system is elliptical and the long axis of the system is in the small sector with sinistral shear, the torque of the entire system is proportional to the size, but inversely to the direction of rotation of the plastic shear, i.e. sinistral. Dextral plastic shear costs energy that is dissipated, that is, it is internal dextral shear. The rotation of the system due to the torque is an external sinistral shear that does not cost any energy. Since the inner and outer rotation balance each other out, the directions are still real. Therefore, the dissipated energy in a plastic dextral simple shear is minimized by an external sinistral rotation, and the energy saved in this way is proportional to the imbalance; so w is _S3 . _p ι = 4.144 - 1.202 = 2.933.

With the same operational field strength, a simple plastic shear therefore costs 26.7% less energy than pure shear. Energetic differences of this magnitude are known from experiments; they occur systematically as a function of the displacement field and are independent of the material (Quartz, salt, plaster; Franssen & Spiers 1990; Franssen, pers. Mitt.).

The method presented here is fully compatible with potential theory and thermodynamic principles. The invention offers a simple, direct correlation from force field to displacement field, which are both vector fields and are controlled by tensors; from a mathematical point of view, this is more systematic than the previous method. The new method also enables predictions that the previous method does not allow, e.g. because it a specific displacement field cannot be derived from a given stress state.

A viscous deformation step for a gas or a liquid can be broken down into a time-independent, reversible, elastic loading step and a time-dependent, irreversible, diffusion-controlled relaxation step. The new method is also suitable for modeling the behavior of viscous liquids, provided that time periods are considered that are short compared to the relaxation rate.

Application example for an elastic deformation of a

Workpiece with a given shape under given boundary conditions

The method presented is based on potential calculations. The thermodynamic continuum is not a continuum of points, but a continuum of systems. Any one

Point Q in the Euclidean space filled with distributed mass serves as the seat of a thermodynamic system; its physical properties - density, chemical composition, temperature, material properties, orientation of the anisotropy - and additionally those of the external ones Boundary conditions are then location functions of Q. Independence from scale then ensures that the extensive quantities per unit mass are valid. Two infinitely close points Qi and Q ₂ represent two different thermodynamic systems V and V ₂ with a finite size, which largely overlap. Their physical properties or the boundary conditions can be subject to gradients in Euclidean space. The systems can be defined to have unit mass, unit volume, or unit radius, all of which are finite. In contrast to the previous method, the invention presented here offers a finite segment that follows its internal logic - the radius of the thermodynamic system - as the basic segment for a wave equation. The spatial properties of the physical quantities can therefore be modeled using a Fourier series approach as a function of the boundary conditions.

The entire force field is divided into the isotropic

Force field f _op and the deviator force field f _dev (see Part I, Koenemann 2000). The latter can be further broken down into the normal component f _n (dev) and the shear component f _s <dev) • Normal forces will shorten or stretch the radius vector assigned to them, depending on the sign. Shear forces, on the other hand, always have an expanding effect, which has the effect of an additional extension component parallel to the extending eigen direction e, which is parallel to x _x . According to Eq. 28 (Koenemann 2000) it was found that the work performed by shear forces applies twice, because one part of the extension in x _{x is performed} by the system itself, the other by the environment. The part performed by the system is a function of the load configuration (load size, external boundary conditions, eg pure shear). The shear dilatation caused by the environment exists in a perfect continuum certainly, but their size must reach zero at interfaces against free space. If all other components are kept constant, it is this component that can vary as a function of the location within a body and is therefore a function of the shape of the body.

The 2D model consists of a rectangular solid. A constant force is exerted on the top and bottom, while the side surfaces are free. The load acts vertically parallel to y (df _y / dx = 0). The loaded surfaces (y = 0, y = 1) are hindered in their freedom due to the friction on the punches and cannot expand in x; on all other y they are free. Therefore, the load is the same throughout the body, there is no gradient, including the points on the side surfaces. Therefore, the sizes of the operative field f _op and the normal component f _n ( _de v) are monotonic in the whole body. They are not considered in detail here.

The boundary conditions for the dilating shear T vary in the body in two ways: (a) The condition that the stressed surfaces must not expand in x achieves its maximum effect on the surfaces y = 0 and y = 1, but is against that Center of the body increasingly ineffective, which allows a progressive expansion in x (here called T_); and

(b) the continuum shear effect (T ₂ ) reaches a minimum - on the free side surfaces because there is no longer an environment; on the exposed areas above and below, because expansion is prevented by condition (a) - but it increases towards the center of the body. The direction of decreasing severity for the

Boundary conditions are towards the middle for Tx and on all sides for T ₂ . So the two boundary conditions have opposite effects and signs (Fig. 14, arrows in the fourth ring for conditions 1 and 2). They must therefore be modeled separately. In the statically stressed state, the system and the environment work together that cannot be measured, because in thermodynamics only the work associated with changes in state can be measured. Time-dependent processes are not observed, system and environment are in balance with each other, and all flows are balanced. The variable T = Tx + T ₂ under discussion is the work done by shear forces, a scalar variable that causes a volume change that is also a scalar variable. It is therefore possible to approach the model using a Laplace equation.

x is treated first. The distribution of a scalar quantity x in the region x (0—> b) y (0 →) with a maximum value tx = M for all points y = 0 and tx = 0 for all points x = 0, x = b and y = d is

(Boas 1983: 546, their equation 2.17), where M is a factor determining the absolute size, b the width of the body in x, d its length in y, and m = 2 n - 1 (m is always odd). The opposite is the case where t ₂ = M for all points y = d, and M = 0 for all points x = 0, x = b, and y = 0,

The common effect is T _x = x + t ₂ . Along x = b / 2 forms Tx is an upward curve with a minimum T _x = -M for y = 0 and y = d and a maximum (ΛfcO) for y = d / 2. Along y = d / 2, T _{λ forms} a downward curve with a maximum Tx for x = 0 and x = b and a minimum at y = d / 2 (which is the maximum point on x = b / 2).

For T ₂ the boundary conditions are as follows: M = 0 for all points x = 0, x = b, y = 0 and y = d, and the graph rises inwards. This happens through

The sum T = T _x + T ₂ is interpreted as the potential for expansion cracks (in Fig.15-Fig.18 above). It is of particular interest to find out where the gradients are greatest for Eq .3. Therefore

and dT _ πx πy

- πcos- d sin (G1. 53) dx T ^~ d

the size of the gradient is determined by

and

given. The derivative d ^ T / dx (the sum of G1.546 and G1.55) of the surface T {b, d) is interpreted as the potential for shear cracks (Fig.15-Fig.18 below). The derivative d ² T / dy ² gives the same results in the sense that the morphologies of the two derivatives are identical, however their absolute sizes depend on the chosen absolute sizes of b and d. For realistic results, the model must be calibrated using observed values.

The approach provides all properties of the voltage as a function of the location within the region of validity bd. Similarly, it should be possible to derive the displacements from the working equation and calculate the shape of the deformed body.

Results

Fig.15-Fig.18 the upper parts show the potential for elongation cracks T, the lower parts show the potential for shear cracks 3 ² Tθx ² . The width of 10 distance units is the same in all figures. The colors indicate relative sizes (purple = minimum, red = maximum).

For thick, short columns (Fig.15, height 2 units) the center is dynamically dead because it is under strong compression. The dilation potential can only be found along the free surfaces, which suggests that a brittle material that is loaded to the elastic limit will split off on the surface. The tear potential is further within the body, which further destabilizes the surface, but is very low in the center. The stress distribution is heavily controlled by the condition that the stressed Do not allow surfaces above and below to slide along the punches.

If the height of the body is chosen to be only slightly larger (Fig. 16, height 5 units), the expansion crack potential is still close to the surface, but the shear crack potential has migrated very quickly into the interior of the workpiece. The two maxima no longer coincide.

In a cube-shaped body (Fig. 17, height 10 units), the expansion crack and the shear crack maximum collapse again, but this time in the center of the workpiece, interpreted to mean that any form of material yield begins here, be it brittle or plastic.

Finally, a long, narrow body is considered (Fig. 18, height 30 units). The expansion crack potential remains in the center of the workpiece, but it spreads along the long axis and only decreases shortly before the loaded surfaces. The shear crack potential has migrated from the center towards the contaminated areas and forms two maxima that run out towards the edges at which the contaminated and the free areas meet and a boundary condition discontinuity exists. The two foothills to the corners and the loaded area enclose a triangular region, which are also only exposed to little stretch; these triangles are practically dead. The calculations allow the interpretation that fragile behavior due to expansion cracks should begin in the center of the workpiece; the

Cracks should continue along the long axis of the test body (which ultimately changes the local boundary conditions within the body to some extent) until the stretching effect disappears; but this is exactly where the shear crack potential reaches its maximum, which is why it is expected that the fracture mode should change from expansion crack to shear crack and the cracks should continue against the edges of the body. The predicted material behavior is indeed observed in brittle materials, for example in ceramics (Winter 1992).

conclusion

It has been shown that the new deformation detection method can be modulated to represent the stress in a discrete body exposed to a particular stress configuration. Compared to the effort for the previous method, the requirements for technical equipment and mathematical framework to achieve the results presented here are extremely low. The calculations included can be carried out on any computer.

The term "computer" is in no way to be understood as limiting. This can be any unit suitable for carrying out calculations, for example a workstation, a personal computer, a microcomputer or a circuit suitable for carrying out calculations.

An important advantage of the invention is that no point network is necessary, but a solution can easily be obtained for any point within the scope, using mathematical methods for which solutions to problems from other field theories have long existed.

In particular, the invention enables deformations without Use of a network of nodes, as shown for example in Fig.01.

The invention thus enables a much easier determination of deformations.

literature

Berthe, D., Choukroune, P., Jegouzo, P. (1979) Orthogneiss, myloni te and non-coaxial deformation of grani tes: the example of the South Armorican Shear Zone. Journal of Structural Geology 1, 31-41

Boas, M.L. (1983) Mathematical methods in the physical sciences. J. Wiley & Sons, 2nd ed., 793pp Bridgman, P.W. (1958) The physics of high pressure. Bell & Sons Ltd, London; 445pp.

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Claims

Claims:

1. A method for determining deformations of a body, in which it is investigated how the action of forces deforms the body, characterized in that, for a part of the body or its entirety, it is determined how a balance of forces between internal forces depends on material properties and external boundary conditions and external forces, and that the deformations of the body are determined in such a way that all acting forces essentially correspond to the following formula:

fext + πigyst + m _A + fg (ext) + I & s (syst) = 0.

2. The method according to claim 1, characterized in that the material properties are determined as a function of a location Q.

3. The method according to one or both of claims 1 or 2, d a d u r c h g e k e n n z e i c h n e t that force field properties are determined taking into account externally controlled limit conditions.

4. The method according to one or more of the preceding claims, characterized in that the total work performed by an external vector field by dividing the work performed by normal and shear components of a vector or vector field according to the equation

r * f = | rχf | + | R-f | = | r | f | = const

is calculated.

5. The method according to one or more of the preceding claims, characterized in that a mechanical effectiveness of the radius of a termodynamic system in the solid is taken into account.

6. The method according to one or more of the preceding claims, characterized in that the radius of the thermodynamic system is taken into account as a lever within a period of time which is short in comparison to the material-dependent diffusion speed.

7. The method according to one or more of the preceding claims, characterized in that the normal extension caused by the shear component of the force fields f _S ( _e χt) and m _S ( _S y _St ) substantially according to the formula

J (f _dev • t) r dθ = 2 sinθcosθΛ / cos ² θ + sin ² θ dθ

is calculated.

8. The method according to one or more of the preceding claims, characterized in that the total shift according to a logarithmic working equation, in particular the equation

is determined.

9. The method according to one or more of the preceding claims, characterized in that the Total shift is determined according to a working equation derived from an equation of state.

10. Computer, characterized in that it comprises at least one calculation means that carries out a method for determining deformations of a body, wherein it is examined how an action of forces deforms the body, wherein it is determined in at least one surface area of the body whether a There is a balance between internal and external forces, that the calculation means can check equilibrium conditions and that the calculation means, in the event that such a balance exists, carries out calculations for carrying out a method according to one or more of claims 1 to 9.