scalar model laws. Closely following the analy-

sis of Croce et al. (1985), this approach is presented

Following the presentation of Langhaar (1951),

here. Two assumptions are made: that soil can

be treated as a continuum and that soil proper-

consider a physical variable of a prototype *f*p(*x*p,

ties are not affected by a change in acceleration.

Dynamic similarity requires that all forces (and

all kinds of forces) have the same scale factor. The

nate measures of a point within the structure, *t *is

forces of interest include the weight force *F*w, the

time, and the subscripts p and m refer to the pro-

external force *F*e, the viscous force *F*v, the inertia

totype and model, respectively. The function *f*m

force *F*i and the seepage force *F*s.

is said to be similar to the function *f*p if the ratio

Scale factors for length *l*, mass *m *and time *t*

are

mologous times. The constant ratio *f *= *f*m/*f*p is

called the scale factor for the function *f*. Scale fac-

,

(A1)

tors can be derived from model laws, i.e. ratios

of physical laws for the model and prototype.

When similarity is achieved, a model's response

is interpreted using the scale factors to infer the

,

(A2)

response of the corresponding prototype.

and

The ideal for physical modeling is complete

similarity, which includes geometric, kinematic,

.

(A3)

thermal and dynamic similarity. Complete simi-

larity is usually not achieved for a model test.

Scale factor conflicts and scale effects, i.e., dis-

Thus the scale factors for area *A *and volume *V*

turbing influences that are associated with the

are

small scale of the model, can limit similarity to a

partial similarity.

(A4)

When a model is made of the material of the

prototype, and when there are no scale factor con-

and

flicts or disturbing scale effects, then the model

material will have the same constitutive behav-

.

(A5)

ior as the prototype material, and prototypes with

general constitutive behavior such as nonlinear

Since the same material is used in the model and

elasticity and plasticity can be modeled.

prototype at the same mass density, the scale factor

for mass is equal to the scale factor for volume:

.

(A6)

For identical effective stresses in the model and

prototype,

σ′

m = σ m = *u*m = 1

The objective of conventional small-scale

(A7)

geotechnical modeling is to achieve a model re-

σ′

σp

p

sponse that is similar to the mechanical behavior

where σ′, σ and *u *are, respectively, the effective

of the prototype, and the approach is to use the

prototype soil in a geometrically similar model.

stress, the total stress and the excess pore water

Thus it is important to ensure that the effective

pressure, including the capillary pressure caused

stresses in the model are the same as in the pro-

by surface tension. Considering eq A4 and A7 and

totype at homologous points and times. The scale

the definition of stress, it follows that the scale

factors for these conditions can be derived using

factor for all forces *F *should be

11

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